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The Many Faces of Maxwell, Dirac and
Einstein Equations.
A Cli¤ord Bundle Approach
(second revised and enlarged edition)
Waldyr Alves Rodrigues Jr. and Edmundo Capelas de Oliveira
October 2015
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Contents
Preface
xxix
1 Introduction
2 Multivector and Extensor Calculus
2.1 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Cotensors . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Multicotensors . . . . . . . . . . . . . . . . . . .
2.1.3 Tensor product of Multicotensors . . . . . . . . .
2.1.4 Involutions . . . . . . . . . . . . . . . . . . . . .
2.2 Scalar Products in V and V . . . . . . . . . . . . . . .
2.3 Exterior and Grassmann Algebras
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V
2.3.1 Scalar Product in V and Hodge Star Operator
2.3.2 Contractions . . . . . . . . . . . . . . . . . . . .
2.4 Cli¤ord Algebras . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Properties of the Cli¤ord Product . . . . . . . .
2.5
2.6
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2.4.2 Universality of C`(V ; g) . . . . . . . . . . . . . . . .
Extensors . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 (p; q)-Extensors . . . . . . . . . . . . . . . . . . . .
2.5.2 Adjoint Operator . . . . . . . . . . . . . . . . . . . .
2.5.3 Properties of the Adjoint Operator . . . . . . . . . .
(1; 1)-Extensors . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Symmetric and Antisymmetric Parts of (1; 1)-Extensors
2.6.2 Exterior Power Extension of (1; 1)-Extensors . . . .
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2.6.3 Properties of t . . . . . . . . . . . . . . . . . . . . .
2.6.4 Characteristic Scalars of a (1; 1)-Extensor . . . . . .
2.6.5 Properties of dett . . . . . . . . . . . . . . . . . . . .
2.6.6 Characteristic biform of a (1; 1)-Extensor . . . . . .
2.6.7 Generalization of (1; 1)-Extensors . . . . . . . . . . .
2.7 Symmetric Automorphisms and Orthogonal Cli¤ord Products
2.7.1 The Gauge Metric Extensor h . . . . . . . . . . . . .
2.7.2 Relation Between dett and the Classical Determinant
det[Tij ] . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.3 Strain, Shear and Dilation Associated with Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Minkowski Vector Space . . . . . . . . . . . . . . . . . . . .
2.8.1 Lorentz and Poincaré Groups . . . . . . . . . . . . .
2.9 Multiform Functions . . . . . . . . . . . . . . . . . . . . . .
2.9.1 Multiform Functions of a Real Variable . . . . . . .
2.10 Multiform Functions of Multiform Variables . . . . . . . . .
2.10.1 Limit and Continuity . . . . . . . . . . . . . . . . .
2.10.2 Di¤erentiability . . . . . . . . . . . . . . . . . . . . .
2.11 Directional Derivatives . . . . . . . . . . . . . . . . . . . . .
2.11.1 Chain Rules . . . . . . . . . . . . . . . . . . . . . . .
2.11.2 Derivatives of Multiform Functions . . . . . . . . .
2.11.3 Properties of @Y F . . . . . . . . . . . . . . . . . . .
2.11.4 The Di¤erential Operator @Y ~ . . . . . . . . . . . .
2.11.5 The A ~ @Y directional derivative . . . . . . . . . .
2.12 Solved Exercices . . . . . . . . . . . . . . . . . . . . . . . .
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3 The Hidden Geometrical Nature of Spinors
3.1 Notes on the Representation Theory of Associative Algebras
3.2 Real and Complex Cli¤ord Algebras and their Classi…cation
3.2.1 Pauli, Spacetime, Majorana and Dirac Algebras . . .
3.3 The Algebraic, Covariant and Dirac-Hestenes Spinors . . .
3.3.1 Minimal Lateral Ideals of Rp;q . . . . . . . . . . . .
3.3.2 Algorithm for Finding Primitive Idempotents of Rp;q .
3.3.3 R?p;q , Cli¤ord, Pinor and Spinor Groups . . . . . . .
3.3.4 Lie Algebra of Spine1;3 . . . . . . . . . . . . . . . . .
3.4 Spinor Representations of R4;1 ; R04;1 and R1;3 . . . . . . . .
3.5 Algebraic Spin Frames and Spinors . . . . . . . . . . . . . .
3.6 Algebraic Dirac Spinors of Type I u . . . . . . . . . . . . .
3.7 Dirac-Hestenes Spinors (DHS) . . . . . . . . . . . . . . . . .
3.7.1 What is a Covariant Dirac Spinor (CDS) . . . . . .
3.7.2 Canonical Form of a Dirac-Hestenes Spinor . . . . .
3.7.3 Bilinear Invariants and Fierz Identities . . . . . . . .
3.7.4 Reconstruction of a Spinor . . . . . . . . . . . . . .
3.7.5 Lounesto Classi…cation of Spinors . . . . . . . . . .
3.8 Majorana and Weyl Spinors . . . . . . . . . . . . . . . . . .
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the Pauli Algebra
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4 Some Di¤erential Geometry
4.1 Di¤erentiable Manifolds . . . . . . . . . . . . . . . . . . . .
4.1.1 Manifold with Boundary . . . . . . . . . . . . . . . .
4.1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . .
4.1.3 Tensor Bundles . . . . . . . . . . . . . . . . . . . . .
4.1.4 Vector Fields and Integral Curves . . . . . . . . . .
4.1.5 Derivative and Pullback Mappings . . . . . . . . . .
4.1.6 Di¤eomorphisms, Pushforward and Pullback when
M =N . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.7 Lie Derivatives . . . . . . . . . . . . . . . . . . . . .
4.1.8 Properties of $v . . . . . . . . . . . . . . . . . . . .
4.1.9 Invariance of a Tensor Field . . . . . . . . . . . . . .
4.2 Cartan Bundle, de Rham Periods and Stokes Theorem . . .
4.2.1 Cartan Bundle . . . . . . . . . . . . . . . . . . . . .
4.2.2 The Interior Product of Forms and Vector Fields . .
4.2.3 Extensor Fields . . . . . . . . . . . . . . . . . . . . .
4.2.4 Exact and Closed Forms and Cohomology Groups .
4.3 Integration of Forms . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Orientation . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Integration of a n-form . . . . . . . . . . . . . . . . .
4.3.3 Chains and Homology Groups . . . . . . . . . . . . .
4.3.4 Integration of a r-form . . . . . . . . . . . . . . . . .
4.3.5 Stokes Theorem . . . . . . . . . . . . . . . . . . . .
4.3.6 Integration of Closed Forms and de Rham Periods .
4.4 Di¤erential Geometry in the Hodge Bundle . . . . . . . . .
4.4.1 Riemannian and Lorentzian Structures on M . . . .
4.4.2 Hodge Bundle . . . . . . . . . . . . . . . . . . . . . .
4.4.3 The Global Inner Product of p-forms . . . . . . . . .
4.5 Pullbacks and the Di¤erential . . . . . . . . . . . . . . . . .
4.6 Structure Equations I . . . . . . . . . . . . . . . . . . . . .
4.6.1 Exterior Covariant Di¤erential of (p + q)-indexed rform Fields . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Bianchi Identities . . . . . . . . . . . . . . . . . . . .
4.6.3 Induced Connections Under Di¤eomorphisms . . . .
4.7 Classi…cation of Geometries on M and Spacetimes . . . . .
4.7.1 Spacetimes . . . . . . . . . . . . . . . . . . . . . . .
4.8 Di¤erential Geometry in the Cli¤ord Bundle . . . . . . . .
4.8.1 Cli¤ord Fields as Sums of Nonhomogeneous Di¤erential Forms . . . . . . . . . . . . . . . . . . . . . . .
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Dotted and Undotted Algebraic Spinors
3.9.1 Pauli Algebra . . . . . . . . . . .
3.9.2 Quaternion Algebra . . . . . . .
3.9.3 Minimal left and Right Ideals in
and Spinors . . . . . . . . . . . .
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4.8.2
4.8.3
4.8.4
4.9
Pullbacks and Relation Between Hodge Star Operators131
Dirac Operators . . . . . . . . . . . . . . . . . . . . 133
Standard Dirac Commutator and Dirac Anticommutator . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.8.5 Geometrical Meanings of the Commutator and Anticommutator . . . . . . . . . . . . . . . . . . . . . . . 136
4.8.6 Associated Dirac Operators . . . . . . . . . . . . . . 138
4.8.7 The Dirac Operator in Riemann-Cartan-Weyl Spaces 140
4.8.8 Torsion, Strain, Shear and Dilation of a Connection 143
4.8.9 Structure Equations II . . . . . . . . . . . . . . . . . 147
4.8.10 D’Alembertian, Ricci and Einstein Operators . . . . 148
4.8.11 The Square of a General Dirac Operator . . . . . . . 153
Some Applications . . . . . . . . . . . . . . . . . . . . . . . 155
4.9.1 Maxwell Equations in the Hodge Bundle . . . . . . 155
4.9.2 Charge Conservation . . . . . . . . . . . . . . . . . 156
4.9.3 Flux Conservation . . . . . . . . . . . . . . . . . . . 157
4.9.4 Quantization of Action . . . . . . . . . . . . . . . . . 157
4.9.5 A Comment on the Use of de Rham Pseudo-Forms
and Electromagnetism . . . . . . . . . . . . . . . . . 158
4.9.6 Maxwell Equation in the Cli¤ord Bundle . . . . . . 159
4.9.7 Einstein Equations and the Field Equations for the a 161
4.9.8 Curvature of a Connection and Bending. The Nunes
Connection of S 2 . . . . . . . . . . . . . . . . . . . . 162
4.9.9 “Tetrad” Postulate? On the necessity of Precise Notations . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5 Cli¤ord Bundle Approach to the Di¤erential Geometry of
Branes
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5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2 Torsion Extensor and Curvature Extensor of a RiemannCartan Connection . . . . . . . . . . . . . . . . . . . . . . . 172
5.3 The Riemannian or Semi-Riemannian Geometry of a Submanifold M of M . . . . . . . . . . . . . . . . . . . . . . . . 181
5.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . 181
5.3.2 Induced Metric and Induced Connection . . . . . . . 183
5.3.3 S(v) = S(v) = @u ^ Pu (v) and S(v? ) = @u yPu (v) . 184
5.3.4 Integrability Conditions . . . . . . . . . . . . . . . . 187
5.3.5 S(v) = S(v) . . . . . . . . . . . . . . . . . . . . . . . 188
5.3.6 d ^ v = @ ^ v + S(v) and dyv = @yv . . . . . . . . . 189
5.3.7 dC = @C + S(C) . . . . . . . . . . . . . . . . . . . . . 190
5.4 Curvature Biform R(u ^ v) Expressed in Terms of the Shape
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
5.4.1 Equivalent Expressions for R(u ^ v) . . . . . . . . . 191
5.4.2 S2 (v) = @ ^ @ (v) . . . . . . . . . . . . . . . . . . 192
5.5 Some Identities Involving P and Pu . . . . . . . . . . . . . 197
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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
6 Some Issues in Relativistic Spacetime Theories
6.1 Reference Frames on Relativistic Spacetimes . . . . . . . . .
6.1.1 Classi…cation of Reference Frames I . . . . . . . . .
6.1.2 Rotation and Fermi Transport . . . . . . . . . . . .
6.1.3 Frenet Frames over
. . . . . . . . . . . . . . . . .
6.1.4 Physical Meaning of Fermi Transport . . . . . . . .
6.1.5 Rotation 2-form, Pauli-Lubanski Spin 1-Form and
Classical Spinning Particles . . . . . . . . . . . . . .
6.2 Classi…cation of Reference Frames II . . . . . . . . . . . . .
6.2.1 In…nitesimally Nearby Observers, 3-Velocities and 3Accelerations . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Jacobi Equation . . . . . . . . . . . . . . . . . . . .
6.3 Synchronizability . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Einstein Synchronization Procedure . . . . . . . . .
6.3.2 Locally Synchronizable Reference Frame . . . . . . .
6.3.3 Synchronizable Reference Frame . . . . . . . . . . .
6.4 Sagnac E¤ect . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Characterization of a Spacetime Theory . . . . . . . . . . .
6.5.1 Di¤eomorphism Invariance . . . . . . . . . . . . . .
6.5.2 Di¤eomorphism Invariance and Maxwell Equations .
6.5.3 Di¤eomorphism Invariance and GRT . . . . . . . . .
6.5.4 A Comment on Logunov’s Objection . . . . . . . . .
6.5.5 Spacetime Symmetry Groups . . . . . . . . . . . . .
6.5.6 Physically Equivalent Reference Frames . . . . . . .
6.6 Principle of Relativity . . . . . . . . . . . . . . . . . . . . .
6.6.1 Internal and External Synchronization Processes . .
6.6.2 External Synchronization . . . . . . . . . . . . . . .
6.6.3 A Non Standard Realization of the Lorentz Group .
6.6.4 Status of the Principle of Relativity . . . . . . . . .
6.7 Principle of Local Lorentz Invariance in GRT? . . . . . . .
6.7.1 LLRF s and the Equivalence Principle . . . . . . . .
6.8 PIRFs on a Friedmann Universe . . . . . . . . . . . . . . .
6.8.1 Mechanical Experiments Distinguish PIRFs . . . . .
6.8.2 LLRF and LLRF 0 are not Physically Equivalent
on a Friedmann Universe. . . . . . . . . . . . . . . .
6.8.3 No Generalization of the Principle of Relativity for
GRT . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Schwarzschild Original Solution and the Existence of Black
Holes in GRT . . . . . . . . . . . . . . . . . . . . . . . . . .
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7 Cli¤ord and Dirac-Hestenes Spinor Fields
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7.1 The Cli¤ord Bundle of Spacetime . . . . . . . . . . . . . . . 257
7.1.1 Details of the Bundle Structure of C`(M; g) . . . . . 258
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7.1.2 Cli¤ord Fields . . . . . . . . . . . . . . . . . . . . .
Spin Structure . . . . . . . . . . . . . . . . . . . . . . . . .
Spinor Bundles and Spinor Fields . . . . . . . . . . . . . . .
7.3.1 Left and Right Spin-Cli¤ord Bundles . . . . . . . . .
7.3.2 Bundle of Modules over a Bundle of Algebras . . . .
7.3.3 Dirac-Hestenes Spinor Fields (DHSF) . . . . . . . .
7.4 Natural Actions on some Vector Bundles Associated with
PSpine1;3 (M ) . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Fiducial Sections Associated with a Spin Frame . . .
7.5 Representatives of DHSF in the Cli¤ord Bundle . . . . . . .
7.6 Covariant Derivatives of Cli¤ord Fields . . . . . . . . . . .
7.7 Covariant Derivative of Spinor Fields . . . . . . . . . . . . .
7.8 The Many Faces of the Dirac Equation . . . . . . . . . . . .
7.8.1 Dirac Equation for Covariant Dirac Fields . . . . . .
7.8.2 Dirac Equation in C`lSpine1;3 (M; g) . . . . . . . . . .
7.8.3 Spin Dirac Operator . . . . . . . . . . . . . . . . . .
7.8.4 Electromagnetic Gauge Invariance of the DEC`l . . .
7.9 The Dirac-Hestenes Equation (DHE) . . . . . . . . . . . . .
7.9.1 Representative of the DHE in the Cli¤ord Bundle .
7.9.2 A Comment about the Nature of Spinor Fields . . .
7.9.3 Passive Gauge Invariance of the DHE . . . . . . . .
7.10 Amorphous Spinor Fields . . . . . . . . . . . . . . . . . . .
7.11 Bilinear Invariants . . . . . . . . . . . . . . . . . . . . . . .
7.11.1 Bilinear Invariants Associated to a DHSF . . . . . .
7.11.2 Bilinear Invariants Associated to a Representative of
a DHSF . . . . . . . . . . . . . . . . . . . . . . . . .
7.11.3 Electromagnetic Gauge Invariance of the DHE . . .
7.12 Commutator of Covariant Derivatives of Spinor Fields and
Lichnerowicz Formula . . . . . . . . . . . . . . . . . . . . .
7.12.1 The Generalized Lichnerowicz Formula . . . . . . . .
7.2
7.3
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8 A Cli¤ord Algebra Lagrangian Formalism in Minkowski
Spacetime
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8.1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 293
8.2 Lagrangians and Lagrangian Densities for Multiform Fields 296
8.2.1 Variations . . . . . . . . . . . . . . . . . . . . . . . . 298
8.3 Stationary Action Principle and Euler-Lagrange Equations 300
8.4 Some Important Lagrangians . . . . . . . . . . . . . . . . . 301
8.4.1 Maxwell Lagrangian . . . . . . . . . . . . . . . . . . 301
8.4.2 Dirac-Hestenes Lagrangian . . . . . . . . . . . . . . 302
8.5 Canonical Energy-Momentum Extensor Field . . . . . . . . 303
8.5.1 Canonical Energy-Momentum Extensor of the Free
Electromagnetic Field . . . . . . . . . . . . . . . . . 306
8.5.2 Canonical Energy-Momentum Extensor of the Free
Dirac-Hestenes Field . . . . . . . . . . . . . . . . . . 309
Contents
8.6
8.7
8.8
Canonical Orbital Angular Momentum and Spin Extensors
8.6.1 Canonical Orbital and Spin Density Extensors for the
Dirac-Hestenes Field . . . . . . . . . . . . . . . . . .
8.6.2 Case L(X; @ x yX) . . . . . . . . . . . . . . . . . . . .
8.6.3 Case L(X; @ x ^X) . . . . . . . . . . . . . . . . . . .
8.6.4 Spin Extensor of the Free Electromagnetic Field . .
The Source of Spin . . . . . . . . . . . . . . . . . . . . . . .
Coupled Maxwell and Dirac-Hestenes Fields . . . . . . . . .
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9 Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
317
9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 317
9.1.1 Functional Derivatives on Jet Bundles . . . . . . . . 317
9.1.2 Algebraic Derivatives . . . . . . . . . . . . . . . . . . 318
9.2 Euler-Lagrange Equations from Lagrangian Densities . . . . 320
9.3 Invariance of the Action Integral under the Action of a Diffeomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 321
9.4 Covariant ‘Conservation’Laws . . . . . . . . . . . . . . . . 322
9.5 When Genuine Conservation Laws Do Exist? . . . . . . . . 326
9.6 Pseudopotentials in GRT . . . . . . . . . . . . . . . . . . . 330
9.6.1 Pseudopotentials are not Uniquely De…ned . . . . . 333
9.7 Is There Any Energy-Momentum Conservation Law in GRT? 333
9.8 Is there any Angular Momentum Conservation law in the GRT338
9.9 Some Non Trivial Exercises . . . . . . . . . . . . . . . . . . 339
10 The DHE on a RCST and the Meaning of Active Local
Lorentz Invariance
347
10.1 Formulation of the DHE on a RCST . . . . . . . . . . . . . 347
10.2 Meaning of Active Lorentz Invariance of the Dirac-Hestenes
Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
11 On the Nature of the Gravitational Field
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.2 Representation of the Gravitational Field . . . . . . . . . .
11.3 Comment on Einstein Most Happy Though . . . . . . . . .
11.4 Field Theory for the Gravitational Field in Minkowski Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.4.1 Legitimate Energy-Momentum Tensor of the Gravitational Field. The Nice Formula . . . . . . . . . . .
11.5 Hamilton Formalism . . . . . . . . . . . . . . . . . . . . . .
11.6 Mass of the Graviton . . . . . . . . . . . . . . . . . . . . . .
11.7 Comment on the Teleparallel Equivalent of GR . . . . . . .
11.8 On Cli¤ord’s Little Hills . . . . . . . . . . . . . . . . . . . .
11.9 A Maxwell Like Equation for a Brane World with a Killing
Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . .
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11.10Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
12 On the Many Faces of Einstein Equations
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . .
12.2.1 Exterior Covariant Di¤erential . . . . . . . . . . . .
12.2.2 Absolute Di¤erential . . . . . . . . . . . . . . . . . .
12.3 Cli¤ord Valued Di¤erential Forms . . . . . . . . . . . . . .
12.4 Fake Exterior Covariant Di¤erential of Cliforms . . . . . . .
12.4.1 The Operator Der . . . . . . . . . . . . . . . . . . .
12.4.2 Cartan Exterior Di¤erential of Vector Valued Forms
12.4.3 Torsion and Curvature Once Again . . . . . . . . . .
12.4.4 FECD and Levi-Civita Connections . . . . . . . . .
12.5 General Relativity as a Sl(2; C) Gauge Theory . . . . . . .
12.5.1 The Non-homogeneous Field Equations . . . . . . .
12.6 Another Set of Maxwell-Like Equations for Einstein Theory
375
375
376
378
380
381
384
384
385
386
387
389
389
391
13 Maxwell, Dirac and Seiberg-Witten
p Equations
13.1 Dirac-Hestenes
Equation
and
i
=
1 . . . . . . . . . . . .
p
13.2 How i =
1 enters Maxwell Theory. . . . . . . . . . . . .
13.2.1 From Spatial Rotation to Duality Rotation . . . . .
13.2.2 Polarization and Stokes Parameters . . . . . . . . .
13.3 ‘Dirac like’Representations of ME . . . . . . . . . . . . . .
13.3.1 ‘Dirac like’Representation of ME on I 0 (M; ) . . . .
13.3.2 Sachs ‘Dirac like’Representation of ME . . . . . . .
13.3.3 Sallhöfer ‘Dirac like’Representation of ME . . . . .
13.3.4 A Three Dimensional Representation of the Free ME
13.4 Mathematical Maxwell-Dirac Equivalence . . . . . . . . . .
13.4.1 Solution of F = 21 ~ . . . . . . . . . . . . . . . . .
13.4.2 A Particular Solution . . . . . . . . . . . . . . . . .
13.4.3 The General Solution . . . . . . . . . . . . . . . . .
13.4.4 The Generalized Maxwell Equation . . . . . . . . . .
13.4.5 The Generalized Hertz Potential Equation . . . . . .
13.4.6 Maxwell Dirac Equivalence of First Kind . . . . . .
13.4.7 Maxwell-Dirac Equivalence of Second Kind . . . . .
13.5 Seiberg-Witten Equations . . . . . . . . . . . . . . . . . . .
13.5.1 Derivation of Seiberg-Witten Equations . . . . . . .
13.5.2 A Possible Interpretation of the Seiberg-Witten Equations . . . . . . . . . . . . . . . . . . . . . . . . . . .
395
395
396
397
397
403
403
404
405
405
407
408
409
410
410
412
414
418
419
420
14 Superparticles and Super…elds
14.1 Spinor Fields and Classical Spinning Particles . . . . . . . .
14.1.1 Spinor Equation of Motion for a Classical Particle on
Minkowski Spacetime . . . . . . . . . . . . . . . . .
14.1.2 Classical ( Nonlinear) Dirac-Hestenes Equation . . .
423
423
421
424
425
Contents
14.1.3 Digression . . . . . .
14.2 The Superparticle . . . . . .
14.3 Superparticle in Minkowski
14.4 Super…elds . . . . . . . . .
. . . . . .
. . . . . .
Spacetime
. . . . . .
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xi
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427
428
432
433
15 Maxwell, Einstein, Dirac and Navier-Stokes Equations
437
15.1 The Conserved Komar Current . . . . . . . . . . . . . . . . 437
15.2 A Maxwell like Equation Encoding Einstein Equation . . . 442
15.3 The Case When K = A is a Killing Vector Field . . . . . . 442
15.4 From Maxwell Equation to a Navier-Stokes Equation . . . . 443
15.4.1 A Realization of Eqs.(15.40) and (15.41) . . . . . . 446
15.4.2 Constraints Imposed by the Nonhomogeneous Maxwell
Like Equation . . . . . . . . . . . . . . . . . . . . . . 447
15.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
16 Magnetic Like Particles and Elko Spinor Fields
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.2 Dictionary Between Covariant and Dirac-Hestenes Spinor
Fields Formalisms . . . . . . . . . . . . . . . . . . . . . . .
16.3 The CSFOPDE Lagrangian for Elko Spinor Fields . . . . .
16.4 Coupling of the Elko Spinor Fields a su(2) ' spin3;0 valued
Potential A . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.5 Di¤erence Between Elko and Majorana Spinor Fields . . . .
16.5.1 Some Majorana Fields are Dual Helicities Objects .
16.5.2 The Majorana Currents J M and J 5M . . . . . . . . .
16.5.3 Making of Majorana Fields that Satisfy the Dirac
Equation . . . . . . . . . . . . . . . . . . . . . . . .
16.6 The Causal Propagator for the K and M Fields . . . . . . .
16.7 A Note on the Anticommutator of Mass Dimension 1 Elkos
According to [6] . . . . . . . . . . . . . . . . . . . . . . . . .
16.7.1 Plane of Nonlocality and Breakdown of Lorentz Invariance . . . . . . . . . . . . . . . . . . . . . . . . .
16.8 A New Representation of the Parity Operator Acting on
Dirac Spinor Fields . . . . . . . . . . . . . . . . . . . . . . .
16.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . .
451
451
A Principal Bundles, Vector Bundles and Connections
A.1 Fiber Bundles . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1.1 Frame Bundle . . . . . . . . . . . . . . . . . . . . . .
A.1.2 Orthonormal Frame Bundle . . . . . . . . . . . . . .
A.1.3 Jet Bundles . . . . . . . . . . . . . . . . . . . . . . .
A.2 Product Bundles and Whitney Sum . . . . . . . . . . . . .
A.3 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3.1 Equivalent De…nitions of a Connection in Principal
Bundles . . . . . . . . . . . . . . . . . . . . . . . . .
475
475
478
479
480
480
481
453
459
460
462
464
464
465
468
469
470
471
472
481
xii
Contents
A.3.2 The Connection on the Base Manifold . . . . . . . .
A.4 Exterior Covariant Derivatives. . . . . . . . . . . . . . . . .
A.4.1 Local Curvature in the Base Manifold M . . . . . .
A.4.2 Transformation of the Field Strengths Under a Change
of Gauge . . . . . . . . . . . . . . . . . . . . . . . .
A.4.3 Induced Connections . . . . . . . . . . . . . . . . . .
A.4.4 Linear Connections on a Manifold M . . . . . . . . .
A.4.5 Torsion and Curvature on M . . . . . . . . . . . . .
A.5 Covariant Derivatives on Vector Bundles . . . . . . . . . . .
A.5.1 Connections on E over a Lorentzian Manifold . . . .
A.5.2 Gauge Covariant Connections . . . . . . . . . . . . .
A.5.3 Curvature Again . . . . . . . . . . . . . . . . . . . .
A.5.4 Exterior Covariant Derivative Again . . . . . . . . .
Acronyms and Abbreviations
484
485
487
488
489
490
491
492
496
497
498
498
501
References
503
Index
533
ABSTRACT In this chapter we describe the contents of all chapters of the
book and their interrelationship.
ABSTRACT This chapter is dedicated to a thoughtful exposition of the
multiform and extensor calculus. Starting from the tensor algebra
V of a real
n-dimensional vector space V we construct the exterior algebra V of V .
Equipping V with a metric tensor g we introduce the Grassmmann algebra and next the Cli¤ord algebra C`(V ; g) associated
to the pair (V ; g).
V
The concept of Hodge dual of elements of V (called nonhomogeneous
multiforms) and of C`(V ; g) (also called nonhomogeneous multiforms or
Cli¤ord numbers) is introduced, and the scalar product and operations of
left and right contractions in these structures are de…ned. Several important formulas and “tricks of the trade” are presented. Next we introduce
the concept of extensors
V
V which are multilinear maps from p subspaces of
V to q subspaces of V and study their properties. Equipped with such
concept we study some properties of symmetric automorphisms and the
orthogonal Cli¤ord algebras introducing the gauge metric extensor (an essential ingredient for theories presented in other chapters). Also, we de…ne
the concepts of strain, shear and dilation associated with endomorphisms.
A preliminary exposition of the Minkoswski vector space is given and the
Lorentz and Poincaré groups are introduced. In the remaining of the chapter we give an original presentation of the theory of multiform functions of
multiform variables. For these objects we de…ne the concepts of limit, continuity and di¤erentiability. We study in details the concept of directional
derivatives of multiform functions and solve several nontrivial exercises to
clarify how to work with these notions, which in particular are crucial for
Contents
xiii
the formulation of Chapter 8 which deals with a Cli¤ord algebra Lagrangian
formalism of …eld theory in Minkowski spacetime.
ABSTRACT This chapter reviews the classi…cation of the real and complex Cli¤ord algebras and analyze the relationship between some particular
algebras that are important in physical applications, namely the quaternion algebra (R0;2 ), Pauli algebra (R3;0 ), the spacetime algebra (R1;3 ), the
Majorana algebra (R3;1 ) and the Dirac algebra (R4;1 ). A detailed and original theory disclosing the hidden geometrical meaning of spinors is given
through the introduction of the concepts of algebraic, covariant and DiracHestenes spinors. The relationship between these kinds of spinors (that
carry the same mathematical information) is elucidated with special emphasis for cases of physical interest. We investigate also how to reconstruct
a spinor from their so-called bilinear invariants and present Lounesto’s classi…cation of spinors. Also, Majorana, Weyl spinors, the dotted and undotted
algebraic spinors are discussed with the Cli¤ord algebra formalism.
ABSTRACT The main objective of this chapter is to present a Cli¤ord
bundle formalism for the formulation of the di¤erential geometry of a manifold M , equipped with metric …elds g 2 sec T20 M and g 2 sec T02 M for
the tangent and cotangent bundles. We start by …rst recalling the standard
formulation and main concepts of the di¤erential geometry of a di¤erential
manifold M . We introduce in M the Cartan bundle of di¤erential forms,
de…ne the exterior derivative, Lie derivatives, and also brie‡y review concepts as chains, homology and cohomology groups, de Rham periods, the
integration of form …elds and Stokes theorem. Next, after introducing the
metric …elds g and g in M we introduce the Hodge bundle presenting the
Hodge star and the Hodge coderivative operators acting on sections of this
bundle. We moreover recall concepts as the pullback and the di¤erential of
maps, connections and covariant derivatives, Cartan’s structure equations,
the exterior covariant di¤erential of (p + q)-indexed r-forms, Bianchi identities and the classi…cation of geometries on M when it is equipped with
a metric …eld and a particular connection. The spacetime concept is rigorously de…ned. We introduce and scrutinized the structure of the Cli¤ord
bundle of di¤erential forms (C`(M; g)) of M and introduce the fundamental
concept of the Dirac operator (associated to a given particular connection
de…ned in M ) acting on Cli¤ord …elds (sections of C`(M; g)). We show that
the square of the Dirac operator (associated to a Levi-Civita connection in
M ) has two fundamental decompositions, one in terms of the derivative and
Hodge codi¤erential operators and other in terms of the so-called Ricci and
D’Alembertian operators. A so-called Einstein operator is also introduced
in this context. These decompositions of the square of the Dirac operator
are crucial for the formulation of important ideas concerning the construction of gravitational theories as discussed in particular in Chapters 9, 11,
15. The Dirac operator associated to an arbitrary (metrical compatible)
connection de…ned in M and its relation with the Dirac operator associated to the Levi-Civita connection of the pair (M; g) is discussed in details
xiv
Contents
and some important formulas are obtained. The chapter also discuss some
applications of the formalism, e.g., the formulation of Maxwell equations
in the Hodge and Cli¤ord bundles and formulation of Einstein equation
in the Cli¤ord bundle using the concept of the Ricci and Einstein operators. A preliminary account of the crucial di¤erence between the concepts
of curvature of a connection in M and the concept of bending of M as a
hypersurface embedded in a (pseudo)-Euclidean space of high dimension
(a property characterized by the concept of the shape tensor, discussed
in details in Chapter 5) is given by analyzing a speci…c example, namely
the one involving the Levi-Civita and the Nunes connections de…ned in a
punctured 2-dimensional sphere. The chapter ends analyzing a statement
referred in most physical textbooks as “tetrad postulate” and shows how
not properly de…ning concepts can produce a lot of misunderstanding and
invalid statements.
ABSTRACT Using the Cli¤ord bundle formalism (CBF) of di¤erential
forms and the theory of extensors acting on C`(M; g) (introduced in Chapter 4) we …rst recall the formulation of the intrinsic geometry of a di¤erential manifold M (a brane) equipped with a metric …eld g of signature
(p; q) and an arbitrary metric compatible connection r introducing the
torsion (2 1)-extensor …eld , the curvature (2 2) extensor …eld R and
(once …xing a gauge) the connection (1 2)-extensor ! and the Ricci operator @ ^ @ (where @ is the Dirac operator acting on sections of C`(M; g))
which plays an important role in this paper. Next, using the CBF we give
a thoughtful presentation of the Riemann or the Lorentzian geometry of
an orientable submanifold M (dim M = m) living in a manifold M (such
that M ' Rn is equipped with a semi-Riemannian metric g with signature
(p; q) and p + q = n and its Levi-Civita connection D) and where there
is de…ned a metric g = i g, where i : M ! M is the inclusion map. We
prove several equivalent forms for the curvature operator R of M . Moreover we show a very important result, namely that the Ricci operator of
M is the (negative) square of the shape operator S of M (object obtained
by applying the restriction on M of the Dirac operator @ of C`(M; g) to
the projection operator P). Also we disclose the relationship between the
(1 2)-extensor ! and the shape biform S (an object related to S). The
results obtained are used in Chapter 11 to give a mathematical formulation
to Cli¤ord’s theory of matter [358].
ABSTRACT The chapter has as main objective to clarify some important
concepts appearing in relativistic spacetime theories and which are necessary of a clear understanding of our view concerning the formulation and
understanding of Maxwell, Dirac and Einstein theories. Using the de…nition
of a Lorentzian spacetime structure (M; g; D; g ; ") presented in Chapter
4 we introduce the concept of a reference frame in that structure which
is an object represented by a given unit timelike vector …eld Z 2 sec T U
(U
M ). We give two classi…cation schemes for these objects, one according to the decomposition of DZ and other according to the concept
Contents
of synchronizability of ideal clocks (at rest in Z). The concept of a coordinate chart covering U and naturally adapted to the reference frame Z
is also introduced. We emphasize that the concept of a reference frame is
di¤erent (but related) from the concept of a frame which is a section of
the frame bundle. The concept of Fermi derivative is introduced and the
physical meaning of Fermi transport is elucidated, in particular we show
the relation between the Darboux biform of the theory of Frenet frames
and its decomposition as an invariant sum of a Frenet biform F (describing Fermi transport) and a rotation biform S such that the contraction of
? S with the velocity …eld v of the spinning particle is directly associated
with the so-called Pauli-Lubanski spin 1-form. We scrutinize the concept
of di¤eomorphism invariance of general spacetime theories and of General
Relativity in particular, discuss what meaning can be given to the concept
of physically equivalent reference frames and what one can understand by
a principle of relativity. Examples are given and in particular, it is proved
that in a general Lorentzian spacetime (modelling a gravitational …eld according to General Relativity) there is in general no reference frame with
the properties (according to the scheme classi…cations) of the inertial referenced frames of special relativity theories. However there are in such a case
reference frames called pseudo inertial reference frames (PIRFs) that have
most of the properties of the inertial references frames of special relativity
theories. We also discuss a formulation (that one can …nd in the literature) of a so-called principle of local Lorentz invariance and show that
if it is interpreted as physical equivalence of PIRFs then it is not valid.
The Chapter ends with a brief discussion of di¤eormorphism invariance
applied to Schwarzschild original solution and the Droste-Hilbert solution
of Einstein equation which are shown to be not equivalent (the underlying
manifolds have di¤erent topologies) and what these solutions have to do
with the existence of blackholes in the “orthodoxof General Relativity.
ABSTRACT This chapter presents an original approach to the theory
of Dirac-Hestenes spinor …elds. After recalling details the structure of the
Cli¤ord bundle of a Lorentzian manifold structure (M; g) we introduce the
concept of spin structures on (M; g), de…ne spinor bundles and spinor …elds.
Left and right spin-Cillford bundles are presented and the concept of DiracHestenes spinor …elds (as sections of a special spin-Cli¤ord bundle denoted
by C`rSpine1;3 (M; g)) is investigated in details disclosing their real nature
and showing how these objects can be represented as some equivalence
classes of even sections of the Cli¤ord bundle C`(M; g). We introduce and
obtain the connection (covariant derivative) acting on spin-Cli¤ord bundles
C`rSpine1;3 (M; g) from a Levi-Civita connection acting on the tensor bundle
of (M; g). Associated with that connection we next introduce the standard
spin-Dirac operator acting on sections of C`rSpine1;3 (M; g) (not to be confused
with the Dirac operator acting on sections of the Cli¤ord bundle de…ned in
Chapter 4). We discuss how to write Dirac equation using the spin-Cli¤ord
bundle formalism and clari…es its properties and many misunderstandings
relating to such a notion that are spread in the literature. We also discuss
xv
xvi
Contents
the notion of what is known as amorphous spinor …elds showing that these
objects cannot represent fermion …elds. Finally, the Chapter also presents a
simple proof of the famous Lichnerowicz formula which relates the square of
the standard spin-Dirac operator to the curvature of the (spin) connection
and also we obtain a generalization of that formula for the square of a
general spin-Dirac operator associated to a general (metric compatible)
Riemann-Cartan connection de…ned in (M; g).
ABSTRACT Using tools introduced in previous chapters, particularly the
concept of Cli¤ord and spin-Cli¤ord bundles (and the representations of
sections of the spin-Cli¤ord bundles as equivalence classes of sections the
Cli¤ord bundle), extensor …elds and the Dirac operator we give an uni…ed and original approach to the Lagrangian …eld theory in Minkowski
spacetime with special emphasis on the Maxwell and Dirac-Hestenes …elds.
We derive for these …elds their canonical energy-momentum extensor …elds
and also their angular momentum and spin momentum extensor …elds. In
particular we show that the antisymmetric part of the canonical energymomentum tensor is the “source” of spin of the …eld. Several nontrivial
exercises are solved with details in order to help the reader to familiarize
with the formalism and to make contact with standard formulations of …eld
theory.
ABSTRACT In this chapter we examine in details the conditions for existence of conservation laws of energy-momentum and angular momentum for
the matter …elds of on a general Riemann-Cartan spacetime (M; g; r; g ; ")
and also in the particular case of Lorentzian spacetimes M = (M; g; D; g ; "
) which as we already know model gravitational …elds in the GRT [38].
Riemann-Cartan spacetimes are supposed to model generalized gravitational …elds in so called Riemann-Cartan theories. In what follows, we
suppose that in (M; g; r; g ; ") (or M) a set of dynamic …elds live and
interact. Of course, we want that the Riemann-Cartan spacetime admits
spinor …elds, and from what we learned in Chapter 7, this implies that the
orthonormal frame bundle must be trivial. This permits a great simpli…cation in our calculations. Moreover, we will suppose, for simplicity that the
dynamic V
…elds A , A = 1; 2; :::; n, are in general distinct r-forms1 , i.e., each
A
2 sec r T M ,! sec C`(M; g), for some r = 0; 1; :::; 4. Before we start
our enterprise we think it is useful to recall some results which serve also
the purpose to …x the notation for this chapter.
ABSTRACT In this chapter we give a formulation of the Dirac-Hestenes
equation on a Riemann-Cartan manifold (M; g; r; g ; ") using the Cli¤ord
and spin-Cli¤ord bundles formalism. We show that the obtained equation
1 This is not a serious restriction in the formalism since we already learned how (given
a spinorial frame) to represent spinor …elds by sums of even multiform …elds.
Contents
xvii
which follows for a properly chosen Lagrangian density (heuristically based
on the principle of minimum coupling) agrees with the one proposed by
some authors using the standard concept of covariant spinor …elds. However, we do more: we show that postulating invariance under active rotational gauge transformations of the Dirac-Hestenes Lagrangian implies in
the equivalence (in a precise sense) of torsion free and non torsion free connections. Such a result suggests that the choice of a particular connection
in order to formulate spacetime …eld theories (which includes the gravitational …eld) is somewhat arbitrary. This issue is deeply investigated in
Chapter 11.
ABSTRACT In this chapter we investigate the nature of the gravitational
…eld. We …rst give a formulation for the theory of that …eld as a …eld in
Faraday’s sense (i.e., as of the same nature as the electromagnetic …eld) on
a 4-dimensional parallelizable manifold M . The gravitational …eld is represented through the 1-form …elds fga g dual to the parallelizable vector
…elds fea g. The ga ’s (a = 0; 1; 2; 3) are called gravitational potentials, and
it is imposed that at least for one of them, dga 6= 0. A metric like …eld
g = ab ga gb is introduced in M with the purpose of permitting the
construction of the Hodge dual operator and the Cli¤ord bundle of di¤erential forms C`(M; g), where g = ab ea eb . Next a Lagrangian density
for the gravitational potentials is introduced with consists of a Yang-Mills
term plus a gauge …xing term and an auto-interacting term. Maxwell like
equations for F a = dga are obtained from the variational principle and a
legitimate energy-momentum tensor for the gravitational …eld is identi…ed
which is given by a formula that at …rst look seems very much complicated.
Our theory does not uses any connection in M and we clearly demonstrate
that representations of the gravitational …eld as Lorentzian, teleparallel
and even general Riemann-Cartan-Weyl geometries depend only on the arbitrary particular connection (which may be or not to be metrical compatible) that we may de…ne on M . When the Levi-Civita connection of g in M
is introduced we prove that the postulated Lagrangian density for the gravitational potentials di¤ers from the Einstein-Hilbert Lagrangian density of
General Relativity only by a term that is an exact di¤erential. The theory
proceeds choosing the most simple topological structure for M , namely that
it is R4 , a choice that is compatible with present experimental data. With
the introduction of a Levi-Civita connection for the structure (M = R4 ; g)
as a mathematical aid we can exhibit a nice short formula for the genuine
energy-momentum of the gravitational …eld. Next, we introduce the Hamiltonian formalism and discuss possible generalizations of the gravitational
…eld theory (as a …eld in Faraday’s sense) when the graviton mass is not
null. Also we show using the powerful Cli¤ord calculus developed in previous chapters that if the structure (M = R4 ; g) possess at least one Killing
vector …eld, then the gravitational …eld equations can be written as a single Maxwell like equation, with a well de…ned current like term (of course,
associated to the energy-momentum tensor of matter and the gravitational
…eld). This result is further generalized for arbitrary vector …elds generating
one-parameter groups of di¤eomorphisms of M in Chapter 14. Chapter 11
xviii
Contents
ends with another possible interpretation of the gravitational …eld, namely
that it is represented by a particular geometry of a brane embedded in a
high dimensional pseudo-Euclidean space. Using the theory developed in
Chapter 5 we are able to write Einstein equation using the Ricci operator in such a way that its second member (of “wood” nature, according
to Einstein) is transformed (also according to Einstein) in the “marble”
nature of its …rst member. Such a form of Einstein equation shows that
the energy momentum quantities T a + 21 T ga (where T a = Tba gb are the
energy momentum 1-form …elds of matter and T = Taa ) which characterize
matter is represented by the negative square of the shape operator (S2 (ga ))
of the brane. Such a formulation thus give a mathematical expression for
the famous Cli¤ord “little hills” as representing matter.
ABSTRACT This chapter gives a Cli¤ord bundle approach to a theory
of the gravitational …eld where this …eld is interpreted as the curvature of
a Sl(2; C) gauge theory. The proposal of this presentation is to emphasize
the many faces of Einstein equation and in the development of the theory
a collection of some non trivial and useful formulas is derived and some
misconceptions in presentations of the theory of the gravitational …eld as
a Sl(2; C) gauge theory is discussed and …xed.
ABSTRACT
p In this Chapter we discuss three important issues. The …rst
is how i =
1 makes its appearance in classical electrodynamics and in
Dirac theory. This issue is important because
p if someone did not really
know the real meaning uncovered by i =
1 in these theories he may
infers nonsequitur results.2 After that we present some ‘Dirac like’ representations of Maxwell equations. Within the Cli¤ord bundle it becomes
obvious why there are so many ‘Dirac like’ representations of Maxwell
equations. The third issue discussed in this chapter are the mathematical Maxwell-Dirac equivalences of the …rst and second kinds and the relation of these mathematical equivalences with Seiberg-Witten equations
in Minkowski spacetime (M; ;D; ; ") which is the arena where we suppose physical phenomena to take place in this chapter. We denote by fx g
coordinates in Einstein-Lorentz-Poincaré gauge associated to an inertial reference frame e0 2 sec T M . Moreover fe = @x@ g 2 sec T M; ( = 0; 1; 2; 3)
is an orthonormalVbasis, with (e ; e ) =
= diag(1; 1; 1; 1) and
f = dx g 2 sec 1 T M ,! sec C`(M; ) is the dual basis of fe g.
ABSTRACT This chapter shows very clearly that the Cli¤ord and spinCli¤ord bundle formalism (introduced in previous chapters) o¤ers a very
simple way to write the equation of motion of a massive spinning particle.
Indeed this equation is immediately derived from Frenet equations for a
2 A clear example of the veracity of that statement is discussed in Section 13.2.1 using
material published in the literature.
Contents
xix
moving frame. Moreover, with addition of very a reasonable hypothesis the
deduced spinor equation for the spinning particle leads directly to a classical (nonlinear) Dirac-Hestenes equation. An additional hypothesis leads
to a linear Dirac-Hestenes equation and suggests automatically a probability interpretation for the Dirac-Hestenes wave function. We also show how
the Cli¤ord and spin-Cli¤ord bundle formalism permit the introduction of
multiform valued Lagrangians and a simple interpretation of the so-called
superparticle theory. Moreover, the Berezin di¤erential and integral calculus is shown to be no more than the result of contractions in an appropriate
Cli¤ord algebra. Also, the nature of super…elds is clearly disclosed.
ABSTRACT In the previous chapters we exhibit several di¤erent faces
of Maxwell, Einstein and Dirac equations. In this Chapter we show that
given certain conditions we can encode the contents
V of Einstein equation in
Maxwell like equations for a …eld F = dA 2 sec 2 T M (see below) , whose
contents can be also encoded in a Navier-Stokes equation. For the particular
cases when it happens that F 2 6= 0 we can also using the Maxwell-Dirac
equivalence of the …rst kind discussed in Chapter 13 to encode the contents
of the
in a Dirac-Hestenes equation for
2
V previousVquoted equations
V
21 ~
:
sec( 0 T M + 2 T M + 4 T M ) such that F =
ABSTRACT This chapter scrutinizes the theory of the so-called Elko
spinor …elds (in Minkowski spacetime) which always appears in pairs and
which from the algebraic point of view are in class …ve in Lounesto classi…cation of spinor …elds. We show how these …elds di¤ers from Majorana …elds
(which also are in class …ve in Lounesto classi…cation of spinor …elds) and
that Elko spinor …elds (as it is the case for Majorana …elds) do not satisfy
the Dirac equation. We discuss the class of generalized Majorana spinor
…elds (objects that are spinor with “components” with take values in a
Grassmann algebra) that satisfy Dirac equations, clarifying some obscure
presentations of that theory appearing in the literature. More important,
we show that the original presentation of the theory of Elko spinor …elds
as having mass dimension 1 leads to breakdown of Lorentz and rotational
invariance by a simple choice of the spatial axes in an inertial reference
frame. We then present a Lagrangian …eld theory for Elko spinor …elds
where these …elds (as it is the case of Dirac spinor …elds) have mass dimension 3=2. We explicitly demonstrate that Elko spinor …elds cannot couple
to the electromagnetic …eld, that they describe pairs of “magnetic” like
particles which are coupled to a short range su(2) gauge potential. Thus
they eventually can serve to model dark matter. The causal propagator
for the 3=2 mass dimension Elko spinor is explicitly calculated with the
Cli¤ord bundle of (multivector) …elds. Taking the opportunity given by the
formalism developed in our theory we present a very nice representation of
the parity operator acting on Dirac-Hestenes spinor …elds.
xx
Contents
ABSTRACT The appendix de…nes …ber bundles, principal bundles and
their associate vector bundles, recall the de…nitions of frame bundles, the
orthonormal frame bundle, jet bundles, product bundles and the Whitney
sums of bundles. Next, equivalent de…nitions of connections in principal
bundles and in their associate vector bundles are presented and it is shown
how these concepts are related to the concept of a covariant derivative in
the base manifold of the bundle. Also, the concept of exterior covariant
derivatives (crucial for the formulation of gauge theories) and the meaning
of a curvature and torsion of a linear connection in a manifold is recalled.
The concept of covariant derivative in vector bundles is also analyzed in
details in a way which, in particular, is necessary for the presentation of the
theory in Chapter12. Propositions are in general presented without proofs,
which can be found, e.g., in [77, 138, 195, 264, 267, 266, 288, 290].
ABSTRACT In this chapter we describe the contents of all chapters of the
book and their interrelationship.
ABSTRACT This chapter is dedicated to a thoughtful exposition of the
multiform and extensor calculus. Starting from the tensor algebra
V of a real
n-dimensional vector space V we construct the exterior algebra V of V .
Equipping V with a metric tensor g we introduce the Grassmmann algebra and next the Cli¤ord algebra C`(V ; g) associated
to the pair (V ; g).
V
The concept of Hodge dual of elements of V (called nonhomogeneous
multiforms) and of C`(V ; g) (also called nonhomogeneous multiforms or
Cli¤ord numbers) is introduced, and the scalar product and operations of
left and right contractions in these structures are de…ned. Several important formulas and “tricks of the trade” are presented. Next we introduce
the concept of extensors
V
V which are multilinear maps from p subspaces of
V to q subspaces of V and study their properties. Equipped with such
concept we study some properties of symmetric automorphisms and the
orthogonal Cli¤ord algebras introducing the gauge metric extensor (an essential ingredient for theories presented in other chapters). Also, we de…ne
the concepts of strain, shear and dilation associated with endomorphisms.
A preliminary exposition of the Minkoswski vector space is given and the
Lorentz and Poincaré groups are introduced. In the remaining of the chapter we give an original presentation of the theory of multiform functions of
multiform variables. For these objects we de…ne the concepts of limit, continuity and di¤erentiability. We study in details the concept of directional
derivatives of multiform functions and solve several nontrivial exercises to
clarify how to work with these notions, which in particular are crucial for
the formulation of Chapter 8 which deals with a Cli¤ord algebra Lagrangian
formalism of …eld theory in Minkowski spacetime.
ABSTRACT This chapter reviews the classi…cation of the real and complex Cli¤ord algebras and analyze the relationship between some particular
algebras that are important in physical applications, namely the quater-
Contents
xxi
nion algebra (R0;2 ), Pauli algebra (R3;0 ), the spacetime algebra (R1;3 ), the
Majorana algebra (R3;1 ) and the Dirac algebra (R4;1 ). A detailed and original theory disclosing the hidden geometrical meaning of spinors is given
through the introduction of the concepts of algebraic, covariant and DiracHestenes spinors. The relationship between these kinds of spinors (that
carry the same mathematical information) is elucidated with special emphasis for cases of physical interest. We investigate also how to reconstruct
a spinor from their so-called bilinear invariants and present Lounesto’s classi…cation of spinors. Also, Majorana, Weyl spinors, the dotted and undotted
algebraic spinors are discussed with the Cli¤ord algebra formalism.
ABSTRACT The main objective of this chapter is to present a Cli¤ord
bundle formalism for the formulation of the di¤erential geometry of a manifold M , equipped with metric …elds g 2 sec T20 M and g 2 sec T02 M for
the tangent and cotangent bundles. We start by …rst recalling the standard
formulation and main concepts of the di¤erential geometry of a di¤erential
manifold M . We introduce in M the Cartan bundle of di¤erential forms,
de…ne the exterior derivative, Lie derivatives, and also brie‡y review concepts as chains, homology and cohomology groups, de Rham periods, the
integration of form …elds and Stokes theorem. Next, after introducing the
metric …elds g and g in M we introduce the Hodge bundle presenting the
Hodge star and the Hodge coderivative operators acting on sections of this
bundle. We moreover recall concepts as the pullback and the di¤erential of
maps, connections and covariant derivatives, Cartan’s structure equations,
the exterior covariant di¤erential of (p + q)-indexed r-forms, Bianchi identities and the classi…cation of geometries on M when it is equipped with
a metric …eld and a particular connection. The spacetime concept is rigorously de…ned. We introduce and scrutinized the structure of the Cli¤ord
bundle of di¤erential forms (C`(M; g)) of M and introduce the fundamental
concept of the Dirac operator (associated to a given particular connection
de…ned in M ) acting on Cli¤ord …elds (sections of C`(M; g)). We show that
the square of the Dirac operator (associated to a Levi-Civita connection in
M ) has two fundamental decompositions, one in terms of the derivative and
Hodge codi¤erential operators and other in terms of the so-called Ricci and
D’Alembertian operators. A so-called Einstein operator is also introduced
in this context. These decompositions of the square of the Dirac operator
are crucial for the formulation of important ideas concerning the construction of gravitational theories as discussed in particular in Chapters 9, 11,
15. The Dirac operator associated to an arbitrary (metrical compatible)
connection de…ned in M and its relation with the Dirac operator associated to the Levi-Civita connection of the pair (M; g) is discussed in details
and some important formulas are obtained. The chapter also discuss some
applications of the formalism, e.g., the formulation of Maxwell equations
in the Hodge and Cli¤ord bundles and formulation of Einstein equation
in the Cli¤ord bundle using the concept of the Ricci and Einstein operators. A preliminary account of the crucial di¤erence between the concepts
of curvature of a connection in M and the concept of bending of M as a
hypersurface embedded in a (pseudo)-Euclidean space of high dimension
xxii
Contents
(a property characterized by the concept of the shape tensor, discussed
in details in Chapter 5) is given by analyzing a speci…c example, namely
the one involving the Levi-Civita and the Nunes connections de…ned in a
punctured 2-dimensional sphere. The chapter ends analyzing a statement
referred in most physical textbooks as “tetrad postulate” and shows how
not properly de…ning concepts can produce a lot of misunderstanding and
invalid statements.
ABSTRACT Using the Cli¤ord bundle formalism (CBF) of di¤erential
forms and the theory of extensors acting on C`(M; g) (introduced in Chapter 4) we …rst recall the formulation of the intrinsic geometry of a di¤erential manifold M (a brane) equipped with a metric …eld g of signature
(p; q) and an arbitrary metric compatible connection r introducing the
torsion (2 1)-extensor …eld , the curvature (2 2) extensor …eld R and
(once …xing a gauge) the connection (1 2)-extensor ! and the Ricci operator @ ^ @ (where @ is the Dirac operator acting on sections of C`(M; g))
which plays an important role in this paper. Next, using the CBF we give
a thoughtful presentation of the Riemann or the Lorentzian geometry of
an orientable submanifold M (dim M = m) living in a manifold M (such
that M ' Rn is equipped with a semi-Riemannian metric g with signature
(p; q) and p + q = n and its Levi-Civita connection D) and where there
is de…ned a metric g = i g, where i : M ! M is the inclusion map. We
prove several equivalent forms for the curvature operator R of M . Moreover we show a very important result, namely that the Ricci operator of
M is the (negative) square of the shape operator S of M (object obtained
by applying the restriction on M of the Dirac operator @ of C`(M; g) to
the projection operator P). Also we disclose the relationship between the
(1 2)-extensor ! and the shape biform S (an object related to S). The
results obtained are used in Chapter 11 to give a mathematical formulation
to Cli¤ord’s theory of matter [358].
ABSTRACT The chapter has as main objective to clarify some important
concepts appearing in relativistic spacetime theories and which are necessary of a clear understanding of our view concerning the formulation and
understanding of Maxwell, Dirac and Einstein theories. Using the de…nition
of a Lorentzian spacetime structure (M; g; D; g ; ") presented in Chapter
4 we introduce the concept of a reference frame in that structure which
is an object represented by a given unit timelike vector …eld Z 2 sec T U
(U
M ). We give two classi…cation schemes for these objects, one according to the decomposition of DZ and other according to the concept
of synchronizability of ideal clocks (at rest in Z). The concept of a coordinate chart covering U and naturally adapted to the reference frame Z
is also introduced. We emphasize that the concept of a reference frame is
di¤erent (but related) from the concept of a frame which is a section of
the frame bundle. The concept of Fermi derivative is introduced and the
physical meaning of Fermi transport is elucidated, in particular we show
the relation between the Darboux biform of the theory of Frenet frames
Contents
xxiii
and its decomposition as an invariant sum of a Frenet biform F (describing Fermi transport) and a rotation biform S such that the contraction of
? S with the velocity …eld v of the spinning particle is directly associated
with the so-called Pauli-Lubanski spin 1-form. We scrutinize the concept
of di¤eomorphism invariance of general spacetime theories and of General
Relativity in particular, discuss what meaning can be given to the concept
of physically equivalent reference frames and what one can understand by
a principle of relativity. Examples are given and in particular, it is proved
that in a general Lorentzian spacetime (modelling a gravitational …eld according to General Relativity) there is in general no reference frame with
the properties (according to the scheme classi…cations) of the inertial referenced frames of special relativity theories. However there are in such a case
reference frames called pseudo inertial reference frames (PIRFs) that have
most of the properties of the inertial references frames of special relativity
theories. We also discuss a formulation (that one can …nd in the literature) of a so-called principle of local Lorentz invariance and show that
if it is interpreted as physical equivalence of PIRFs then it is not valid.
The Chapter ends with a brief discussion of di¤eormorphism invariance
applied to Schwarzschild original solution and the Droste-Hilbert solution
of Einstein equation which are shown to be not equivalent (the underlying
manifolds have di¤erent topologies) and what these solutions have to do
with the existence of blackholes in the “orthodoxof General Relativity.
ABSTRACT This chapter presents an original approach to the theory
of Dirac-Hestenes spinor …elds. After recalling details the structure of the
Cli¤ord bundle of a Lorentzian manifold structure (M; g) we introduce the
concept of spin structures on (M; g), de…ne spinor bundles and spinor …elds.
Left and right spin-Cillford bundles are presented and the concept of DiracHestenes spinor …elds (as sections of a special spin-Cli¤ord bundle denoted
by C`rSpine1;3 (M; g)) is investigated in details disclosing their real nature
and showing how these objects can be represented as some equivalence
classes of even sections of the Cli¤ord bundle C`(M; g). We introduce and
obtain the connection (covariant derivative) acting on spin-Cli¤ord bundles
C`rSpine1;3 (M; g) from a Levi-Civita connection acting on the tensor bundle
of (M; g). Associated with that connection we next introduce the standard
spin-Dirac operator acting on sections of C`rSpine1;3 (M; g) (not to be confused
with the Dirac operator acting on sections of the Cli¤ord bundle de…ned in
Chapter 4). We discuss how to write Dirac equation using the spin-Cli¤ord
bundle formalism and clari…es its properties and many misunderstandings
relating to such a notion that are spread in the literature. We also discuss
the notion of what is known as amorphous spinor …elds showing that these
objects cannot represent fermion …elds. Finally, the Chapter also presents a
simple proof of the famous Lichnerowicz formula which relates the square of
the standard spin-Dirac operator to the curvature of the (spin) connection
and also we obtain a generalization of that formula for the square of a
general spin-Dirac operator associated to a general (metric compatible)
Riemann-Cartan connection de…ned in (M; g).
xxiv
Contents
ABSTRACT Using tools introduced in previous chapters, particularly the
concept of Cli¤ord and spin-Cli¤ord bundles (and the representations of
sections of the spin-Cli¤ord bundles as equivalence classes of sections the
Cli¤ord bundle), extensor …elds and the Dirac operator we give an uni…ed and original approach to the Lagrangian …eld theory in Minkowski
spacetime with special emphasis on the Maxwell and Dirac-Hestenes …elds.
We derive for these …elds their canonical energy-momentum extensor …elds
and also their angular momentum and spin momentum extensor …elds. In
particular we show that the antisymmetric part of the canonical energymomentum tensor is the “source” of spin of the …eld. Several nontrivial
exercises are solved with details in order to help the reader to familiarize
with the formalism and to make contact with standard formulations of …eld
theory.
ABSTRACT In this chapter we examine in details the conditions for existence of conservation laws of energy-momentum and angular momentum for
the matter …elds of on a general Riemann-Cartan spacetime (M; g; r; g ; ")
and also in the particular case of Lorentzian spacetimes M = (M; g; D; g ; "
) which as we already know model gravitational …elds in the GRT [38].
Riemann-Cartan spacetimes are supposed to model generalized gravitational …elds in so called Riemann-Cartan theories. In what follows, we
suppose that in (M; g; r; g ; ") (or M) a set of dynamic …elds live and
interact. Of course, we want that the Riemann-Cartan spacetime admits
spinor …elds, and from what we learned in Chapter 7, this implies that the
orthonormal frame bundle must be trivial. This permits a great simpli…cation in our calculations. Moreover, we will suppose, for simplicity that the
dynamic V
…elds A , A = 1; 2; :::; n, are in general distinct r-forms3 , i.e., each
A
2 sec r T M ,! sec C`(M; g), for some r = 0; 1; :::; 4. Before we start
our enterprise we think it is useful to recall some results which serve also
the purpose to …x the notation for this chapter.
ABSTRACT In this chapter we give a formulation of the Dirac-Hestenes
equation on a Riemann-Cartan manifold (M; g; r; g ; ") using the Cli¤ord
and spin-Cli¤ord bundles formalism. We show that the obtained equation
which follows for a properly chosen Lagrangian density (heuristically based
on the principle of minimum coupling) agrees with the one proposed by
some authors using the standard concept of covariant spinor …elds. However, we do more: we show that postulating invariance under active rotational gauge transformations of the Dirac-Hestenes Lagrangian implies in
the equivalence (in a precise sense) of torsion free and non torsion free connections. Such a result suggests that the choice of a particular connection
in order to formulate spacetime …eld theories (which includes the gravi3 This
is not a serious restriction in the formalism since we already learned how (given
a spinorial frame) to represent spinor …elds by sums of even multiform …elds.
Contents
xxv
tational …eld) is somewhat arbitrary. This issue is deeply investigated in
Chapter 11.
ABSTRACT In this chapter we investigate the nature of the gravitational
…eld. We …rst give a formulation for the theory of that …eld as a …eld in
Faraday’s sense (i.e., as of the same nature as the electromagnetic …eld) on
a 4-dimensional parallelizable manifold M . The gravitational …eld is represented through the 1-form …elds fga g dual to the parallelizable vector
…elds fea g. The ga ’s (a = 0; 1; 2; 3) are called gravitational potentials, and
it is imposed that at least for one of them, dga 6= 0. A metric like …eld
g = ab ga gb is introduced in M with the purpose of permitting the
construction of the Hodge dual operator and the Cli¤ord bundle of di¤erential forms C`(M; g), where g = ab ea eb . Next a Lagrangian density
for the gravitational potentials is introduced with consists of a Yang-Mills
term plus a gauge …xing term and an auto-interacting term. Maxwell like
equations for F a = dga are obtained from the variational principle and a
legitimate energy-momentum tensor for the gravitational …eld is identi…ed
which is given by a formula that at …rst look seems very much complicated.
Our theory does not uses any connection in M and we clearly demonstrate
that representations of the gravitational …eld as Lorentzian, teleparallel
and even general Riemann-Cartan-Weyl geometries depend only on the arbitrary particular connection (which may be or not to be metrical compatible) that we may de…ne on M . When the Levi-Civita connection of g in M
is introduced we prove that the postulated Lagrangian density for the gravitational potentials di¤ers from the Einstein-Hilbert Lagrangian density of
General Relativity only by a term that is an exact di¤erential. The theory
proceeds choosing the most simple topological structure for M , namely that
it is R4 , a choice that is compatible with present experimental data. With
the introduction of a Levi-Civita connection for the structure (M = R4 ; g)
as a mathematical aid we can exhibit a nice short formula for the genuine
energy-momentum of the gravitational …eld. Next, we introduce the Hamiltonian formalism and discuss possible generalizations of the gravitational
…eld theory (as a …eld in Faraday’s sense) when the graviton mass is not
null. Also we show using the powerful Cli¤ord calculus developed in previous chapters that if the structure (M = R4 ; g) possess at least one Killing
vector …eld, then the gravitational …eld equations can be written as a single Maxwell like equation, with a well de…ned current like term (of course,
associated to the energy-momentum tensor of matter and the gravitational
…eld). This result is further generalized for arbitrary vector …elds generating
one-parameter groups of di¤eomorphisms of M in Chapter 14. Chapter 11
ends with another possible interpretation of the gravitational …eld, namely
that it is represented by a particular geometry of a brane embedded in a
high dimensional pseudo-Euclidean space. Using the theory developed in
Chapter 5 we are able to write Einstein equation using the Ricci operator in such a way that its second member (of “wood” nature, according
to Einstein) is transformed (also according to Einstein) in the “marble”
nature of its …rst member. Such a form of Einstein equation shows that
the energy momentum quantities T a + 21 T ga (where T a = Tba gb are the
xxvi
Contents
energy momentum 1-form …elds of matter and T = Taa ) which characterize
matter is represented by the negative square of the shape operator (S2 (ga ))
of the brane. Such a formulation thus give a mathematical expression for
the famous Cli¤ord “little hills” as representing matter.
ABSTRACT This chapter gives a Cli¤ord bundle approach to a theory
of the gravitational …eld where this …eld is interpreted as the curvature of
a Sl(2; C) gauge theory. The proposal of this presentation is to emphasize
the many faces of Einstein equation and in the development of the theory
a collection of some non trivial and useful formulas is derived and some
misconceptions in presentations of the theory of the gravitational …eld as
a Sl(2; C) gauge theory is discussed and …xed.
ABSTRACT
p In this Chapter we discuss three important issues. The …rst
is how i =
1 makes its appearance in classical electrodynamics and in
Dirac theory. This issue is important because
p if someone did not really
know the real meaning uncovered by i =
1 in these theories he may
infers nonsequitur results.4 After that we present some ‘Dirac like’ representations of Maxwell equations. Within the Cli¤ord bundle it becomes
obvious why there are so many ‘Dirac like’ representations of Maxwell
equations. The third issue discussed in this chapter are the mathematical Maxwell-Dirac equivalences of the …rst and second kinds and the relation of these mathematical equivalences with Seiberg-Witten equations
in Minkowski spacetime (M; ;D; ; ") which is the arena where we suppose physical phenomena to take place in this chapter. We denote by fx g
coordinates in Einstein-Lorentz-Poincaré gauge associated to an inertial reference frame e0 2 sec T M . Moreover fe = @x@ g 2 sec T M; ( = 0; 1; 2; 3)
is an orthonormalVbasis, with (e ; e ) =
= diag(1; 1; 1; 1) and
f = dx g 2 sec 1 T M ,! sec C`(M; ) is the dual basis of fe g.
ABSTRACT This chapter shows very clearly that the Cli¤ord and spinCli¤ord bundle formalism (introduced in previous chapters) o¤ers a very
simple way to write the equation of motion of a massive spinning particle.
Indeed this equation is immediately derived from Frenet equations for a
moving frame. Moreover, with addition of very a reasonable hypothesis the
deduced spinor equation for the spinning particle leads directly to a classical (nonlinear) Dirac-Hestenes equation. An additional hypothesis leads
to a linear Dirac-Hestenes equation and suggests automatically a probability interpretation for the Dirac-Hestenes wave function. We also show how
the Cli¤ord and spin-Cli¤ord bundle formalism permit the introduction of
multiform valued Lagrangians and a simple interpretation of the so-called
superparticle theory. Moreover, the Berezin di¤erential and integral calcu4 A clear example of the veracity of that statement is discussed in Section 13.2.1 using
material published in the literature.
Contents
xxvii
lus is shown to be no more than the result of contractions in an appropriate
Cli¤ord algebra. Also, the nature of super…elds is clearly disclosed.
ABSTRACT In the previous chapters we exhibit several di¤erent faces
of Maxwell, Einstein and Dirac equations. In this Chapter we show that
given certain conditions we can encode the contents
V of Einstein equation in
Maxwell like equations for a …eld F = dA 2 sec 2 T M (see below) , whose
contents can be also encoded in a Navier-Stokes equation. For the particular
cases when it happens that F 2 6= 0 we can also using the Maxwell-Dirac
equivalence of the …rst kind discussed in Chapter 13 to encode the contents
of the
in a Dirac-Hestenes equation for
2
V
V previousVquoted equations
21 ~
:
sec( 0 T M + 2 T M + 4 T M ) such that F =
ABSTRACT This chapter scrutinizes the theory of the so-called Elko
spinor …elds (in Minkowski spacetime) which always appears in pairs and
which from the algebraic point of view are in class …ve in Lounesto classi…cation of spinor …elds. We show how these …elds di¤ers from Majorana …elds
(which also are in class …ve in Lounesto classi…cation of spinor …elds) and
that Elko spinor …elds (as it is the case for Majorana …elds) do not satisfy
the Dirac equation. We discuss the class of generalized Majorana spinor
…elds (objects that are spinor with “components” with take values in a
Grassmann algebra) that satisfy Dirac equations, clarifying some obscure
presentations of that theory appearing in the literature. More important,
we show that the original presentation of the theory of Elko spinor …elds
as having mass dimension 1 leads to breakdown of Lorentz and rotational
invariance by a simple choice of the spatial axes in an inertial reference
frame. We then present a Lagrangian …eld theory for Elko spinor …elds
where these …elds (as it is the case of Dirac spinor …elds) have mass dimension 3=2. We explicitly demonstrate that Elko spinor …elds cannot couple
to the electromagnetic …eld, that they describe pairs of “magnetic” like
particles which are coupled to a short range su(2) gauge potential. Thus
they eventually can serve to model dark matter. The causal propagator
for the 3=2 mass dimension Elko spinor is explicitly calculated with the
Cli¤ord bundle of (multivector) …elds. Taking the opportunity given by the
formalism developed in our theory we present a very nice representation of
the parity operator acting on Dirac-Hestenes spinor …elds.
ABSTRACT The appendix de…nes …ber bundles, principal bundles and
their associate vector bundles, recall the de…nitions of frame bundles, the
orthonormal frame bundle, jet bundles, product bundles and the Whitney
sums of bundles. Next, equivalent de…nitions of connections in principal
bundles and in their associate vector bundles are presented and it is shown
how these concepts are related to the concept of a covariant derivative in
the base manifold of the bundle. Also, the concept of exterior covariant
derivatives (crucial for the formulation of gauge theories) and the meaning
of a curvature and torsion of a linear connection in a manifold is recalled.
xxviii
Contents
The concept of covariant derivative in vector bundles is also analyzed in
details in a way which, in particular, is necessary for the presentation of the
theory in Chapter12. Propositions are in general presented without proofs,
which can be found, e.g., in [77, 138, 195, 264, 267, 266, 288, 290].
ABSTRACT Speci…cally, we …rst show in Section 15.1 how each LSTS
(M; g; D; g ; ") which, as we already know, is a model of a gravitational
…eld generated by T 2 sec T20 M (the matter plus non gravitational …elds
energy-momentum momentum tensor) in Einstein GRT is such that for
any K 2 sec T M which is a vector …eld generating a one parameter group
of di¤eomorphisms of M we can encode Einstein equation in Maxwell like
equations satis…ed by F = dK where K = g(K; ) with a well determined
current term named the Komar current JK , whose explicit form is given.
Next we show in Section 15.2 that when K = A is a Killing vector …eld, due
to some noticeable results (Eqs.(15.28) and (15.29)) the Komar current acquires a very simple form and is then denoted JA . Next, interpreting, as in
Chapter 11 the Lorentzian spacetime structure (M; g; D; g ; ") as no more
than an useful representation for the gravitational …eld represented by the
gravitational potentials fga g which live in Minkowski spacetime (here denoted by (M = R4 ; g; D; g ; ")) we show in Section 15.3 that we can …nd
a Navier-Stokes equation which encodes the contents of the Maxwell like
equations (already encoding Einstein equations) once a proper identi…cation is made between the variables entering the Navier-Stokes equations
and the ones de…ning A = g(A; ) and F = dA, objects clearly related (see
Eq.(15.49)) to A and F = dA. We also explicitly determine also the constraints imposed by the nonhomogeneous Maxwell like equation F = JA
g
on the variables entering the Navier-Stokes equations and the ones de…ning
A (or A).
This is page xxix
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Preface
In this second edition of our book we rewrote some sections and added new
ones to some chapters for better intelligibility, as e.g., the new section 6.9
titled “Schwarzschild Original Solution and the Existence of Black Holes”.
Three new chapters (in this edition, Chapters 5, 15 and 16 ) are included.
Chapter 5 gives a Cli¤ord bundle approach to the Riemannian or semiRiemannian di¤erential geometry of branes understood as submanifolds of
a Euclidean or pseudo-Euclidean space of large dimension. We introduce
the important concept of the projection operator, study its properties and
introduce several other operators associated to it, as the shape operator and
the shape biform, crucial objects to de…ne the bending of a submanifold
equipped with a Euclidean or pseudo-Euclidean embedded in a Euclidean
or pseudo-Euclidean space of large dimension. We give several di¤erent expressions for the curvature biform operator in terms of derivatives of the
projection operator and of the shape operator and prove the remarkable
formula S2 (v) = @ ^ @(v), which says that the square of the shape operator applied to a 1-form …eld v is equal to the negative of the Ricci operator
(introduced tin Chapter 4) applied to a 1-form …eld v. Such result is used
in Chapter 11 to show how to transform in “marble”the “wood”part of
Einstein equation. By this we mean that we can express its second member
containing a phenomenological energy-momentum tensor in a purely geometrical term involving the square of the shape operator. More, Chapter
11 besides including the results just mentioned has been completely rewritten and is now titled “On the Nature of the Gravitational Field ”. We hope
that this chapter leaves clear that the interpretation of the gravitational
…eld as the geometry of a Lorentzian spacetime structure is only one among
xxx
Preface
di¤erent possible choices, not a necessary one. We added to this second
edition Chapter 15 titled — “Maxwell, Einstein, Dirac and Navier-Stokes
Equations”— which besides reveling some surprising relations concerning
the many faces of Maxwell, Dirac, Einstein and the Navier-Stokes equations also clarify the meaning of the so called Komar currents and …nd
their explicit form in General Relativity theory. There is now also Chapter
16 that analyzes the similarities and main di¤erences between Dirac, Majorana and elko spinor …elds, a subject that is receiving a lot of attention
in the last few years. We present an alternative theory for elko spinor …elds
of mass dimension 3=2 (instead of mass dimension 1 as originally proposed
in [6]) and show that our elko spinor …elds can be used to describe electric
neutral particles carrying “magnetic like”charges charges with short range
interaction mediated by a su(2)-valued gauge potential. We also change (in
relation to the …rst edition) some symbols for better clarity on the typing
of some formulas. Of course, in a book of this size it is almost impossible
not to use a same symbol to represent (at di¤erent places) di¤erent objects.
We tried to minimize such occurrences and a list of the principal symbols
is given at the end of the book. There the reader will …nd also a list of
the main references we used, list of acronyms and an index. A detailed
description of the contents of the chapters is given in Chapter 1.
We are particularly grateful to the many helpful discussions on the subjects presented in this book that we had for years with P. Anglès, G. Bruhn,
F. W. Hehl, R. F. Leão, (the late) P. Lounesto, (the late) Ian Porteous,
R. A. Mosna, A. M. Moya, E. A. Notte Cuello, R. da Rocha, J. E. Maiorino, Z. Oziewicz, F. Grangeiro Rodrigues, Q. A. G. Souza, J. Vaz, Jr. and
S. A. Wainer. We are also grateful to the many readers of the …rst edition that pointed to us misprints and some errors that (we hope) have
been corrected in the present edition. Of course, they are not guilty for
any remaining error or misconception that the reader may eventually …nd.
Moreover, the authors will be grateful to anyone who inform them of any
additional correction that should be made to the text.
Campinas, August 2015
Waldyr Alves Rodrigues Jr. and Edmundo Capelas de Oliveira.
This is page 1
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1
Introduction
Maxwell, Dirac and Einstein’s equations are certainly among the most important equations of XX-th century Physics and it is our intention in this
book to investigate some of the many faces1 of these equations and their
relationship and to discuss some foundational issues involving some of the
theories where they appear. To do that, let us brie‡y recall some facts.
Maxwell equations which date back to the XIX-th century encodes all
classical electromagnetism, i.e., they describe the electromagnetic …elds
generated by charge distributions in arbitrary motion. Of course, when
Maxwell formulated his theory the arena where physical phenomena were
supposed to occur was a Newtonian spacetime, a structure containing a
manifold which is di¤eomorphic to R R3 , the …rst factor describing Newtonian absolute time [342] and the second factor the Euclidean space of
our immediate perception2 . In his original approach Maxwell presented his
equations as a system of eight linear …rst order partial di¤erential equations involving the components of the electric and magnetic …elds [237]
generated by charge and currents distributions with prescribed motions in
vacuum3 . It was only after Heaviside [168], Hertz and Gibbs that those
1 By many faces of Maxwell, Dirac and Einstein’s equations, we mean the many
di¤erent ways in which those equations can be presented using di¤erent mathematical
theories.
2 For details on the Newtonian spacetime structure see, e.g., [342].
3 Inside a material the equations involve also other …elds, the so-called polarization
…elds. See details in [237].
2
1. Introduction
equations were presented using vector calculus, which by the way, is the
form they appear until today in elementary textbooks on Electrodynamics
and Engineering Sciences. In the vector calculus formalism Maxwell equations are encoded in four equations involving the well known divergent and
rotational operators. The motion of charged particles under the action of
prescribed electric and magnetic …elds was supposed in Maxwell’s time to
be given by Newton’s second law of motion, with the so-called Lorentz
force acting on the charged particles. It is a well known story that at the
beginning of the XX-th century, due to the works of Lorentz, Poincaré and
Einstein [449], it has been established that the system of Maxwell equations
was compatible with a new version of spacetime structure serving as arena
for physical phenomena, namely Minkowski spacetime (to be discussed in
details in the following chapters). It became also obvious that the classical
Lorentz force law needed to be modi…ed in order to leave the theory of
classical charged particles and their electromagnetic …elds invariant under
spacetime transformations in Minkowski spacetime de…ning a representation of the Poincaré group. Such a condition was a necessary one for the
theory to satisfy the Principle of Relativity.4
Modern presentations of Maxwell equations make use of the theory of
di¤erential forms and succeed in writing the original system of Maxwell
equations as two equations involving the exterior derivative operator and
the so-called Hodge star operator.5
Now, one of the most important constructions of human mind in the
XX-th century has been Quantum Theory. Here, we need to recall that in
a version of that theory which is also in accordance with the Principle of
Relativity, the motion of a charged particle under the action of a prescribed
electromagnetic …eld is not described by the trajectory of the particle in
Minkowski spacetime predicted by the Lorentz force law once prescribed
initial conditions are given. Instead (as it is supposed well known by any
reader of this book) the state of motion of the particle is described by a
(covariant) Dirac spinor …eld (Chapter 7), which is a section of a particular spinor bundle. In elementary presentations6 of the subject a Dirac
spinor …eld is simply a mapping from Minkowski spacetime (expressed in
global coordinates in the Einstein-Lorentz-Poincaré gauge) to C4 . At any
given spacetime point, the space of Dirac spinor …elds is said to carry7 the
D(1=2;0) D(0;1=2) representation of Sl(2; C), which is the universal covering group of the proper orthochronous Lorentz group. Moreover the Dirac
4 The reader is invited to study Chapter 6 in detail to know the exact meaning of
this statement.
5 These mathematical tools are introduced in Chapter 4.
6 The elementary approach is related with a choice of a global spin coframe (Chapter
7) in spacetime.
7 The precise mathematical meaning of this statement can be given only within the
theory of spin-Cli¤ord bundles, as described in Chapter 6.
1. Introduction
3
spinor …eld in interaction with a prescribed electromagnetic potential is
supposed to satisfy a linear di¤erential equation called the Dirac equation.
Such theory was called relativistic quantum mechanics (or …rst quantized
relativistic quantum mechanics)8 .
Now, the covariant Dirac spinor …eld used in quantum electrodynamics
is, at …rst sight, an object of a mathematical nature very di¤erent from
that of the electromagnetic …eld, which is described by a 2-form …eld (or
the electromagnetic potential that is described by a 1-form …eld). As a
consequence we cannot see any relationship between these …elds or between
Maxwell and Dirac equations.
It would be nice if those …elds which are the dependent variables in
Maxwell and Dirac equations, could be represented by objects of the same
mathematical nature. As we are going to see, this is indeed possible. It turns
out that Maxwell …elds can be represented by appropriate homogeneous
sections of the Cli¤ord bundle of di¤erential forms C`(M; g) and Dirac
…elds can be represented (once we …x a spin coframe) by a sum of even
homogeneous sections of C`(M; g). These objects are called representatives
of Dirac-Hestenes spinor …elds9 . Once we arrive at this formulation, which
requires of course the introduction of several mathematical tools, we can see
relationships between those equations that are not apparent in the standard
formalism (Chapter 13). Finally, we can also easily see the meaning of the
many Dirac-like presentations of Maxwell equations that appeared in the
literature during the last century. Moreover, we will see that our formalism
is related in an intriguing way to formalisms used in modern theories of
Physics, like the theories of superparticles and super…elds (Chapter 14).
Besides providing mathematical unity to the theory of Maxwell and Dirac
…elds, we give also a Cli¤ord bundle approach to the di¤erential geometry
of a Riemann-Cartan-Weyl spacetime. This is done with the objective of
…nding a description of the gravitational …eld as a set of sections of an appropriate Cli¤ord bundle over Minkowski spacetime, which moreover satisfy equations equivalent to Einstein’s equations on an e¤ective Lorentzian
spacetime. That enterprise is not just a mathematical game. There are serious reasons for formulating such a theory. To understand the most impor8 Soon it became clear that the interaction of the electromagnetic …eld with the Dirac
spinor …eld could produce pairs (electrons and positrons). Besides that it was known
since 1905 that the classical concept of the electromagnetic …eld was not in accord with
experience and that the concept of photons as quanta of the said …eld needed to be
introduced. The theory that deals with the interaction of photons and electrons (and
positrons) is a particular case of a second quantized renormalizable quantum …eld theory
and is called quantum electrodynamics. In that theory the electromagnetic and the Dirac
spinor …elds are interpreted as operator valued distributions [53] acting on the Hilbert
space of the state vectors of the system. We shall not discuss further this theory in this
book, but will return to some of its issues in a sequel volume [283].
9 The details are given in Chapter 7 whose intelligibility presupposes that the reader
has studied Chapter 3.
4
1. Introduction
tant reason (in our opinion), let us recall some facts. First, keep in mind
that in Einstein’s General Relativity theory (GRT) a gravitational …eld
is modeled by a Lorentzian spacetime10 (which is a particular RiemannCartan-Weyl spacetime). This means that in GRT the gravitational …eld
is an object of a di¤erent physical nature from the electromagnetic and
Dirac-Hestenes …elds, which are …elds living in a spacetime. This distinct
nature of the gravitational …eld implies (as will be proved in Chapter 10)
the lack of conservation laws of energy-momentum and angular momentum
in that theory. But how to formulate a more comprehensible theory?
A possible answer is provided by the theory of extensors and symmetric automorphisms of Cli¤ord algebras developed in Chapter 2, which,
together with the Cli¤ord bundle formalism of Chapter 4, suggests naturally to interpret Einstein theory as a …eld theory for cotetrad …elds (de…ning a coframe) in Minkowski spacetime. How this is done is described in
the appropriate chapters that follow and which are now summarized.
Chapter 2, called Multiform and Extensor Calculus, gives a detailed
introduction to exterior and Grassmann algebras and to the Cli¤ord algebra of multiforms. The presentation has been designed in order for any
serious student to acquire quickly the necessary skill which will permit him
to reproduce the calculations done in the text. Thus, several exercises are
proposed and many of them are solved in detail. Also in Chapter 2 we
introduce the concept of extensor, which is a natural generalization of the
concept of tensor and which plays an important role in our developments.
In particular extensors appear in the formulation of the theory of symmetric automorphisms and orthogonal Cli¤ord products, which permit to see
di¤erent Cli¤ord algebras associated with vector spaces of the same dimension, but equipped with metrics of di¤erent signatures, as deformations of
each other. A theory of multiform functions and of the several di¤erent
derivatives operators acting on those multiform functions is also presented
in Chapter 2. Such a theory is the basis for the presentation of Lagrangian
…eld theory, a subject discussed in Chapters 8 and 9. There, we see also
the crucial role of extensor …elds when representing energy-momentum and
angular momentum in …eld theory.
Chapter 3 describes the hidden geometrical nature of spinors, and it is
our hope to have presented a fresh view on the subject. So, that chapter
starts recalling some fundamental results from the representation theory of
associative algebras and then gives the classi…cation of real and complex
Cli¤ord algebras, and in particular discloses the relationship between the
spacetime algebra and the Majorana, Dirac and Pauli algebras, which are
the most important Cli¤ord algebras in our study of Maxwell, Dirac and
Einstein’s equations. Next, the concepts of left, right and bilateral ideals
on Cli¤ord algebras are introduced, and the notion of algebraically and
1 0 Indeed,
by an equivalence class of di¤eomorphic Lorentzian spacetimes.
1. Introduction
5
geometrically equivalent ideals is given. Equipped with these notions we
give original de…nitions of algebraic and covariant spinors. For the case of
a vector space equipped with a metric of signature (1; 3) we show that it
is very useful to introduce the concept of Dirac-Hestenes spinors. The hidden geometrical nature of these objects is then disclosed. We think that
the concept of Dirac-Hestenes spinor and of Dirac-Hestenes spinor …elds
(introduced in Chapter 6) are very important and worth to be known
by every physicist and mathematician. Indeed, we shall see in Chapter
12 how the concept of Dirac-Hestenes spinor …elds permits us to …nd unsuspected mathematical relations between Maxwell and Dirac equations
and also between those equations and the Seiberg-Witten equations (in
Minkowski spacetime). Chapter 3 discuss also Majorana, Weyl and dotted and undotted algebraic spinors. Bilinear invariants, Fierz identities and
the notion of boomerangs are also introduced since, in particular, the Fierz
identities play an important role in the interpretation of Dirac theory.
Chapter 4 discusses aspects of di¤erential geometry that are essential
for a reasonable understanding of spacetime theories and in our opinion necessary to avoid wishful thinking concerning mathematical possibilities and
physical reality. Our main purpose is to present a Cli¤ord bundle approach
to the geometry of a general Riemann-Cartan-Weyl space M (a di¤erentiable manifold) carrying a metric g 2 sec T20 M of signature (p; q), a connection r, an orientation g . Such a structure is denoted by (M; g; r; g ).
The pair (r; g) de…nes a geometry for M . When M is 4-dimensional (and
satis…es some other requirements to be discussed later) and g has signature
(1; 3), the pair (M; g) is called a Lorentzian manifold. Moreover, endow M
with a general connection r, with a spacetime orientation g and with
a time orientation " (see De…nition 273). Such a general structure will be
denoted by (M; g; r; g ; ") and is said to be a Riemann-Cartan-Weyl spacetime. Lorentzian spacetimes are structures (M; g;D; g ; ") restricted by the
condition that the connection D is metric compatible, i.e., Dg = 0 and that
the torsion tensor of that connection is zero, i.e., [D] = 0. A connection
satisfying these two requirements is called a Levi-Civita connection11 (and
it is unique). Moreover, if in addition to the previous requirements, M ' R4
and the Riemann curvature tensor of the connection is null (i.e., R[D] = 0)
the Lorentzian spacetime is called Minkowski spacetime. That structure is
the ‘arena’for what is called special relativistic theories.
It is necessary that the reader realize very soon that a given n-dimensional
manifold M may eventually admit many di¤erent metrics12 and many
di¤erent connections. Thus, there is no meaning in saying that a given
1 1 Of course that denomination holds for any manifold M , dim M = n equipped with
a metric of signature (p; q):
1 2 The possible types of di¤erent metrics depend on some topological restrictions. This
will be discussed at the appropriate place.
6
1. Introduction
manifold has torsion and/or curvature. What is meaningful is to say that
a given connection on a given manifold has torsion and/or curvature. A
classical example [265], is the Nunes connection on the punctured sphere
M = S 2 = fS 2 n north + south polesg, discussed in Section 4.9.8 Indeed,
let (S 2 ; g; D) be the usual Riemann structure for a punctured sphere. That
means that D is the Levi-Civita connection of g, the metric on S 2 which it
inherits (by pullback) from the ambient three dimensional Euclidean space.
According to that structure, as it is well known, the Riemann curvature
tensor of D on S 2 is not null and the torsion of D on S 2 is null. However,
we can give to the punctured sphere the structure (S 2 ; g; rc ) where rc is
the Nunes connection. For that connection the Riemann curvature tensor
R[rc ] = 0, but its torsion [rc ] 6= 0. So, according to the connection
D the punctured sphere has (Riemann) curvature di¤erent from zero, but
according to the Nunes connection it has zero (Riemann) curvature.
If the above statements looks odd to you, it is because you always thought
of the sphere as being a curved surface (bent) living in Euclidean space.
Notice, however, that there may exist a surface S that is also bent as
a surface in the three dimensional Euclidean space, but is such that the
structure (S; g; D) is ‡at. As an example, take the cylinder S = S 1 R
with g the usual metric on S 1 R that it inherits from the ambient threedimensional Euclidean space, and where D is the Levi-Civita connection of
g, then R[D] = 0 and [D] = 0.
The above discussion makes clear that we cannot confuse the Riemann
curvature of a connection on a manifold M , with its bending 13 , i.e., with
the topological fact that M may be realized as a (hyper)surface embedded
on an Euclidean ( or pseudo-Euclidean) space of su¢ ciently high dimension14 . Keeping in mind these (simple) ideas is important for appreciating
the theory (and the nature) of the gravitational …eld, to be discussed in
Chapter 11.
As we already said the main objective of Chapter 4 is to introduce a
Cli¤ ord bundle formalism, which can e¢ ciently be used in the study of the
di¤erential geometry of manifolds an also to give an uni…ed mathematical
description of the Maxwell, Dirac and gravitational …elds.
1 3 Bending of a manifold viewed as submanifold of a Euclidean or pseudo-Euclidean
space of large dimension is characterized by the shape operator, a concept introduced
in Chapter 5.
1 4 Any manifold M; dim M = n, according to Whitney’s theorem, can be realized as
a submanifold of Rm , with m = 2n. However, if M carries additional structure the
number m in general must be greater than 2n. Indeed, it has been shown by Eddington
[109] that if dim M = 4 and if M carries a Lorentzian metric g, which moreover satis…es
Einstein’s equations, then M can be locally embedded in a (pseudo)Euclidean space R1;9 .
Also, isometric embeddings of general Lorentzian spacetimes would require a lot of extra
dimensions [82]. Indeed, a compact Lorentzian manifold can be embedded isometrically
in R2;46 and a non-compact one can be embedded isometrically in R2;87 !
1. Introduction
7
So, we introduce the Cartan, Hodge and Cli¤ ord bundles15 and present
the relationship between them and we also recall Cartan’s formulation of
di¤erential geometry, extending it to a general Riemann-Cartan-Weyl space
or spacetime (hereafter denoted RCWS).
We study the geometry of a RCWS in the Cli¤ord bundle C`(M; g) of
the cotangent bundle16 .
First we introduce, for a given manifold M , a structure (M; g; D), where
D is the Levi-Civita connection of an arbitrary …ducial metric g (which is
supposed to be compatible with the structure of M ) and call such a structure the standard structure. Next we introduce the concept of the standard
Dirac operator @j acting on sections of the Cli¤ord bundle C`(M; g) of di¤erential forms. Using @j, we de…ne the concepts of standard Dirac commutator
and anticommutator and we discuss the geometrical meaning of those operators. With the theory of symmetric automorphisms of a Cli¤ord algebra
(discussed in Chapter 2) we introduce in…nitely many other Dirac-like
operators given (M ;D), one for each non-degenerated bilinear form …eld
g 2 sec T20 M that can be de…ned on the standard structure (M; g; D).
_
_
Such new Dirac-like operators are denoted by @j . Using @j we introduce the
analogous of the concepts of Dirac commutator and anticommutator, which
are …rst introduced using @j and explore their geometrical meaning. In particular, the standard Dirac commutator permits us to give the structure of a
local Lie algebra to the cotangent bundle, in analogy with the way in which
the bracket of vector …elds de…nes a local Lie algebra structure for the tangent bundle. The coe¢ cients of the standard Dirac anticommutator— called
the Killing coe¢ cients— are related to the Lie derivative of the metric by
a very interesting relation (see Eq.(4.160)).
Subsequently, we introduce, besides the structure (M; g; D) on M , also a
general RCWS structure (M; g; r), r 6= D, on the same manifold M and
study its geometry with the help of C`(M; g). Associated with that structure
we introduce a Dirac operator @ and again using the theory of symmetric
automorphisms of a Cli¤ord algebra we introduce in…nitely many other
_
Dirac-like operators @ , one for each non-degenerated bilinear form …eld
g 2 sec T20 M that can be de…ned on the structure (M; g; r).
1 5 Spin-Cli¤ord
bundles are introduced in Chapter 7.
this book, the metric of the tangent bundle is always denoted by a boldsymbol
letter, e.g., g 2 secT20 M . The corresponding metric of the cotangent bundle is always
represented by a typewriter symbol, in this case, g 2 sec T02 M . Moreover, we represent
by g : T M ! T M the endomorphism associated with g. We have g(u; v) = g (u) v,
gE
¯
¯
for any u; v 2 sec T M , where g E is an appropriate Euclidean metric on T M . The inverse
of the endomorphism g is denoted g 1 . We represent by g : T M ! T M the endo¯ to g. Finally,
¯
morphism corresponding
the inverse of g is denoted by g 1 : See details
in Section 2.7.
1 6 In
8
1. Introduction
With respect to the structures (M; g; r) and (M; g; D) we also obtain
new decompositions of a general connection r. It is then possible to exhibit
some tensor quantities which are not well known, and have been …rst introduced (for the best of our knowledge) in the literature in [399] and exhibits
interesting relations between the geometries of the structures (M; g; D)
and (M; g; r). These results are used latter to shed a new light on the ‡at
space17 formulations of the theory of the gravitational …eld (Chapter 11)
and on the theory of spinor …elds in RCWS, as discussed in Chapter 10.
We also show that the square of the standard Dirac operator is (up to
a signal, which depends on convention) equal to the Hodge Laplacian of
the standard structure. The Hodge Laplacian maps p-forms on p-forms.
However, the square of the Dirac operator @ in a general Riemann-Cartan
space does not maps p-forms on p-forms and as such cannot play the role
of a wave operator in such spaces. The role of such an operator must be
played by an appropriate generalization of the Hodge Laplacian in such
spaces. We have identi…ed such a wave operator L+ , which (apart from a
constant factor) is the relativistic Hamiltonian operator that describes the
theory of Markov processes, as used, e.g., in [313].
Chapter 4 presents also some applications of the formalism, namely
Maxwell equations in the Hodge and Cli¤ord bundles, ‡ux and action quantization. The concepts of Ricci and Einstein operators acting on a set of
cotetrad …elds a de…ning a coframe are also introduced. Such operators are
used to write ‘wave’equations for the cotetrad …elds which for Lorentzian
spacetimes are equivalent to Einstein’s equations.
Chapter 5 gives a Cli¤ord bundle approach to the Riemannian or semiRiemannian di¤erential geometry of branes understood as submanifolds of
a Euclidean or pseudo-Euclidean space of large dimension. We introduce
the important concept of the projection operator and de…ne some other
operators associated to it, as the shape operator and the shape biform. The
shape operator is essential to de…ne the concept of bending of a submanifold
(as introduced above) and to leave it clear that a surface can be bended
and yet the Riemann curvature of a connection de…ned in it may be null
(as already mentioned for the case of the Nunes connection).
We give several di¤erent expression for the curvature biform operator
in terms of derivatives of the projection operator and the shape operator
and prove the remarkable formula S2 (v) = @ ^ @(v), which says that the
square of the shape operator applied to a 1-form …eld v is equal to the
negative of the Ricci operator (introduced tin Chapter 4) applied to a 1form …eld v. Such result is used in Chapter 11 to show how to transform
in marble the “wood” part of Einstein equation. By this we mean that
we can express its second member containing a phenomenological energy1 7 The word ‡at here refers to formulations of the gravitational …eld, in which this …eld
is a physical …eld, in the sense of Faraday, living on Minkowski spacetime.
1. Introduction
9
momentum tensor in a purely geometrical term involving the square of the
shape operator.
Chapter 6 introduces concepts and discusses issues that are in our opinion crucial for a perfect understanding of the Physics behind the theories
of Special and General Relativity18 . To ful…ll our goals it is necessary to
give a mathematically well formulated statement of the Principle of Relativity.19 . This requires a precise formulation (using the mathematical tools
introduced in previous chapters and some new ones) of an ensemble of essential concepts as, e.g., observers, reference frames20 , physical equivalence
of reference frames, naturally adapted coordinate chart to a given reference
frame (among others) which are rarely discussed in textbooks or research
articles. Using the Cli¤ord bundle formalism developed earlier, Chapter
6 presents a detailed discussion of the concept of local rotation as detected
by an observer and the Fermi-Walker transport. After that, a mathematical
de…nition of reference frames (which are modeled by timelike vector …elds
in the spacetime manifold) and their classi…cations according to two complementary schemes are presented. In particular, one classi…cation refers
to the concept of synchronizability. Some simple examples of the formalism, including a discussion of the Sagnac e¤ ect, are given. We de…ne the
concepts of covariance and invariance of theories based on the spacetime
concept, discussing in details and with examples (including Maxwell and
Einstein’s equations) what we think must be understood by the concept of
di¤ eomorphism invariance of a physical theory. Next, the concept of the
indistinguishable group of a class of reference frames is introduced and the
Principle of Relativity for theories which have Minkowski spacetime as part
of their structures is rigorously formulated. We brie‡y discuss the empirical status of that principle, which is also associated with what is known
as Poincaré invariance of physical laws. We show also the not well known
fact that in a general Lorentzian spacetime, modeling a gravitational …eld
according to the GRT, inertial reference frames as the ones that exist in
Minkowski spacetime do not exist in general. Ignorance of this and other
facts, e.g., that distinct reference frames are in general not physically equivalent in GRT generated a lot of confusion for decades, in particular lead
many people to believe in the validity of a “General Principle of Relativity”. The reference frames in a general Lorentzian spacetime which more
closely resembles inertial reference frames are the pseudo-inertial reference
1 8 We presuppose that the reader of our book knows Relativity Theory at least at the
level presented at the classical book [203].
1 9 A perfect understanding of the Principle of Relativity is also crucial in our forthcoming book [283] which discusses ‘superluminal wave phenomena’
2 0 It is important to distinguish between the concept of a frame (which are sections of
the frame bundle) introduced in Appendix A.1.1 with the concept of a reference frame
to be de…ned in Chapter 6.
10
1. Introduction
frames (PIRFs)21 and the local Lorentz reference frame associated with
(LLRF )22 . With the help of these concepts we prove that the so-called
‘Principle of Local Lorentz Invariance’23 of GRT is not a true law of nature, despite statements in contrary by many physicists. In general a single
PIRF is selected as preferred in reasonable cosmological models in a precise
sense discussed in Section 6.8. Such a selected PIRF V is usually identi…ed
with the reference frame where the cosmic background radiation is isotropic
(or the comoving frame of the galaxies). It is important to keep this point
in mind, for the following reason. Suppose that physical phenomena occur in Minkowski spacetime, and that there is some phenomenon breaking
Lorentz invariance, as, e.g., would be the case if genuine superluminal motion existed. In that case, the phenomenon breaking Lorentz invariance
could be used to identify a preferred inertial reference frame I0 , as has
been shown by many authors24 . However, the identi…cation (as done, e.g.,
in [316, 317, 318]) of I0 with V, the reference frame where the cosmic background radiation is isotropic and which is supposed to exist in a Lorentzian
spacetime (solution of Einstein’s equations for some distribution of energymomentum) cannot be done, because I0 is an inertial reference frame and V
is a PIRF and these are concepts belonging to di¤erent theories. Chapter
6 gives also a short account of the Schwarzschild original solution to Einstein’s equation and the notion of black holes, emphasizing that the global
topology of a given solution to Einstein equations obtained, of course in
an open set U of a not yet de…ned manifold M is most the time added by
hand in the process of obtaining the maximal extension of that solution.
In Chapter 7 a thoughtful presentation of the theory of a Dirac-Hestenes
spinor …eld (DHSF) on a general RCST (M; g; r; g ; ") is given, together
with a clari…cation of its ontology.25 A DHSF is a section of a particular
spinor bundle (to be described below), but the important fact that we
shall explore is that any DHSF has representatives on the even subbundle
of the Cli¤ord bundle C`(M; g) of di¤erential forms. Each representative is
relative to a given spin coframe.
In order to present the details of our theory we scrutinize the vector bundle structure of the Cli¤ord bundle C`(M; g), de…ne spinor structures, spin frame bundles, spinor bundles, spin manifolds, and introduce
Geroch’s theorem. Particularly important for our purposes are the left
(C`lSpine1;3 (M; g)) and right (C`rSpine1;3 (M; g)) spin-Cli¤ord bundles on a spin
2 1 See
de…nition 428.
de…nition 430. is a timelike geodesic in the Lorentzian manifold representing
spacetime.
2 3 Here, this principle is a statement about indistinguishable of LLRF . It is not to
be confused with the imposition of (active) local Lorentz invariance of Lagrangians and
…eld equations discussed in Section 10.2.
2 4 This issue is discussed in details in [283].
2 5 For the genesis of these objects we quote [350].
2 2 See
1. Introduction
11
manifold (M; g; g ; "). We study in details how these bundles are related
with C`(M; g). Left algebraic spinor …elds and Dirac-Hestenes spinor …elds
(both …elds are sections of C`lSpine1;3 (M; g)) are de…ned and the relation between them is established. Then, we show that to each DHSF
2 sec
C`lSpine1;3 (M ) and to each spin coframe 2 sec PSpine1;3 (M ) there is a well
de…ned sum of even multiform …elds (EMFS)
2 sec C`(M; g) associated
with . Such an EMFS is called a representative of the DHSF on the given
spin coframe. Of course, such an EMFS (the representative of the DHSF)
is not a spinor …eld, but it plays a very important role in calculations.
Indeed, with this crucial distinction between a DHSF and their EMFS representatives, we …nd useful formulas for calculating the derivatives of both
Cli¤ord …elds and representatives of DHSF on C`(M; g) using the general
theory of covariant derivatives of sections of a vector bundle, brie‡y recalled
in the Appendix A.5. This is done by introducing an e¤ ective spinorial
connection26 for the derivation of representatives of a DHSF on C`(M; g).
We thus provide a consistent theory for the covariant derivatives of Clifford and spinor …elds of all kinds. Besides that, we introduce the concepts
of curvature and torsion extensors of a (spin) connection and clarify some
misunderstandings appearing in the literature.
Next, we introduce the Dirac equation for a DHSF 2 sec C`lSpine1;3 (M; g)
(denoted DEC`l ) on a Lorentzian spacetime27 . Then, we obtain a representation of the DEC`l in the Cli¤ord bundle. It is that equation that we call
the Dirac-Hestenes equation (DHE) and which is satis…ed by even Cli¤ord
…elds
2 sec C`(M; g):
We study also the concepts of local Lorentz invariance and the electromagnetic gauge invariance. We show that for the DHE such transformations
are of the same mathematical nature, thus suggesting a possible link between them. Chapter 7 also discusses the concept of amorphous spinor
…elds, which are ideal sections of the Cli¤ord bundle C`(M; g) and which
have been some times confused with true spinor …elds (see also Chapter
12).
Chapter 8 deals with the Lagrangian formalism of classical …eld theory
in Minkowski spacetime. Recall that in Chapters 4 and 7 we show how
to represent Maxwell, Dirac and Einstein equations in several di¤erent formalisms. We hope to have convinced the reader that studied those chapters(
and will study Chapters 9 and 11) of the elegance and conciseness of the
representation of the electromagnetic and gravitational …elds as appropriate homogeneous sections of the Cli¤ord bundle C`(M; g) (Cli¤ord …elds)28 .
2 6 The
same as that used in [350].
case of Dirac-Hestenes equation on a Riemann-Cartan manifold is discussed in
Section 10.1.
V
2 8 For a description of the gravitational …eld by a set of 1-forms ga 2 sec 1 T M ,
a = 0;1; 2; 3 see Chapter 11.
2 7 The
12
1. Introduction
Also, we hope to have convinced the reader that the Dirac-Hestenes spinor
…elds— which are sections of the left spin-Cli¤ord bundle, C`lSpine1;3 (M; g)—
can be represented, once we …x a spin coframe, as even sections of C`(M; g),
i.e., a sum of non homogeneous di¤erential forms.
Taking into account these observations Chapter 8 is dedicated to the
formulation of the Lagrangian formalism for (interacting) Cli¤ord …elds
and representatives of Dirac-Hestenes …elds living on Minkowski spacetime
using the theory of multiform functions and extensors developed in Chapter 2. We show that it is possible to exhibit trustful conservation laws of
energy-momentum and angular momentum for such …elds. Several exercises are proposed and many solved in details, in order to help the reader
to achieve a complete domain on the mathematical methods employed. A
thoughtful discussion is given of non-symmetric energy-momentum extensors, since in any case the non-symmetric part is responsible for the spin
of the …eld (see Section 8.7). The cases of the electromagnetic and DHSF
are studied in details.
Chapter 9 is dedicated to the study of conservation laws on RiemannCartan and Lorentzian spacetimes. We already observed that the nature of
the gravitational …eld and the nature of other …elds (e.g., the electromagnetic and Dirac …elds) in GRT are very distinct. The former, according to
the orthodox interpretation, is to be identi…ed with some aspects of the
geometry of the world manifold (spacetime) while the latter are physical
…elds, in the sense of Faraday living in a background spacetime. This crucial
distinction implies that there are no genuine conservation laws of energy,
momentum and angular momentum in GRT. A proof of this statement is
one of the main objectives of Chapter 9. To motivate the reader for the
importance of the issue we quote page 98 of Sachs&Wu [367]:
“As mentioned in section 3.8, conservation laws have a great predictive power. It is
a shame to lose the special relativistic total energy conservation law (Section 3.10.2) in
general relativity. Many of the attempts to resurrect it are quite interesting; many are
simply garbage."
The problem of the conservations laws in GRT is a particular case of the
problem of the conservation laws of energy-momentum and angular momentum for …elds living in a general Riemann-Cartan spacetime (M; g; r; g ; ").
This latter problem is also relevant in view of the fact that recently a geometric alternative formulation of the theory of gravitational …eld (called the
teleparallel equivalent of GRT [227]) is being presented (see, e.g., [99, 12])
as one that solves the issue of the conservation laws. This statement must
be quali…ed and it is discussed in Section 11.6, after we prove in Chapter 9 that for any …eld theory describing a set of interacting …elds living
in a background Riemann-Cartan spacetime there are conservation laws
involving only the energy-momentum and angular momentum tensors of
the matter …elds, if and only if, the Riemann-Cartan spacetime time has
special global vector …elds that besides being Killing vectors satisfy also
1. Introduction
13
additional constraints, a fact that unfortunately is not well known as it
should be. In the teleparallel equivalent of GRT (with null or non null cosmological constant) it is possible to …nd a true tensor corresponding to the
energy of the gravitational …eld and thus to have an energy-momentum
conservation law for the coupled system made of the matter and gravitational …elds. However, this result as we shall see is a triviality once we
know how to formulate a gravitational …eld in Minkowski spacetime, an
issue also discussed in Chapter 11. In Chapter 9 we treat in details
only the case where each one ofV the …elds A , A = 1; 2; :::n, is a homor
geneous Cli¤ord …eld A 2 sec
T M ,! sec C`(M; g). Moreover, we restrict ourselves to the case where the Lagrangian density is of the form,
L^ ( )
L j^ ( ) = L^ (x; ; d ). This case is enough for our purposes,
which refers to matters of principles. However, the results are general and
can be easily extended for nonhomogeneous Cli¤ord …elds, and thus includes the case of the representatives in the Cli¤ord bundle of DHSF on a
general Riemann-Cartan spacetime. We remind also that Chapter 9 ends
with a series of non trivial exercises (with detailed solutions) including one
that gives the detailed derivation of Einstein’s equation from a Lagrangian
density written directly for the cotetrad …elds a , a = 0; 1; 2; 3.
Chapter 10 gives a presentation of the theory of DHSFs on a general
Riemann-Cartan spacetime. Such a theory reveals a hidden problem, the
one of knowing the exact meaning of active local Lorentz invariance of the
DHSF Lagrangian and of the DHE. We show that a rigorous mathematical
meaning to that concept can be implemented with the concept of generalized gauge connections introduced in Appendix A.5.2 and surprisingly
implies in a “gauge equivalence”between spacetimes with di¤erent connections which have di¤erent torsion and curvature tensors.
Chapter 1129 in this second edition has been completely rewritten and
is now titled On the Nature of the Gravitational Field. It …rst presents
a theory of the gravitational …eld where
V1this …eld is described by global
gravitational potentials fga g, ga 2 sec
T M , a = 0; 1; 2; 3, living on a
parallelizable manifold M . Using the ga to introduce a metric like …eld on
M , namely g = ab ga gb makes the Lorentzian structure (M; g) a spin
manifold. With the …eld g we introduce a Hodge star operator ? which is
g
then used in the writing of a postulated Lagrangian density for the gravitational potentials ga in interaction with matter …elds. Such a Lagrangian
does not involves any connection de…ned in M , which thus permit us to
present convincing arguments that the geometrical interpretation of the
gravitational …eld modelled by a Lorentzian spacetime structure (LSTS)
(M; g; D; g ; ") according to GRT, is no more than a possible choice.
2 9 Chapter
10 in the …rst edition.
14
1. Introduction
This geometrical interpretation becomes almost obvious once we prove
that the …eld equations for ga are equivalent to Einstein equations describing the gravitational …eld by the LSTS (M; g; D; g ; "). However , our
approach makes clear that there are other geometrical structures such as,
e.g., a teleparallel spacetime structure (M; g; r; g ; ") (and yet other geometrical structures) that can equally well represent the gravitational …eld.
Recalling that we learned in Chapter 9 that in GRT there are no genuine
energy-momentum and angular momentum for the gravitational …eld and
no genuine conservations laws for the matter and the gravitational …elds30 ,
we mention yet that:
(a) Einstein’s gravitational equation has as second member the energymomentum tensor of matter …elds, and this tensor is symmetric. However,
we learned in Chapter 8 that the canonical energy-momentum tensors
of the electromagnetic …eld and of the Dirac …eld are not symmetric, this
fact being associated with the existence of spin. If this fact is taken into
account (even if we leave out quantum theory from our considerations) it
immediately becomes clear that Einstein’s gravitational theory, must be
an approximation to a more complete theory. Science does not yet know,
which is the correct theory of the gravitational …eld, and indeed many alternatives have been and continues to be investigated. We are particularly
sympathetic with the view that gravitation is a low energy manifestation
of the quantum vacuum, as described, e.g., in Volovik31 [440]. This sympathy comes from the fact that, as we shall see, it is possible to formulate
theories of the gravitational …eld in Minkowski spacetime in such a way
that a gravitational …eld results as a kind of plastic distortion of the physical quantum vacuum. However we do not discuss such a theory here, the
interested reader may consult32 [127].
(b) The crucial distinction between the gravitational …eld and the other
physical …elds, mentioned above, has made it impossible so far to formulate
a well-de…ned and satisfactory quantum theory for the gravitational …eld,
despite the e¤orts of a legion of physicists and mathematicians. Eventually,
as a …rst step in arriving at such a quantum theory we should promote the
gravitational …eld described by the potentials ga to the same status of
all other physical …elds, i.e., a physical …eld living in Minkowski spacetime which satis…es …eld equations such that genuine conservation laws of
3 0 There are, of course, other serious problems with the formulation of a quantum
theory of Einstein’s gravitational …eld, that we are not going to discuss in this book.
The interested reader should consult on this issue, e.g., [212, 213].
3 1 Honestly, we think that gravitation is an emergent macroscopic phenomenon which
need not to be quantized and which will eventually …nd its correct description in a theory
about the real structure of the physical vacuum as suggested, e.g., in [440]. However, we
are not going to discuss such a possibility in this book.
3 2 On this issue, see also the book by Kleinert [194], which however describes plastic
distortions by means of multivalued functions.
1. Introduction
15
energy-momentum and angular momentum hold. Theories of this type are
indeed possible and it seems, at least to the authors, that they are more
satisfactory than GRT, and indeed many presentations have been devised
in a form or another, by several eminent physicists in the past. In Chapter
11 our version of such a possible theory is given. In it is possible to formulate a genuine energy-momentum tensor for the gravitational …eld. The
formula (Eq.(11.9)) derived by standard methods directly from the postulated Lagrangian density is a very complicated one involving many terms.
However, using some results of the Cli¤ord bundle formalism and taking
into account that we can use D, the Levi-Civita connection of the metric
like …eld g as no more than a mathematical device to simplify eventual
calculations, we show that it is possible to represent the energy momentum
1-form …elds for the gravitational …eld by a very nice and short formula
[356] Eq.(11.35)) worth to be registered.
We mention again that in our theory the …eld g obeys Einstein’s gravitational equation in an e¤ ective Lorentzian spacetime. However, g is considered a physical …eld (in the sense of Faraday) living in Minkowski spacetime (the true arena where physical phenomena takes place), thus being
an object of analogous nature as the electromagnetic …eld and the other
physical …elds we are aware of. The geometrical interpretation, i.e., the
orthodox view that the gravitational …eld is the geometry described by a
LSTS (M; g; D; g ; ") is a simple coincidence, as emphasized by Weinberg
[447], valid as an approximation. Such view is a useful one, because the
motion of probe particles and photons can be described with a very good
approximation by geodetic motions in the e¤ective Lorentzian spacetime
generated by a ‘big’ source of energy-momentum. However, keep in mind
that as mentioned above other possible representations of the gravitational
…eld are also possible. Chapter 11 has also a section on Einstein’s most
happy thought, i.e., the equivalence principle (EP). We remark (see also
the discussions in Chapter 6) that the interpretations of the equivalence
principle in GRT (as originally suggested by Einstein) is subject to criticisms. It has been recently suggested that the original Einstein suggestion
for the meaning of the EP seems more reasonable described in the teleparallel interpretation of GRT. However, even that interpretation, as will be
shown, is subject to criticism. We mention also that Chapter 11 discuss
also the Hamiltonian formalism for our theory and how it relates to the
ADM energy concept in GRT. Finally, despite our view that gravitation is
well described by a …eld theory in Minkowski spacetime we present (using
results proved in Chapter 5) in Sections 11.7 and 11.8 a mathematical formulation of Cli¤ord’s idea of matter as curvature in a brane world,
in particular showing how the “wood” part of Einstein equation (i.e., the
second member with the phenomenological energy-momentum tensor) can
be transformed in “marble”, i.e., can be given a geometrical formulation in
terms of the square of the shape tensor.
16
1. Introduction
In Chapter 12, On the Many Faces of Einstein’s Equations, we show
how the orthodox Einstein theory can be formulated in a way that resembles the gauge theories of particle physics, in particular a gauge theory with
gauge group Sl(2; C). This exercise will reveal yet another face of Einstein’s
equations, besides the ones already discussed in previous chapters. For our
presentation we introduce new mathematical objects, namely, the Cli¤ord
valued di¤erential forms (cliforms) and a new operator, D, which we called
the fake exterior covariant di¤ erential (FECD) and an associated operator
Der acting on them. Moreover, with our formalism we show that Einstein’s
equations can be put in a form that apparently resembles Maxwell equations. Chapter 12 also clari…es some misunderstandings appearing in the
literature concerning that Maxwell-like form of Einstein’s equations.
In Chapter 13, calledp Maxwell, Dirac and Seiberg-Witten Equations,
1 enters Dirac theory since complex numbers
we …rst discuss how i =
do not appear in the equivalentpDirac-Hestenes description of fermionic
…elds. Next we discuss how i =
1 enters classical Maxwell theory and
give (using Cli¤ord bundle methods) a detailed presentation of the theory of polarization and Stokes parameters. Our approach leaves the reader
equipped to appreciate the nonsense that sometimes even serious publishing houses leave to appear, as e.g., [117, 118, 119, 120, 121, 122, 123]. We
also present several Dirac-like representations of Maxwell equations. These
Dirac-like forms of Maxwell equations (which are trivial within the Cli¤ord
bundle formalism) really use amorphous spinor …elds and do not seem to
have any real importance until now. A three dimensional Majorana-like
representation of Maxwell equations is also easily derived and it looks like
Schrödinger equation. After that, we exhibit mathematical equivalences of
the …rst and second kinds between Maxwell and Dirac equations. We think
that the results presented are really nice and worth to be more studied.
The chapter ends showing how the Maxwell-Dirac equivalence of the …rst
kind plus a reasonable ansatz can provide an interpretation for the SeibergWitten equations in Minkowski spacetime.
In Chapter 14 we explore the potential of the mathematical methods developed previously. We show that we can describe the motion of
classical charged spinning particles (CCSP) when free or in interaction
with the electromagnetic …eld using DHSF. We show that in the free case
there is a unitary DHSF describing the motion of the CCSP which satis…es a linear Dirac-Hestenes equation. When the CCSP interacts with an
electromagnetic …eld, a non linear equation that we called the classical
Dirac-Hestenes equation is satis…ed by a DHSF describing the motion of
the particle. We study the meaning of the nonlinear terms and suggests a
possible conjecture: that this nonlinear term may compensate the term due
to radiation reaction, thus providing the linear Dirac-Hestenes equation introduced in Chapter 7. Moreover, we show that our approach suggests
by itself an interpretation for the Dirac-Hestenes wave function, namely,
1. Introduction
17
as a probability distribution. This may be important as it concerns the
interpretation of quantum theory, but that issue will not be discussed in
this book. After that we introduce (using the multiform calculus developed
in Chapter 2 and some generalizations of the Lagrangian formalism developed in Chapter 8) the dynamics of the superparticle. This consists in
showing that it is possible to give a Lagrangian formulation to the Frenet
equations describing a CCSP and that the resulting equations are in one
to one correspondence with the famous Berezin-Marinov equations for the
superparticle. We recall moreover how the Cli¤ord-algebraic methods previously developed suggest a very simple interpretation for Berezin calculus
and provides alternative geometric intelligibility for super…elds, a concept
appearing in modern physics theories such as supergravity and string theory. We show moreover that super…elds may be correctly interpreted as non
homogeneous sections of a particular Cli¤ord bundle over spacetime, and
we observe that we already had contact with objects of this kind in previous chapters, namely the representatives of DHSF introduced in Chapter
7 and the generalized potential (Section 13.4.5) appearing in the theory
of the Hertz potential. Our main intention in presenting these issues is to
induce some readers to think about the ontology of abstract objects being
used in advanced theories in alternative ways.
We exhibit in Chapters 4-14 several di¤erent, mathematical faces of
Maxwell, Einstein and Dirac equations. In Chapter 15 titled — “Maxwell,
Einstein, Dirac and Navier-Stokes Equations”— we show that given certain
conditions we can encode the contents
like
V2of Einstein equations in Maxwell
V2
equations for a …eld F = dA 2 sec T M (or F = dA 2 sec T M )33 ,
whose contents can be also encoded in a Navier-Stokes equation. For the
particular cases when it happens that F 2 6= 0 we can also using the
Maxwell-Dirac equivalence of the …rst kind discussed in Chapter 13 to encode the contents of the
equations
in a Dirac-Hestenes like
V0 previousVquoted
V4
2
21 ~
equation for 2 sec( T M + T M + T M ) such that F =
.
4
Speci…cally, Section 15.1 shows how each LSTS (M = R ; g; D; g ; "
) which, which as we already mentioned, is a model of a gravitational
…eld generated by T 2 sec T20 M (the matter plus non gravitational …elds
energy-momentum momentum tensor) in Einstein GRT is such that for any
K 2 sec T M — which is a vector …eld generating a one parameter group of
di¤eomorphisms of M — we can encode Einstein equations in Maxwell like
equations satis…ed by F = dK where K = g(K; ) with a well determined
current term named the Komar current JK =
K, whose explicit form
g
is given.
3 3 A = g(A; ) and A = g(A; ) with g and g the metrics of Minkowski spacetime denoted in Chapter 15 by (M = R4 ; g; D; g ; ") and of the structure (M = R4 ; g; D; g ; ")
descrybing an e¤ective Lorentzian spacetime.
18
1. Introduction
Next we show in Section 15.2 that when K = A is a Killing vector …eld,
due to some noticeable results (Eqs.(15.28) and (15.29)) the Komar current
acquires a very simple form and is then denoted JA . Then, interpreting, as
in Chapter 11 the Lorentzian spacetime structure (M = R4 ; g; D; g ; ") as
no more than an useful representation for the gravitational …eld represented
by the gravitational potentials fga g which lives in Minkowski spacetime we
show in Section 15.3 that we can …nd a Navier-Stokes equation which
encodes the contents of of the Maxwell like equations (encoding Einstein
equations) once a proper identi…cation is made between the variables entering the Navier-Stokes equations and the ones de…ning A and F , objects
clearly related to A and F = dA. We also explicitly determine also the constraints imposed by the nonhomogeneous Maxwell like equation F = JA
g
on the variables entering the Navier-Stokes equations and the ones de…ning A (or A). Finally we comment on relations between Einstein and the
Navier-Stokes equations living in spacetimes of di¤erent dimensions found
by other authors.
Chapter 16 analyzes the similarities and main di¤erences between Dirac,
Majorana and elko spinor …elds and the equations satis…ed by these …elds,
a subject that is receiving a lot of attention in the last few years. We
present an alternative theory for elko spinor …elds of mass dimension 3=2
(instead of mass dimension 1 as originally proposed in [6]) and show that
our elko spinor …elds can be used to describe electric neutral particles carrying “magnetic like” charges with short range interaction
mediated by
V1
a su(2)-valued gauge potential A = Ai
T M
spin3;0 ,!
i 2 sec
sec C`(M; ) R1;3 . Some crucial criticisms to the mass dimension 1 elko
spinor …eld theory are also given, since that theory breaks Lorentz and
rotational symmetries in a very odd way as shown in Section 16.7.
The book contains an Appendix where we review some of the main de…nitions and concepts of the theory of principal bundles and their associated
vector bundles, including the theory of connections in principal and vector
bundles, exterior covariant derivatives, etc., which are needed in order to
introduce the Cli¤ord and spin-Cli¤ord bundles and to discuss some other
issues in the main text. Jet bundles are also de…ned. We believe that the
material presented in the Appendix is enough to guide our reader permitting him to follow the most di¢ cult passages of the text, and in particular
to see the reason for our use of many eventually sloppy notations.
Having resumed the contents of our book, the following observations
are necessary. First, it is not a Mathematics book, despite the format of
the presentation in some sections, a format that has been used simply
because it is in our opinion the most e¢ cient one, for quotations. Even
though it is not a Mathematics book, several mathematical theories (some
sophisticated, indeed) have been introduced and we hope that they do not
scare a potential reader. We are sure that any reader (be him a student,
a physicist or even a mathematician) who will spend the appropriate time
1. Introduction
19
studying our book will really bene…t from its reading, since he (or she) will
start to see under a di¤erent point of view some of the foundational issues
associated with the theories discussed. Hopefully this will give to some
readers new insights on several subjects, a necessary condition to advance
knowledge.
Second, we recall that we mentioned in the introduction of the …rst edition that it was our intention to discuss the question of the arbitrary velocities solutions of the relativistic wave equations and that due to the size
attained by that …rst edition with its thirteen chapters plus an Appendix,
it has been decided to discuss that subject in a sequel volume entitled Subluminal, Luminal and Superluminal Wave Motion. Well, unfortunately the
planned new book is not ready yet as the beginning of 2014, being still
being written.
We tried to quote all papers and books that we have studied and that
in‡uenced our work, and we o¤er our apologies to any author not cited
who feels that some of his writings should be quoted.
20
1. Introduction
This is page 21
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2
Multivector and Extensor Calculus
2.1 Tensor Algebra
We recall here some basic facts of tensor algebra. Let V be a vector space
over the real …eld R of …nite dimension, i.e., dimV = n, n 2 N. By V = V
we denote the dual space of V. Recall that dim V = dim V . The elements of
V are called vectors and the elements of V are called covectors or 1-forms.
2.1.1 Cotensors
De…nition 1 We call space of k-cotensors (denoted Tk V ) the set of all
k-linear mappings k such that
k
V
: |
V::: V
{z
} ! R:
k-copies
Remark 2 In what follows we identify T0 V
R, and T1 V
(2.1)
V.
2.1.2 Multicotensors
P1
De…nition
3 Consider the (exterior ) direct sum T V := k=0 Tk V
L1
multicotensor of order M 2 N is an element of T V of
k=0 Tk V . AP
Mr
the form = k=0
k , k 2 Tk V , such that all the components k 2
Tk V of are null for k > M . T V is called the space of multicotensors.
22
2. Multivector and Extensor Calculus
We can easily show that T V is a vector space over R. We have that the
order of + , ; 2 T V is the greatest of the orders of or , and of
course, the order
PM of a , a 2 R, 2 T V is equal to the order of . The
set TM V =
Tk V is clearly a subspace of T V . Sometimes it is
k=0
convenient to denote an element of TM V by = ( 0 ; 1 ; :::; k ; :::; M ).
De…nition 4 The k-part operator is a mapping hik : T V ! Tk V such
that for all j 2 N, j 6= k we have that
hh ik ij = 0;
(2.2)
where hh ik ij 2 Tj V .
Then, if
= ( 0;
1 ; :::; k ; :::; M )
2 TM V we have that
h ik = (0; :::;
De…nition 5 A multicotensor
k if and only if = h ik :
k ; :::; 0):
2 T V is said to be homogeneous of grade
Of course, the set of multicotensors of grade k is a subspace of T V which
is isomorphic to Tk V and, we can write any 2 TM V as
XM
=
h ik :
(2.3)
k=0
2.1.3 Tensor product of Multicotensors
De…nition 6 The tensor product of multicotensors is a mapping : T V
T V ! T V such that:
(i) if a; b 2 T0 V = R, a b = ab,
(ii) if a 2 R and 2 Tp V , p 1, then a
=
a=a ,
(iii) if 2 Tj V and 2 Tk V , with j; k 1 then
2 Tj+k V and is
such that for v1 ; ::::; vj ; vj+1 ; :::; vj+k 2 V we have
(v1 ; ::::; vj ; vj+1 ; :::; vj+k ) = (v1 ; ::::; vj ) (vj+1 ; :::; vj+k );
(2.4)
(iv) the tensor product satis…es the distributive laws on the right and on
the left and it is associative, i.e., for a; b 2 R, 2 Tj V , 2 Tk V ; 2 Tl V
( + )
=
+
( + )=
(
(
+
)
=
)k =
XK
(v) If ; 2 T V then
;
(
J=0
j
;
):
(2.5)
k j;
(2.6)
where j
k j is the tensor product of the j component of
component of .
by the (k
j)
2.1 Tensor Algebra
23
Of course, from (iii) we may verify that the tensor product of cotensors
is not commutative in general.
Remark 7 We recall that the tensor product of multitensors is de…ned
in complete analogy to the previous one. The reader may easily …ll in the
details. We recall also that it is possible to extend the de…nition of tensor
product by allowing tensor products of a r-tensor by a s-cotensor. We then
denote, as usual, by
Tsr V = V
V V
V
V
V
V
V
(2.7)
the space of the r-contravariant and s-covariant tensors. P 2 Tsr V is a
(r + s)-multilinear map
V
P: |
::: V
{z
}
s-copies
V
V
|
::: V
{z
}
r-copies
V
!R
:
(2.8)
Remark 8 If P 2 Tsr V and S 2 Tq p V we de…ne the tensor product of P by
r+p
S as the multilinear mapping R S 2 Ts+q
V. Note that in general, P S
PM
6= S P. The (exterior ) direct sum T V = r=0;s=0 Ts r V equipped with
the tensor product is a real vector space over R, called
L1the kgeneral tensor
algebra of V. Note also that we can de…ne T V =
k=0 T V, the space
of multitensors in complete analogy to De…nition 3. Note moreover that
Ts 0 V = Ts V .
2.1.4 Involutions
De…nition 9 The main involution or grade involution is an automorphism ^ : T V ! T V such that: (i) if 2 R, ^ = ;
(ii) if a1
ak 2 Tk V , k 1, (a1
ak )^ = ( 1)k a1
ak ;
(iii) if a; b 2PR and ; 2 T V then (a + b ) ^ = a^ + b^;
(iv) if =
k , k 2 Tk V then
Xn
^=
^k ;
(2.9)
k=0
De…nition 10 The reversionP
operator is the anti-automorphism ~ : T V 3
7! ~ 2T V such that if = k k , k 2 Tk V then (i) if 2 R, ~ = ;
(ii) if a1
ak 2 Tk V , k 1, (a1
ak )~ = ak
a1 ;
~
(iii) if a; b 2PR and ; 2 T V then (a + b ) = a~ +b~;
(iv) if = k k , k 2 Tk V then
Xn
~=
~k ;
(2.10)
k=0
where ~ is called the reverse of .
24
2. Multivector and Extensor Calculus
De…nition 11 The composition of the grade evolution with the reversion
operator, denoted by the symbol is called by some authors the conjugation
and, is said to be the conjugate of . We have = (~)^ = (^) .
2.2 Scalar Products in V and V
Let V be real vector space, dim V = n.
De…nition 12 A metric tensor is a 2-cotensor g 2 T2 V which is symmetric and non degenerated. A basis fek g of V is said to be orthonormal if
g(ek ; ek ) is equal to +1 or equal to 1 and g(ek ; ej ) = 0 for j 6= k: We
have
8
i; j = 1; 2; :::p
< 1 if
1 if i; j = p + 1; p + 2; :::; p + q ;
g(ei ; ej ) = g(ej ; ei ) = gij =
:
0 if
i 6= j
(2.11)
with p+q = n. The signature of g is de…ned by the di¤ erence (p q) and it is
usual to say that the metric has signature (p; q). We denote g(v; w) v w
and called the dot a scalar product in V.
De…nition 13 A metric g induces a fundamental isomorphism between V
and V given by V 3 v 7! ]v = g(v; ) 2 V such that given any w 2 V, we
have
]v(w) = g(v; w):
(2.12)
De…nition 14 A general metric in V is a 2-tensor g 2 T 2 V, i.e., a mapping g : V V ! R which is symmetric and non degenerated.
Let f"k g be the basis of V dual to a basis fek g of V, i.e., "k (ej ) = jk .
We are particularly interested in a metric g 2 T 2 V such that if f"k g is the
basis of V dual to an arbitrary basis fek g of V then
g("i ; "j ) = g("j ; "i ) = g ij ;
(2.13)
and g ij gik = ki , i.e., the matrix with elements g ij is the inverse of the
matrix with elements gij in Eq.(2.11). Recall also that the inverse of the
isomorphism ] is the mapping ] 1 : V 3 7! ] 1 = g( ; ) 2 V.
De…nition 15 The scalar product of ; 2 V equipped with the metric
g given by Eq.(2.13) is denoted (to emphasize the relation between the
components of g and g)
g
= g( ; ) = g(]
1
;]
1
):
(2.14)
2.3 Exterior and Grassmann Algebras
Remark 16 Take notice that v w = g(v; w) = g(]v; ]w) and that
will be denoted simply by
25
g
when the context is clear.
De…nition 17 Let f"k g be the basis of V dual to the basis fek g of V.
A basis fek g of V is said to be the reciprocal basis of fek g if and only
if ek = ] 1 "k , for all k = 1; 2; ::; n. Also, a basis f"k g of V is called the
reciprocal basis of f"k g if and only if "k = ]ek .
The reader may verify that ek ej =
k
j
and "k "j =
k
j.
Exercise 18 Let fek g and f"k g be bases of V and V , such that "k (ej ) =
k
1
k
p1
"pk ; ::::; "p1
"pM )g1 k M
j . Show that the set f(1; " ; :::; " ; :::; "
is a basis of TM V and dim TM V =
nM +1 1
n 1 .
2.3 Exterior and Grassmann Algebras
De…nition 19 The exterior algebraof V is the quotient algebra
^
T (V )
;
J
V =
(2.15)
where J
T V is the bilateral ideal 1 in T V generated by
V the elements of
the form u v + v u, with u; v 2 V . The elements of V will be called
multiforms 2 .
V
V
Let : T VV! V be the canonical projection ofVT V onto
V . MultiV
plication in V will be denoted as usually by ^ : V ! V and called
exterior product. We have
V
De…nition 20 For every A; B 2 V ,
A ^ B = (A
where
B);
(2.16)
: T V ! T V is the usual tensor product.
Remark 21 Note that if u; v 2 V we can write
u
v=
1
(u
2
v
v
1
u) + (u
2
v+v
u);
(2.17)
and then,
(u
v) = u ^ v =
1
(u
2
v
v
u):
(2.18)
1 Given an associative algebra A, a bilateral ideal I is a subalgebra of A such that for
any a; b 2 A and for y 2 I, ay 2 I, yb 2 I and ayb 2 I. More on ideals on Chapter 3.
2 If we do the analogous construction of the exterior algebra using V instead of V
V , then the elements of the resulting space are called multivectors.
26
2. Multivector and Extensor Calculus
V
We can easily show that V is a 2n -dimensional associative algebra
with unity. In addition, it is a Z-grade algebra, i.e.,
^
and
^r
V =
n ^
M
r
V;
r=0
^s
^r+s
V;
Vr
n
r; s
0, where
V = (Tr V ) is the r -dimensional subspace of the
V0
V1
Vr
Vr
r-formson V . ( V = R;
V =V;
V = f0g if r > n). If A 2
V
for some …xed r (r = 0; : : : ; n), then A is said to be homogeneous. For any
such multivectors we have:
Vr
Vs
V ^
V
A ^ B = ( 1)rs B ^ A;
(2.19)
A2
V, B2
V.
The exterior algebra (as can easily be veri…ed) inherits the associativity
of the tensor algebra, a very important property. It satis…es also, of course,
the distributive laws (on the left and on the right), i.e.,
(A + B) ^ C = A ^ C + B ^ C;
A ^ (B + C) = A ^ B + A ^ C:
(2.20)
De…nition 22V The antisymmetrization operator A is the linear mapping
k
A : Tk V !
V such that (i) for all
2 R : A = , (ii) for all
v 2 V : Av = v, (iii) for all X1 X2
Xk 2 Tk V ; with k 2,
1 X
( )X (1) X (2)
X (k) ; (2.21)
A(X1 X2
Xk ) =
k!
2Sk
where : f1; 2; :::kg ! f (1); (2); ::: (k)g is a permutation of k elements
f1; 2; :::kg. Of course, the composition of permutations is a permutation
and the set of all permutations is Sk , the symmetric group.
Exercise 23 (a) For 2 Tk V a general k-cotensor and v1 ; : : : ; vk 2 V
show that
1
i1
ik
A (v1 ; : : : ; vk ) =
(2.22)
i
i (v ; : : : ; v );
k! 1 k
where i1 :::ik is the permutation symbol of order k;
8
ik is an even permutation of 1 : : : k
< 1; if i1
1;
if
i
ik is an odd permutation of 1 : : : k
=
1
i1 ik
:
0;
otherwise
Vp
Vq
(b) Show that if Ap 2
V and Bq 2
V then
Ap ^ Bq = A(Ap
Bq ):
(2.23)
(2.24)
2.3 Exterior and Grassmann Algebras
27
Remark 24 Some authors de…ne the exterior product Ap ^ Bq by
Ap ^ B q =
(p + q)!
A(Ap
p!q!
Bq ):
(2.25)
This de…nition is more used by di¤ erential geometers, whereas the de…nition given by Eq.(2.24) is more used by algebraists. Additional material
on this issue may be found in [129]. See also Exercise 31
2.3.1 Scalar Product in
V
V and Hodge Star Operator
Now let us suppose that V and V are metric vector spaces that is, they are
endowed with nondegenerate metric tensors which we denote conveniently
in what follows by g 2 T2 V and g 2 T 2 V of signature (p; q) and such
that gij g jk = ik , with gij = g(ei ; ej ) and g jk = g("j ; "j ), where fei g is
j
a basis for V and f"j g a basis for V with "j (e
We can use those
V i) = i . V
metric tensors toVinduce scalar products on V and V . We give the
construction for V .
V
De…nition
V
V 25 The (…ducial ) scalar product in V is the linear mapping
: V
V ! R given by
g
A B = det(g(ui ; vj ));
(2.26)
g
Vr
for homogeneous
multivectors A = u1 ^
^ ur 2
V and B = v1 ^
Vr
^ vV
V , ui ; vi 2 V , i = 1; : : : ; r. This scalar product is extended
r 2
Vr to
all of V due to linearity and orthogonality and A B = 0 if A 2
V,
g
Vs
V0
B2
V , r 6= s. We shall agree that if a; b 2
V
R, then a b = ab.
If the metric vector space (V ; g) is also endowed
Vn with an orientation,
i.e., a metric volume n-covector denoted by g 2
V such that
g
=
p
jdet gj "1 ^
~g
g
^ "n = p
g
1
jdet gj
= ( 1)q ;
"1 ^
^ "n ;
(2.27)
then we can
De…nition 27) a natural isomorphism between
Vrintroduce
Vn(see
r
V and
V.
the spaces
Remark 26 When the metric g used in the de…nition of the scalar product
is obvious we use from now on only the symbol in order to simplify the
formulas.
28
2. Multivector and Extensor Calculus
De…nition
star operator (or Hodge dual ) is the linear mapVr 27 The
Vn Hodge
r
ping ? :
V !
V such that
g
A ^ ?B = (A B ) g ;
g
(2.28)
Vr
for every A; B 2 V V . OfVcourse, this operator is naturally extended
Vn r to an
isomorphism ? : V ! V by linearity. The inverse ? 1 :
V !
g
g
Vr
V of the Hodge star operator is given by:
?
1
g
A = ( 1)r(n
r)
sgng ? A;
g
(2.29)
Vn r
for A 2
V and where sgn g = det g=j det gj denotes the sign of the
determinant of the matrix with entries gij where fei g is an arbitrary basis
of V.
Note that ? is a linear isomorphism but is not an algebra isomorphism.
g
Remark 28 When the metric g used in the de…nition of the Hodge star
operator is obvious we use only the symbol ? in order to simplify the formulas.
V
Exercise 29 Show that for any X; Y 2 V
~ i0 = hX Y~ i0 = Y X
X Y = hXY
(2.30)
Exercise 30 Let f"i g and orthonormalVbasis of V and f"j g is reciprocal
basis, i.e., "i "k = ik . Then, any Y 2 V can be written as
1 j1
Y
p!
1
= Yj1
p!
Y =
Show that:
(a)
(b)
Y j1
Yj1
jp
jp
jp
"j1 ^
jp "
j1
^
= Y ("j1 ^
= Y ("j1 ^
^ "jp
(2.31)
^ "jp :
^ "jp );
^ "jp ):
(2.32)
V2
1 ij
Y "i ^ "j 2
V (Y ij =
Exercise 31 Verify that, e.g., that if Y = 2!
ji
ij
i j
3
i j
Y , of course) then Y 6= Y (e ; e ). Find Y (e ; e ; ) in terms of Y ij .
V
1 ij
_ j 2 2 V then Y("i ; "j ) =
On the other hand show that if Y = 2!
Y "i ^"
Y ij .
3 Recall again that when the exterior product is de…ned (as, e.g., in [129, 138]) by
Eq.(2.25), then Xj1 :::jp in Eq.(2.32) really means X("j1 ; : : : ; "jp ). So, care is need when
reading textbooks or articles in order to avoid errors.
2.3 Exterior and Grassmann Algebras
29
V
The algebra V inherits the operators ^ (main automorphism), ~ (reversion) and (conjugation) of the tensor algebra T V , and we have
^
(AB)^ = A^B;
~ A;
~
(AB)~ = B
(2.33)
~ = (A)
^
(A) = (A)
^
~
V
for all A; B 2 V , with A^ = A if A 2 R, A^ =
A 2 R or A 2 V .
A if A 2 V and A~ = A if
2.3.2 Contractions
V
De…nition 32 For arbitrary multiforms X; Y; Z 2 V
V theVleft (y)Vand
V ! V,
right (x) contractions of X and Y are the mappings y : V
g
V
V
V
x: V
V ! V such that
g
(XyY ) Z = Y
g
g
~ ^ Z);
(X
(XxY ) Z = X (Z ^ Y~ ):
g
(2.34)
g
These contracted products y and x are internal laws on
g
g
V
V . Sometimes
the contractions are called interior products. Both contract products satisfy the distributive laws (on the left and on the right) but they are not
associative.
Remark 33 When the metric g used in the de…nitions of the left and right
contractions is obvious from the context we use the symbols y and x.
V
De…nition 34 The vector space V endowed with each one of these contracted products (either y or x) is a non-associative V
algebra. We call Grassmann algebra of multiforms
the
algebraic
structure
(
V ; ^; y; x), which will
V
simply be denoted by ( V ; g):
We present now some of the most important properties of the contractions:
V
(i) For any a; b 2 R; and Y 2 V
ayb = axb = ab (product in R),
ayY = Y xa = aY (multiplication by scalars).
(2.35)
(ii) If a; b1 ; :::; bk 2 V then
ay(b1 ^
^ bk ) =
k
X
j=1
( 1)j+1 (a bj )b1 ^
^ bj ^
^ bk ;
(2.36)
30
2. Multivector and Extensor Calculus
where the symbol bj means that the bj factor did not appear in the j-term
of the sum.
Vj
Vk
(iii) For any Yj 2
V and Yk 2
V with j k
Yj yYk = ( 1)j(k
Vj
Vk
(iv) For any Yj 2
V and Yk 2
V
j)
Yk xYj :
(2.37)
Yj yYk = 0; if j > k;
Yj xYk = 0; if j < k:
(v) For any Xk ; Yk 2
Vk
(2.38)
V
~ k Yk = Xk Y~k ;
Xk yYk = Yk xYk = X
(2.39)
Vk
where Y~k the reverse of Yk 2
V is the antiautomorphism given by
k
Y~k = ( 1) 2 (k
V
(vi) For any v 2 V and X; Y 2 V
1)
Yk :
^ ^ (vyY ):
vy(X ^ Y ) = (vyX) ^ Y + X
(2.40)
(2.41)
2.4 Cli¤ord Algebras
De…nition 35 The Cli¤ ord algebra C`(V ; g) of a metric vector space (V ; g)
is de…ned as the quotient algebra4
C`(V ; g) =
T (V )
;
Jg
where Jg
T V is the bilateral ideal of T V generated by the elements of
the form u v + v u 2g(u; v), with u; v 2 V
TV .
Cli¤ord algebras generated by symmetric bilinear forms are sometimes
referred as orthogonal, in order to be distinguished from the symplectic
Cli¤ord algebras, which are generated by skew-symmetric bilinear forms
(see, e.g., [92]).
Let g : T V ! C`(V ; g) be the natural projection of T V onto the
quotient algebra C`(V ; g). Multiplication in C`(V ; g) will be denoted as
usually by juxtaposition and called Cli¤ ord product. We have:
AB :=
g (A
B);
(2.42)
4 For other possible de…nitions of Cli¤ord algebras see, e.g., [2, 218, 220]. In this
chapter we shall be concerned only with Cli¤ord algebras over real vector spaces, induced
by nondegenerate bilinear forms.
2.4 Cli¤ord Algebras
31
A; B 2 C`(V ; g). The subspaces R; V
T V are identi…ed with their images
in C`(V ; g). In particular, for u; v 2 V
C`(V ; g), we have:
u
v=
1
(u
2
v
v)
uv =
v
1
u) + g(u; v) + f(u
2
v+v
u
2g(u; v)g; (2.43)
and then
(u
1
(u
2
v
v
u) + g(u; v) = u ^ v + g(u; v):
(2.44)
From here we get the standard relation characterizing the Cli¤ord algebra
C`(V ; g),
uv + vu = 2g(u; v):
(2.45)
2.4.1 Properties of the Cli¤ord Product
With some work the reader can prove [129] the following rules satis…ed by
the Cli¤ord product of multiforms:
V
(i) For all a 2 R and Y 2 V : aY = Y a equals multiplication of
multiform Y by scalar a.
V
(ii) For all v 2 V and Y 2 V :
vY = vyY + v ^ Y and Y v = Y xv + Y ^ v:
(2.46)
V
(iii) For all X; Y; Z 2 V : X(Y Z) = (XY )Z.
V
The Cli¤ord product is an internal law on V . It is associative (by
(iii)) and satis…es the distributive laws (on the left and on the right). The
distributive laws follow from the corresponding distributive laws of the
contracted and exterior products.
Recall that the Cli¤ord product is associative but it is not commutative
(as follows from (ii)) .
Note that since the ideal Jg T V is V
nonhomogeneous, of even grade, it
induces a parity grading in the algebra V , i.e.,
C`(V ; g) = C`0 (V ; g)
C`1 (V ; g);
(2.47)
with
C`0 (V ; g) =
g(
M1
T2r V )
r=0
M1
C`1 (V ; g) = g (
T2r+1 V ):
r=0
(2.48)
We say that C`(V ; g) is Z2 graded algebra. The elements of C`0 (V ; g) form
a subalgebra of C`(V ; g), called even subalgebra of C`(V ; g). Note that
C`1 (V ; g) is not a Cli¤ord algebra.
32
2. Multivector and Extensor Calculus
2.4.2 Universality of C`(V ; g)
We now quote a standard theorem concerning real Cli¤ord algebras [92]:
Theorem 36 If A is a real associative algebra with unity, then each linear
mapping : V ! A such that:
( (u))2 = g(u; u)
(2.49)
for every u 2 V , can be extended in an unique way to a homomorphism
C : C`(V ; g) ! A, satisfying the relation:
=C
g:
(2.50)
Let (V ; g) and (V 0 ; g 0 ) be two metric vector spaces and
linear mapping satisfying:
g 0 ( (u); (v)) = g(u; v)
: V ! V0 a
(2.51)
for every u; v 2 V .
We denote by
the natural extension of , i.e., the linear mapping
V
V 0
: V ! V called exterior power extension (see more details in
Section 2.6 ) such that:
V0
V
(i) for s 2
V
V,
(s) = s;
(2.52)
V
V
V
(ii) for any homogenous multiform 1 ^
^ r 2 V
V, i 2 V
we have
^r
^
( 1^
^ r ) = ( 1) ^
^ ( r) 2
V0
V 0;
(2.53)
(iii) If A =
Mn
j=0
Aj 2
Vj
(A) =
V , then
Mn
(Aj ) 2
j=0
^j
V0
^
V 0:
(2.54)
Then, using Theorem 36 we can show that there exists a homomorphism
C : C`(V ; g) ! C`(V 0 ; g 0 ) between their Cli¤ord algebras such that:
C
g
=
g0
:
(2.55)
Moreover, if V and V are metrically isomorphic5 vector spaces, then
their Cli¤ord algebras are isomorphic. In particular, two Cli¤ord algebras
C`(V ; g) and C`(V ; g 0 ) with the same underlying vector space V are isomorphic if and only if the bilinear forms g and g 0 have the same signature.
5 We call metric isomorphism a vector space isomorphism satisfying Eq.(2.51). The
term isometry will be reserved to designate a metric isomorphism from a space onto
itself.
2.4 Cli¤ord Algebras
33
Therefore, there is essentially one Cli¤ord algebra for each signature on a
given vector space V .6
De…nition 37 Let Rp;q (p + q = n) denote the vector space Rn endowed
with a metric tensor of signature (p; q). We denote by Rp;q the Cli¤ ord
algebra of Rp;q :
Natural Embedding
V
V ,! C`(V ; g)
Another important (indirect) consequence of the universality is thatVC`(V ; g)
is isomorphic, as a vector space over R, to the Grassmann algebra ( V ; g).
Let the symbol A ,! B means V
that A is embedded in B and A B:There
is a natural embedding [206] V ,! C`(V ; g). Then C`(V ; g) is a 2n dimensional vector space and given A 2 C`(V ; g) we can write:
A=
n
X
hAir ;
(2.56)
r=0
Vr
with hAir 2
V V
,! C`(V ; g) the projection of A (see De…nition 4) in the
V
r
V subspace of V .
De…nition 38 The elements of C`(V ; g) will also be called multiforms and
sometimes also called “Cli¤ ord numbers”. Furthermore, if A = hAir for
some …xed r, we sayVthat A is homogeneous of grade r. In that case, we
r
also write A = Ar 2
V ,! C`(V ; g).
V
Since V ,! C`(V ; g), C`(V ; g) inherits the main antiautomorphism,
theVreversion and the conjugation operators that we de…ned
V (see Eq.(2.33)
in V . Note moreover that C`(V ; g) also inherits from ( V ; g) the scalar
and contracted products of multiforms.
Exercise 39VShow that:
(i) If A 2 V ,! C`(V ; g) then
1
~ r = ( 1) 2 r(r 1) hAir :
(2.57)
hAi
Vr
Vs
(ii) If a 2 V ,! C`(V ; g) , Ar 2
V , Bs 2
V , r; s 0: (see [173])
hAir = ( 1)r hAir ,
ay(Ar Bs )
a ^ (Ar Bs )
=
=
=
=
(ayAr )Bs + ( 1)r Ar (ayBs )
(a ^ Ar )Bs ( 1)r Ar (a ^ Bs );
(a ^ Ar )Bs ( 1)r Ar (ayBs )
(ayAr )Bs + ( 1)r Ar (a ^ Bs ):
(2.58)
6 However, take into account that Cli¤ord algebras C`(V; g) and C`(V; g0 ) over the
same vector space may be isomorphic as algebras (but not as graded algebras) even if
:
g and g0 do not have the same signature. The reader may verify the validity of this
statement by inspecting Table 2 in Chapter 3 and …nding examples.
34
2. Multivector and Extensor Calculus
(iii) if a 2 V ,! C`(V ; g) and A 2
ayA =
V
V ,! C`(V ; g) then
Axa, a ^ A = A ^ a:
V
(iv) if a 2 V ,! C`(V ; g) and A 2 V ,! C`(V ; g) then
(v) if A; B 2
V
(2.59)
ay(A ^ B) = (ayA) ^ B + A^ ^ (ayB):
(2.60)
V ,! C`(V ; g) then
E
D
E
1D ~
e
~ A = ByA
AB B
;
2
1
1
E
D
E
1D ~
e
~ A = AyB
AB + B
:
2
1
1
V
(vi) if a 2 V ,! C`(V ; g) and A; B 2 V ,! C`(V ; g) then
(vii) if A; B; C 2
(viii) if A; B; C 2
(ix) if A; B; C 2
(x) if Ar 2
Vr
Ar Bs = hAr Bs ijr
V
(2.62)
(ayA) B = A (a ^ B):
(2.63)
V ,! C`(V ; g) then
V
V
(2.61)
Ay(ByC) = (A ^ B)yC;
(2.64)
(AxB)xC = Ax(B ^ C):
(2.65)
V ,! C`(V ; g) then
(AyB) C = B (A~ ^ C);
~
(BxA) C = B (C ^ A):
(2.66)
(2.67)
V ,! C`(V ; g) then
~
(AB) C = B (AC);
~
(BA) C = B (C A):
V ,! C`(V ; g); Bs 2
sj +hAr Bs ijr
Vs
sj+2 +
(2.68)
(2.69)
V ,! C`(V ; g) then [173] :
+hAr Bs ir+s =
m
X
k=0
hAr Bs ijr
(2.70)
with m = 21 (r +Vs jr sj).
Vs
r
(xi) if Ar 2
V ,! C`(V ; g); Bs 2
V ,! C`(V ; g) and r
that
Ar yBs = hAr Bs ijr
sj+2k ;
s show
Bs xAr :
(2.71)
V2
V1
V ,! C`(V ; g)
(xii) Show that for "a ; "b 2
V ,! C`(V ; g) and B 2
sj
= ( 1)r(s
r)
hBs Ar ijr
sj
= ( 1)r(s
h("a ^ "b )Bi2 = "a ^ ("b yB) + "a y("b ^ B):
r)
(2.72)
2.5 Extensors
35
Exercise 40 De…ne the commutator product of A; B 2 C`(V ; g) by
A
B=
1
[A; B]:
2
(2.73)
Show that the commutator product satisfy the Jacobi identity, i.e. for A; B; C 2
C`(V ; g)
A
(B
C) + B
(C A) + C (A B) = 0
V2
Exercise 41 Show that if F 2
V ,! C`(V ; g), then
F2 =
F F + F ^ F:
(2.74)
(2.75)
If the metric vector space (V ; g) is oriented by g , then we can also
extend the Hodge star operator de…ned in the Grassmann algebra to the
Cli¤ord algebra C`(V ; g), by letting ? : C`(V ; g) ! C`(V ; g) be given by:
X
?A =
?hAir :
(2.76)
r
Exercise 42 Show that for any Ar 2
Vr
V and Bs 2
Vs
V , r; s
Ar ^ ?Bs = Bs ^ ?Ar ; r = s
Ar ?Bs = Bs ?Ar ; r + s = n
Ar ^ ?Bs = ( 1)r(s 1) ? (A~r yBs ); r s
Ar y ? Bs = ( 1)rs ? (A~r ^ Bs ); r + s n
?Ar = A~r y g = A~r g
? g = sgng; ?1 = g :
0:
(2.77)
Exercise 43 Let f" g be a basis of V (a n-dimensional real vector space)
dual to a basis fe g of V. Let g be the metric of V such that g(e ; e ) = g
and let g be a metric for V such that g(" ; " ) = g and g g = .
^ " p , " p+1 n = " p+1 ^
^ " n we
Show that writing " 1 ::: p = " 1 ^
have
p
1
jdet gjg 1 1
g p p 1 n " p+1 n :
(2.78)
?" 1 ::: p =
(n p)!
All identities in the above exercises are very important for calculations.
They are part of the tricks of the trade.
2.5 Extensors
V
V
In what follows V V denotes an arbitrary subspace of V , called
V a subspace
part
of
V
.
Consider
the
arbitrary
subspace
parts
1 V ,...,
V
V
V
V
and
V
of
V
.
m
36
2. Multivector and Extensor Calculus
De…nition 44 Any linear mapping t :
an extensor [130] over V .
V
1
V
:::
V
m
V !
V
V is called
V
V
The setVof extensors over V , with domain 1 V
:::
m V and
codomain
V has a natural structure of a vector space over R and will be
called EXT-(V ). In what follows we shall need to study mainly theVproperties of extensors where the domain
V is a single
V subspace
V part, say 1 V .
The space of that extensors where 1 V =
V = V will be denoted
by ext-(V ).
2.5.1
(p; q)-Extensors
De…nition
45 Let p; q 2 N
Vp
Vqwith 0 p; q n: A extensor over V with domain
V and codomain
V is called a (p; q)-extensor over
VpV . The
Vq real
vector space of the (p; q)-extensors over V is denoted by ext ( V ;
V ).
Vp
Vq
Note that if dim V = n then dim ext( V ;
V ) = np nq :
2.5.2 Adjoint Operator
De…nition 46 Let (f"j g; f"j g) be a pair of arbitrary reciprocal basis for
V (i.e., "j "i = ij ). The linear operator
^p
^q
^q
^p
y
:ext(
V;
V ) 3 t 7! ty 2 ext(
V;
V)
such that
1
(t("j1 ^
p!
1
= (t("j1 ^
p!
ty (Y ) =
^ "jp ) Y )"j1 ^
^ "jp
^ "jp ) Y )"j1 ^
^ "jp ;
(2.79)
is called the adjoint operator relative to the scalar product de…ned by g. ty
is called the adjoint of t.
The adjoint operator is well de…ned since the sums in the second members
of the above equations are independent of the chosen pair (f"j g; f"j g):
2.5.3 Properties of the Adjoint Operator
(i) The operator
y
is involutive, i.e.,
(ty )y = t:
(2.80)
Vq
Vp
Vp
Vq
(ii) Let t 2 ext( V ;
V ). Then, for any X 2
V and Y 2
V
t(X) Y = X ty (Y ):
(2.81)
2.6 (1; 1)-Extensors
37
Vq
Vr
Vp
Vq
(iii) Let t 2 ext( V ;
V ) and V
u 2 ext(
V;
V ). Then, composiV
p
r
tion of u with t denoted t u 2 ext( V ;
V ) and we have
(t u)y = uy ty :
(2.82)
2.6 (1; 1)-Extensors
2.6.1 Symmetric and Antisymmetric Parts of (1; 1)-Extensors
V1
V1
De…nition 47 An extensor t 2 ext( V ;
V ) is said adjoint symmetric relative to the scalar product de…ned by g (adjoint antisymmetric) if,
and only if t = ty (t = ty ).
V1
V1
V ); there are two (1; 1)It is easy to see that for any t 2 ext( V ;
extensors over V , say t+ and t ; such that t+ is adjoint symmetric (i.e.,
t+ = ty+ ) and t is adjoint antisymmetric (i.e., t = ty ) and
t(a) = t+ (a) + t (a):
(2.83)
1
(t(a)
2
(2.84)
Moreover,
t (a) =
ty (a)):
De…nition 48 t+ and t are called the adjoint symmetric and adjoint
antisymmetric parts of t.
2.6.2 Exterior Power Extension of (1; 1)-Extensors
V1
V1
V
De…nition 49 Let t 2 ext( V ;
V )) and Y 2 V . The linear operator
^1
^1
(2.85)
: ext(
V;
V ) ! ext-(V );
t 7! t;
such that for any
t(Y ) = 1 Y +
=1 Y +
n
X
1
k=1
n
X
k=1
(("j1 ^
^ "jk ) Y )t("j1 ) ^
^ t("jk )
(2.86)
1
(("j1 ^
k!
^ "jk ) Y )t("j1 ) ^
^ t("jk );
(2.87)
k!
is called the (exterior power ) extension operator relative to the scalar product de…ned by g. We read t as the extended of t:
The extension operator is well de…ned since the sums in the second members of the above equations is independent of the chosen pair (f"j g; f"j g).
TakeVinto account also V
that the extension operator preserve grade, i.e., if
k
k
Y 2
V then t(Y ) 2
V:
38
2. Multivector and Extensor Calculus
2.6.3 Properties of t
V1
V1
(i) For any t 2 ext( V ;
V ); and any
have
t(v1 ^
2 R; v; v1 ; : : : ; vk 2
V1
V we
t( ) = ;
(2.88)
t(v) = t(v);
(2.89)
^ vk ) = t(v1 ) ^
^ t(vk ):
(2.90)
An obvious corollary of the last property is
t(X ^ Y ) = t(X) ^ t(Y );
(2.91)
t u = t u:
(2.92)
V
for any X; Y 2 V .
V1
V1
(ii) For all t; u 2 ext( V ;
V ) it holds
V1
V1
V1
V1
(iii) Let t 2 ext( V ;
V ) with inverse t 1 2 ext( V ;
V ) (i.e.,
V1
V1
t t 1 = t 1 t = id ; where id 2 ext( V ;
V ) is the identity extensor).
Then
(2.93)
(t) 1 = (t 1 ) t 1 :
V1
V1
(iv) For any t 2 ext( V ;
V)
(ty ) = (t)y
ty :
V1
V1
V
(v) Let t 2 ext( V ;
V ); for any X; Y 2 V
Xyt(Y ) = t(ty (X)yY ):
(2.94)
(2.95)
2.6.4 Characteristic Scalars of a (1; 1)-Extensor
V1
V1
De…nition 50 The trace of t 2 ext( V ;
V ) relative to the scalar
product de…ned by g is a mapping
tr : ext(
such that
^1
V;
^1
V)!R
tr(t) = t("j ) "j = t("j ) "j :
(2.96)
Note that the de…nition is independent of the chosen pair (f"j g; f"j g).
V1
V1
Also, for any t 2 ext( V ;
V );
tr(ty ) = tr(t):
(2.97)
2.6 (1; 1)-Extensors
39
V1
V1
De…nition 51 The determinant of t 2 ext( V ;
V ) relative to the
V1
V1
scalar product de…ned by g is the mapping det : ext( V ;
V ) ! R such
that
1
t("j1 ^
n!
1
= t("j1 ^
n!
det t =
^ "jn ) ("j1 ^
^ "jn )
(2.98)
^ "jn ) ("j1 ^
^ "jn ):
(2.99)
Remark 52 Note that the de…nition is independent of the chosen pair
(f"j g; f"j g). When it becomes necessary to explicitly specify the metric g
relative to which the determinant of t is de…ned we will write dett instead
g
of dett.
Remark 53 For the relation between the de…nition of dett and the classical
determinant of a square matrix representing t in a given basis see Exercise
58.
Using the combinatorial formulas v j1 ^
^ v jn = j1 :::jn v 1 ^
^ vn
j1 jn
and j1 jn are the
and vj1 ^
^ vjn = j1 :::jn v1 ^
^ vn ; where
permutation symbols of order n and v 1 ; : : : ; v n and v1 ; : : : ; vn are linearly
independent covectors, we can also write
4
dett = t("1 ^
^ "n ) ("1 ^
^ "n ) = t( " ) "
= t("1 ^
^ "n ) ("1 ^
^ "n ) = t( " ) " ;
4
where we used: " = "1 ^
4
4
4
^ "n and " = "1 ^
4
(2.100)
(2.101)
^ "n .
2.6.5 Properties of dett
(i) For any t 2 ext(
V1
V;
V1
V ),
detty = dett:
V1
V1
Vn
(ii) For any t 2 ext( V ;
V ); and I 2
V we have
(iii) For any t; u 2 ext(
V1
t(I) = Idett:
V1
V;
V ),
(2.102)
(2.103)
det(t u) = dett detu:
(2.104)
V1
V1
V ) with inverse t 1 2 (1; 1)-ext(V ) (i.e.,
(iv) Let t 2 ext( V ;
t t 1 = t 1 t = id ; where id 2 (1; 1)-ext(V ) is the identity extensor),
then
dett 1 = (dett) 1 :
(2.105)
40
2. Multivector and Extensor Calculus
In what follows we use the notation det 1 t meaning dett 1 or (dett) 1 .
V1
V1
(v) If t 2 ext( V ;
V ) is non degenerated (i.e., dett 6= 0), then it
V1
V1
has an inverse t 1 2 ext( V ;
V ), given by
t
where I 2
Vn
1
(a) = det
1
tty (aI)I
1
;
(2.106)
V is any non null pseudoscalar.
2.6.6 Characteristic biform of a (1; 1)-Extensor
V1
V1
De…nition 54 The 2-form [131] of t 2 ext( V ;
V ) is
bif (t) = t("j ) ^ "j = t("j ) ^ "j 2
^2
V:
(2.107)
Note that bif (t) is indeed a characteristic of t since the de…nition is
independent of the chosen pair (f"j g; f"j g):
Properties of bif (t)
V1
V1
(i) Let t 2 ext( V ;
V ); then
bif (ty ) =
bif (t):
(2.108)
V1
V1
(ii) The adjoint antisymmetric part of any t 2 ext( V ;
V ) can be
‘factored’by the formula [130]
t (a) =
where
1
bif (t)
2
a;
(2.109)
means the commutator product (see Eq.(2.73)).
2.6.7 Generalization of (1; 1)-Extensors
De…nition
V 55 Let G : ext(
any Y 2 V
V1
V;
V1
V ) 3 t 7! T 2 ext-(V ) such that for
Gt (Y ) = T (Y ) = t("j ) ^ ("j yY ) = t("j ) ^ ("j yY );
(2.110)
The linear operator G is called the generalization operator of t relative to
the scalar product de…ned by g, T is read as the generalized t.
Note that G is well de…ned since it does not depend on the choice
Vk of the
pair (f"j g; f"j g). Note also that G preserves grade, i.e., if Y 2
V ; then
Vk
T (Y ) 2
V:
2.7 Symmetric Automorphisms and Orthogonal Cli¤ord Products
Properties of G
(i) For any
2 R and v 2
(ii) For any X; Y 2
V
V1
41
V we have
T ( ) = 0;
(2.111)
T (v) = t(v):
(2.112)
V we have
T (X ^ Y ) = T (X) ^ Y + X ^ T (Y ):
(2.113)
(iii) G commutes with the adjoint operator. Thus, T y means either the
adjoint of the generalized as well as the generalized of the adjoint.
(iv) The adjoint antisymmetric part of the generalized of t is equal to
the generalized of the antisymmetric adjoint part of t; and can be factored
as
1
T (Y ) = bif (t) Y;
(2.114)
2
V
for any Y 2 V .
2.7 Symmetric Automorphisms and Orthogonal
Cli¤ord Products
Besides the “natural” Cli¤ord product of C`(V ; g), we can introduce in…nitely many other Cli¤ord-like products on this same algebra, one for each
symmetric automorphism of its underlying vector space. In what follows
we are going to construct such new Cli¤ord products.7
There is a one-to-one correspondence between the endomorphisms of
(V ; g) and the bilinear forms over V . Indeed, to each endomorphism g :
V ! V we can associate a bilinear form g : V V ! R, by the relation:
g(u; v) = g(g (u) ; v)
(2.115)
for every u; v 2 V .
As we know (recall Eq.(2.79)) the adjoint of the extensor g : V ! V is
the extensor (linear mapping) gy : V ! V such that for any u; v 2 V
g(g (u) ; v) = g (u) v = u gy (u) :
7 The possibility of introducing di¤erent Cli¤ord products in the same Cli¤ord algebra
was already established by Arcuri [23]. A complete study of that issue is given in [132].
The relation between di¤erent Cli¤ord products is an essential tool in the theory of
the gravitational …eld as presented in [127], where this …eld is represented by the gauge
metric extensor h (Section 2.7.1).
42
2. Multivector and Extensor Calculus
De…nition 56 An endomorphism G : V ! V is said to be symmetricor
skew-symmetric if its associated bilinear form G : V
V ! R is, respectively, symmetric or skew-symmetric. In the more general case we can write
a bilinear form G as:
G = G+ + G ;
(2.116)
with G (u; v) = 21 [G(u; v)
G(v; u)], for every u; v 2 V .
Then, correspondingly, its associated endomorphism G will be written
as the sum of a symmetric and a skew-symmetric endomorphism, i.e.,
G = G+ + G ;
(2.117)
with G+ ; G : V ! V standing for the endomorphisms associated to
the bilinear forms G+ and G , respectively. We see immediately that for a
symmetric automorphism, i.e., G = G+ we have G+ Gy .
If g g+ = gy is a symmetric automorphism of (V ; g), the bilinear form
g 2 T 2 V associated to it has all the properties of a metric tensor on V
and in that case g can be used to de…ne a new Cli¤ord algebra C`(V ; g)
associated to the pair (V ; g).
This is done by associating to the bilinear form g ([173]) a new scalar
V1
product
of vectors in the algebra in the space V
V related to
g
the old scalar product by
u v := g(u; v) = g(u) v
(2.118)
for every u; v 2 V .
Vp
Also, given Ap = u1 ^ ::: ^ up ; Bp = v1 ^ ::: ^ vp 2
V ,! C`(V ; g) we
de…ne in analogy with De…nition 25
Ap Bp = det(g(ui ; vj )) = det(g(ui ) vj )
(2.119)
V
This new scalar product is extended
to linearity and
Vr to all of VVdue
s
orthogonality and A B = 0 if A 2
V and B 2
V , r 6= s. Also, we
V0
agree that if a; b 2 R
V then a b = ab.
2.7.1 The Gauge Metric Extensor h
If it can be easily proved that C`(V ; g) will be isomorphic to the original
Cli¤ord algebra C`(V ; g) if and only if there exists an automorphism h :
V ! V (called the gauge metric extensor) such that:
g(u) v = h(u) h(v);
(2.120)
for every u; v 2 V . We will say that h is the square root of g (or by abuse
of language of g) even when h 6= hy ,
2.7 Symmetric Automorphisms and Orthogonal Cli¤ord Products
43
Eq.(2.120) implies that
g = hy h:
(2.121)
We can prove (see below) that every positive symmetric transformation
possesses at most 2n square roots, all of them being symmetric transformations, but only one being itself positive (see, e.g., [132]).
If Eq.(2.120) is satis…ed, we can reproduce the Cli¤ord product of C`(V ; g)
into the algebra C`(V ; g) de…ning an operation
_
g
: C`(V ; g)
A_B
C`(V ; g) ! C`(V ; g);
1
A B=h
g
((h(A)h(B));
(2.122)
for every A; B 2 C`(V ; g), V
where hV1 is the inverse of the automorphism
of h, and h : C`(V ; g) - V ! V ,! C`(V ; g) is the extended of h
de…ned according to Eq.(2.85). In particular, if u; v 2 V ,! C`(V ; g) are
covectors, then
u v = u v + u ^ v:
g
In addition, the product _ : C`(V ; g) C`(V ; g) ! C`(V ; g) satis…es all
the properties of a Cli¤ord product which we have stated previously.
We introduce also the contractions y : C`(V ; g) C`(V ; g) ! C`(V ; g)
g
and x : C`(V ; g) C`(V ; g) ! C`(V ; g) induced by _ : C`(V ; g) C`(V ; g) !
g
C`(V ; g), in complete analogy to De…nition 32, and we left to the reader to
…ll in the details. Relations analogous to those given in the earlier section
will be obtained, with the usual contraction product “y” replaced by this
new one.
Furthermore, if we perform aVchange in the volume scale by introducing
n
another volume n-vector g 2
V such that
~g
g
= ( 1)q ;
then we can also de…ne the analogous of the Hodge duality operation for
this new Cli¤ord product, by letting ? : C`(V ; g) ! C`(V ; g) be given by:
g
Ar ^ (?Br ) = (A~r
Br ) g ;
g
(2.123)
Vr
for every Ar ; Br 2
V ,! C`(V ; g), r = 0; : : : ; n. Of course, the operator
? just de…ned satis…es relations that are analogous to those satis…ed by the
g
operator ?(= ?) (see Eq.(2.28)), and we have:
g
? = sgn(deth)h
1
sgn(deth)h
1
g
as can be veri…ed (see Exercise 60)
? h
g
?h
(2.124)
44
2. Multivector and Extensor Calculus
Remark 57 For the case of two metrics, say g and of the same Lorentz
signature (1; 3) such that g may be continuously deformed into we have
deth > 0. In this case, we can write
?=h
1
? h:
g
which we abbreviate as ? = h
1
g
(2.125)
? h.
2.7.2 Relation Between dett and the Classical Determinant
det[Tij ]
Let V be a real vector space of …nite dimension n and V its dual. Using
notations introduced previously, let gE and g E be metrics of Euclidean
signature and g and g be metrics of signature (p; q) in V and V which
are related as explained in Section 2.2. Let moreover g : V V ! R be
an arbitrary nondegenerate symmetric bilinear form on V and g : V
V
! R a bilinear form on V such that if f"j g is a basis of V dual to fej g
( an arbitrary basis of V), then gij g jk = jk , where gij = g(ei ; ej ) and
g jk = g("j ; "k ). Let also f"j g be the reciprocal basis of f"j g with respect
V1
V1
to the Euclidean metric g E . Let also t 2 ext( V ;
V ) be the unique
V1
2
extensor that corresponds to T 2 T V, such that for any a; b 2
V we
V1
V1
have T (a; b) = t(a)
b and t 2 ext( V ;
V ) be the unique extensor
gE
V1
that corresponds to T 2 T 2 V, such that for any a; b 2
V we have
T (a; b) = t(a) b. De…ne Tij = T ("i ; "j ).
g
Exercise 58 Show that the classical determinant of the matrix [Tij ] denoted det[Tij ] and dett and dett are related by:
gE
g
det[Tij ] = dett("1 ^
gE
= dett("1 ^
g
^ "n )
("1 ^
gE
^ "n ) ("1 ^
g
^ "n )
: ^ "n ):
(2.126)
Solution: We show the …rst line of Eq.(2.126). Recall that for any
v1 ; v2 ; :::; w1 ; :::; wn 2 V we have
(v1 ^
^ vk )
gE
(w1 ^
and the property t(v1 ^
classical determinant of n
det[Tjk ] =
=
s1 :::sn
t("1 )
gE
s1
"s1
sn
= t("1 ^
v1
gE
ws1
vk
gE
wsk (2.127)
^ vk ) = t(v1 ) ^
^ t(vk ). By de…nition of
n real matrix we have
T1s1
t("n )
s1 :::sk
^ wk ) =
Tnsn =
gE
s1 :::sn
T ("1 ; "s1 )
"sn = (t("1 ) ^
^ t("n ))
^ "n )
^ "n );
gE
("1 ^
T ("n ; "sn )
gE
("1 ^
^ "n )
(2.128)
2.7 Symmetric Automorphisms and Orthogonal Cli¤ord Products
45
hence, by de…nition of dett, i.e., t(I) = Idett for all non-null pseudoscalar
gE
gE
I, Eq.(2.126) follows.
Remark 59 Note, in particular, for future reference that writing det g =
det[g(ei ; ej )], det g = det[g("j ; "k )] we have,
(det g)
1
= det g = detg("1 ^
^ "n )
gE
= detg("1 ^
gE
^ "n ) ("1 ^
g
("1 ^
^ "n )
^ "n ):
g
(2.129)
V1
V1
where g 2 ext( V ;
V ) is the unique extensor that corresponds to g 2
V1
V1
b and g 2 ext( V ;
V ) is the unique
T 2 V such that g(a; b) = g(a)
gE
extensor that corresponds to g 2 T 2 V such that g(a; b) = g(a) b.
g
Exercise 60 Prove Eq.(2.124).
Solution:
To prove Eq.(2.124) we need to take into account that V
for each invertible
(1; 1)-extensor h such that g = hy h and for any X; Y 2 V the following
identity holds
h(XyY ) = h(X)yh(Y );
(2.130)
g
g
By de…nition
~ g;
?X = Xy
g
~ g:
?X = Xy
g
Also, since jdet gj = 1, we have
g
g
=
p
(2.131)
g
jdet gj g , det g = (deth)
h( g ) = (deth) g :
2
and
(2.132)
Then
~
Xy
g
g
=
=
=
p
p
p
~
jdet gjXy
g
g
jdet gjh
1
~ h( g )
h(X)y
g
jdet gj(deth)h
= sgn(deth)h
1
1
~ ( g)
h(X)y
g
?h(X)
(2.133)
2.7.3 Strain, Shear and Dilation Associated with
Endomorphisms
Recall that every linear transformation can be expressed as composition of
elementary transformations of the types Ra : V ! V and Sab : V ! V ,
46
2. Multivector and Extensor Calculus
de…ned by: (see, e.g., [173]).
Ra (u) = aua 1
Sab (u) = u + (u a)b;
(2.134)
for every u 2 V , where a; b 2 V are non-zero (co)vectors parametrizing the
transformation and a 1 = a=a2 = a=(a a). Transformations of the type
Ra are called elementary re‡ections. Recall also that it is a classical result
proved in any good book of linear algebra that any isometry of (V ; g) can
always be written as the composite of at most n such transformations. The
skew-symmetric part of a transformation of the type “Sab ”will be denoted
by S[ab] . We have:
S[ab] (u) =
1
(Sab (u)
2
Sba (u)) =
1
uy(a ^ b);
2
(2.135)
for every u 2 V . It is again an example of an extensor, this time mapping
biforms in forms.
The symmetric part of a transformation of the type “Sab ” is called a
strain; it is a shear in the a ^ b-plane if a b = 0, or a dilation along a, if
a ^ b = 0. Obviously, a dilation along a direction a can be written more
simply as:
a
(2.136)
Sa (u) = u + (u a) 2 ;
a
for every u 2 V , where 2 R, > 1, is a scalar parameter. If = 0,
then Sa is the identity map of V , for any a 2 V . If 6= 0, then Sa is a
contraction ( 1 < < 0) or a dilation ( > 0), in the direction of a, by a
factor 1 + .
Now, we can show that every positive symmetric transformation can
always be written as the composite of dilations along at most n orthogonal
directions. To see this, it is su¢ cient to remember that for any symmetric
transformation g associated to g we can …nd an orthonormal basis f" g of
V for which (aligned indices are not to be summed over)
g(" ) =
( )"
;
where ( ) 2 R is the eigenvalue of g associated to the eigenvector "
( = 1; : : : ; n). Then, de…ning
S" (u) = u +
for every u 2 V , with
( )
=
( )
g = S1
( ) (u
" )
"
;
"2
(2.137)
1, we get:
Sn :
(2.138)
If the symmetric transformation g is in addition positive, i.e., g = hhy ,
we have " g(" ) = h(" ) h(" ) = ( ) " " . Then, since the signature
2.7 Symmetric Automorphisms and Orthogonal Cli¤ord Products
47
of a bilinear form is preserved by linear transformations (Sylvester’s law of
inertia), we conclude that:
( )
=
h(" ) h(" )
> 0:
" "
(2.139)
This means that in Eq.(2.137), ( ) = ( ) 1 > 1 and therefore it
satis…es the de…nition of dilation given by Eq.(2.136). Note also that the
1=2
1=2
positive “square root” of g is given by h = S1
Sn , with
S"1=2 (u) = u +
( ) (u
" )
"
;
"2
(2.140)
p
for every u 2 V , where ( ) = 1 +
( ).
With these results it is trivial to give an operational form to the product
de…ned through Eq.(2.122), although eventually this may demand a great
deal of algebraic manipulation.
We emphasize that although we have considered only the positive symmetric transformations in the developments above, the formalism can be
adapted to more general transformations (see, e.g. [132]).
We give as example of the above formalism the Table 1, where we listed
the g’s relating all metrics of signature (p; q) in a vector space V , with
dim V = p + q = 4, relative to a standard metric of Lorentzian signature.
Note that if f" g is a basis of V with " " =
= diag(1; 1; 1; 1)
then
g(v) = R("
1
:"
r)
v;
(2.141)
where R(" 1 ::::" r ) is a product of r-re‡ections, each one in relation to the
hyperplane orthogonal to " i . Then, we have, e.g., R"0 (v) = "0 v"0 ; R"i (v) =
"i v"i ; i = 1; 2; 3. Also, the last column of the Table 1 list the matrix algebra
to which the corresponding Cli¤ord algebra Rp;q is isomorphic (see Chapter
3 for details).
g
R(0)
R(1)
R(01)
R(12)
R(012)
R(123)
R(0123)
g(v)
"0 v"0
"1 v"1
0 1 1 0
" " v" "
"1 "2 v"2 "1
"0 "1 "2 v"2 "1 "1
"1 "2 "3 v"3 "2 "1
0 1 2 3 3 2 1 0
" " " " v" " " "
g 00
1
+1
1
+1
1
+1
1
g 11
1
+1
+1
+1
+1
+1
+1
g 22
1
1
1
+1
+1
+1
+1
g 33
1
1
1
1
1
+1
+1
(p; q)
(0; 4)
(2; 2)
(1; 3)
(3; 1)
(2; 2)
(4; 0)
(3; 1)
Rp;q
H(2)
R(4)
H(2)
R(4)
R(4)
H(2)
R(4)
Table 1. Endomorphisms generating all Cli¤ord algebras in 4-dimensions.
48
2. Multivector and Extensor Calculus
2.8 Minkowski Vector Space
We give now some details concerning the structure of Minkowski vector
space which plays a fundamental role in the theories presented in this book.8
De…nition 61 Let V be a real four dimensional space over the real …eld
R and : V V !R a metric of signature 9 2. The pair (V; ) is called
Minkowski vector space and is denoted by R1;3 .
We recall that the above de…nition implies that there exists an orthonormal basis b = fe0 ; e1 ; e2 ; e3 g of R1;3 such that
8
= = 0;
< 1 if
1 if
= = 1; 2; 3;
(e ; e ) = (e ; e ) =
=
(2.142)
:
0 if
6= :
We use in what follows the short notation
The notation
= diag(1; 1; 1; 1).
(u; w) = u w = w u = (w; u);
(2.143)
where the symbol as usual is called a scalar product will also be used.
Also, for any u 2 R1;3 , we write u u = u2 .
We recall that given a basis b = fe g of V the basis b = fe g of V such
that
1 if
= ;
e e =
=
(2.144)
0 if
6= :
is called the reciprocal basis of b:
Note that e e =
, with the matrix with entries
being the
matrix diag(1; 1; 1; 1).
The dual space of V in the de…nition of the Minkowski vector space is as
usual denoted by V . The dual basis of a basis b = fe g of R1;3 is the set
b = f" g such that " (e ) = . We introduce in V a metric of signature
2, : V V ! R, such that
(" ; " ) =
:
(2.145)
The pair (V ; ) will be denoted by R 1;3 . We write (u; v) = u v. Recall
also that the reciprocal basis of the basis b = f" g of V is the basis
b = f" g of V such that
" " = :
(2.146)
The metric
induces an isomorphism between V and V given by
V 3 a 7!a = (a; ) 2 V ;
8 The
9 The
proofs of the propositions of this section are left to the reader.
signature of a metric tensor is sometimes de…ned as the number (p
(2.147)
q).
2.8 Minkowski Vector Space
49
such that for any w 2 V we have
a(w) = (a; w):
(2.148)
Of course the structures R1;3 = (V; ) and R 1;3 = (V ; ) are also
isomorphic. To see this it is enough to verify that if v; w 2 V and v = (v; ),
w = (w; ) 2 V then, (u; v) = (v; w).
De…nition 62 Let v 2 V, then we say that v is spacelike if v2 < 0 or
v = 0, that v is lightlike if v2 = 0 and v 6= 0, and that v is timelike if
v2 > 0. This classi…cation gives the causal character of v. An analogous
terminology are used to classify the elements of V .
p
When v 2V is timelike, we denote by jjvjj = v v the norm of v.
Remark 63 From nowhere when no confusion arises we write u 2 V as
u 2 R1;3 . That notation emphasize that u is an element of V and it can
be classi…ed as a spacelike or lightlike or timelike vector. Also, within the
same spirit we write when no confusion arises u 2 V as u 2 R 1;3 . The
same notation will be used also for subspaces S of V (S of V ), i.e., we
write S V (S V ) as S R1;3 (S R 1;3 ).
De…nition 64 Let S
R 1;3 be a subspace. We say that S is spacelike
if all its covectors are spacelike, that S is lightlike if it contains a lightlike
covector but no timelike vector, and S is timelike if it contains a timelike
covector.
De…nition 64 establish that a given subspace S
R 1;3 is necessarily spacelike or lightlike or timelike. We now present without proofs some
propositions that will give us some insight on the linear algebra of R 1;3 .
Proposition 65 A subspace S
onal complement is spacelike.
R
1;3
is timelike if and only if its orthog-
Proposition 66 Let S R 1;3 be a lightlike subspace. Then its orthogonal
complement S ? is lightlike and S \ S ? 6= f0g.
Proposition 67 Two lightlike covectors, n1 ; n2 2 R
and only if they are proportional.
1;3
are orthogonal if
Proposition 68 There are only two orthogonal spacelike covectors that are
orthogonal to a lightlike covector.
Proposition 69 The unique way to construct an orthonormal basis for
R 1;3 is with one timelike covector and three spacelike covectors.
The next proposition shows how to divide the set T R 1;3 of all timelike
covectors in two disjoint subsets T+ and T , which are called respectively
future pointing and past point timelike vectors.
50
2. Multivector and Extensor Calculus
Proposition 70 Let u; v 2 T. The relation " on the set T T such that
u " v if and only if u v>0 is an equivalence relation and it divides T in
two disjoint equivalence classes T+ and T .
Remark 71 Such equivalence relation is used to de…ne a time orientation
in spacetime structures (see Chapter 4).
Proposition 72 T+ and T
are convex sets.
Recall that T+ to be convex means that given any u; w 2 T+ and a 2
(0; 1) and b 2 [0; 1) then (au+bw) 2 T+ .
Proposition 73 Let u; v 2 T+ . Then they satisfy the anti-Schwarz inequality, i.e., ju vj jjujj jjvjj and the equality only occurs if u and v are
proportional.
We end this section presenting the anti-Minkowski inequality, which is
the basis for a trivial solution [336] of the “twin paradox”, once we introduce
a postulate (see Axiom 374, Chapter 6) concerning the behavior of ideal
clocks.
Proposition 74 Let u; v 2 T+ . Then we have
jju + vjj
jjujj + jjvjj ;
(2.149)
which will be called anti-Minkowski inequality.
Proposition 75 Let u; v be spacelike covectors such that the span[u; v]
is spacelike. Then the usual Schwarz and Minkowski inequalities ju vj
jjujj jjvjj and jju+vjj jjujj + jjvjj hold. If the equalities hold then u and
v are proportional.
Proposition 76 Let u; v be spacelike covectors such that the span[u; v] is
timelike. Then the anti-Schwarz inequality ju vj
jjujj jjvjj holds. If the
equality holds then u and v are proportional. If u v
0 and u + v is
spacelike then the anti-Minkowski inequality jju + vjj
jjujj + jjvjj holds,
the equality holding if u; v are proportional.
Exercise 77 Prove Propositions 65-76.
2.8.1 Lorentz and Poincaré Groups
De…nition 78 A Lorentz transformation L :V !V acting on the Minkowski
(co)vector space R 1;3 = (V ; ) is a Lorentz extensor, i.e., an element of
V1
V1
ext( V ;
V ) ' V V such that for any u; v 2 V we have
(Lu; Lv) = (u; v):
(2.150)
2.8 Minkowski Vector Space
51
The set of all Lorentz transformations has a group structure under composition of mappings. We denote such group by O1;3 . As it is well known
these transformations can be represented by 4 4 real matrices, once a
basis in V is chosen. Eq.(2.150) then implies that
det L =
(2.151)
1:
Transformations such that det L = 1 are called improper and the ones
with det L =1 are called proper. Improper transformations, of course do
not close in a group. The subset of proper Lorentz transformations de…ne
a group denoted SO1;3 . Finally, the subset of SO1;3 connected with the
identity is also a group. We denote it by SOe1;3 . This group also denoted by
L"+ is called the proper orthochronous Lorentz group. The reason for the
adjective orthochronous becomes obvious once you solve the next exercise.
Exercise 79 Show that if L 2 SOe1;3 and u 2 T+
show that if v2T , then Lv"v.10
V , then Lu"u. Also
De…nition 80 Let L be a Lorentz transformation and let a 2 V . A Poincaré
transformation acting on Minkowski vector space R1;3 is a mapping P :V !V ,
v7! Pv= Lv+a.
A Poincaré transformation is denoted by P = (L;a). The set of all Poincaré
transformations de…ne a group denoted by P— called the Poincaré group—
under the multiplication rule (group composition law) given by
(L1 ; a)(L2 ; b)v= (L2 L1 ; L1 b+a)v = L2 L1 v + L1 b+a:
(2.152)
This composition law says that the Poincaré group is the semi direct
product ( ) of the Lorentz group O1;3 by the translation group. We have
T = f(1;a); 1 2 O1;3 ; a2V g;
(2.153)
"
and we write P = O1;3 T . We also denote by P " = SO1;3 T and by P+
e
= SO1;3 T .
The theory of representations of the Lorentz and Poincaré groups is an
essential ingredient of relativistic quantum …eld theory. We shall need only
some few results of these theories in this book and that results are present
at the places where they are needed.
Exercise 81 Show that if u; v 2 V , and P = (L; a) is a Poincaré transformation then P(u v) P(u v) = (u v) (u v).
1 0 If
the reader has any di¢ culty in solving that exercice he must consult, e.g., [339].
52
2. Multivector and Extensor Calculus
2.9 Multiform Functions
2.9.1 Multiform Functions of a Real Variable
De…nition 82 A mapping Y : I !
function of real variable [259].
V
V (with I
R) is called a multiform
For simplicity when the image of Y is a scalar (or 1-form, or 2-form ,
: : : ; etc.) the multiform function 7! Y ( ) is said respectively scalar, (or
1-form, or 2-form , : : : ; etc.) function of real variable.
Limit and continuity
The notions of limit and continuity for multiform functions can be easily introduced, following a path analogous to the one used in the case of
ordinary real functions.
V
De…nition 83 A multiform L 2 V is said to be the limit of Y ( ) for
2 I approaching 0 2 I; if and only if for any " > 0 there exists > 0
such that 11 kY ( ) Lk < "; if 0 < j
0 j < : We write lim Y ( ) = L:
!
0
We prove without any di¢ culty that:
lim (Y ( ) + Y ( )) = lim Y ( ) + lim Y ( ):
!
0
!
0
!
0
!
0
!
0
!
0
lim (Y ( ) ~ Y ( )) = lim Y ( ) ~ lim Y ( );
(2.154)
(2.155)
where ~ is any one of the product of multiforms, i.e., (^); ( ); (y; x) or
(Cli¤ ord product):
Y ( ) is said continuous at 0 2 I if and only if lim Y ( ) = Y ( 0 ):
!
0
Then, sum and products of any continuous multiform functions are also
continuous.
De…nition 84 The derivative of Y in
Y 0(
0)
= lim
!
0
is
Y( )
Y(
0)
:
(2.156)
0
0
The following rules are valid:
(Y + Y )0 = Y 0 + Y 0
(2.157)
(Y ~ Y ) = Y ~ Y + Y ~ Y (Leibniz’s rule)
0
(Y
0
0
) = (Y
0
)
0
0
(2.158)
(chain rule),
(2.159)
p
V
that the norm of a multiform X 2 V is de…ned by kXk = X X, where
the symbol refers to the Euclidean scalar product.
1 1 Recall
2.10 Multiform Functions of Multiform Variables
53
where ~ is any one of the product of multiforms, i.e., (^); ( ); (y; x) or
(Cli¤ ord product) and is an ordinary real function ( 0 is the derivative
d
of ). We also write sometimes
Y ( ) = Y 0 ( ):
d
We can generalize easily all ideas above for the case of multiform functions of several real variables, and when need such properties will be used.
2.10 Multiform Functions of Multiform Variables
De…nition 85 A mapping F :
of multiform variable.
V
V !
V
V ; is said a multiform function
If F (Y ) is a scalar (or 1-form, or 2-form, : : :) then F is said scalar (or
1-form, or 2-form,...) function of multiform variable.
2.10.1 Limit and Continuity
V
V
De…nition
86 M 2 V is said the limit of F for Y 2
V approaching
V
Y0 2
V if and only if for any " > 0 there exists
> 0 such that
kF (Y ) M k < "; if 0 < kY Y0 k < . We write lim F (Y ) = M:
Y !Y0
The following properties are easily proved:
lim (F (Y ) + G(Y )) = lim F (Y ) + lim G(Y ):
(2.160)
lim (F (Y ) ~ G(Y )) = lim F (Y ) ~ lim G(Y );
(2.161)
Y !Y0
Y !Y0
Y !Y0
Y !Y0
Y !Y0
Y !Y0
where ~ is any one of the product of multiforms, i.e., (^); ( ); (y; x) or
(Cli¤ ord product):
V
V
De…nition
87 Let F :
V ! V . F is said to be continuous at Y0 2
V
V if and only if lim F (Y ) = F (Y0 ):
Y !Y0
Of course, the sum Y 7! (F + G)(Y ) = F (Y ) + G(Y ) or any product
Y 7! (F ~ G)(Y ) = F (Y ) ~ G(Y ) of continuous multiform functions of
multiform variable are continuous.
2.10.2 Di¤erentiability
V
V
V
De…nition 88 F :
V ! V is said to be di¤ erentiable at Y0 2
V
if and only if there exists an extensor over V , say, fY0 2 ext-(V ); such
that
F (Y ) F (Y0 ) fY0 (Y Y0 )
= 0:
(2.162)
lim
Y !Y0
kY Y0 k
54
2. Multivector and Extensor Calculus
Remark 89 It is possible to show [261] that if such fY0 exists, it must be
unique.
De…nition 90 fY0 is called the di¤ erential (extensor ) of F at Y0 :
2.11 Directional Derivatives
Let F be any multiform function of multiform variable which is continuous
at Y0 . Let 6= 0 be any real number and A an arbitrary multiform. Then,
there exists the multiform
F (Y0 + hAiY0 ) F (Y0 )
lim
:
(2.163)
!0
De…nition 91 The limit in Eq.(2.163) denoted by FA0 (Y0 ) or (A @Y )F (Y0 )
is called the directional derivative of F at Y0 in the direction of the multivector A [261]. The operator A @Y is said the directional derivative with
respect to A and this notation, i.e., A @Y is justi…ed in Remark 102. We
write
F (Y0 + hAiY0 ) F (Y0 )
FA0 (Y0 ) = (A @Y )F (Y0 ) = lim
;
(2.164)
!0
or more conveniently
FA0 (Y0 ) = (A @Y )F (Y0 ) =
d
F (Y0 + hAiY0 )
d
:
(2.165)
=0
The directional derivative of a multiform function of multifunction variable is linearVwith respect to the multiform direction, i.e., for all ; 2 R
and A; B 2 V
F 0 A+
B (Y
) = FA0 (Y ) + FB0 (Y );
(2.166)
or
( A + B) @Y F (Y ) = A @Y F (Y ) + B @Y F (Y ):
(2.167)
The following propositions are true and their proofs may be found in
[261].
Remark 92 When there is no risk of confusion we write (A @Y )F (Y )
simply as A @Y F .
Proposition 93 Let Y 7! F (Y ) and Y 7! G(Y ) be any two di¤ erentiable
multiform functions. Then, the sum Y 7! (F + G)(Y ) = F (Y ) + G(Y )
and the products Y 7! (F ~ G)(Y ) = F (Y ) ~ G(Y ); where ~ means (^);
( ); (y; x) or (Cli¤ ord product) of di¤ erentiable multiform functions. Also,
(F + G)0A (Y ) = FA0 (Y ) + G0A (Y ):
(F ~
G)0A (Y
)=
FA0 (Y
) ~ G(Y ) + F (Y ) ~
(2.168)
G0A (Y
) (Leibniz’s rule).
(2.169)
2.11 Directional Derivatives
55
2.11.1 Chain Rules
Proposition 94 Let Y 7! F (Y ) and Y 7! G(Y ) be any two di¤ erentiable multiform functions. Then, the composite multiform function Y 7!
(F G)(Y ) = F (hG(Y )iY ); with Y 2 domF; is a di¤ erentiable multiform
function and
(F G)0A (Y ) = FG0 0 (Y ) (hG(Y )iY ):
(2.170)
A
Proposition 95 Let Y 7! F (Y ) and 7! Y ( ) be two di¤ erentiable multiform functions, the …rst one a multiform function of multivector variable
and the second one a multiform function of real variable. Then, the composite function 7! (F Y )( ) = F (hY ( )iY ); with Y 2 domF; is a multiform
function of real variable and
(F
Y )0 ( ) = FY0 0 ( ) (hY ( )iY ):
(2.171)
V
Proposition 96 Let : R ! R and
:
V ! R be respectively an
ordinary di¤ erentiable real
function
and
a
di¤
erentiable
multiform function.
V
The composition
:
V ! R, (
)(Y ) = ( (Y )) is a di¤ erentiable
scalar function of multiform variable and
(
)0A (Y ) =
0
( (Y ))
0
A (Y
):
(2.172)
The above formulas can also be written
A @Y (F + G)(Y ) = A @Y F (Y ) + A @Y G(Y ):
(2.173)
A @Y (F ~ G)(Y ) = A @Y F (Y ) ~ G(Y ) + F (Y ) ~ A @Y G(Y ): (2.174)
A @Y (F
G)(Y ) = A @Y G(Y ) @Y F (hG(Y )iY );
d
(F
d
A @Y (
2.11.2
with Y 2 domF:
(2.175)
d
Y ( ) @Y F (hY ( )iY ); with Y 2 domF: (2.176)
d
d
)(Y ) =
( (Y ))A @Y (Y ):
(2.177)
d
Y )( ) =
Derivatives of Multiform Functions
Let F be any multiform function of multiform variable, which is di¤erentiable at Y . Let (f"j g; f"j g) be an arbitrary pair of reciprocal basis for
V (i.e., "j "i = ij ). Then, there exists a well de…ned multiform (i.e.,
independent of the pair (f"j g; f"j g) and depending only on F )
X
J
1 J 0
" F"J (Y )
(J)!
X
J
1
"J F"0J (Y ):
(J)!
(2.178)
56
2. Multivector and Extensor Calculus
The symbol J denotes collective indices. Recall, e.g., that
"J = 1; "j1 ; :::; "j1 j2
J
j1
" = 1; " ; :::; "
j1 j2
jn
jn
= "j1 ^ "j2 ^
j1
j2
=" ^" ^
and (J) = 0; 1; 2; ::: for J = ?; j1 ; j1 j2
run from 1 to n.
"jn ;
^ "jn
(2.179)
jn ; ::: where all indices j1 ; j2 ; :::; jn
De…nition 97 The multiform given by Eq.(2.178) will be denoted by F 0 (Y )
or @Y F (Y ) and is called [132, 259] the derivative of F at Y . We write
X 1
F 0 (Y ) = @Y F (Y ) =
"J F"0J (Y ):
(2.180)
(J)!
J
2.11.3 Properties of @Y F
Proposition 98 (a) Let Y !
7
(Y ) be a scalar function of multiform
variable. Then,
A @Y (Y ) = A (@Y (Y )):
(2.181)
(b) Let Y 7! F (Y ) and Y 7! G(Y ) be di¤ erentiable multiform functions.
Then,
@Y (F + G)(Y ) = @Y F (Y ) + @Y G(Y ):
(2.182)
@Y (F G)(Y ) = @Y F (Y )G(Y ) + F (Y )@Y G(Y ) ( Leibniz’s rule).
(2.183)
Remark 99 In …eld theories formulated in terms of di¤ erential formsV(see
r
Chapter 9) theVdirectional derivative of a multiform
3
Vr function, say F :
n
X 7! F (X) 2
in the direction of W = X 2
V is written as
F := X ^
@F
@X
(2.184)
@F
called the variational derivative of F and @X
is called the algebraic derivative of F . Now, since F = W @X we can, e.g., verify that for F = X ^ ?X
we have the identi…cation
X @X F = F = X ^
r
@F
= ( 1) 2 (r
@X
1)
X ^ @X F:
(2.185)
More details may be found in [127].
2.11.4 The Di¤erential Operator @Y ~
De…nition 100 Given a multiform function of multiform variable, we de…ne the di¤ erential operator [261] @Y ~ by
X 1
@Y ~ F (Y0 ) =
"J ~ "J @Y F (Y0 );
(2.186)
(J)!
J
2.11 Directional Derivatives
57
where ~ is any one of the products (^); ( ); (y; x) or (Cli¤ ord product).
@Y (the derivative operator ) is also called the generalized gradient operator and @Y F is said to be the generalized gradient of F . Also @Y ^ is
called the generalized rotational operator and @Y ^ F is said the generalized
rotational of F . Finally the operators @Y and @Y y are called generalized
divergent operators and @Y F is said the generalized scalar divergent of F
and @Y yF is said the generalized contracted divergent of F:
Note that these di¤erential operators are well de…ned since the right
side of Eq.(2.186) depends only on F and are independent of the pair of
reciprocal bases used.
2.11.5 The A ~ @Y directional derivative
De…nition 101 The A ~ @Y directional derivative of a di¤ erentiable multiform function F (Y ) is
A ~ @Y F (Y ) =
X
J
1
(A ~ "J )F"0J (Y )
(J)!
(2.187)
where ~ is any one of the four products (^), ( ), (y), (x) or the Cli¤ ord
product. The symbol J denotes collective indices.
Remark 102 We already introduced above the A @Y directional derivative. The other derivatives will appear in our formulation of the Lagrangian
formalism in Chapters 7 and 8. We see moreover that the symbol A @Y
for the directional derivative is well justi…ed. Indeed, from Eq.(2.187) using
in the place of ~ we have
A @Y F (Y ) =
X
J
=
X
J
0
= FP
1
(A "J )F"0J (Y )
(J)!
1
AJ F"0J (Y )
(J)!
(Y )
1
AJ "J
(J)!
J
0
= FA (Y ):
(2.188)
We now solve some exercises dealing with calculations of directional
derivatives and derivatives of some multiform functions that will be used
in the main text, in particular in the Lagrangian formulation of multiform
and extensor …elds over a Lorentzian spacetime (see Chapters 8 and 9).
58
2. Multivector and Extensor Calculus
2.12 Solved Exercices
V
V
Exercise
103 Let
V 3 Y 7! Y Y 2 R; where
V is some subspace
V
of V . Find A @Y (Y Y ) and @Y (Y Y ).
Solution:
d
(Y + hAiY ) (Y + hAiY )
d
=0
d
(Y Y + 2 hAiY Y + 2 hAiY hAiY )
=
d
A @Y (Y Y ) =
; (2.189)
=0
A @Y (Y Y ) = 2 hAiY Y = 2A Y:
P 1 J
P 1 J
@Y (Y Y ) =
" "J @Y (Y Y ) =
" 2("J Y ) = 2Y:
(J)!
J (J)!
J
(2.190)
Exercise 104 Let
Y ) and @Y (B Y ):
V
V 3 Y 7! B Y 2 R; with B 2
V
V . Find A @Y (B
Solution:
A @Y (B Y ) =
d
B (Y + hAiY )
d
=0
= B hAiY = A hBiY ;
(2.191)
@Y (B Y ) =
P
J
P 1 J
1 J
" "J @Y (B Y ) =
" ("J hBiY ) = hBiY :
(J)!
J (J)!
(2.192)
V
V
Exercise 105 Let
V 3 Y 7! (BY C) Y 2 R, with B; C 2 V : Find
A @Y ((BY C) Y ) and @Y ((BY C) Y ).
Solution:
d
(B(Y + hAiY )C) (Y + hAiY )
d
=0
d
=
((BY C) Y + (B hAiY C) Y + (BY C) hAiY )
d
+ 2 (B hAiY C) hAiY ) =0
A @Y ((BY C) Y ) =
= (B hAiY C) Y + (BY C) hAiY
e C);
e
= hAiY (BY C + BY
D
E
e C
e
A @Y ((BY C) Y ) = A BY C + BY
:
Y
(2.193)
2.12 Solved Exercices
59
e B).
e Then,
where we used that (AY B) Y = Y (AY
P
1 J
" "J @Y (BY C Y )
J (J)!
D
E
P 1 J
e C
e );
=
" ("J BY C + BY
Y
J (J)!
@Y ((BY C) Y ) =
and …nally
D
E
e C
e
@Y ((BY C) Y ) = BY C + BY
:
Y
(2.194)
V1
V2
Exercise 106 Let x ^ B, with x; a 2
V and B 2
V . Let also
V1
a2
V . Find a @x (x ^ B), the divergence @x y(x ^ B); the rotational
@x ^ (x ^ B) and the gradient @x (x ^ B):
Solution: Taking into account that x ^ B 2
V3
V we have
d
= a ^ B;
(x + a) ^ B
d
=0
n
n
P
P
@x y(x ^ B) =
"j y"j @x (x ^ B) =
"j y("j ^ B) = (n
a @x (x ^ B) =
j=1
@x ^ (x ^ B) =
@x (x ^ B) =
=
n
P
2)B:
j=1
"j ^ "j @x (x ^ B) =
j=1
n
P
"j "j @x (x ^ B) =
j=1
n
P
(2.195)
n
P
"j ^ ("j ^ B) = 0:
(2.196)
(2.197)
j=1
n
P
j=1
"j ("j ^ B)
("j y("j ^ B) + "j ^ ("j ^ B));
(2.198)
j=1
@x (x ^ B) = (n
2)B:
(n = dim V )
(2.199)
V1
V2
V1
Exercise 107 Let x 2
V and B 2
V and consider xy B 2
V.
Find a @x (xyB), @x y(xyB), @x ^ (xyB) and @x (xyB):
60
2. Multivector and Extensor Calculus
Solution:
d
= ayB:
(2.200)
(x + a)yB
d
=0
n
n
n
P
P
P
@x y(xyB) =
"j y"j @x (xyB) =
"j y("j yB) =
("j ^ "j )y Bk = 0;
a @x (xyB) =
j=1
j=1
j=1
(2.201)
@x ^ (xyB) =
@x (xyB) =
@x (xyB) =
n
P
j=1
n
P
j=1
n
P
j=1
"j ^ "j @x (xyB) =
"j "j @x (xyB) =
n
P
"j ^ ("j yB) = 2B:
(2.202)
j=1
n
P
"j ("j yB)
(2.203)
j=1
("j y("j yB) + "j ^ ("j yB)) = 2B:
(n = dim V )
(2.204)
V1
Vk
V
Exercise 108 Let a 2
V and Bk 2
V and A 2 V . Then, a^Bk 2
Vk+1
V and we can de…ne a (k + 1)-multiform function of vector variable
^1
V 3 a 7! a ^ Bk 2
^k+1
V:
(2.205)
Calculate A @a a ^ Bk , the divergence @a y(a ^ Bk ), the rotational @a ^
(a ^ Bk ) and the gradient @a (a ^ Bk ).
Solution:
A @a a ^ Bk =
d
(a + hAia ) ^ Bk
d
@a y(a ^ Bk ) =
X
@a y(a ^ Bk ) =
n
X
@a ^ (a ^ Bk ) =
@a (a ^ Bk ) =
J
j=1
n
X
j=1
n
X
j=1
=0
= hAia ^ Bk = hAi1 ^ Bk ;
(2.206)
X 1
1 J
" y"J @a y(a ^ Bk ) =
"J y(h"J i1 ^ Bk );
v(J)!
v(J)!
J
(2.207)
"j y("j ^ Bk ) = (n
k)Bk ;
"j ^ ("j ^ Bk ) = 0;
"j ("j ^ Bk ) =
= @a y(a ^ Bk ) = (n
(2.208)
(2.209)
n
X
["j y("j ^ Bk ) + "j ^ ("j ^ Bk )]
j=1
k)Bk :
(2.210)
2.12 Solved Exercices
61
V1
Vk
V
Exercise 109 Let a 2
V and Bk 2
V , 1 k n, and A 2 V .
Vk 1
Then, ayBk 2
V and we can de…ne a (k 1)-multiform function of
vector variable
^1
^k 1
V 3 a 7! ayBk 2
V:
(2.211)
Calculate A @a ayBk , the divergence @a y(ayBk ), the rotatitonal @a ^ (ayBk )
and the gradient @a (ayBk ).
Solution:
d
= hAia yBk = hAi1 yBk ; (2.212)
(a + hAia )yBk
d
=0
X 1
X 1
@a y(ayBk ) =
"J y"J @a y(ayBk ) =
"J y(h"J i1 yBk );
v(J)!
v(J)!
A @a ayBk =
J
J
(2.213)
@a y(ayBk ) =
n
X
j=1
n
X
"j y("j yBk ) =
("j ^ "j )yBk = 0;
(2.214)
j=1
@a ^ (ayBk ) =
@a (ayBk ) =
=
n
X
j=1
n
X
j=1
n
X
j=1
"j ^ ("j yBk ) = kBk
(2.215)
"j ("j yBk )
(2.216)
["j y("j yBk ) + "j ^ ("j yBk )]
= @a ^ (ayBk ) = kBk :
Exercise
Given two multiform functions F :
Vr
V110
q
V !
V abbreviated F (Y ), G(Y ) show that
(2.217)
Vr
V !
Vp
V, G :
@Y [F (Y ) ^ G(Y )] = @Y F (Y ) ^ G(Y ) + ( 1)pq F (Y )^@Y G(Y ):
(2.218)
(a) @x [@a (ayX) Y ] = (@x yX) Y + X (@x ^ Y )
(b) @x [@a (a ^ X) Y ] = (@x ^ X) Y + X (@x yY )
(c)
@x [@a (aX) Y ] = (@x X) Y + X (@x Y )
(2.219)
V1
V
Exercise 111 Let a 2
V and Y; Y :
V ! V be di¤ erentiable funcV1
tions of the position form x = xk "k 2
V . Prove the following identities
V1
Solution:
62
2. Multivector and Extensor Calculus
Note that if we prove (a) and (b) then (c) follows by summing the identities (a) and (b).
V1
(a) Using V
the algebraic identity (byB) C = B (b ^ C) valid for b 2
V
and B; C 2 V we can write the second member of (a) as
(@x yX) Y + X (@x ^ Y ) = ("k y"k @x X) Y + X ("k ^ "k @x Y )
= "k @x ("k yX) Y + "k yX ("k @x Y ):
(2.220a)
Now, taking into account Eq.(2.216) we can write the …rst member of
identity (a) in of Eq.(2.219) as follows:
@x [@a (ayX) Y ] = @x f"k [("k yX) Y ]g
= "l "k @l [("k yX) Y ]
=
kl
[("k y@l X) Y + ("k yX) @l Y ]
= (" @k X) Y + ("k yX) @k Y
k
= "k @x ("k yX Y ) + ("k yX) ("k @x Y ):
(2.221)
Comparing Eq.(2.220a) and Eq.(2.221) identity (a) is proved.
(b) The second member of identity (b) in Eq.(2.219) can be written as
(@x ^ X) Y + X (@x ^ Y ) = Y (@x ^ X) + (@x ^ Y ) X
= @x [@a (ayX) Y ]:
(2.222)
Taken into account the algebraic identity (b ^ B) C = B (byC) valid for
V1
V
b2
V and B; C 2 V we can write
@x [@a (ayX) Y ] = @x [@a X (a ^ Y )]
= @x [@a (a ^ X) Y ];
(2.223)
and identity (b) in Eq.(2.219) is proved.
(c) As we already said summing up identities (a) and (b), identity (c)
in Eq.(2.219) follows. Nevertheless we give another simple proof of that
identity. Indeed, we can work the …rst member of identity (c) as
@x [@a (aX) Y ] = "k "l @k ["l X Y ]
=
lk
[("l @k X) Y + ("l X) (@k Y )]
^
= ("k @k X) Y + h("
l X)(@k Y )i0
e
= (@Y ) Y + hX"l @k Y )i0
e )i0
= (@Y ) Y + hX@Y
= (@X) Y + X (@Y ):
(2.224)
This is page 63
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3
The Hidden Geometrical Nature of
Spinors
3.1
Notes on the Representation Theory of
Associative Algebras
To achieve our goal mentioned in Chapter 1 of disclosing the real secret
geometrical meaning of Dirac spinors, we shall need to brie‡y recall some
few results of the theory of representations of associative algebras. Propositions are presented without proofs and the interested reader may consult
[39, 92, 206, 220, 302, 303] for details.
Let V be a …nite dimensional linear space over K (a division ring).
Suppose that dimK V = n, where n 2 Z. We are interested in what follows
in the cases where K = R; C or H. In this case we also call V a vector space
over K. When K = H it is necessary to distinguish between right or left
H-linear spaces and in this case V will be called a right or left H-module.
Recall that H is a division ring (sometimes called a noncommutative …eld
or a skew …eld) and since H has a natural vector space structure over the
real …eld, then H is also a division algebra.
De…nition 112 Let V be a vector space over R and dimR V = 2m = n.
A linear mapping
J:V!V
(3.1)
such that
J2 =
s called a complex structure mapping.
IdV ;
(3.2)
64
3. The Hidden Geometrical Nature of Spinors
De…nition 113 Let V be as in the previous de…nition. The pair (V; J) is
called a complex vector space structurepand denote by VC if the following
product holds. Let C 3 z = a + ib (i =
1)and let v 2 V. Then
zv = (a + ib)v = av + bJv:
It is obvious that dimC =
(3.3)
m
2.
De…nition 114 Let V be a vector space over R. A complexi…cation of V
is a complex structure associated with the real vector space V V. The
resulting complex vector space is denoted by V C . Let v; w 2 V. Elements
of VC are usually denoted by c = v + iw, and if C 3 z = a + ib we have
zc = av
bw + i(aw + bv):
(3.4)
Of course, we have that dimC VC = dimR V.
De…nition 115 A H-module is a real vector space S carrying three linear
transformation, I, J and K each one of them satisfying
I 2 = J2 =
IJ =
IdS ,
JI = K; JK =
Exercise 116 Show that K2 =
KJ = I; KI =
IK = J:
(3.5a)
IdS
In what follows A denotes an associative algebra on the commutative
…eld F = R or C and F A.
De…nition 117 Any subset I
A such that
a 2 I; 8a 2 A; 8 2 I;
+
2 I; 8 ;
2I
(3.6)
is called a left ideal of A.
Remark 118 An analogous de…nition holds for right ideals where Eq.(3.6)
reads a 2 I; 8a 2 A; 8 2 I, for bilateral ideals where in this case Eq.(3.6)
reads a b 2 I; 8a; b 2 A; 8 2 I.
De…nition 119 An associative algebra A is simple if the only bilateral
ideals are the zero ideal and A itself.
Not all algebras are simple and in particular semi-simple algebras are
important for our considerations. A de…nition of semi-simple algebras requires the introduction of the concepts of nilpotent ideals and radicals.
To de…ne these concepts adequately would lead us to a long incursion on
the theory of associative algebras, so we avoid to do that here. We only
3.1 Notes on the Representation Theory of Associative Algebras
65
quote that semi-simple algebras are the direct sum of simple algebras and
of course simple algebras are semi simple. Then, for our objectives in this
chapter the study of semi-simple algebras is reduced to the study of simple
algebras.
De…nition 120 We say that e 2 A is an idempotent element if e2 = e.
An idempotent is said to be primitive if it cannot be written as the sum
of two non zero annihilating (or orthogonal ) idempotent, i.e., e 6= e1 + e2 ,
with e1 e2 = e2 e1 = 0 and e21 = e1 ; e22 = e2 .
We give without proofs the following theorems valid for semi-simple (and
thus simple) algebras A:
Theorem 121 All minimal left (respectively right) ideals of semi-simple A
are of the form J = Ae (respectively eA), where e is a primitive idempotent
of A.
Theorem 122 Two minimal left ideals of a semi-simple algebra A, J =
Ae and J = Ae0 are isomorphic, if and only if, there exist a non null
Y 0 2 J 0 such that J 0 = JY 0 .
Let A be an associative and simple algebra on the …eld F (R or C), and
let S be a …nite dimensional linear space over a division ring K F and let
E = EndK S = HomK (S; S) be the endomorphism algebra of S.1
De…nition 123 A representation of A in S is a K algebra homomorphism 2
: A ! E = EndK S which maps the unit element of A to IdE . The
dimension K of S is called the degree of the representation.
De…nition 124 The addition in S together with the mapping A S ! S,
(a; x) 7! (a)x turns S in a left A-module 3 , called the left representation
module.
Remark 125 It is important to recall that when K = H the usual recipe
for HomH (S; S) to be a linear space over H fails and in general HomH (S; S)
is considered as a linear space over R, which is the centre of H.
Remark 126 We also have that if A is an algebra on F and S is an Amodule, then S can always be considered as a vector space over F and if
a 2 A, the mapping : a ! a with a (s) = as; s 2 S, is a homomorphism
1 Recall
that HomK (V; W) is the algebra of linear transformations of a …nite dimensional vector space V over K into a …nite vector space W over K. When V = W the set
End K V =HomK (V; V) is called the set of endomorphisms of V.
2 We recall that a K-algebra homomorphism is a K-linear map
such that 8X; Y 2 A;
(XY ) = (X) (Y ).
3 We recall that there are left and right modules, so we can also de…ne right modular
representations of A by de…ning the mapping S A! S, (x; a) 7! x (a). This turns S
in a right A-module, called the right representation module.
66
3. The Hidden Geometrical Nature of Spinors
A ! EndF S, and so it is a representation of A in S. The study of A modules
is then equivalent to the study of the F representations of A.
De…nition 127 A representation
is faithful if its kernel is zero, i.e.,
(a)x = 0; 8x 2 S ) a = 0. The kernel of is also known as the annihilator
of its module.
De…nition 128 is said to be simple or irreducible if the only invariant
subspaces of (a); 8a 2 A, are S and f0g.
Then, the representation module is also simple. That means that it has
no proper submodules.
De…nition 129 is said to be semi-simple, if it is the direct sum of simple
modules, and in this case S is the direct sum of subspaces which are globally
invariant under (a); 8a 2 A.
When no confusion arises (a)x may be denoted by a x or ax.
De…nition 130 Two A-modules S and S0 (with the “exterior” multiplication being denoted respectively by and ) are isomorphic if there exists a
bijection ' : S ! S0 such that,
'(x + y) = '(x) + '(y); 8x; y 2 S;
'(a x) = a '(x); 8a 2 A;
and we say that the representations
modules are isomorphic.
and
0
(3.7)
of A are equivalent if their
This implies the existence of a K-linear isomorphism ' : S ! S0 such
that ' (a) = 0 (a) '; 8a 2 A or 0 (a) = ' (a) ' 1 . If dim S = n,
then dim S0 = n.
De…nition 131 A complex representation of A is simply a real representation : A ! HomR (S; S) for which
(Y ) J = J
This means that the image of
by fIdS ; Jg C.
(Y ); 8Y 2 A:
(3.8)
commutes with the subalgebra generated
De…nition 132 A quaternionic representation of A is a representation
: A ! HomR (S; S) such that
(Y ) I = I
(Y );
(Y ) J = J
(Y );
(Y ) K = K
(Y ); 8Y 2 A: (3.9)
3.2 Real and Complex Cli¤ord Algebras and their Classi…cation
This means that the representation
morphic to the quaternion ring.
The following theorem is crucial:
67
has a commuting subalgebra iso-
Theorem 133 (Wedderburn). If A is simple algebra over F then A is
isomorphic to D(m), where D(m) is a matrix algebra with entries in D (a
division algebra), and m and D are unique (modulo isomorphisms).
3.2 Real and Complex Cli¤ord Algebras and their
Classi…cation
Now, it is time to specialize the previous results to the Cli¤ord algebras
on the …eld F = R or C. We are particularly interested in the case of real
Cli¤ord algebras. In what follows we take V = Rn . We denote as in the
previous Chapter by Rp;q (n = p + q) the real vector space Rn endowed
with a nondegenerate metric g : Rn Rn ! R. Let fEi g ; (i = 1; 2; :::; n)
be an orthonormal basis of Rp;q ,
8
< +1; i = j = 1; 2; :::p;
1; i = j = p + 1; :::; p + q = n;
(3.10)
g(Ei ; Ej ) = gij = gji =
:
0; i 6= j:
We recall (De…nition 37 that the Cli¤ord algebra Rp;q = C`(Rp;q ) is the
Cli¤ord algebra over R, generated by 1 and the fEi g ; (i = 1; 2; :::; n) such
that Ei2 = g(Ei ; Ei ), Ei Ej = Ej Ei (i 6= j), and E1 E2 :::En 6= 1.
Rp;q is obviously of dimension 2n and as a vector space it is the direct
Vk n
sum of vector spaces
R of dimensions nk ; 0
k
n. The canonical
Vk n
basis of
R is given by the elements V
eA = E 1
E k; 1
1 < ::: <
n n
n. The element eJ = E1
En 2
R ,! Rp;q commutes (n odd)
k
V1 n
or anticommutes (n even) with all vectors E1 ; :::; En 2
R
Rn . The
V0 n
V0 n V0 n
center Rp;q is
R
R if n is even and it is the direct sum
R
R
if n is odd.4
All Cli¤ord algebras are semi-simple. If p + q = n is even, Rp;q is simple
and if p + q = n is odd we have the following possibilities:
(a) Rp;q is simple $ e2J =
isomorphic to C;
1$p
q 6= 1 (mod 4) $ center of Rp;q is
(b) Rp;q is not simple (but is a direct sum of two simple algebras) $
e2J = +1 $ p q = 1 (mod 4) $ center of Rp;q is isomorphic to R R.
4 For
a proof see [302].
68
3. The Hidden Geometrical Nature of Spinors
Table 2. Representation of the Cli¤ord algebras Rp;q as matrix algebras.
p q
mod 8
Rp;q
0
1
2
3
4
R(2 )
R(2 )
R(2 )
C(2 )
H(2
R(2 )
5
1)
6
H(2
1
) H(2
H(2
1
)
7
1)
C(2 )
Now, for Rp;q the division algebras D are the division rings R, C or H.
The explicit isomorphism can be discovered with some hard but not di¢ cult
work. It is possible to give a general classi…cation of all real (and also the
complex) Cli¤ord algebras and a classi…cation table can be found, e.g., in
[302]. One convenient table is the following one (where = [n=2] means
the integer part of n=2).
We denoted by R0p;q the even subalgebra of Rp;q and by R1p;q the set of
odd elements of Rp;q . The follwing very important result holds true
Proposition 134 R0p;q ' Rp;q
1
and also R0p;q ' Rq;p
1:
Now, to complete the classi…cation we need the following theorem:
Theorem 135 (Periodicity).5 We have
Rn+8
Rp+8;q
=
=
Rn;0
Rp;q
R8;0
R8;0
R0;n+8
Rp;q+8
= R0;n
= Rp;q
R0;8
R0;8 :
(3.11)
Remark 136 We emphasize here that since the general results concerning
the representations of simple algebras over a …eld F applies to the Cli¤ ord
algebras Rp;q we can talk about real, complex or quaternionic representation
of a given Cli¤ ord algebra, even if the natural matrix identi…cation is not
a matrix algebra over one of these …elds. A case that we shall need is that
R1;3 ' H(2). But it is clear that R1;3 has a complex representation, for
any quaternionic representation of Rp;q is automatically complex, once we
restrict C H and of course, the complex dimension of any H-module must
be even. Also, any complex representation of Rp;q extends automatically to
a representation of C Rp;q .
Remark 137 , C Rp;q is isomorphic to the complex Cli¤ ord algebra
C`p+q . The algebras C and Rp;q are subalgebras of C`p+q
5 See
[302].
3.3 The Algebraic, Covariant and Dirac-Hestenes Spinors
69
3.2.1 Pauli, Spacetime, Majorana and Dirac Algebras
For the purposes of our book we shall need to have in mind that:
R0;1 ' C;
R0;2 ' H;
R3;0 ' C(2);
R1;3 ' H(2);
R3;1 ' R(4);
R4;1 ' C(4):
(3.12)
R3;0 is called the Pauli algebra, R1;3 is called the spacetime algebra, R3;1
is called Majorana algebra and R4;1 is called the Dirac algebra. Also, the
following particular results, which can be easily proved, will be used many
times in what follows:
R01;3 ' R03;1 = R3;0 ;
R4;1 ' C
R3;1 ;
R04;1 ' R1;3 ;
R4;1 ' C
R3;1 :
R01;4 ' R1;3 ;
(3.13)
In words: the even subalgebras of both the spacetime and Majorana algebras is the Pauli algebra. The even subalgebra of the Dirac algebra is the
spacetime algebra and …nally the Dirac algebra is the complexi…cation of
the spacetime algebra or of the Majorana algebra.
Eqs.(3.13) show moreover, in view of Remark 137 that the spacetime
algebra has also a matrix representation in C(4). Obtaining such a representation is very important for the introduction of the concept of a DiracHestenes spinor, an important ingredient of the present work.
3.3 The Algebraic, Covariant and Dirac-Hestenes
Spinors
3.3.1 Minimal Lateral Ideals of Rp;q
We now give some results concerning the minimal lateral ideals of Rp;q .
Theorem 138 The maximum number of pairwise orthogonal idempotents
in K(m) (where K = R, C or H) is m .
The decomposition of Rp;q into minimal ideals is then characterized by
a spectral set {epq;j } of idempotents elements of Rp;q such that:
n
P
(a)
epq;j = 1;
i=1
(b) epq;j epq;k = jk epq;j ;
(c) the rank of epq;j is minimal and non zero, i.e., is primitive.
70
3. The Hidden Geometrical Nature of Spinors
V
By rank of epq;j we mean the rank of the Rp;q morphism, epq;j : 7 !
epq;j . Conversely, any 2 Ipq;j can be characterized by an idempotent
epq;j of minimal rank 6= 0, with = epq;j .
We now need to know the following theorem [217]:
Theorem 139 A minimal left ideal of Rp;q is of the type
Ipq = Rp;q epq ;
(3.14)
where
epq =
1
(1 + e 1 )
2
1
(1 + e
2
k
)
(3.15)
is a primitive idempotent of Rp;q and where e 1 ; :::; e k are commuting
elements in the canonical basis of Rp;q (generated in the standard way
through the elements of a basis (E1;::: ; Ep ; Ep+1 ; :::Ep+q ) of Rp;q ) such that
(e i )2 = 1; (i = 1; 2; :::; k) generate a group of order 2k , k = q rq p
and ri are the Radon-Hurwitz numbers, de…ned by the recurrence formula
ri+8 = ri + 4 and
i
ri
0
0
1
1
2
2
3
2
4
3
5
3
6
3
7
:
3
(3.16)
Recall that Rp;q is a ring and the minimal lateral ideals are modules over
the ring Rp;q . They are representation modules of Rp;q , and indeed we have
(recall the above table) the following theorem [217]:
Theorem 140 If p + q is even or odd with p
q 6= 1 (mod 4), then
Rp;q = HomK (Ipq ; Ipq ) ' K(m);
(3.17)
where (as we already know ) K = R, C or H. Also,
dimK (Ipq ) = m;
(3.18)
K ' eK(m)e;
(3.19)
and
where e is the representation of epq in K(m).
If p + q = n is odd, with p q = 1 (mod 4), then
Rp;q = HomK (Ipq ; Ipq ) ' K(m)
K(m);
(3.20)
with
dimK (Ipq ) = m
(3.21)
3.3 The Algebraic, Covariant and Dirac-Hestenes Spinors
71
and
eK(m)e ' R
R
eK(m)e ' H
H:
or
(3.22)
With the above isomorphisms we can immediately identify the minimal
left ideals of Rp;q with the column matrices of K(m).
3.3.2 Algorithm for Finding Primitive Idempotents of Rp;q .
With the ideas introduced above it is now a simple exercise to …nd primitive
idempotents of Rp;q . First we look at Table 2 and …nd the matrix algebra
to which our particular Cli¤ord algebra Rp;q is isomorphic. Suppose Rp;q
is simple6 . Let Rp;q ' K(m) for a particular K and m. Next we take an
element e 1 2 feA g from the canonical basis feA g of Rp;q such that
e2 1 = 1:
(3.23)
Next we construct the idempotent epq = (1 + e 1 )=2 and the ideal Ipq =
Rp;q epq and calculate dimK (Ipq ). If dimK (Ipq ) = m, then epq is primitive.
If dimK (Ipq ) 6= m, we choose e 2 2 feA g such that e 2 commutes with e 1
and e2 2 = 1 and construct the idempotent e0pq = (1 + e 1 )(1 + e 2 )=4. If
0
) = m, then e0pq is primitive. Otherwise we repeat the procedure.
dimK (Ipq
According to the Theorem 139 the procedure is …nite.
3.3.3 R?p;q , Cli¤ord, Pinor and Spinor Groups
The set of the invertible elements of Rp;q constitutes a non-abelian group
which we denote by R?p;q . It acts naturally on Rp;q as an algebra homomor^ or adjoint represenphism through its twisted adjoint representation (Ad)
tation (Ad)
^ : R?p;q ! Aut(Rp;q ); u 7! Adu ; with Adu (x) = ux^
Ad
u 1;
(3.24)
Ad : R?p;q ! Aut(Rp;q ); u 7! Adu ; with Adu (x) = uxu
1
(3.25)
De…nition 141 The Cli¤ ord-Lipschitz group is the set
p;q
p;q
= u 2 R?p;q
8x 2 Rp;q ; ux^
u
n
?(1)
= u 2 R?(0)
p;q [ Rp;q
or
1
2 Rp;q ;
8x 2 Rp;q ; uxu
1
2 Rp;q
(3.26a)
o
;
(3.26b)
6 Once we know the algorithm for a simple Cli¤ord algebra it is straightfoward to
devise an algorithm for the semi-simple Cli¤ord algebras.
72
3. The Hidden Geometrical Nature of Spinors
?(0)
?(1)
Note in Eq.(3.26b) the restriction to the even (Rp;q ) and odd (Rp;q )
parts of R?p;q .
De…nition 142 The set
group.
0
p;q
=
0
p;q \Rp;q
is called special Cli¤ ord-Lipschitz
De…nition 143 The Pinor group Pinp:q is the subgroup of
Pinp;q = fu 2
p;q
j N (u) =
p;q
such that
1g ;
(3.27)
where
N : Rp;q ! Rp;q ; N (x) = hxxi0 :
(3.28)
De…nition 144 The Spin group Spinp;q is the set
Spinp;q = u 2
0
p;q
j N (u) =
1 :
(3.29)
It is easy to see that Spinp;q is not connected.
De…nition 145 The Special Spin Group Spinep;q is the set
Spinep;q = u 2 Spinp;q j N (u) = +1 :
(3.30)
The superscript e, means that Spinep;q is the connected component to the
identity. We can prove that Spinep;q is connected for all pairs (p; q) with the
exception of Spine (1; 0) ' Spine (0; 1).
We recall now some classical results [242] associated with the pseudoorthogonal groups Op;q of a vector space Rp;q (n = p+q) and its subgroups.
Let G be a diagonal n n matrix whose elements are Gij
G = [Gij ] = diag(1; 1; :::; 1; 1; :::
1);
(3.31)
with p positive and q negative numbers.
De…nition 146 Op;q is the set of n
n real matrices L such that
LGLT = G; det L2 = 1:
(3.32)
Eq.(3.32) shows that Op;q is not connected.
De…nition 147 SOp;q , the special (proper ) pseudo orthogonal group is the
set of n n real matrices L such that
3.3 The Algebraic, Covariant and Dirac-Hestenes Spinors
LGLT = G; det L = 1:
73
(3.33)
When p = 0 (q = 0) SOp;q is connected. However, SOp;q (for, p; q 6= 0) is
not connected and has two connected components for p; q 1.
De…nition 148 The group SOep;q , the connected component to the identity
of SOp;q will be called the special orthochronous pseudo-orthogonal group 7 .
Theorem 149 AdjPinp;q : Pinp;q ! Op;q is onto with kernel Z2 .
AdjSpin : Spinp;q ! SOp;q is onto with kernel Z2 . AdjSpine : Spinep;q !
p;q
p;q
SOep;q is onto with kernel Z2 .
We have,
Op;q =
Spinep;q
Spinp;q
Pinp;q
; SOp;q =
; SOep;q =
:
Z2
Z2
Z2
(3.34)
The group homomorphism between Spinep;q and SOe (p; q) will be denoted
by
(3.35)
L : Spinep;q ! SOep;q :
The following theorem that …rst appears in [302] is very important.
Exercise 150 (Porteous). Show that for p+q
j u~
u = 1g.
4, Spine (p; q) = fu 2 Rp;q
Solution: We must show that for any u 2 R0p;q , N (u) = 1 and x 2 Rp;q
^ u (x) = uxu 1 . We
^ u (x) 2 Rp;q . But when u 2 R0p;q , Ad
we have that Ad
must then show that
y = uxu 1 2 Rp;q .
Since u 2 R0p;q we have that y 2 R1p;q . Let ei ; i = 1; 2; 3; 4 an orthonormal
basis of Rp;q , p + q = 4. Now, y = (uyu 1 )^~ = uxu 1 = y. Writing
1 ijk
y ei ej ek ,
3!
y i ; y ijk 2 R;
y = y i ei +
we get
y=
y i ei;
from which follows that y 2 Rp;q .
7 This nomenclature comes from the fact that SO e (1; 3) = L" is the special (proper)
+
orthochronous Lorentz group. In this case the set is easily de…ned by the condition
L00 +1. For the general case see [242].
74
3. The Hidden Geometrical Nature of Spinors
3.3.4 Lie Algebra of Spine1;3
It can be shown [218, 220, 455] that for each u 2 Spine1;3 it holds u = eF ;
V2 1;3
F 2
R ,! R1;3 and F can be chosen in such a way to have a positive
sign in Eq.(3.33), except in the particular case F 2 = 0 when u = eF .
From Eq.(3.33) it follows immediately that the Lie algebra of Spine1;3 is
V2 1;3
generated by the bivectors F 2
R ,! R1;3 through the commutator
product.
Exercise 151 Show that when F 2 = 0 we must have u =
eF .
3.4 Spinor Representations of R4;1 ; R04;1 and R1;3
We investigate now some spinor representations of R4;1 ; R04;1 and R1;3
which will permit us to introduce the concepts algebraic, Dirac and DiracHestenes spinors in the next section.
Let b0 = fe0 ; e1 ; e2 ; e3 g be an orthogonal basis of R1;3 ,! R1;3 , such
that e e + e e = 2 , with
= diag(+1; 1; 1; 1). Now, with
the results of the previous section we can verify without di¢ culties that
the elements e, e0 ; e00 2 R1;3
1
(1 + e0 )
2
1
e0 = (1 + e3 e0 )
2
1
e00 = (1 + e1 e2 e3 )
2
e=
(3.36)
(3.37)
(3.38)
are primitive idempotents of R1;3 . The minimal left ideals8 , I = R1;3 e,
I 0 = R1;3 e0 , I 00 = R1;3 e00 are right two dimension linear spaces over the
quaternion …eld (He = eH = eR1;3 e).
An elements 2 R1;3 21 (1+e0 ) has been called by Lounesto [219] a mother
spinor9 . Let us see the justice of this denomination. First recall from the
Spin
Pin
general result of the previous section that Z21;3 ' O1;3 , Z21;3 ' SO1;3 ,
Spine1;3
Z2
L"+
' SOe1;3 , and Spine1;3 ' Sl(2; C) is the universal covering group of
SOe1;3 , the special (proper) orthochronous Lorentz group. We can
show [134, 135] that the ideal I = R1;3 e carries the D(1=2;0) D(0;1=2)
representation of Sl(2; C). Here we need to know [134, 135] that each
8 According to De…nition 158 these ideals are algebraically equivalent. For example,
e0 = ueu 1 , with u = (1 + e3 ) 2
= 1;3 .
9 Elements of I 0 are sometimes called Hestenes ideal spinors.
3.4 Spinor Representations of R4;1 ; R04;1 and R1;3
75
can be written as
=
1e
+
2 e3 e1 e
+
3 e3 e0 e
+
4 e1 e0 e
=
X
i si ;
(3.39)
s4 = e1 e0 e
(3.40)
i
s1 = e;
s2 = e3 e1 e;
s3 = e3 e0 e;
and where the i are formally complex numbers, i.e., each i = (ai +bi e2 e1 )
with ai ; bi 2 R and the set fsi ; i = 1; 2; 3; 4g is a basis in the mother spinors
space.
Exercise 152 Prove Eq.(3.39).
Now we determine an explicit relation between representations of R4;1
and R3;1 . Let ff0 ; f1 ; f2 ; f3 ; f4 g be an orthonormal basis of R4;1 with
f02 = f12 = f22 = f32 = f42 = 1;
fA fB =
fB vA , A 6= B and A; B = 0; 1; 2; 3; 4:
De…ne the pseudo-scalar
i = f0 f1 f2 f3 f4 ;
i2 =
1;
ifA = fA i;
A = 0; 1; 2; 3; 4:
(3.41)
Put
E = f f4 ;
(3.42)
we can immediately verify that
E E +E E =2
:
(3.43)
Taking into account that R1;3 ' R04;1 we can explicitly exhibit here this
isomorphism by considering the map j: R1;3 ! R04;1 generated by the linear
extension of the map j# : R1;3 ! R04;1 , j# (e ) = E = f f4 , where E , ( =
0; 1; 2; 3) is an orthogonal basis of R1;3 . Note that j(1R1;3 ) = 1R04;1 , where
1R1;3 and 1R04;1 (usually denoted simply by 1) are the identity elements in
R1;3 and R04;1 . Now consider the primitive idempotent of R1;3 ' R04;1 ,
e041 = j(e) =
1
(1 + E0 )
2
0
= R04;1 e041 .
and the minimal left ideal I4;1
0
The elements Z 2 I4;1 can be written analogously to
as,
X
Z=
zi si
(3.44)
2 R1;3 21 (1+ e0 )
(3.45)
where
s1 = e041 ; s2 = E1 E3 e041 ; s3 = E3 E0 e041 ; s4 = E1 E0 e041
and where
zi = ai + E2 E1 bi ;
(3.46)
76
3. The Hidden Geometrical Nature of Spinors
are formally complex numbers, ai ; bi 2 R.
Consider now the element f 2 R4;1
1
f = e041 (1 + iE1 E2 )
2
1
1
= (1 + E0 ) (1 + iE1 E2 );
(3.47)
2
2
with i de…ned as in Eq.(3.41).
Since f R4;1 f = Cf = f C it follows that f is a primitive idempotent of
R4;1 . We can easily show that each 2 I = R4;1 f can be written
X
=
i fi ;
i 2 C;
i
f1 = f; f2 =
E1 E3 f; f3 = E3 E0 f; f4 = E1 E0 f:
(3.48)
With the methods described in ([134, 135]) we …nd the following representation in C(4) for the generators E of R4;1 ' R1;3
E0 7!
0
=
12
0
0
12
$ Ei 7!
i
=
0
i
i
0
;
(3.49)
where 12 is the unit 2 2 matrix and i , (i = 1; 2; 3) are the standard
Pauli matrices. We immediately recognize the -matrices in Eq.(3.49) as
the standard ones appearing, e.g., in [50].
The matrix representation of 2 I will be denoted by the same letter
in boldface, i.e., 7! 2 C(4)f , where
p
1
1
f = (1 + 0 ) (1 + i 1 2 ); i =
1:
(3.50)
2
2
We have
1
0
1 0 0 0
B 2 0 0 0 C
C
=B
(3.51)
i 2 C:
@ 3 0 0 0 A;
4 0 0 0
Eqs.(3.49,3.50,3.51) are su¢ cient to prove that there are bijections between the elements of the ideals R1;3 21 (1+ e0 ), R04;1 12 (1+E0 ) and R4;1 21 (1+
E0 ) 12 (1 + iE1 E2 ).
We can easily …nd that the following relation exist between
2 R4;1 f
and Z 2 R04;1 12 (1 + E0 );
1
= Z (1 + iE1 E2 ):
(3.52)
2
Decomposing Z into even and odd parts relative to the Z2 -graduation of
R04;1 ' R1;3 , Z = Z 0 + Z 1 we obtain Z 0 = Z 1 E0 which clearly shows that
all information of Z is contained in Z 0 . Then,
1
1
= Z 0 (1 + E0 ) (1 + iE1 E2 ):
2
2
(3.53)
3.5 Algebraic Spin Frames and Spinors
77
1
Now, if we take into account that R04;1 12 (1+E0 ) = R00
4;1 2 (1+E0 ) where the
1
00
00
0
symbol R4;1 means R4;1 ' R1;3 ' R3;0 we see that each Z 2 R00
4;1 2 (1 + E0 )
can be written
1
0
Z=
(1 + E0 )
2 R00
(3.54)
4;1 ' R1;3 :
2
Then putting Z 0 = =2, Eq.(3.54) can be written
1
1
(1 + E0 ) (1 + iE1 E2 )
2
2
01
= Z (1 + iE1 E2 ):
2
=
(3.55)
The matrix representation of
and Z in C(4) (denoted by the same
letter in boldface) in the matrix representation generated by the spin basis
given by Eq.(3.48) are
0
1
0
1
0 0
1
3
1
2
4
2
B 2
C
B 2
0 0 C
4
1
3 C
1
C : (3.56)
=B
; Z=B
@ 3
A
@
0 0 A
1
3
4
2
4
0 0
4
2
4
3
1
3
3.5 Algebraic Spin Frames and Spinors
We introduce now the fundamental concept of algebraic spin frames 10 . This
is the concept that will permit us to de…ne spinors (steps (i)-(vii)).11
(i) In this section (V; ) refers always to Minkowski vector space.
(ii) Let SO(V; ) be the group of endomorphisms of V that preserves
and the space orientation. This group is isomorphic to SO1;3 but there is
no natural isomorphism. We write SO(V; ) ' SO1;3 . Also, the connected
component to the identity is denoted by SOe (V; ) and SOe (V; ) ' SOe1;3 .
Note that SOe (V; ) preserves besides orientation also the time orientation.
(iii) We denote by C`(V; ) the Cli¤ord algebra12 of (V; ) and by
Spine (V; ) ' Spine1;3 the connected component of the spin group Spin(V; )
' Spin1;3 . Consider the 2 : 1 homomorphism L : Spine (V; ) ! SOe (V; ),
u 7! L(u) Lu . Spine (V; ) acts on V identi…ed as the space of 1-vectors
of C`(V; ) ' R1;3 through its adjoint representation in the Cli¤ord algebra
1 0 The
name spin frame will be reserved for a section of the spinor bundle structure
PSpine1;3 (M ) which will be introduced in Chapter 7.
1 1 This section follows the developments given in [350].
1 2 We reserve the notation R
n
p;q for the Cli¤ord algebra of the vector space R equipped
with a metric of signature (p; q), p + q = n. C`(V; g) and Rp;q are isomorphic, but there
is no canonical isomorphism. Indeed, an isomorphism can be exhibit only after we …x
an orthonormal basis of V.
78
3. The Hidden Geometrical Nature of Spinors
C`(V; ) which is related with the vector representation of SOe (V; ) as
follows13 :
Spine (V; ) 3 u 7! Adu 2 Aut(C`(V; ))
Adu jV : V ! V; v 7! uvu
1
= Lu
v:
(3.57)
In Eq.(3.57) Lu v denotes the standard action Lu on v and where we
identi…ed Lu 2 SOe (V; ) with Lu 2 V V and
(Lu
v; Lu
v) =
(v; v) :
(3.58)
(iv) Let B be the set of all oriented and time oriented orthonormal basis14
of V. Choose among the elements of B a basis b0 = fb0 ; ::::; b3 g, hereafter
called the …ducial frame of V. With this choice, we de…ne a 1 1 mapping
: SOe (V; ) ! B;
(3.59)
given by
Lu 7!
(Lu ) :=
Lu
= Lb0
(3.60)
where Lu = Lu b0 is a short for fe1 ; ::::; e3; g 2 B, such that denoting the
action of Lu on bi 2 b0 by Lu bi we have
ei = Lu bi := Lji bj ,
i; j = 0; :::; 3:
(3.61)
In this way, we can identify a given vector basis b of V with the isometry
Lu that takes the …ducial basis b0 to b. The …ducial basis b0 will be also
denoted by L0 , where L0 = e, is the identity element of SOe (V; ).
Since the group SOe (V; ) is not simple connected their elements cannot
distinguish between frames whose spatial axes are rotated in relation to the
…ducial vector frame L0 by multiples of 2 or by multiples of 4 . For what
follows it is crucial to make such a distinction. This is done by introduction
of the concept of algebraic spin frames.
De…nition 153 Let b0 2 B be a …ducial frame and choose an arbitrary
u0 2 Spine (V; ). Fix once and for all the pair (u0 ; b0 ) with u0 = 1 and
call it the …ducial algebraic spin frame.
De…nition 154 The space Spine (V; ) B = f(u; b); ubu 1 = u0 b0 u0 1 g
will be called the space of algebraic spin frames and denoted by S.
Remark 155 It is crucial for what follows to observe here that the De…nition 154 implies that a given b 2 B determines two, and only two, algebraic
spin frames, namely (u; b) and ( u; b), since ub( u 1 ) = u0 b0 u0 1 .
1 3 Aut(C`(V; g))
1 4 We
denotes the (inner) automorphisms of C`(V; g).
will call the elements of B (in what follows) simply by orthonormal basis.
3.5 Algebraic Spin Frames and Spinors
79
(v) We now parallel the construction in (iv) but replacing SOe (V; ) by
its universal covering group Spine (V; ) and B by S. Thus, we de…ne the
1 1 mapping
Spine (V; ) !
u
7
!
:
S;
(u): =
u=
(3.62)
(u; b),
where ubu 1 = b0 .
The …ducial algebraic spin frame will be denoted in what follows by
It is obvious from Eq.(3.62) that ( u)
u = ( u; b) 6= u .
0.
De…nition 156 The natural right action of a 2 Spine (V; ) denoted by
on S is given by
a
u
Observe that if
u0
u
1
(u; b) = (ua; Ada 1 b) = (ua; a
=a
= (u0 ; b 0 ) = u0
1 0
= (u
u)
u
Note that there is a natural 2
and
0
= (u0 ; u0
u
1
ba):
= (u; b) = u
ubu
(3.63)
0
then,
1 0
u ):
1 mapping
s : S!B;
1
7! b = ( u
u
)b0 ( u);
(3.64)
(s(
(3.65)
such that
s((u
1 0
u)
u ))
= Ad(u
1 u0 )
1
u )):
Indeed,
s((u
1 0
u)
u ))
1 0
= s((u
0 1
=u
u)
ub(u
= Ad(u
(u; b))
0 1
1 u0 )
1
u)
1
= b0
b = Ad(u
1 u0 )
1
(s(
u )):
(3.66)
This means that the natural right actions of Spine (V; ), respectively on S
and B, commute. In particular, this implies that the algebraic spin frames
u;
u 2 S, which are, of course distinct, determine the same vector
frame Lu = s( u ) = s( u ) = L u . We have,
Lu
=
L
u
= Lu
1u
0
L u0 ;
Lu
1u
0
2 SOe1;3 :
(3.67)
Also, from Eq.(3.65), we can write explicitly
; u 2 Spine (V; g);
(3.68)
where the meaning of Eq.(3.68) of course, is that if Lu = L u = b =
fe0 ; ::::; e3; g 2 B and Lu0 = b0 2 B is the …ducial frame, then
u0
Lu0 u0
1
=u
Lu u
1
; u0
Lu0 u0
1
= ( u)
u0 bj u0 1 = ( u)ej ( u
L
1
u
):
( u)
1
(3.69)
80
3. The Hidden Geometrical Nature of Spinors
In resume, we can say that the space S of algebraic spin frames can be
thought as an extension of the space B of vector frames, where even if two
vector frames have the same ordered vectors, they are considered distinct
if the spatial axes of one vector frame is rotated by an odd number of 2
rotations relative to the other vector frame and are considered the same
if the spatial axes of one vector frame is rotated by an even number of
2 rotations relative to the other frame. Even if the possibility of such a
distinction seems to be impossible at …rst sight, Aharonov and Susskind
[4] claim that it can be implemented physically in a spacetime where the
concept of algebraic spin frame is enlarged to the concept of spin frame
used for the de…nition of spinor …elds. See Chapter 7 for details.
(vi) Before we proceed an important digression on the notation used below is necessary. We recalled above how to construct a minimum left (or
right) ideal for a given real Cli¤ord algebra once a vector basis b 2 B for
V ,! C`(V; g) is given. That construction suggests to label a given primitive idempotent and its corresponding ideal with the subindex b. However,
taking into account the above discussion of vector and algebraic spin frames
and their relationship we …nd useful for what follows (specially in view
of the De…nition 157 and the de…nitions of algebraic and Dirac-Hestenes
spinors (see De…nitions 159 and 161 below)) to label a given primitive idempotent and its corresponding ideal with a subindex u . This notation is
also justi…ed by the fact that a given idempotent is according to de…nition
159 representative of a particular spinor in a given algebraic spin frame u .
(vii) Next we recall Theorem 139 which says that a minimal left ideal of
C`(V; ) is of the type
I u = C`(V; )e u
(3.70)
where e u is a primitive idempotent of C`(V; ).
It is easy to see that all ideals I u = C`(V; )e u and I
such that
e u0 = (u0 1 u)e u (u0 1 u) 1
u0
= C`(V; )e
u0
(3.71)
u; u0 2 Spine (V; ) are isomorphic. We have the
De…nition 157 Any two ideals I u = C`(V; )e u and I u0 = C`(V; )e u0
such that their generator idempotents are related by Eq.(3.71) are said geometrically equivalent.
Remark 158 If u is simply an element of the Cli¤ ord group, then the
ideals are said to be algebraically equivalent.
But take care, no equivalence relation has been de…ned until now. We
observe moreover that we can write
I
u0
=I
u
(u0
1
u)
1
;
an equation that will play a key role in what follows.
(3.72)
3.6 Algebraic Dirac Spinors of Type I
3.6 Algebraic Dirac Spinors of Type I
u
81
u
Let fI u g be the set of all ideals geometrically equivalent to a given minimal
I uo as de…ned by Eq.(3.72). Let be
T = f(
u;
u
) j u 2 Spine (V; ),
Let u ; u0 2 S,
u 2 I
relation E on T by setting
(
u
u;
,
2 I
u0
u
)
(
2 S,
u
u0
u0 ;
u
2I
u
g:
(3.73)
. We de…ne an equivalence
u0
);
(3.74)
if and only if and
(i) us(
u )u
(ii)
u0
1
u0
= u0 s(
1
=
0 1
;
u0 )u
u
1
:
(3.75)
)] 2 T=E
(3.76)
u
De…nition 159 An equivalence class
u
= [(
u;
u
is called an algebraic spinor of type I u for C`(V; ).
2 I u is said to
u
be a representative of the algebraic spinor
in
the
algebraic
spin frame
u
.
u
We observe that the pairs ( u ;
are equivalent, but the pairs ( u ;
are not. This distinction is essential
space (over the real …eld) to the set
on T is given by
a[(
u;
)] + b[(
u;
(a + b)[(
u;
u
0
)
)
u
in
T.
u
u
)] = [(
u
)] = a[(
and ( u ;
u;
u) = (
u)
and ( u ;
)
=
(
;
)
u
u
u
order to give a structure of linear
Indeed, a natural linear structure
u; a
u;
u
)] + [(
u
)] + b[(
u0 ; b
u;
0
u
u
)];
)]:
(3.77)
Remark 160 The de…nition just given is not a standard one in the literature ([74, 92]). However, the fact is that the standard de…nition (licit as
it is from the mathematical point of view ) is not adequate for a comprehensive formulation of the Dirac equation using algebraic spinor …elds or
Dirac-Hestenes spinor …elds which will be introduced in Chapter 7.
We end this section recalling that as observed above a given Cli¤ord
algebra Rp;q may have minimal ideals that are not geometrically equivalent since they may be generated by primitive idempotents that are related
by elements of the group R?p;q which are not elements of Spine (V; ) (see
Eqs.(3.36, 3.37, 3.38) where di¤erent, non geometrically equivalent primitive ideals for R1;3 are shown). These ideals may be said to be of di¤erent
82
3. The Hidden Geometrical Nature of Spinors
types. However, from the point of view of the representation theory of the
real Cli¤ord algebras all these primitive ideals carry equivalent (i.e., isomorphic) modular representations of the Cli¤ord algebra and no preference
may be given to any one15 . In what follows, when no confusion arises and
the ideal I u is clear from the context, we use the wording algebraic Dirac
spinor for any one of the possible types of ideals.
The most important property concerning algebraic Dirac spinors is a
coincidence given by Eq.(3.78) below. It permit us to de…ne a new kind of
spinors.
3.7 Dirac-Hestenes Spinors (DHS)
Let
u
2 S be an algebraic spin frame for (V;
u)
s(
) such that
= fe0 ; e1 ; e2 ; e3 g 2 B:
Then, it follows from Eq.(3.54) that
I
= C`(V; )e
u
when
u
2I
u
u
= C`0 (V; )e
u
;
1
(1 + e0 ):
2
can be written as
e
Then, each
u
=
u
u
e
(3.79)
=
u
;
u
(3.78)
2 C`0 (V; ):
(3.80)
From Eq.(3.75) we get
u0
u0
1
ue
u
=
u
e
u
;
u
;
2 C`0 (V; ):
u0
(3.81)
A possible solution for Eq.(3.81) is
u0
u0
1
=
u
u
1
:
(3.82)
Let S C`(V; ) and consider an equivalence relation E such that
(
if and only if
u0
and
u;
u
u
)
(
u0 ;
u0
) (mod E);
(3.83)
are related by
u0
u0
1
=
u
u
1
:
(3.84)
This suggests the following
1 5 The fact that there are ideals that are algebraically, but not geometrically equivalent
seems to contain the seed for new Physics, see [255, 256].
3.7 Dirac-Hestenes Spinors (DHS)
83
De…nition 161 The equivalence classes [( u ; u )] 2 S C`(V; )=E are
the Hestenes spinors.
Among the Hestenes spinors, an important subset is the one consisted of
Dirac-Hestenes spinors where [( u ; u )] 2 (S C`0 (V; ))=E.
We say that u ( u ) is a representative of a Hestenes (Dirac-Hestenes)
spinor in the algebraic spin frame u .
3.7.1 What is a Covariant Dirac Spinor (CDS)
Let L0 : S ! B and let L0 ( u ) = fE0 ; E1 ; E2 ; E3 g and L0 ( u0 ) = fE00 ; E10 ; E20 ; E30 g
with L0 ( u ) = uL0 ( o )u 1 , L( u0 ) = u0 L0 ( o )u0 1 be two arbitrary basis
for R1;3 ,! R4;1 .
As we already know f 0 = 12 (1+E0 ) 12 (1+iE1 E2 ) (Eq.(3.48)) is a primitive
idempotent of R4;1 ' C(4). If u 2 Spin (1; 3)
Spin(4; 1) then all ideals
I u = I 0 u 1 are geometrically equivalent to I 0 . From Eq.(3.49) we can
write
X
X
0 0
I u3
=
=
(3.85)
i fi ; and I 0u 3
i fi ;
u
u
where
f1 = f
u
;
f2 =
E1 E3 f
u
f10 = f
u
;
f3 = E3 E0 f
u
;
f20 =
E10 E30 f
u
;
f4 = E1 E0 f
u
;
f30 = E30 E00 f
u
;
f4 = E10 E00 f
u
and
Since
u0
0
u
=
=
X
i
(u0
1
u)
1
, we get
X
0 1
u) 1 fi0 =
Sik [(u
i (u
u
1 0
u )] i fk =
i;k
Then
k
:
=
X
Sik (u
X
k fk :
k
1 0
u ) i;
(3.86)
i
where Sik (u 1 u0 ) are the matrix components of the representation in C(4)
of (u 1 u0 ) 2 Spine1;3 . As proved in ([134, 135]) the matrices S(u) correspond
to the representation D(1=2;0) D(0;1=2) of Sl(2; C) ' Spine1;3 .
Remark 162 We remark that all the elements of the set fI u g of the ideals
geometrically equivalent to I 0 under the action of u 2 Spine1;3
Spine4;1
have the same image I = C(4)f where f is given by Eq.(3.47), i.e.,
f=
where
;
1
(1 +
2
0
)(1 + i
1 2
);
i=
p
1;
= 0; 1; 2; 3 are the Dirac matrices given by Eq.(3.50). Then, if
: R4;1 ! C(4)
End(C(4));
x 7! (x) : C(4)f ! C(4)f
(3.87)
84
3. The Hidden Geometrical Nature of Spinors
it follows that
(E ) = (E 0 );
(f ) = (f 0 )
(3.88)
for all fE g, fE 0 g such that E 0 = (u0 1 u)E (u0 1 u) 1 . Observe that all
information concerning the geometrical images of the algebraic spin frames
0
u,
u0 ; :::; under L disappear in the matrix representation of the ideals
I u ; I u0 ; ::::; in C(4) since all these ideals are mapped in the same ideal
I = C(4)f .
Taking into account Remark 162 and taking into account the de…nition
of algebraic spinors given above and Eq.(3.86) we are lead to the following
De…nition 163 A covariant Dirac spinor for R1;3 is an equivalence class
of pairs ( m
), where m
u ;
u is a matrix algebraic spin frame associated to
1
1
the algebraic spin frame u through the S(u 1 ) 2 D( 2 ;0) D(0; 2 ) representation of Spine1;3 ; u 2 Spine1;3 . We say that ; 0 2 C(4)f are equivalent
and write
0
( m
) ( m
);
(3.89)
u ;
u0 ;
if and only if,
0
= S(u0
1
u) ;
us(
u )u
1
= u0 s(
0 1
:
u0 )u
(3.90)
Remark 164 The de…nition of CDS just given agrees with that given in
[77] except for the irrelevant fact that there, as well as in the majority of
Physics textbook’s, authors use as the space of representatives of a CDS a
complex four-dimensional space C4 instead of I = C(4)f .
3.7.2 Canonical Form of a Dirac-Hestenes Spinor
Let v 2 R1;3 ,! R1;3 be a non lightlike vector, i.e., v2 = v v 6= 0 and
consider a linear mapping
L : R1;3 ! R1;3 , v 7! z = v ~,
z2 = v 2
(3.91)
~
with 2 R1;3 and 2 R+ . Now, recall that if R 2 Spine1;3 then w = RvR
2
2
is such that w = v . It follows that the most general solution of Eq.(3.91)
is
= 1=2 e 2 e5 R;
(3.92)
V4 1;3
where 2 R is called the Takabayasi angle and e5 = e0 e1 e2 e3 2
R ,!
R1;3 is the pseudoscalar of the algebra. Now, Eq.(3.92) shows that
2
R01;3 ' R3;0 : Moreover, we have that ~ 6= 0 since
~= e
e5
=
= cos ;
+ e5 !;
! = sin :
(3.93)
3.7 Dirac-Hestenes Spinors (DHS)
85
The Secret
Now, let
be a representative of a Dirac-Hestenes spinor (De…nition
u
161) in a given spin frame u . Since
2 R01;3 ' R3;0 we have disclosed
u
the real geometrical meaning of a Dirac-Hestenes spinor. Indeed, a Dirac~ 6= 0 induces the linear mapping given
Hestenes spinor such that
u
u
by Eq.(3.91), which rotates a vector and dilate it. Observe, that even if we
started our considerations with v 2 R1;3 ,! R1;3 and v2 6= 0, the linear
mapping (3.91) also rotates and ‘dilate’a light vector.
3.7.3 Bilinear Invariants and Fierz Identities
De…nition 165 Given a representative
in the algebraic
u of a DHS
16
spin frame …eld V
the
bilinear
invariants
associated
with
it are the obu
V4 1;3
0
jects:
?! 2 ( R1;3 +
R ) ,! R1;3 , J = J e 2 R1;3 ,! R1;3 ,
V2 1;3
S = 12 S e e 2
R ,! R1;3 , K = K e 2 R1;3 ,! R1;3 such that
u
and where ?! =
~
u
=
u
e1 e2 ~
u
e3 ~
u
?!
u
=S
=K
u
e0 ~
u
e0 e3 ~
u
e0 e1 e2 ~
u
= J;
u
= ?S;
u
(3.94)
= ?K:
e5 sin .
The bilinear invariants satisfy the so called Fierz identities, which are
J2 =
2
+ !2 ;
J K = 0;
J2 =
K 2;
J ^ K = (!
8
SxK = !J;
< SxJ = !K
(?S)xJ =
K
(?S)xK =
J;
:
~ 0 = 2 !2
S S = hS Si
(?S) S = 2 !:
8
JS = (! ? )K;
>
>
>
>
> SJ = (! + ? )K;
>
<
SK = (! ? )J;
> KS = (! + ? )J;
>
>
2
>
2 (?!);
> S 2 = !2
>
:
1
S = KSK=J 4 :
? )S (3.95)
(3.96)
(3.97)
Exercise 166 Prove the Fierz identities.
1 6 In Physics literature the components of J, S and K when written in terms of covariant Dirac spinors are called bilinear covariants :
86
3. The Hidden Geometrical Nature of Spinors
3.7.4 Reconstruction of a Spinor
The importance of the bilinear invariants is that once we know !, , J, K
and F we can recover from them the associate covariant Dirac spinor (and
thus the DHS ) except for a phase. This can be done with an algorithm due
to Crawford [90] and presented in a very pedagogical way in [218, 219, 220].
Here we only give the result for the case where and/or ! are non null.
De…ne
p the object B 2 C R1;3 ' R4;1 called boomerang and given by
1)
(i =
B = + J + iS ie5 K + e5 !
(3.98)
Then, we can construct
= Bf 2 R4;1 f , where f is the idempotent
given by Eq.(3.47) which has the following matrix representation in C(4)
(once the standard representation of the Dirac gamma matrices are used)
0
1
1 0 0 0
B
C
^ =B 2 0 0 0 C
(3.99)
@ 3 0 0 0 A
4 0 0 0
Now, it can be easily veri…ed that = Bf determines the same bilinear
covariants as the ones determined by
. Note however that this spinor
u
is not unique. In fact, B determines a class of elements B where is an
arbitrary element of R4;1 f which di¤ers one from the other by a complex
phase factor.
3.7.5 Lounesto Classi…cation of Spinors
A very interesting classi…cation of spinors have been devised by Lounesto
[218, 219, 220] based on the values of the bilinear invariants. He identi…ed
six di¤erent classes and proved that there are no other classes based on
distinctions between bilinear covariants. Lounesto classes are:
1.
6= 0;
! 6= 0.
2.
6= 0;
! = 0.
3.
= 0;
! 6= 0.
4.
= 0 = !;
K 6= 0;
S 6= 0.
5.
= 0 = !;
K = 0;
S 6= 0.
6.
= 0 = !;
K 6= 0;
S = 0.
The current density J is always non-zero. Type 1, 2 and 3 spinor are denominated Dirac spinor for spin-1/2 particles and type 4, 5, and 6 are
singular spinors respectively called ‡ag-dipole, ‡agpole and Weyl spinor.
3.8 Majorana and Weyl Spinors
87
Majorana spinor is a particular case of a type 5 spinor. It is worthwhile to
point out a peculiar feature of types 4, 5 and 6 spinor: although J is always
non-zero, we have due to Fierz identities that J 2 = K 2 = 0.
Spinors belonging to class 4 have not previously been identi…ed in the
literature. For the applications we have in mind we are interested (besides
Dirac spinors which belong to classes 1 or 2 or 3) in spinors belonging to
classes 5 and 6, respectively the Majorana and Weyl spinors.
Remark 167 In [6] Ahluwalia-Khalilova and Grumiller introduced from
physical considerations a supposedly new kind of spinors representing dark
matter that they dubbed ELKO spinors. The acronym stands for the German word Eigenspinoren des Ladungskonjugationsoperators. It has been
proved in [325] that from the algebraic point of view ELKO spinors are
simply class 5 spinors. In [6] it is claimed that di¤ erently from the case of
Dirac, Majorana and Weyl spinor …elds which have mass dimension 3=2,
ELKO spinor …elds must have mass dimension 1 and thus instead of satisfying Dirac equation satisfy a Klein-Gordon equation. A thoughtful analysis
of this claim is given in Chapter 16.
3.8 Majorana and Weyl Spinors
Recall that for Majorana spinors = 0, ! = 0, K = 0, S 6= 0, J; 6= 0
Given a representative of an arbitrary Dirac-Hestenes spinor we may
construct the Majorana spinors
1
(
2
Note that de…ning an operator C:
have
C M =
M
e01 ) :
=
7!
(3.100)
e01 (charge conjugation) we
M;
(3.101)
i.e., Majorana spinors are eigenvectors of the charge conjugation operator.
Majorana spinors satisfy
~
M
M
=
M
~
M
= 0:
(3.102)
For Weyl spinors = ! = S = 0 and K 6= 0, J 6= 0.
Given a representative of an arbitrary Dirac-Hestenes spinor we may
construct the Weyl spinors
W
=
1
(
2
e5 e21 ) :
(3.103)
Weyl spinors are ‘eigenvectors’of the chirality operator e5 = e0 e1 e2 e3 ,
i.e.,
e5 W =
(3.104)
W e21 :
88
3. The Hidden Geometrical Nature of Spinors
We have also,
~
W
W
=
W
~
W
= 0:
(3.105)
For future reference we introduce the parity operator acting on the space
of Dirac- Hestenes spinors. The parity operator P in this formalism [217]
is represented in such a way that for 2 R01;3
P
=
e0 e0 :
(3.106)
The following Dirac-Hestenes spinors are eigenstates of the parity operator with eigenvalues 1:
P
P
where
:=
"
#
=+
=
"
#
;
;
"
#
= e0
= e0
e0
+ e0
+
+
;
;
(3.107)
W
3.9 Dotted and Undotted Algebraic Spinors
Dotted and undotted covariant spinor …elds are very popular subjects in
General Relativity. Dotted and undotted algebraic spinor …elds may be
introduced using the methods of Chapter 7 and are brie‡y discussed in
Exercise 500. A preliminary to that job is a deep understanding of the
algebraic aspects of those concepts, i.e., the dotted and undotted algebraic
spinors which we now discuss. Their relation with Weyl spinors will become
apparent in a while.
Recall that the spacetime algebra R1;3 is the real Cli¤ord algebra associated with Minkowski vector space R1;3 , which is a four dimensional real
vector space, equipped with a Lorentzian bilinear form
: R1;3
R1;3 ! R:
(3.108)
Let fe0 ; e1 ; e2 ; e3 g be an arbitrary orthonormal basis of R1;3 , i.e.,
(e ; e ) =
;
(3.109)
where the matrix with entries
is the diagonal matrix diag(1; 1; 1; 1).
Also, fe0 ; e1 ; e2 ; e3 g is the reciprocal basis of fe0 ; e1 ; e2 ; e3 g, i.e., (e ; e ) =
. We have in obvious notation
(e ; e ) =
;
where the matrix with entries
is the diagonal matrix diag(1; 1; 1; 1).
The spacetime algebra R1;3 is generate by the following algebraic fundamental relation
e e +e e = 2
:
(3.110)
3.9 Dotted and Undotted Algebraic Spinors
89
As we already know (Section 3.6.1) the spacetime algebra R1;3 asVa vector
space over the real …eld is isomorphic to the exterior algebra R1;3 =
M 4 Vj
V
R1;3 of R1;3 . We code that information writing R1;3 ,! R1;3 .
j=0
V0 1;3
V1 1;3
Also, we make the following identi…cations:
R
R and
R
R1;3 .
Moreover, we identify the exterior product of vectors by
e ^e =
1
(e e
2
e e );
(3.111)
and also, we identify the scalar product of vectors by
(e ; e ) =
1
(e e + e e ) :
2
(3.112)
Then we can write
e e = (e ; e ) + e ^ e :
(3.113)
Now, an arbitrary element C 2 R1;3 can be written as sum of nonhomogeneous multivectors, i.e.,
1
1
C=s+c e + c e e + c
2
3!
e e e + pe5
(3.114)
where s; c ; c ; c ; p 2 R and c ; c
are completely antisymmetric in
all indices. Also e5 = e0 e1 e2 e3 is the generator of the pseudoscalars. Recall
also that as a matrix algebra we have that R1;3 ' H(2), the algebra of the
2 2 quaternionic matrices.
3.9.1 Pauli Algebra
Next, we recall (again) that the Pauli algebra R3;0 is the real Cli¤ord
algebra associated with the Euclidean vector space R3;0 , equipped as usual,
with a positive de…nite bilinear form. As a matrix algebra we have that
R3;0 ' C (2), the algebra of 2 2 complex matrices. Moreover, we recall
that R3;0 is isomorphic to the even subalgebra of the spacetime algebra,
(0)
(1)
i.e., writing R1;3 = R1;3 R1;3 we have,
(0)
R3;0 ' R1;3 :
(3.115)
The isomorphism is easily exhibited by putting
Indeed, with ij = diag(1; 1; 1), we have
i
j
+
j
i
=2
ij
;
i
= ei e0 , i = 1; 2; 3.
(3.116)
which is the fundamental relation de…ning the algebra R3;0 . Elements of
the Pauli algebra will be called Pauli numbers17 . As a vector space over
1 7 Sometimes they are also called ‘complex quaternions’. This last terminology will
become obvious in a while.
90
3. The Hidden Geometrical Nature of Spinors
the real …eld, we have that R3;0 is isomorphic to
any Pauli number can be written as
P = s + pi
where s; pi ; pij ; p 2 R and pij =
i=
Note that i2 =
trivially verify
i
j
R3;0 ,! R3;0
+ pi;
R1;3 . So,
(3.117)
pji and also
1
i=
2
3
= e5 :
(3.118)
1 and that i commutes with any Pauli number. We can
i
[
1
+ piij
2
i
V
i
;
j
j
= i"ik j
]=
i
k
j
ij
+
j
i
;
(3.119)
i
=2
(0)
j
^
=
2i"ik j
k
:
(1)
In that way, writing R3;0 = R3;0 + R3;0 , any Pauli number can be written
as
(0)
P = Q1 +iQ2 ;
(1)
Q1 2 R3;0 ;
iQ2 2 R3;0 ;
(3.120)
with
k
Q1 = a0 + ak (i
);
a0 = s;
k
Q2 = i b0 + bk (i
);
1 ij
" pij ;
2 k
bk = pk :
ak =
b0 = p;
(3.121)
3.9.2 Quaternion Algebra
Eqs.(3.121) show that the quaternion algebra R0;2 = H can be identi…ed
as the even subalgebra of R3;0 , i.e.,
(0)
R0;2 = H ' R3;0 :
(3.122)
The statement is obvious once we identify the basis f1;^{;^|; ^k g of H with
f1, i
1
,i
2
,i
3
g;
(3.123)
(0)
which are the generators of R3;0 . We observe moreover that the even subalgebra of the quaternions can be identi…ed (in an obvious way) with the
(0)
complex …eld, i.e., R0;2 ' C. Returning to Eq.(3.117) we see that any
P 2 R3;0 can also be written as
P = P1 + iL2 ;
(3.124)
where
P1 = (s + pk
iL2 = i(p + ilk
k
)2
k
^0
)2
R3;0
^2
R3;0
^1
R3;0
^3
R3;0 ;
R
^1
R3;0 ;
(3.125)
3.9 Dotted and Undotted Algebraic Spinors
91
with lk = "ik j pij 2 R. The important fact that we want to emphasize
V1 3;0
V2
V3 3;0
here is that the subspaces (R
R ) and ( R3;0
R ) do not close
V1 3;0
separately any algebra. In general, if A; C 2 (R
R ) then
^1
^2
AC 2 R
R3;0
R3;0 :
(3.126)
To continue, we introduce
i=
i
ei e0 =
;
i = 1; 2; 3:
(3.127)
^ of H can be identi…ed with
Then, i =
{ ; |^; kg
1 2 3 and the basis f1; ^
f1, i 1 , i 2 , i 3 g.
Now, we know that R3;0 ' C (2). This permit us to represent
the Pauli
p
numbers by 2 2 complex matrices, in the usual way (i =
1). We write
R3;0 3 P 7! P 2 C(2), with
1
7!
1
2
7!
2
=
3
7!
3
=
=
0
1
0
i
1
0
1
0
i
0
0
1
;
;
(3.128)
:
3.9.3 Minimal left and Right Ideals in the Pauli Algebra and
Spinors
(0)
The elements e = 21 (1 + 3 ) = 12 (1 + e3 e0 ) 2 R1;3 ' R3;0 , e2 = e
are minimal idempotents of R3;0 . They generate the minimal left and right
ideals
(0)
(0)
I = R1;3 e ; R = e R1;3 :
(3.129)
From now on we write e = e+ . It can be easily shown (see below) that,
e.g., I = I+ has the structure of a 2-dimensional vector space over the
complex …eld [134, 217], i.e., I ' C2 . The elements of the vector space I
are called representatives of algebraic contravariant undotted spinors18 and
the elements of C2 are the usual contravariant undotted spinors used in
1
physics textbooks. They carry the D( 2 ;0) representation of Sl(2; C) [242].
If ' 2 I we denote by ' 2 C2 the usual matrix representative19 of ' is
'=
'1
'2
;
'1 ; '2 2 C:
(3.130)
1 8 We omit in the following the term representative and call the elements of I simply by
algebraic contravariant undotted spinors. However, the reader must aways keep in mind
that any algebraic spinor is an equivalence class, as de…ned and discussed in Section 4.6.
1 9 The matrix representation of the elements of the ideals I; I,
_ are of course, 2 2
complex matrices (see, [134], for details). It happens that both columns of that matrices
have the same information and the representation by column matrices is enough here
for our purposes.
92
3. The Hidden Geometrical Nature of Spinors
(0)
Denoting by I_ = eR1;3 the space of the algebraic covariant dotted
spinors, we have the isomorphism, I_ ' (C2 )y ' C2 , where y denotes Hermitian conjugation. The elements of (C2 )y are the usual contravariant
1
spinor …elds used in physics textbooks. They carry the D(0; 2 ) represen_ then its matrix representation in (C2 )y is
tation of Sl(2; C) [242]. If 2 I,
a row matrix usually denoted by
_=
1_
;
2_
1_ ; 2_
2 C:
(3.131)
The following representation of 2 I_ in (C2 )y is extremely convenient.
We say that to a covariant undotted spinor there corresponds a covariant
dotted spinor _ given by
7! _ = " 2 (C2 )y ;
I_ 3
1; 2
2 C;
(3.132)
with
0
1
"=
1
0
:
(3.133)
_ Indeed, since I = R(0) e we have
We can easily …nd a basis for I and I.
1;3
that any '2 I can be written as
' = '1 #1 +'2 #2
where
#1 = e;
#2 =
1
1 e;
2
' = a + ib;
' = c + id;
a; b; c; d 2 R:
(3.134)
2 I_ can be written as
Analogously we …nd that any
1_ 1_
s +
=
_
2_
2_ s ;
_
s1 = e;
s2 = e
1:
(3.135)
De…ning the mapping
(0)
I_ !R1;3 ' R3;0 ;
:I
('
)=' ;
(3.136)
we have
1
0
_
(s1
_
s2 + s2
1
=
2
= [i(s1
s2
3
=
s1
(s1
_
s1 + s2
= (s1
_
_
s2 );
_
s1 );
_
s2
s1 )];
s2
s2 ):
_
(3.137)
3.9 Dotted and Undotted Algebraic Spinors
93
From this it follows the identi…cation
(0)
R3;0 ' R1;3 ' C(2) =I
_
I;
C
(3.138)
and then, each Pauli number can be written as an appropriate sum of
Cli¤ord products of algebraic contravariant undotted spinors and algebraic
covariant dotted spinors. And, of course, a representative of a Pauli number
in C2 can be written as an appropriate Kronecker product of a complex
column vector by a complex row vector.
Take an arbitrary P 2R3;0 such that
1 k1 k2
p
j!
P=
kj
k1 k2
kj ;
and
0
(3.139)
where pk1 k2 :::kj 2 R and
k1 k2 :::kj
=
kj ;
k1
(0)
With the identi…cation R3;0 ' R1;3 ' I
C
1 2 R:
_ we can also write
I,
_
_
P = PAB_ (sA
(3.140)
sB ) = PAB_ sA sB ;
(3.141)
where the PAB_ = XAB_ + iYAB_ , XAB_ ; YAB_ 2 R.
Finally, the matrix representative of the Pauli number P 2 R3;0 is P 2
C(2) given by
_
(3.142)
P = P AB_ sA sB ;
with P AB_ 2 C and
1
0
s1 =
_
s1 =
1
;
0
0
1
s2 =
;
_
s2 =
0
;
1
(3.143)
:
It is convenient for our purposes to introduce also covariant undotted spinors and contravariant dotted spinors. Let ' 2 C2 be given as in
Eq.(3.130). We de…ne the covariant version of undotted spinor ' 2 C2 as
' 2 (C2 )t ' C2 such that
'A sA ;
' = ('1 ; '2 )
'A = 'B "BA;
s1 =
1
0
'B = "BA 'A ;
;
s2 =
0
1
;
(3.144)
where20 "AB = "AB = adiag(1; 1). We can write due to the above identi…cations that there exists " 2 C(2) given by Eq.(3.133) which can be written
2 0 The
symbol adiag means the antidiagonal matrix.
94
3. The Hidden Geometrical Nature of Spinors
also as
" = "AB sA
where
sB = "AB sA
0
1
sB =
1
0
=i
(3.145)
2
denotes here the Kronecker product of matrices. We have, e.g.,
s1
s2 =
1
0
s1
s1 =
1
0
1
0
0
=
1
0
1
=
0
1
0
1
1
=
0
0
0
=
1
0
1
0
;
0
0
: (3.146)
We now introduce the contravariant version of the dotted spinor
_=
1_
as being _ 2 C2 such that
B_
2_
_ _
= "B A
s1_ =
!
1_
_ =
A_ ;
1
0
2 C2
2_
A_
=
A_
sA_ ;
B_
= "B_ A_
; s2_ =
0
1
;
;
(3.147)
_ _
where "A_ B_ = "AB = adiag(1; 1). Then, due to the above identi…cations
we see that there exists "_ 2 C(2) such that
_ _
"_ = "AB sA_
_
sB_ = "A_ B_ sA
0
1
s_ B =
1
0
= ":
(3.148)
_
Also, recall that even if fsA g,fsA_ g and fsA g,fsA g are bases of distinct
spaces, we can identify their matrix representations, as it is obvious from
_
the above formulas. So, we have sA
sA_ and also sA = sA . This is the
reason for the representation of a dotted covariant spinor as in Eq.(3.132).
Moreover, the above identi…cations permit us to write the matrix representation of a Pauli number P 2 R3;0 as, e.g.,
P = PAB sA
sB
(3.149)
besides the representation given by Eq.(3.142).
2
Exercise 168 Consider the ideal I = R1;3 21 (1 e0 e3 ). Show that
I is a representative in a spin frame u of a covariant Dirac spinor 21
2 1 Recall
that
are the Dirac matrices de…ned by Eq.(3.49).
3.9 Dotted and Undotted Algebraic Spinors
95
2 C(4) 21 (1 + 0 )(1 + i 1 2 ) . Let
be the representative (in the same
spin frame u ) of a Dirac-Hestenes spinor, associated to a mother spinor
2 R1;3 12 (1 + e0 ) by = 12 (1 + e0 ). Show that 2 I can be written as
= 12 (1 + e0 ) 12 (1 e0 e3 ).
(a) Show that e5 = e21 :
(b) Weyl spinors are de…ned as eigenspinors of the chirality operator,
= i
. Show that Weyl spinors corresponds to the even and
i.e., 5
odd parts of :
(c) Relate the even and odd parts of to the algebraic dotted and undotted
spinors.22
2 2 As
a suggestion for solving the above exercise the reader may consult [134].
96
3. The Hidden Geometrical Nature of Spinors
This is page 97
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4
Some Di¤erential Geometry
4.1 Di¤erentiable Manifolds
In this section we brie‡y recall, in order to …x our notations, some results
concerning the theory of di¤erentiable manifolds, that we shall need in the
following.
De…nition 169 A topological space is a pair (M; U) where M is a set and
U a collection of subsets of M such that
(i) ?; M 2 U .
(ii) U contains the union of each one of its subsystems.
(iii) U contains the intersection of each one of its …nite subsystems.
We recall some more terminology1 . Each U 2 U ( belongs to an index
set which eventually is in…nite) is called an open set. Of course we can give
many di¤erent topologies to a given set by choosing di¤erent collections
of open sets. Given two topologies for M , i.e., the collections of subsets
U1 and U2 if U1
U2 we say that U1 is coarse than U2 and U2 is …ner
than U1 . Given two coverings fU g and fV g of M we say that fV g is a
re…nement of fU g if for each V there exists an U such that V
U .
A neighborhood of a point x 2 M is any subset of M containing some
1 In general we are not going to present proofs of the propositions, except for a few
cases, which may considered as exercices. If you need furhter details, consult e.g., [77,
138, 265].
98
4. Some Di¤erential Geometry
(at least one) open set U 2 U. A subset X
M is called closed if its
complement is open in the topology (M; U). A family fU g; U 2 U is
called a covering of M if [ U = M . A topological space (M; U) is said
to be Hausdor¤ (or separable) if for any distinct points x; x0 2 M there
exists open neighborhoods U and U 0 of these points such that U \ U 0 = ?.
Moreover, a topological space (M; U) is said to be compact if for every open
covering fU g; U 2 U of M there exists a …nite subcovering, i.e., there
exists a …nite subset of indices, say = 1; 2; :::m, such that [m=1 U = M .
A Hausdor¤ space is said paracompact if there exists a covering fV g of M
such that every point of M is covered by a …nite number of the V , i.e., we
say that every covering has a locally …nite re…nement.
De…nition 170 A smooth di¤ erentiable manifold M is a set such that
(i) M is a Hausdor¤ topological space.
(ii) M is provided with a family of pairs (U ; ' ) called charts, where
fU g is a family of open sets covering M , i.e., [ U = M and being fV g
a family of open sets covering Rn , i.e., [ V = Rn , the ' : U ! V are
homeomorphisms. We say that any point x 2 M has a neighborhood which
is homeomorphic to Rn . The integer n is said the dimension of M , and we
write dim M = n.
(iii) Given any two charts (U; ') and (U 0 ; '0 ) of the family described in
(ii) such that U \U 0 6= ? the mapping = ' ' 1 : '(U \U 0 ) ! '(U \U 0 )
is di¤ erentiable of class C r .
The word smooth means that the integer r is large enough for all statements that we shall done to be valid. For the applications we have in mind
we will suppose that M is also paracompact. The whole family of charts
f(U ; ' )g is called an atlas.
The coordinate functions of a chart (U; ') are the functions xi = ai ' :
U ! R, i = 1; 2; :::; n where ai : Rn ! R are the usual coordinate functions
of Rn (see Figure 4.1). We write xi (x) = xi and call the set (x1 ; ::; xn )
(denoted fxi g) the coordinates of the points x 2 U in the chart (U; '), or
brie‡y, the coordinates2 . If (U 0 ; '0 ) is another chart of the maximal atlas of
M with coordinate functions x0i such that x 2 U \ U 0 we write x0i (x) = x0i
and
x0j (x) = f j (x1 (x); :::; xn (x));
(4.1)
and we use the short notation x0j = f j (xi ), i; j = 1; :::; n. Moreover, we
often denote the derivatives @f j =@xi by @x0j =@xi .
Let (U; ') be a chart of the maximal atlas of M and h : M ! M , x 7!
y = h(x) a di¤eomorphism such that x; y 2 U \ h(U ). Putting xi (x) = xi
2 We remark that some authors (see, e.g., [265] call sometimes the coordinate function
x i simply by coordinate. Also, some authors (see, e.g., [138]) call sometimes fxi g a
coordinate system (for U
M ). We eventually also use these terminologies.
4.1 Di¤erentiable Manifolds
99
FIGURE 4.1. Coordinate Chart (U; ), coordinate functions x : U ! R and
coordinates xi (x) = xi
and y j = xj (h(x)) we write the mappings hj : (x1 ; :::; xn ) 7! (y 1 ; :::; y n ) as
y j = hj (xi );
(4.2)
and often denote the derivatives @hj =@xi of the functions hj by @y j =@xi .
Observe that in the chart (V; {), V
h(U ) with coordinate functions
fyj g such that xj = yj h, xj (x) = xj = yj (y) = y j and @y j =@xi = ji .
4.1.1 Manifold with Boundary
In the de…nition of a n-dimensional (real) manifold we assumed that each
coordinate neighborhoods, U 2 M is homeomorphic to an open set of Rn .
We now give the
De…nition 171 A n-dimensional (real ) manifold M with boundary is a
topological space covered by a family of open sets fU g such that each one
is homeomorphic to an open set of Rn+ = f(x1 ; :::; xn ) 2 Rn j xn 0g.
De…nition 172 The boundary of M is the set @M of points of M that are
mapped to points in Rn with xn = 0.
Of course, the coordinates of @M are given by (x1 ; :::; xn 1 ; 0) and thus
@M is a (n 1)-dimensional manifold. of the same class (C r ) as M .
4.1.2 Tangent Vectors
Let C r (M; x) be the set of all di¤erentiable functions of class C r (smooth
functions) which domain in some neighborhood of x 2 M . Given a curve
in M , : R I ! M , t 7! (t) we can construct a linear function
(t) : C r (M; x) ! R;
(4.3)
100
4. Some Di¤erential Geometry
such that given any f 2 C r (M; x),
(t)[f ] =
Now,
rule:
d
[f
dt
](t):
(4.4)
(t) is a derivation, i.e., a linear function that satisfy the Leibniz’s
(t)[f g] =
(t)[f ]g + f
(t)[g];
(4.5)
r
for any f; g 2 C (M; x):
This linear mapping has all the properties that we would like to impose
to the tangent to at (t) as a generalization of the concept of directional
derivative of the calculus on Rn . It can shown that to every linear derivation
it is associated a curve (indeed, an in…nity of curves) as just described, i.e.,
curves ; : R I ! M are equivalent at x0 = (0) = (0) provided
d
d
](t) t=0 = dt
[f
](t) t=0 for any f 2 C r (M; x0 ). This suggests
dt [f
the
De…nition 173 A tangent to M at the point x 2 M is a mapping vjx :
C r (M; x) ! R such that for any f; g 2 C r (M; x), a; b 2 R,
(i)
(ii)
vjx [af + bg] = avx [f ] + b vjx [g];
:
vjx [f g] = vjx [f ]g + f vjx [g]:
(4.6)
As can be easily veri…ed the tangents at x form a linear space over the
real …eld. For that reason a tangent at x is also called a tangent vector to
M at x.
De…nition 174 The set of all tangent vectors at x is denoted by Tx M
and called the tangent space at x. The dual space of Tx M is denoted by
r
M is the space of
Tx M and called the cotangent space at x. Finally Tsx
r-contravariant and s-covariant tensors at x.
De…nition 175 Let fxi g be the coordinate functions of a chart (U; '). The
partial derivative at x with respect to xi is the representative in the given
@
chart of the tangent vector denoted @x
@i jx such that
i
x
@
@xi
f :=
x
=
@
[f
@xi
'
1
]
;
'(x)
@f i
(x );
@xi
(4.7)
with
f (x) = f
'
1
(x1 (x); :::; xn ) = f (x1 ; :::; xn ):
(4.8)
@
Remark 176 Eventually we should represent the tangent vector @x
by
i
x
@
a di¤ erent symbol, say @ x i x . This would cause less misunderstandings.
@
However, @x
is almost universal notation and we shall use it. We note
i
x
4.1 Di¤erentiable Manifolds
101
moreover that other notations and abuses of notations are widely used, in
particular f ' 1 is many times denoted simply by f and then f (xi ) is
@f
@
denoted simply by f (xi ) and also we …nd @x
[f ]
i
@xi (x), (or worse)
x
@f
@
@xi x [f ]
@xi . We shall use these (and other ) sloppy notations, which
are simply to typewrite when no confusion arises, in particular we will use
j
@xj
@
j
the sloppy notations @x
@xi (x) or @xi for @xi x [x ], i.e.
@
@xi
[xj ]
x
@xj
(x)
@xi
@xj
=
@xi
j
i:
If fxi g are the coordinate functions of a chart (U; ') and vjx 2 Tx M ,
then we can easily show that
vjx = vjx [xi ]
@
@xi
= vi
x
@
@xi
;
(4.9)
x
with vjx [xi ] = v i : U ! R.
As a trivial consequence we can verify that the set of tangent vectors
@
@xi x ; i = 1; 2; :::; n is linearly independent and so dim Tx M = n.
@
2 Tx U and its
Remark 177 Have always in mind that vjx = v i @x
i
x
@
n
representative in T '(x) R is the tangent vector vj'(x) =: v i @x
such
i
'
(x)
that v i
@
@xi x
f = vj'(x) f .
De…nition 178 The tangent vector …eld to a curve
denoted by (t) or ddt .
: R
I ! M is
This means that
(t) = ddt (t) is the tangent vector to the curve
the point (t). Note that (t) has the expansion
(t) = v i ( (t))
@
@xi
;
at
(4.10)
x= (t)
where, of course,
v i ( (t)) =
(t)[xi ] =
dxi
(t)
dt
=
d i (t)
;
dt
(4.11)
with i = xi
. We then see, that given any tangent vector vjx 2 Tx M ,
the solution of the di¤erential equation, Eq.(4.11) permit us to …nd the
components i (t) of the curve to which vjx is tangent at x. Indeed, the
theorem of existence of local solutions of ordinary di¤erential equations
warrants the existence of such a curve. More precisely, since the theorem
holds only locally, the uniqueness of the solution is warranted only in a
neighborhood of the point x = (t) and in that way, we have in general
many curves through x to which vjx is tangent to the curve at x.
102
4. Some Di¤erential Geometry
4.1.3 Tensor Bundles
S
In what follows we denote respectively by T M = x2M Tx M and T M =
S
and cotangent bundles3 of M and more generally,
x2M Tx M the tangentS
r
we denote by Tsr M = x2M Tsx
M the bundle of r-contracovariant and
s-covariant tensors. A tensor …eld t of type (r; s) is a section of the Tsr M
bundle and we write4 t 2 sec Tsr M . Also, T00 M M R is the module of
real functions over M and T10 M T M , Tsr M = T M .
4.1.4 Vector Fields and Integral Curves
Let : I ! M a curve and v 2 sec T M a vector …eld which is tangent
to each one of the points of . Then, taking into account Eq.(4.11) we can
write that condition as
d (t)
:
(4.12)
v( (t)) =
dt
De…nition 179 A curve : I ! M satisfying Eq.(4.12) is called an integral curve of the vector …eld v:
4.1.5
Derivative and Pullback Mappings
Let M and N be two di¤erentiable manifolds, dim M = m, dim N = n and
: M ! N a di¤erentiable mapping of class C r .
is a di¤eomorphism of
1
class C r if is a bijection and if and
are of class C r .
De…nition 180 The reciprocal image or pullback of a function f : N ! R
is the function f : M ! R given by
f =f
:
(4.13)
De…nition 181 Given a mapping : M ! N , (x) = y and v 2 Tx M ,
the image of v under is the vector w such that for any f : N ! R
w[f ] = v[f
]:
(4.14)
The mapping
jx : sec Tx M ! sec Ty N is called the di¤ erential or derivative (or pushforward) mapping of at x. We write w =
jx v.
Remark 182 When the point x 2 M is left unspeci…ed (or is arbitrary),
we sometimes write
instead of
jx .
The image a vector …eld v 2 sec T M at an arbitrary point x 2 M is
v[f ](y) = v[f
](x):
(4.15)
3 In Appendix we list the main concepts concerning …ber bundle theory that we need
for the purposes of this book.
4 See details in Notation 631 in the Appendix.
4.1 Di¤erentiable Manifolds
Note that if (x) = y, and if
says that or
1
is invertible, i.e., x =
v[f ](y) = v[f
](x) = v[f
1
](
103
(y) then Eq.(4.15)
(y)):
(4.16)
This suggests the
De…nition 183 Let : M ! N be invertible mapping. Let v 2 sec T M .
The image of v under is the vector …eld v 2 sec T N such that for any
f :N !R
1
v[f ] = v[f
] = v[f
]
:
(4.17)
In this case we call
: sec T M ! sec T N;
(4.18)
the derivative mapping of .
Remark 184 If v 2 sec T M is a di¤ erentiable …eld of class C r over M
and is a di¤ eomorphism of class C r+1 , then
v 2 sec T N is a di¤ erentiable vector …eld of class C r over N . Observe however, that if is not
invertible the image of v under
is not in general a vector …eld on N
[77]. If is invertible, but not di¤ erentiable the image is not di¤ erentiable.
When the image of a vector …eld v under some di¤ erentiable mapping is
a di¤ erentiable vector …eld, v is said to be projectable. Also, v and v are
said -related.
Remark 185 We have denoted by (t) the tangent vector to a curve :
I ! M . If we look for the de…nition of that tangent vector and the de…nition
of the derivative mapping we see that the rigorous notation that should be
d
used for that tangent vector is
jt [ dt
], which is really cumbersome, and
thus avoided, unless some confusion arises. We will also use sometimes the
simpli…ed notation
to refer to the tangent vector …eld to the curve .
De…nition 186 Given a mapping
the mapping
?
: M ! N , the pullback mapping is
: sec T N ! sec T M;
!(v) = !(
v)
;
for any projectable vector …eld v 2 sec T M . Also,
the pullback of ! (Figure 2 ).
?
(4.19)
! 2 sec T N is called
Remark 187 Note that di¤ erently from what happens for the image of
vector …elds, the formula for the reciprocal image of a covector …eld does
1
not use the inverse mapping
. This shows that covector …elds are more
interesting than vector …elds, since ? ! is always di¤ erentiable if ! and
are di¤ erentiable.
104
4. Some Di¤erential Geometry
4.jpg
FIGURE 4.2. (a) The derivative ampping
(b) The pullback mapping
4.1 Di¤erentiable Manifolds
105
Remark 188 From now, we assume that : M ! N is a di¤ eomorphism,
unless explicitly said the contrary and generalize the concepts of image and
reciprocal images de…ned for vector and covector …elds for arbitrary tensor
…elds.
De…nition 189 The image of a function f : M ! R under a di¤ eomorphism : M ! N is the function ? f : N ! R such that
?f
1
=f
(4.20)
The image of a covector …eld
2 sec T M under a di¤ eomorphism
:
M ! N is the covector …eld
such that for any projectable vector …eld
v 2 sec T M; w = v 2 sec T N , we have
(w) = ( 1 w), or
1
=(
)
1 ; :::
r;
v 1 ; :::;
(4.21)
S 2 sec Tsr N by
For S 2 sec Tsr M we de…ne its image
S(
:
v s ) = S(
1 ; ::: r ; v 1 ; :::; v s );
(4.22)
for any projectable vector …elds v i 2 sec T M , i = 1; 2; :::; s and covector
…elds j 2 sec T M , j = 1; 2; :::; r.
If fei g is any basis for T U , U
then
:::ir j1
S = Sji11 :::j
s
M and f i g is the dual basis for T U ,
js
ei1
(4.23)
e ir
and
:::ir
S = (Sji11 :::j
s
1
j1
)
js
e i1
eir :
(4.24)
De…nition 190 Let S 2 sec Tsr N , and 1 ; 2 ; :::; r 2 sec T M and v 1 ; :::; v s 2
sec T M be projectable vector …elds:The reciprocal image (or pullback ) of S
is the tensor …eld S 2 sec Tsr M such that
S(
1 ; :::; r ;v 1 ; :::;v s )
= S(
1 ; :::;
r;
v 1 ; :::;
v s );
(4.25)
and
:::ir
S = (Sji11 :::j
s
)
j1
js
1
e i1
1
eir :
(4.26)
Let xi be the coordinate functions of the chart (U; ') of U
M and
f@=@xj g; fdxi g; i; j = 1; :::; m dual5 coordinate bases for T U and T U , i.e.,
dxi (@=@xj ) = ji . Let moreover yl be the coordinate functions of (V; ), V
N and f@=@y k g,fdy l g, k; l = 1; :::; n dual bases for T V and T V . Let x 2
:::kr 1
M; y 2 N with y = (x) and xi (x) = xi ; yl (y) = y l . If Slk11:::l
(y ; :::; y n )
s
5 See
Remark 209 for the reason of the notation dxi .
106
4. Some Di¤erential Geometry
:::kr j
Slk11:::l
(y ) are the components of S at the point y in the chart (V; ), then
s
the components S0 = S in the chart (U; ') at the point x are
:::kr j
:::ir
(xi ) = Slk11:::l
(y (xi ))
Sj0i11:::j
s
s
@y l1 @y ls @xi1
:::
@xj1 @xjs @y k1
@xir
;
@y kr
:::ir
i
1 :::ir
(xi ) = (h? S)0i
Sj0i11:::j
j1 :::js (x ):
s
(4.27)
4.1.6 Di¤eomorphisms, Pushforward and Pullback when
M =N
De…nition 191 The set of all di¤ eomorphisms in a di¤ erentiable manifold
M de…ne a group denoted by GM and called the manifold mapping group.
Let A; B M . Let GM 3 h : M ! M be a di¤eomorphism such that h :
A ! B; e 7! he. The di¤eomorphism
h induces two important mappings in
M
r
the tensor bundle T M =
Ts M , the derivative mapping h , in this
r;s=0
case known as pushforward, and the pullback mappings h . The de…nitions
of these mappings are the ones given above.
We now recall how to calculate, e.g., the pullback mapping of a tensor
…eld in this case.
Suppose now that A and h(A)
B can be covered by a local charts
(U; ') and (V; ) of the maximal atlas of M (with A; h(A) U \ V ) with
respective coordinate functions fx g and fy g de…ned by6
x (e) = x ; x (h(e)) = y ; y (e) = y .
(4.28)
We then have the following coordinate transformation
y = x (h(e)) = h (x ):
(4.29)
Let f@=@x g and f@=@y g be a coordinate bases for T (U \V ) and fdx g and
fdy g the corresponding dual basis for T (U \ V ).
sec T M in the coorThen, if the local representation of S 2 sec Tsr M
dinate chart fy g at any point of U \ V is S 2 sec Tsr Rn ,
S=S
1 ::: r
1 :::: s
(y j )dy
1
:::
dy
s
@
@y
:::
1
@
;
@y r
(4.30)
we have that the representative of S0 = h S in Tsr Rn at any point e 2 U \ V
is given by
::: r
h S = S 0 11::::
(xj )dx
s
::: r
S 0 11::::
(xj ) = S
s
6 Note
1 :::
1 ::::
1
:::
dx
@y 1 @y
r
:::
(y i (xj ))
s
@x 1 @x
that in general y (h(e)) 6= y .
s
s
s
@x
@y
@
@x
1
@x
:::
1
@y
1
:::
@
@x
r
r
r
:
(4.31)
4.1 Di¤erentiable Manifolds
107
Remark 192 Another important expression for the pullback mapping can
be found if we choice charts with the coordinate functions fx g and fy g
de…ned by
x (e) = y (h(e))
(4.32)
Then writing
x (e) = x ; y (h(e)) = y ,
(4.33)
we have the following coordinate transformation
y = h (x ) = x ;
(4.34)
from where it follows that in this case
::: r
S 0 11::::
(xj ) = S
s
1 ::: r
1 :::: s
(y i (xj )):
(4.35)
4.1.7 Lie Derivatives
De…nition 193 Let M be a di¤ erentiable manifold. We say that a mapping : M R !M is a one parameter group if
(i) is di¤ erentiable,
(ii) (x; 0) = x, 8x 2 M ,
(iii) ( (x; s); t) = (x; s + t), 8x 2 M , 8s; t 2 R.
These conditions may be expressed in a more convenient way introducing
the mappings t : M ! M such that
t (x)
= (x; t):
(4.36)
For each t 2 R, the mapping t is di¤erentiable, since t =
lt , where
lt : M ! M R is the di¤erentiable mapping given by lt (x) = (x; t).
Also, condition (ii) says that 0 = idM . Finally, condition (iii) implies,
as can be easily veri…ed that
t
s
=
s+t :
(4.37)
Observe also that if we take s = t in Eq.(4.37) we get t
t = idM .
It follows that for each t 2 R, the mapping t is a di¤eomorphism and
( t) 1 = t:
De…nition 194 We say that a family ( t ; t 2 R) of mappings
is a one-parameter group of di¤ eomorphisms G1 of M .
De…nition 195 Given a one-parameter group
x 2 M , we may construct the mapping
x
x
: R !M;
(t) = (x; t);
: M
t
:M !M
R !M for each
(4.38)
which in view of condition (ii) is a curve in M , called the orbit (or trajectory) of x generate by the group. Also, the set of all orbits for all points of
M are the trajectories of G1 .
108
4. Some Di¤erential Geometry
It is possible to show, using condition (iii) that for each point x 2 M pass
one and only one trajectory of the one-parameter group. As a consequence
it is uniquely determined by a vector …eld v 2 sec T M which is constructed
by associating to each point x 2 M the tangent vector to the orbit of the
group in that point, i.e.,
v(
x
(t)) =
d
dt
x
(t) :
(4.39)
De…nition 196 The vector …eld v 2 sec T M determined by Eq.(4.39) is
called a Killing vector …eld relative to the one parameter group of di¤ eomorphisms ( t ; t 2 R):
Remark 197 It is important to have in mind that in general, given a
vector …eld v 2 sec T M it does not de…ne a group (even locally) of diffeomorphisms in M . In truth, it will be only possible, in general, to …nd
a local one-parameter pseudo-group that induces v. A local one parameter
pseudo-group means that t is not de…ned for all t 2 R, but for any x 2 M ,
there exists a neighborhood U (x) of x, an interval I(x) = ( "(x); "(x))
R and a family ( t ; t 2 I(x)) of mappings t : M ! M , such that the
properties (i-iii) in De…nition 195 are valid, when jtj < "(x), jsj < "(x) and
jt + sj < "(x).
De…nition 198 Taking into account the previous remark, the vector …eld
v 2 sec T M is called the in…nitesimal generator of the one parameter local
pseudo-group ( t ; t 2 I(x)) and the mapping : M I(x)!M is called the
‡ow of the vector …eld .
Of course, given v 2 sec T M we obtain the one parameter local pseudogroup that induces v by integration of the di¤erential equation Eq.(4.39).
From that, we see that the trajectories of the group are also the integral
lines of the vector …eld v.
De…nition 199 Let ( t ; t 2 I(x)) a one-parameter local pseudo group of
di¤ eomorphisms of M that induces the vector …eld v and let S 2 sec Tsr M .
The Lie derivative of S in the direction of v is the mapping
$v : sec Tsr M ! sec Tsr M;
S S
:
$v S = lim t
t!0
t
(4.40)
Remark 200 It is possible to de…ne the Lie derivative using the pushforward mapping, the results that follows are the same. In this case we have
$v S = limt!0 S t t S :
4.1 Di¤erentiable Manifolds
109
FIGURE 4.3. The Lie Derivative
4.1.8 Properties of $v
(i) $v is a linear mapping and preserve contractions.
0
(ii) Leibniz’s rule. If S 2 sec Tsr M , S 0 2 sec Tsr0 M , we have
$v (S
S 0 ) = $v S
S0 + S
$v S 0 :
(4.41)
(iii) If f : M ! R, we have
$v f = v(f ):
(4.42)
(iv) If v; w 2 sec T M , we have
$v w = [v; v];
(4.43)
where [v; w] is the commutator of the vector …elds v and w, such that
[v; w](f ) = v(w(f ))
w(v(f )):
(4.44)
(v) If ! 2 sec T M , we have
$v ! = v(!k ) + !i ek (v i )
cijk !i v j
k
;
(4.45)
where v j and !i are the components of in the dual basis fej g and f i g and
the cijk are called the structure coe¢ cients of the frame fej g , and
[ej ; ek ] = cijk ei :
(4.46)
Exercise 201 Show that if v 1 ; v 2 ; v 3 2 sec T M , then they satisfy Jacobi’s
identity, i.e.,
[v 1 ; [v 2 ; v 3 ]] + [v 2 ; [v 3 ; v 1 ]] + [v 3 ; [v 1 ; v 2 ]] = 0:
(4.47)
Exercise 202 Show that for u; v 2 sec T M
$[u;v] = [$u ; $v ]:
(4.48)
110
4. Some Di¤erential Geometry
4.1.9 Invariance of a Tensor Field
The concept of Lie derivative is intimately associated to the notion of invariance of a tensor …eld S 2 sec Tsr M .
De…nition 203 We say that S is invariant under a di¤ eomorphism h :
M ! M , or h is a symmetry of S, if and only if
h Sjx = Sjx :
(4.49)
We extend naturally this de…nition for the case in which we have a local
one-parameter pseudo-group t of di¤ eomorphisms. Observe, that in this
case, it follows from the de…nition of Lie derivative, that if S is invariant
under t , then
$v S = 0
(4.50)
More properties of Lie derivatives of di¤erential forms that we shall need
in future chapters, will be given at the appropriate places.
Remark 204 A correct concept for the Lie derivative of spinor …elds is as
yet a research subject and will not be discussed in this book. A Cli¤ ord bundle approach to the subject which we think worth to be known is presented
in [207]
4.2 Cartan Bundle, de Rham Periods and Stokes
Theorem
In this section, we brie‡y discuss the processes of di¤erentiation in the
Cartan bundle and the concept of de Rham periods and Stokes theorem.
4.2.1 Cartan Bundle
De…nition 205 The Cartan bundle over the cotangent bundle of M is the
set
n ^
^
[ ^
[ M
r
T M=
Tx M =
Tx M;
(4.51)
V
x2M r=0
x2M
where Tx M , V
x 2 M , is V
the exterior algebra of the vector space Tx M .
r
The sub-bundle
T M
T M given by:
^r
T M=
[ ^r
x2M
is called the r-forms bundle (r = 0; : : : ; n).
Tx M
(4.52)
4.2 Cartan Bundle, de Rham Periods and Stokes Theorem
111
De…nition 206 The exterior derivative is a mapping
^
^
d : sec T M ! sec T M;
satisfying:
(i)
(ii)
(iii)
(iv)
V
for every A; B 2 sec T
d(A + B) = dA + dB;
d(A ^ B) = dA ^ B + A^ ^ dB;
(4.53)
df (v) = v(f );
2
d = 0;
V0
M , f 2 sec
T M and v 2 sec T M .
Vp
Exercise 207 Show that for A 2 sec
T M and v 0 ; v 1 ; :::; v p 2 sec T M;
dA(v 0 ; v 1 ; :::; v p ) =
p
X
( 1)i v i (A(v 0 ; v 1 ; :::; v i ; :::; v p ))
i=1
+
X
( 1)i+j A([v i ; v j ]v 0 ; v 1 ; :::; v i ; :::; v j ; :::v p ):
0 i<j p
(4.54)
Remark 208 Note that due to property (ii) the exterior derivative does
not satisfy the Leibniz’s rule, and as such it is not a derivation. In fact the
technical term is antiderivation (see [77]).
Remark 209 Let xi be coordinate functions of a chart (U; ') of an atlas
of M . A coordinate basis for T U in that chart is denoted f@=@xi g. This
means that for each x 2 U , @=@xi x is a basis of Tx U . As we already
know, the dual (coordinate) basis for T U is denoted7 fdxj g. This means
that dxj x is a basis for Tx U . We have (indeed ) that
dxj (@=@xi )
x
= @xj =@xi
x
=
j
i:
(4.55)
4.2.2 The Interior Product of Forms and Vector Fields
Another important antiderivation is the so called interior product (sometimes also called inner product).
De…nition 210 Given a vector …eld
Vp v 2 secT M we de…ne the interior
product extensor of v with 2 sec
T M as the mapping
sec T M
sec
^p
T M ! sec
^p
1
T M;
(v; ) 7! iv ;
7 Eventually
a more rigorously notation for a basis of T U should be fdx i g:
(4.56)
112
4. Some Di¤erential Geometry
where iv : sec
(i) For any
Vp
Vp 1
T M!
T M satisfy
Vsec
, 2 sec T M and a; b 2 R;
iv (a + b ) = aiv + biv :
V0
(4.57)
(ii) if f 2 sec
T M is a smooth function, then iv f = 0,
(iii) If fei g is an arbitrary basis for T U , U M , and f i g its dual basis,
iek
j1
jp
^ :::: ^
jk
where as usual
=
p
X
( 1)r+1
jr j1
k
r=1
means that the term
^ ::::
jk
^ ::: ^
jp
;
(4.58)
jk
is missing in the expression.
Vp
Vq
From Eq.(4.58) it follows that for Ap 2 sec T M and Bq 2 sec T M
we have
iv (Ap ^ Bq ) = iv Ap ^ Bq + ( 1)pq Ap ^ iv Bq
(4.59)
and we usually say that iv is an antiderivation.
Exercise 211 If fxi g are coordinate functions of a local chart of M , and
@
i
i
v = v i @x
i ; show that iv dx = v .
Exercise 212 Properties of iv . Show that
i2v = 0;
(4.60)
div + iv d = $v ;
(4.61)
[$v ; iw ] = $v iw
iw $v = i[v;w];
(4.62)
$v d = d$v :
(4.63)
Eq.(4.61) is sometimes called Cartan’s magical formula. It is really, a
very important formula in the formulation of conservation laws, as we shall
see in Chapter 9.
4.2.3 Extensor Fields
Let f i g be an arbitrary basis for sec T U , U
M . Let
= i i 2
V1
Vr
1
i1
ir
sec
^
^
2 sec
T M , r = 1; 2; :::; n.
T M and ! = r! !i1 :::ir
V1
V1
De…nition 213 A (1; 1)-extensor …eld t : sec
T M ! sec
T M and
V1
V1
its extension t : sec
T M ! sec
T M are the linear operators given
by
t( ) = t(
i
i
)=
1
t(!) = t( !i1 :::ir
r!
for all
i t(
i1
i
^
);
^
ir
)=
1
!i :::i t(
r! 1 r
and !, r = 1; 2; :::n. Moreover, if f 2 sec
i1
)^
V0
^ t(
ir
)
(4.64)
T M , we put t(f ) = f .
4.2 Cartan Bundle, de Rham Periods and Stokes Theorem
113
4.2.4 Exact and Closed Forms and Cohomology Groups
Vr
De…nition 214 A r-form Gr 2 sec
T MVis called closed (or a cocycle)
r
if and only if dGr = 0. A r-form Fr 2 sec
T M is called exact (or a
Vr 1
coboundary) if and only if Fr = dAr 1 , with Ar 1 2 sec
T M.
De…nition 215 The space of closed r-forms is called the r-cocycle group
and denoted by Z r (M ). The space of exact r-forms is called the r-coboundary
group and denoted by B r (M ).
We recall that the sets Z r (M ) and B r (M ) have the structures of vector
spaces over the real …eld R. Since according to Eq.(4.53 iv) d2 = 0 it
follows that B r (M )
Z r (M ). Then if Fr = dAr 1 ) dFr = 0, but in
Vr 1
general dGr = 0 ; Gr = dCr 1 , with Cr 1 2 sec
T M.
De…nition 216 The space H r (M ) = Z r (M )=B r (M ) is the r-de Rham
cohomology group of the manifold M . Obviously, the elements of H r (M )
are equivalent classes of closed forms, i.e., if Fr ; Fr0 2 sec H r (M ), then
Vr 1
Fr Fr0 = dWr 1 , Wr 1 2 sec
T M.
As a vector space over the real …eld, H r (M ) is called the r-de Rham
vector space group of the manifold M .
De…nition 217 The dimension of the r-homology 8 (respectively cohomology) group is called the Betti number br (respectively br ) of M .
A very important result is the
Proposition 218 (Poincaré Lemma).
If U
Vr
then any closed r-form Fr 2 sec
T U (r
U is also exact.
M is di¤ eomorphic to Rn
1) which is di¤ erentiable on
Proof. For a proof see , e.g., [265].
Note that if U
M is di¤eomorphic to Rn then U is contractible to a
point p 2 M . Also, from Poincaré’s lemma it follows that the Betti numbers
of U , br = 0; r = 1; 2; :::; r.
Any closed form is exact at least locally and the non triviality of de
Rham cohomology group is an obstruction to the global exactness of closed
forms.
Remark 219 It is V
very important to observe that Poincaré’s lemma does
r
not hold if Fr 2 sec
T M is not di¤ erentiable at certain points of Rn ,
since in that case the manifold where Fr is di¤ erentiable is not homeomorphic to Rn . The ‘classical’ example according to Spivack [402] is A 2
V1
sec
T R2 ;
ydx + xdy
y
A=
= d(arctan ):
(4.65)
2
2
x +y
x
8 See
De…nition 233
114
4. Some Di¤erential Geometry
Observe that A is di¤erentiable on R2 f0g, but despite the third member
of Eq.(4.65) A is not exact on R2 , because arctan xy is not a di¤erentiable
function on R2 :
4.3 Integration of Forms
In what follows we brie‡y recall some concepts related to the integration of
forms on orientable manifolds. First we introduce the de…nition of the integral of a n-form in an n-dimensional manifold M and next the integration
of a r-form Ar 2 sec T M which is realized over a r-chain.
4.3.1 Orientation
Let M be an n-dimensional connected manifold and U ; U
M , U \U 6=
?. Let (U ; ' ), (U ; ' ) be coordinate charts of the maximal atlas of M
with coordinate functions fxi g and fxj g, i; j = 1; 2:::; n. Let e 2 U \ U .
The natural ordered bases f @x@i
e
the same orientation if J = det
g and f @x@ i
i
@x
@xi
e
g of Te M are said to have
> 0. If J < 0 the bases are said
e
to have opposite orientations. An orientation at e 2 U \ U is a choice of
an ordered basis (not necessarily a coordinate one) for Te M .
Now, suppose that the basis f @x@i g is declared positive (a right-handed
e
basis). A orientation in Te M induces naturally an orientation in Te M as
follows. Let f i e g be an ordered basis of Te M . Let e = 1 e ^
^ n je .
Then,
e(
2
6
6
1
6
=
det 6
6
n!
4
If
@
e ( @x1
e
e
@
@x1
e
@
@x1
1
2
:::
n
je
; :::;
@
@x1
@
@xn
e
@
@x1
e
e
e
@
@x2
2
e
:::
n
e
e
@
@x2
1
je
@
)
@xn e
; :::;
@
@x2
e
e
e
:::
1
:::
2
e
@
@xn
e
@
@xn
:::
:::
:::
n
je
@
@xn
3
7
7
7
7 . (4.66)
7
5
e
e
e
) > 0 we say that the ordered basis f
i
e
g of Te M
is positive. If e ( @x@1 ; :::; @x@n ) < 0 we say that the ordered basis f i e g
e
e
of Te M is negative.
h ii
@x
Suppose that for all e 2 U \ U we have J = det @xi > 0. In this
case we de…ne that on U \ U that the bases f @x@i g and f @x@ i g of T U
4.3 Integration of Forms
and T U
have the same orientation. If J = det
bases have opposite orientation on U \ U .
h
@xi
@xi
i
115
< 0 we say that the
De…nition 220 Let fU g be a covering for M , an n-dimensional connected manifold. We say that M is orientable if for any two overlapping
charts U and U there exist coordinate functions fxi g, fxj g for U and
h ii
@x
U such that det @xi > 0.
Remark 221 From what has been said V
above it is clear that if M is orin
entable, there exists an n-form 2 sec
T M called a volume element
which is never null.
Thus, we have the alternative (equivalent) de…nition of an orientable
manifold.
De…nition 222 A connected n-dimensional
if
Vn manifold M 0 is orientable
Vn
there exists a non null global section of
T M and ; 2 sec
T M
de…ne the same orientation (respectively opposite orientation) if there exV0
ists a global function 2 sec T M such that > 0 (respectively < 0)
such that 0 = :
Remark 223 Of course, a given orientable manifold M admits two inequivalent orientations, one is declared right-handed, and the other lefthanded. It is quite obvious that there are manifolds which are not orientable,
the classical example is the Möbius strip, which may be found in almost all
books in di¤ erential geometry, as, e.g.,[77, 265].
4.3.2 Integration of a n-form
In what follows we suppose that M is orientable9 . Let (U; ') be a chart of
the maximal atlas of M and fxi g the coordinate functions of the chart. Let
V0
h 2 sec VT M be a Lebesgue integrable function and10 = dx1 ^
^
n
dxn 2 sec
T M.
Vn
De…nition 224 The integral of h 2 sec
T M in A U M is
Z
Z
h :=
h ' 1 (xi )dx1
dxn
(4.67)
A
'(A)
where in the second member of Eq.(4.67) is the ordinary multiple integral
of a Lebesgue integrable function h = h ' 1 (xi ) of n variables.
9 In Chapter 6 we will learn that a spacetime manifold admitting spinor …elds must
necessarily be orientable.
1 0 Of course, we should write
= ' (dx1 ^ ^dxn ) since dxi are 1-forms in T' (U ) Rn .
So, ours is a sloppy (universally used) notation.
116
4. Some Di¤erential Geometry
Let be A
U \ V and (V; ) another chart of the maximal
h atlas
i of M
@xi
0j
with coordinate functions fx g and suppose that J = det @x0j > 0 on
U \ V: Then we can write that
h =h
and
1
(x0i )Jdx01 ^
Z
^dx0n = h
Z
h =
A
h
1
1
(x0i ) jJj dx01 ^
(x0i )) jJj dx01
^dx0n (4.68)
dx0n ;
(4.69)
(A)
which corresponds to the classical formula for a change of variables in a
multiple integral.
Now, if M is paracompact, i.e., there is an open covering fU g of M such
that each e 2 M is covered by a …nite number of the U a partition of the
unity associated to the covering fU g is a family of di¤erentiable functions
p : M ! R such that: (a) 0 p
1; (b) p (e) = 0 for all e 2
= U ; (c) If
k is the …nite number of U covering e then for any e 2 M we have that
Xk
p (e) = 1. It is obvious that we can write
=1
h(e) =
We then have the
Xk
=1
p (e)h(e) =
De…nition 225 The integral of h 2 sec
Z
X Z
X Z
h :=
h =
h
M
U
Xk
Vn
=1
h (e):
(4.70)
T M in M is
'
1
(xi )dx1
dxn
(4.71)
' (U )
We may verify that the de…nition is independent of the choice of atlas
used for M (and thus of the partition of the unity used) if the new atlas
has the same orientation as the previous one.
4.3.3 Chains and Homology Groups
Orientation of Subspaces
Let (u1 ; :::; un ) be a right handed coordinate system for Rn . For any Rr
Rn (u1 ; :::; ur ) is a naturally right handed coordinate system for Rr , which
is supposed to be coherently oriented with Rn .
De…nition 226 A r-rectangle P r in Rr
Rn is a naturally positive orir
i
i
i
u
b , i = 1; :::; r. The boundary of
ented subset of R such that a
the rectangle P r is the set @P r of 2r rectangles P r 1 2 Rr 1 de…ned by the
faces ui = ai and ui = bi of P r . We suppose that the boundary @P r is
coherently oriented with P r . That means that any face has the orientation
(u1 ; :::ui ; :::ur ) if ui = ai ; i is even and ui = bi ; i is odd and the opposite
orientation if ui = ai ; i is odd and ui = bi ; i is even.
4.3 Integration of Forms
117
Next we introduce the concept of elementary chain in M .
De…nition 227 An elementary r-chain or cr in a n-dimensional connected
manifold M is a pair (P r ; f ), with f : Rr
U ! M a di¤ erentiable
mapping. The image of the P r rectangle is denoted by suppcr . When f is
a di¤ eomorphism suppcr is called an elementary r-domain of integration.
De…nition 228 The boundary of an elementary r-chain is the image of
@P r .
De…nition 229 A r-chain on M is a formal linear
P combination of elementary r-chains crj with real coe¢ cients Cr =
j aj crj . The space of
r-chains in M forms a vector space over the real …eld. It is denoted by
Cr (M ) and called the r-chain group.
Remark 230 We are in general interested in formal locally …nite linear
combinations with aj = 1, in which case Cr is said a domain of integration
on M: More generally, in algebraic topology the coe¢ cients aj are in many
applications elements of a …nite group. In that case Cr (M ) is a group, but
it is not a vector space. That is the reason why Cr (M ) has been called the
r-chain group.
De…nition 231 The boundary operator @ is a mapping
@ : Cr (M ) ! Cr 1 (M )
P
such that for any r-chain Cr = j aj crj
X
@Cr =
aj @crj ;
j
(4.72)
(4.73)
where @crj is the image under f of an elementary Pjr -rectangle.
The boundary operator @ has the fundamental property
@ 2 = 0;
(4.74)
a formula that will be proved below.
De…nition 232 A …nite r-chain Cr is said to be a cycle if and only if
@Cr = 0. The space of cycles is denoted Zr (M ). Also, a …nite r-chain
Cr is said to be a boundary if and only if Cr = @Cr 1 and the space of
boundaries is denoted by Br (M ).
Since @ 2 = 0 it follows that Br (M )
Zr (M ). We then have
De…nition 233 The quotient set Hr (M ) = Zr (M )=Br (M ) is called the
r-homology group of M.
Remark 234 Recall that the dimension of the r-homology group is called
the Betti number br of M .
118
4. Some Di¤erential Geometry
In what follows we use the standard convention that Z 0 (M ) is the space
of di¤erentiable functions h such that dh = 0. Also, we agree that B 0 (M ) =
?. Finally, we agree that Z0 (M ) = C0 (M ) and that B0 (M ) = ?.
4.3.4 Integration of a r-form
Vr
De…nition 235 The integration of Fr 2 sec
T M over suppCr is
Z
Z
Z
X
X
(4.75)
Fr =
aj
Fr =
aj f Fr ;
j
j
Pjr
crj
Cr
where f is the pullback mapping induced by f .
When Fr is continuous and Cr is …nite the integral is always de…ned. The
integral is also always de…ned if Fr has compact support and Cr is locally
…nite. In what follows we suppose that this is the case. The de…nition 235
shows very clearly that it is bilinear in Fr and Cr and suggests
the de…nition
Vr
of a non degenerated inner product hi : Cr (M ) sec
M ! R given
Z
hCr ; Fr i = Fr :
(4.76)
Cr
With the aid of that de…nition we can say that two chains Cr and Cr0
are equal if and only if hCr ; Fr i = hCr0 ; Fr i. This observation is important
because the decomposition of a chain into elementary chains is not unique.
Recall that given a manifold, say M with boundary, its boundary is
denoted by @M . The manifold M is called triangulable if it can be decomposed as a union of adjacent n-domains of integration with orientation
compatible with the orientation of M:
4.3.5 Stokes Theorem
Vr
Theorem 236 (Stokes). For any Fr 2 sec
T M and Cr 2 Cr (M ) it
holds
Z
Z
dFr =
Fr :
(4.77)
Cr
@Cr
Proof. For a proof, see, e.g., [265].
Stokes formula can be written in the suggestive way
hCr ; dFr i = h@Cr ; Fr i
(4.78a)
Proposition 237 The boundary operator @ has the fundamental property
@ 2 = 0:
(4.79)
4.3 Integration of Forms
119
Proof. It follows directly from the fact that d2 = 0 and Stokes theorem.
Indeed,
h@ 2 Cr ; Fr i = h@Cr ; dFr i = hCr ; d2 Fr i = 0;
which proves the proposition.
4.3.6 Integration of Closed Forms and de Rham Periods
Vr
We now investigate integration in the case when Gr 2 sec
T M is closed.
The inner product introduced by Eq.(4.76) permit us to de…ne a mapping
from the space of closed (cocycles) forms Z r (M ) into the (dual) space of
cycles Zr (M ), by
I : Z r (M ) ! Zr (M );
(4.80)
Vr
T M and zr 2 Zr (M );
such that for any Gr 2 sec
I(Gr )(zr ) = hzr ; Gr i:
(4.81)
Note now that
hzr + @c; Gr i = hzr ; Gr i + h@c; Gr i = hzr ; Gr i + hc; dGr i = hzr ; Gr i; (4.82)
because Gr is closed. This implies that I(Gr ) can be considered as a linear
function on the equivalent class of zr modulus Br (M ), i.e., it de…nes a
mapping
I : Z r (M ) ! Hr (M ):
(4.83)
Also, I(Gr +dGr
transformation
1)
I(Gr ), so it is obvious that I really de…nes a linear
I : H r (M ) ! Hr (M ):
(4.84)
Theorem 238 (de Rham 1). The mapping I : H r (M ) ! Hr (M ) is an
isomorphism. If Hr (M ) is …nite dimensional as when M is compact and
(b)
(1)
if zr ; :::zr (with b = the r-Betti number ) is a r-cycle basis of Hr (M )
and if 1 ; :::; r 2 R are arbitrary numbers then there is a closed r-form
Gr 2 Z r (M ) such that
hzr(i) ; Gr i =
i,
i = 1; :::; r:
(4.85)
Proof. See, e.g., [265].
De…nition 239 The number
(i)
form Gr on the cycle zr .
r
in Eq.(4.85) is called the period of the
Vr
Corollary 240 (de Rham 2) If for a closed form Gr 2 sec
T M and for
(i)
(i)
any zr 2 Hr (M ) we have V
hzr ; Gr i = 0 then Gr is exact, i.e., Gr = dGr 1
r
for some form Gr 1 2 sec
T M:
120
4. Some Di¤erential Geometry
Note also, that when M is compact the spaces Hr (M ) and H r (M ) are
…nite dimensional and dim H r (M ) = bp . Thus de Rham theorem justi…es
writing
H r (M ) = (Hr (M )) ;
(4.86)
and the nomenclature: homology and cohomology groups for Hr (M ) and
H r (M ).
4.4 Di¤erential Geometry in the Hodge Bundle
4.4.1 Riemannian and Lorentzian Structures on M
Next we introduce on M a smooth metric …eld g 2 sec T20 M and gives the
De…nition 241 A pair (M; g), dim M = n is a n-dimensional Riemann
structure (or Riemann manifold ) if g 2 sec T20 M is a smooth metric of
signature (n; 0). If g has signature (p; q) with p + q = n, p 6= n or q 6= n
then the pair (M; g) is called a pseudo Riemannian manifold. When g has
signature (1; n 1) the pair (M; g) is called an hyperbolic manifold. When
dim M = 4 and g has signature (1; 3) the pair (M; g) is called a Lorentzian
manifold.11
We already de…ned the concept of oriented manifold. Thus, we say that a
Riemannian (or pseudo Riemannian or Lorentzian) manifold is orientable
if and
element …eld g 2
Vn only if it admits a continuous metric volume
sec
T M given in local coordinate functions fxi g covering U M by
p
jdet gjdx1 ^ ::: ^ dxn ;
(4.87)
g =
where
det g = det g(
@
@
; j) :
i
@x @x
(4.88)
Proposition 242 Any C r manifold M , dim M = n admits a C r 1 Riemannian metric g (signature (n; 0)) if and only if it is paracompact.
Proof. For a proof see, e.g., [77].
Let us consider now a smooth oriented metric manifold M = (M; g; g ),
dimVM = n, where g is a smooth metric …eld of signature (p; q) and g 2
n
sec
T M . We denote by g 2 sec T02 M the metric tensor
V of the cotangent
bundle. Also we denote the scalar product induced on T M by the metric
1 1 When Lorentzain manifolds serve as models of spacetimes it is also imposed that M
is noncompact. See section 4.7.1.
4.4 Di¤erential Geometry in the Hodge Bundle
121
V
V
V0
tensor g 2 sec T02 M by12 : sec T M sec T M ! sec
T M . If
g
^p
A; B 2 sec
T M we have (recall Eq.(2.123)
(A B)
g
g
= A ^ ?B
(4.89)
g
4.4.2 Hodge Bundle
De…nition 243 The Hodge bundle of the structure M is the triple
^
^
(M) = ( T M; ; g ):
(4.90)
g
The importance of the Hodge bundle is that besides the exterior derivative operator, we can now introduce a new di¤erential operator called the
Hodge codi¤erential. Equipped with these two operators we can write, e.g.,
Maxwell equations (with currents) in a di¤ eomorphism invariant way13 (see
Section 4.9.1). This is a very important fact, which is often not well known
as it should be.
De…nition
244 The HodgeVcodi¤ erential operator
in the Hodge bundle of
V
V
(M) is the mapping : sec T M ! sec T M , given, for homogeneous
g
multiforms, by:
g
= ( 1)r ?
g
1
d?;
(4.91)
g
where ? is the Hodge star operator associated to the scalar product
g
g
.
De…nition 245 The Hodge Laplacian operator is the mapping
^
^
: sec T M ! sec T M
g
given by:
g
=
(d + d):
g
(4.92)
g
The exterior derivative, the Hodge codi¤erential and the Hodge Laplacian satisfy the relations:
dd =
gg
= 0;
d = d;
g
g
?=(
gg
=
= (d
gg
g
= ( 1)r d?;
?d = d?;
g g
)2 ;
;
gg
gg
1)r+1 ?d; ?
g
gg
d ? = ? d;
gg
g
g g
(4.93)
g
? = ?:
gg
gg
1 2 When there is no chance of confusion we eventually used the symbol instead of the
symbol in order to simplify the notation.
g
1 3 For
the exact meaning of the concept of di¤eomorphism invariance of a spacetime
physical theory (as used in this text) see Section 6.6.3.
122
4. Some Di¤erential Geometry
Remark 246 When it is clear from the context which metric …eld is involved we use the symbols ?, and in place of the symbols ?, and in
g
g
g
order to simplify the writing of equations.
4.4.3 The Global Inner Product of p-forms
^p
De…nition 247 Let A; B 2 sec
T M and suppose that the support of
A or B is compact. The global inner product of these p-forms is
Z
hA; Bi = A ^ ?B:
(4.94)
M
^q
T M ! sec
T M be a (p; q) extensor
^p
…eld acting on the sections of
T M of compact support. We de…ne the
metric transpose of T as the the (q; p) extensor …eld T t such that
De…nition 248 Let T : sec
^p
hT A; Bi = hA; T t Bi
Exercise 249 Show that d and
(4.95)
are metric transposes of each other i.e.,
hdA; Bi = hA; Bi;
h A; Bi = hA; dBi
(4.96)
Are the formulas given in Eq.(4.96) true for a compact manifold with
boundary?
4.5 Pullbacks and the Di¤erential
Proposition 250 Let
: M ! N be aVdi¤ erentiable mapping and let h
be the pullback mapping. Let A; B 2 sec T M . Then
(A ^ B) =
A^
B:
(4.97)
Proof. It is a simple algebraic manipulation.
Proposition 251 Let : M ! N beVa di¤ erentiable mapping and let
be the pullback mapping. Let A 2 sec T M . Then,
dA = d(
A)
(4.98)
Proof. Since an arbitrary form is a …nite sum of exterior products of functions and di¤erential of functions, we see that it is only necessary to prove
4.6 Structure Equations I
the theorem for a 0-form and an exact 1-form
because,
dg = d(g
123
. The …rst case is true
)
= d(
g)
(4.99)
where we used the de…nition of reciprocal image. Now, if
exact, we have
d = ddg = 0:
= dg, i.e.,
is
Also,
d(
) = d(
g)] = d2
dg) = d[d(
g = 0;
(4.100)
and the proposition is proved.
Proposition 251 is also very much important in proving the invariance of
some exterior di¤erential system of equations under di¤eomorphisms.
4.6 Structure Equations I
Let us now endow the metric manifold (M; g), with an arbitrary linear
connection r obtaining the structure (M; g; r).
De…nition 252 The torsion and curvature operations and the torsion and
curvature tensors of a connection r, are respectively the mappings 14 :
: sec(T M
T M ) ! sec T M;
: sec(T M
T M ) ! EndT M
(u;v) = ru v
(u; v) = ru rv
rv u
[u;v];
r v ru
r[u;v]
(4.101)
(4.102)
and
( ; u;v) =
( (u;v)) ;
R( ; w; u;v) = ( (u;v)w);
for every u;v;w 2 sec T M and
1 4 EndT M
2 sec
V1
T M.
means the set of endomorphisms T M ! T M
(4.103)
(4.104)
124
4. Some Di¤erential Geometry
Exercise 253 Show that for any di¤ erentiable functions f; g and h we
have
(gu;hv) = gh (u;v);
(gu;hv)f w = ghf (u;v):
(4.105)
Given an arbitrary moving frame fe g on T M , let f g be the dual frame
of fe g (i.e., (e ) =
). We write:
[e ;e ] = c
re e
= L
e ;
e ;
(4.106)
where c
are the structure coe¢ cientsof the frame fe g and L
are the
connection coe¢ cients in this frame. Then, the components of the torsion
and curvature tensors are given, respectively, by:
T
= e (L
:=
)
( ; e ;e ) = L
L
R
:= R( ; e ; e ;e )
e (L ) + L L
L L
We also have:
d = 21 c
re
= L
^
;
c
;
(4.107)
c
L
:
;
(4.108)
V1
V2
where ! 2 sec
T M are the connection 1-forms,
2 sec
T M
V2
are the torsion 2-forms and R 2 sec
T M are the curvature 2-forms,
given by:
!
R
:= L
1
:= T
2
1
:= R
2
;
^
;
^
(4.109)
:
Multiplying Eqs.(4.107) by 21 ^
and using Eqs.(4.108) and (4.109),
we get the Cartan’s structure equations:
d +! ^
=
;
d! + ! ^ ! = R :
(4.110)
Exercise 254 Show that the torsion tensor can be written as
=e
a1 :::ar
Exercise 255 Put
Show that when
d
d?
g
a1 :::ar
a1 :::ar
a
a1
=
= 0 we have
=
=
! ab1 ^
! ab1
^
b:::ar
^?
g
b:::ar
(4.111)
^
ar
and ?
a1 :::ar
g
g
! abr ^
! abr
= ?(
a1 :::b
^?
g
;
a1 :::b
a1
^
^
ar
).
(4.112)
:
(4.113)
4.6 Structure Equations I
125
4.6.1 Exterior Covariant Di¤erential of (p + q)-indexed
r-form Fields
De…nition 256 Suppose that X 2 sec Tpr+q M and let
X
1 :::: p
1 :::: q
(v 1 :::; v r ) 2 sec
such that
X
1 :::: p
1 :::: q
^r
T M;
(v 1 :::; v r ) = X(v 1 :::; v r ; e 1 ; ::::e q ;
for v 1 :::; v r 2 sec T M . The X
1 :::: p
1 :::: q
1
(4.114)
; :::;
p
):
are called (p + q)-indexed r forms.
De…nition 257 The exterior covariant di¤ erential15 D of X
manifold with a general connection r is the mapping:
D : sec
such that16
(r + 1)DX
=
r
X
1 :::: p
1 :::: q
^r
T M ! sec
(4.115)
^r+1
T M, 0
r
1 :::: p
1 :::: q
4;
on a
(4.116)
(v 0 ; v 1 :::; v r )
( 1) re X(v 0 ; v 1 :::; v ; :::v r ; e 1 ; :::: e q ;
1
; :::;
p
)
=0
0
X
;&
( 1)
+&
X( (v ; v & ); v 0 ; v 1 :::; v ; :::; v & ; :::; er ; e 1 ; ::::e q ;
1
; :::;
r
(4.117)
Then, we may verify that
DX
1 :::: p
1 :::: q
= dX
!
s
1
1 :::: p
1 :::: q
^X
+!
1 ::::
s ::::
1
s
^X
s :::: p
1 :::: q
!
p
q
1
s
+
^X
+ !
1 ::::
1 ::::
p
s
1
s
^X
1 :::: p
1 :::: q
(4.118)
:
Remark 258 Sometimes, Eqs.(4.110) are written by some authors [421]
as:
D
“D!
:=
;
:= R :”
(4.119)
V
V
and D : sec T M ! sec T M is said to be the exterior covariant derivative related to the connection r. Whereas the equation D :=
is well
de…ned, we see that the equation “D! := R :” is an equivocated one.
1 5 Sometimes
also called exterior covariant derivative.
usual the inverted hat over a symbol (in Eq.(4.117)) means that the corresponding
symbol is missing in the expression.
1 6 As
p
):
126
4. Some Di¤erential Geometry
Indeed if Eq.(4.118) is applied on the connection 1-forms ! we would get
D! = d! + ! ^ !
! ^ ! . So, we see that the symbol D!
given by the second formula in Eq.(4.119), supposedly de…ning the curvature 2-forms is to be avoided. The reason for the failure of Eq.(4.118) in
that case is that there do not exist a tensor …eld ! 2 sec T12 M which satisfy
the corresponding Eq.(4.115). More details on this issue may be found in
Appendix A.3
Vr
Vs
Exercise 259 Show that if X J 2 sec
T M and Y K 2 sec
T M are
sets of indexed forms17 , then
D(X J ^ Y K ) = DX J ^ Y K + ( 1)rs X J ^ DY K :
Vr
Exercise 260 Show that if X 1 :::: p 2 sec
T M then
DDX
1 :::: p
= dX
1 :::: p
+R
1
s
^X
s :::: p
+
R
p
s
^X
(4.120)
1 :::: s
: (4.121)
Exercise 261 Show that for any metric-compatible connection r if g =
g
then,
Dg = 0:
(4.122)
Since we are dealing with a metric manifold, we must complete Cartan’s
structure equations with the equations stating the relation between the
connection and the metric. For this, following the usual nomenclature [388,
458, 29] we give the
De…nition 262 The nonmetricity tensor …eld of the structures (M; g; r)
is the tensor …eld Q 2 sec T30 M with components18 in the basis f g given
by
Q
:= r g = e (g ) + g L + g L :
(4.123)
Correspondingly, we introduce the nonmetricity 2-forms, by:
Q :=
where Q [ ] = g (Q
Q
using Eq.(4.110a), we get:
D
where f
1
Q
2 [
^
]
;
(4.124)
). Multiplying Eq.(4.123) by
d
!
^
=
;
g is the reciprocal frame of f g is the (i.e.,
=
^
and
(4.125)
=g
) and
Q :
Eq.(4.125) can be used as the complement of Cartan’s structure equations
for the case of a metric manifold.
1 7 Multi
indices are here represented by J and K.
:::
:::
use the notation r t :::
(re t) :::
(rt) :::::: for the components of the
:::
covariant derivative of a tensor …eld t. This is not to be confused with re t :::
:::
e (t :::
), the derivative of the components of t in the direction of e .
1 8 We
4.6 Structure Equations I
127
4.6.2 Bianchi Identities
Di¤erentiating Eqs.(4.110) and Eq.(4.125) we obtain the Bianchi identities:19
(a) D
=d
(b) DR
(c) D
+! ^
=R ^
;
= dR
R ^ !v + ! ^ R
=d
!
^
=
R ^
= 0;
(4.126)
:
4.6.3 Induced Connections Under Di¤eomorphisms
Let M and N be two di¤erentiable manifolds, dim M = m, dim N = n.
De…nition 263 Let r be a connection on N and X; Y 2 sec T N and
T 2 sec Tsr N , f : N ! R and h:M ! N a di¤ eomorphism. The induced
connection h r on M is de…ned by
h rh
1
Xh
T = h (rX T):
(4.127)
Example 264 Let f : N ! R and Y 2 sec T N . Then,
h rh
1
Xh
Y = h (rX Y);
from where it follows (taking into account that for any vector …eld V 2
sec T N , h N = h 1 N) that
h rh
1
Xh
Y f
e
h = h (rX Y)je f
h = rX Yjh(e) f; 8e 2 M:
Remark 265 Now, suppose that M = N and h: M ! M a di¤ eomorphism. Suppose that D is the Levi-Civita connection of g, then h D = D0
is the Levi-Civita connection of h g = g 0 since using Eq.(4.127) we infer
that
h Dh
1
Xh
g
e
= Dh0
1
X
h g
e
= h (DX g)je , 8e 2 M:
(4.128)
Taking into account that20 h [X; Y] = [h X;h Y] we have for X; Y 2
sec T M;
h (DX Y DY X [X; Y]) = 0:
(4.129)
Remark 266 Eq.(4.127) applied to the case M = N also implies, as the
reader may verify the important fact that the curvature tensor of h D will
be null if the curvature tensor of D is null.
1 9 To our knowledge, Eqs.(4.125) and Eq.(4.126c) are not found anywhere in the literature, although they appear to be the most natural extension of the structure equations
for metric manifolds.
2 0 See, e.g., page 135 of [77].
128
4. Some Di¤erential Geometry
4.7 Classi…cation of Geometries on M and
Spacetimes
De…nition 267 Given a triple (M; g; r):
(a) it is called a Riemann-Cartan geometry21 if and only if
rg = 0
and
[r] 6= 0:
(4.130)
(b) it is called Weyl geometry if and only if
rg 6= 0
and
[r] = 0:
(4.131)
(c) it is called a Riemann geometry if and only if
rg = 0
and
[r] = 0;
(4.132)
and in that case the pair (r; g) is called Riemannian structure.
(d) it is called Riemann-Cartan-Weyl geometry if and only if
rg 6= 0
and
[r] 6= 0:
(4.133)
(e) it is called a (Riemann) ‡at geometry if and only if
rg = 0
and
R[r] = 0;
(f) it is called teleparallel geometry if and only if
rg = 0;
[r] 6= 0 and R[r] = 0:
(4.134)
For each metric tensor de…ned on the manifold M there exists one and
only one connection in the conditions of Eq.(4.132). It is called Levi-Civita
connection of the metric considered, and is denoted by D. If in a given
context it is necessary to distinguish between the Levi-Civita connections
of two di¤erent metric tensors g and g on the same manifold, we write D,
D.
Remark 268 When dim M = 4 and the metric g has signature (1; 3) we
sometimes substitute the word Riemann by the word Lorentzian in the previous de…nitions.
4.7.1 Spacetimes
From nowhere besides the constraints already imposed (Hausdor¤ and
paracompact) on M , we suppose also that it is connected and noncompact [163, 367]. We now introduce the concept of time orientability on an
oriented Lorentzian manifold structure (M; g; g ), which plays a key role
in physical theories.
2 1 Or
Riemann space.
4.7 Classi…cation of Geometries on M and Spacetimes
129
S
De…nition 269 Let (M; g) be a Lorentzian manifold , T M = e2M Te M
its tangent bundle and
: T M ! M the canonical projection (see Appendix). The causal character of (e; v) 2 T M is the causal character of v
(De…nition 62).
De…nition 270 A line element at x 2 M is a one-dimensional subspace
of Tx M:
Proposition 271 Let M be a C 1 paracompact and Hausdor¤ manifold,
dim M = 4. Then the existence of a continuous line element …eld on M is
equivalent to the existence of a Lorentzian structure on M .
Proof. For a proof see [77].
Proposition 272 The set T T M of timelike points is an open manifold
and it has either one (connected ) component or two.
Proof. A proof of this important result can be found in [367].
De…nition 273 A connected Lorentzian manifold (M; g) is said to be time
orientable if and only if T has two components and one of the components
is labeled the future T+ and the other component T is labelled the past.
We denote by " the time orientability of a Lorentzian manifold.
De…nition 274 A spacetime is a pentuple (M; g; r; g ; ") where (M; g) is
a Lorentzian oriented and time oriented manifold and r is an arbitrary
covariant derivative operator on M:
De…nition 275 When (M; g; r; g ; ") is a spacetime and r = D is the
Levi-Civita connection of g the spacetime is said to be Lorentzian. When
rg = 0 and (r) 6= 0 we call the structure M = (M; g; r; g ; ") a
Riemann-Cartan spacetime. The particular Riemann-Cartan spacetime for
which R(D) = 0, [r] 6= 0 is called a teleparallel spacetime (also called
Weintzenböck spacetime according to [268]).
De…nition 276 A Lorentzian spacetime structure M = (M; ; D; ; ") is
said to be Minkowski spacetime if and only if M ' R4 and R(D) = 0.
Remark 277 We just establish that any Lorentzian manifold admits a
continuous element …eld. If it is also time orientable, we can choose a
direction for the continuous element …eld, and say that it is a timelike vector
…eld pointing to the future. This is a nontrivial result and very important
for our discussion of the Principle of Relativity (Chapter 6).
130
4. Some Di¤erential Geometry
4.8
Di¤erential Geometry in the Cli¤ord Bundle
It is well known [288] that the natural operations on metric vector spaces,
such as, e.g., direct sum, tensor product, exterior power, etc., carry over
canonically to vector bundles with metric tensors. Then we give the
De…nition 278 The Cli¤ ord bundleof di¤ erential forms of the metric manifold (M; g) is:
C`(M; g) =
[
TM
=
C`(Tx M; gx );
Jg
(4.135)
x2M
where T M denotes the (covariant) tensor bundle of M , Jg
T M is the
bilateral ideal of T M generated by the elements of the form
+
2g( ; ), with ; 2 sec T M T M and C`(Tx M; gx ) is the Cli¤ ord
algebra of the metric vector space structure (Tx M; gx ).
It will be shown in Chapter 7 that the Cli¤ord bundle C`(M; g) (as
de…ned by Eq.(4.135)) is a vector bundle associated to the principal bundle
of orthonormal frames PSOe p;q , i.e.,
C`(M; g) = PSOe p;q
Ad
Rp;q :
(4.136)
In Eq.(4.136) Ad is the adjoint representation of Spinep;q , i.e., Ad : SOep;q !
Aut(Rp;q ), u 7! Adu , with Adu A = A u 1 , 8u 2 SOep;q , 8A 2 Rp;q '
C`(Tx M; gx ). Details on these groups may be found in Chapter 3. In Chapter 7 we scrutinize the vector bundle structure of the Cli¤ord bundle of
di¤erential forms over a general Riemann-Cartan manifold modelling spacetime.
4.8.1 Cli¤ord Fields as Sums of Nonhomogeneous Di¤erential
Forms
De…nition 279 Sections of C`(M; g) are called Cli¤ ord …elds.
We recall some notations and conventions. By F (U ) we denote the frame
bundle (see Appendix A.3) of U M . A section of F (U ) will be denoted by
fe g 2 sec F (U ): The dual frame of a frame fe g will be denoted by f g,
where
2 sec T U
T M . When fe g is a coordinate frame associated
to the coordinate functions fx g of a local chart covering U we use instead
of e the notation e = @ and in this case
= dx . When fe g refers
to an orthonormal frame we use instead of e the notation ea and instead
of
the notation a .
Recall that as V
a vector space over R, C`(Tx M; gx ) is isomorphic to the
exterior algebra Tx M of the cotangent space and
^
Mn ^
k
Tx M =
Tx M;
(4.137)
k=0
4.8 Di¤erential Geometry in the Cli¤ord Bundle
131
Vk
where
Tx M is theVnk -dimensional space of k-forms. Then, there is a
natural embedding 22 T M ,! C`(M; g) [206] and sections of C`(M; g)—
Cli¤ord …elds (De…nition 279)— can be represented as a sum of non homogeneous di¤erential forms. Let fea g be an orthonormal basis for T U T M ,
i.e., g(ea ; eb ) = ab ; where the matrix with entries ab is the diagonal
matrix, diag(1; 1; ::::: 1; :::; 1) and (a; b; i; j; ::: = 1; 2; :::; n). Moreover, let
V1
f a g 2 sec
T M ,! sec C`(M; g) such that the set f a g is the dual basis
of fea g. We denote by f i g be the reciprocal basis of f i g, i.e., i j = ij .
g
For the particular case of a 4 dimensional spacetime, of course, the
range of the bold labels are a; b; i; :: = 0; 1; 2; 3. Recall that the fundamental
Cli¤ ord product is generated by
i j
+
j i
=2
ij
:
(4.138)
If C 2 sec C`(M; g) is a Cli¤ord …eld, we have:
where
5
=
i
1
bij
2!
i j
i j k
+p
5
(4.139)
is the volume element and
^0
s; vi ; bij ; tijk ; p 2 sec
T M ,! sec C`(M; g):
(4.140)
+
+
1
tijk
3!
;
C = s + vi
0 1 2 3
4.8.2 Pullbacks and Relation Between Hodge Star Operators
Let M be a n-dimensional manifold and g, g 2 sec T20 M two metrics of
the same signature with corresponding metrics (for the cotangent bundle)
g; g 2 sec T02 M . Let g and g be the extensor …elds associated to g and g.Let
h: M ! M be a di¤eomorphism such that
g = h g:
(4.141)
From the algebraic results of Section 2.7 we easily infer that there exists
a metric gauge extensor …eld h such that
g(a) b = h(a) h(b)
g
g
(4.142)
V1
for any a; b 2 sec T M and we write g = hy h. Then, as in the purely
algebraic case discussed in Section 2.7 we can also show that we have the
following relation between the Hodge star operators associated to g and g
?=h
g
2 2 Recall
1
? h:
g
(4.143)
again that the symbol A ,! B means that A is embedded in B and A
B.
132
4. Some Di¤erential Geometry
Remark 280 In this case we say that the metric gauge extensor h is related to the pullback mapping h and describes an elastic distortion. However, keep in mind that in general given a h it does not implies the existence
of h such that Eq.(4.141) holds. In this case h is said to generate a plastic
distortion. More details in [127].
We now show the
Proposition 281 Let h : M ! M a di¤ eomorphism. LetVg, g 2 sec T20 M
p
two metrics of the same signature. Then for any ! 2 sec T M we have
?h ! = h ? !
g
(4.144)
g
Proof. As in Remark 192 take two charts (U; ') and (V; ), U , h(U ); V
with coordinate functions xi and yi such that and xi (e) = yi (h(e)), i.e.,
calling xi (e) = xi , yi (h(e)) = y i we have @y i =@xj = ji , dxi = dy i . Let also
g(@=@y k ; @=@y k ) = gkl (y j ). Then it follows that gkl (xi ) = gkl (y j (xi )) =
1
!i1 :::ip xi dxi1 ^ ::: ^ dxip ; we can
gkl (xi ) and det g = det g. Now, if ! = p!
Vp
Vp
write (taking into account that
T M ,! C`(M; g) and also
T M ,!
C`(M; g))
?h ! = hg
!y
g
g
g
= hg
!
g
p
1
^
!i1 :::ip y i (xj )dxi1 ^
::: ^ dxip j det gjdx1 ^ ::: ^ dxn
p!
p
1
^
::: ^ dy ip j det gjdy 1 ^ ::: ^ dy n
= !i1 :::ip (y i (xj ))dy i1 ^
p!
p
1
^
= !i1 :::ip y i dy i1 ^
::: ^ dy ip y j det gjdy 1 ^ ::: ^ dy n
g
p!
= h ? !;
=
g
and the proposition is proved.
Remark 282 When g = h g , there exists an associated metric gauge
extensor …eld h such satisfying Eq.(4.142), i.e., g = hy h. The relation
?h ! = h ? ! and ? = h 1 ? h permit us to write the suggestive operator
g
g
g
g
identity
?hh ! = hh ? !:
g
g
(4.145)
Exercise 283 Consider any di¤ eomorphism h : M ! M; and two metrics
g and g such that g = h g. Show that
?d ? h ! = h ? d ? !;
g
for any ! 2
V
T M:
g
g
g
(4.146)
4.8 Di¤erential Geometry in the Cli¤ord Bundle
133
Solution: The …rst member of Eq.(4.146) can be writing successively
using Eq.(4.144) as
?d ? h ! = ?dh ? !
g
g
g
g
= ?h d ? !
g
g
= h ? d ? !:
g
g
4.8.3 Dirac Operators
We now equip the Riemannian (pseudo Riemannian, or Lorentzian) manifold (M; g) with a standard structure (M; g; D), where D is the Levi-Civita
connection of g.
We are going to introduce in the Cli¤ord bundle of di¤erential forms
C`(M; g) a di¤erential operator @j, called the standard Dirac operator23 ,
which is associated to the Levi-Civita connection of the structure (M; g; D)
and we study the properties of that operator. Next we de…ne new Diraclike operators associated with a connection di¤erent from the Levi-Civita
one, i.e., to connections r de…ning a general Riemann-Cartan-Weyl geometry (M; g; r). Moreover, making use of the results developed in section
2.7, we show that it is possible to introduce in…nitely many others Diraclike operators, one for each bilinear form …eld de…ned on the manifold M
of the structure (M; g; D). These constructions enable us to formulate the
geometry of a Riemann-Cartan-Weyl space in the Cli¤ord bundle C`(M; g).
Some interesting geometrical concepts, like the Dirac commutator and anticommutator, are introduced. Moreover, we show a new decomposition of a
general linear connection, identifying some new relevant tensors which are
important for a clear understanding of any formulation of the gravitational
theory in ‡at Minkowski spacetime (Chapter 11) and other related subjects
appearing in the literature.
The Standard Dirac Operator
Vr
Given u 2 sec T M and A 2 secV T M ,! sec C`(M; g) consider the tenr
sorial mapping A 7! Du A 2 sec T M ,! sec C`(M; g). Since Du Jg Jg ,
where Jg is the ideal used in the de…nition of C`(M; g), we see immediately that the notion of covariant derivative (related to the Levi-Civita
connection24 ) pass to the quotient bundle C`(M; g), i.e., given A; B 2
2 3 It is crucial to distinguish the Dirac operators introduced in this chapter and which
act on sections of Cli¤ord bundles with the spin Dirac operator introduced in Chapter
7 and which act on sections of spin-Cli¤ord bundles.
2 4 And more generally, to any metric compatible connection.
134
4. Some Di¤erential Geometry
Vr
sec T M ,! sec C`(M; g) we have taking into account the fact that
Du g = 0 = Du g that
1
Du (AB) = Du [ (A
2
1
= Du [ (A
2
B
B
B
B
A) + g(A; B)]
A)] + (Du g)(A; B) + g(Du A; B) + g(A; Du B)
= Du (A)B + ADu (B):
(4.147)
Before continuing we agree that the scalar and contracted products induced
by g will be denoted simply by the symbols and y instead of the symbol
and y .
g
g
De…nition 284 The standard Dirac operator acting on sections of C`(M; g)
is the …rst order di¤ erential operator
@j =
For A 2 sec C`(M; g),
@j A =
(De A) =
De :
(4.148)
y(De A) +
^ De A)
and then we de…ne:
@jyA
@j ^A
in order to have:
y(De A);
^ (De A);
=
=
@j = @jy + @j ^.
Remark 285 Note moreover that for A 2 sec
can also write
@jyA = @j A:
(4.149)
V1
T M ,! sec C`(M; g) we
(4.150)
Exercise 286 Verify that the operators @jy and @j ^ satisfy the following
identities:
(a)
(b)
(c)
@j ^(A ^ B) = (@j ^A) ^ B + A^ ^ (@j ^B);
j@y(Ar yBs ) = (@j ^Ar )yBs + A^r y(@jyBs ); r + 1
@jy? = ( 1)r ? @j ^ ; ? @jy = ( 1)r+1 @j ^ :
s;
(4.151)
In addition to these identities, we have the important result [337, 222]
Proposition 287 The standard Dirac derivative @j is related to the exterior
derivative d and to the Hodge codi¤ erential by:
@j = d
that is, we have @j ^ = d and @jy =
.
;
(4.152)
4.8 Di¤erential Geometry in the Cli¤ord Bundle
135
Proof. If f is a function, @j ^f =
^ De f = e (f ) = df and @jyf =
yDe f =
De f = 0. For the 1-form …elds
of a moving frame on
T M , we have @j ^ =
^ De
=
^
= !
^ ^
=d .
1
Now, for a r-forms …eld ! = r!
! 1 ::: r 1 ^ : : : ^ r , we get, using
Eq. (4.151a),
1
(d! 1 ::: r ^
r!
+
+ ( 1)r+1 !
@j ^! =
1
^
1 :::
^
1
r
= d!:
^
r
+!
^
d
1 :::
r
r
^d
1
1
^
r
2
^
^
r
)
Finally, using Eq. (4.93c) and Eq.(4.151c), we get @jy! =
!.
Note also that given an arbitrary coordinate moving frame f = dx g on
M (x : U ! R, U M , are coordinate functions), we have the following
interesting relations:
p
p
(a) @jy
@j
= g
= j(det g)j@ ( j(det g) 1 jg )
p
p
(4.153)
(b) @jy
@j
=
= j(det g)j@ ( j(det g) 1 j);
where f@
@=@x g is the dual frame of f g. Note that det g = (det g)
V1
Exercise 288 Verify that if ; 2 sec
T M ,! sec C`(M; g) then
@j(
)=(
@j) + (
@j)
y(@j ^ )
y(@j ^ ):
1
.
(4.154)
4.8.4 Standard Dirac Commutator and Dirac
Anticommutator
V1
De…nition 289 Given the 1-form …elds ; 2
T M ,! sec C`(M; g)
and @j, the standard Dirac operator of the manifold, the operators [[; ]] and
f; g given by
[[ ; ]] = ( @j)
( @j)
(4.155)
f ; g = ( @j) + ( @j) ;
are called, respectively, the Standard Dirac commutator (or Lie bracket)
and the Standard Dirac anticommutator of the 1-form …elds and .
We have the identities:
[[ ; ]] =
f ; g =
@jy( ^ )
@j ^(
)
(@j ) ^
(@j ^ )x
^ (@j )
y(@j ^ ) :
(4.156)
The algebraic meaning of these equations is clear: they state that the Dirac
commutator and the Dirac anticommutator measure the amount by which
the operators @jy =
and @j ^ = d fail to satisfy the Leibniz’s rule when
applied, respectively, to the exterior and to the dot product of 1-form …elds.
136
4. Some Di¤erential Geometry
Now, let fe g be an arbitrary moving frame on T M , f g its dual frame
on T M and f g the reciprocal frame of f g. From Eqs.(4.155) we obtain,
respectively:
[[
;
]] = De
De
=(
=c
)
;
(4.157)
and
f
;
g = De
+ De
=(
+
=b
;
;
)
(4.158)
where
are the components of the Levi-Civita connection D of g, c
are the structure coe¢ cients of the frame fe g and where we introduce
the notation b
=
+
. The meaning of these coe¢ cients will be
discussed below.
Clearly, Eq.(4.157) states that the Dirac commutator is the analogous of
the Lie bracket of vector …elds. These operations have similar properties.
In particular, the Dirac commutator satis…es the Jacobi identity:
[[ ; [[ ; !]]]] + [[ ; [[!; ]]]] + [[!; [[ ; ]]]] = 0;
(4.159)
V1
; ;! 2
T M ,! sec C`(M; g). Therefore it gives to the cotangent
bundle of M the structure of a local Lie algebra.
4.8.5 Geometrical Meanings of the Commutator and
Anticommutator
The geometrical meanings of the Dirac commutator and the Dirac anticommutator are easily discovered from Eqs.(4.157) and (4.158). Indeed,
Eq.(4.157) means that the Dirac commutator measures the amount by
which the vector …elds ea = g( ; ) and eb = g( ; ) and their in…nitesimal lifts (e0a = g( 0 ; ), e0 = g(( 0 ; )) along their integral lines fail to form
a parallelogram. By its turn, Eq.(4.158) means that the Dirac anticommutator measures the rate of deformation of the frame f g, i.e., f ; g
gives the rate of dilation of the vector …eld g( ; ) under dislocations along
its own integral lines, while f ; g, 6= , gives the rate of variation of
the angle between g( ; ) and g( ; ) under dislocations in the direction of
each other.
We state now another interesting result:
4.8 Di¤erential Geometry in the Cli¤ord Bundle
137
FIGURE 4.4. Geometrical Interpretation of the: (a) Standard Commutator
[[ ; ]] and (b) Standard Anticommutator f ; g
Proposition 290 The coe¢ cients b
moving frame f g are given by:
b
=
of the Dirac anticommutator of a
($e g)
;
(4.160)
where $e denotes the Lie derivative in the direction of the vector …eld e
and fe g is the dual frame of f g .
Proof. The coe¢ cients
by: (e.g.,[77])
1
g
2
1
+ g
2
=
Hence,
b
=g
h
e (g
of the Levi-Civita connection of g are given
[e (g ) + e (g
h
g c +g c
) + e (g
)
e (g
)
)
e (g
g
g
c
c
)]
i
:
g
(4.161)
c
i
(4.162)
and the r.h.s. of Eq.(4.162) is just the negative of the components of the
Lie derivative of the metric tensor in the direction of e = g e .
Killing coe¢ cients
In view of the result stated by Eq.( 4.160), the attempt to …nd (if existing)
a moving frame for which b
= 0 is equivalent to solve, locally, the Killing
equations for the manifold. Because of this we shall refer to these coe¢ cients
as the Killing coe¢ cients of the frame. Of course, since the solutions of the
Killing equations are restricted by the structure of the metric as well as by
the topology of the manifold, it will not be possible, in the more general
case, to …nd any moving frame for which these coe¢ cients are all null.
138
4. Some Di¤erential Geometry
4.8.6 Associated Dirac Operators
Besides the standard Dirac operator we have just analyzed, we can also
introduce in the Cli¤ord bundle C`(M; g) in…nitely many other Dirac-like
operators, one for each nondegenerate symmetric bilinear form …eld that
can be de…ned on the structure (M; g; D).
Let g 2 sec T20 M be an arbitrary nondegenerate positive symmetric bilinear form …eld on M . To g corresponds g 2 sec T02 M as already introduced.
We denoted by g : sec T M ! sec T M the associated extensor …eld to g
and by h : sec T M ! sec T M the …eld of linear transformations which
induces g, i.e., have:
g( ; ) =
g( ) = h( ) h( )
= g(h( ); h( ))
(4.163)
V1
for every ; 2 sec
T M ,! sec C`(M; g).
We also denote by _
: C`(M; g) C`(M; g) ! C`(M; g) the “Cli¤ord
g
product” induced on C`(M; g) by the bilinear form …eld g and by
g
:
C`(M; g) C`(M; g) ! C`(M; g) the “dot product” associated to the new
Cli¤ord product “_.”
De…nition 291 Let f g be a moving frame on T M , dual to the moving
frame fe g on T M . We call Dirac operator associated to the bilinear form
g 2 sec T02 M the operator:
_
@j
@j _ = (
g
De )
(
_ De ):
(4.164)
We also de…ne
_
@j y =
g
yDe ;
g
(4.165)
where y is the contracted product with respect to g.Then,
g
_
_
_
_
@j = @j y + @j ^ = @j y + @j ^;
g
g
(4.166)
_
because the exterior part of the operator @j coincides with the exterior part
of the operator @j.
_
Of course, the properties of the operator @j di¤er from those of the standard Dirac operator @j. It is enough to state the properties of the operator
_
@j y, which are obtained from the following proposition:
g
4.8 Di¤erential Geometry in the Cli¤ord Bundle
139
_
Proposition 292 The operators @j y and @jy are related by:
g
_
@j y! = @jy! + sy!;
(4.167)
g
for every ! 2 sec C`(M; g), where s = g D g
2 sec T M ,! sec C`(M; g)
is called the dilation 1-form of the bilinear form g.
Proof. Given a r-forms …eld ! =
we have
1
De ! = (D !
r!
1
r! !
1 :::
r
^
1
1 :::
r
)
^
^
1
^
r
2 sec C`(M; g),
;
r
with
D !
1 :::
r
= e (!
1 :::
r
)
1
!
2 :::
r
r
!
1 :::
r
: (4.168)
1
Then,
1
D ! 1 ::: r y(
g
r!
1
= D ! 1 ::: r (g 1
r!
+ ( 1)r+1 g r 1 ^
yDe ! =
g
^
1
2
^
^
^
^
r
)
r
1
r
+
);
or
_
@j y! =
g
1
(r
1)!
g D !
2
2 :::
r
^
^
r
:
(4.169)
Now, taking into account that
g D !
2 :::
g
r
D g
and recalling also that g
_
@j y! =
g
+
Thus, writing !
the Eq.(4.167).
1
(r
1)!
1
(r
2 :::
1)!
r
= D (g !
2 :::
=
;
g Dg
r
)
(D g )!
2 :::
r
;
= g g , we conclude that
g (D g !
g (g
=g !
2 :::
r
)
D g )g !
2 :::
r
2
^
2 :::
and s = g
^
2
r
^
r
^
r
:
D g , we …nally obtain
140
4. Some Di¤erential Geometry
4.8.7 The Dirac Operator in Riemann-Cartan-Weyl Spaces
We now consider the structure (M; g; r) where r is an arbitrary linear
connection. In this case, the notion of covariant derivative does not pass to
the quotient bundle C`(M; g) [92]. Despite this fact, it is still a well de…ned
operation and in analogy with the earlier section, we can associate to it,
acting on the sections of C`(M; g), the operator:
@=
where f
TM.
re ;
g is a moving frame on T M , dual to the moving frame fe g on
De…nition 293 The operator @ is called the Dirac operator (or Dirac
derivative, or sometimes gradient).
We also de…ne:
@yA
@^A
y(re A);
^ (re A);
=
=
(4.170)
for every A 2 sec C`(M; g), so that:
@ = @y + @^ .
(4.171)
The operator @^ satis…es, for every A; B 2 sec C`(M; g):
@ ^ (A ^ B) = (@ ^ A) ^ B + A^ ^ (@ ^ B);
(4.172)
what generalizes Eq.(4.151a). By its turn, Eq.( 4.151c) is generalized according to the following proposition:
Proposition 294 Let Q be the nonmetricity 2-forms associated with the
connection r in an arbitrary moving frame f g and re e = L e .
Then we have, for homogeneous multiforms,
(a) ( 1)r ? 1 @y?
= @^ + Q ^ i ;
(b) ( 1)r+1 ? 1 @^? = @y Q yj ;
where i A =
(4.173)
yA and j A =
^ A, for every A 2 sec C`(M; g).
Vr
1
! 1 ::: r 1 ^ ^ r 2 sec
T M ,! sec C`(M; g) be a
Proof. Let ! = r!
r-form …eld on M . We have ( 1 ^ ^ r ) ^ ! = (( 1 ^ ^ r ) !) g =
! 1 ::: r g and it follows that re (( 1 ^
^ r ) ^ !) = e (! 1 ::: r ) g .
But on the other hand, we also have
re (
1
^
^
r
) ^ ?! =
1
+ (L
(Q
^v^
1
!
!
1
r
^ re ? !
2 ::: r
+
+L
2 ::: r
+
+Q
r
!
!
r
1 ::: r
1 ::: r
1
1
)
)
g
g
4.8 Di¤erential Geometry in the Cli¤ord Bundle
141
and therefore we get, after some algebraic manipulation:
re ? ! = ?re ! + Q
?(
^ ( y!));
(4.174)
from which Eqs.(4.173) follow immediately.
Taking into account the result stated in the above proposition and the
de…nition of the Hodge codi¤erential (Eq.(4.91), we are motivated to introduce in the Cli¤ord bundle the Dirac coderivative operator, given, for
homogeneous multiforms, by:
@ = ( 1)r ?
1
@? .
(4.175)
Of course, we have:
@ = ( 1)r ?
1
@y ? + ( 1)r ?
1
@^?
(4.176)
and we can, then, de…ne:
@y + Q yj
@y := ( 1)r ?
1
@^? =
@^ := ( 1)r ?
1
@y ? @ ^ + Q ^ i ;
(4.177)
so that:
@ = @ ^ +@y .
(4.178)
The following identities are trivially established:
@ = ( 1)r+1 ?
1
?@ = ( 1)r+1 @?;
@ @? = ?@@;
@?
?@ = ( 1)r @?
(4.179)
?@ @ = @@?
2
?@ =
(@)2 ?;
?(@)2 =
@2 ? .
In addition, we note that the Dirac coderivative permit us to generalize
Eq.(4.151b) in a very elegant way. In fact, in consequence of Proposition
294 we have:
Vr
Vs
Corollary 295 For Ar 2 sec
T M ,! sec C`(M; g), Bs 2 sec
T M ,!
sec C`(M; g), with r + 1 s, it holds:
@y(Ar yBs ) = (@ ^ Ar )yBs + ( 1)r Ar y(@yBs ):
(4.180)
V1
Vs
Proof. Given a 1-form …eld 2
T M and a s-form …eld ! 2 sec
T M,
we have, from Eq.(4.174), that re ? ( y!) = ?re ( y! + Q
?[ ^
( y( y!))].
142
4. Some Di¤erential Geometry
We also have that
re ? ( y!) = ( 1)s+1 re ( ^ ?!)
)y! + y(re ! + Q
= ?[(re
^ ( y!))];
(
where we have used Eq.(4.174) once again. It follows that:
re ( y!) = (re
)y! + y(re !) + Q
y!:
(4.181)
Then, recalling that
Vs ( 1 ^ : : : ^ r )y! = 1 y : : :y r y!, with 1 ; : : : ; r 2
sec T M; ! 2 sec
T M; r s + 1, and applying Eq.(4.181) successively
in this expression, we get Eq.(4.180).
Another very important consequence of Proposition 294 states the relation
between the operators @ and @j:
Proposition 296 Let
=
Q , where
and Q denote, respectively, the torsion and the nonmetricity 2-forms of the connection r in an
arbitrary moving frame f g. Then:
(a) @^
(b) @y
= @j ^
= @jy
^i
yj
;
(4.182)
.
Proof. If f is a function, @ ^ f =
^ re f = e (f ) = df and @yf =
yre f = 0. For the 1-form …eld
of a moving frame on T M , we have
@^ =
^ re
= L
^
= ! ^
=d
.
1
! 1 ::: r 1 ^ : : : ^ r , we get
Now, for a r-form …eld ! = r!
1
(d! 1 ::: r ^ 1 ^
^ r + ! 1 ::: r d
r!
+
+ ( 1)r+1 ! 1 ::: r 1 ^
^ r 1 ^d
1
(! 1 ::: r 1 ^ 2 ^
^ r+
r!
r
+ ( 1)r+1 ! 1 ::: r 1 ^
^ r 1^
)
1
^ (! 2 ::: r 2 ^
^ r+
= d!
r!
1
+ ( 1)r+1 ! 1 ::: r 1
^
^ r 1)
@^! =
= d!
1
r
^
)
^i !
and Eq.(4.182a) is proved.
Finally, from Eq.(4.173b) and Eq.(4.182a) we obtain
@ ^ ?! = ( 1)r+1 ?@y!
= @ ^ ?!
r+1
= ( 1)
( 1)r+1 ?Q yj !
^ ?!
? @jy!
( 1)r+1 ?
yj !:
2
^
^
r
4.8 Di¤erential Geometry in the Cli¤ord Bundle
Therefore, @y! = @jy!
143
yj !, and Eq.(4.182b) is proved.
From Eqs.(4.182) we obtain the expressions of @y and @^ in terms of @jy
and @j ^:
@jy +
@y
=
@^
= @j ^
yj
(4.183)
^i :
Obviously, the Dirac coderivative associated to the standard Dirac operator
is given by:
@jy = d + :
@ = @j ^
(4.184)
We observe …nally that we can still introduce another Dirac operator,
obtained by combining the arbitrary a¢ ne connection r with the algebraic
structure induced by the generic bilinear form …eld g 2 sec T02 M . With
respect to an arbitrary moving frame f g on T M , this operator has the
expression:
@_ =
_ re :
(4.185)
It is clear that in the particular case where r = D is the Levi-Civita
connection of g, the operator @— which in this case is the standard Dirac
operator associated to g— will satisfy the properties of Section 4.8.3, with
the usual Cli¤ord product exchanged by the new Cli¤ord product “_.” In
addition, for a more general connection we can apply the results of Section
4.8.6, once again with all the occurrences of g replaced by g. (In particular,
the standard Dirac operator associated to g is replaced by that associated
with g.)
4.8.8 Torsion, Strain, Shear and Dilation of a Connection
In analogy with the introduction of the Dirac commutator and the Dirac
anticommutator, let us de…ne the operations:
V1
De…nition 297 Given ; 2 sec
T M the Dirac commutator and anticommutator of these 1-form …elds are
(a)
(b)
[[ ; ]]
ff ; gg
= (
= (
@)
(
@) + (
[[ ; ]]
f ; g:
@)
@)
(4.186)
We have subtracted the Dirac commutator and the Dirac anticommutator in the r.h.s. of these expressions in order to have objects which are
independent of the structure of the …elds on which they are applied.
If f g is the reciprocal of an arbitrary moving frame f g on T M , we
get, from Eq.(4.186a):
[[
;
]] = (T
Q[
])
;
(4.187)
144
4. Some Di¤erential Geometry
where T are the components of the usual torsion tensor (Eq. (4.107)).
Note from this last equation that the operation de…ned through Eq.(4.186a)
does not satisfy the Jacobi identity. Indeed we have:
X
X
(4.188)
Q [ ]) ;
[[[[ ; ]]; ]] =
(T
Q [ ] )(T
[
]
[
]
where the summation in this equation is to be performed on the cyclic
permutations of the indices ; and .
From Eq.(4.186b), we get:
ff
where Q (
)
;
:= g (Q
gg = (S
+Q
S
Q(
))
;
) and we have written:
=L
+L
b
:
(4.189)
It can be easily shown that the object having these components is also a
tensor. Using the nomenclature of the theories of continuum media [378,
398] we will call it the strain tensor of the connection. Note that it can be
further decomposed into:
S
=S
+
2
s g
n
(4.190)
where S
is its traceless part, which will be called the shear of the connection, and
1
s = g S
(4.191)
2
is its trace part, which will be called the dilation of the connection.
It is trivially established that:
L
=
1
+ T
2
1
+ S
2
:
(4.192)
where
= 12 (b + c ) are the components of the Levi-Civita connec25
tion of g.
Eq.(4.192) can be used to relate the covariant derivatives with respect
to the connections D and r of any tensor …eld on the manifold. In particular, recalling that D g = e (g ) g
g
= 0, we get the
2 5 We note that the possibility of decomposing the connection coe¢ cients into rotation
(torsion), shear and dilation has already been suggested in a Physics paper by Baekler et
al. [29] but in their work they do not arrive at the identi…cation of a tensor-like quantity
associated to these last two objects. The idea of the decompositions already appeared
in [388].
4.8 Di¤erential Geometry in the Cli¤ord Bundle
145
expression of the nonmetricity tensor of r in terms of the torsion and the
strain, namely,
Q
=
1
(g
2
T
+g
1
) + (g
2
T
S
+g
S
):
(4.193)
Eq.(4.193) can be inverted to yield the expression of the strain in terms of
the torsion and the nonmetricity. We get:
S
= g (Q
+Q
Q
)
g (g
T
+g
T
):
(4.194)
From Eq.(4.193) and Eq.(4.194) it is clear that nonmetricity and strain can
be used interchangeably in the description of the geometry of a RiemannCartan-Weyl space. In particular, we have the relation:
Q
+Q
where S
=g S
the relation:
+Q
=S
+S
+S
;
(4.195)
. Thus, the strain tensor of a Weyl geometry satis…es
S
+S
+S
= 0:
In order to simplify our next equations, let us introduce the notation:
K
=L
1
(T
2
=
+S
):
(4.196)
From Eq.(4.194) it follows that:
K
=
1
g (r g
2
1
g (g
2
T
+r g
+g
r g
T
g
T
)
);
(4.197)
where we have used that Q
= r g . Note the similarity of this
equation with that which gives the coe¢ cients of a Riemannian connection
(Eq. 4.161). Note also that for rg = 0, K
is the so-called contorsion
tensor.26
Returning to Eq.(4.192), we obtain now the relation between the curvature tensor R
associated with the connection r and the Riemann
curvature tensor R
of the Levi-Civita connection D associated with
the metric g. We get, by a simple calculation:
R
=R
+J
[
];
(4.198)
2 6 Eqs.(4.196) and (4.197) have appeared in the literature in two di¤erent contexts:
with rg = 0, they have been used in the formulations of the theory of the spinor …elds
in Riemann-Cartan spaces [170, 445] and with [r] = 0 they have been used in the
formulations of the gravitational theory in a space endowed with a background metric
[361, 362, 157, 216, 105].
146
4. Some Di¤erential Geometry
where:
J
=D K
K
K
=r K
K
1
2
Multiplying both sides of Eq.(4.198) by
K
^
+K
K
: (4.199)
we get:
=R +J ;
R
(4.200)
where we have written:
J
=
1
J
2
[
^
]
:
(4.201)
From Eq.(4.198) we also get the relation between the Ricci tensors of the
connections r and D. We de…ne the Ricci tensor by
Ricci = R
R
dx
:= R
dx ;
:
(4.202)
Then, we have
R
=R
+J
;
(4.203)
with
J
=D K
D K
+K
K
K
K
=r K
r K
K
K
+K
K
:
(4.204)
Observe that since the connection r is arbitrary, its Ricci tensor will be
not be symmetric in general. Then, since the Ricci tensor R
of D is
necessarily symmetric, we can split Eq.(4.203) into:
R[
]
= J[
)
=R
];
(4.205)
R(
+ J(
):
Now we specialize the above results for the case where the general connection r = D is the Levi-Civita connection of a bilinear form …eld
g 2 sec T20 M , i.e.,
= 0 and rg = 0. The results that we show next
generalize and clear up those found in the formulations of the gravitational
theory in a background metric space [361, 362, 157, 216].
First of all, note that the connection D plays with respect to the tensor
…eld g a role analogous to that played by the connection r with respect
to the metric tensor g and in consequence we shall have similar equations
relating these two pairs of objects. In particular, the strain of D with respect
to g equals the negative of the strain of r with respect to g, since we have:
S
=L
+L
b
=
(
+
d
)=S
;
4.8 Di¤erential Geometry in the Cli¤ord Bundle
147
where b
=
+
and d
= L
+L
denote the Killing
coe¢ cients of the frame with respect to the tensors g and g respectively.
Furthermore, in view of Eq.(4.197), we can write K
= 12 S
as:
K
=
=
1
g (r g
2
1
g (D g
2
+r g
+D g
r g
D g
)
):
(4.206)
Introducing the notation:
{=
s
det g
;
det g
(4.207)
we have the following relations:
K
g
K
g
K
1
1
g r g = g
2
2
1
=
D ({g );
{
1
r ({ 1 g ):
=
{ 1
=
D g
=
1
e ({);
{
(4.208)
Another important consequence of the assumption that r is a LeviCivita connection is that its Ricci tensor will then be symmetric. In view
of Eqs.(4.205), this will be achieved, if and only if, the following equivalent
conditions hold:
D K =D K ;
(4.209)
r K =r K :
4.8.9 Structure Equations II
With the results stated above, we can write down the structure equations
of the RCWS structure de…ned by the connection r in terms of the Riemannian structure de…ned by the metric g. For this, let us write Eq.(4.192)
in the form:
! =!
^ +w =!
^ +
+
;
(4.210)
and
with ! = L
, !
^ =
, w = K
,
= 12 T
1
= 2S
. Then, recalling Eq.(4.200) and the structure equations for
both the RCWS and the Riemannian structures, we easily conclude that:
w ^
=
;
w ^
=
;
Dw + w ^ w = J ;
(4.211)
148
4. Some Di¤erential Geometry
where D is the exterior covariant di¤erential (of indexed form …elds) associated to the Levi-Civita connection D of g. The third of these equations
can also be written as:
=J ;
w ^w
Dw
(4.212)
where D is the exterior covariant di¤erential (of indexed form …elds) associated to the connection r.
Now, the Bianchi identities for the RCWS structure are easily obtained
by di¤erentiating the above equations. We get:
(a) D
(b) D
(c) DJ
=J ^
w ^
;
=J ^ +w ^ ;
=R ^w
w ^R ;
(4.213)
or equivalently,
D
D
=J ^
=J ^
;
;
DJ = R ^ w
(4.214)
w ^R
:
4.8.10 D’Alembertian, Ricci and Einstein Operators
As we have seen in the Section 4.8.3 given the structure (M; D; g) we can
construct the Cli¤ord algebra C`(M; g) and the standard Dirac operator @j
given by (Eq. 4.152)
@j = d
:
(4.215)
We investigate now the square of the standard Dirac operator. We shall
see that this operator can be separated in some interesting parts that are
related to the D’Alembertian, Ricci and Einstein operators of (M; D; g).
De…nition 298 The square of standard Dirac operator @j is the operator,
Vp
Vp
2
@j = @j @j : sec
T M ,! sec C`(M; g) ! sec
T M ,! sec C`(M; g) given
by:
@j 2 = (d
)(d
) = (d + d):
(4.216)
2
We recognize that @j
is the Hodge Laplacian of the manifold introduced by (Eq. 4.92). On the other hand, remembering also that Eq.(4.148)
@j =
De ;
where f g is an arbitrary reference frame on the manifold and D is the
Levi-Civita connection of the metric g, we have:
@j
2
=(
De )(
=g
(De De
De ) =
(
De De + (De
De ) +
^
(De De
)De )
De ):
4.8 Di¤erential Geometry in the Cli¤ord Bundle
149
Then de…ning the operators:
@j @j = g (De De
@j ^ @j =
^ (De De
(a)
(b)
De );
De );
(4.217)
we can write:
= @j
2
or,
@j
2
= @j @j + @j ^ @j
(4.218)
= (@jy + @j ^)(@jy + @j ^)
= @jy @j ^ + @j ^ @jy :
(4.219)
Remark 299 It is important to observe that the operators @j @j and @j ^ @j do
not have anything analogous in the formulation of the di¤ erential geometry
in the Cartan and Hodge bundles.
Vr
Remark 300 Moreover we write for ! 2 sec
T M ,! sec C`(M; g),
@j @j ! and @j ^ @j ! to mean respectively (@j @j)! and (@j ^ @j)!. The parenthesis
will be included in a formula only if there is a risk of confusion.
The operator @j @j can also be written as:
h
1
@j @j = g
De De + De De
2
b
Applying this operator to the 1-forms of the frame f
@j @j
1
g
2
=
M
i
De
:
(4.220)
g, we get:
;
(4.221)
where:
M
=e (
)+e (
b
)
: (4.222)
The proof that an object with these components is a tensor is a consequence
of the following proposition:
Vr
1
Proposition 301 For every r-form …eld ! 2 sec
T M , ! = r!
! 1 ::: r 1 ^
r
, we have:
::: ^
@j @j ! =
1
g
r!
Proof. We have De ! =
D !
1 :::
r
= e (!
1 :::
D D !
1
r! D
r
!
1
1 :::
)
1 :::
r
1
!
1
r
^
^ ::: ^
2 :::
^
r
r
:
(4.223)
, with
r
r
!
1 :::
Observe moreover that we have
D D !
1 :::
r
= e (D !
1
1 :::
D !
r
)
2 :::
1
r
:::
D !
r
2 :::
D !
r
1 :::
r
1
r
1
:
150
4. Some Di¤erential Geometry
but
1
De De ! = De ( D ! 1 :::
r!
1
(e (D ! 1 ::: r )
r!
D ! 1 ::: r
r
1
r
^
1
1
)
^
r
D !
1
^
2 :::
^
)
r
r
:
Thus we conclude that:
(De De
De )! =
1
D D !
r!
Finally, multiplying this equation by g
the Eq.(4.223).
1
1 :::
r
^
^
r
:
and using the Eq.(4.217a), we get
In view of Eq.(4.223), we give the
= @j @j is called (covariant) D’Alembertian.
De…nition 302 The operator
Note that the D’Alembertian of the 1-forms
@j @j
=g
D D
=
1
g
2
can also be written as:
(D D
+D D
)
and therefore, taking into account the Eq.(4.221), we conclude that:
M
=
(D D
+D D
);
(4.224)
what proves our assertion that M
are the components of a tensor.
j
j
By its turn, the operator @ ^ @ can also be written as:
@j ^ @j =
1
2
h
De De
^
De De
c
De
Applying this operator to the 1-forms of the frame f
@j ^ @j
=
1
R
2
(
^
)
=
R
i
:
(4.225)
g, we get
;
(4.226)
where R
are the components of the curvature tensor of the connection
D. From Eq.(2.46), we get:
R
=R x
+R ^
:
The second term in the r.h.s. of this equation is identically null because
of the Bianchi identity given by Eq.( 4.213a) for the particular case of a
4.8 Di¤erential Geometry in the Cli¤ord Bundle
151
symmetric connection (
= 0). Using Eq.(2.35) and Eq.(2.37) we can
write the …rst term in the r.h.s. as:
1
R x = R
( ^ )x
2
1
R
y( ^ )
=
2
1
R
(
)
=
2
= R
=R
;
(4.227)
where R are the components of the Ricci tensor of the Levi-Civita connection D of g. Thus we have:
@j ^ @j
=R ;
(4.228)
where R = R
are the Ricci 1-forms of the manifold. Because of this
relation, we give the
De…nition 303 The operator @j ^ @j is called the Ricci operator of the manifold associated to the Levi-Civita connection D of g.
The proposition below shows that the Ricci operator can be written in
a purely algebraic way:
Proposition 304 The Ricci operator @j ^ @j satis…es the relation:
@j ^ @j = R ^ i + R
where (keep in mind ) R
:= g
R
^i i ;
= 21 g
R
(4.229)
^
.
1
Proof. The Hodge Laplacian of an arbitrary r-form …eld ! = r!
! 1 ::: r 1 ^
: : : ^ r is given by: (e.g., [77]— recall that our de…nition di¤ers by a sign
2
1 j2
from that given there) ! = @j ! = r!
(@ !) 1 ::: r 1 ^ : : : ^ r , with:
( !)
1 :::
r
= g D D ! 1 :::
X
( 1)p R: : p !
r
1 :::
p :::
r
p
2
X
( 1)p+q R
q
p
!
1 :::
p :::
q :::
r
;
(4.230)
p;q
p<q
where the notation means that the index was exclude of the sequence.
The …rst term in the r.h.s. of this expression are the components of the
D’Alembertian of the …eld !.
Now, recalling that i ! = y!, we obtain:
"
#
1 X
p
1
R ^i ! =
( 1) R p ! 1 ::: p ::: r
^
^ r
r! p
152
4. Some Di¤erential Geometry
and also,
R
^i i ! =
2
2 4X
( 1)p+q R
r! p;q
q
p
!
1 :::
p :::
q :::
r
p<q
3
5
1
^
^
r
:
Hence, taking into account Eq.(4.218), we conclude that:
(@j ^ @j)! = R ^ i ! + R
^ i i !;
for every r-form …eld !.
Observe that applying the operator given by the second term in the r.h.s.
of Eq.(4.229) to the dual of the 1-forms , we get:
R
^i i ?
=R
=
R
y(
y
^ ?(
^
?
= ?(R y(
^
))
)
^
(4.231)
));
where we have used the Eqs.(2.77). Then, recalling the de…nition of the
curvature forms and using the Eq.(2.36), we conclude that:
R
^
y
y?
=
2 ? (R
1
R
2
)=
2?G ;
(4.232)
where R is the scalar curvature of the manifold and the G may be called
the Einstein 1-form …elds. That observation motivate us to give the
De…nition 305 The Einstein operator of the Levi-Civita connection D of
g on the manifold M is the mapping : sec C`(M; g) ! sec C`(M; g) given
by:
1 1
=
? (R ^ i i ) ? :
(4.233)
2
Obviously, we have:
1
R :
(4.234)
=G =R
2
In addition, it is easy to verify that ? 1 (@j ^ @j)? = @j ^ @j and ?
i )? = R yj . Thus we can also write the Einstein operator as:
=
1
(@j ^ @j R yj ):
2
1
(R ^
(4.235)
Another important result is given by the following proposition:
Proposition 306 Let ! be the Levi-Civita connection 1-forms …elds in
an arbitrary moving frame f g on (M; D; g). Then:
(a)
(b)
@j @j
@j ^ @j
=
=
(@j !
(@j ^!
! ! )
! ^! ) ;
(4.236)
4.8 Di¤erential Geometry in the Cli¤ord Bundle
153
that is,
@j
2
=
(@j !
! ! ) :
(4.237)
Proof. We have
!
=(
(@j !
!
and !
=g
@j ! =
De (
=
(e (
=g
(e (
)
)
)
)
)
)=g
) (
. Then,
!v )
(e (
)
)
1
g (e (
2
= @j @j :
=
)+e (
)
b
)
Eq.(4.236b) is proved analogously.
Exercise 307 Show that
R is the curvature scalar.
(
^
)yR
= R(
^
) R
= R; where
4.8.11 The Square of a General Dirac Operator
Consider the structure (M; r; g), where r is an arbitrary Riemann-CartanWeyl connection and the Cli¤ord algebra C`(M; g). Let us now compute
the square of the (general) Dirac operator @ = tr(uru ). As in the earlier
section, we have, by one side,
@ 2 = (@y + @^)(@y + @^)
= @y@y + @y@^ + @^@y + @@^@^
and we write @y@y
@ 2 y, @ ^ @^
@ 2 ^ and
L+ = @y@ ^ +@ ^ @y;
(4.238)
@ 2 = @ 2 y@ + L+ @ + @ 2 ^ .
(4.239)
so that:
The operator L+ when applied to scalar functions corresponds, for the
case of a Riemann-Cartan space, to the wave operator introduced in [313].
Obviously, for the case of the standard Dirac operator, L+ reduces to the
usual Hodge Laplacian of the manifold, which preserve graduation of forms.
154
4. Some Di¤erential Geometry
Now, a similar calculation for the product @ @ of the Dirac derivative and
the Dirac coderivative yields:
@ @ = @ y@y + L + @ ^ @^;
(4.240)
L = @y@ ^ +@ ^ @y .
(4.241)
with
On the other hand, we have also:
=(
re )(
=g
(re re
re ) =
L
(
re re + (re
re ) +
^
)re )
(re re
L
re )
and we can then de…ne:
@ @
@^@
= g (re re
L re )
=
^ (re re
L re )
(4.242)
@ 2 = @@ = @ @ + @ ^ @ .
(4.243)
in order to have:
The operator @ @ can also be written as:
1
2
1
= g
2
(re re
@ @=
L
re ) +
[re re + re re
1
2
(L
(re re
+L
L
re )
)re ]
or,
@ @=
1
g
2
(re re + re re
b
re )
s re ;
(4.244)
where s has been de…ned in Eq.(4.191).
By its turn, the operator @ ^ @ can also be written as:
@^@ =
1
2
=
1
2
^
(re re
L
^
[re re
r e re
re ) +
1
2
^
(L
(re re
L
L
re )
(4.245)
)re ]
or,
@^@ =
1
2
^
(re re
r e re
c
re )
re :
(4.246)
Exercise 308 Prove that the Ricci and Einstein operators are (1; 1)-extensor
V1
…elds on a Lorentzian spacetime, i.e., for any A 2 sec
T M ,! sec C`(M; g)
we have
@ ^ @ A = @ ^ @ (A
A=
(A
)=A @^@
)=A
:
;
(4.247)
4.9 Some Applications
155
Solution: We prove the …rst formula, since after proving it the second
one is obvious. We choose for simplicity an orthonormal cobasis f a g for
T M dual to the basis fea g for T M , such that [ea ; eb ] = cdab ed . Let r
be a connection on a Riemann-Cartan-Weyl spacetime, such that rea eb =
Ldab ed . Recalling (Eq.(4.245)) we have
@^@ A
1 a
=
^
2
1 a
^
=
2
1
= T dab
2
b
f[ea ; eb ](Ak )
b
fcdab
a
^
b
Ldab
Ldab ed (Ak )
Ldba g
+ Ak @ ^ @
k
k
Ldba ed (Ak )g
+ Ak @ ^ @
= Ak @ ^ @
k
k
g + Ak @ ^ @
k
k
;
since for a Lorentzian spacetime the torsion tensor (with components T dab )
is null.
Exercise 309 Show that for any A 2 sec
V1
T M ,! sec C`(M; g) we have
@ ^ @ A = @j ^ @j A + J
where A := A
given by
, A := g
g
A and J
A;
:= g
(4.248)
J
, where J
is
4.9 Some Applications
4.9.1 Maxwell Equations in the Hodge Bundle
The system of Maxwell equations has many faces27 . Here we show how
to express that system of equations in the Hodge bundle and then in the
Cli¤ord bundle. To start, let (M; g; g ) be an oriented Lorentzian manifold.
Maxwell equations on (M; g; g ) refers to an exterior system of di¤erenV2
tial equations given by a closed 2-form F 2 sec
T M and a exact 3-form
V3
V2
J e 2 sec
T M . Then there exists G 2 sec
T M such that
dF = 0 and dG =
J e:
(4.249)
It is postulated that in vacuum there is a relation between G and F (said
constitutive relation) given by
G = ?F:
2 7 Besides
(4.250)
the ones presented in this chapter, others will be exhibited in Chapter 13.
156
4. Some Di¤erential Geometry
V1
In that case putting J e = ?Je , Je 2 sec
T M and taking into account
Eq.(4.91) we can write the system 4.249 as28
dF = 0 and F =
Je :
(4.251)
F is called the Faraday …eld and Je is called the electric current.
4.9.2 Charge Conservation
Of course, Je = 0, which means that charge is conserved. Indeed, let C3
be a three dimensional volume contained in a space slice, i.e., in a spacelike
surface. Then the electric ‡ux contained in C2 = @C3 is
Z
Z
Z
Q=
?Je =
dG =
?F:
(4.252)
C3
C3
@C3
It is an empirical fact that all observable free charges are integer multiple
of the electron charge. This phenomenon is called charge quantization. 0n
the other hand consider a 4-volume C4 with boundary given by @C4 =
(2)
(1)
(2)
C3
C3 + S where with the condition Je jS = 0 and where C3 and
(1)
C3 are three dimensional volumes contained in two di¤erent space slices.
Then,
Z
Z
Z
?Je =
dG =
d2 G = 0;
(4.253)
@C4
from where it follows that
@C4
Z
(1)
?Je =
C3
C4
Z
(2)
?Je :
(4.254)
C3
We postulate that F is closed but it may be (eventually) not exact. In
that case it may have period integrals according to de Rham theorem, i.e.,
Z
F = g(i) ;
(4.255)
(i)
z2
(i)
where z2 2 H2 (M ) are cycles. It seems to be an empirical fact that all
g(i) = 0, at least for cycles in the region of the universe where men already
did experiments. This means that F is exact,
V1 i.e., it is possible to de…ne
globally a di¤erentiable potential A 2 sec
T M such that F = dA. This
also means that there are no magnetic monopoles in nature29 . Indeed, if z2
is a cycle (a closed surface) then we have
Z
Z
F =
dA = h@z2 ; Ai = h0; Ai = 0:
(4.256)
z2
z2
2 8 Thirring [422] said that the two equations in Eq.(4.251) is the XX Century presentation of Maxwell equations.
2 9 See however the news in [314] where it is claimed that magnetic monopoles have
been observed in a synthetic magnetic …eld.
4.9 Some Applications
157
4.9.3 Flux Conservation
Of V
course, A is only de…ned modulus a gauge, i.e., A + A0 , with A0 2
1
sec
T M a closed form. The period integrals of A0 according to de Rham
theorem are
Z
A0 = (i) :
(4.257)
(i)
zi
Now, it is an empirical fact that (i) is quantized in some (but not all )
physical systems, like, e.g., in superconductors [171]. The phenomenon is
then called ‡ux quantization. In appropriate units
Z
A0 = nh=2e;
(4.258)
z1
where n is an integer and h is Planck constant and e is the electron charge.
Note also that from J e = dG in Eq.(4.249) it follows that G is de…ned also only modulus a closed form G0 . The period integrals of G0 may
eventually correspond to topological charges. Another possibility of having
‘charge without charge’coming from statistical distributions of quantized
‡ux loops has been investigated in [187, 188]. We shall not discuss these
interesting issues in this book.
4.9.4 Quantization of Action
Finally we mention the following. As we shall see in Chapter 7 the Lagrangian density of the electromagnetic …eld in free space is given by
L(A) =
1
F ^ ?F:
2
(4.259)
1
dK:
2
(4.260)
Calling K = A ^ ?F , we can write
L(A) =
Now, it seems an empirical fact that action is quantized, i.e., we have
Z
a=
L(A)
C
Z 4
=
K = nh:
(4.261)
C3 =@C4
R
Remark 310 We observe that C3 =@C4 K has been introduced by Kiehn
(see [192]). However he called A ^ ?F the topological spin, which is not a
good name (and identi…cation of observable) in our opinion. The reason is
that according to the Lagrangian formalism (see Chapter 8, Eq.(8.124))30
3 0 See
also [441]
158
4. Some Di¤erential Geometry
the spin density is proportional to A ^ F . This result and the other period
integrals discussed above suggests that quantization may be linked to topology in a way not suspected by contemporary physicists. On this issue, see
also [305].
4.9.5 A Comment on the Use of de Rham Pseudo-Forms and
Electromagnetism
Besides the forms we have been working until now, in a famous book, de
Rham [100] introduces also the concept of impair forms31 in a n-dimensional
manifold M , which is essential for the formulation of a theory of integration
in a non orientable manifold.
De…nition 311 An impair p-form in M is a pair of p-forms such that if
its representative in a given A M in a cobasis f i g for T U (U A) is
declared as being
!jU =
1
!i :::i
p! 1 p
i1
^
then its representative !jV in A
for T V (V \ U A) is
!jV =
1
!j :::j
p! 1 p
i1
with
!j1 :::jp =
det
det
^
i
j
i
j
2 sec
^p
T M
M in a cobasis f i g,
V
^
ip
^
ip
!i1 :::ip
2 sec
i1
j1
^p
T M;
i1
j1 :
i
=
i j
j ;
(4.262)
(4.263)
The introduction of impair forms leads to the question of exterior (and
interior) multiplication of forms of di¤erent parities (i.e., even and odd).
The rule introduced by de Rham [100] is that the product of two forms of
the same parity is a form, whereas the product of two forms of di¤erent parities is an impair form. Also de Rham introduces the rule that application
of the di¤erential operator d to a form preserves its parity.
n
X
V
Vp
We can verify that if we denote by impair T M =
impair T M the
p=0
real vector space of the pseudo formsVwe can give
V a structure of associative
algebra to the (exterior) direct sum T M
impair T M equipped with
the exterior product satisfying the de Rham rules mentioned above.
Having introduced the concept of de Rham pseudo forms we call the
reader’s attention to the following remarks.
3 1 Also
called by some authors pseudo forms.
4.9 Some Applications
159
Remark 312 In our brief presentation above of Maxwell equations
V1 we introduced the electromagnetic current as Je = ? 1 J e , Je 2 sec
T M.
Since until that point we have not introduced the concept of impair forms
its is clear that we supposed that J e is 3-form. This certainly means that
the theory as presented presupposes that we use always bases with the same
orientation in order to calculate the charge in a certain three dimensional
volume contained in a given space slice (Eq.(4.253)). The use of bases with
the same orientation presupposes that spacetime is an orientable manifold.
As will be discussed in Chapter 7 orientability of a spacetime manifold is
a necessary condition for the existence of spinor …elds. Since these objects
seems to be an essential tool for the understanding of the world we live in,
we restrict all our considerations to orientable manifolds. Eventually, if is
discovered some of these days that our universe cannot be represented by
an orientable manifold, then it will be necessary to study deeply the theory
of impair forms.
Remark 313 If the spacetime manifold is orientable we do not need to
consider, as some authors claim (e.g., [192, 305]) that J e and G must be
considered as pseudo forms. A thoughtful discussion of this issue may be
found in [331].
4.9.6 Maxwell Equation in the Cli¤ord Bundle
Let now, (M; g; D; g ; ") be a Lorentzian spacetime and let C`(M; g) be the
Cli¤ord bundle of di¤erential forms. Since D is the Levi-Civita connection
of g we know
Vp(Eq.(4.152)) that the action of the Dirac operator @ on
any P 2 sec
T M ,! C`(M; g) is @P = (d
)P . So, let us suppose
that the Faraday …eld
and
the
electric
current
are
sections
of the Cli¤ord
V2
V1
bundle, i.e., F 2 sec
T M ,! C`(M; g), Je 2 sec
T M ,! C`(M; g).
In that case, it is licit two sum the equations dF = 0 and F = Je , which
according to Eqs.(4.251) represent the system of Maxwell equations in the
Hodge bundle. We get, of course, the single equation
@F = Je ;
(4.264)
which we be call Maxwell equation. Parodying Thirring [422] we may say
that Eq.(4.264) the XXI century representation of Maxwell system of equations.
Exercise 314 Show that in Minkowski spacetime (M; ; D; ; ") (De…nition 276) Eq.(4.264) is equivalent to the standard vector form of Maxwell
equations, that appears in elementary electrodynamics textbooks.
Solution:
We recall (see Table 2 in Chapter 3) that for any x 2 M , C`(Tx M; x ) '
R1;3 ' H(2), is the so called spacetime algebra. The even elements of R1;3
160
4. Some Di¤erential Geometry
close a subalgebra called the Pauli algebra. That subalgebra is denoted
by R01;3 ' R3;0 ' C(2). Also, H(2) is the algebra of the 2 2 quaternionic
matrices and C(2) is the algebra of the 2 2 complex matrices. As in section
3.9.1 a convenient isomorphism R01;3
R3;0 is easily exhibited. Choose a
global orthonormal tetrad coframe f g,
= dx ,
= 0; 1; 2; 3, and let
f g be the reciprocal tetrad of f g, i.e.,
=
. Now, put
i
=
i 0;
0 1 2 3
i=
5
=
:
(4.265)
Observe thatpi commutes with bivectors and thus
S acts like the imagi1 in the subbundle C`0 (M; ) = x2M C`0 (Tx M; x ) ,!
nary unity i =
C`(M; ), which we call Pauli bundle. Now, the electromagnetic …eld is repV2
resented in C`(M; ) by F = 21 F
^
2 sec T M ,! sec C`(M; ))
with
0
1
0
E1
E2
E3
B E1
0
B3 B2 C
C ;
F =B
(4.266)
@ E2 B3
0
B1 A
E3
B2 B1
0
where (E1 ; E2 ; E3 ) and (B1 ; B2 ; B3 ) are the usual Cartesian components of
the electric and magnetic …elds. Then, as it is easy to verify we can write
~ + iB;
~
F =E
P3
P3
~ =
~
with , E
i=1 Ei i , B =
i=1 Bi i .
For the electric current density Je =
0 Je
=
0
~j =
(4.267)
+ Ji
i
we can write
J i i:
(4.268)
For the Dirac operator we have
3
0@ =
X
@
+
0
@x
i=1
i @i
=
@
+ r:
@t
Multiplying both members of Eq.(4.264) on the left by
=
0 Je ;
@
~ + iB)
~ =
+ r)(E
@t
~j
0 @F
(
(4.269)
0
we obtain
(4.270)
From Eq.(4.270) we obtain
~ + i@0 B
~ +r E
~ +r^E
~ + ir B
~ + ir ^ B
~ =
@0 E
~j:
(4.271)
~ 2 C`0 (M; ) ,! C`(M; ) we de…ne the rotational
For any ‘vector …eld’ A
operator r by
~ = ir ^ A:
~
r A
(4.272)
4.9 Some Applications
161
This relationP
follows once we realize
P3 that the usual vector product of two
3
vectors ~a = i=1 ai i and ~b = i=1 bi i can be identi…ed with the dual
of the bivector ~a ^~b through the formula ~a ~b = i~a ^~b. Finally we obtain
from Eq.(4.271) by equating terms with the same grades (in the Pauli
subbundle )
(a)
(c)
~ = ;
r E
~
~ = 0;
r E + @0 B
(b)
(d)
r
~ @0 E
~ = ~j;
B
~
r B = 0;
(4.273)
which we recognize as the system of Maxwell equations in the usual vector
notation.
We just exhibit three equivalent presentations of Maxwell systems of
equations, namely Eqs.(4.251), Eq.(4.264) and Eqs.(4.273). They are some
of the many faces of Maxwell equations. Other faces exist as we shall see
in Chapter 11.
4.9.7 Einstein Equations and the Field Equations for the
a
As, it is the case of Maxwell equations, also Einstein equations have many
faces. Here we exhibit an interesting one which is possible once we have at
our disposal the Cli¤ord bundle formalism. So, let now (M; g; D; g ; ") be a
Lorentzian spacetime (De…nition 275) modelling a gravitational …eld in the
general theory of Relativity [367]. Let fea g be an arbitrary orthonormal
basis of T U (a tetrad32 ) and f b g of T M its dual basis (a cotetrad), with
a; b = 0; 1; 2; 3. We recall that Einstein’s equations relating the distribution
of matter energy represented by the energy-momentum tensor T = Tba b
ea 2 sec T11 U sec T11 M can be written (in appropriated units)
Rba
1
2
a
bR
Tba ;
=
(4.274)
where Rba is the Ricci tensor and R is the scalar curvature. Multiplying both
members of Eq.(4.274) by b and taking into account Eq.(4.228) de…ning
the Ricci 1-forms in terms of the Ricci operator @ ^ @ (with @ = a Dea )
we can write after some trivial algebra
@^@
a
+
T
2
a
=
T a;
(4.275)
V1
where33 T a = Tba b 2 sec
T M ,! sec C`(M; g) are the energy-momentum
1-form …elds and T = Taa .
3 2 We shall see in Chapter 6 that any Lorentzian spacetime admmiting spinor …elds
must have a global tetrad.
3 3 Sometimes in the written of some formulas in the next chapters it is convenient to
use the notation T a = T a .
162
4. Some Di¤erential Geometry
Now, taking into account Eq.(4.218) and Eq.(4.219) we can write
@ @
a
+ @ ^ (@
a
) + @y(@ ^
a
1
)+ T
2
a
=
T a:
(4.276)
Now, let fx g be the coordinate functions of a local chart of the maximal
atlas of M covering U M . When a is an exact di¤erential, and in that
case we write a 7!
= dx and if the coordinate functions are harmonic
[137], i.e.,
= @y = g
= 0, Eq.(4.276) becomes
1
R
2
where
=T ;
(4.277)
is the covariant D’Alembertian operator (De…nition 302).
4.9.8 Curvature of a Connection and Bending. The Nunes
Connection of S 2
Consider the manifold S 2 = fS 2 nnorth pole + south poleg
R3 , it is
an sphere of radius R = 1 excluding the north and south poles. Let g 2
sec T20 S 2 be a metric …eld for S 2 , which is the pullback on it of the metric
of the ambient space R3 . Now, consider two di¤erent connections on S 2 ,
D— the Levi-Civita connection— and rc , a connection— here called the
Nunes34 (or navigator) connection35 — de…ned by the following parallel
transport rule: a vector is parallel transported along a curve, if at any
x 2 S 2 the angle between the vector and the vector tangent to the latitude
line passing through that point is constant during the transport (see Figure
4.5 ).
3 4 Pedro Salacience Nunes (1502–1578) was one of the leading mathematicians and cosmographers of Portugal during the Age of Discoveries. He is well known for his studies
in Cosmography, Spherical Geometry, Astronomic Navigation, and Algebra, and particularly known for his discovery of loxodromic curves and the nonius. Loxodromic curves,
also called rhumb lines, are spirals that converge to the poles. They are lines that maintain a …xed angle with the meridians. In other words, loxodromic curves directly related
to the construction of the Nunes connection. A ship following a …xed compass direction
travels along a loxodromic, this being the reason why Nunes connection is also known as
navigator connection. Nunes discovered the loxodromic lines and advocated the drawing
of maps in which loxodromic spirals would appear as straight lines. This led to the celebrated Mercator projection, constructed along these recommendations. Nunes invented
also the Nonius scales which allow a more precise reading of the height of stars on a
quadrant. The device was used and perfected at the time by several people, including Tycho Brahe, Jacob Kurtz, Christopher Clavius and further by Pierre Vernier who in 1630
constructed a practical device for navigation. For some centuries, this device was called
nonius. During the 19th century, many countries, most notably France, started to call it
vernier. More details in http://www.mlahanas.de/Stamps/Data/Mathematician/N.htm.
3 5 Some authors call the Columbus connection the Nunes connection. Such name is
clearly unappropriated.
4.9 Some Applications
163
FIGURE 4.5. Characterization of the Nunes Connection
Exercise 315 (i) Show that the structure (S 2 ; g; D) is a Riemann geometry of constant curvature and;
(ii) that the structure (S 2 ; g; rc ) is a teleparallel geometry, with zero
Riemann curvature tensor, but non zero tensor tensor.
Solution:
The …rst part of the exercise is a standard one and can be found in
many good textbooks on di¤erential geometry. Here, we only show (ii).
We clearly see from Figure 4.5(a) that if we transport a vector along the
in…nitesimal quadrilateral pqrs composed of latitudes and longitudes, …rst
starting from p along pqr and then starting from p along psr the parallel
transported vectors that result in both cases will coincide. Using the definition of the Riemann curvature tensor, we see that it is null. So, we see
that S 2 considered as part of the structure (S 2 ; g; rc ) is ‡at!
Let (x1 ; x2 ) = (#; ') 0 < # < , 0 < ' < 2 , be the standard spherical
coordinates of a S 2 or unitary radius, which covers all the open set U which
is S 2 with the exclusion of a semi-circle uniting the north and south poles.
Introduce …rst the coordinate bases
f@ = @=@x g; f
= dx g
(4.278)
for T U and T U .
Introduce next the orthonormal bases fea g; f a g for T U and T U with
1
@2 ;
sin x1
= sin x1 dx2 :
e1 = @1 , e2 =
1
= dx1 ,
2
(4.279)
(4.280)
Then,
[ei ; ej ] = ckij ek ;
c212 =
c221 =
(4.281)
cot x1 ;
164
4. Some Di¤erential Geometry
and
g = dx1
=
1
dx1 + sin2 x1 dx2
1
+
2
2
dx2
:
(4.282)
Now, it is obvious from what has been said above that our teleparallel
connection is characterized by
rcej ei = 0:
(4.283)
Then taking into account the de…nition of the curvature operator (de…nition 4.104), we have
R( a ; ek ; ei ; ej ) =
a
h
rcei rcej
rcej rcei
i
rc[ei ;ej ] ek = 0:
(4.284)
Also, taking into account the de…nition of the torsion operation (de…nition 4.103) we have
(ei ; ej ) = rcej ei
rcei ej
[ei ; ej ]
= [ei ; ej ];
(4.285)
and T 221 = T 212 = cot #.
If you still need more details, concerning this last result, consider Figure
4.5(b) which shows the standard parametrization of the points p; q; r; s
in terms of the spherical coordinates introduced above. According to the
geometrical meaning of torsion, we determine its value at a given point by
calculating the di¤erence between the (in…nitesimal)36 segments (vectors)
pr1 and pr2 determined as follows. If we transport the vector pq along ps
1
we get (recalling that R = 1) the vector ~v = sr1 such that jg(~v ; ~v )j 2 =
sin #4'. On the other hand, if we transport the vector ps along pr we get
the vector qr2 = qr. Let w
~ = sr. Then,
jg(w;
~ w)j
~ = sin(#
4#)4' ' sin #4'
cos #4#4';
(4.286)
1 @
), u = jg(~u; ~u)j = cos #4#4'
sin # @'
(4.287)
Also,
~u = r1 r2 =
u(
Then, the (Riemann-Cartan) connection rc of the structure (S 2 ; g; rc ; g )
has a non null torsion tensor . Indeed, the component of ~u = r1 r2 in the
3 6 This wording, of course, means that this vectors are identi…ed as elements of the
appropriate tangent spaces.
4.9 Some Applications
165
'
direction @=@' is precisely T#'
4#4'. So, we get (recalling that rc @j @i =
k
ji @k )
T '#' =
'
#'
'
'#
=
cot :
(4.288)
c
To complete the exercise we must show that r g = 0. We have,
0 = rcec g(ei ; ej ) = (rcec g)(ei ; ej ) + g(rcec ei ; ej ) + g(ei ; rcec ej )
= (rcec g)(ei ; ej ):
(4.289)
Remark 316 This exercise, shows clearly that we cannot mislead the Riemann curvature tensor of a connection with the fact that the manifold where
that connection is de…ned may be bend as a surface in an Euclidean manifold where it is embedded. Bending is characterized by the shape operator 37
(a fundamental concept in di¤ erential geometry that will be presented in
Chapter 5 using the Cli¤ ord bundle formalism). Neglecting this fact may
generate a lot of wishful thinking. Taking it into account may suggest new
formulations of the gravitational …eld theory as we will show in Chapter
11.
4.9.9 “Tetrad” Postulate? On the necessity of Precise
Notations
Given a di¤erentiable manifold M , let X; Y 2
sec T M be vector …elds
L1
and C 2 sec T M a covector …eld . Let T M = r;s=0 Tsr M be the tensor
bundle of M and P 2 sec T M a general tensor …eld. We already introduced
in M a rule for di¤erentiation of tensor …elds, namely the Lie derivative.
Taking into account Appendix A4 we introduce three covariant derivatives
operators, r+ ;r and r, de…ned as follows:
r+ : sec T M
(X; Y ) 7!
r+
XY
r : sec T M
(X; C) 7! rX C;
r : sec T M
(X; P) 7! rX P;
;
sec T M ! sec T M;
sec T M ! sec T M;
sec M ! sec T M;
(4.290)
(4.291)
(4.292)
Each one of the covariant derivative operators introduced above satisfy
the following properties: Given, di¤erentiable functions f; g : M ! R,
vector …elds X; Y 2 sec T M and P; Q 2 sec T M we have
3 7 See,
e.g., [173, 286, 397, 358] for details.
166
4. Some Di¤erential Geometry
rf X+gY P = f rX P+grY P;
rX (P + Q) = rX P + rX Q;
rX (f P) = f rX (P)+X(f )P;
rX (P
Q) = rX P
Q+P
rX Q:
(4.293)
The absolute di¤ erential of P 2 sec Tsr M is given by the mapping
r
r: sec Tsr M ! sec Ts+1
M;
rP(X;X 1 ; :::; X s ;
1 ; :::;
= rX P(X 1 ; :::; X s ;
X 1 ; :::; X s 2 sec T M;
1 ; ::: r
r)
1 ; :::;
r );
2 sec T M:
(4.294)
To continue we must give the relationship between r+ ;r and r. Let
U
M and consider a chart of the maximal atlas of M covering U
coordinate functions fx g. Let g 2 sec T20 M be a metric …eld for T M
and g 2 sec T02 M the corresponding metric for T M (as introduced previously). Let f@ g be a basis for T U , U
M and let f
= dx g be
the dual basis of f@ g. The reciprocal basis of f g is denoted f g, and
. Introduce next a set of di¤erentiable functions
we have g( ; ) =
ha ; hb : U ! R such that :
ha q b =
b
a,
ha ha =
.
(4.295)
De…ne
eb = hb @
where the set fea g is an orthonormal basis38 for T U , i.e., g(ea; eb ) = ab .
The reciprocal basis of fea g is fea g and g(ea; eb ) = ba The dual basis of T U
is f a g, with a = ha dx and g( a ; b ) = ab . Also, f b g is the reciprocal
basis of f a g, i.e. g( a ; b ) = ba . It is trivial to verify the formulas
g
ab
= ha hb
ab ,
= ha hb g ;
g
ab
= ha hb
ab
a b
:
=h h g
;
(4.296)
The connection coe¢ cients associated to the respective covariant derivatives in the respective bases are denoted as:
38 P
(M )
SOe
1;3
is the orthonormal frame bundle (see Appendix A.1.2.)
4.9 Some Applications
r+
@ @ =
c
r+
ea eb = ! ab ec ;
@ ; r@ @ =
b
r+
ea e =
r@ dx =
b
re a
!abc =
d
ad ! bc
=
c
re a
!cba ;
b
! ba c
=
!cab
bk
=
=
b
; r@
!kal
a
!b a
=
c
(4.297)
c
r+
@ eb = ! b ec ;
! bac ec ;
dx ; r@
! bac
=
@ ;
(4.298)
;
(4.299)
;
(4.300)
;
cl
167
(4.301)
;
! ba c
=
! cab
etc...
(4.302)
(4.303)
To understood how r works, consider its action, e.g., on the sections of
T11 M = T M T M . For that case, if X 2 sec T M , C 2 sec T M , we have
that
r = r+ IdT M + IdT M r ;
(4.304)
and
r(X
C) = (r+ X)
C +X
r C:
(4.305)
The general case, of r acting on sections of T M is an obvious generalization of the previous one, and details are left to the reader.
For every vector …eld V 2 sec T U and a covector …eld C 2 sec T U we
have
+
r+
@ V = r@ (V @ );
r@ C = r@ (C
)
(4.306)
and using the properties of a covariant derivative operator introduced
above, r+
@ V can be written as:
+
+
r+
@ V = r@ (V @ ) = (r@ V ) @
= (@ V )@ + V r+
@ @
=
@V
@x
@ := (r+ V )@ ;
+V
where it is to be kept in mind that the symbol r+ V
for
r+ V := (r+
@ V) :
(4.307)
is a short notation
(4.308)
Also, we have
r@ C = r@ (C
=
@C
@x
:= (r C )
) = (r@ C)
C
;
;
(4.309)
168
4. Some Di¤erential Geometry
where it is to be kept in mind that39 that the symbol r C
notation for
r C := (r@ C) :
is a short
(4.310)
Remark 317 When there is no possibility of confusion, we shall use only
the symbol r to denote any one of the covariant derivatives introduced
above. However, the standard practice of many Physics textbooks of representing, r+ V and r+ V by r V should be avoided whenever possible
in order to not produce misunderstandings (see Exercise below ).
Exercise 318 Calculate r ha := (r@ a ) = (r@ ha @ ) and r+ ha :=
+ b
a
a
a
+ a
(r+
@ @ ) = (r@ h eb ) . Show that in general r h 6=r h 6= 0 and that
@ ha + ! a b hb
a
b
bh
= 0:
(4.311)
Exercise 319 De…ne the object
e = ea
a
= ha @
dx 2 sec T11 M;
(4.312)
which is clearly the identity endomorphism acting on sections of T U . Show
that
a
a
b
r ha := (r@ e) = @ ha + ! a b hb
(4.313)
b h = 0:
Remark 320 Eq.(4.313) is presented in many textbooks (see., e.g., [69,
156, 364]) under the name ‘tetrad postulate’. In that books, since authors do
not distinguish clearly the derivative operators r+ ; r and r, Eq.(4.313)
becomes sometimes misunderstood as meaning r ha or r+ ha , thus generating a big confusion. For a discussion of this issue see [353].
3 9 Recall
C
:
that other authors prefer the notations (r@ V ) := V: and (r@ C)
. What is important is always to have in mind the meaning of the symbols.
:=
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5
Cli¤ord Bundle Approach to the
Di¤erential Geometry of Branes
5.1 Introduction
In this Chapter we use the Cli¤ord bundle formalism (CBF) developed
previously in order to analyze the Riemann or the Lorentzian geometry of
an orientable submanifold M (dim M = m) living in a manifold M such
that M ' Rn is equipped with a semi-Riemannian metric g (with signature
(p; q) and p + q = n) and its Levi-Civita connection D.
In order to achieve our objectives and exhibit some nice results that
are not well known (and which, e.g., may possibly be of interest for the
description and formulation of branes theories [229] and string theories [37])
we …rst recall in Section 5.2 how to formulate using the CBF the intrinsic
di¤erential geometry of a structure hM; g; ri where r is a general metric
compatible Riemann-Cartan connection, i.e., rg = 0 and the Riemann
and torsion tensors of r are non null. We recall that we introduced in
Chapter 4 (once we …x a gauge in the frame bundle) a (1; 2)-extensor …eld
V1
V2
! : sec T M !
T M closed related with the connection 1-forms
which permits to write a very nice formula for the covariant derivative (see
Eq.(5.30)) of any section of the Cli¤ord bundle of the structure hM; g; ri.
V1
V2
It will be shown that ! is related to S : sec T M !
T M the shape
operator biform of the manifold.
170
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
Then in Section 5.3, we suppose that M is a proper submanifold1 of M
which i : M 7! M the inclusion map. Introducing
natural global coordinates
Pn
i
i
(x1 ; :::; xn ) for M ' Rn we write g =
dxj
ij dx
i;j=1 ij dx
dxj and equip M with the pullback metric g := i g. We then …nd the
relation between the Levi-Civita connection D of g and D, the Levi-Civita
connection of g. We suppose that g is non degenerated of signature (p; q)
with p + q = m.
In this Chapter C`(M; g) and C`(M; g) denote respectively the Cli¤ord
bundles of di¤erential forms of M and M . Moreover, in what follows
g=
Pn
ij
i;j=1
@
@xj
@
@xi
ij
@
@xi
@
@xj
(5.1)
is the metric of the cotangent bundle. The Dirac operators2 of C`(M; g)
and C`(M; g) will be denoted by @ and @. Let l = n m and
fe1 ; e2 ; :::; em ; em+1 ; :::; em+l g
an orthonormal basis for T U (U
M ) such that
fe1 ; e2 ; :::; em g = fe1 ; e2 ; :::; em g
is a basis for T U (U
(5.2)
(5.3)
U ) and if
f
1
;
2
; :::;
m
;
m+1
; :::;
m+l
g
(5.4)
is the dual basis of the fei g we have that f 1 ; 2 ; :::; m g = f 1 ; 2 ; :::; m g
is a basis for T U dual to the basis fe1 ; e2 ; :::; em g of T U . We have, as well
known from Chapter 4
Pn
Pm i
i
@ = i=1 i Dei = i Dei , @ =
Dei ;
(5.5)
i=1 Dei =
Remark 321 Take notice that the bold face sub and superscripts are used
to denote bases fei g and f i g of the tangent and cotangent space of M .
This notation is conveniently used in what follows.
@
The dual basis of the natural coordinate basis f @x
i g is denoted in what
i
i
i
follows by f g where, of course,
= dx . Moreover, fe1 ; e2 ; :::; em g denotes the reciprocal frame of fei g, i.e., g(ei ; ej ) = ji and by f i g the
1 By a proper (or regular) submanifold M of M we mean a subset M
M such that
for every x 2 M in the domain of a chart (U; ) of M such that : M \ U ! Rn flg;
(x) = (x1 ; :::; xn ; l1 ; :::; lm n ), where l = (l1 ; :::; lm n ) 2 Rn m .
2 Take notice that the Dirac operators used in this Chapter are acting on sections
of the Cli¤ord bundle. It is not to be confused with the Dirac operator which acts on
sections of the spinor bundle (see details in [278]). According to [328]. this last operator
can also be used to probe the topology of the brane.
5.1 Introduction
reciprocal basis off i g, i.e., g( i ;
j)
i
:=
j
i
j
i
=
i
171
Moreover, take into
account that for i; j = 1;
; m it is g( ; j ) = g( ; j ) So we will write
also g( i ; j ) = i j = ji . The representation of the Dirac operator @ in
Pn i @
the natural coordinate basis of M is of course, @ = i=1 @x
i . Note that
= 0, i.e., the f m+1 ; :::;
= 0; :::; m+l
we have m+1
M
M
vector …eld a 2 sec T U and d = 1; :::; m + l we have
m+d
M
m+l
g for any
(a) = 0:
We denote moreover
d=@
M
:=
i
i
@=
Pm
i=1
i
Dei =
i
Dei
(5.6)
the restriction of @ on the submanifold M . The projection operator P
(an extensor …eld) on M and the shape operator S = dP: sec C`(M; g) !
V1
sec C`(M; g) and shape biform operator of the manifold M , S : sec T M 7!
V2
m
T M; S(a) := (a dIm )Im1 (where g = Im = 1 2
is the volume
3
form on U M ) are fundamental objects in our study. The de…nition of
those objects are given in Section 5.3 and the main algebraic properties of
P; S and S besides all identities necessary for the present paper are given
and proved at the appropriate places.
Section 5.4 is dedicated to …nd several equivalent expressions for the
curvature biform R(u; v) in terms of the shape operator. We recall that the
square of the Dirac operator @ acting on sections of the Cli¤ord bundle has
two di¤erent decompositions, namely
@2 =
(d + d) = @ @ + @ ^ @;
(5.7)
where d and are respectively the exterior derivative and the Hodge coderivative and @ @ + @ ^ @ are respectively the covariant Laplacian and the
Ricci operator. The explicit forms of @ @ and @ ^ @ are given in Chapter
4 where it is shown moreover that @ ^ @ is an extensorial operator and
@^@
i
= Ri ;
(5.8)
V1
where the objects Ri = Rji j 2 sec T M ,! sec C`(M; g) with Rji the
components of the Ricci tensor associated with D are called the Ricci 1form …elds.
One of the objectives of the present Chapter is to give (Section 5.5) a
detailed proof of the remarkable equation
@ ^ @ (v) =
3 The
g
S2 (v);
(5.9)
1 2
volume g for on U
M will be denoted by In =
m m+1
m+l = I
on U
M will be denoted In = 1 2
m
m
. The volume form
m+l .
m m+1
172
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
which says that the shape biform operator is the negative square root of
the Ricci operator 4 . We moreover …nd the relation between S(v) and !(v)
thus providing a very interesting geometrical meaning for the connection 1forms ! ij of the Levi-Civita connection D, namely as the angular ‘velocity’
with which the pseudo scalar Im when it slides on M .
In section Section 5.5 we prove some identities involving the projection
operator and its covariant derivative which are necessary, in particular, to
prove Proposition 367.
In Section 5.6 we present our conclusions.
5.2 Torsion Extensor and Curvature Extensor of a
Riemann-Cartan Connection
V1
Let u; v; t; z 2 sec T M and u; v; t; z 2 sec
T M ,! sec C`(M; g) the
physically equivalent 1-forms, i.e., u = g(u;), etc. Let moreover fea g be
V1
an orthonormal basis for T M and f a g, a 2 sec
T M ,! C`(M; g)
the corresponding dual basis and consider the Riemann-Cartan structure
(M; g; r).
De…nition 322 The form derivative of M is the operator
g : sec C`(M; g) ! sec C`(M; g);
gC : =
a
gea C
(5.10)
where gea is the Pfa¤ derivative of form …elds
Pm
gea C : = r=0 gea hCir
(5.11)
a
such that if hCir is expanded
Vr in the basis generated by f g, i.e., hCir =
1
i1 ir
2 sec
T M ,! sec C`(M; g) it is
Cr = r! Ci1 ir
1
1
ea (Ci1 ir i1 ir ) = ea (Ci1 ir ) i1 ir :
(5.12)
r!
r!
Given two di¤erent pairs of basis fea ; a g and fe0a ; 0a g we have that
gea hCir :=
a
since for all Cr
g0 Cr =
0a 0
gea Cr
=
0a 0
ea (
gea C =
1 0
C
r! i1
ir
0i1
0a 0
gea C;
ir
)=
(5.13)
a
1
ea ( Ci1
r!
ir
i1
ir
): (5.14)
4 This result appears (with a positive sign on the second member of Eq.(5.9) in [172].
See also [397]. However, take into account that the methods used in those references use
the Cli¤ord algebra of multivectors and thus, comparison of the results there with the
standard presentations of modern di¤erential geometry using di¤erential forms are not
so obvious, this being probably one of the reasons why some important and beautiful
results displayed in [172] are unfortunately ignored.
5.2 Torsion Extensor and Curvature Extensor of a Riemann-Cartan Connection
173
V2
Remark 323 V
We recall also that any biform B 2 sec T M ,! sec C`(M; g)
r
and any Ar 2
T M ,! sec C`(M; g) with r 2 it holds that
BAr = ByAr + B
Ar + B ^ Ar .
where for any C; D 2 sec C`(M; g) it is C
V1
that for v 2
T M ,! sec C`(M; g) it is
B
v = Bxv =
D=
(5.15)
1
2 (CD
DC):We observe
vyB:
(5.16)
O
Call @ := a rea the Dirac operator associated with r, a general RiemannCartan connection. In Chapter
V1 4 we introduced the Dirac commutator of
two 1-form …elds u; v 2 sec
T M ,! sec C`(M; g) associated with r by
^1
^1
^1
[[ ; ]] : sec
T M
sec
T M ! sec
T M
O
O
[[u; v]] = (u @)v
(v @)u
[[u; v]]
where
[[u; v]] = (u @)v
(v @)u;
(5.17)
5
is the Lie bracket of 1-form …elds .
De…nition 324 For a metric compatible connection r, recalling the de…nition of the torsion operator we conveniently write
(u; v) = [[u; v]] ;
(5.18)
which we call the (form) torsion operator.
Remark 325 We recall (see Chapter 2) the action of the operator @u (u 2
V1
V1
sec V T M ! sec C`(M; g)) acting on an extensor …eld F : sec
T M!
r
sec
T M , u 7! F (u). If u = ui i , @u := k @u@ k acting on F (u) is given
by
@
@
F (ui i ) := k k ui F ( i )
@uk
@u
= k F ( k ) = k yF ( k ) + k ^ F ( k ):
@u F (u) :=
k
(5.19)
Also the action of the operator @u ^ @v (u = ui i „v = v i i ) acting on
V1
V1
Vr
an extensor …eld G : sec
T M sec
T M ,! sec
T M , (u; v) 7!
G(u; v) is given by
@
@
^ l l um un G(
@uk
@u
= k ^ l G( k ; l ):
@u ^ @v G(u; v) =
5 Recall
k
that if [ea ; eb ] = cdab ed , then [
a ; b ]]
= cdab
d.
m
;
n
)
(5.20)
174
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
De…nition 326 The mapping
^2
^1
t : sec
T M ! sec
T M;
t(B) =
1
B (@u ^ @v ) (u; v):
2
(5.21)
is called the (2-1)-extensorial torsion …eld and
t(u ^ v) = (u; v):
(5.22)
Indeed, from Eq.(5.21) we have taking B = a ^ b
t(a ^ b) =
1
(a ^ b) (@u ^ @v ) (u; v):
2
(5.23)
Now,
(@u ^ @v ) (u; v) = (
Then,
t(a ^ b) =
1
(a ^ b) (
2
k
^
k
^
l
l
) ( k ; l ):
) ( k ; l ) = (a; b):
De…nition 327 The extensor mapping
^1
^2
: sec
T M ! sec
T M;
(c) =
(5.24)
1
(@u ^ @v ) (u; v) c;
2
(5.25)
is called the Cartan torsion …eld.
We have that
and if rea
b
:=
! bac
t(u ^ v) = @c (u ^ v)
c
(c)
then
z t(u ^ v) = zd ua v b T dab ;
T cab = ! cab
! cba
ccab :
(5.26)
De…nition 328 The connection (1-2)-extensor …eld ! in a given gauge is
given by (v = g(v; ))
^1
^2
! : sec
T M ! sec
T M;
v 7! !(v) =
1 c ab
v !c
2
a
^
b:
(5.27)
We also introduce the operator
^1
^2
! : sec
T M ! sec
T M;
v 7! !(v) = !v :=
1 c ab
v !c
2
a
^
b
(5.28)
5.2 Torsion Extensor and Curvature Extensor of a Riemann-Cartan Connection
and it is clear that
!(v) = !v :
(5.29)
We recall that it is proved in Chapter 7 (Section 7.5) using the theory
of covariant derivatives in vector bundles that for any C 2 sec C`(M; g) we
have
1
rv C = gv C + [!v ; C]
2
= gv C + !v C;
where !v
bundle.
C : = 12 (!v C
(5.30)
C!v ) is the commutator of sections of the Cli¤ord
Exercise 329 Choose a basis for C`(M; g) and verify by direct computation the validity of Eq.(5.30).
Remark 330 Note for future reference that if v = g(v; ) then
v
C = vyC:
Also take notice that
(5.31)
O
rv C = v @
(5.32)
De…nition 331 The form curvature operator is the mapping
: sec (
^1
T M
^1
O
T M ) ! End
O
(u; v) = [u @; v @]
= [ru ; rv ]
^1
T M;
O
[[u; v]] @
r[u;v]
with u = g(u; ), v = g(v; ), u; v 2 sec T U
sec T M
De…nition 332 The form curvature extensor is the mapping
: sec (
^1
T M
^1
T M
O
^1
O
(u; v; w) = [u @; v @]w
= [ru ; rv ]w
T M ) ! sec
O
[[u; v]] @w
^1
T M;
r[u;v] w
with u = g(u; ), v = g(v; ), w = g(w; ), u; v; w 2 sec T U T M
It is obvious that for any Riemann-Cartan connection we have
(u; v; w) =
(v; u; w);
(5.33)
175
176
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
One can easily verify that for a Levi-Civita connection we have
(u; v; w) + (v; w; u) + (w; u; v) = 0:
(5.34)
Note however that Eq.(5.34) is not true for a general connection.
De…nition 333 The mapping
R : sec (
^1
T M )4 ! sec
R(a; b; c; w) =
^0
T M;
(a; b; c) w;
(5.35)
a = g(a;); b = g(b;); c = g(c;); w = g(w;)
with a; b; c; w 2 sec T U
T M is called the curvature tensor.
One can verify that for the connection r
R(a; b; c; w) =
R(b; a; c; w);
(5.36)
R(a; b; c; w) =
R(a; b; w; c);
(5.37)
and that for a Levi-Civita connection
R(a; b; c; w) = R(c; w; a; b);
(5.38)
R(a; b; c; w) + R(b; c; a; w) + R(c; a; b; w) = 0;
(5.39)
Equation 5.39 is known as the …rst Bianchi identity.
Proposition 334 There exists a smooth (2-2)-extensor …eld,
R : sec
V2
T M!
V2
T M;
B 7! R(B)
called the curvature biform such that for any a; b; c; d 2 sec
R(a; b; c; d) = R(a ^ b) (c ^ d) =
(5.40)
V1
(c ^ d)yR(a ^ b)
T M we have
(5.41)
Such B 7! R(B) is given by
R(B) =
1
B (@a ^ @b )@c ^ @d (a; b; c) d;
4
(5.42)
and we also have
R(a ^ b) =
1
@c ^ @d (a; b; c) d:
2
(5.43)
5.2 Torsion Extensor and Curvature Extensor of a Riemann-Cartan Connection
Proof. First, we verify that Eq.(5.42) and Eq.(5.43) are indeed equivalent.
Indeed, Eq.(5.42) implies Eq.(5.43) since we have
R(a ^ b) =
=
=
=
=
1
(a ^ b) (@p ^ @q )@c ^ @d (p; q; c) d
4
1
a @p a @q
det
@c ^ @d (p; q; c) d
b @p b @q
4
1
(a @p b @q a @q b @p ) @c ^ @d (p; q; c) d
4
1
(a @p b @q ) @c ^ @d (p; q; c) d
2
1
@c ^ @d (a; b; c) d :
2
Also, Eq.(5.43) implies Eq.(5.42) since taking into account that
1
B (@a ^ @b )a ^ b
2
B=
we have
1
R(B) = R( B (@a ^ @b )a ^ b)
2
1
= B (@a ^ @b )R(a ^ b)
2
1
=
B (@a ^ @b )@c ^ @d (a; b; c) d :
4
Now, we show the validity of Eq.(5.41). We have taking into account
Eq.(5.43)
R(a ^ b)y(c ^ d) =
=
=
=
=
1
(c ^ d) (@p ^ @q ) (a; b; p) q
2
1
c @p c @q
det
(a; b; p) q
d @p d @q
2
1
(c @p d @q c @q d @p ) (a; b; p) q
2
c @p d @q (a; b; p) q
(a; b; c) d = R(a; b; c; d);
and the proposition is proved.
Proposition 335 The curvature biform R(u ^ v) is given by 6
R(u ^ v) = u g! (v)
v g! (u) + !(u)
!(v):
(5.44)
6 Recall that in Eq.(5.45) [u;v] is the standard Lie bracket of vector …elds u and v
and [u; v]] := u @v v @u is the commutator of the 1-form …elds u and v.
177
178
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
Proof. The proof is given in three steps (a), (b) and (c)
(a) We …rst show that Eq.(5.44) can be written as
O
O
R(u ^ v) = u @! v
= ru ! v
1
[! u ; ! v ]
2
v @! u
rv ! u
1
[! u ; ! v ]
2
! [u;v]
! [u;v] ;
(5.45)
with u = g(u; ), v = g(v; ). Indeed,we have
O
1
u @(!(v)) = u g(!(v))+ [!(u);!(v)]:
2
(5.46)
and recalling the de…nition of the derivative of an extensor …eld, it is:
(u g!) (v)
u g! (v) := u g(!(v)) !(u gv):
(5.47)
we have,
u g(!(v)) v g(!(u)) = u g !(v) v g !(u)+!(u gv) !(v gu)
= u g !(v) v g !(u)+!([[u; v]])
= u g !v
v g ! u + ! [u;v] ;
(5.48)
and using the above equations in Eq.(5.44) we arrive at Eq.(5.45)
(b) Next we show (by …nite induction) that for any C 2 sec C`(M; g) we
have
1
(5.49)
([ru ; rv ] r[u;v] )C = [R(u ^ v); C],
2
with R(u ^ v) given by Eq.(5.45). Given that any C P
2 sec C`(M; g) is a
n
sum of nonhomogeneous di¤erential forms, i.e. C =
p=0 Cp with Cp 2
Vr
1
sec T M ,! sec C`(M; g) and taking into account that Cp = r!
Ci1 ip i1
it is enough to verify the formula V
for p-forms. We …rst verify the validity of
1
the formula for a 1-form i 2 sec
T M ,! sec C`(M; g). Using Eq.(5.45)
and the Jacobi identity
[[! v ; [[! u ; i ]]]] + [[! u ; [[ i ; ! v ]]]] + [[ i ; [[! v ; ! u ]]]] = 0;
ip
(5.50)
we have that
1
[R(u ^ v); i ]
2
1
=
(ru ! v i
2
1
ru ! v ;
=
2
= ru rv
i
(5.51)
1
ru ! v + [[! v ; ! u ]; i ] (rv ! u i + i rv ! u
2
1
1
+ [! v ; [! u ; i ]] rv ! v ; i
[! u ; [! v ; i ]]
2
2
i
i
r v ru
i
r[u;v] i :
[! [u;v] ; i ]
[! [u;v] ; i ]
(5.52)
5.2 Torsion Extensor and Curvature Extensor of a Riemann-Cartan Connection
179
Now, suppose the formula is valid for p-forms. Let us calculate the …rst
ir+1
member of Eq.(5.49) for the (r + 1)-form i1 ir+1 = i1 i2
. We
have .
ru rv (
i1
ir+1
= ru ((rv
(r[u;v]
)
)
i2
ir+1
ir+1
+ rv
i2
ir+1
)
i2
ir+1
(ru rv
i2
ir+1
i1
i1
i1
i1
)
i1
+
i1
)
ir+1
(r[u;v]
i1
i2
i2
(rv ru
i1
i1
r v ru (
)
= (ru rv
=
i1
)
ru
i1
)
ir+1
i2
rv
i1
i1
ru
rv
r[u;v]
i1
r[u;v] (
ir+1
r[u;v]
r v ru
i1
)
)
rv ((ru
i2
ir+1
i2
ir+1
i2
ir+1
i2
ir+1
i2
ir+1
i1
i2
ir+1
i1
+ ru
i1
rv
r[u;v]
)
i1
rv
ru
i2
i2
)
ir+1
ir+1
i2
ir+1
i2
ir+1
i1
+
+
i1
i1
ru
i2
r u rv
rv ru
ir+1
i2
ir+1
i2
ir+1
)
ir+1
r v ru
r[u;v] )
+(ru rv
1
1
= i1 ( [R(u ^ v); i2 ir+1 ]) + ( [R(u ^ v);
2
2
1
= [R(u ^ v); i1 ir+1 ];
2
i1
])
i2
ir+1
(5.53)
where the last line of Eq.(5.53) is the second member of Eq.(5.49) evaluated
ir+1
.
for i1 i2
(c) Now, it remains to verify that
R(u; v; t; z) = (t ^ z) R(u ^ v);
(5.54)
with R(u ^ v) given by Eq.(5.45). Indeed, from a well known identity, we
V1
V2
have that for any t; z 2 sec T M , and R(u ^ v) 2 sec T M it is
(z ^ t) R(u ^ v) =
zy(tyR(u ^ v))
= z (R(u ^ v)xt)
1
= z [R(u; v); t]
2
Eq.(5.49)
=
z (ru rv t
r v ru t
r[u;v] t)
and the proposition is proved.
In particular we have:
R(u; v; z; t) = zc td ua v b Rcdab ;
Rdcab = ea (! dbc )
eb (! dac ) + ! dak ! kbc
! dbk ! kac
ckab ! dkc :
(5.55)
and
R( a ;
b
;
a; b)
=(
where R is the curvature scalar.
a
^
b
) R(
a
^
b)
= R;
)
(5.56)
180
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
Proposition 336 For any v 2 sec
[rea ; reb ]v = R(
^
a
V1
T M ,! sec C`(M; g)
c
(Tab
b )xv
! cab + ! cba )rec v:
(5.57)
Proof. From Eq.(5.49) we can write
1
[R(
2
[rea ; reb ]v =
= R(
a
^
b )xv
= R(
+ r([ea ;eb ]
^
a
= R(
b )xv
^
a
b ); v]
+ r[ea ;eb ] v
rea eb +reb ea ) v
+ rrea eb v
a
^
T cab ec v
+r
(T cab
b )xv
r!cba ec v
+ r!cab ec v
! cab
+
rreb ea v
! cba )rec v
which proves the proposition.
Proposition 337
a
R(
^
= Rab = d! ab + ! ac ^ ! cb
b)
(5.58)
Proof. Recall that using
([rek ; rel ]
r[ek ;al ] )
([rek ; rel ]
r[ek ;al ] )
j
=
j
= Rijkl i ;
(5.59)
= v m Rimab i :
(5.60)
= (ek ;el )
j
Rjikl i ;
j
= (ek ;el )
we have
R(
a
^
b )xv
= v m (ea ; eb )
m
On the other hand, for a general connection, we must write
Rab :=
1
Rklab
2
k
^
l
(5.61)
and then
Rab xv =
1 m
v Rklab (
2
k
^ l )x
m
=
v m Rmlab
l
= v m Rlmab
l
= v m Rlmab
and the proposition is proved.
Proposition 338 The Ricci 1-forms7 Rd := Rbd b and the curvature biform R( a ^ b ) for the Levi-Civita connection D of g are related by,
Rd =
1
(
2
a
^
b
)(R(
a
^
b )x
d
)
(5.62)
Proof. Recalling that the Ricci operator is given by
@^@
7 The
a :=
Rb
d
=
1
(
2
ca Rk
ckb are
a
^
b
) [Dea ; Deb ]
d
ccab Dec
the components of the Ricci tensor.
d
(5.63)
l
5.3 The Riemannian or Semi-Riemannian Geometry of a Submanifold M of M
and moreover taking into account that by the …rst Bianchi identity it is
Rcd ^ c = 0, we have
1
(
2
a
^
b
) [Dea ; Deb ]
d
ccab Dec
d
1 a
( ^ b )Rdcab c = Rcd c
2
cd
= Rcd y c + Rcd ^ c =
c yR
1
a
=
^ b )Rcdab
c y(
2
= Rcdcb b = Rd
(5.64)
=
which proves the proposition.
Proposition 338 suggests the
De…nition 339 The Ricci extensor is the mapping
R : sec
^1
T M ! sec
^1
T M;
R(v) = @u R(u ^ v):
Remark 340 Of course, we must have R(
d
(5.65)
) = Rd . Moreover, we have
@
@
R(uk k ^ v) = b b uk R(
b
@u
@u
= b R( bk k ^ v) = b R( b ^ v)
So,
@u R(u ^ v) =
b
=
b
yR(
b
=
b
yR(
b
^ v) +
^ v):
b
^ R(
b
k
^ v)
^ v)
@u R(u ^ v) = @u yR(u ^ v) and @u ^ R(u ^ v) = 0:
(5.66)
5.3 The Riemannian or Semi-Riemannian
Geometry of a Submanifold M of M
5.3.1 Motivation
Any manifold M; dim M = m, according to Whitney’s theorem (see, e.g.,
[27]), can be realized as a submanifold of Rn , with n = 2m. However, if M
carries additional structure the number n in general must be greater than
2m. Indeed, it has been shown by Eddington [109] that if dim M = 4 and
if M carries a Lorentzian metric g and which moreover satis…es Einstein’s
equations, then M can be locally embedded in a (pseudo)Euclidean space
R1;9 . Also, isometric embeddings of general Lorentzian spacetimes would
require a lot of extra dimensions [82]. Indeed, a compact Lorentzian manifold can be embedded isometrically in R2;46 and a non-compact one can be
181
182
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
embedded isometrically in R2;87 ! In particular this last result shows that
the spacetime of M-theory [107, 179] may not be large enough to contain
4-dimensional branes (representing Lorentzian spacetimes) with arbitrary
metric tensors. In what follows we show how to relate the intrinsic di¤erential geometry of a structure (M; g; D) where g is a metric of signature
(p; q), D is its Levi-Civita connection and M is an orientable proper submanifold of M , i.e., there is de…ned on M a global volume element g = Im
whose expression on U M is given by
Im =
1 2
m
:
(5.67)
We suppose moreover that M ' Rn and that it is equipped with a metric
g of signature (p; q) = n. However, take notice that our presentation in the
form of a local theory is easily adapted for a general manifold M .
Projection Operator P
Pn
Vr
De…nition 341 Let C = r=0 Cr , with Cr 2 sec T M ,! sec C`(M; g).
The Projection operator on M is the extensor …eld
P : sec C`(M; g) ! sec C`(M; g);
P(C) = (CyIm )Im1 :
(5.68)
Vk
Remark 342 Note that for all Ck 2 sec T M ,! sec C`(M; g),Vif k > m
r
then P(Ck ) = 0, but of course, it may happen that even if Ar 2 sec T M ,!
sec C`(M; g) with r m we may have P(Ar ) = 0:
We de…ne the complement of P by
P? (C) = C
(5.69)
P(C)
and it is clear that P? (C) have only components lying outside C`(M; g).
It is quite clear also that any C with components not all belonging to
sec C`(M; g) will satisfy CyIm = 0:
Having introduced in Section 5.1 the derivative operators @ and its restriction d = @
(Eq. (5.5) and Eq.(5.6)) we extend the action of P to
M
act on the operator d, de…ning:
Pm
Pm
P(d) = k=1 P( k Dek ) := k=1 P(
k
)Dek =
Shape Operator S
Pm
k=1
k
Dek = d:
(5.70)
De…nition 343 Given C 2 sec C`(M; g) we de…ne the shape operator
S : sec C`(M; g) ! sec C`(M; g);
S(C) = dP (C) = d(P(C))
P(dC):
(5.71)
5.3 The Riemannian or Semi-Riemannian Geometry of a Submanifold M of M
183
5.3.2 Induced Metric and Induced Connection
For any C 2 sec C`(M; g) and v 2 sec T M we write as usual [77, 175]
Dv C = (Dv C)k + (Dv C)?
(5.72)
where (Dv C)k 2 sec C`(M; g) and (Dv C)? 2 sec[C`(M; g)]? where [C`(M; g)]?
is the orthogonal complement of C`(M; g) in C`(M; g).
As it is very well known [77, 175] if g := i g and v 2 sec T M (and
v = g(v; )) and C 2 sec C`(M; g) the Levi-Civita connection D of g := i g
is given by
Dv C := (Dv C)k
(5.73)
and of course
Dv C := (v @C)k
(5.74)
Moreover, note that we can write for any C 2 sec C`(M; g)
v @C = (v @C)k = P(v @C)
(5.75)
(Dv C)? := P? (v @C)
(5.76)
Also, writing
we have
v @ = P(v @) = (v @)k = (v d)k = v @ P? (v @) = v d P? (v d): (5.77)
So, it is
v dIm Im1 =
Pm
j=1
1
(Dv
j
+ P? (v d j ))
= Dv Im Im1 + P? (v d j ) ^
j
:
m
Im1
(5.78)
Vm
Now, Dv Im 2 sec
T M ,! sec C`(M; g) is a multiple of Im and since
2
= 1 depending on the signature of the metric g we have that Dv Im =
Im
0.
Indeed,
2
0 = Dv Im
= 2(Dv Im )Im
(5.79)
and so
0 = (Dv Im )Im Im1 = Dv Im :
In any Cli¤ord algebra bundle, in particular C`(M; g) we can build multiples of the Im not only multiplying it by an scalar function, but also
multiplying it by a an appropriated biform. This result will be used below
in the de…nition of the shape biform.
184
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
5.3.3 S(v) = S(v) = @u ^ Pu (v) and S(v? ) = @u yPu (v)
For any C 2 sec C`(M; g) it is C = P(C) we have (with u 2 sec
sec C`(M; g))
d(P(C)) = dP (C)
V1
T M ,!
P(dC)
= @u Pu (C)
P(dC)
= @u ^ Pu (C) + @u yPu (C)
P(dC)
(5.80)
where
Pu (C) := u dP (C) = u d(P(C)) (P(u dC)):
V1
Recall that for v 2 sec T M ,! sec C`(M; g) we can write
S(v) = dP (v) = @u ^ Pu (v) + @u yPu (v)
(5.81)
(5.82)
where we used that for any C 2 sec C`(M; g) it is
@u Pu (C) :=
Pm
i
i=1
Pm
@
u dP (C) = i=1 i Dei P (C) = dP (C)
@ui
(5.83)
Putting v = vk + v? = v + v? we have the
Proposition 344
S(v) = S(v) = @u ^ Pu (v);
S(v? ) = @u yPu (v):
(5.84)
Proof. Indeed,
S(v) = S(vk ) + S(v? ) = @u ^ Pu (v) + @u yPu (v)
(5.85)
and so, it is enough to show that
@u yPu (vk ) = 0 and @u ^ Pu (v? ) = 0;
(5.86)
From P2 (v) = P(v) we get
Pu P(vk ) + PPu (vk ) = Pu (vk );
(5.87)
So, for vk and v? it is
PPu (vk ) = 0
and PPu (v? ) = Pu (v? ):
(5.88)
Since PPu (vk ) = 0 we have that
@u yPPu (vk ) = P(@u )yPu (vk ) = @u yPu (vk ) = 0:
(5.89)
From PPu (v? ) = Pu (v? ) we can write
@u ^Pu (v? ) = @u ^PPu (v? ) = P(@u )^PPu (v? ) = P(@u ^Pu (v? )): (5.90)
5.3 The Riemannian or Semi-Riemannian Geometry of a Submanifold M of M
Now, take t; y 2 sec
V1
T M ,! sec C`(M; g). We have
(t ^ y) (@u ^ PPu (v? )) = (t ^ y) (@u ^ Pu (v? ))
= ty((y @ u ^ Pu (v? )
= ty(y d(P(v? ))
i
P(y dv? )
= ty(P(y dv? )
= ty(Dy v? )
@ u ^ ((yyPu (v? ))
^ (yy(
i
i
i
^ (yyP(
d(P(v? ))
P(
i
dv? )))
dv? )))
i
^ (yy(Dei v? ))) = 0
(5.91)
from where it follows that
@u ^ Pu (v? ) = 0
(5.92)
and the proposition is proved.
Shape Biform S
De…nition 345 The shape biform (a (1; 2)-extensor …eld ) is the mapping
V1
V2
S : sec T M !
T M;
(5.93)
v 7! S(v);
such that
v dIm =
S(v)Im :
(5.94)
From Eq.(5.15) it follows that
S(v)yIm = 0 and S(v) ^ Im = 0:
(5.95)
Since S(v)yIm = 0 it follows from Remark 342 that
P(S(v)) = 0:
(5.96)
Now, using the fact that Dv Im = 0 and Eq.(5.94) it follows from Eq.(5.78)
that
v dIm = (P? (v d j ) ^ j )Im = S(v)Im ;
i.e.,
S(v) =
P? (v d j ) ^
j
:
(5.97)
Proposition 346 For any C 2 sec C`(M; g) we have
Dv C = v dC + S(v)
C = Dv C + S(v)
C:
Proof. Taking into account Eq.(5.97) we have for any v; w 2 sec
sec C`(M; g)
vyS(w) =
=
vy(P? (w d j )^ j )
(vy(P? (w d j ))
j
T M ,!
+ v j P? (w d j )
= v j P? (w d j ) = P? (w dv)
= P? (w dv):
(5.98)
V1
P? [(w dv j ) j ]
(5.99)
185
186
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
So,
v dw = P(v dw) + P? (v dw)
= Dv w + wyS(v)
= Dv w
Now, for v; w 2 sec
V1
S(v)xw:
(5.100)
T M ,! sec C`(M; g) we have
Dv (wu) = (Dv w)u + wDv u = (Dv w)u + (S(v)
= (Dv wu) + (S(v)
w)u
= (Dv wu) + (S(v)
wu);
w(u
w)u + wDv u + w(S(v)
S(v))
(5.101)
from where the proposition follows trivially by …nite induction.
Of course, it is
Dei C = Dei C + S(v) C
Now, recalling Eq.(5.30) we have
Dei C = gei C + !ei C
Pn
j
where for i; j = 1; :::; m; Dei j = Dei j =
k=1 ! ik
!v =
(5.102)
8
1 c ab
v !c
2
a
^
b
(5.103)
k
, it is
(5.104)
So, we get
Dv C = v dC + S(v)
C
= dv C + (!v + S(v))
C
(5.105)
and in particular
Dei C = gei C + (!ei + S(ei ))
C:
(5.106)
Comparison of Eq.(5.106) with Eq.(5.28) (valid for any metric compatible
connection) implies the important result
!v = (!v + S(v))
(5.107)
We can easily …nd by direct calculation that in a gauge where !v 6= 0,
!v = P(!v )
(5.108)
which is consistent with the fact that from Eq.(5.96) it is P(S(v)) = 0.
Let (x1 ; :::; xm ; :::; xn ) be the natural orthogonal coordinate functions of
M ' Rn :
8 Take notice that this formula being gauge dependent is not valid if e 7! x where
i
i
the xi coordinate vector …elds. See Corollary 347.
u)
5.3 The Riemannian or Semi-Riemannian Geometry of a Submanifold M of M
Corollary 347 For C 2 sec C`(M; g)
Dv C = v i
@
C + S(v)
@xi
Proof. Taking into account that D
!
@
@xi
=
1
(
2
@
@xi
C:
(5.109)
dxj = 0 follows that
kil )dx
k
Using this result in Eq.(5.98) with ei 7!
^ dxl = 0:
@
@xi
(5.110)
gives the desired result.
Remark 348 Comparison of Eq.(5.105) and Eq.(5.109) shows that S(v)
cannot always be identi…ed with !(v) which is a gauge dependent operator.
5.3.4 Integrability Conditions
Remark 349 Take into account that the commutator of Pfa¤ derivatives
acting on any C 2 sec C`(M; g) is in general non null, i.e.,
Pm
[gei ; gej ]C = r=0 ckij ek (Ci1 im ) i1 im 6= 0;
(5.111)
unless ei are coordinate vector …elds, i.e., ei 7! xi .
Remark 350 Also, since the torsion of D is null we have in general
[
i
d;
j
d]C = [ i ; j ] dC = ckij
dC = ckij Dek C 6=0;
k
(5.112)
unless ei are coordinate vector …elds. Moreover, for the case of orthonormal
vector …elds
[ i d; j d]C 6= [gei ; gej ]C:
(5.113)
Remark 351 The integrability condition for the connection D is expressed,
given the previous results, by
@^@ =0
(5.114)
which means that for any C 2 sec C`(M; g) it is
@^@ C =0
For the manifold M recalling that xi
for any C 2 sec C`(M; g)
(gxi gxj
gxi is a Pfa¤ derivative we have
gxj gxi )C = 0:
(5.115)
If we recall the de…nition of the form derivative (Eq.(5.10)), putting
g := #i gxi
(5.116)
187
188
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
we can express the ‘integrability’ condition in M by
g^g = 0:
(5.117)
Finally recalling Eqs.(5.63) and (5.64) for v 2 sec
it is
@ ^ @ v = vi Ri
V1
T M ,! sec C`(M; g)
(5.118)
where Ri are the Ricci 1-form …elds.
5.3.5 S(v) = S(v)
Proposition 352 Let C = v 2 sec
V1
T M ,! sec C`(M; g) we have
S(v) = S(v):
(5.119)
Proof. We have
S(v) = d(P(v))
= dv
P(dv)
P(dv):
(5.120)
Now, dv = d^v + dyv and since P(dyv) = dyv we have
S(v) = d^v
P(d^v)
(5.121)
It is only necessary due to the linearity S of to show Eq.(5.119) for v =
d ,d =1; :::; m. We then evaluate
Pm
Pm Pn
d^ d = k=1 k Dek d = k=1 t=1;t6=k !tkd k ^ t
Pm Pm
Pm Pm+l
= k=1 t=1;t6=k !tkd k ^ t + k=1 t=m+1 !tkd k ^ t ; (5.122)
from where it follows that
S(
d)
=
Pm Pm+l
k=1
t=m+1 !tkd
k
^
t
=
On the other hand
d
dIm =
11
mm
Ded (
1
= Ded ( 1
m)
Pm Pn
=
t=1 !tdk
k=1
=
+
1 Pm Pm+l
(!tkd
2 k=1 t=m+1
^
1
Pm Pm
k=1
t=1 !tdk 1
Pm Pm+l
k=1
t=m+1 !tdk 1
^
!ktd )
k
^
t
(5.123)
m)
t
|{z}
m
k-p osition
t
|{z}
m
k-p osition
t
|{z}
k-p osition
m
(5.124)
5.3 The Riemannian or Semi-Riemannian Geometry of a Submanifold M of M
and now we can easily see that
S(
and it follows that S(
We also have the
d)
d)
= S(
Im =
d
dIm
(5.125)
d ).
Proposition 353 Let v; w 2 sec
V1
T M ,! sec C`(M; g) we have
v S(w) = w S(v)
(5.126)
Proof. Recalling Eq.(5.123) we can write
v S(w) =
=
Pm
i;d=1 v
i
wd i y
Pm Pm+l
k=1
1 Pm
v i wd (!tid
2 i;d=1
t=m+1 !tkd
!itd )
t
k
^
t
= w S(v)
and the proposition is proved.
5.3.6 d ^ v = @ ^ v + S(v) and dyv = @yv
We …rst observe that since torsion is null for the Levi-Civita connection D
we have for any u; v 2 sec T M ,! sec C`(M; g) we have
u @v = v @u + [[u; v]]
(5.127)
from where it follows P? ([[u; v]]) = 0 when u; v 2 sec T M ,! sec C`(M; g)
since calculating [[u; v]] with @ expressed in the natural coordinates of M
we …nd that [[u; v]] 2 sec T M ,! sec C`(M; g). From this it follows that
P? (u dv) = P? (v du):
(5.128)
Then we can write
d^v =
=
m
P
r
m;k=1
m
P
r;k=1
r
^ Der (v k
k)
^ fer (v k ) + v k
=@^v+
=@^v+
=@^v+
=@^v+
m
P
m;k=1
m
P
m;k=1
r
r
m
P
Lsrk s )g +
s=1
r;k=1
r
^ v k P? (Dr
r
^ v k P? (Dk r )
^ P? (Dv r )
^ P? (v d r )
m
P
k)
r
^ vk
m+l=n
P
s=m+1
Lsrk
s
189
190
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
P? (v d r )^
and recalling that S(v) =
r
we …nally have
d ^ v = @ ^ v + S(v)
(5.129)
and the proposition is proved.
Also, from (Eq.(5.86)) we know that @ u yPu (vk ) = 0. So,
dv = d(P(v)) = dP (v) + P(dv)
= @u ^Pu (v) + @u yPu (v) + @v
= @u ^Pu (v) + @ ^ v + @yv
= S(v) + @ ^ v + @yv
(5.130)
and thus we see that
dyv = @yv
(5.131)
dv = @v + S(v):
(5.132)
We then can write
5.3.7 dC = @C + S(C)
We can generalize Eq.(5.132), i.e., we have the
Proposition 354 For any C 2 sec C`(M; g) we have
dC = @C + S(C);
d ^ C = @ ^ C + S(C);
dyC = @yC:
(5.133)
Proof. (i) From the fact that for any A; B 2 sec C`(M; g) it is P(A ^ B) =
V1
P(A) ^ P(B) we have di¤erentiating with respect to u 2 sec T M ,! sec
C`(M; g)
Pu (A ^ B) = Pu (A) ^ B + A ^ Pu (B)
(5.134)
and of course
Pu (A? ^ Bk ) = Pu (A? ) ^ Bk ;
Pu (A? ^ B? ) = 0;
Pu (Ak ^ Bk ) = Pu (Ak ) ^ Bk + Ak ^ Pu (Bk ):
(5.135)
(ii) For C 2 sec C`(M; g) it is C = P(C) and we have using Eq.(5.71)
dC = dP (C)
P(dC)
= d^P (C) + dyP (C) + @C
(5.136)
(iii) Now, we can verify recalling that S(C) = S(Ck + C? ) = S(Ck ) + S(C? )
and following steps analogous to the ones used in the proof of Proposition
344 that
S(Ck ) = S(P(Ck )) = d^P (C k )
S(C? ) = P(S(C? )) = dyP (C k ): (5.137)
5.4 Curvature Biform R(u^v) Expressed in Terms of the Shape Operator
191
(iv) Using Eq.(5.137) in Eq.(5.135) we have
d^C + dyC = d^P (C) + dyP (C) + @^C + @yC
or
d^C + dyC = S(C) + S(C? ) + @^C + @yC
= S(C) + @^C + @yC;
(5.138)
which provides the proof of the proposition.
Proposition 355 For any C 2 sec C`(M; g) we have:
@C = P(dC)
(5.139)
Proof. From Eq.(5.71) when C 2 sec C`(M; g) it is
P(S(C)) = 0:
Then, applying P to both members of the …rst line of Eq.(5.133) we have
P(dC) =(dC)k = P(@C) + P2 (S(C)) = P(@C) = @C
(5.140)
and the proposition is proved.
5.4 Curvature Biform R(u ^ v) Expressed in Terms
of the Shape Operator
5.4.1 Equivalent Expressions for R(u ^ v)
In this section we suppose that the structure (M; g; D) is such that is M
a submanifold of M ' Rn and D the Levi-Civita connection of g = i g.
We obtained in Section 5.1 a formula (Eq.(5.44)) for the curvature biform
R(u ^ v) of a general Riemann-Cartan connection. Of course, taking into
account the fact that R is an intrinsic object, the evaluation of R(u ^ v)
does not depend on the coordinate chart and basis for vector and form
…elds used for its calculation. In what follows we take advantage of this
@
i
fact choosing the basis f @x
i ; dx g as introduced above for which !(u) = 0.
Thus, we have, recalling Eq.(5.44) and Eq.(5.45) that
R(u ^ v) = Du !(v)
Dv !(u) + !(u)
= Du !(v)
Dv !(u) + !(u)
!(v)
!(u)
= Du !(v)
!(u)
!(v)
Dv !(u) + !(u)
!(v)
! [u;v]
!(v)
! [u;v]
!(v)
! [u;v] :
(5.141)
192
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
On the other hand since in the gauge where !(u) = 0 we have that !(u) =
S(u) and thus we can also write
R(u ^ v) = Du !(v)
Dv !(u) + S(u)
S(v)
S([u; v]):
(5.142)
Now, putting xi = @=@xi we have
Du !(v)
=
=
=
Dv !(u)
i j
u v fDxi S(#j )
i j
u v fDxi (Dxj Im Im1 )
=
=
=
=
=
Dxj (Dxi Im Im1 )g
ui v j fDxi Dxj Im ) Im1 ) + (Dxj Im ) (Dxi Im1 )
(Dxj Dxi Im ) Im1 )
=
DxJ S(#i )g
(Dxi Im ) (Dxj Im1 )g
ui v j f(D[xi ;xj ] Im Im1 )
(Dxj Im ) (Dxi Im1 )
ui v j f (Dxj Im ) (Dxi Im1 )
(Dxi Im ) (Dxj Im1 )g
ui v j f((Dxi Im )Im1 ) ((Dxj Im1 )Im )
ui v j f((Dxi Im )Im1 ) ((Dxj Im )Im1 )
ui v j S(#i )S(#j ) + ui v j S(#j )S(#i )
S(u)S(v) + S(v)S(u) =
(Dxi Im ) (Dxj Im1 )g
2S(u)
((Dxi Im )Im1 ) ((Dxj Im1 )Im )g
((Dxj Im )Im1 ) ((Dxi Im )Im1 )g
S(v):
(5.143)
Thus,we get
R(u ^ v) =
S(u)
S([u; v]):
(5.144)
V2
Now take into account that since R(u^v) 2 sec T M ,! sec C`(M; g) we
must have, of course, S(u) S(v) S([u; v]) = P( S(u) S(v) S([u; v]))
and since Eq.(5.96) tell us that P(S([u; v]) = 0 we have the nice formula9
R(u ^ v) =
S(v)
P(S(u)
S(v))
(5.145)
which express the curvature biform in terms of the shape biform.
5.4.2 S2 (v) =
@ ^ @ (v)
In this subsection we want to show the
V1
Proposition 356 Let v 2 sec T M ,! sec C`(M; g). Then,
S2 (v) =
@ ^ @ (v):
(5.146)
9 Note that in [172, 397] the second member of Eq.(5.145) is the negative of what we
found. Our result agrees with the one in [104].
5.4 Curvature Biform R(u^v) Expressed in Terms of the Shape Operator
193
Eq.(5.146) tell us that the square of the shape operator applied to a 1form …eld v is equal to the Ricci operator applied to v. This result will play
an important role in Chapter 11 where we give a formulation to Cli¤ord
intuition that matter is represented by little hills in a brane.[358]
Now, to prove the Proposition 356 we need the following lemmas:
V1
Lemma 357 Let C 2 sec C`(M; g) and v 2 sec T M ,! sec C`(M; g).Then
Pv (C) = P(C)
S(v)
S(v)):
P(C
(5.147)
Proof. Indeed,
Pv (C) = Dv (P(C))
= Dv (P(C))
= Dv C
= P(C)
S(v)
S(v)
P(Dv C)
S(v)
P(C)
P(C
P(C)
P(Dv C
Dv C + P(S(v)
S(v)
S(v))
C)
C)
(5.148)
which proves the lemma.
Lemma 358 Let C 2 sec C`(M; g) and v 2 sec
V1
T M ,! sec C`(M; g).Then
Dv C = Dv C Pv (C):
(5.149)
Proof. Follows from the …rst line in Eq.(5.148).
Lemma 359 Let C 2 sec C`(M; g) and u; v 2 sec
V1
T M .,! sec C`(M; g).Then
Du Dv C = P(Du Dv C) + Pu Pv (C):
(5.150)
Proof. Using Eq.(5.149) we have
Du (Dv C) = Du (Dv C Pv (C))
= Du Dv C Pu (Dv C)
Du (Pv (C)) + Pu Pv (C):
(5.151)
On the other hand we have
P(Du Dv C) =
Pu (Dv C) + Du (P(Dv C))
=
Pu (Dv C) + Du (Dv C)
=
Pu (Dv C) + Du (Dv C Pv (C))
= Du Dv C Pu (Dv C)
Du (Pv (C)):
(5.152)
Putting Eq.(5.152) in Eq.(5.150) gives the desired result.
V1
Lemma 360 Let C 2 sec C`(M; g) and u; v 2 sec T M ,! sec C`(M; g).Then
194
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
R(u ^ v)
C = [Pu ; Pv ]C:
(5.153)
Proof. Using Eq.(5.150) we have
[Du ; Dv ]C = P([Du ; Dv ]C) + [Pu ; Pv ]C
= P(D[u;v] C) + [Pu ; Pv ]C
= D[u;v] C + [Pu ; Pv ]C
(5.154)
Thus we get that
([Du ; Dv ]
D[u;v] )C = [Pu ; Pv ]C
(5.155)
Recalling now Eq.(5.49) we have
R(u ^ v)
C = [Pu ; Pv ]C
and the lemma is proved
Lemma 361 Let C 2 sec C`(M; g) and u; v 2 sec
Then,
R(u ^ v)
C=
V1
T M .,! sec C`(M; g).
S(v))
P(S(u)
C
(5.156)
Proof. This follows directly from Eq.(5.145).
Remark 362 We shall now evaluate directly the …rst member of Eq.(5.157)
to get Eq.(5.160) which when compared with Eq.(5.145) will furnish identities given by Eq. (5.161).
([Du ; Dv ]
D[u;v] )C = R(u ^ v)
C:
(5.157)
Given the linearity of R(u ^ v) we calculate the …rst member of Eq.(5.157)
for the case u = xi , v = xj . Taking into account that Du C = Du C + S(u)
C we get with calculations analogous to the ones in Eq.(5.143) that
[Dxi ; Dxj ]C =
S(#i )
S(#j )
C
(5.158)
and taking into account that it is [xi ; xj ] = 0 we can write the last equation
as
([Dxi ; Dxj ]
D[xi ;xj ] )C = R(#i ^ #j )
C=
S(#i )
S(#j )
C (5.159)
and so it follows that
R(u ^ v)
C=
S(u)
S(v)
C:
(5.160)
Of course, we must have P(S(u) S(v) C) 2 sec C`(M; g). Since S(u)
S(v) = P(S(u) S(v)) + P? (S(u) S(v)) and we already know that
5.4 Curvature Biform R(u^v) Expressed in Terms of the Shape Operator
R(u ^ v) = P(S(u)
moreover we get that
S(v)) it follows that P? (S(u)
195
S(v)) = 0 and
P(S(u) S(v) C) = P(S(u) S(v)) C = P(S(u) S(v)) P(C): (5.161)
Lemma 363 Let C 2 sec C`(M; g) and u; v 2 sec
R(u ^ v) = Pv (S(u))
V1
T M ,! 2 sec C`(M; g)
(5.162)
Proof. Taking C = S(u) in Eq.(5.148) and recalling Eq.(5.96) P(S(u)) =
0:we get
Pv (S(u)) = P(S(u) S(v))
(5.163)
which proves the lemma.
Remark 364 From Eq.(5.163) we immediately have
Pu (S(v)) =
P(S(v) S(u)) = P(S(u) S(v)) =
Pv (S(u)) =
Pv (S(u))
(5.164)
where the last term follows from the fact that S(u) = S(u).
Proof. (of Proposition 356) We know that R(v) = @u R(u ^ v). Thus using
Eq.(5.164) and recalling Eq.(5.71) we can write
R(v) = @u Pv (S(u)) =
=
dP (S(v)) =
@u Pu (S(v))
S(S(v)) =
(5.165)
2
S (v):
Since we already showed that R(v) = @ ^ @ (v) we get
@ ^ @ (v) =
S2 (v)
and the proposition is proved.
Remark 365 Take notice that whereas S(v) is a section of C`(M; g), S2 (v) 2
V1
sec T M ,! sec C`(M; g):
V1
Proposition 366 Let u; v; w 2 sec T M ,! sec C`(M; g). Then,
1
@w ^ [Pv ; Pu ](w):
(5.166)
2
V1
Proof. From Eq.(5.153) with C = w 2 sec T M ,! sec C`(M; g) we have
R(u ^ v) =
R(u ^ v)
w = [Pu ; Pv ](w):
Now, the …rst member of Eq.(5.167) is
R(u ^ v)
w=
wyR(u ^ v)
(5.167)
196
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
Now, writing R(u ^ v) = 21 ui v j Rijkl
1 i j
uv
2
k
@w ^ (R(u ^ v)
r
^ ( r yRijkl
k
^
l
we have
r
w) =
^ l) =
^ ( r yR(u ^ v))
2R(u ^ v):
(5.168)
Taking into account Eq.(5.167) and Eq.(5.168) the proof follows.
We can also prove the proposition as follows: directly from Eq.(5.155)
we can write
[Pu ; Pv ](w) = uk v l ([Dek ; Del ]
D[ek ;el ] )w = uk v l wj Rijkl i :
(5.169)
Thus
1 m @
1
^ (uk v l wj Rijkl i )
@w ^ [Pu ; Pv ](w) =
2
2 @wm
1
= uk v l Rimkl m ^ i
2
= uk v l Rkl = R(u ^ v)
(5.170)
and the proof is complete.
Proposition 367 Let u; v; w 2 sec
V1
T M ,! sec C`(M; g). Then,
R(u ^ v) = @w ^ Pv Pu (w):
(5.171)
Proof. Recall that we proved (Eq.(5.162)) that R(u ^ v) = Pv (S(u)).
Also Eq.(5.84) says that S(u) = S(u) = @w ^ Pw (u) for any u; w 2
V1
sec T M ,! sec C`(M; g). Now from Eq.(5.179) we have
Pw (u) = Pu (w) = u dP (w) = Du P (w) = Du (P(w)) P(Du w) = (Du w)?
(5.172)
which means that
Pw (u) = (Pw (u))? :
(5.173)
Then, we have that
R(u ^ v) = Pv (S(u)) = Pv (@w ^ Pw (u))
= Pv (@w ^ Pu (w)) = Pv ((@w )k ^ (Pu (w))? )
Eq:(5:182)
=
@w ^ Pv Pu (w))?
and the proof is complete.
Remark 368 Since R(u^v) =
V1
sec T M ,! sec C`(M; g)
R(v^u), Eq.(5.171) implies that u; v; w 2
@w ^ Pv Pu (w) =
@w ^ Pu Pv (w);
thus exhibiting the consistency of Eq.(5.171) with Eq.(5.166).
(5.174)
5.5 Some Identities Involving P and Pu
197
5.5 Some Identities Involving P and Pu
Here we derive some identities involving the projection operator that has
been used in the above sections. The projection operator P has been de…ned
by Eq.(5.68) and its covariant derivative Pu := u dP has been de…ned by
Eq.(5.81). Let C; D 2 sec C`(M; g). Since
P(C ^ D) = P(C) ^ P(D);
we have for any u 2 sec
V1
(5.175)
T M ,! sec C`(M; g) that
Pu (C ^ D) = Pu (C) ^ P(D) + P(C) ^ Pu (D):
(5.176)
From P2 (C) = P(C) we have that
Pu P(C) + PPu (C) = Pu (C):
(5.177)
Now, we easily verify that
Pu (w) = P? (u dw);
Pw (u) = P? (w du):
(5.178)
Now, we already know from Eq.(5.128) that P? (u dw) and P? (w du)
are equal and thus
Pu (w) = Pw (u):
(5.179)
From this equation also follows immediately that
Pu P(w) = Pw P(u):
(5.180)
Now given that each X 2 sec C`(M; g) can be written as X = Xk + X? ,
with Xk = P(X ) we get from Eq.(5.177) that
PPu (Xk ) = 0,
PPu (X? ) = Pu (X? ):
(5.181)
Also, from Eq.(5.176) we have immediately taking into account Eq.(5.181)
for any C; D 2 sec C`(M; g) that
Pu (Ck ^ D? ) = Ck ^ Pu (D? ):
(5.182)
5.6 Conclusions
We presented a thoughtful presentation of the geometry of vector manifolds using the Cli¤ord bundle formalism, hopping to provide a useful text
for people (who know the Cartan theory of di¤erential forms) 10 and are
1 0 This
includes people, we think, interested in string and brane theories and GRT.
198
5. Cli¤ord Bundle Approach to the Di¤erential Geometry of Branes
interested in the di¤erential geometry of submanifolds M (of dimension m
equipped with a metric11 g = i g of signature (p; q) and its Levi-Civita
connection D) of a manifold M ' Rn (of dimension n and equipped with a
metric g of signature (p; q) and its Levi-Civita connection D). We proved
in details several equivalent expressions for the curvature biforms R(u ^ v)
and moreover proved that the Ricci operator @ ^ @ when applied to a 1form …eld v is such that @ ^ @ (v) = R(v) = S2 (v) (R(v) = Rba b ) is
the negative of the square of the shape operator S. It will be shown in
Chapter 11 that when this result is applied to GRT it permits to give a
mathematical realization to Cli¤ord’s theory of matter.
We observe that our methodology in this Chapter di¤ers considerably
[172, 174, 397]. Indeed, we use in our approach the Cli¤ord bundle of differential forms C`(M; g)) and give detailed and (we hope) intelligible proofs
of all formulas, clarifying some important issues, presenting, e.g., the precise relation between the shape biform S evaluate at v (a 1-form …eld) and
the connection extensor ! evaluated at v (Eq.(5.107)). In particular, our
approach also generalizes for a general Riemann-Cartan connection the results in [174] which are valid only for the Levi-Civita connection D of a
Lorentzian metric of signature (1; 3). Moreover our approach makes rigorous the results in [174] which are valid only for 4-dimensional Lorentzian
spacetimes admitting a spinor structure 12 , since in [174] it is postulated
that the frame bundle of M has a global section (there called a …ducial
frame).
11 i
: M ! M is the inclusion map.
result follows from Geroch theorem (Chapter 7). See [146, 147] for details.
1 2 This
This is page 199
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6
Some Issues in Relativistic Spacetime
Theories
6.1 Reference Frames on Relativistic Spacetimes
In this Chapter M = (M; g; D; g ; ") denotes a Lorentzian spacetime and
M = (M; ; D; ; ") denotes Minkowski spacetime. We adopt, as before,
natural units such that the value of the velocity of light is c = 1.
Proposition 369 Let Q 2 sec T U
sec T M be a time-like vector …eld
such that g(Q; Q) = 1. Then, there exist, in a coordinate neighborhood U ,
three space-like vector …elds ei which together with Q form an orthogonal
moving frame for x 2 U .
Proof. Suppose that the metric of the manifold in a chart (U; ) with
coordinate functions fx g is g = g dx
dx . Let Q = (Q @=@x ) 2
sec T M be an arbitrary reference frame and Q = g(Q; ) = Q dx ; Q =
g Q . Then, g (x)Q Q = 1. Now, de…ne
0
=
=
( Q ) dx
= Q dx ;
Q Q
g :
(6.1)
Then the metric g can be written due to the hyperbolicity of the manifold
as
200
6. Some Issues in Relativistic Spacetime Theories
g=
3
X
a
a
a
ab
=
b
;
(x)dx
dx :
(6.2)
a=1
Now, call e0 = Q and take ea such that a (eb ) = ba . It follows immediately that g(ea ; eb ) = ab , a; b = 0; 1; 2; 3:
Before we proceed we need to know precisely how the metric g relates
tangent space magnitudes to magnitudes on the manifold. Let
: R
I ! M , be a smooth curve, i.e., is C 0 and piecewise C 1 . We denote the
inclusion function I ! R by u, and the distinguished vector …eld on I by
d=du. For each u 2 I; u denotes the tangent vectors at (u) 2 M . Thus,
u
where
=
(
d
)
du
(u)
2T
(u) M;
(6.3)
denotes the derivative mapping of the mapping .
De…nition 370 A curve is said timelike (respectively lightlike, or respectively spacelike) if for all u 2 I, g( u ; u ) > 0 (respectively g( u ; u ) =
0, or respectively g( u ; u ) < 0).
De…nition 371 The path length between events e1 = (a) and e2 = (b)
along the curve , such that for all u 2 I, g( u ; u ) has the same signal
at all points along (u) is the quantity
Zb
du [jg(
u;
u )j]
1
2
:
(6.4)
a
Observe that taking the point (a) as a reference point we can use
Eq.(6.4) to de…ne a function with domain (I) by
s : (I) ! R; s( (u)) = s(u) =
Zu
du0 [jg(
u0 ;
u0 )j]
1
2
:
(6.5)
r
With Eq.(6.5) we can calculate the derivative of the function s (after
introducing a local chart fx g covering the domain of interest)
ds
= [jg(
du
u0 ;
u0 )j]
1
2
=
g
dx
du
dx
du
1
2
(6.6)
From Eq.(6.6) old books on di¤erential geometry and almost all books
on General Relativity writes the equation
(ds)2 = g dx dx ;
(6.7)
6.1 Reference Frames on Relativistic Spacetimes
201
which is supposed to represent the square of the length of the ‘in…nitesimal’
arc determined by the coordinate displacement
x
(r) 7! x
(r) +
dx
du
(r)";
(6.8)
where " << 1 is an ‘in…nitesimal’number.
The above mathematically correct notation is somewhat cumbersome,
and when no confusion arises x
(r) is denoted simply by x (r).
We recall that a moving frame at x 2 M is a basis for the tangent space
Tx M . An orthonormal frame at x 2 M is a basis of orthonormal vectors
for Tx M .
De…nition 372 An observer in a spacetime M is a future pointing timelike curve : R I ! M such that g( ; ) = 1. The timelike curve is
said to be the world line of the observer.
Bradyons and Luxons
One of the important ingredients of physical theory is the concept of particle. Roughly speaking a particle is de…ned by the attributes it carries and
a classi…cation of particles can only be given in the context of a particular theory. However, one of the attributes of a particle is its inertial mass,
modelled classically by m 2 R+ + f0g. We can give the
De…nition 373 A massive scalar particle is a pair (m; ), with m > 0 and
: I ! M a timelike curve pointing to the future such that g( ; ) = 1.
A scalar particle as above de…ned is sometimes called a scalar bradyon.
Relativistic quantum …eld theory admits the existence in nature zero mass
particles, also called luxons, as, e.g., photons. Now, particles with m > 0 or
m = 0 may also carry intrinsic spin, and the classical description of that
property in the case m > 0 will be given in Section 6.1.5 and Chapter 6.
Photons also carry spin, but in that case, a coherent description of their
properties can only be given in the context of relativistic quantum …eld
theory [385], which is a subject that will not be discussed in this book.
However, a crucial property of photons is that their paths in spacetime are
null geodesics. These null geodesics are also supposed in classical GRT to
be the paths followed in spacetime by light rays. This assumption can be
justi…ed classically once we suppose light is a wave phenomenon described
by Maxwell equations, and de…ne light rays as normals to wave fronts. For
details see, e.g., [77, 139].
Axiom 374 (standard clock postulate) Let be an observer. Then, there
exists standard clocks that can be ‘carried by ’ and such that they register (in ) proper-time, i.e., the inclusion parameter u of the de…nition of
observer.
202
6. Some Issues in Relativistic Spacetime Theories
It seems that modern atomic clocks are standard clocks [15], and indeed
they are used as such clocks in the GPS system [14, 25]. However we must
call the readers attention that the earlier experiments by Hafelle-Keating
always quoted in textbooks, like e.g., in [81] were not precise enough to
verify any claim [116, 190]. In [81] it is also claimed that the lifetime of
unstable elementary particles, like, e.g., the lifetimes of muons traveling
at near the speed of light in a storage ring [31] are compatible with the
standard clock postulate. However, as observed by Apsel [22], there are
indeed some small discrepancies. On this issue see also [336, 339].
De…nition 375 An instantaneous observer is an element of T M , i.e., a
pair (x; Q); where z 2 M , and Q 2 Tx M is a future pointing unit timelike
vector. SpanQ Tx M is the local time axis of the observer and Q? is the
observer rest space.
Of course, Tx M = SpanQ Q? , and we denote in what follows SpanQ =T
and Q? = H, which is called the rest space of the instantaneous observer.
If : R
I ! M is an observer, then ( u; u) is said to be the local
observer at u and write
T
uM
= Tu
Hu ; u 2 I:
De…nition 376 The orthogonal projections are the mappings
pu = T
uM
! Hu ; q u : T
uM
! Tu :
(6.9)
Then if Y is a vector …eld over then pY and qY are vector …elds over
given by
(pY)u = pu (Yu ); (qY)u = qu (Yu ):
(6.10)
De…nition 377 Let (x; Q) be a instantaneous observer and px : Tx M !
H the orthogonal projection. The projection tensor is the symmetric bilinear mapping h : T M T M ! R such that for any U; W 2 Tx M we
have:
hx (U; W) = g x (pU; pW)
(6.11)
Let fx g be coordinates of a chart covering U
Q = g x (Q; ). We have the properties:
(a)
(b)
(c)
(d)
(e)
(f)
M , with x 2 U and
hx = g x
Q
Q
hjQ? = g x jQ?
h(Q; ) = 0
h(U; ) = g(U; ) , g(U;Q) = 0
p = h @x@ x dx jx
trace(h @x@ x dx jx ) = 3
The Proposition 369 together with the above de…nitions suggests:
(6.12)
6.1 Reference Frames on Relativistic Spacetimes
203
De…nition 378 A reference frame for U M in a spacetime M is a timelike vector …eld which is a section of T U such that each one of its integral
lines is an observer.
De…nition 379 Let Q 2 sec T M; be a reference frame. A chart in U M
of an oriented atlas of M with coordinate functions fx g such that @=@x0 2
sec T U is a timelike vector …eld and the @=@xi 2 sec T U (i = 1; 2; 3) are
spacelike vector …elds is said to be a possible naturally adapted coordinate
chart to the frame Q (denoted (nacs|Q) in what follows) if the space-like
components of Q are null in the natural coordinate basis f@=@x g of T U
associated with the chart.
Note that such chart always exist [48].
6.1.1 Classi…cation of Reference Frames I
An arbitrary reference frame Q 2 sec T U sec T M for a general Lorentzian
spacetime may be classi…ed according to: (i) a decomposition of DQ given
by Eq.(6.42) below or, (ii) according to its synchronizability. In order to
present these concepts we shall need to introduce some additional mathematical tools, as the Fermi-Walker connection and in…nitesimally nearby
observers. However, even without that concepts we already may give the
following de…nitions.
De…nition 380 A reference frame Q 2 secT U
geodetic) if and only if aQ = D Q Q = 0.
sec T M is in free fall (or
De…nition 381 A reference frame Q 2 secT U
0; Q is said an inertial reference frame (IRF).
sec T M such that DQ =
Remark 382 IRFs exist in Minkowski spacetime, as it is easy to verify.
However, we can also easily show that in the GRT where each gravitational
…eld is modelled by a Lorentzian spacetime there is in general no frame
Z 2 sec T M satisfying DZ = 0. Indeed, existence of an IRF is possible
[367] only in a spacetime where the Ricci tensor satis…es Ricci(Z; F) = 0,
for any F 2 sec T M . This excludes, e.g., Einstein-de Sitter spacetime.
So, no IRF exist in many models of GRT considered to be of interest by
one reason or another by ‘professional relativists’. The best we can have
are some reference systems maintaining some of the characteristics of an
IRF. These reference frames are the pseudo inertial reference frames and
the locally inertial reference frames associated to an observer in free fall.
These important concepts will be discussed below (Section 6.9).
6.1.2 Rotation and Fermi Transport
Let
over
: R
I ! M , 7! ( ) be an observer. Let Y be a vector …eld
. As well known [367] in order for the observer to decide when a
204
6. Some Issues in Relativistic Spacetime Theories
unitary vector Y 2 ( )? has the same spatial direction of the unitary
vector Y0 2 ( 0 )? ( 0 =
6
), he has to introduce the concept of the FermiWalker connection.
Proposition 383 There exists one and only one connection F over ,
such that
FX Y = [p( D)X p + q( D)X q]Y
(6.13)
for all vector …elds X on I and for all vector …elds Y over :
In Eq.(6.13) D is the induced connection over of the Levi-Civita connection D and F is called the Fermi-Walker connection over , and we shall
use the notations F ; F=d or F"0 (see below) when convenient. We also
will write (by abuse of language) only D, as usual, for D in what follows.
Proof. The proof follows at once from the general properties concerning
the behavior of connections under pullback mappings [48].
De…nition 384 A moving (orthonormal ) frame f a g over is an orthonormal basis for T (I) M with 0 =
. The set f"a g; "a 2 sec T (I) M
,! sec C`(M; g) is the dual comoving frame on , i.e., "a ( b ) = ba :The set
f"a g, "a 2 sec T (I) M ,! sec C`(M; g), with "a "b = ba is the reciprocal
frame of f"a g.
Let X and Y be vector …elds over and Y = g(X; ); Y = g(Y; ) 2
sec T (I) M ,! C`(M; g) the physically equivalent 1-form …elds.
Proposition 385 Let be X; Y be form …elds over , as de…ned above. The
Fermi-Walker connection F satis…es the properties
(a)
F o Y = D 0 Y ("0 Y )a + (a Y )"0 ;
where a = D 0 "0 ; is the (1-form) acceleration.
d
(b)
d (X Y ) = F 0 X Y + X F 0 Y
(c)
F 0 "0 = 0
(d)
If X; Y are vector …elds on such that X u ; Yu 2 Hu 8u 2 I
then1 g(F 0 X; )ju and g(F 0 Y; )ju 2 Hu , 8u and
F 0X Y = D 0X Y
(6.14)
Proof. We prove only (a). The other properties are trivial. First we prove
that
F 0 Y = D 0 Y ( 0 Y)a + (a Y) 0
(6.15)
with a =D
0
0,
and where we wrote
0
Y = g ( 0 ; Y), a Y = g(a; Y):
1 Recall the notations introduced in Chapter 4, where g denote the metric in the
cotangent bundle.
6.1 Reference Frames on Relativistic Spacetimes
205
From Eq.(6.13) and the abuse of notation mentioned above and taking
into account that q(D 0 0 ) = 0, we can write
F o Y = p[D 0 (pY)] + q[D 0 (qY)]
(6.16)
Now,
q[D 0 (qY)] = qfD 0 (Y
= qf(D 0 Y)
= [(D 0 Y)
0) 0g
0 )"0
0] 0
+ Y (D
+ (Y a)
0
0) 0
+ (Y
0 )D
0
0g
0
= q(D 0 Y) + (Y a) 0 :
(6.17)
Also,
p[D 0 (pY)] = p[D 0 Y
D 0 (qY)]
= p(D 0 Y)
pfD 0 (Y
= p(D 0 Y)
pf[(D 0 Y)
= p(D 0 Y) (Y
0) 0g
0] 0
+ (Y a)
0
+ (Y
0 )a:
0 )ag
(6.18)
Summing Eq.(6.17) and Eq.(6.18) we get Eq.(6.15). Then (a) follows at
once we take into account that for any X; Y 2 sec T M;
g(D 0 Y;) = D 0 [g(Y; )] = D 0 Y;
(6.19)
which proves the result.
Now, let Y0 2 sec T 0 M . Then, by a well known property of connections
there exists one and only one 1-form …eld Y over such that F o Y = 0
and Y ( 0 ) = Y0 . So, if f "a j 0 g is an orthonormal basis for T 0 M ( "a j 0 2
T 0 M , a = 0; 1; 2; 3) we have that the "a ’s such that F o "a = 0 are
orthonormal for any , as follows from (b) in proposition 385. We then,
give the
De…nition 386 Let Y1 2 H 1 and Y2 2 H 2 are said to have the same
spatial direction if and only if Y1 = ai "i j 1 , Y2 = ai "i j 2 .
This suggests the following de…nition:
De…nition 387 We say that Y 2 sec T M is transported without rotation
(Fermi transported ) if and only if F 0 Y = 0.
In that case we have
D 0 Y = "0 @Y
D
Y = (Y "0 )a (a Y )"0
d
= Y y("0 ^ a) = (a ^ "0 )xY:
(6.20)
206
6. Some Issues in Relativistic Spacetime Theories
6.1.3 Frenet Frames over
Proposition 388 If f"a g is a comoving coframe over (De…nition 384),
then there exists a unique biform …eld D over , called the angular velocity
(Darboux biform) such that the "a satisfy the following system of di¤ erential
equations,
D 0 "a =
D x"a ;
(6.21)
D
=
1
!ab "a ^ "b =
2
1
(D 0 "b ) ^ "b :
2
Proof. It follows at once if we take into account that "a "b =
Eq.(2.36) and Eq.(2.37).
b
a
and use
Corollary 389 If the comoving coframe f"a g is Fermi transported then
the angular velocity is F ;
F
= a ^ "0 :
(6.22)
Proof. The result follows from Eq.(6.20) and Eq.(6.21).
6.1.4 Physical Meaning of Fermi Transport
Suppose that a comoving coframe f"a g is Fermi transported along a timelike curve , ‘materialized’by some particle. The physical meaning associated to Fermi transport is that the spatial axis of the tetrad g("i ; ) =
i ; i = 1; 2; 3 are to be associated to the orthogonal spatial directions of
three ‘small’gyroscopes carried along :
De…nition 390 A Frenet coframe ffa g over
such that f0 = g( ; ) = g( 0 ; ) = "0 and
D 0 fa =
D
=
is a moving coframe over
D xfa ;
0f
1
^ f0 +
1f
2
^ f1 +
2f
3
^ f 2;
(6.23)
where i ; i = 0; 1; 2 is the i-curvature, which is the projection of
f i+1 ^ f I plane.
D
in the
De…nition 391 We say that a 1-form …eld Y over
is rotating if and
only if it is rotating in relation to gyroscopes axis, i.e., if F 0 Y 6= 0.
6.1.5 Rotation 2-form, Pauli-Lubanski Spin 1-Form and
Classical Spinning Particles
From Eq.(6.23) taking into account that a = D 0 f0 =
we can write
D = a ^ f0 + S :
0f
1
and Eq.(6.22)
(6.24)
6.2 Classi…cation of Reference Frames II
207
We now show that the 2-form S over is directly related (a dimensional
factor apart) with the spin 2-form of a classical spinning particle. More, we
show that the Hodge dual of S is associated with the Pauli-Lubanski 1form. To have a notation as closely as possible the usual ones of physical
textbooks, let us put f0 = v.
Call ? S the Hodge dual of S . It is the 2-form over given by
?
S
=
Sf
5
; f 5 = f 0f 1f 2f 3:
De…ne the rotation 1-form S over
S=
Since
S xv
(6.25)
by
?
S xv:
(6.26)
= 0, we have immediately that
S v=
?
S x(v
^ v) = 0:
Now, since D 0 (S v) = 0 we have that (D 0 S) a =
D 0S =
(6.27)
S a and then
Sy(a ^ v);
(6.28)
and it follows that
F 0 S = 0:
(6.29)
It is intuitively clear that we must associate S with the spin of a classical
spinning particle which follows the worldline . And, indeed, we de…ne the
Pauli-Lubanski spin 1-form by
W = k~S; ~ = 1;
(6.30)
where k > 0 is a real constant and ~ is Planck constant, which is equal to 1
in the natural system of units used here. We recall that as it is well-known
D 0 W = W y(a ^ v) is the equation of motion of the intrinsic spin of a
classical spinning particle which is being accelerated by a force producing
no torque [447].
6.2 Classi…cation of Reference Frames II
6.2.1 In…nitesimally Nearby Observers, 3-Velocities and
3-Accelerations
Let Q 2 sec T U sec T M be a reference frame, s its ‡ux and an integral
line of Q and suppose that we have a parametrization : R
I ! M,
g( ; ) = 1. Then is an observer.
208
6. Some Issues in Relativistic Spacetime Theories
De…nition 392 An in…nitesimally nearby observer to is a vector …eld
W : R I ! T (I) M such that there exists a vector …eld W0 over such
that W = pW0 and which is Lie parallel with respect to Q, i.e., $Q W0 = 0.
W is sometimes called a Jacobi …eld.
Proposition 393 Suppose that Q is a geodesic frame, i.e., D Q Q = 0.
Then, $Q W = 0:
Proof. Recall that in a Lorentzian manifold, where the torsion tensor is null
the equation $Q W0 = 0 implies that D W0 Q = D Q W0 . Now, W = W0
g(W0 ; Q)Q and then, [W; Q] [g(W0 ; Q)Q; Q] = 0. We need then to prove
that [g(W0 ; Q)Q; Q] = 0. We have,
[g(W0 ; Q)Q; Q] = D g(W0 ;Q)Q Q D Q g(W0 ; Q)Q
= [D Q g(W0 ; Q)]Q:
But, D Q g(W0 ; Q) = g(D Q W0 ; Q) = g(D W0 Q; Q) = D W0 g(Q; Q) = 0,
and the proposition is proved.
Now, let us examine following [367] the geometrical meaning of an in…nitesimally nearby observer. Recall that for s 2 I there is a neighborhood
E = (s "; s + ") of s and a neighborhood U of (s) and a vector …eld
V 2 sec T U such that W = V on E and $Q V = 0. Let (U; ) be a map
of the maximal atlas of M with coordinates2 fx g covering (U ) such that
QjU = @@x0 U . We write WjE = w ( @ @x
). We may write for p 2 U;
E
0
1
2
3
(p) = f(x (p); x (p); x (p); x (p)) j jx (p)j < "; 8 g and assume that for
p = (s), ( (s)) = (s; 0; 0; 0). There is then a congruence of integral
curves of Q determined by
(s; u) 7! (s + w0 u; w1 u; w2 u; w3 u):
(6.31)
Note that u = 0 gives jE and u times a constant gives another curve
of the congruence in U where the parametrization given by fx g holds.
Now, when WjE and Q are linearly independent, di¤erent curves have
di¤erent images. This uniquely determines WjE as its transversal vector
…eld, i.e., for any f : U ! R and s 2 E,
(Wf )(s) =
@
f (s + w0 u; w1 u; w2 u; w3 u)
@u
.
(6.32)
u=0
Conversely, once WjE is given, the family is determined up to …rst order
in x0 in the sense of a Taylor series.
In the coordinates fx g where V = V @ @x , the equation $Q V = 0 implies that Q(V ) = 0. Let fwu g be real constants such that Wj 0 =
) for some 0 2 ". Then putting V = w @ @x we have
w ( @ @x
0
2 This
chart always exist [48].
6.2 Classi…cation of Reference Frames II
209
$Q V = 0. Then, the interpretation of W becomes clear. It represents
the linearized version of a one parameter family of integral curves of Q
near . Note however that in an arbitrary chart chart (U; 0 ) with coordinates fx g covering 0 (U ), where V = V @x@ , the equation $Q V = 0
implies that
@Q
V ;
(6.33)
Q[V ] =
@x
which is nonzero in general.
De…nition 394 Let be an observer on Q and W be an in…nitesimally
nearby observer to
and F the Fermi-Walker connection over . Then,
F W and F 2 W = F (F W ) are called respectively the 3-velocity and
the 3-acceleration of the in…nitesimally nearby observer W relative to .
We observe that we can show (see below) that for any s 2 I, (F W)u 2
Hs and (F 2 W) 2 Hs , thus justifying the names 3-velocity and the 3acceleration. The meaning of these concepts become clear with the aid of
the following propositions.
Proposition 395 Let Q 2 sec T U sec T M be a reference frame and :
I ! M an observer in Q. Let also Y 2 Hs . Then, the mapping Y 7!D Y Q
de…nes a linear transformation AQ : Hs ! Hs , such that if W is any
in…nitesimally nearby observer to we have
F W = AQ W = D W Q:
Proof. First we need to show that AQ Hs
since 8Y 2 Hs we have
g(AQ Y; Q) = g(D Y Q; Q) =
(6.34)
Hs , 8s 2 I. But this is trivial
1
D Y g(Q; Q) = 0:
2
(6.35)
Now, we need to show that 8s 2 I F s W = DWs Q. To show that is, of
course, equivalent to show that 8Y 2 Hs ;
g(F
S
W; Y) = g(D Ws Q; Y):
(6.36)
Now, recall (e) of proposition 385 which says that
g(F
S
W; Y) = g(D
s
Q; Y):
Then, we need only to prove that
g(D
S
W; Y) = g(D
s
Q; Y); 8Y 2 Hs :
(6.37)
Since W is an in…nitesimally nearby observer to
we know that there
exists W0 , a vector …eld over , such that the projection pW0 = W and
such that $Q W0 = 0. Of course, W0 = W f for some smooth function
210
6. Some Issues in Relativistic Spacetime Theories
f on . Now, let V0 and F be respectively a vector …eld and a smooth
function de…ned on a su¢ ciently small open set U of (s) such that
V0
= W0 ;
[V0 ; Q] = 0;
F
= f:
(6.38)
Moreover, de…ne
V = V0
FQ
(6.39)
such that V
= W. Now, since the torsion of a Lorentzian spacetime is
zero, we have DV Q DQ V [V; Q] = 0, and we can write
D V Q = D Q V + Q(F )Q:
df
At s, we have D Vs Q =D s W+ ds
(s) s . This implies that g(D
g(D s Q; Y); 8Y 2 Hs and the proposition is proved.
(6.40)
S
W; Y) =
Now, recall that since g(W; Q) = 0 then W has only spatial components
relative to an orthonormal frame fea g 2 sec PSOe1;3 (U ), with e0 = Q and
Fe 0 ea = 0. Then from Eq.(6.34) we have (i; j = 1; 2; 3)
d
Wi (s) = (D e j
ds
Q )i W
j
(s);
(6.41)
where Q = g(Q; ). So, the …rst member of Eq.(6.41) clearly shows that the
Fermi derivative F W is a measure of the rate of change of the coordinates
of an in…nitesimally nearby observer W to on Q. To interpret the second
member of Eq.(6.41) we need the
Proposition 396 Given an arbitrary reference frame Q 2 sec T U sec T M
for a Lorentzian spacetime [367] there exists a unique decomposition of
Q = g(Q; ) as
D
= aQ
Q
Q
+ !Q +
Q
1
+ EQ h;
3
(6.42)
where h 2 sec T20 M is the projection tensor (De…nition 377), aQ is the
(form) acceleration of Q, !Q is the rotation tensor (or vortex ) of Q,
Q is the shear of Q and EQ is the expansion ratio of Q . In a coordinate
chart (U; ) with coordinates x covering (U ), writing Q = Q @=@x and
h = (g
Q Q )dx
dx we have
aQ = g(D Q Q; );
!Q
Q
= Q[
; ]h
= [Q(
; )
EQ = Q ; :
h ;
1
EQ h ]h h ;
3
(6.43)
6.2 Classi…cation of Reference Frames II
211
Proof. It left as exercise.
Clearly, !Q measures the rotation that one of the in…nitesimally nearby
curves to had in an in…nitesimal lapse of propertime with relation to an
orthonormal basis Fermi transported by the observer at . Also, Q represents the ratio of change of the separation between and an in…nitesimally
nearby curves. Finally, as it is easy to verify, EQ = Q gives the fractional
ratio expansion of the 3-dimensional volume element de…ned by and its
nearby curves.
Remark 397 A preliminary classi…cation of reference frames in RiemannCartan spacetimes is given in [149].
Exercise 398 Compute aQ ; !Q ;
2 PSOe1;3 (U ), with e0 = Q and F
Q
0
and EQ in the orthonormal frame fei g
i = 0, a = ea j .
Exercise 399 Show that
Q
^d
Q
= 0 , !Q = 0.
(6.44)
6.2.2 Jacobi Equation
Lemma 400 Let Q 2 sec T U sec T M be a reference frame and : I !
M an observer in M . Let also Y 2 Hs . If is the Riemann curvature
operator (see Eq.(4.102)) then the mapping
Y 7! (Q;Y)Q
(6.45)
de…nes a mapping Hs ! Hs :
Proof. The result follows once we verify that g(Q; (Q;Y)Q) = 0, which
is a simple consequence of the symmetries of the Riemann tensor.
Proposition 401 Let Q be a free fall frame,
an in…nitesimally nearby observer to :Then,
F2 W = (
; w)
an observer on Q and W
;
(6.46)
Proof. Recall that Proposition 393 says that $Q W = 0 when DQ Q = 0.
Now, let s 2 I and let V be a vector …eld de…ned in some su¢ ciently
small neighborhood of (s) and such that [V; Q] = 0 and W = V .
Then, taking into account that in a Lorentzian manifold the torsion tensor
is zero, which means that D Q V D V Q [Q; V] = 0, and that D Q Q = 0
by hypothesis, we can write
D 2Q V = D Q (D Q V) = D Q fD V Q + [Q; V]g = D Q D V Q
= D Q D V Q D VDQ Q D [Q;V] Q
= (Q;V)Q:
(6.47)
212
6. Some Issues in Relativistic Spacetime Theories
Restricting Eq.(6.47) to and taking into account that F = D since
is geodesic, the Eq.(6.46) follows.
Eq.(6.46) is known as the geodesics separation equation or Jacobi equation.
Example 402 Consider a Friedmann-Robertson-Walker universe which is
modelled by a Lorentzian spacetime with metric given by
g = dt
dt
R(t)2
dx1
dx1 + dx2 dx2 + dx3
(1 + Kr2 =4)2
dx3
;
(6.48)
p
where r =
(x1 )2 + (x2 )2 + (x3 )2 , K is a constant and R(t) a smooth
@
(the comoving reference
function (t > 0). The reference frame Q = @t
frame) in a Friedmann-Robertson-Walker universe is supposed to be realized
by a dust of ‘particles’ representing the matter of the universe. Choose an
integral line of Q. As an in…nitesimally nearby observer to take, e.g.,
the vector …eld over given by
W=
@
@x1
:
(6.49)
(I)
Let us calculate F 2 W. It is clear that we must take
= Qj (I) and
V = @=@x1 . We get:
•
R
(6.50)
F 2 W = W;
R
which means that nearby observers, even if they have constant coordinates
in the chart ft; xi g naturally adapted to Q shows a 3-acceleration in that
• 6= 0. Recall however that the acceleration of any observer
universe if R
on Q is D
= 0.
6.3 Synchronizability
We can now give a classi…cation of reference frames according to their
synchronizability and discuss carefully the meaning of that concept.
De…nition 403 Let Q 2 sec T U sec T M be an arbitrary reference frame
and Q = g(Q; ). We say that Q is locally synchronizable i¤ Q ^d Q = 0.
The reference frame Q is said to be locally proper time synchronizable if
and only if d Q = 0. Q is said to be synchronizable if and only if there are
C 1 functions h; t : M ! R such that Q = hdt and h > 0. Q is proper
time synchronizable if and only if Q = dt.
At …rst sight, the classi…cation of reference frames according to their
synchronizability does not look intuitive, so it is a good idea to see from
where it came from.
6.3 Synchronizability
213
FIGURE 6.1. Einstein Synchronization of Standard Clocks
6.3.1 Einstein Synchronization Procedure
To start, let Q 2 sec T U
sec T M be an arbitrary reference frame, and
let (U; ') be a chart covering U with coordinate functions fx g which are
used by the observers in Q to label events.
Now, consider two clocks A and B at rest on Q 2 sec T U
sec T M ,
and let and be their world lines (which are integral lines of Q) and
which moreover we suppose to be worldlines of two nearby observers in the
sense of De…nition 392. According to the standard clock postulate (Axiom
374) the events on and can be ordered. Let e1 ; e; e2 be three events on
ordered as
e1 < e < e2 ;
(6.51)
and let e0 be an event on 0 (see Figure 5.1) At event e1 a light signal
is sent from clock A to clock B where it arrives at the event e0 and is
instantaneously re‡ected back to clock A where it arrives at event e2 .
The question arises: which event e on is simultaneous to the event e0 ?
The answer to that question depends on a de…nition as realized by Einstein
[110] and long before him by Poincaré [299]. Let3
'(e1 ) = (x0e1 ; x1 ; x2 ; x3 );
'(e) = (x0e ; x1 ; x2 ; x3 );
'(e2 ) = (x0e2 ; x1 ; x2 ; x3 );
'(e0 ) = (x0e0 ; x1 +
x1 ; x2 +
x2 ; x3 +
x3 );
(6.52)
be the coordinates of the pertinent events in chart (U; '). The following
de…nition is known as Einstein’s synchronization procedure (or Einstein’s
convention).
3 We
are using an obvious sloppy notation.
214
6. Some Issues in Relativistic Spacetime Theories
De…nition 404 The event e in which is simultaneous to the event e0 in
is the one such that its timelike coordinate is given by
x0e = x0e1 +
1
2
x0e0
x0e1 + (x0e2
x0e0 ) :
(6.53)
Remark 405 The factor 1=2 in Eq.(6.53) is purely conventional. Non
standard synchronizations can be used as explained in details, e.g., by Reichenbach [315]. In a non standard synchronization we change the factor
1=2 by any function of the coordinates such that 0 < < 1.
Our next task is to determine the relation between the timelike coordinates of events e and e0 as functions of 4xi and the metric coe¢ cients.
Recall then that, as already observed in Section 6.2.1, in Relativity Theory
the motion of a ray of light is supposed to take place along a null geodesic.
So, let ` : R I ! M , u 7! `(u) be a null geodesic passing through events
e1 and e0 and e2 (see Figure 6.1). Then,
g(` ; ` ) = 0:
(6.54)
Calling 4x01 = x0e0 x0e1 , 4x02 = x0e2 x0e0 , and taking into account that the
‘in…nitesimal’arcs e1 e0 and e0 e2 in ` can be represented by the vectors
W1 = ( 4x01 ; 4x1 ; 4x2 ; 4x3 ); W2 = (4x02 ; 4x1 ; 4x2 ; 4x3 );
(6.55)
their arc length in are given by g(W 1 ; W1 ) = g(W 2 ;W 2 ) = 0. Solving
that equations for 4x01 and for 4x02 , we get
x0e = x0e0 +
gi0
4xi :
g00
(6.56)
6.3.2 Locally Synchronizable Reference Frame
Let us now explain the genesis of the de…nition of a locally synchronizable
frame Q 2 sec T M , for which Q ^ d Q = 0. We need to recall …rst some
purely mathematical results related to Frobenius theorem [26].
(i) A 3-direction vector …eld Hx in a 4-dimensional manifold M for x 2 M
is a 3-dimensional vector subspace Hx of Tx M which satis…es a di¤ erentiability condition. This condition is usually expressed as one of the two
equivalent propositions that follows:
Proposition 406 For each point x0 2 M there is a neighborhood U M
of x0 such that there exist three di¤ erential vector …elds, Yi 2 sec T M
(i = 1; 2; 3) such that Y1 (x); Y2 (x); Y3 (x), is a basis of Hx ; 8x 2 U:
Proof. see [26].
Proposition 407 For each point x0 2 M there is a neighborhood U M
of x0 such that there exist a one form 2 sec T M such that for all Y 2
Hx , (Y) = 0.
6.3 Synchronizability
215
Proof. see [26].
(ii) We now recall Frobenius theorem:
Proposition 408 (Frobenius) Let be x0 2 M and let be U a neighborhood
of x0 . In order that, for each x0 2 M there exist a 3-dimensional manifold
x0 3 x0 (called the integral manifold through x0 ) of the neighborhood U ,
tangent to Hx for all x 2 x0 it is necessary and su¢ cient that [Yi ; Yj ]x 2
Hx , (for all i; j = 1; 2; 3 and 8x 2 x0 ) if we consider the condition in
Proposition 406. If we consider the condition on Proposition 407 then a
necessary and su¢ cient condition for the existence of x0 is that
^ d = 0:
(6.57)
(iii) Now, let us apply Frobenius theorem for the case of a Lorentzian
manifold. Let Q 2 sec T M be a reference frame for which Q ^ d Q = 0.
Then, from the condition for the existence of a integral manifold through
x0 2 M we can write,
Q (Y1 )
=
Q (Y2 )
=
Q (Y3 )
= 0:
(6.58)
Now, since Q is a time like vector …eld, Eq.(6.58) implies that the Yi
(i = 1; 2; 3) are spacelike vector …elds. It follows that the vector …eld Q is
orthogonal to the integral manifold x0 which is in this case a spacelike
surface.
Now the meaning of a synchronizable reference frame (which is given by
Eq.(6.57)) becomes clear. Observers in such a frame can locally separate
any neighborhood U of x0 (where they “are”) in time space.
6.3.3 Synchronizable Reference Frame
Now, suppose that Q = Q @=@x 2 T U is synchronizable, i.e., there exists
a function t : U ! R such that Q = hdt. In this case we can choose a
(nacsjQ), with x0 = t as the time like coordinate in U . We have
(
whose solution is Q = p
Q)
= g Q dx = hdx0
1
@=@x0
g00 (x)
with g00 (x) = h2 and that gi0 (x) = 0.
This can be seen as follows.
When Q = p 1 @=@x0 , we have from Proposition 369 that
g00 (x)
0
= g(Q; )
can be written as
0
=
p
gi0 (x)
g00 (x)dx0 + p
dxi ;
g00 (x)
(6.59)
216
6. Some Issues in Relativistic Spacetime Theories
and also
3
X
i
i
=
(x)dx
dx
i=1
=
gi0 (x)gj0 (x)
g00 (x)
gij (x) dxi
dxj ; i; j = 1; 2; 3.
(6.60)
Now, since Q = 0 the metric g in the coordinates fx g must be diagonal with g00 (x) = h2 and that gi0 (x) = 0: In that case Eq.(6.56) implies
in that case
(6.61)
x0e = x0e0 ;
which justi…es the de…nition of a synchronizable frame.
It is an appropriate time to work out some examples of the above formalism, which, we admit, may look very abstract at …rst sight.
6.4 Sagnac E¤ect
As an important application of the concepts introduced above, we give in
this section a kinematical analysis of the Sagnac e¤ect [370]. Our description illustrates typical problems that arises when we deal with reference
frames that are not locally synchronizable (i.e., frames Q for which Q ^
d Q 6= 0). To go directly to the essentials, in this section (M; ; D; ; ")
is Minkowski spacetime.
We recall that according to De…nition 381, a reference frame I 2 sec T M
on Minkowski spacetime is inertial if and only if DI = 0.
Now, let (t; r; ; z) be cylindrical coordinates of a chart (U; ) for M such
that I = @=@t. Then
= dt
dt
dr
r2 d
dr
d
dz
dz:
(6.62)
@
@
(6.63)
Let P 2 T M be another reference frame on M given by
!2 r2 )
P = (1
1=2
@
+ !(1
@t
P is well de…ned in an open set U
!2 r2 )
1=2
M de…ned through the coordinates by
(U ) = f 1 < t < 1; 0 < r < 1=!; 0
< 2 ; 1 < z < 1g: (6.64)
Then,
P
= (P; ) = (1
!2 r2 )
1=2
and
P
^d
P
=
(1
dt
!r2 (1
!2 r2 )
2!r
dt ^ dr ^ d
!2 r2 )
1=2
d
(6.65)
(6.66)
6.4 Sagnac E¤ect
217
The rotation (or vortex) vector can be calculated from the formula wQ =
(!Q ; ) = ( ( P ^ d P ); )], being here the cotangent bundle Minkowski
metric. We have
@
wQ = !(1 ! 2 r2 ) 1 :
(6.67)
@z
Since wQ 6= 0, for 0 < r < 1=!. This means that P is rotating with
(classical) angular velocity ! (as measured) according to I in the z direction.
Now P can be realized on U M by a rotating platform, but is obvious
that at the same ‘time’on U , I cannot be realized by any physical system.
P is a typical example of a reference frame for which it does not exist a
(nacsjP) such that the time like coordinate of the frame has the meaning
of proper time registered by standard clocks at rest on P.
According to the classi…cation of reference frames given above it is indeed
trivial to see that P is not proper time synchronizable or even locally
synchronizable.
This fact is not very well known as it should be and leads some people
from time to time to claim that optical experiments done on a rotating
platform disproves the Special Relativity Theory (SRT). Recent claims of
this kind has been done by Selleri and collaborators on a series of papers
[379, 380, 91, 152] and by Vigier [439]. This last author wrongly stated
that in order to explain the e¤ects observed it is necessary to attribute a
non-zero mass to the photon and that this fact, by its turn, implies in the
existence of a fundamental reference frame.
There are also some other misleading papers that claim that in order
to describe the Sagnac e¤ect it is necessary to build a “non-abelian electrodynamics” [32, 33, 34, 35, 36, 21]. This is only a small sample of papers containing very wrong statements concerning the Sagnac experiment.4
Now, the Sagnac e¤ect is a well established fact (used in the technology of
the gyro-ring) that the transit time employed for a light ray to go around
a closed path enclosing a non-null area in a non inertial reference frame
depends on the sense of the curve followed by the light ray.
Selleri arguments that such a fact implies that the velocity of light as
determined by observers on the rotating platform cannot be constant, and
must depend on the direction, implying that nature realizes a synchronization of the clocks which are at rest on the platform di¤erent from the one
given by the Einstein method.
Selleri arguments also that each small segment of the periphery of the
rotating platform of radius R can be thought as at rest on an inertial
reference frame moving with speed !R relative to the laboratory (here
modeled by I = @=@t). In that way, he argues that if the synchronization
is done à l’Einstein between two clocks at neighboring points of a small
segment, the resulting measured value of the one way velocity of light must
4 Of
course, there are good papers on the Sagnac e¤ect, and [304] is one of them.
218
6. Some Issues in Relativistic Spacetime Theories
result constant in both directions (c = 1), thus contradicting the empirical
fact demonstrated by the Sagnac e¤ect.
Now, we have already said that P is not proper time synchronizable, nor
is P locally synchronizable, as can be veri…ed.
However, for two neighboring clocks at rest on the periphery of a uniformly rotating platform an Einstein’s synchronization can be done. Let us
see what we get.
First, let fb
x g be coordinate functions for U such that5
b
t = t;
In these coordinates
= (1
! 2 rb2 )db
t
= g d^
x
d^
x :
rb = r;
b=
2!b
r2 d b
dt
is written as
db
t
!t;
db
r
db
r
zb = z
r2 d b
(6.68)
db
db
z
db
z
(6.69)
Now take two standard clocks A and B, at rest on P. Suppose they follow
the world lines and 0 which are in…nitesimally close.
As we know (Axiom 374) the events on or 0 can be ordered. Let
e1 ; e; e2 2 with e1 < e < e2 , where
(e1 ) = (^
x0e1 ; x
^1 ; x
^2 ; x
^3 );
(e) = (^
x0e ; x
^1 ; x
^2 ; x
^3 );
(e2 ) = (^
x0e2 ; x
^1 ; x
^2 ; x
^3 );
(e0 ) = (^
x0e0 ; x
^1 + x
^1 ; x
^2 + x
^2 ; x
^3 + x
^3 ); (6.70)
with e1 is the event on when a light signal is sent from clock A to clock
B, e0 is the event when the light signal arrives at clock B on 0 and is
(instantaneously) re‡ected back to clock A where it arrives at event e2
and …nally e is the event simultaneously on to the event e0 according to
Einstein’s convention introduced above. Then according to Eq.(6.56)
x
^0e = x
^0e0 +
g0i
g00
x
^i 6= x
^0e0
(6.71)
We emphasize that Eq.(6.71) does not mean that we achieved a process
permitting the synchronization of two standard clocks following the world
lines and 0 , because standard clocks in general do not register the “‡ow”
of the time-like coordinate x0 . However, in some particular cases such as
when g is independent of x0 and for the speci…c case where the clocks are
very near (see below) and at rest on the periphery of a uniformly rotating
platform this can be done.
This is so because two standard clocks at rest at the periphery of a
uniformly rotating platform “tick tack”at the same ratio relative to I. Once
synchronized they will remain synchronized. It follows that the velocity of
light measured by these two clocks will be independent of the direction
5 Note
that f^
x g is a (nacs jP) since in these coordinates P =(1
!2 r2 )
1
2
@=@ t^.
6.4 Sagnac E¤ect
219
followed by the light signal and will result to be c = 1 every time that the
measurement will be done. This statement can be trivially veri…ed [348]
and is in complete disagreement with a proposal of [76]. We now analyze
with more details what will happen if we try the impossible task (since P
is not proper time synchronizable, as already said above) of synchronizing
standard clocks at rest at the ring of a rotating platform which is the
material support of the reference frame P.
Suppose that we synchronize (two by two) a series of standard clocks
(such that any two are very close 6 ) at rest and living on a closed curve
along the periphery of a rotating platform. Let us number the clocks as
0; 1; 2; :::; n. Clocks 0 and 1 are supposed “be”at the same point p1 and are
the beginning of our experiment synchronized. After that we synchronize,
clock 1 with 2, 2 with 3,... and …nally n with 0. From Eq.(6.71) we get
immediately that at the end of the experiment clocks 0 and 1 will not be
synchronized and the coordinate time di¤erence between them will be
I
I
!R2
g0i i
b
db
x =
d b:
(6.72)
t=
g00
1 ! 2 R2
For !R << 1 we have b
t = 2!S where S is the area of the rotating
platform and the signals refer to the two possible directions in each we
can follow around the rotating platform.
The correct relativistic explanation of the Sagnac experiment is as follows. Suppose (accepting the validity of the geometrical optics approximation) that the world line of a light signal that follows the periphery of the
rotating platform of radius R is the curve : R I ! M such that
is
a null vector. Using the coordinate b
t as a curve parameter we have
(
;
Then
) = (1
! 2 R2 )(db
t
)2
2!R2 (d b
)(db
t
)
R2 (d b
db
= ! 1=R:
db
t
It follows that the coordinate times for a complete round are
Tb = 2 R=1
!R;
)2 = 0:
(6.73)
(6.74)
(6.75)
where the signals refer to the two possible paths around the periphery,
with when the signal goes in the direction of rotation and + in the other
case. It is quite obvious that Tb can be measured by a single clock.
Length of the Periphery of a Rotating Disk
How observers on board of a rotating platform will de…ne the length of the
periphery of their disc? We think that a reasonable answer is that they will
6 Very close means that l=R
1, where l is the distance between the cloks and R is
the radius of the plataform, both distances being determined in the frame P.
220
6. Some Issues in Relativistic Spacetime Theories
de…ne such a length as an appropriate sum of the in…nitesimal distances
determined by them at a given coordinate time t (instantaneous observers).
This means the following. Each observer is equipped with a standard ruler,
device realized by a clock and a light emitter and receiver at rest on the
platform, which permits the determination of the time of ‡ight of a light
signal that are emitted from the position of the standard clock, go to the
in…nitesimal point in the disc whose distance is to be measured and is
re‡ected back to the clock. At coordinate time t all observers make measurements of the in…nitesimal distances that they are supposed to measure
and then one of them collect the results and e¤ectuates the appropriate
sum. From the above calculations it is clear that the metric of the rest
space corresponds to the projection tensor h (Eq.(6.11)) and we get7
L = 2 R(1
! 2 R2 )
1=2
:
(6.76)
Being
the proper times measured by standard clock at rest at the periphery of the rotating platform, corresponding to Tb , we have
= (1
! 2 R2 )1=2 Tb = L(1
!R):
(6.77)
This equation explains trivially the Sagnac e¤ect according to Special Relativity.
Selleri [379, 380, 91, 152] calls the quantities c
L
1
=
c
(6.78)
the global velocities of light in the rotating platform for motions of light in
the directions of rotation ( ) and in the contrary sense (+). He then argues
that these values must also be the local values of the one-way velocities
of light, i.e., the values that an observer would necessarily measure for
light going from a point p1 in the periphery of a rotating platform to a
neighboring point p2 . He even believes to have presented an ontological
argument that implies that Special Relativity is not true. Well, what is
wrong with Selleri’s argument is the following. Although it is true that the
global velocities c can be measured with a single clock, the measurement of
the local one way velocity of light transiting between p1 and p2 requires two
standard clocks synchronized at that points. Local Einstein synchronization
is possible and as described above gives a local velocity equal to c = 1, and
this fact leads to no contradiction.
Before concluding this section it is very much important to recall that a
reference frame …eld as introduced above is a mathematical instrument. It
7 This result follows at once from the de…nition Note that the claim by Klauber [193]
that the space geometry of the disc is ‡at is in contradiction with the way measurements
are realized in Relativity Theory.
6.5 Characterization of a Spacetime Theory
221
did not necessarily need to have a material substratum (i.e., to be realized
as a material physical system) in the points of the spacetime manifold where
it is de…ned. More properly, we state that the integral lines of the vector
…eld representing a given reference frame do not need to correspond to
world lines of real particles. If this crucial aspect is not taken into account
we may incur in serious misunderstandings.
Exercise 409 Do solid rulers on board of a rotating platform su¤ ers the
Lorentz contraction? Suppose that the platform and the small solid rulers
on board are of the same material. Suppose that when the platform is at
rest the maximum integral number of the small rulers around the periphery
of the disc is p0 and the maximum integral number of small rulers around
the diameter is d0 . Let p and q be the respective number of rulers when the
platform is rotating. How did you model the platform in both cases? Is p=q
= p0 =d0 ?
6.5 Characterization of a Spacetime Theory
In this section we de…ne what we mean by a general relativistic spacetime
theory [348]. In our approach a physical theory z is characterized by:
(i) a theory of a certain species of structure in the sense of Boubarki [52];
(ii) its physical interpretation;
(iii) its present meaning and present applications.
We recall that in the mathematical exposition of a given physical theory z, the postulates or basic axioms are presented as de…nitions. Such
de…nitions mean that the physical phenomena described by z behave in a
certain way. Then, the de…nitions require more motivation than the pure
mathematical de…nitions. We call coordinative de…nitions the physical definitions, a term introduced by Reichenbach [315]. Also, according to Sachs
and Wu [367] it is necessary to make clear that completely convincing and
genuine motivations for the coordinative de…nitions cannot be given, since
they refer to nature as a whole and to the physical theory as a whole.
The theoretical approach to Physics behind (i), (ii) and (iii) above is
then to admit the mathematical concepts of the species of structure de…ning z as primitives, and de…ne coordinately the observational entities from
them. Reichenbach assumes that “physical knowledge is characterized by
the fact that concepts are not only de…ned by other concepts, but are also
coordinated to real objects”. However, in our approach, each physical theory, when characterized as a species of structure, contains some implicit
geometric objects, like some of the reference frame …elds de…ned above,
that cannot in general be coordinated to real objects. Indeed, it would
be an absurd to suppose that the in…nity number of IRFs that (mathematically) exist in a Minkowski spacetime are simultaneously realized as
physical systems.
222
6. Some Issues in Relativistic Spacetime Theories
De…nition 410 A general relativistic spacetime theory is a theory of a
species of structure such that:
(i) If Mod z is the class of models of z, then each 2 Mod z contains as
substructure a Lorentzian spacetime M = (M; D; g; g ; "). We recall here
that g is a Lorentz metric and D is the Levi-Civita connection of g on M .
More precisely, we have
= ((M; D; g;
g ; ");
1; : : : ;
m)
;
(6.79)
(ii) The objects Si ; i = 1; 2, (S1 = g; S2 = D) of the substructure M characterize the geometry of a spacetime. The i 2 sec T M (the tensor bundle)8 ,
i = 1; : : : ; m are (explicit) geometrical objects de…ned in U M characterizing the physical …elds and particle trajectories that cannot be geometrized
in the theory. Here, to be geometrizable means to be a metric …eld or a connection on M or objects derived from these concepts as, e.g., the Riemann
tensor (or the torsion tensor in more general spacetime theories).
(iii) The mathematical objects oi = ( S1 ; S2 ; k , k = 1; : : : ; m) describing
any particular model 2 Mod z are supposed to satisfy the proper axioms
of the theory z, also called the equations of motion (or dynamical laws) of
z. The equations of motion for all spacetime theories analyzed in this work
are intrinsic equations, i.e., they do not need the introduction of a particular chart of the manifold to be presented 9 . Here we write the equations
of motion symbolically as Ooi = f (o1 ; ::oj ) where O is some (di¤ erential )
operator and f a ‘function’ of the oi :
(iv) Each spacetime theory has some implicit geometrical objects that do
not appear explicitly in Eq.(6.79). These objects are the reference frame
…elds (which we already de…ned above).
Remark 411 We shall investigate some crucial issues associated with the
fact that the equations of motion of the principal physical theories possesses
the mathematical property of being di¤ eomorphically invariant, a concept
to be introduced below.
6.5.1 Di¤eomorphism Invariance
De…nition 412 Let ; 0 2 Mod z,
= ((M; g; D; g ; "); 1 ; : : : ; m )
0
0
0
0
0
and
= ((M; g ; D ; g0 ; "); 1 ; : : : ; m ) with the g, D, g and the i ,
i = 1; : : : ; m de…ned in U
M and g 0 , D 0 , g0 and the 0i , i = 1; : : : ; m
8 Some
of the i may be sections of spinor bundles. See Chapter 6.
intrinsic equations, for any coordinate chart of the maximal atlas of M with
coordinate functions fx g translate as a set of partial di¤erential equations, which may
be nonlinear.
9 The
6.5 Characterization of a Spacetime Theory
223
de…ned in V
M . We say that and 0 are mathematically equivalent 10
0
(and denotes
) if
(i ) there exists h 2 GM such that 0 = h ; i.e., V
h(U ) and
D 0 = h D;
g 0 = h g;
0
1
=h
1 ; :::;
0
m
=h
m;
(6.80)
The models and 0 are:
(ii) free boundary solutions of the equations of motion of z, or (more
important)
(iii) solutions of well posed initial and boundary valued problems to the
equations of motion of z. This means that if the mathematical objects de…ning satisfy the equations of motion of the theory with initial and boundary
conditions B(U ) in U M , then the mathematical objects de…ning 0 satisfy di¤ eomorphically equivalent equations of motion (to the ones satis…ed
by ) and verify the initial and boundary conditions h B(U ) in h(U ) V:
In Physics literature any spacetime theory satisfying De…nition 412 is
said to be di¤ eomorphism invariant or generally covariant.
However, in the literature of GRT it is also stated that ; 0 2 Mod
z do represent the same physical model. Moreover, due to the acceptance
of that statement sometimes it is stated that general covariance is to be
identi…ed with a Principle of General Relativity. What are the meaning of
these statements? Let us analyze them in detail.
6.5.2 Di¤eomorphism Invariance and Maxwell Equations
To understand the meaning to be attached to di¤eomorphism invariance let
us start with a simple example, i.e., zLM E , the Lorentz-Maxwell electrodynamics (LME) in Minkowski spacetime. We are here interested in knowing
under which conditions and in which sense any two ; 0 2 Mod zLM E
related by a di¤eomorphism do represent the same physical model. Now,
zLM E has as model
LM E
= (M; ;
; D; "; F; J; f i ; mi ; qi g);
(6.81)
where (M; ; ; D; ") is Minkowski spacetime, f i ; mi ; qi g, i = 1; 2; : : : ; N
is a set of all charged particles (generating J), mi and qi being their
masses and
I ! M being their world
V1 lines. Also,
V2 charges with i : R
F 2 sec T M is the electromagnetic …eld and J 2 sec T M is the
electromagnetic current. Denoting v(i) = ( i ; ), the proper axioms of the
1 0 In fact we can be a little bit more general and de…ne as equivalent
=
((M; g; D; g ; "); 1 ; : : : ; m ) and 0 = ((M 0 ; g 0 ; D0 ; g0 ; "); 01 ; : : : ; 0m ), where h: M !
M 0 is any di¤eomorphism between two manifolds M and M 0 .
224
6. Some Issues in Relativistic Spacetime Theories
theory are
dF = 0;
mi D
i
F =
J;
(6.82)
v(i) = qi v(i) yF;
(6.83)
Consider the substructure (M; ) and a general di¤eomorphism h : M !
M . Under this di¤eomorphism 7! h = g.
The metric …eld g in M , de…nes a Lorentzian manifoldV(M; g). Now,
r
since
T M, K 2
Vpg = h , we know that in this case for any L 2 sec
sec
T M , r p, we have
dh K = h dK;
?d ? h K = h ? d ? K;
g
g
h Lyh K = h (LyK);
(6.84)
g
where the …rst two lines in the above equation follows directly from Eq.(4.146)
and the last line from a convenient use of the second identity in Eq.(2.130)
and from Eq.(4.145).
Thus, from a mathematical point of view it is a trivial result that zLM E
has the following important property.
Proposition 413 If Eqs.(6.82) and (6.83) have a solution (F; J; ( i ; mi ; qi ))
in h(U ) M in the ‘universe’(M; ; ; D; "), then (h F; h J; (h i ; mi ; qi ))
is also a solution of modi…ed equations of motion
dh F = 0;
mi h Dh
1
in U in the ‘universe’ (M; g;
i
g
h F =
h v(i) = qi h v(i) yh F;
g
g; h
h J;
(6.85)
(6.86)
D; ")11 .
Proof. It follows trivially from Eqs.(6.84).
V1
To …x the ideas imagine an electromagnetic current J 2 sec
T M with
has support only in a region U M . Let C U be a Cauchy surface and
@C its boundary, where Cauchy data and boundary conditions are given.
Consider a series of di¤eormorphisms fhg such that each h : M ! M is
equal to the identity in the region U M and nonzero in the region M nU .
V2
Then h J = J. Let F 2 sec
T M be a solution of the proper axioms of
1 1 We observe also that although we restrict our example to Minkowski spacetime it
is easy to prove that Maxwell equations are di¤eomorphism invariant in any Lorentzian
spacetime.
6.5 Characterization of a Spacetime Theory
225
the theory12 , i.e., dF = 0, F = J, for a well posed initial and boundary
valued problem obtained in a global coordinate chart (M; ) of an atlas of
M with coordinate functions fx g in Einstein-Lorentz-Poincaré gauge. As
well known, since Maxwell equations are a hyperbolic system of di¤erential
equations a well posed problem consists in the Cauchy problem (initial
data on a Cauchy surface) with appropriate boundary conditions given at
the boundary of the Cauchy surface [178]. It is obvious that any h F 2
V2
sec
T M will satisfy the transformed Maxwell equations (Eq.(6.85)) in
the coordinate chart (M; ) with coordinate functions fx g in EinsteinLorentz-Poincaré gauge. Moreover h F will satisfy the same initial and
boundary conditions in C and @C, since h F jC = F jC and h F j@C =
F j@C This is no contradiction because the solution h F refers to a …eld
which moves in another structure, i.e., the structure (M; g), not the original
Minkowski structure (M; ).
So, we claim that for an arbitrary di¤eomorphism h : M ! M , the structures (F; J; ( i ; mi ; qi )) and (h F; h J; (h i ; mi ; qi )) do represent the same
model if we identify the ‘universes’(M; ; D; ; ") and (M; g; h D; g ; "),
and indeed, more generally, we identify all universes that are related by
di¤eomorphisms of the type described above. This is a perfect procedure
from the mathematical point of view. However we need to investigate if
what is mathematically perfect is also correct from the physical point of
view. So we need to have an answer for the following question:
Question 1: Is our identi…cation of ‘universes’ related by di¤eomorphisms in the case of electromagnetic phenomena also compatible with
other known physical phenomena taking place in these ‘universes’? This
will be investigated in the next section. However, …rst we introduce yet
another important question, which is presented after the following remark.
Remark 414 Return to h F . It is obvious that in general h F is not
a solution of the original Maxwell equations (Eq.(6.82)) in the universe
(M; ; ; D; "), for indeed if h F would be a solution we would have two
di¤ erent solutions h F and F satisfying the originally given initial and
boundary conditions in that universe. This is impossible as a consequence
of the uniqueness theorem for the solutions of hyperbolic di¤ erential equations satisfying Cauchy conditions and appropriated boundary conditions,
once we exclude, on physical grounds advanced solutions [253].
Question 2: When do F and h F under the above conditions do represent possible physical phenomena in the same ‘universe’(M; ; ; D; ")?
To answer that question we introduce the concept of Poincaré di¤eomorphisms.
1 2 We
write for simplicity
= , y =y.
226
6. Some Issues in Relativistic Spacetime Theories
Poincaré Di¤eomorphisms
Since the di¤eomorphism invariance of Maxwell equations is true for any
"
h 2 GM , it is true for ` 2 P+
GM , i.e., for any Poincaré di¤eomorphism.
We have
` = , ` D = D:
(6.87)
In this case, (` F; ` J; (` 'i ; mi ; ei )) can be considered a solution of the
equations of motion (Eqs.(6.82)) in the universe (M; ; ; D; ") but the coordinate representations of the original equations of motion in the EinsteinLorentz-Poincaré coordinates fx g must satisfy appropriate transformed
initial and boundary conditions (of course).
Remark 415 Take into account that (` F; ` J; (` 'i ; mi ; ei )) will represent for observers at rest in an inertial reference frame I a phenomenon
distinct from the one described by (F; J; ('i ; mi ; ei )). A typical example is
the …eld of an electric charge at rest in I and the one of a charge moving
at constant velocity (relative to I). See Example 427.
Exercise 416 Let (M; g 1 ) and (N; g 2 ) be two 4-dimensional Lorentzian
manifolds. A di¤ eomorphism h : M ! N is called a conformal map if
2
h g2 =
g1 ;
(6.88)
where 2 : M ! R+ . If h is a orientation-preserving conformal map show
that:
V2
(a) If F 2 sec T M , then
h ( ? F ) = ? (h F )
g2
(6.89)
g1
(b) Put M = N ' R4 and g 1 = g 2 = , a Minkowski metric. Show that
the free Maxwell equations are invariant under conformal transformations,
i.e., dF = 0, F = 0 ) dh F = 0, h F = 0.
(c) Discuss the initial value problem for the situation described in (b).
Enter the Electromagnetic Potential A
The homogeneous Maxwell equation
dF = 0 in Minkowski spacetime imV1
plies that there exists A 2 sec T M such that
F = dA
As already discussed A is de…ned modulus a closed form
terms of A, Eq.(6.82) read
A + d A = J;
(6.90)
(d
= 0). In
(6.91)
6.5 Characterization of a Spacetime Theory
227
where = ( d + d ) is the Hodge Laplacian. Eq.(6.91) is di¤eomorphism
invariant (in the sense explained above), since the corresponding equation
for h A (see Eq.(6.85) is
( d + d )h A + d h A = h J:
g
g
g
(6.92)
Note that even the Lorenz gauge A = 0 is di¤eomorphism invariant, since
?d ? h A = h ? d ? A = 0:
g
g
(6.93)
Taking into account Eq.(4.93) we have from Eq.(6.91), taking into account
that J = 0, that
A + d A = 0:
(6.94)
This equation shows that we indeed have only three independent equations for the components of A in any given coordinate chart. The missing degree of freedom, corresponds to gauge invariance. This means that
Eq.(6.91) is not enough to determine A for a Cauchy problem, unless we …x
the gauge, a procedure that eliminates one degree of freedom. In particular,
if we work in the Lorenz gauge A = 0, Eq.(6.91) is the wave equation,
which has a unique solution for a Cauchy problem.
Remark 417 Sometimes we …nd in the literature the statement that if A
is solution of Eq.(6.91) so is (A + ). Well, this is true only for boundary
free solutions and if
is harmonic, i.e.,
= 0. The same is not true
for a Cauchy problem, for indeed, if (A + ) is to satisfy the same initial
conditions as A on a Cauchy surface, it is necessary that
satis…es homogeneous boundary conditions on that Cauchy surface, and since solves
the wave equation we must have that = 0.
6.5.3 Di¤eomorphism Invariance and GRT
Let us now analyze a problem in GRT which is analogous to the one in
electrodynamics just discussed in the previous section. This will permit
us to give an answer to the question just introduced above. Here, one of
the Si is g, which is a dynamic variable determined by the distribution of
the energy momentum tensor T of all …elds and particles in M through
Einstein’s equations,
1
gR = T :
(6.95)
Ricci
2
This equation is manifestly di¤eomorphism invariant (De…nition 412). Consider then a series of di¤eormorphisms h (where belongs to a given index set) such that each h is equal to the identity in the region U
M
where the energy momentum tensor of matter is di¤erent of zero and
nonzero in the region M nU . If the above conditions are satis…ed we have
that h T = T If we accept that
= ((M; g;D; g ; "); T ) and 0 =
228
6. Some Issues in Relativistic Spacetime Theories
((M; h g; h D; h g ; "); h T ) do represent the same physical model (as
suggested by the di¤eomorphism invariance of the theory) we see that it is
necessary that the di¤eomorphism invariance of Einstein’s equations implies that if the metric …eld g solves a well posed initial value and boundary problem for Einstein’s equations13 for a given T , then h g must also
solve the di¤eomorphic initial and boundary valued problem for Einstein’s
equations for the same T = h T .14 This may only be true, of course,
if the mathematical nature of the Einstein’s equations do not allow the
complete determination of the ten functions g (x) once that equations
are written in a coordinate chart fx g of the maximal atlas of M (covering a region big enough for our considerations to make sense). And indeed, that is the case15 , since we have ten equations for the Einstein tensor
1
T , but they are not independent since we have
G =R
2g R =
four constraints, coming from D G = 0.
This implies according to the majority view that in GRT any particular gravitational …eld must be described by an element of the quotient
space LorM=GM where LorM is the space of all spacetimes associated to
Lorentzian manifolds (M; g) for all possible Lorentzian metrics g on M
and GM is the group of di¤eomorphisms of M . More on this issue may be
found in [306].
6.5.4 A Comment on Logunov’s Objection
The above arguments are so simple and appealing that is hard to imagine
any objection. However, according to Logunov [212, 213] in practice, when
we introduce an arbitrary chart for U M in order to …nd a gravitational
…eld for a given matter distribution, we have a problem. Indeed, Logunov
argues: which h g do we choose as solution for Einstein’s equation? He
argues that GRT has no internal answer for that question and this is true.
In GRT [163, 249, 447] a unique h g is speci…ed only by arbitrarily selecting four unknown components of the metric tensor for a given problem.
Experts in GRT say that that procedure corresponds in …xing a coordinate
gauge. Other authors are more subtle and only say that a ‘gauge condition’ must be given. What is the meaning of such statements? Hawking
and Ellis, e.g., [163] select the ‘gauge’ by …xing a background metric g
and imposing a condition on the covariant derivative of h g relative to
the Levi-Civita connection determined16 by g. In particular they choose
1 3 The initial value problem (Cauchy problem) in GRT is very subtle one and di¤ucult
one, and the reader interested in details must consult, e.g., [163, 78].
1 4 This situation is known as Einstein’s hole argument. A very detailed discussion of
the argument is given in [321], where many important references can be found.
1 5 As known since a long time ago. See, e.g., [213, 249, 447] for details.
1 6 This of course, implies in four conditions to be satisi…ed by the components of the
metric h g:
6.5 Characterization of a Spacetime Theory
229
the harmonic gauge. Such a gauge has been used by Fock [137] for the
particular case where g is a constant Minkowski metric on a manifold M
di¤eomorphic17 to R4 . So, the above procedure furnishes a solution g which
satis…es Einstein’s equations, the given initial and boundary valued conditions (supposed physically realizable in nature) and the gauge condition. To
see the importance of the above remarks, let us analyze a simple problem,
namely the solutions of Einstein’s equations for a static and spherically
symmetric distribution of matter with its energy momentum tensor having
support in an open set U M . Introducing a chart (U; ') with coordinates
(x0 ; x1 ; x2 ; x3 ) = (t; r; ; ') for '(U ), and being the image of U under the
coordinate mapping given by f 1 < t < 1; (0 < r < r0 )[ (r0 < r < 1)g
the form of the metric for r > r0 must be18
g = g00 dt
dt + 2g01 dt
dr + g11 dr
dr + g22 d
d + g33 d'
d'; (6.96)
where all the metric coe¢ cients can be functions only of the radial coordinate. An analysis of the possible solutions of Einstein’s equations for
the above problem has been given by several authors, the presentation of
[212, 213] being particularly well done and careful. The conclusion is that
there are in…nite possible metrics that have the same asymptotic behavior
when r ! 1. Two of them are:
gs =
1
2m
r
dt dt
1
2m
r
1
dr dr r2 (d
d +sin2 d' d');
(6.97)
and
gi =
r m
r+m
dt dt
r+m
r m
dr dr (r+m)2 (d
d +sin2 d' d');
(6.98)
where m is a parameter with the meaning of the mass of the system in the
natural systems of units.
Eq.(6.97) is (wrongly) known (see Section 6.9) as Schwarzschild solution
and Eq.(6.98) is another valid solution, for which
p
D ( det g i g i ) = 0;
(6.99)
where D = D is the Levi-Civita covariant derivative of a Minkowski metric
= g, which in the coordinates (t; r; ; ') is written as
g = dt
dt
dr
dr
r2 (d
d + sin2 d'
d'):
(6.100)
1 7 You may argue that in so doing we have …xed the topology of the spacetime. Well,
this is true, but do not forget that any solution of Einstein’s equations is obtained in a
local chart of an abstract manifold, and in general there are many di¤erent topologies
consistent with the metric obtained in the particular chart. This means that in truth,
the topology of a solution to Einstein’s equations is …xed by hand. This will become
clear in Section 6.9
1 8 See, e.g., [203].
230
6. Some Issues in Relativistic Spacetime Theories
Eq.(6.99)19 is not satis…ed by g s . So, which one are we going to use in
order to compare empirical data with predictions of the theory? Any expert
on GRT will, of course, answer that question saying that both metrics are
permitted, because, the meaning of the coordinates in each one are di¤ erent
(even if they are presented by the same letter). The reasoning behind that
answer is that we can know what the spacetime labels mean, only after we
…x a metric on it. With this answer, di¤eomorphism invariance holds, if we
do not include a particular gauge …xing equations as part of the theory.
Indeed, we can perform a coordinate transformation in Eq.(6.97) which
makes it in the new variables to have the appearance of Eq.(6.98).
However, according to Logunov, to not know a priori the meaning of the
coordinates leads to ambiguities in the predictions of experiments, e.g., in
the time delay of radio signals in the solar system20 . The reason is that
the time delay of a radio signal that goes, e.g., from Earth to Mercury and
comes back ‡ying in the background gravitational …eld generated by the
Sun, is de…ned as the amount of extra time (as measured with a clock on
Earth) to do the same path in the absence of the Sun’s gravitational …eld.
Now, both metrics (g s and g i ) reduce to Minkowski metric in the absence
of the Sun’s gravitational …eld and so, we have two di¤ erent predictions
using the labels (t; r; ; '), if we suppose that in g s and g i they have the
same meaning.
We think that only meaningful question that can be done at this point
is the following: what is the meaning attributed by astronomers to the
coordinate functions (t; r; ; ')?
This is an important question since once we know that answer only one
of the predictions, using g s or g i will agree with empirical data once we
use the astronomers’interpretation of (t; r; ; '). Logunov [212, 213] claims
that under these conditions only the metric g i is compatible with experimental data. More, according to him, his theory (with a zero graviton
mass) predicts that data because in it there is an analogous to Einstein’s
equation plus Eq.(6.99), which is an integral part of his theory.
Logunov’s conclusion would be correct only if one can claim that the
astronomers meaning of the coordinates (t; r; ; ') for the time delay experiment is the one that those coordinates have in a ‡at Minkowski spacetime.
But this can only be con…rmed if: someone can do the time delay radio signal experiment putting o¤ the gravitational …eld, and this no astronomer
can do, of course.
According to our view, what can be inferred is the following: if the data
favors the use of g i , this only says that the procedure astronomers use to
1 9 Eq.(6.99)
is similar but not identical to harmonic gauge condition.
Scharf [386] also puts in doubt the validity of di¤eomorphism invariance.
We will not comment on his paper here.
2 0 Recently
6.5 Characterization of a Spacetime Theory
231
put labels to spacetime events make the coordinates (t; r; ; ') to have the
meaning that they have as encoded in g i .
Remark 418 Logunov emphasizes that there are compelling reasons to
claim that the Minkowski structure of spacetime manifests itself in some
identi…able way. Indeed, in his theory it is claimed that manifestation of
the Minkowski spacetime structure arises in the empirical validity of the
energy-momentum and angular-momentum conservation laws for all physical phenomena. In Chapter 9 we show in details that in GRT there are no
genuine energy-momentum and angular-momentum conservation laws.
Remark 419 It is also said that di¤ eomorphism invariance is a crucial
requirement that GRT must satisfy in order to avoid indeterminism [108,
306, 321]. We prefer not to go on the that discussion here. The reason is
that as it will become clear in Chapter 9, where we study the shameful problem of the ‘energy-momentum conservation’ in GRT, we do not think that
this theory, with its orthodox interpretation, is one worthy to be preserved
anymore. One way to have trustful energy-momentum conservation is to
suppose that the arena of physical phenomena is Minkowski spacetime and
that the gravitational …eld is a …eld in the Faraday sense, which no special
distinction in relation to the other …elds. This, of course do not implies
that we necessarily need to have Logunov’s theory as the unique possibility.
However, whatever theory we decide to use it must give the results predicted
by GRT in the case that we know these predictions are good ones. In that
case, we may say that the geometrical description “à la General Relativity”
is a coincidence, which may be valid as a …rst approximation (see Chapter
11). We mention yet, that di¤ eomorphism invariance is also said to play a
crucial role in some recent tentatives of formulation of a quantum theory
of gravity, like loop gravity. Here is not the place to venture on that subject.
The interested reader may consult [364] and the careful analysis presented
in [321] of many of Rovelli’s claims.
6.5.5 Spacetime Symmetry Groups
Let h 2 GM . We recall (De…nition 203) that if for a geometrical object T
we have h T = T or equivalently
h T=T
(6.101)
then h is said to be a symmetry of T.
De…nition 420 The set of all fh 2 GM g such that Eq.(6.101) holds is
said to be the symmetry group of T.
De…nition 421 Let ; 2 Mod z;
= h(M; g; D; g ; "); T1 ; : : : ; Tm i,
= h(M; h D; h g; h g ; h ");h T1 ; : : : ; h Tm i with the Ti , i = 1; : : : ; m
232
6. Some Issues in Relativistic Spacetime Theories
de…ned in U
such that
M and T0i , i = 1; : : : ; m de…ned in V
D = h D;
Then
h(U )
g = h g:
is said to be the h-deformed version of
M and
(6.102)
.
Remark 422 Take notice that— as will be clear in a while— (T1 ; : : : ; Tm )
and (h T1 ; : : : ; h Tm ) in general will correspond to di¤ erent phenomena as
registered by observers at rest in a given arbitrary reference frame Q 2 sec T M .
De…nition 423 Let Q 2 sec T U
sec T M; Q 2 sec T V
sec T M , U \
V 6= ; and let fx g, fx g (the coordinate functions associated respectively
to the charts (U; ') and (V; ')) be respectively a (nacsjQ) and a (nacsjQ)
and suppose that x = x h 1 : h(U ) ! R. Thus, Q = h Q and Q is said
to be a h-deformed version of Q.
Let ; 2 Mod z be as in De…nition 421. Call o = (D; g; :::; T1 ; : : : ; Tm )
and o = (D; g; ; :::; h T1 ; : : : ; h Tm ). Now, o is such that it solves a set of
di¤erential equations in '(U ) R4 with a given set of boundary conditions
denoted bofx g , which we write as
Dfx g (ofx g )e = 0 ; bofx
and o de…ned in h(U )
g
;
e 2 U;
(6.103)
V solves
Dfx g (h ofx g )jh e = 0 ; bh
ofx g
;
h e 2 h(U )
V:
(6.104)
In Eqs.(6.103) and (6.104) Dfx g and Dfx g , = 1; 2; : : : ; m, mean sets of
di¤erential equations in R4 .
How can observers living on the universe (M; g; D; g ; ") discover that
; 2 Mod z are deformed versions of each other? In order to answer this
question we need some additional de…nitions.
6.5.6 Physically Equivalent Reference Frames
De…nition 424 Let Q; Q be as in De…nition 423. We say that Q and Q
are physically equivalent or indistinguishable according to theory z (and
we denote Q
Q) if and only if there exist a (nacsjQ) and a (nacsjQ)
such that: (i) the functions D Q and D Q have the same functional
form and (ii) the system of di¤ erential equations (6.103) have the same
functional form as the system of di¤ erential equations (6.104) and bh ofx g
must be relative to fx g the same as bofx g is relative to fx g and if bofx g
is physically realizable then bh ofx g must also be physically realizable.
De…nition 425 Given a reference frame Q 2 sec T U
sec T M the set
of all di¤ eomorphisms fh 2 GM g such that h Q Q forms a subgroup of
GM called the equivalence group of the class of reference frames of kind Q
according to the theory z.
6.6 Principle of Relativity
233
Remark 426 In the next section we establish what is the meaning of a
Principle of Relativity for a spacetime theory based on Minkowski spacetime. One of the meanings of this principle is that all inertial reference
frames are physically equivalent. It is very important also to realize that
general covariance is not to be identi…ed with a Principle of General Relativity. Indeed, if such a principle is to have the meaning that all arbitrary
reference frames are physically equivalent then it does not hold in GRT as
will become clear in Section 6.8.2.
6.6 Principle of Relativity
In this section the arena where physical phenomena occur is supposed to
be Minkowski spacetime M = (M; ; D; ; "). Let then I; I0 2 sec TM be
two distinct IRF in M, which we recall are frames such that DI = DI0 = 0.
According to De…nition 403 any IRF is proper time synchronizable. A global
@
(nacsjI) fx g is said to be in the Einstein-Lorentz-Poincaré gauge if I = @x
0
@
@
@
and the set f @x g is an orthonormal frame, i.e., ( @x ; @x ) =
. It is
crucial to have in mind that, given the natural system of units used in
this book where the light velocity c = 1, the coordinate x0 = x(e) has the
meaning of proper time as measured by clocks at rest in I and synchronized
à l’Einstein. Also, the spatial coordinates xi = xi (e), i = 1; 2; 3 have the
meaning of proper distances as measured along the spatial axis from a …xed
origin. Let fx0 g be a (nacsjI0 ) also in the Einstein-Lorentz-Poincaré gauge.
Then the coordinates x = x(e) and x0 = x0 (e) of any event e 2 M are
related by
x0 = L x + a ;
(6.105)
where L are the matrix components of a Lorentz transformation L 2 L"+
and a are real constants. When a = 0, Eq.(6.105) is a special orthochronous Lorentz mapping. According to Eq.(4.29) the set of all coordinate
transformations of the form given by Eq.(6.105) can be associated to a
"
"
subset of the di¤eomorphism group, namely P+
= f`g GM . Any ` 2 P+
is a Poincaré di¤ eomorphism. If T 2 sec T M (or is a connection) we call
` T a Poincaré deformed version of T. When, ` induces only a Lorentz
transformation, then ` T is called a Lorentz deformed version of T.
We can verify that
`
= ;
` D = D;
"
8` 2P+
;
(6.106)
i.e., according to De…nition 420 the Poincaré group (and in particular its
e
) is a symmetry group of and D.
subgroup L"+ = SO1;3
Now, the following statement denoted PR1 is usually presented as the
Principle of Relativity in active form [340].
234
6. Some Issues in Relativistic Spacetime Theories
"
PR1 : Let ` 2 P+
GM . If for any possible physical theory z, if 2 z,
= ((M; ; D; ; "); T1 ; : : : ; Tm ) is a possible phenomenon, then 0 =
`
is also a possible physical phenomenon.
The following statement denoted PR2 is known as the Principle of Relativity in passive form [340].
PR2 : All inertial reference frames are physically equivalent or indistinguishable for any possible physical theory z.
PR1 and PR2 are equivalent statements of the Principle of Relativity
and we believe that they capture the ideas of Poincaré [301] and Einstein
[110] (see also on this respect [214, 241, 449]). The existence of a Principle
of Relativity for a physical theory permit us to …nd nontrivial solutions
for the equations of motion of the theory once very simple solutions are
known. We illustrate this case in the following
Example 427 As in Section 6.5 let zLM E be classical electrodynamics
taking place Minkowski spacetime. More precisely, consider Lorentz-Maxwell
electrodynamics (LME) zLM E as a theory of a species of structure. We already know that LME has as model
LM E
= (M; ;
; D; "; F; J; f i ; mi ; ei g):
(6.107)
We already know from Section 5.7.1 the precise sense in which LM E may
be considered di¤ eomorphism invariant for h 2 GM :
"
Now, let us study the case where h = ` 2 P+
GM , i.e., for a Poincaré
di¤ eomorphism for which
`
= , ` D = D:
(6.108)
In this case, as we learned in Section 6.5 (` F; ` J; (` 'i ; mi ; ei )) can be
considered a solution of the equations of motion (Eqs.(6.82)) in the universe
(M; ; ; D; "), but with appropriate transformed initial and boundary conditions (of course).
The explicit form of Poincaré di¤ eomorphisms is introduced as follows.
Let fx g and fx0 g be two coordinate charts covering M naturally adapted
to the global IRFs I = @=@x0 and I0 = @=@x00 , such that, e.g.,
@
vi
@
+p
;
(6.109)
0
1 v2 @x
1 v2 @xi
X3
where v = (v 1 ; v 2 ; v 3 ) and v2 =
(v i )2 . Then, a Poincaré mapping
i=1
` : e 7! `e is de…ned by the following coordinate transformation
I0 = p
1
x0 (e) = x (`e)
x (e) + a ;
(6.110)
Suppose we have a charge at rest at the origin of the I0 frame and let be
F = 21 F (x0 )dx0 ^ dx0 be the electromagnetic …eld generated by this
6.6 Principle of Relativity
235
charge. As it is well known,
ex0i
F0i (x0 ) = r
X3
i=1
Then, we have
; Fij (x0 ) = 0:
(6.111)
(xi )2
1
F (x )
dx ^ dx ;
2
= F (x0 (x ))
:
F := ` F =
F (x )
(6.112)
To simplify the calculations, let v = (v; 0; 0). Writing, as usual, in Cartesian notation, F = (E; 0) and F = (E; B), we have
e
p
x1 vx0 ; x2 ; x3 ; B = v E(x );
1 v2
r
(x1 vx0 )
+ (x2 )2 + (x3 )2
R=
1 v2
E(x )=
R3
(6.113)
The …eld F = ` F describes the …eld of a charge moving with constant
velocity along the x-axis. We have obtained this result in a practically trivial
way using PR1 . The reader will certainly appreciate the power of PR1 once
he tries to solve directly Maxwell equation for that problem (see, e.g., [221]).
6.6.1 Internal and External Synchronization Processes
We recall once more that a IRF on Minkowski spacetime is mathematically
described by a unit time like vector …eld I 2 sec T M such that DI = 0.
We also de…ned a set fx g of coordinate functions covering M which is
moreover a (nacsjI) in the Einstein-Lorentz-Poincaré gauge, i.e., in these
coordinates I = @=@x0 and ( @x@ ; @x@ ) =
. We recall that x0 = x(e)
has the meaning of proper time at e 2 M as measured by a clocks at
rest in the frame and which have been synchronized by Einstein’s method
described above. Such a procedure is an internal operation in the I frame,
and as such it is more properly called an internal synchronization process21 .
Of course, Einstein’s method using light signals, is not the only possible
internal synchronization procedure, and indeed, several other methods are
known, like, e.g., the ones described in [183, 185, 230, 231, 454].
As discussed in [345] the validity of the Principle of Relativity implies
that any possible internal synchronization procedure of clocks must be
equivalent to Einstein’s method and no method can realize absolute synchronization relative to the time of a chosen ‘preferred’ reference frame.
2 1 It is crucial to have in mind that the synchronabilty to which de…ntion 403 refers is
internal synchronization.
236
6. Some Issues in Relativistic Spacetime Theories
This statement means the following. Let I0 = @=@ x0 be a ‘preferred’ inertial reference frame and fx g be a (nacsjI0 ) in the Einstein-LorentzPoincaré gauge and where it is experimentally veri…ed that all internal
synchronization procedures agree with the one obtained through Einstein’s
synchronization procedure. Let, e.g.,
I =p
1
1
v2
@
@ x0
p
v
1
v2
@
@ x1
(6.114)
be another inertial reference frame. Choose a (nacsjI) fx g in the EinsteinLorentz-Poincaré gauge such that
x0 vx1
x1 vx0
x0 = p
; x1 = p
; x2 = x2 , x3 = x3 :
2
1 v
1 v2
(6.115)
Let A and B clocks at rest in I following world lines A and B parametrized by x0 . When the standard clocks at I0 reads x0 the spatial coori
i
dinates of the clocks are xi A = xi
A (x0 ) and x B = x
A (x0 ). Then
from Eq.(6.115) it follows that the readings of the standard clocks A and
B are (according to the observers in I0 ) out of phase by
x0A
x0B =
p
v
1
v2
(x1 A
x1 B ) =
p
v
1
v2
4x1
(6.116)
in relation to the standard clocks synchronized “à l’Einstein” in I0 . The
phase di¤erence depends explicitly on the relative velocity of the frames.
Suppose now that it would be possible with an internal synchronization
procedure, di¤ erent from Einstein’s method, to synchronize clocks in I such
that the time registered by standard clocks (say at A and B) at rest in that
frame are given by coordinates x0A and x0B and such that when the standard
clocks at I0 reads x0 we have
x0A
x0B = 0:
(6.117)
The coordinate x0 may be named absolute synchronization as de…ned by
I0 .
Then, by comparing set of clocks synchronized with the di¤erent synchronization procedures it would be possible to determine the velocity of
the I reference frame relative to the I0 reference frame. If for all possible
inertial frames it would be possible to …nd coordinate functions that realize absolute synchronization in the sense of Eq.(6.117), this will select
I0 as a preferred one, and we would have a breakdown of the Principle
of Relativity. This is because, the phenomenon involved in the alternative
synchronization procedure in I will not be a Lorentz deformed version of
the same phenomenon in I0 :
6.6 Principle of Relativity
237
6.6.2 External Synchronization
Suppose that we eventually identify in the universe we live a given IRF, say
I0 as having some cosmic signi…cance. Let fx g be a (nacsjI0 ) in EinsteinLorentz-Poincaré coordinate gauge. Let I (given by Eq.(6.114)) be another
IRF whose observers have determined the velocity of their frame I relative to I0 by realizing experiments involving some phenomena generated
external p
to that frame. Of course, nothing prevents those observers to use
x0 = x0 1 v 2 as time coordinate function representing the reading of
standard clocks at rest in I which are synchronized according to the readings of the clocks at rest in I0 . A natural set of global coordinates which
could be used by observers in I are then
x 0 = x0
p
1
x1 vx0 2
; x = x2 , x3 = x3 ;
v 2 ; x1 = p
1 v2
(6.118)
which we call the absolute gauge coordinates, and which have been used
by many authors in the past, as, e.g., in [72, 236, 232, 233, 318, 319, 320,
345, 418]. Now, the coordinate functions fx g such that x (e) = x is a
(nacsjI), since in that coordinates we have
I=
@
:
@x0
(6.119)
Also, in these coordinates the Minkowski metric tensor reads,
= dx0 dx0 2vdx0 dx1 (1 v 2 )dx1 dx1 dx2 dx2 dx2 dx2 ; (6.120)
being non diagonal.
Recall that the set of naturally adapted coordinates fx g (EinsteinLorentz-Poincaré gauge) and fx g (absolute gauge) to I are related by
xi = xi ; x0 = x0
vx1 :
(6.121)
Some authors, as e.g., [50, 84, 232, 233, 316, 317, 318, 320] claims that
I0 may be identi…ed as the reference frame where the cosmic background
radiation is isotropic. However, if the reference frame where the cosmic
background radiation is isotropic is to be understood as a reference frame on
a Lorentzian spacetime modeling a cosmological model according to GRT,
then this identi…cation is not possible since as already said in Remark 382
there are in general no inertial frames in a general Lorentzian spacetime.
The use of the coordinates fx g will be a useful one indeed only in the
case that we can identify a preferred IRF by internal experiments breaking
Lorentz invariance in the frame I. More on this issue will be discussed below
and in our book yet in preparation [283], where we shall need the results
of the next section.
238
6. Some Issues in Relativistic Spacetime Theories
6.6.3 A Non Standard Realization of the Lorentz Group
Let
@
@
vi
+p
;
0
@x
1 v @ xi
1
v 0i
1
@
@
+q
;
I0 = q
0
@
x
@
xi
2
2
1 (v 0 )
1 (v 0 )
I= p
1
v2
(6.122)
be two IRFs. Suppose that observers at the IRF I and I0 can measure their
velocities relative to a given preferred frame I0 . Observers in the frames I
and I0 may use coordinates in the Einstein-Lorentz-Poincaré gauge, denoted
respectively by fx g and fx0 g or they can use coordinates in the absolute
gauge, respectively fx g and fx0 g. If they use absolute gauge coordinates,
the velocity of the preferred frame as measured by those observers can be
written as
@
@
I0 = u0 0 + ui i
@x
@x
@
@
I0 = u00 00 + u0i 0i :
(6.123)
@x
@x
Also, the velocity of frame I0 as determined by the observers in the I
frame will be written as
@
@
I0 = w 0 0 + w i i
(6.124)
@x
@x
We write xt = (x0 ; x1 ; x2 ; x3 ), xt = (x0 ; x1 ; x2 ; x3 ), xt = (x0 ; x1 ; x2 ; x3 ),
x = (x00 ; x01 ; x02 ; x03 ), ut = (u0 ; ui ) and u0t = (u00 ; u0i ), wt = (w0 ; wi )
Then the set of coordinates in the absolute gauge are related by
0t
x=
(L; u)x; x0 =
x0 =
(L; u)x;
0
(L ; u0 )x ,
(6.125)
and
u0 =
(L; u)u:
(6.126)
0
In Eqs.(6.124), L; L ; L are elements of the Lorentz group, u and u0 are,
of course, the components (i.e., the velocity) of the I0 frame as determined
by I and I (see Eq.(6.124)) in absolute gauge coordinates. Finally, (L; u)
is a 4 4 matrix, whose explicit form (has been found in [236] and with
more details in [316, 317, 318, 320]) is22 :
(a) For rotations we have
(R; u) =
2 2 We
1
0
0
R
;
use for future reference the notations of [316, 317, 318, 320].
(6.127)
6.6 Principle of Relativity
239
where R 2 SO3 is a standard rotation matrix.
(b) For boosts we have
(Lw ; u) =
(w0 ) 1
w
!
0
13 +
t
w
pw
1+ 1+jw2
u0 w
wt
(6.128)
For boosts (with “parallel”space axis) the coordinates introduced above
are related as follows:
1 0
x ; x = x u x0
u0
1
u0 = p
; u = u0
2
1
x0 =
1+
u x
p
1 + u2
;
(6.129)
1 0
x ; x 0 = x u x0
u00
1
u00 = p
; u0 = u00 0
02
1
u0 x
p
1 + 1 + u2
1 0
x ; x0 = x w x0
w0
u0
(u0 + u00 )(u
w0 = 00 ; w =
u
1 + u0 u00 (1 +
w x
p
1 + 1 + w2
u0 )
i j
ij u u )
x00 =
;
(6.130)
and
x00 =
The metric tensor reads, e.g., in coordinates fx g as
with
1
u0 ut
g =
0
u u
13 + (u0 )2 u ut
;
= g dx
:
(6.131)
dx ,
(6.132)
Also, as a generalization of Eq.(6.121) we also have
x0 = x0
ij
i j
x ; xi = xi :
(6.133)
which gives the relation between coordinates in the Lorentz and absolute
coordinate gauges.
Finally, the nonstandard realization of the Lorentz group is given by the
following rules:
(L2 ; (L1 ; u)u) (L1 ; u) =
1
(L; u) =
(L2 L1 ; u);
(L
(L; u) = 14 :
1
; (L; u)u);
(6.134)
240
6. Some Issues in Relativistic Spacetime Theories
6.6.4 Status of the Principle of Relativity
The Principle of Relativity according to textbooks and in particular in
Roberts review [322] titled: ‘What is the Experimental Basis of Special
Relativity?’ is supposed to be one of the most well tested principles of
Physics (see also [457]). High Energy physicists proudly claimed during all
the XX century that PR1 is routinely veri…ed in any high energy physical
laboratory23 and that PR2 is routinely used when data of di¤erent laboratories are compared. The validity of the Principle of Relativity is known
also as Poincaré invariance of physical laws. According to the de…nitions
given above the reason for that is obvious.
Despite all enthusiasm, the question arises? Is the Principle of Relativity a true law on nature valid under all conditions? Well, if the GRT is
a correct description of the nature of the gravitational …eld the answer is
no. The reason is that in this theory there are no inertial reference frames
(in general) as we already remarked. Even a so called Principle of Local
Lorentz Invariance (PLLI), stated in many books and articles (see references below) is not true, as it will be proved in Section 6.8 below. Finally,
even if we could formulate (as indeed we can, see, e.g., Chapter 11, and also
[133, 214, 340, 389, 447]) a theory of the gravitational …eld in Minkowski
spacetime, (satisfying the Principle of Relativity), no one can warrant that
this Principle is a true law of nature valid under all conditions, for we do
not know all laws of nature. Having said that the reader must be informed
that:
(a) From time to time there are claims in the literature that certain very
low energy experiments involving the propagation of light and the rototranslational motion of solid bodies violate Lorentz invariance. We quote
in that class of experiments24 [232, 233] and [196, 425, 426]. The data
described in [425] can be explained with a very simple model, where it
is postulated a breakdown of PR1 for solids in roto-translational motion
[345]. However it seems that the data in [426] is more compatible with a null
result. Breakdown of Lorentz invariance in the roto-translational motion of
solid bodies as described in [345] may also explain the data in [232, 233].
However, these experiments (to the best of the authors’ knowledge) have
not been duplicated and we have serious doubts about those results. In this
respect see also the thoughtful analysis in [223, 424].
(b) Also, recently Cahill [60, 61] and also Consoli [85] and Consoli and
Constanzo [86, 87] claim that the small, but not null results in MichelsonMorley like experiments done in the past can be accounted by explicitly
2 3 Which they wrongly suppose are inertial reference frames. Indeed, all known high
energy laboratories are located on the Earth, which is not an inertial reference frame.
2 4 We must say that we do not know if these experiments have been duplicated. In
[345] it is analyzed a possible breakdown of PR 1 for solids in rototranslational motion
which accounts for the results of the Kolen-Torr experiment [425, 196].
6.6 Principle of Relativity
241
postulating that the propagation of light in a medium breaks Lorentz invariance. For the case of the anomalous experimental results in the Brillet
and Hall experiment (which is a Michelson-Morley like experiment done
in vacuum) Cahill postulated that the Schwarzschild gravitational …eld of
the Earth accounts for an e¤ective medium. With that hypothesis his calculations at …rst sight seems to explain those ‘anomalous’results. On the
other hand, Klauber [193] claims to explain the anomalous Brillet and Hall
[57] experimental results by postulating that the devices used in the experiment does not su¤er Lorentz contraction when turned. Also it is worth
to quote here that a reanalysis of the data of Miller’s classical experiment
[243] by Allais25 [13], show the existence of correlations that are hard to
believe to have the simple explanations given by Shakland [381]. All these
claims need, of course, to be more carefully analyzed, but in particular it
is necessary to take into account that Earth is not an inertial reference
frame according to both the SRT and the GRT. Recall that inertial reference frames did not exist (in general) in GRT as already remarked several
times. This fact, it seems to us, has not been properly taken into account
by those authors in their analysis of the classical experiments. In particular, in any real Michelson-Morley like experiment the light paths enclose a
…nite area and then, those experiments are indeed analogous to a Sagnac
experiment, and a non null (but very small) phase shift may be predict
for them. This observation has already been remarked by Post [305, 304].
However, if the authors quoted above are correct, since along time ago a
breakdown of PR1 has already been found. This would be disturbing to say
the less26 .
(c) There are also many conjectures that a possible breakdown of Lorentz
invariance will happen in phenomena involving low and very high energies
(see, e.g., [5, 17, 18, 19, 198, 199, 200, 225, 440] and references therein27 )
and/or cosmological scales [272]
(d) Also, there are claims of a possible breakdown of Lorentz invariance
in the phenomenon of the wave packet reduction of an entangled quantum
state of two identical particles (see, e.g., [162, 297, 298, 58, 316, 317, 318,
319, 320, 406, 412, 456] and references therein)28 .
2 5 Maurice Allais is a French physicist that won Noble Prize in Economics. It is worth
to visit his home page at: http://allais.maurice.free.fr/Science.htm.
2 6 We advise the reader keep an open eye on this issue following the articles on the
subject appearing at the arXiv.
2 7 In fact, a google search for "breakdown of Lorentz invariance" will show several
hundreed of serious articles on the subject. Of course, we do have the opportunity to
analyze all such posibilities in our book.
2 8 Cases (c) and (d) will discussed in our planeed book [283], whose proposal is among
others to show that none of the experiments claiming superluminal propagation (as e.g.:
(A) Superluminal group velocities of voltage and currents con…gurations propagating in
wires [244, 245, 191, 293, 324]; (B 1 ) Superluminal group velocities of electromagnetic
…eld con…gurations propagating in dispersive media with absorption or gain [75, 407];
242
6. Some Issues in Relativistic Spacetime Theories
For the rest of this book we assume the validity of Principle of Relativity
(PR1 or PR2 ) in our discussions.
6.7 Principle of Local Lorentz Invariance in GRT?
From time to time we …nd in the literature the statement that the existence
of the cosmic background radiation de…nes a kind of preferred inertial reference frame. However, as we already know, in GRT in a general Lorentzian
spacetime modelling a gravitational …eld generated by a given energy momentum tensor, in general, inertial reference frames do not exist.
In view of that fact, the following question arises naturally: which characteristics a reference frame on a GRT spacetime model must have in order to
re‡ect as much as possible the properties of an IRF that exists in Minkowski
spacetime?
The answer to the question is that there are two kind of frames in GRT,
namely PIRFs (de…nition 428) and LLRFs (de…nition 430)], such that each
frame in one of these classes share some important aspects of the IRFs
of SRT. Both concepts are useful and it is worth to distinguish between
them in order to avoid misunderstandings. A thoughtful discussion of these
concepts was presented in [348] and we follow that exposition.
De…nition 428 A reference frame I 2 sec T U; U
M is said to be a
pseudo inertial reference frame (PIRF) if D I I = 0 and I ^ d I = 0, with
I = g(I; ).
This de…nition means that a PIRF is in free fall and is non rotating. It
means also that it is at least locally synchronizable.
De…nition 429 A chart (U; ') of an oriented atlas of M with coordinates
is said to be a local Lorentzian coordinate chart (LLCC) and f g are
said to be local Lorentz coordinates (LLC) in p0 2 U if and only if
g(@=@
(
) jp0 = 0;
;
(
; @=@
) jp =
) j p0 =
1
(R
3
(
;
)+R
(6.135)
(
)) jp ; p 6= p0 :
(6.136)
(B 2 ) Superluminal group velocities of tunneling microwaves [113, 114, 115, 169, 269,
270, 271]; (B 3 )Superluminal group velocities of a single tunneling photon [408]; (B 4 )
Superluminal group velocities of tunneling electrons [208]; (C 1 ) Superluminal group
velocities of microwaves launched and received by non axially aligned horn antennas
[182, 310, 311, 309, 148, 181]; (C 2 ) Direct measurements of superluminal velocities
of peaks of …nite aperture approximations to electromagnetic Bessel beams (X -waves),
both in the optical and in the microwave region [366, 262]) do not imply in any violation
of the Principle of Relativity.
6.7 Principle of Local Lorentz Invariance in GRT?
243
Let (V; ) (V \ U 6= ;) be an arbitrary chart with coordinates fx g.
Then, supposing that p0 is at the origin of both coordinate systems the
following relations holds (approximately)
=x +
x =
1
2
1
2
(p0 )x x ;
(p0 )
;
(6.137)
where in Eqs.(6.137)
(p0 ) are the values of the connection coe¢ cients
at p0 expressed in the coordinates fx g.
The coordinates f g are also known as Riemann normal coordinates
and the explicit methods for obtaining them are described in many texts
of Riemannian geometry as, e.g., in [265] and of GRT, as e.g., in [447].
Let 2 U
M be the world line of an observer in geodetic motion in
spacetime, i.e., D
= 0. Then as it is well known we can introduce in U
LLC f g such that for every p 2 we have
@
@ 0
(
)
=
jp ;
p2
p2
=g
g(@=@
g(@=@
; @=@
)jp2 =
; D @=@ @=@
)
p2
;
= 0:
(6.138)
Take into account for future reference that if the f g are LLC then it
( ) jp 6= 0 for all p 2
= .
is clear from de…nition 429 that in general
De…nition 430 Given a geodetic line
U
M and LLCC (U; ) we
say that a reference frame L = @=@ 0 2 sec T U is a Local Lorentz Reference
Frame Associated to (LLRF ) if and only if
Ljp2 =
L
^d
L jp2
@
@ 0
= 0:
=
p2
jp ;
(6.139)
Moreover, we say also that the Riemann normal coordinate functions or
Lorentz coordinate functions (LLC) f g are associated with the LLRF :
Remark 431 It is very important to have in mind that for a LLRF L
in general D L Ljp2= 6= 0 (i.e., only the integral line of L in free fall in
general ), and also eventually L ^ d L jp2= 6= 0, which may be a surprising
result for many readers. In contrast, a PIRF I such that Ij = Lj has all
its integral lines in free fall and the rotation of the frame is always null in
all points where the frame is de…ned. Finally its is worth to recall that both
I and L may eventually have shear and expansion even at the points of the
geodesic line that they have in common [348].
244
6. Some Issues in Relativistic Spacetime Theories
De…nition 432 Let be a geodetic line as in De…nition 430. A section s
e U; U
of the orthogonal frame bundle PSO1;3
M is called an inertial moving
tetrad along (IMT ) when the set
s = f( 0 (p);
1 (p); 2 (p); 3 (p)); p
2
\ Ug
s;
(6.140)
it such that 8p 2
0 (p)
=
jp ;
g(
;
)jp2 =
(6.141)
with
(p) = g
;
g(
(p); D "
(p)
(p)) = 0:
(6.142)
e U satisfying the above conditions can
The existence of s 2 sec PSO1;3
be easily proved. Introduce coordinates f g for U such that at p0 2
;
0 (p0 )
=
@
@ 0
po
=
jp0 ,
and
i (p0 )
=
@
@ i
po
; i = 1; 2; 3 (three ortho-
normal vectors) satisfying Eq.(6.138) and parallel transport the set (p0 )
along . The set (p0 ) will then also be Fermi transported since is a
geodesic and as such they de…ne the standard of no rotation along :
Remark 433 Let I 2 sec T V be a PIRF and
U V one of its integral
lines and let f g; U
M be a LLC through all the points of the world
line such that
= Ij . Then, in general f g is not a (nacsjI) in U ,
0
i.e., Ijp2= 6= @=@ p2= even if Ijp2 = @=@ 0 p2 .
6.7.1 LLRF s and the Equivalence Principle
There are many presentations of the EP and even very strong criticisms
against it, the most famous being probably the one o¤ered by Synge [415].
We are not going to bet on this particular issue. Our intention here is
to prove that there are models of GRT where the so called Principle of
Local Lorentz Invariance (PLLI) which according to several authors (see
below) follows from the Equivalence Principle is not valid in general. Our
strategy to prove this strong statement is to give a precise mathematical
wording to the PLLI (which formalizes the PLLI verbally introduced by
several authors) in terms of a physical equivalence of LLRF s (see below)
and then prove that PLLI is a false statement according to GRT. We start
by recalling formulations and comments concerning the EP and the PLLI.
According to Friedman [140] the
“Standard formulation of the EP characteristically obscure [the] crucial
distinction between …rst order laws and second order laws by blurring the
distinction between in…nitesimal laws, holding at a single point, and local
laws, holding on a neighborhood of a point”....
According to our point of view, in order to give a mathematically precise formulation of Einstein’s EP besides the distinctions mentioned above
6.7 Principle of Local Lorentz Invariance in GRT?
245
between in…nitesimal and local laws, it is also necessary to distinguish between some very di¤erent (but related) concepts, namely,29
(i) The concept of an observer (De…nition 372);
(ii) The general concept of a reference frame in a Lorentzian spacetime
(De…nition 378);
(iii) The concept of a natural adapted coordinate chart to a reference
frame (De…nition 379);
(iv) The concept of PIRFs (de…nition 428) and LLRF s (De…nition 430)
on U M ;
(v) The concept of an inertial moving observer carrying a tetrad along
(a geodetic curve), a concept we abbreviate by calling it an IMT (De…nition 432).
Einstein’s EP is formulated by Misner, Thorne and Wheeler (MTW)[247]
as follows:
“in any and every Local Lorentz Frame (LLF), anywhere and anytime
in the universe, all the (non-gravitational ) laws of physics must take on
their familiar special relativistic forms. Equivalently, there is no way, by
experiments con…ned to small regions of spacetime to distinguish one LLF
in one region of spacetime from any other LLF in the same or any other
region”.
We comment here that those authors30 did not give a formal de…nition
of a LLF. They try to make intelligible the EP by formulating its wording
in terms of a LLCC (see de…nition 429) and indeed these authors as many
others do not distinguish the concept of a reference frame Z 2 sec T M
from that of a (nacsjZ). This may generate misunderstandings. The mathematical formalization of a LLF used by MTW (and many other authors)
corresponds to the concept of LLRF introduced in De…nition 430.
In [81] Ciufolini and Wheeler call the above statement of MTW the
medium strong form of the EP. They introduced also what they called the
strong EP as follows:
“in a su¢ ciently small neighborhood of any spacetime event, in a locally
falling frame, no gravitational e¤ ects are observable”.
Again, no mathematical formalization of a locally falling frame is given,
the formulation uses only LLCC. Worse, if local means in a neighborhood
of a given spacetime event this principle must be false. For, e.g., it is well
known fact that the Riemann tensor couples locally with spinning particles.
Moreover, the neighborhood must be at least large enough to contain an
experimental physicist and the devices of his laboratory and must allow
for enough time for the experiments. With a gradiometer built by Hughes
2 9 These concepts are in general used without distinction by di¤erent authors leading
to misunderstandings and misconceptions.
3 0 For the best of our knowledge no author gave until now the fomal de…nition of a
LLRF as in De…nition 430.
246
6. Some Issues in Relativistic Spacetime Theories
corporation which has an area of approximately 400 cm2 any researcher can
easily discover if he is leaving in a region of spacetime with a gravitational
…eld or if he is living in an accelerated frame in a region of spacetime free
with a zero gravitational …eld.
Following [81, 247] recently several authors as, e.g., Will [450], Bertotti
and Grishchuk [45] and Gabriel and Haugan [143] (see also Weinberg [447])
claim that Einstein EP requires a sort of local Lorentz invariance. This
concept is introduced in, e.g., [45] with the following arguments.
To start we are told that to state the Einstein EP we need to consider
a laboratory that falls freely through an external gravitational …eld. Moreover, such a laboratory must be shielded, from external non-gravitational
…elds and must be small enough such that e¤ects due to the non homogeneity of the …eld are negligible through its volume. Then, they say, that the
local non-gravitational test experiments are experiments performed within
such a laboratory and in which self-gravitational interactions play no signi…cant part. They de…ne:
“The Einstein EP states that the outcomes of such experiments are independent of the velocity of the apparatus with which they are performed
and when in the universe they are performed”.
This statement is then called the Principle of Local Lorentz Invariance
(PLLI) and ‘convincing’ proofs of its validity are o¤ered, and there is no
need to repeat that ‘convincing’proofs here. Prugovecki [306] endorses the
PLLI and also said that it can be experimentally veri…ed. In his formulation
he translates the statements of [140, 247, 81, 450, 45, 143, 447] in terms
of Lorentz and Poincaré covariance of measurements done in two di¤erent
IMF (see de…nition 432). Based on these past tentatives of formalization
31
we give the following one.
Einstein EP: Let
be a timelike geodetic line on the world manifold M . For any LLRF (see de…nition 430) all non gravitational laws
of physics, expressed through the coordinates f g which are LLC associated with the LLRF (de…nition 430) should at each point along be
equal (up to terms in …rst order in those coordinates) to their special relativistic counterparts when the mathematical objects appearing in these
special relativistic laws are expressed through a coordinates in EinsteinLorentz-Poincaré gauge naturally adapted to an arbitrary inertial frame
I 2 sec T M 0 , where M 0 = R4 is the manifold of a Minkowski spacetime
structure M = (M 0 ; ; D; ; ").
Also, if the PLLI would be a true law of nature it could be formulated
as follows:
Principle of Local Lorentz Invariance: Any two LLRF and LLRF 0
associated with the timelike geodetic lines and 0 of two observers such
3 1 See
[273] for a history of the subject.
6.8 PIRFs on a Friedmann Universe
that \
event p.
0
247
= p are physically equivalent (according to De…nition 424) at
Of course, if PLLI is correct, it must follow that from experiments done
by observers inside some LLRF 0 — say L0 that is moving relative to another
LLRF L— there is no means for that observers to determine that L0 is in
motion relative to L.
Unfortunately the PLLI is not true. To show that it is only necessary
to …nd a model of GRT where the statement of the PLLI is false. Before
proving this result we shall need to prove that there are models for GRT
were PIRFs are not physically equivalent also.
6.8 PIRFs on a Friedmann Universe
Recall that GRT is a theory of the gravitational …eld [367] where a typical
model z 2 M odzE is of the form
z = ((M; g; D;
g ; "); T; (m;
));
(6.143)
where M = (M; g; D; g ; ") is a relativistic spacetime and T 2 secT20 M is
called the energy-momentum tensor. The tensor T represents the material
and energetic content of spacetime, including contributions from all physical …elds (with exception of the gravitational …eld and test particles). For
what follows we do not need to know the explicit form of T. Also, (m; )
represents a test particle, whose world line is . The proper axioms of zE
are:
1
Rg = T;
(6.144)
Dg = 0, (D) = 0, G = Ricci
2
where
is the torsion tensor, G is the Einstein tensor, Ricci is the Ricci
tensor and r is the Ricci scalar. The equation of motion of the test particle
(m; ) that moves only under the in‡uence of gravitation is:
D
= 0:
(6.145)
M is in general not ‡at, which implies that in general (see Remark 382)
there is no IRF I, i.e., a reference frame such that DI = 0.
Now, the physical universe we live in is reasonably described by metric
tensors of the Robertson-Walker-Friedmann type [367]. In particular, a very
simple spacetime structure M that represents the main properties observed
(after the big-bang) is formulated as follows: Let M = R3 I, I R and
R : I ! (0; 1); t ! R(t) and de…ne gin M (considering M as subset of
R4 ) by:
X
g = dt dt R(t)2
dxi dxi ; i = 1; 2; 3:
(6.146)
Then g is a Lorentzian metric in M and V = @=@t is a time-like vector
…eld in (M; g). Let M be oriented in time by @=@t and spacetime oriented
by dt ^ dx1 ^ dx2 ^ dx3 . Then M is a relativistic spacetime for I = (0; 1).
248
6. Some Issues in Relativistic Spacetime Theories
Now, V = @=@t is a reference frame. Taking into account that the connection coe¢ cients in a (nacsjV) given by the coordinate system in Eq.(6.146)
are
_
R
i
0
k
k
_
kl = 0;
kl = RR kl ,
0l =
R l
i
0
0
(6.147)
00 =
0l =
00 = 0;
we can easily verify that V is a PIRF ( according to de…nition 6) since
DV V= 0 and d V ^ V = 0, V = g(V; ). Also, since V = dt, V is
proper time synchronizable.
Proposition 434 In a spacetime de…ned by Eq.(6.146) which is a model
of zE there exists a PIRF Z 2 sec T U which is not physically equivalent
to V = @=@t.
Proof : Let Z 2 sec T U be given by
Z=
u
(R2 + u2 )1=2
@=@t + 2 @=@x1
R
R
(6.148)
where in Eq.(6.148) u 6= 0 is a real constant.
Since DZ Z = 0 and d Z ^ Z = 0,
Z = g(Z; ), it follows that Z
is a PIRF32 . All that is necessary in order to prove our proposition is to
show that Z and V are not equivalent. It is enough to prove that the
expansion ratios EZ 6= EV . Indeed, Eq.(6.43) gives
_
EV = 3R=R;
h
i
_ + 2R(R
_ 2 + u2 )1=2
RR
EZ =
;
1=2
R2 (R2 + u2 )
where
v = R(
d 1
x
dt
= u(1 + u2 )
)
(6.149)
1=2
(6.150)
t=0
is the initial metric velocity of Z relative to V, since we choose in what
follows the coordinate function t such that R(0) = 1, t = 0 being taken
as the present epoch where the experiments are done. Then, EV (p0 ) = 3a;
and for v << 1; EZ (p0 ) = 3a av 2 .
6.8.1 Mechanical Experiments Distinguish PIRFs
The question arises: can mechanical experiments (distinct from the one
designed to measure the expansion ratio) distinguish between the PIRFs
V and Z? The answer is yes. To prove our statement we proceed as follows.
3 2 Introducing the (nacsjZ) given by eq.(6.153) we can show that
follows that is also proper time synchronizable.
Z
= dt0 and it
6.8 PIRFs on a Friedmann Universe
(i) We start by …nding a (nacsjZ). To do that we note if
curve of Z, we can write
Zj = [
d
(x
ds
)
249
is an integral
@
]j
@x
(6.151)
where s is the proper time parameter along . Then, we can write [taking
into account Eqs.(6.147)] its parametric equations as
d 1
x
dt
d 1
( ds
x
=
)
d
( ds
t
)
=
R(R2
u
;
+ u2 )1=2
x2
= 0;
x3
= 0 (6.152)
(The direction x1
= 0 is obviously arbitrary). We then choose as a
(nacsjZ) the coordinate functions (t0 ; x01 ; x02 ; x03 ) such that:
Z t
1
01
1
dr
; x02 = x2 ;
x =x
u
2 (r) + u2 ]1=2
R(r)[R
0
Z t
[R2 (r) + u2 ]1=2
x03 = x3 ; t0 =
ux1
(6.153)
dr
R(r)
0
We then get:
0
0
g = dt
0 2
dt
R(t )
( h
1 v 2 (1 R(t0 ) 2 )
1 v2
02
02
i
dx01 dx01
dx + dx03 dx03
+dx
)
;
(6.154)
and the connection coe¢ cients in the (nacsjZ) are,
:
:
2
RR
0
kl
=
i
kl
= 0,
2
1
(R + u2 ) 2
i
00
=
0
0l
:
2
RR
kl ,
1
01
=
=
0
00
= 0:
2
3
(R + u2 ) 2
,
2
02
=
3
03
=
R
2
1
(R + u2 ) 2
;
(6.155)
where R(t0 ) = R(t(t0 )) and v given by Eq.(6.150) is the initial metric velocity of Z relative to V, since we choose in what follows the coordinate
function t such that R(0) = 1, t = 0 being taken as the present epoch where
the experiments are done. Z = @=@t0 is a proper time synchronizable reference frame and we can verify that t0 is the time shown by standard clocks
at rest in the Z reference frame and which are synchronized à l’Einstein.
Notice that an observer at rest in Z does not know a priori the value of v.
He can discover this value as follows:
(ii) The solution of the equation of motion for a free particle (m; ) in
V with the initial conditions at p0 = (0; xi
(0) = 0); i = 1; 2; 3 and
d i
d i
i
x
(0)
=
u
for
a
…xed
i
and
x
(0)
=
0, j 6= i, is given by an
dt
dt
equation analogous to Eq.(6.152). The accelerations are such that
d2 j
x
ds2
(t))
p0
= 0; j 6= i:
(6.156)
250
6. Some Issues in Relativistic Spacetime Theories
(iii) The equation of motion for a free particle (m; 0 ) in Z , can be written
d2 01
d2 01 0
0 0
as (we write for simplicity in what follows ds
(t )
2x
ds2 x (t )
2
d
01
ds2 x , etc...)
:
d2 x01
=
ds2
2
2
dx 01 dt0 2
( ) ;
3
2
0
ds
(R + u2 ) 2 dt
RR
:
d2 x0i
=
ds2
d2 t0
=
ds2
dx 0i dt0
( 2 ); i = 2; 3;
2
0
(R + u2 ) dt ds
"
:
2
2
dx02
RR
dx01
+
2 2
1
0
dt
dt0
(R + u2 ) 2
R
2
1
2
"
dt0
b2
= 1+R
ds
dx01
dt0
2
+R
2
dx02
dt0
2
+
2
2
+R
2
dx03
dt0
dx03
dt0
2
#
#
1
2
(6.157)
where the dot over R in Eq.(6.157) means derivative with respect to t0 and
b denotes the square root of the coe¢ cient of dx01 dx01 term in Eq.(6.154).
R
From these equations it is easy to verify that the two situations :
(a) motion in the (x01 ; x02 ) plane with initial conditions at p0 with coordinates (t0 = 0; x01 = x02 = 0 = x03 ) given by
dx01 (t0 )
dt0
= v10 ;
p0
dx02 (t0 )
dt
= 0;
(6.158)
p0
and
(b) motion in the (x01 ; x02 ) plane with initial conditions at p0 with coordinates (t0 = 0; x01 = x02 = 0 = x03 ) given by
dx01 (t0 )
dt0
= 0;
p0
dx02 (t0 )
dt0
= v20 ;
(6.159)
p0
produce asymmetrical outputs for the measured accelerations along x01 and
x02 . The explicit values depends of course of the function R(t). If we take
R(t) = 1 + at, the asymmetrical accelerations will be given in terms of
a << 1 and v. This would permit in principle for the eventual observers
living in the PIRF Z to infer the value of u (or v).
6.8.2 LLRF and LLRF 0 are not Physically Equivalent on a
Friedmann Universe.
Proposition 435 33 There are models of GRT for which two Local Lorentz
Reference Frames are not physically equivalent.
3 3 The suggestion of the validity of a proposition like the one formalized by proposition
435 has been …rst proposed by Rosen [363]. However, he has not been able to identify
6.8 PIRFs on a Friedmann Universe
251
Proof. Take as model of GRT the one just described above where g is given
by Eq.(6.146) and take as before, R(t) = 1 + at. Consider two integral lines
and 0 of V and Z such that \ 0 = p.
We can associate with these two integral lines the LLRF L and the
LLFR 0 L0 as in de…nition 429. Observe that Vj = Lj and Zj 0 = L0 j 0 .
De…nition 425 says that if L and L0 are physically equivalent then we
must have DL = DL0 . However, a simple calculation shows that in general
DL 6= DL0 even at p! Indeed, we have
EL =
3t
_
R
R
!2
;
(6.160)
:
:
2
2 1=2
EL0 = 2R(R + u )
:
2
_
+R
:
2
4
2R R
(R +
u2 )3
01
tx
2R
2
(R +
RR
:
2
2
(R + u2 )3=2
2R
2
(R + u2 )1=2
:
2
u2 )
02
tx
2R
2
(R +
2
u2 )
tx03 :
(6.161)
From Eq.(6.160) and Eq.(6.161) we see that the expansions ratios EL and
EL0 are di¤erent in our model and then our claim is proved. At p, we have
EL (p) = 0 and EL0 (p) = 2av 2 .
Remark 436 Proposition 435 establishes that in a Friedmann universe
there is a LLRF (say L) whose expansion ratio at p is zero. Any other
LLRF 0 (say L0 ) at p will have an expansion ratio at p given by 2av 2 ,
where a
1 and v is the metric velocity of L0 relative to L at p. This
expansion ratio can in principle be measured and this is the reason for the
non validity of the PLLI as formulated by many contemporary physicists
and formalized above. Note that all experimental veri…cations of the PLLI
mentioned by the authors that endorse the PLLI have been obtained for
LLRF s moving with v << 1, and have no accuracy in order to contradict
the result we found. We do not know of any experiment that has been done
on a LLRF which enough precision to verify the e¤ ect. Anyway the non
physical equivalence between L and L0 is a prediction of GRT and must be
accepted if this theory is right. In conclusion, PLLI is only approximately
valid.
the true nature of the V and Z which he thought as representing ‘inertial’ frames. He
tried to show the validity of the proposition by analyzing the output of mechanical and
optical experiments done inside the frames V and Z. We present below a simpli…ed
version of his suggested mechanical experiment. It is important to emphasize here that
from the validity of the proposition 435 Rosen suggested that it implies in a breakdown
of the PLLI. Of course, the PLLI refers to the physical equivalence of LLRF s. Also the
proof of proposition 435 given above appeared originally in [349].
252
6. Some Issues in Relativistic Spacetime Theories
We recall that Friedman [140] formulates the PLLI by saying that if
(U; ), (U 0 ; ) (U \ U 6= ;) are LLCC adapted to the L and L0 respectively, then the PLLI implies that two experiments whose initial conditions
read alike in terms of f g and f g will also have the same outcome in
terms of these coordinates .
Friedman’s statement is not correct, of course, in view of Proposition
435 above, for measurement of the expansion ratio of a real reference
frame which has material support is something objective and, of course,
can be in principle, be determined with an appropriate physical experiment. However, for experiments di¤erent from this one (measurement of
the expansion ratio) we can accept Friedman’s formulation of the PLLI as
an approximately true statement.
Recall the expansion ratios calculated for V; Z; L; L0 . Now, a << 1.
Then, if v << 1 the LLRF L and the LLRF 0 L0 will be almost ‘rigid’
whereas the V and Z are expanding. In other words, the L and L0 frames
can be thought as being physically materialized in their domain by real
solid bodies and thus correspond to small real laboratories, the one used
by physicists. On the other hand it is well known that the V frame is
an idealization, since only the center of mass of the galactic clusters are
supposed to be comoving with the V frame, i.e., each center of mass of a
galactic cluster follows some particular integral line of V. Concerning the
Z frame, in order for it to be realized as a physical system it must be build
with a special matter that su¤ers in all points of its domain an expansion a
little bit greater than the cosmic expansion. Of course, such a frame would
be a very arti…cial one, and we suspect that such a special matter cannot
be prepared in our universe.
6.8.3 No Generalization of the Principle of Relativity for
GRT
In the previous sections we presented a careful analysis of the concept of
a reference frames in GRT. These objects have been modelled as certain
unit timelike vector …elds. We gave physically motivated and mathematical
rigorous de…nition of physically equivalent reference frames in a relativistic
spacetime theory. We investigate which are the reference frames in GRT
which share some of the properties of the inertial reference frames of the
Special Theory of Relativity. We found that in GRT there are two classes
of frames that have some of the properties of the inertial frames of Special Theory of Relativity. These are the class of the pseudo inertial reference frames (PIRFs) and the class of the local Lorentz reference frames
(LLRF s). We showed that LLRF s are not physically equivalent in general and this implies that the so called Principle of Local Lorentz invariance
(PLLI) which several authors state as meaning that LLRF s are equivalent
is false. It can only be used as an approximation in experiments that do not
6.9 Schwarzschild Original Solution and the Existence of Black Holes in GRT
253
have enough accuracy to measure the e¤ect we found. We prove moreover
that there are models of GRT where PIRFs are not physically equivalent
also. Our results show without any doubt that there is no generalization of
the Principle of Relativity, understood as a generalization of PR2 , i.e., that
there exists physical equivalence of all reference frames in GRT. Indeed, in
the structure (M; g; D; g ; ") given two arbitrary reference frames Z; Z0 it
is not the even the case in general that DZ = DZ0 and so they are not
physically equivalent.
6.9 Schwarzschild Original Solution and the
Existence of Black Holes in GRT
1. Schwarzschild [382] looked for a solution of Einstein equations supposing
a priori that the spacetime manifold where a point mass and the gravitational …eld it generated live is M = R R3 where time takes values in R
and R3 denotes the usual three-dimensional Euclidean space. Indeed, he
equipped R R3 with coordinates (t; x; y; z) explicitly saying that (x; y; z)
are rectangular Cartesian coordinates. After that, as a second step he introduced usual polar coordinate functions in the game. In so doing, he correctly
left out from R3 the origin O = (0; 0; 0) where the point mass particle is
supposed to be located at any instant of coordinate time t and start solv3
ing Einstein equations in the manifold R (R
O)
R (0; 1) S 2 ,
introducing standard spherical coordinates (r; #; ). However, since those
coordinate functions do not satisfy the Einstein coordinate gauge34 that
Schwarzschild chose to use, he introduced spherical coordinates with determinant 1. After some mathematical tricks, he found as solution of his
problem a metric …eld g os , which has a unique singularity at O and as such
his solution does not imply in any black hole (de…ned in 4 below).35
Schwarzschild original solution reads
g os = h(r)dt
dt
h(r)
1 0
f (r)dr
dr
F (r)(d
d + sin2 d'
d');
(6.162)
where
h(r) =
1
2m
f (r)
;
f (r) = (r3 + 8m3 )1=3
(6.163)
2. However, in fact Schwarzschild, in order to determine one of the integration constants of the di¤erential equations he was solving and that
leads him to Eq.(6.162), needed for his calculations to use the manifold
p
coordinate gauge …xes jdet gj = 1.
can easily verify that the coordinate time for a radial light ray to go from any
r > 0 to a point such that 2m < r < 1 is …nite.
3 4 Einstein
3 5 One
254
6. Some Issues in Relativistic Spacetime Theories
with boundary 36 R [0; 1) S 2 and thus his original mass point supposed
to be located for any instant of time at the point O 2 R3 ended to be represented by the manifold37 f0g S 2 (something obviously odd that Hilbert
elegantly, without criticizing Schwarzschild, observed in a footnote of his
paper on the Schwarzschild solution).
3. Before proceeding, and in order to avoid any confusion note that
despite the fact that the original manifold postulated as model of spacetime by Schwarzschild is R R3 this does not imply that this manifold or
the manifold R [0; 1) S 2 equipped with the Levi-Civita connection D
of g S (that solves Einstein equation) is ‡at. In fact, the connection D for
Schwarzschild problem is curved, this statement meaning, of course, that
its Riemann curvature tensor is non null. Please, take always the following
statement into account [127]:
“Manifolds do not have curvature, it is the connection imposed on a
manifold that may or may not have non null curvature (and/or non null
torsion, non null nonmetricity). Some manifolds may be bended surfaces
in a Euclidean (or pseudo-Euclidean) space of appropriate dimension. But
to be bended (a property described by the shape operator introduced in
Chapter 5 ) has in general nothing to do with the fact that a connection
de…ned in the manifold is curved.”
4. Droste [106] and Hilbert [176] found independently another solution
of Einstein equations based on di¤erent assumptions than the ones used by
Schwarzschild38 . Modern relativists39 (following Droste and Hilbert) …nd as
solution40 with rotational symmetry of Einstein equation in vacuum a metric …eld g DH (at least C 2 ) de…ned in the manifold R (0; 2m)[(2m; 1) S 2 .
Relativists say that the “part”R (0; 2m) S 2 where the solution is valid
de…nes a black hole [367].
5. It is crucial to have in mind that the quasi spherical coordinates
functions (r; #; ) used by modern relativists are such that the coordinate
function r is not the Schwarzschild spherical coordinate function r, i.e.,
r 6= r:
3 6 For
the use of the concept of manifold with boundary to present singularities in
General Relativity, see, [163, 383, 384, 144]. For some skifull commentaries on the peril
of using boundary manifolds in General Relativity without taking due care see [411]
3 7 f0g denotes the set whose unique element is 0 2 [0; 1) , i.e., the boundary of the
semi-line [0; 1).
3 8 In the words of Synge [414]: Schwarzschild imposed spherical symmetry, whereas
Droste and Hilbert imposed rotational symmetry, a subtle but crucial detail.
3 9 See, e.g., [163, 287]
4 0 Eventually, it would better to say, as did O’Neill in his book [287] that we start
looking for one solution of a problem and ended with two solutions.
6.9 Schwarzschild Original Solution and the Existence of Black Holes in GRT
255
6. Schwarzschild wrote his …nal formula for g os using a function f (r)
which is formally identical to the Droste-Hilbert formula for g DH if f (r) is
read as the coordinate function r. However, Schwarzschild solution is valid
only for f (r) > 2m whereas the Droste-Hilbert solution is valid for any
r 2 (0; 2m) [ (2m; 1).
7. Of course, there is no sense in supposing that space-time has a disconnected topology. Thus, under the present ideology of …nding maximal
extension of manifolds equipped with Lorentzian metrics as the true representatives of gravitational …elds, relativists maximally extend the solution
g DH to a solution g valid in a connected manifold called the Kruskal (sometimes, Kruskal- Szekeres) spacetime [201, 417]. The total Kruskal manifold
which has an exotic topologyis usually assocoated to a hypothetical object
called the wormhole. The …nal solution g is presented as a function of coordinate functions (u; v; #; ') and r which (keep this in mind ) becomes an
implicit function of the coordinate functions (u; v).
8. It is assumed by relativists that a connected “part” of the Kruskal
manifold describes a black-hole where g has a real singularity only at the
place de…ned by the function r(u; v) = 0.
9. In conclusion, Schwarzschild original solution and the Kruskal extension of the Droste-Hilbert solution de…ne space-times with very di¤ erent
topologies, so they are not the same solution of Einstein equation. In the
former the topology of the manifold has been …xed a priori, in the latter
the topology of the manifold has been …xed a posteriori by the process of
maximal extension.
10. There are some published papers that do not properly distinguish
these two di¤erent solutions41 , moreover, there are some authors stating
(explicitly, or in a disguised way) that it is possible to extend the Schwarzschild original solution that was written in terms of the function f for the
domain 0 f (r) < 1, but this idea is, of course, a logical non sequitur,
since for Schwarzschild the manifold is …xed a priori. Any appropriate discussion of the mathematical aspect of the back-hole solution of Einstein
equations clearly requires a reasonable understanding of di¤erential geometry, and of course, of topology42 . And it is also important, to advise that
everyone that wants to discuss the black-hole issue and did not read the
original Schwarzschild paper (or its English version, available at the arXiv)
must do that in a hurry.
4 1 Take notice that there are also some non sequitur mathematical statements in some
of those papers. See http://www.physicsforums.com/showthread.php?t=124277.
4 2 No more than what may be found in [163] or [287]. A thoughtful mathematical
discusssion (and historical review) of the Schwarzschild,the Droste-Hilbert and KruskalSzekeres solutions may be found in [248].
256
6. Some Issues in Relativistic Spacetime Theories
11. Failing to properly understand the di¤erent topologies of the two
solutions mentioned above (Schwarzschild and the maximal extension of
the Droste-Hilbert) is thus making some people (including some that say
to be relativist physicists) not to discuss contemporary GRT, but some
other things, believing to be the same thing.
12. The question if black holes exist or not is, of course, not a mathematical one 43 , it is a physical question and presently at least one of the present
authors believe that they do not exist, leaving this clear in [127], where
it is argued that it is necessary to construct a theory of the gravitational
…eld where that …eld is to be regarded as a …eld in the sense of Faraday
(like the electromagnetic …eld and the weak and strong force …elds) “living” in Minkowski space-time (see also Chapter 11). Thus, that “part” of
the maximal extension of the Droste-Hilbert solution of Einstein equations
(describing a black hole) probably does not describe anything real in the
physical world.
13. It is indeed out of question, the fact that Einstein equations (according to the contemporary interpretation of GRT) have solutions describing black holes. Of course, this does not leave everyone happy and
many physicists have proposed and are proposing alternative solutions of
Einstein equations capable of describing the …nal stage of super dense stars
and which according to them looks more “realistic”. See, e.g., [238].
Exercise 437 (i) Show that under the transformation
t0 = t; r = f (r) = (r3 + 8m3 )1=3 ;
0
= ; '0 = '
(6.164)
the expression of g os is the one given by g s in Eq.(6.97).
(ii) Are g os and g s di¤ eomorphically equivalent?
4 3 We mean that black holes exist as legitimate solutions of the Einstein equation,e.g.,
the ones described by a “part”[287] of the Kruskal-Szekeres solution.
This is page 257
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7
Cli¤ord and Dirac-Hestenes Spinor
Fields
7.1 The Cli¤ord Bundle of Spacetime
In this Chapter, M refers (unless otherwise stated) to a four dimensional,
real, connected, paracompact and non-compact manifold. We recall that
in Section 4.7.1 of Chapter 4 we de…ned a Lorentzian manifold as a pair
(M; g), where g 2 sec T20 M is a Lorentzian metric of signature (1; 3), i.e.,
for all x 2 M , Tx M ' Tx M ' R1;3 , where R1;3 is the vector Minkowski
space. Moreover, we also de…ned in section 4.7.1 a spacetime M as pentuple (M; g; r; g ; "), where (M; g; g ; ") is an oriented Lorentzian manifold
(oriented by g ) and time oriented by " , and r is a linear connection on
M such that rg = 0. Also, T and R are respectively the torsion and curvature tensors of r. If T(r) = 0, then M has been called in Chapter 4 a
Lorentzian spacetime. The particular Lorentzian spacetime where M ' R4
and such that R(r) = 0 has been called Minkowski spacetime and will be
denoted by M. We recall also that when rg = 0, R(r) 6= 0 and T(r) 6= 0,
M has been called a RCST. The particular RCST such that R(r) = 0 has
been called a teleparallel spacetime.
In what follows PSOe1;3 (M ) (PSOe1;3 (M )) denotes the principal bundle
of oriented Lorentz tetrads (cotetrads). We assume that the reader is acquainted with the structure of PSOe1;3 (M ) (PSOe1;3 (M )) (as recalled, e.g., in
Appendix A.1.1) whose local sections in U M are the time oriented and
oriented orthonormal frames (coframes) in U , or simply frame (coframe)
when no confusion arises [77, 138, 265, 264, 288]. Also, since we work with
258
7. Cli¤ord and Dirac-Hestenes Spinor Fields
the Cli¤ord bundle of nonhomogeneous di¤erential forms, given a choice of
a frame
2 sec PSOe1;3 (M ) we immediately choose for almost all considerations in this chapter the corresponding dual frame 2 sec PSOe1;3 (M ).
We recall also that PSOe1;3 (M ) ' PSOe1;3 (M ). We already de…ned the Clifford bundle of di¤erential forms on a structure (M; g) on Chapter 4. We
recall that the Cli¤ord bundle of di¤erential forms of a Lorentzian manifold
(M; g) is the bundle of algebras
C`(M; g) =
[
x2M
C`(Tx M; gx );
(7.1)
where C`(Tx M; gx ) ' R1;3 the spacetime algebra introduce in Chapter 3
and g is the metric of the cotangent bundle such that gx is related to g x
as introduced in Chapter 4..
7.1.1 Details of the Bundle Structure of C`(M; g)
We mention in Chapter 4 that a Cli¤ord bundle is a vector bundle associated to the principal bundle PSOe1;3 (M ) of orthonormal frames. More
speci…cally we have that
C`(M; g) = PSOe1;3 (M )
Ad0
R1;3 ' P SOe1;3 (M )
Ad0
R1;3
We now give the details of the bundle structure.
(i) Let c : C`(M; g) ! M be the canonical projection of C`(M; g)
and let fU g be an open covering of M . There are trivialization mappings
1
R1;3 of the form i (p) = ( c (p); i;x (p)) = (x; i;x (p)).
i : c (Ui ) ! Ui
If x 2 Ui \ Uj and p 2 c 1 (x), then
i;x (p)
= hij (x)
j;x (p)
(7.2)
for hij (x) 2 Aut(R1;3 ), where hij : Ui \ Uj ! Aut(R1;3 ) are the transition
mappings of C`(M; g). We know from Chapter 3 that every automorphism
of R1;3 is inner. Then,
hij (x)
j;x (p)
= gij (x)
i;x (p)gij (x)
1
(7.3)
for some gij (x) 2 R?1;3 , the group of invertible elements of R1;3 .
(ii) Now, as we learned in Chapter 3, the group SOe1;3 has a natural extension in the Cli¤ord algebra R1;3 . Indeed we know that R?1;3 acts naturally
on R1;3 as an algebra automorphism through its adjoint representation. A
set of lifts of the transition functions of C`(M; g) is a set of elements fgij g
7.1 The Cli¤ord Bundle of Spacetime
259
R?1;3 such that if1
Ad : g 7! Adg ;
Adg (a) = gag
1
; 8a 2 R1;3 ;
(7.4)
then Adgij = hij in all intersections.
(iii) Also = AdjSpine1;3 de…nes a group homeomorphism : Spine1;3 !
SOe1;3 which is onto with kernel Z2 . We have that Ad 1 = identity, and so
Ad : Spine1;3 ! Aut(R1;3 ) descends to a representation of SOe1;3 . Let us call
Ad0 this representation, i.e., Ad0 : SOe1;3 ! Aut(R1;3 ). Then we can write
Ad0 (g) a = Adg a = gag 1 .
(iv) It is clear then, that the structure group of the Cli¤ord bundle
C`(M; g) is reducible from Aut(R1;3 ) to SOe1;3 . Thus the transition maps
of the principal bundle of oriented Lorentz cotetrads PSOe1;3 (M ) can be
(through Ad0 ) taken as transition maps for the Cli¤ord bundle. We then
have [206]
C`(M; g) = PSOe1;3 (M ) Ad0 R1;3 ;
(7.5)
i.e., the Cli¤ord bundle is an associated vector bundle to the principal
bundle PSOe1;3 (M ) of orthonormal Lorentz coframes.
7.1.2 Cli¤ord Fields
also a vector space over R which is
Recall that C`(Tx M; gx ) is V
V isomorphic to the exterior algebra Tx M of the cotangent space and Tx M =
M4 V
Vk
k
Tx M , where
Tx M is the k4 -dimensional space of k-forms.
k=0
Then,
as we already saw in Section 4.6.1, there is a natural embedding
V
T M ,! C`(M; g) [206] and sections of C`(M; g)— Cli¤ord …elds (de…nition 279)— can be represented as a sum of non homogeneous di¤erential
forms. Let fea g 2 sec PSOe1;3 (M ) (the orthonormal frame bundle) be a
tetrad basis for T U
T M , i.e., g(ea ; eb ) = ab = diag(1; 1; 1; 1)
and (a; b = 0; 1; 2; 3). Moreover, let f a g 2 sec PSOe1;3 (M ). Then, for each
V1
a = 0; 1; 2; 3, a 2 sec
T M ,! sec C`(M; g), i.e., f b g is the dual basis of
V1
fea g. Finally, let f a g, a 2 sec
T M ,! sec C`(M; g) be the reciprocal
basis of f b g, i.e., a b = ab :
Recall also that the fundamental Cli¤ ord product is generated by
ab
+
ba
=2
ab
:
(7.6)
If C 2 sec C`(M; g) is a Cli¤ord …eld, we have:
1 Recall that Spine = fa 2 R+ : a~
a = 1g ' Sl(2; C) is the universal covering group
1;3
1;3
of the restricted Lorentz group SOe1;3 .
260
7. Cli¤ord and Dirac-Hestenes Spinor Fields
1
1
bij i j + tijk i j k + p 5 ;
2!
3!
= 0 1 2 3 is the volume element and
^0
s; vi ; bij ; tijk ; p 2 sec
T M ,! sec C`(M; g):
C = s + vi
where
5
i
+
(7.7)
(7.8)
7.2 Spin Structure
De…nition 438 A spin structure on M consists of a principal …bre bundle
e
s : PSpine1;3 (M ) ! M (called the Spin Frame Bundle) with group Spin1;3
and a map
s : PSpine1;3 (M ) ! PSOe1;3 (M )
(7.9)
satisfying the following conditions:
(i) (s(p)) = s (p) 8p 2 PSpine1;3 (M ); is the projection map of the
bundle PSOe1;3 (M ).
(ii) s(pu) = s(p)Adu ; 8p 2 PSpine1;3 (M ) and Ad : Spine1;3 ! Aut(R1;3 );
Adu : R1;3 3 x 7! uxu 1 2 R1;3 .
De…nition 439 Any section of PSpine1;3 (M ) is called a spin frame …eld (or
simply a spin frame).
Remark 440 We de…ne also a Spin Coframe Bundle PSpine1;3 (M ) by substituting in Eq.(7.9) PSOe1;3 (M ) by PSOe1;3 (M ). Of course, PSpine1;3 (M ) '
PSpine1;3 (M ). Any section of PSpine1;3 (M ) is called a spin coframe …eld (or
simply a spin coframe). We shall use the symbol
2 sec PSpine1;3 (M ) to
denoted a spin coframe dual to a spin frame 2 sec PSOe1;3 (M ).
Recall that we learned in Chapter 3 that the minimal left (right) ideals
of Rp;q are left (right) modules for Rp;q There, covariant, algebraic and
Dirac-Hestenes spinors (when (p; q) = (1; 3)) were de…ned as certain equivalence classes in appropriate sets. We are now interested in de…ning algebraic Dirac spinor …elds and also Dirac-Hestenes spinor …elds, on a general
RCST, as sections of appropriate vector bundles (spinor bundles) associated
to PSpine1;3 (M ). The compatibility between PSpine1;3 (M ) and PSOe1;3 (M ), as
captured in de…nition 438 (and taking into account Remark 440), is essential for that matter.
It is therefore natural to ask: When does a spin structure exist on an
oriented manifold M ? The answer, which is a classical result [28, 40, 206,
92, 138, 146, 147, 246, 265, 264, 266, 295, 288], is that a necessary and
su¢ cient condition for the existence of a spin structure on M (given the
already assumed properties of M ) is that the second Stiefel-Whitney class
7.2 Spin Structure
261
w2 (M ) of M is trivial. Moreover, when a spin structure exists, one can show
that it is unique (modulo isomorphisms), if and only if, H 1 (M; Z2 ), the deRham cohomology group with values in Z2 is trivial. We now, introduce
without proof a theorem that is crucial for our theory.
Theorem 441 For a Lorentz manifold (M; g), a spin structure exists if
and only if PSOe1;3 (M ) is a trivial bundle.
Proof. See Geroch [146].
Remark 442 Recall that global sections
2 sec PSOe1;3 (M ) are Lorentz
frames and global sections
2 sec PSpine1;3 (M ) are spin frames. We recall from Appendix A.1.3 that each 2 sec PSOe1;3 (M ) is a basis for T M ,
which is completely speci…ed once we give an element of the Lorentz group
for each x 2 M and …x a …ducial frame. Each 2 sec PSpine1;3 (M ) is also
a basis for T M and is completely identi…ed once give an element of the
Spine1;3 for each x 2 M and …x a …ducial frame. Note that two ordered
basis for T M when considered as spin frames are to be considered di¤erent
even if consisting of the same vector …elds, which are related by multiples
of a 2 rotation. Also, two ordered basis for T M are considered equal when
considered as spin frames, if they consist of the same vector …elds related
by multiples of a 4 rotation. Although we recognize that this mathematical construction seems at …rst sight impossible of experimental detection,
Aharonov and Susskind [4] warrants that with clever experiments the spinor
structure can be detected.
Remark 443 Recall that a principal bundle is trivial, if and only if, it admits a global section. Therefore, Geroch’s result says that a (non-compact)
spacetime admits a spin structure, if and only if, it admits a (globally
de…ned ) Lorentz frame. In fact, it is possible to replace PSOe1;3 (M ) by
PSpine1;3 (M ) in Remark 441 (see footnote 25 in [146] ). In this way, when a
(non-compact) spacetime admits a spin structure, the bundle PSpine1;3 (M )
is trivial and, therefore, every bundle associated to it is also trivial. This
result is indeed a very important one, because it says to us that the real
spacetime of our universe (that, of course, is inhabited by several di¤ erent
types of spinor …elds) must have a topology that admits a global tetrad …eld,
which is de…ned only modulus a local Lorentz transformation. A dual cotetrad has been associated to the gravitational …eld in Chapter 4 (see more
details in Chapters 9 and 11), where we wrote wave equations for them.
In a certain sense that cotetrad …eld is a representation of the substance of
physical spacetime.
De…nition 444 A oriented manifold endowed with a spin structure will be
called a spin manifold.
Exercise 445 Show that Minkowski and de Sitter spacetimes are spin manifolds.
262
7. Cli¤ord and Dirac-Hestenes Spinor Fields
7.3 Spinor Bundles and Spinor Fields
We now present the most usual de…nitions of spinor bundles appearing
in the literature2 and next we …nd appropriate vector bundles such that
particular sections are LIASF (De…nition 453) or DHSF (De…nition 464)
De…nition 446 A real (left) spinor bundle for M is a vector bundle
S(M; g) = PSpine1;3 (M )
l
M
(7.10)
where M is a left module for R1;3 and l is a representation of Spine1;3 on
End(M) given by left multiplication by elements of Spine1;3 .
De…nition 447 The dual bundle S ? (M; g) is a real (right) spinor bundle
S ? (M; g) = PSpine1;3 (M )
r
M?
(7.11)
where M? is a right module for R1;3 and r is a representation Spine1;3 in
End(M) given by right multiplication by (inverse) elements of Spine1;3 . By
right multiplication we mean that given a 2 M ? ; r (u)a = au 1 , then
r (uu)a
= (uu)
1
= u
1
u
1
=
r (u) r (u)a:
(7.12)
De…nition 448 A complex spinor bundle for M is a vector bundle
Sc (M; g) = PSpine1;3 (M )
c
Mc
(7.13)
where Mc is a complex left module for C R1;3 ' R4;1 ' C(4), and where
e
c is a representation of Spin1;3 in End(Mc ) given by left multiplication
e
by elements of Spin1;3 .
De…nition 449 The dual complex spinor bundle for M is a vector bundle
Sc? (M; g) = PSpine1;3 (M )
c
M?c
(7.14)
where M?c is a complex right module for C R1;3 ' R4;1 ' C(4), and where
e
c is a representation of Spin1;3 in End(Mc ) given by right multiplication
e
by (inverse) elements of Spin1;3 .
Taking, e.g., Mc = C4 and c the D(1=2;0) D(0;1=2) representation
of Spine1;3 = Sl(2; C) in End(C4 ), we immediately recognize the usual
de…nition of the (Dirac) covariant spinor bundle of M , as given, e.g.,
in [77, 138, 265].
2 We recall that there are some other (equivalent) de…nitions of spinor bundles that
we are not going to introduce in this book as, e.g., the one given in [51] in terms of
mappings from PSpine1;3 to some appropriate vector space.
7.3 Spinor Bundles and Spinor Fields
263
7.3.1 Left and Right Spin-Cli¤ord Bundles
We saw in Chapter 3 that besides the ideal I = R1;3 12 (1 + E0 ), other ideals
exist in R1;3 that are only algebraically equivalent to this one. In order to
capture all possibilities we recall that R1;3 can be considered as a module
over itself by left (or right) multiplication. We are thus led to the
De…nition 450 The left real spin-Cli¤ ord bundle of M is the vector bundle
C`lSpine1;3 (M; g) = PSpine1;3 (M )
l
R1;3
(7.15)
where l is the representation of Spine1;3 on R1;3 given by l(a)x = ax. Sections of C`lSpine1;3 (M; g) are called left spin-Cli¤ ord …elds.
Remark 451 C`lSpine1;3 (M; g) is a ‘principal R1;3 - bundle’, i.e., it admits
a free action of R1;3 on the right [206], which is denoted by Rg , g 2 R1;3 .
This will be discussed in detail below.
Remark 452 There is a natural embedding PSpine1;3 (M ) ,! C`lSpine1;3 (M; g)
which comes from the embedding Spine1;3 ,! R01;3 . Hence (as we shall see
in more details below ), every real left spinor bundle (see De…nition 450)
for M can be captured from C`lSpine1;3 (M; g), which is a vector bundle very
di¤ erent from C`(M; g). Their relation is presented below, but before that
we give the
De…nition 453 Let I(M; g) be a subbundle of C`lSpine1;3 (M; g) such that
there exists a primitive idempotent e of R1;3 with
Re
=
(7.16)
for all
2 sec I(M; g) ,! sec C`lSpine1;3 (M; g): Then, I(M; g) is called a
subbundle of ideal left algebraic spinor …elds. Any
2 sec I(M; g) ,!
sec C`lSpine1;3 (M; g) is called a left ideal algebraic spinor …eld (LIASF).
I(M; g) can be thought of as a a real spinor bundle for M such that M
in Eq.(7.13) is a minimal left ideal of R1;3 :
De…nition 454 Two subbundles I(M; g) and I(M; g) of LIASF are said
to be geometrically equivalent if the idempotents e; e0 2 R1;3 (appearing in
the previous de…nition) are related by an element u 2 Spine1;3 , i.e., e0 =
ueu 1 .
De…nition 455 The right real spin-Cli¤ ord bundle of M is the vector bundle
C`rSpine1;3 (M; g) = PSpine1;3 (M ) r R1;3 :
(7.17)
Sections of C`rSpine1;3 (M; g) are called right spin-Cli¤ ord …elds.
264
7. Cli¤ord and Dirac-Hestenes Spinor Fields
In Eq.(7.17) r refers to a representation of Spine1;3 on R1;3 , given by
r(a)x = xa 1 . As in the case for the left real spin-Cli¤ord bundle, there
is a natural embedding PSpine1;3 (M ) ,! C`rSpine1;3 (M; g) which comes from
the embedding Spine1;3 ,! R01;3 . There exists also a natural left La action
of a 2 R1;3 on C`rSpine1;3 (M; g). This will be proved in Section 7.4.
De…nition 456 Let I ? (M; g) be a subbundle of C`rSpine1;3 (M; g) such that
there exists a primitive idempotent element e of R1;3 with
Le
=
(7.18)
for any
2 sec I ? (M; g) ,! sec C`rSpine1;3 (M; g)g. Then, I ? (M; g) is called
a subbundle of right ideal algebraic spinor …elds. Any 2 sec I ? (M; g) ,!
sec C`rSpine1;3 (M; g) is called a RIASF. I ? (M; g) can be thought of as a real
spinor bundle for M such that M? in Eq.(7.14) is a minimal right ideal of
R1;3 .
De…nition 457 Two subbundles I (M; g) and I (M; g) of RIASF are
said to be geometrically equivalent if the idempotents e;e0 2 R1;3 (appearing in the previous de…nition) are related by an element u 2 Spine1;3 , i.e.,
e0 = ueu 1 .
Proposition 458 In a spin manifold, we have
C`(M; g) = PSpine1;3 (M )
Ad
R1;3 :
Proof. Remember once again that the representation
Ad : Spine1;3 ! Aut(R1;3 )
Adu a = uau
1
u 2 Spine1;3
is such that Ad 1 = identity and so Ad descends to a representation Ad0 of
SOe1;3 which we considered above. It follows that when PSpine1;3 (M ) exists
C`(M; g) = PSpine1;3 (M ) Ad R1;3 .
7.3.2 Bundle of Modules over a Bundle of Algebras
Proposition 459 S(M; g) (or C`lSpine1;3 (M; g)) is a bundle of (left) modules over the bundle of algebras C`(M; g). In particular, the sections of the
spinor bundle (S(M; g) or C`lSpine1;3 (M; g)) are a module over the sections
of the Cli¤ ord bundle.
Proof. For the proof, see [206].
Corollary 460 Let ; 2 sec C`lSpine1;3 (M; g) and
ists 2 sec C`(M; g) such that
=
:
6= 0. Then there ex(7.19)
7.3 Spinor Bundles and Spinor Fields
265
Proof. It is an immediate consequence of Proposition 459.
So, the corollary allows us to identify a correspondence between some
sections of C`(M; g) and some sections of I(M; g) or C`lSpine1;3 (M; g) once
we …x a section on C`lSpine1;3 (M; g). This and other correspondences are
essential for our theory of Dirac-Hestenes spinor …elds. Once we clari…ed
what is the meaning of a bundle of modules S(M; g) over a bundle of
algebras C`(M; g); we can give the following
De…nition 461 Two real left spinor bundles are equivalent, if and only if,
they are equivalent as bundles of C`(M; g) modules.
Remark 462 Of course, geometrically equivalently real left spinor bundles
are equivalent.
Remark 463 In what follows we denote the complexi…ed left spin Cli¤ ord
bundle and the complexi…ed right spin Cli¤ ord bundle by
C`lSpine1;3 (M ) = PSpine1;3 (M )
l
C
R1;3
PSpine1;3 (M )
r
R4;1 ;
C`rSpine1;3 (M )
r
C
R1;3
PSpine1;3 (M )
r
R4;1 :
= PSpine1;3 (M )
(7.20)
7.3.3 Dirac-Hestenes Spinor Fields (DHSF)
Let ea , a = 0; 1; 2; 3 be the canonical basis of R1;3 ,! R1;3 which generates
the algebra R1;3 . They satisfy the basic relation ea eb + eb ea = 2 ab . We
recall that we showed in Chapter 3 that
e=
1
(1 + e0 ) 2 R1;3
2
(7.21)
is a primitive idempotent of R1;3 and
f=
1
1
(1 + e0 ) (1 + i e2 e1 ) 2 C
2
2
R1;3
(7.22)
is a primitive idempotent of C R1;3 . Now, let I = R1;3 e and IC = C R1;3 f
be respectively the minimal left ideals of R1;3 and C R1;3 generated by e
and f. Let = e 2 I and
= f 2 IC . Then, any 2 I can be written
as
= e
(7.23)
with
2 R01;3 . Analogously, any
=
with
2 R01;3 .
2 IC can be written as
1
e (1 + ie2 e1 );
2
(7.24)
266
7. Cli¤ord and Dirac-Hestenes Spinor Fields
Recall moreover that C R1;3 ' R4;1 ' C(4), where C(4) is the algebra
of the 4 4 complexes matrices. We can verify that
0
1
1 0 0 0
B 0 0 0 0 C
B
C
(7.25)
@ 0 0 0 0 A
0 0 0 0
is a primitive idempotent of C(4) which is a matrix representation of f.
In that way as proved in Chapter 3 there is a bijection between column
spinors, i.e., elements of C4 (the complex 4-dimensional vector space) and
the elements of IC . All that, plus the de…nitions of the left real and complex
spin bundles and the subbundle I(M; g) suggests the
2 sec I(M; g) ,! sec C`lSpine1;3 (M; g) be as in De…n-
De…nition 464 Let
ition 453, i.e.,
Re
=
e=
A DHSF associated with
that
; e2 = e =
1
(1+e0 ) 2 R1;3 :
2
is an even section
=
(7.26)
of C`lSpine1;3 (M; g) such
e:
(7.27)
Remark 465 An equivalent de…nition of a DHSF is the following. Let
2 sec C`lSpine1;3 (M; g) be such that
Rf
=
f=
; f2 = f =
Then, a DHSF associated to
that
1
1
2
(1 + e0 ) (1 + ie e1 ) 2 C R1;3 :
2
2
is an even section
=
f:
(7.28)
of C`lSpine1;3 (M; g) such
(7.29)
Remark 466 In what follows, when we refer to a DHSF
we omit for
simplicity the wording associated with
(or ). It is very important to
observe that a DHSF is not a sum of even multivector …elds although,
under a local trivialization,
which is a section of C`lSpine1;3 (M; g) is for
each x 2 M mapped on an even element3 of R1;3 . We emphasize that a
DHSF is a particular section of a spinor bundle, not of the Cli¤ ord bundle.
However, we show below a very important fact, namely that DHSFs have
representatives in the Cli¤ ord bundle.
3 Note
that it is meaningful to speak about even (or odd) elements in C`lSpine (M )
since Spine1;3
1;3
R01;3 .
7.4 Natural Actions on some Vector Bundles Associated with PSpine1;3 (M )
267
7.4 Natural Actions on some Vector Bundles
Associated with PSpine1;3 (M )
Recall that, when M is a spin manifold:
(i) The elements of C`(M; g) = PSpine1;3 (M ) Ad R1;3 are equivalence
classes [(p; a)] of pairs (p; a); where p 2 PSpine1;3 (M ), a 2 R1;3 and (p; a)
(p0 ; a0 ) , p0 = pu 1 ; a0 = uau 1 , for some u 2 Spine1;3 ;
(ii) The elements of C`lSpine1;3 (M; g) are equivalence classes of pairs (p; a),
where p 2 PSpine1;3 (M ), a 2 R1;3 and (p; a) (p0 ; a0 ) , p0 = pu 1 ; a0 = ua,
for some u 2 Spine1;3 ;
(iii) The elements of C`rSpine1;3 (M; g) are equivalence classes of pairs (p; a),
where p 2 PSpine1;3 (M ), a 2 R1;3 and (p; a)
(p0 ; a0 ) , p0 = pu 1 ; a0 =
au 1 , for some u 2 Spine1;3 .
In this way, it is possible to de…ne the following natural actions on these
associated bundles.
Proposition 467 There is a natural right action of R1;3 on C`lSpine1;3 (M; g)
and a natural left action of R1;3 on C`rSpine1;3 (M; g).
2 C`lSpine1;3 (M; g); select a representative
Proof. Given b 2 R1;3 and
(p; a) for and de…ne b := [(p; ab)] : If another representative (pu 1 ; ua)
is chosen for ; we have (pu 1 ; uab) (p; ab) and thus b is a well de…ned
element of C`lSpine1;3 (M; g).
Denote the space of R1;3 -valued smooth functions on M by F(M; R1;3 ).
Then, the above proposition immediately yields the following
Corollary 468 There is a natural right action of F(M; R1;3 ) on the sections of C`lSpine1;3 (M; g) and a natural left action of F(M; R1;3 ) on the sections of C`rSpine1;3 (M; g).
Proposition 469 There is a natural left action of sec C`(M; g) on the sections of C`lSpine1;3 (M; g) and a natural right action of sec C`(M; g) on sections of C`rSpine1;3 (M; g).
Proof. Given
2 sec C`(M; g) and
2 sec C`lSpine1;3 (M; g); select repre-
1
sentatives (p; a) for (x) and (p; b) for (x) (with p 2
(x)) and del
…ne ( )(x) := [(p; ab)] 2 C`Spine1;3 (M; g): If alternative representatives
(pu
(pu
1
1
; uau 1 ) and
; ub) are chosen for (x) and (x); we have
(pu
and thus (
1
; uau
1
ub) = (pu
1
; uab)
(p; ab)
)(x) is a well de…ned element of C`lSpine1;3 (M; g).
268
7. Cli¤ord and Dirac-Hestenes Spinor Fields
Proposition 470 There is a natural pairing
sec C`lSpine1;3 (M; g)
sec C`rSpine1;3 (M; g) ! sec C`(M; g):
2 sec C`lSpine1;3 (M; g) and
Proof. Given
2 sec C`rSpine1;3 (M; g); select rep-
1
resentatives (p; a) for (x) and (p; b) for (x) (with p 2
(x)) and de…ne
( )(x) := [(p; ab)] 2 C`(M; g). If alternative representatives (pu 1 ; ua)
and (pu 1 ; bu 1 ) are chosen for (x) and (x); we have (pu 1 ; uabu 1 )
(p; ab) and thus ( )(x) is a well de…ned element of C`(M; g).
Proposition 471 There is a natural pairing
sec C`rSpine1;3 (M; g)
sec C`lSpine1;3 (M; g) ! F(M; R1;3 ):
2 sec C`rSpine1;3 (M; g) and
Proof. Given
2 sec C`lSpine1;3 (M; g); select
1
representatives (p; a) for (x) and (p; b) for (x) (with p 2
(x)) and
de…ne ( )(x) := ab 2 R1;3 . If alternative representatives (pu 1 ; au 1 )
and (pu 1 ; ub) are chosen for (x) and (x); we have au 1 ub = ab and
thus ( )(x) is a well de…ned element of R1;3 .
7.4.1 Fiducial Sections Associated with a Spin Frame
We start by exploring the possibility of de…ning “unit sections” on the
various vector bundles associated with the principal bundle PSpine1;3 (M ).
Let
i
1
:
Spine1;3 ;
(Ui ) ! Ui
j
1
:
(Uj ) ! Uj
Spine1;3
be two local trivializations for PSpine1;3 (M ); with
i (u)
= ( (u) = x;
i;x (u));
j (u)
= ( (u) = x;
j;x (u)):
Recall that the transition function on gij : Ui \ Uj ! Spine1;3 is then given
by
1
gij (x) = i;x
;
j;x
which does not depend on u.
Proposition 472 C`(M; g) has a naturally de…ned global unit section.
Proof. For the associated bundle C`(M; g), the transition functions corresponding to local trivializations
i
:
1
c
(Ui ) ! Ui
R1;3 ,
j
:
1
c
(Uj ) ! Uj
R1;3 ;
(7.30)
are given by hij (x) = Adgij (x) . De…ne the local sections
1i (x) =
1
i
(x; 1);
1j (x) =
1
j
(x; 1);
(7.31)
7.4 Natural Actions on some Vector Bundles Associated with PSpine1;3 (M )
269
where 1 is the unit element of R1;3 . Since hij (x) 1 = Adgij (x) (1) =
gij (x)1gij (x) 1 = 1, we see that the expressions above uniquely de…ne
a global section 1 for C`(M; g) with 1jUi = 1i .
Remark 473 It is clear that such a result can be immediately generalized for the Cli¤ ord bundle C`p;q (M; g), of any n-dimensional manifold endowed with a metric of arbitrary signature (p; q) (where n = p + q). Now,
we observe also that the left (and also the right) spin-Cli¤ ord bundle can
be generalized in an obvious way for any spin manifold of arbitrary …nite
dimension n = p + q, with a metric of arbitrary signature (p; q). However, another important di¤ erence between C`(M; g) and C`lSpinep;q (M; g) or
C`rSpine1;3 (M; g)) is that these latter bundles only admit a global unit section
if they are trivial.
Proposition 474 There exists an unit section on C`rSpinep;q (M; g) (and also
on C`lSpinep;q (M; g)), if and only if, PSpinep;q (M ) is trivial.
Proof. We show the necessity for the case of C`rSpinep;q (M; g),4 the su¢ ciency is trivial. For C`rSpinep;q (M; g), the transition functions corresponding
to local trivializations
i
:
1
sc (Ui )
! Ui
Rp;q ,
j
:
1
sc (Uj )
! Uj
Rp;q ;
(7.32)
are given by kij (x) = Rgij (x) , with Ra : Rp;q ! Rp;q ; x 7! xa 1 . Let 1 be
the unit element of R1;3 . An unit section in C`rSpinep;q (M; g) – if it exists –
is written in terms of these two local trivializations as
1ri (x) =
1
i
(x; 1);
1rj (x) =
1
j
(x; 1);
(7.33)
and we must have 1ri (x) = 1rj (x) 8x 2 Ui \ Uj . As i (1ri (x)) = (x; 1) =
r
r
r
1
,
j (1j (x)), we have 1i (x) = 1j (x) , 1 = kij (x) 1 , 1 = 1gij (x)
gij (x) = 1. This proves the proposition.
Remark 475 For general spin manifolds, the bundle PSpinep;q (M ) is not
necessarily trivial for arbitrary (p; q), but Geroch’s theorem (Remark 441)
warrants that, for the special case (p; q) = (1; 3) with M non-compact,
PSpine1;3 (M ) is trivial. The above proposition implies that C`rSpine1;3 (M; g)
and also C`lSpinep;q (M; g) have global “unit sections”. It is most important
to note, however, that each di¤ erent choice of a (global ) trivialization i
on C`rSpine1;3 (M; g) (respectively C`lSpinep;q (M; g)) induces a di¤ erent global
unit section 1ri (respectively 1li ). Therefore, even in this case there is no
canonical unit section on C`rSpine1;3 (M; g) (respectively on C`lSpine1;3 (M; g)).
4 The
proof for the case of C`lSpine (M ) is analogous.
p;q
270
7. Cli¤ord and Dirac-Hestenes Spinor Fields
By Remark 443, when the (non-compact) spacetime M is a spin manifold,
the bundle PSpine1;3 (M ) admits global sections. With this in mind, let us …x
a spin frame
and its dual spin coframe for M . This induces a global
trivialization for PSpine1;3 (M ) and of course of PSpine1;3 (M ). We denote the
trivialization of PSpine1;3 (M ) by
: PSpine1;3 (M ) ! M
Spine1;3 ;
1
with
(x; 1) = (x). We recall that a spin coframe 2 sec PSpine1;3 (M )
(Remark 642) can also be used to induce certain …ducial global sections on
the various vector bundles associated to PSpine1;3 (M ):
(i) C`(M; g)
Let fea g be a …xed orthonormal basis of R1;3 R1;3 (which can be thought
of as the canonical basis of R1;3 ). We de…ne basis sections in C`(M; g) =
PSpine1;3 (M ) Ad R1;3 by a (x) = [( (x); ea )]. Of course, this induces a
multiform basis f I (x)g for each x 2 M . Note that a more precise notation
( )
for a would be, for instance, a .
(ii) C`lSpine1;3 (M; g)
Let 1l be a section of C`lSpine1;3 (M; g) de…ned by 1l (x) = [( (x); 1)].
Then the natural right action of R1;3 on C`lSpine1;3 (M; g) leads to 1l (x)a =
[( (x); a)] for all a 2 R1;3 . It follows from Corollary 468 that an arbitrary section
of C`lSpine1;3 (M; g) can be written as
= 1l f , with
f 2 F(M; R1;3 ).
(iii) C`rSpine1;3 (M; g)
Let 1r be a section of C`rSpine1;3 (M; g) de…ned by 1r (x) = [( (x); 1)].
Then the natural left action of R1;3 on C`rSpine1;3 (M; g) leads to a1r (x) =
[( (x); a)] for all a 2 R1;3 . It follows from Corollary 468 that an arbitrary section
of C`rSpine1;3 (M; g) can be written as
= f 1r , with
f 2 F(M; R1;3 ).
Now recall (De…nition 438) that a spin structure on M is a 2-1 bundle map s : PSpine1;3 (M ) ! PSOe1;3 (M ) such that s(pu) = s(p)Adu ; 8p 2
PSpine1;3 (M ); u 2 Spine1;3 , where Ad : Spine1;3 ! SOe1;3 ; Adu : x 7! uxu 1 .
We see that the speci…cation of the global section in the case (i) above is
compatible with the Lorentz coframe f a g = s( ) assigned by s. More precisely, for each x 2 M , the element s( (x)) 2 PSOe1;3 (M ) is to be regarded
as proper isometry s( (x)) : R1;3 ! Tx M , so that a (x) := s(p) ea yields a
Lorentz coframe f a g on M , which we denoted by s( ). On the other hand,
7.4 Natural Actions on some Vector Bundles Associated with PSpine1;3 (M )
271
C`(M; g) is isomorphic to PSpine1;3 (M ) Ad R1;3 , and we can always arrange
things so that a (x) is represented in this bundle as a (x) = [( (x); ea )] : In
fact, all we have to do is to verify that this identi…cation is covariant5 under
a change of coframes. To see that, let 0 be another section of PSpine1;3 (M ),
i.e., another spin coframe on M . From the principal bundle structure of
PSpine1;3 (M ), we know that, for each x 2 M , there exists (an unique)
u(x) 2 Spine1;3 such that 0 (x) = (x)u 1 (x). If we de…ne, as above,
0
0
0
1
(x)) ea = s( (x))Adu(x) ea =
a (x) = s( (x)) ea , then a (x) = s( (x)u
1
( (x); Adu(x) ea ) = ( (x)u (x); ea ) = [( 0 (x); ea )], which proves
our claim.
Proposition 476
(i)
ea = 1r (x) a (x)1l (x), 8x 2 M;
l r
(ii) 1 1 = 1; global unity section of C`(M; g);
(iii)
1r 1l = 1 2 R1;3 :
Proof. This follows from the form of the various actions de…ned in Propositions 467-471. For example, for each x 2 M; we have 1r (x) a (x) =
[( (x); 1ea )] = [( (x); ea )], a section of C`rSpine1;3 (M; g) (from proposition
469). Then, it follows from Proposition 471 that 1r (x) a (x)1l (x) = ea 1 =
ea , 8x 2 M .
Let us now consider how the various global sections de…ned above transform when the spin coframe
is changed. Let 0 2 sec PSpine1;3 (M ) be
0
another spin coframe with (x) = (x)u(x), where u(x) 2 Spine1;3 . Let a ,
1r , 1l and a0 , 1r 0 , 1l 0 be the global sections respectively de…ned by and
0
(as above). We then have
Proposition 477 Let and 0 be two spin coframes (sections of PSpine1;3 (M ))
related by 0 = u 1 , where u : M ! Spine1;3 . Then
(i)
0
a
=U
aU
1
(ii) 1l 0 = 1l u = U 1l ;
(iii) 1r 0 = u
1 r
1 = 1r U
1
;
(7.34)
where U 2 sec C`(M; g) is the Cli¤ ord …eld associated to u by U (x) =
[( (x); u(x))]. Also, in (ii) and (iii), u and u 1 respectively act on 1l 2
sec C`lSpine1;3 (M; g) and 1r 2 sec C`rSpine1;3 (M; g) according to Proposition
468.
5 Being covarint means here that the identi…cation does not depend on the choice of
the coframe used in the computations.
272
7. Cli¤ord and Dirac-Hestenes Spinor Fields
Proof. (i) We have
0
a (x)
= [( 0 (x); ea ] = [( (x)u(x); ea )]
= [( (x); u(x)ea u
1
(x))]
= [( (x); u(x))][( (x); ea )][( (x); u
1
= U (x) a (x)U
1
(x))]
(x):
(7.35)
(iii) It follows from Proposition 469 that
1r 0 (x) = [( 0 (x); 1)] = [( (x)u(x); 1)]
= ( (x); 1u(x)
1
) = ( (x); u
1
(x)) = u
1
(x)1r (x);
(7.36)
where in the last step we used Proposition 468 and the fact that 1r (x) =
[( (x); 1)].
To prove that 1r 0 = 1r U 1 we observe that
u
1
(x)1r (x) = ( (x); u(x)
1
1
= ( (x); 1u(x)
r
= 1 (x)U
1
)
) = [( (x); 1)] ( (x); u(x)
1
)
(x);
(7.37)
for all x 2 M: It is important to note that in the last step we have a product
between an element of C`rSpine1;3 (M; g) (i.e. [( (x); 1)]) and an element of
C`(M; g) (i.e. ( (x); u(x) 1 ) ).
We emphasize that the right unit sections associated with spin coframes
are not constant in any covariant way. To understand the meaning of this
statement we now investigate the theory of the covariant derivatives of
Cli¤ord and spinor …elds.
7.5 Representatives of DHSF in the Cli¤ord
Bundle
Let
l
, a section of C`0l
Spine1;3 (M; g) be a Dirac-Hestenes spinor …eld asso-
ciated to
l
= [( ;
l
l
a (ideal) section of I(M; g) (De…nition 464). Recalling that
)] and Corollary 460 we give the
De…nition 478 For a given spin frame , a representative of
Cli¤ ord bundle is the even section
of C`(M; g) such that
l
:=
1l
l
in the
(7.38)
Recalling Proposition 476 which says that 1l 1r = 1, the global unity
section of C`(M; g) we have that
=
l r
1 :
(7.39)
7.6 Covariant Derivatives of Cli¤ord Fields
273
Given another spin frame 0 = u the representative in C`(M; g) of
is is the even section
of C`(M; g) such that
0
To …nd the relation between
0
(x) =
l
l r
=
1 0:
and
0
(7.40)
we recall Eq.(7.34) and write
r
r
(x)1 0 (x) = [( (x);
(x))][( 0 (x); 1)]
= [( (x);
r
(x))][( (x)u(x); 1)]
= [( (x);
r
(x))][( (x); u
= [( (x);
r
(x))][( (x); 1)][( (x); u
= [( (x);
l
=
((x)U
r
1
(x))]1 U
1
l
1
(x))]
1
(x))]
(x)
(x):
(7.41)
r
C`rSpine1;3 (M; g)
Remark 479 A Right DHSF
, a section of
associated
to
a (ideal) section of I (M; g) also has representatives in the Cli¤ ord
bundle and we have in obvious notation
:= 1l
r
0
:= 1l 0
0
0
and two di¤ erent representatives
7.6
;
and
=U
r
(7.42)
are related by
:
(7.43)
Covariant Derivatives of Cli¤ord Fields
Since the Cli¤ord bundle of di¤erential forms is C`(M; g) = T M=Jg ; it
is clear that any linear connection r on T M which is metric compatible
(rg = 0) passes to the quotient T M=Jg , and thus de…ne an algebra bundle
connection [92]. In this way, the covariant derivative of a Cli¤ord …eld
A 2 sec C`(M; g) is completely determined.
We now prove a very important formula for the covariant derivative
of Cli¤ord …elds and of DHSF using the general theory of connections
in principal bundles and covariant derivatives in associate vector bundles,
which we recalled in Appendix A.
Proposition 480 The covariant derivative (in a given gauge) of a Cli¤ ord
…eld A 2 sec C`(M; g), in the direction of the vector …eld V 2 sec T M is
given by
1
(7.44)
rV A = gV (A) + [!V ; A];
2
V2
where !V is the usual ( T M -valued ) connection 1-form written in the
basis f a g and, if A = Aa a ; then gV is the (Pfa¤ ) derivative operator
such that gV (A) := V (AI ) I introduced in Chapter 5.6
6I
denotes collective indexes of a basis of C`(M; g) (see Eq.(7.7)).
274
7. Cli¤ord and Dirac-Hestenes Spinor Fields
Proof. Writing A(t) = A( (t)) in terms of the multiform basis f I g of
sections associated to a given spin coframe, as in Section 7.4.1, we have
A(t) = AI (t) I (t) = AI (t)[( (t); eI )] = [( (t); AI (t)eI )] = [( (t); a(t))];
with a(t) := AI (t)eI 2 R1;3 . If follows from item (ii) De…nition 679 in
Appendix A.5 that
A0jjt = [( (0); g(t)a(t)g(t)
for some g(t) 2
Spine1;3 ,
1
g(t)a(t)g(t)
t!0 t
)]
(7.45)
with g(0) = 1. Thus
1
lim
1
dg
ag
dt
a(0) =
1
+g
da
g
dt
1
= a(0)
_
+ g(0)a(0)
_
+ ga
dgt 1
dt
t=0
a(0)g(0)
_
=
= V (AI )EI + [g(0);
_
a(0)];
where g(0)
_
2 spine1;3 ('
with
V2
R1;3 ) is the Lie algebra of Spine1;3 . Therefore
!V = [(p; g(0))]
_
=
=
1 c ab
V !c
2
a
1 a
!
2 Vb
^
b
=
b
^
a
=
1 c
V !acb
2
we have
rV A = V (AI )
1 a
!
2 V
a
^
b
b
a
^
b
(7.46)
:
1
+ [!V ; A];
2
I
which proves the proposition.
Remark 481 In particular, calculating the covariant derivative of the basis 1-covector …elds a yields 12 [!ec ; a ] = ! bca b . Note that
!acb =
d
ad ! cb
=
!bca
(7.47)
and
! ac b =
ka
!kcl
lb
=
! bc a
(7.48)
V2
In this way, ! : V 7! !V is the usual ( T M -valued ) connection 1-form
on M written in a given gauge (i.e., relative to a spin coframe).
Remark 482 Eq.(7.44) shows that the covariant derivative preserves the
degree of an homogeneous Cli¤ ord …eld, as can be easily veri…ed.
The general formula Eq. (7.44) and the associative law in the Cli¤ord
algebra immediately yields the
Corollary 483 The covariant derivative rV on C`(M; g) acts as a derivation on the algebra of sections, i.e., for A; B 2 sec C`(M; g) and V 2
sec T M , it holds
rV (AB) = (rV A)B + A(rV B):
(7.49)
7.7 Covariant Derivative of Spinor Fields
275
Proof. It is a trivial calculation using Eq.(7.44).
Corollary 484 Under a change of gauge (local Lorentz transformation)
!V , transforms as
1
1
!V 7! U !V U
2
2
1
+ (rV U )U
1
;
(7.50)
Proof. Using Eq.(7.44) we can calculate rV A in two di¤erent gauges as
or
1
rV A = gV (A) + [!V ; A];
2
(7.51)
1
rV A = g0V (A) + [!V0 ; A];
2
(7.52)
where by de…nition gV (A) = V (AI ) I and g0V (A) = V (A0I )
observe that since 0I = U I U 1 , we can write
U gV (U
1
A) = U gV (U
=
Now, U gV (U
1
g0V
1
0I
. Now, we
A0I "0I ) = V (A0I )"0I + A0I "0I U dV (U
(A) + AU gV (U
A) = gV A + U (dV U
g0V (A) = gV (A)
1
1
1
)
):
)A and it follows that
1
[(gV U ) U
; A]:
Then, we see comparing the second members of Eq.(7.51) and Eq.(7.52)
that
1
1
!V = !V0 + U gV U
2
2
1 0
1
! = !V + (gV U ) U
2 V
2
1
1
;
:
(7.53)
Finally, we have
1
1
1
1 0
!V = !V + rV U
!V U + U !V U
2
2
2
2
1
= U !V U 1 + (rV U )U 1 ;
2
1
(7.54)
and the proposition is proved.
7.7 Covariant Derivative of Spinor Fields
The spinor bundles introduced in Section 5.5, like I(M; g) = PSpine1;3 (M ) `
I, I = R1;3 21 (1 + E0 ), and C`lSpine1;3 (M; g), C`rSpine1;3 (M; g) (and subbundles)
276
7. Cli¤ord and Dirac-Hestenes Spinor Fields
are vector bundles. Thus, as in the case of Cli¤ord …elds we can use the
general theory of covariant derivative operators on associate vector bundles
to obtain formulas for the covariant derivatives of sections of these bundles.
Given
2 sec C`lSpine1;3 (M; g) and
2 sec C`rSpine1;3 (M; g), we denote the
corresponding covariant derivatives by7 rsV
Proposition 485 Given
we have,
rsV
rsV
and rsV .
2 sec C`lSpine1;3 (M; g) and
2 sec C`rSpine1;3 (M; g)
1
= gV ( ) + !V ;
2
1
!V :
= gV ( )
2
(7.55)
(7.56)
Proof. It is analogous to that of Proposition 480, with the di¤erence that
Eq. (7.45) should be substituted by 0jjt = [( (0); g(t)a(t))] and 0jjt =
[( (0); a(t)g(t) 1 )].
Proposition 486 Let r be the connection on C`(M; g) to which rs is related. Then, for any V 2 sec T M , A 2 sec C`(M; g), 2 sec C`lSpine1;3 (M; g)
and 2 sec C`rSpine1;3 (M; g);
rsV (A ) = A(rsV
rsV
( A) =
) + (rV A) ;
(rV A)+(rsV
(7.57)
)A:
(7.58)
Proof. Recalling that C`lSpine1;3 (M; g) (C`rSpine1;3 (M; g)) is a module over
C`(M; g); the result follows from a simple computation.
Finally, let
2 sec C`lSpine1;3 (M; g) be such that e=
R1;3 is a primitive idempotent. Then, since e = ,
rsV
where e2 = e 2
1
=rsV ( e) = gV ( e) + !V e
2
1
s
= [gV ( ) + !V ]e = (rV )e;
2
(7.59)
from where we verify that the covariant derivative of a LIASF is indeed a
LIASF.
Finally we can prove a statement done at the end of the previous section:
that the right unit sections associated with spin coframes are not constant
in any covariant way. In fact, we have the
7 Recall
that I l (M; g) ,! C`lSpine (M; g) and I r (M; g) ,! C`rSpine (M; g).
1;3
1;3
7.7 Covariant Derivative of Spinor Fields
277
Proposition 487 Let 1r 2 sec C`rSpine1;3 (M; g) be the right unit section
associated to the spin coframe . Then
1 r
1 !ea .
2
rsea 1r =
(7.60)
Proof. It follows directly from Eq.(7.56).
Exercise 488 Calculate the covariant derivative of 2 sec C`lSpine1;3 (M; g)
in the direction of the vector …eld V 2 sec T M and con…rm the validity of
Eq.(7.50).
Solution:
Let u : M ! Spine1;3
R1;3 be such that in the spin coframe
2
(0)
1
e
sec PSpin1;3 (M ) we have for U 2 sec C` (M; g), U U
= 1, U (x) =
[( (x); u(x))]:We can write the covariant derivative of 2 sec C`lSpine1;3 (M; g)
in two di¤erent gauges ; 0 2 sec PSpine1;3 (M ) as
1
= gV ( ) + !V ;
2
1
= g0V ( ) + !V0 :
2
rsV
rsV
(7.61)
(7.62)
Now,
=
I
Is
0 0I
Is ;
=
where sI ; s0I are the following equivalent classes in C`lSpine1;3 (M; g) :
sI =
(x); S I
`
; s0I =
0
(x); S I
`
;
with S I a spinor basis in an minimal left ideal in R1;3 . Now, if 0 = u
we can write using the fact that C`lSpine1;3 (M; g) is a module over C`(M; g)
s0I =
0
(x); S I
`
=
= [( (x); u(x))]C`
(x)u(x); S I
(x); S
I
`
`
=
(x); uS I
`
:
Recalling that U = [( (x); u(x))]C` 2 sec C`(M; g) we can write
s0I = U sI :
(7.63)
Moreover, since
gV ( ) = V (
I
I )s ;
g0V ( ) = V (
0
0I
I )s ;
(7.64)
we get
g0V ( ) = gV ( ) + U (gV U
1
) :
(7.65)
278
7. Cli¤ord and Dirac-Hestenes Spinor Fields
Indeed, since
)]` = [( 0 ;
= [( ;
[( 0 ;
0
0
)]` = [( u;
0
)]` we can write
)]` = [( ; u
0
)]` ;
i.e.,
0
=u
1
:
(7.66)
Then,
gV ( ) = V (
I)
; SI
g0V
0
I)
0
( )=V(
= [( ; ugV (
;S
0
`
I
`
=
;V (
0
=
)] =
I )S
`
0
I
I )S
;V (
; ugV (u
I
= [( ; gV (
`
)]
0
= [( ; gV (
1
0
(7.67)
)]
(7.68)
from where Eq.(7.65) follows. Also,
rsV
rsV
1
= gV ( ) + !V
2
1
= g0V ( ) + !V0
2
1
) + wV
2
1
0
; gV ( 0 ) + wV0
2
=
; gV (
=
;
(7.69)
`
0
;
(7.70)
`
with
!V = [( ; wV )]C` ;
!V0
= [(
0
; wV0
(7.71)
)]C` :
(7.72)
Then we can write using Eq.(7.70),
rsV
=
=
=
1
) + wV0 u 1
2
`
1
; ugV (u 1 ) + uwV0 u 1
2
`
1
; gV
+ ugV (u 1 )
+ uwV0 u
2
u; gV (u
1
1
(7.73)
`
Comparing Eqs.(7.69) and (7.70) we get
1 0
1
wV = u
2
2
1
wV u
gV (u
1
)u:
(7.74)
7.8 The Many Faces of the Dirac Equation
279
We now must verify that Eqs.(7.74) and (7.50) are compatible. To do
this, we use Eq.(7.72) to write
1 0
1
1
!V = [( 0 ; wV0 )]C` = U !V U 1 + (rV U )U
2
2
2
1
1
= !V + (gV U )U
2
1
=
; wV + gV (u)u 1
2
C`
0
1 1
=
u ; wV + gV (u)u 1
2
C`
0 1
1
1
=
; u wV u + u gV (u)
:
2
C`
1
(7.75)
Comparing Eqs.(7.75) and (7.74), we see that Eqs.(7.74) and (7.50) are
indeed compatible.
7.8 The Many Faces of the Dirac Equation
As it is the case of Maxwell and Einstein equations also the Dirac equation
has many faces. In this section we exhibit two of them.
7.8.1 Dirac Equation for Covariant Dirac Fields
As well known [77], a covariant Dirac spinor …eld is a section
2
sec Sc (M; g) = PSpine1;3 (M ) l C4 . Let (U = M; ); ( ) = (x; j (x)i) be
a global trivialization corresponding to a spin coframe (Remark (442)),
such that for f a g 2 PSOe1;3 (M ) and fea g 2 PSOe1;3 (M ) it is
s( ) = f a g,
a b
+
a
b a
2 sec C`(M; g);
=2
ab
a
(eb ) =
, a; b = 0; 1; 2; 3:
a
b
(7.76)
The usual Dirac equation in a Lorentzian spacetimeVfor the spinor …eld in
1
interaction with an electromagnetic …eld A 2 sec
T M ,! sec C`(M; g)
is then
(7.77)
i a (rsea + iqAa )j (x)i mj (x)i = 0;
where
a
2 C(4), a = 0; 1; 2; 3 is a set of constant Dirac matrices satisfying
a b
and j (x)i 2 C4 for x 2 M .
+
b a
=2
ab
(7.78)
280
7.8.2
7. Cli¤ord and Dirac-Hestenes Spinor Fields
Dirac Equation in C`lSpine1;3 (M; g)
Due to the one-to-one correspondence between ideal sections of the vector
bundles C`lSpine1;3 (M; g), C`lSpine1;3 (M; g) and of Sc (M; g) as explained above,
we can translate the Dirac Eq. (7.77) for a covariant spinor …eld into an
equation for a spinor …eld, which is a section of C`lSpine1;3 (M; g), and …nally
write an equivalent equation for a DHSF 2 sec C`lSpine1;3 (M; g). In order
to do that we introduce the spin-Dirac operator.
7.8.3 Spin Dirac Operator
De…nition 489 The spin Dirac operator acting on sections of C`lSpine1;3 (M; g)
is the …rst order di¤ erential operator [206]
@s =
a
rsea .
(7.79)
where f a g is a basis as de…ned in Eq.(7.76).
Remark 490 It is crucial to keep in mind the distinction between the
Dirac operator @ introduce through De…nition 293 and the spin Dirac operator @ s just de…ned.
We now write Dirac equation in C`lSpine1;3 (M; g), denoted DEC`l . It is
@ s e21
where
m e0
qA = 0
(7.80)
2 sec C`lSpine1;3 (M; g) is a DHSF and the ea 2 R1;3 are such that
ea eb + eb ea = 2 ab . Multiplying Eq.(7.80) on the right by the idempotent
f = 21 (1+e0 ) 12 (1+ie1 e2 ) 2 C R1;3 we get after some simple algebraic manipulations the following equation for the (complex) ideal left spin-Cli¤ord
…eld f = 2 sec C`lSpine1;3 (M; g),
i@ s
m
qA
= 0:
(7.81)
Now we can easily show, using the methods of Chapter 3, that given
any global trivializations (U = M; ) and (U = M; ); of C`(M; g) and
C`lSpine1;3 (M; g); there exists matrix representations of the f a g that are
equal to the Dirac matrices a (appearing in Eq.(7.77)). In that way the
correspondence between Eqs.(7.77), (7.80) and (7.81) is proved.
Remark 491 We must emphasize at this point that we call Eq.(7.80)
the DEC`l . It looks similar to the Dirac-Hestenes equation (on Minkowski
spacetime) discussed, e.g., in [350], but it is indeed very di¤ erent regarding
its mathematical nature. It is an intrinsic equation satis…ed by a legitimate spinor …eld, namely a DHSF
2 sec C`lSpine1;3 (M; g). The question
7.9 The Dirac-Hestenes Equation (DHE)
281
naturally arises: May we write an equation with the same mathematical information of Eq.(7.80) but satis…ed by objects living on the Cli¤ ord bundle
of an arbitrary Lorentzian spacetime, admitting a spin structure? Below we
show that the answer to that question is yes. But before we prove that result
let us recall how to formulate the electromagnetic gauge invariance for the
DEC`l .
7.8.4 Electromagnetic Gauge Invariance of the DEC`l
Proposition 492 The DEC`l is invariant under electromagnetic gauge
transformations
7!
0
21
= eqe
;
(7.82)
A 7! A + @ ;
(7.83)
!ea 7! !ea
;
0
2
(7.84)
sec C`lSpine1;3 (M )
A 2 sec
^1
(7.85)
T M ,! sec C`(M; g)
(7.86)
with ; 0 distinct DHSF, @ is the Dirac operator and
is a gauge function.
: M ! R ,!R1;3
Proof. The proof is obtained by direct veri…cation.
Remark 493 It is important to note that although local rotations and electromagnetic gauge transformations look similar, they are indeed very di¤ erent mathematical transformations, without any obvious geometrical link between them, di¤ erently of what seems to be the case for the Dirac-Hestenes
equation which we introduce in the next section.
7.9 The Dirac-Hestenes Equation (DHE)
We obtained above a Dirac equation, which we called DEC`l , describing
the motion of spinor …elds represented by sections
of C`lSpine1;3 (M; g) in
interaction with an electromagnetic …eld A 2 sec C`(M; g);
@ s e21
qA
= m e0 ;
(7.87)
where @ s = a rsea , f a g is given by Eq.(7.76), rsea is the natural spinor
covariant derivative acting on sec C`lSpine1;3 (M; g) and fea g 2 R1;3 R1;3 is
such that ea eb +eb ea = 2 ab . As we already mentioned, although Eq.(7.87)
is written in a kind of Cli¤ord bundle (i.e., C`lSpine1;3 (M; g)), it does not suffer from the inconsistency of representing spinors as pure di¤erential forms
282
7. Cli¤ord and Dirac-Hestenes Spinor Fields
and, in fact, the object
behaves as it should under Lorentz transformations.
As a matter of fact, Eq.(7.87) can be thought of as a mere rewriting of
the usual Dirac equation, where the role of the constant gamma matrices is
undertaken by the constant elements fea g in R1;3 and by the set f a g. In
this way, Eq.(7.87) is not a kind of Dirac-Hestenes equation as discussed,
e.g., in [350]. It su¢ ces to say that (i) the state of the electron, represented
by , is not a Cli¤ ord …eld and (ii) the ea , a = 0; 1; 2; 3 are just constant
elements of R1;3 and not sections of 1-form …elds in C`(M; g). Nevertheless,
as we show in the following, Eq.(7.87) does lead to a multiform Dirac
equation once we carefully employ the theory of right and left actions on
the various Cli¤ord bundles introduced earlier. It is that multiform equation
to be derived below that we call the DHE.
7.9.1 Representative of the DHE in the Cli¤ord Bundle
Let fea g be, as before, a …xed orthonormal basis of R1;3 R1;3 : Remember
that these objects are fundamental to the Dirac equation (7.87) in terms
of sections
of C`lSpine1;3 (M; g):
@ s e21
qA
= m e0 :
Let 2 sec PSpine1;3 (M ) be a spin frame and 2 sec PSpine1;3 (M ) its dual
spin coframe on M and de…ne the sections 1l ; 1r and "a , respectively
on C`lSpine1;3 (M; g); C`rSpine1;3 (M; g) and C`(M; g), as above. Now we can
use Proposition 476 to write the above equation in terms of sections of
C`(M; g) :
r
(7.88)
(@ s )1 21 1l
qA = m 1r "0 1l :
Right-multiplying by 1r yields, using Proposition 476,
a
(rsea )1
r
21
qA 1r = m 1r
0
:
(7.89)
It follows from Proposition 470 and 486 that
r
(rsea )1 = rea ( 1r )
= re a ( 1 r ) +
1
2
rsea (1r )
1r !ea ;
(7.90)
where Proposition 487 was employed in the last step. Therefore
a
rea ( 1r ) +
1
2
1r !ea
21
qA( 1r ) = m( 1r )
0
:
(7.91)
To proceed we recall De…nition 478 which says that
:=
1r
(7.92)
7.9 The Dirac-Hestenes Equation (DHE)
is a representative in C`(M; g) of
then have
a
re a
+
1
2
283
associated to the spin coframe . We
!ea
21
qA
0
=m
:
(7.93)
7.9.2 A Comment about the Nature of Spinor Fields
A comment about the nature of spinors is in order. As we repeatedly said
in the previous sections, spinor …elds should not be ultimately regarded
as …elds of multiforms, for their behavior under Lorentz transformations is
not tensorial (they are able to distinguished between 2 and 4 rotations).
So, how can the identi…cation above be correct? The answer is that the
de…nition in Eq.(7.92) is intrinsically spin coframe dependent. Clearly, this
is the price one ought to pay if one wants to make sense of the procedure
of representing spinors by di¤erential forms.
Note also that the covariant derivative acting on
in Eq.(7.93) is the
tensorial covariant derivative rV on C`(M; g), as it should be. However, we
see from the expression above that rV acts on
together with the term
1
!
.
Therefore,
it
is
natural
to
de…ne
an
“e¤ective
covariant derivative”
V
2
(s)
rV acting on
by
r(s)
ea
:= rea
+
1
2
!ea :
(7.94)
Then, Proposition 480 yields
r(s)
ea
1
) + !ea
2
= dea (
,
(7.95)
which emulates the spinorial covariant derivative8 , as it should. We observe
moreover that if C 2 sec C`(M; g) and if
2 sec C`(M; g) is a representative of a Dirac-Hestenes spinor …eld then
r(s)
ea (C
) = (rea C)
+ Cr(s)
ea
:
(7.96)
With this notation, we …nally have the DHE for the Cli¤ord …eld representative
2 sec C`(M; g) relative to the spin coframe on a Lorentzian
spacetime9 of a DHSF
2 sec C`lSpine1;3 (M; g) :
@ (s)
21
qA
=m
0
;
(7.97)
where
@ (s) =
8 This
9 The
a
re(s)
a
is the derivative used in [350], there introduced in an ad hoc way.
DHE on a Riemann-Cartan spacetime is discussed in Chapter 10.
(7.98)
284
7. Cli¤ord and Dirac-Hestenes Spinor Fields
will be called the representative of the spin-Dirac operator in the Cli¤ord
bundle.
Let us …nally show that this formulation recovers the usual transformation properties characteristic of the Hestenes’s formalism as described, e.g.,
in [350]. For that matter, consider two spin coframes ; 0 2 sec PSpine1;3 (M );
with 0 (x) = (x)u(x), where u(x) 2 Spine1;3 . We already know that
0 =
U 1 . Therefore, the various spin coframe dependent Cli¤ord …elds
from Eq.(7.93) transform as
0
a
0
=U
=
1
aU
U
1
;
(7.99)
:
These are exactly the transformation rules one expects from …elds satisfying
the Dirac-Hestenes equation.
7.9.3 Passive Gauge Invariance of the DHE
Exercise 494 Show that if
0a
(s)
re0a
0
021
qA
0
=m
0
00
then if the connection !V transforms as in Eq.(7.50) then
a
r(s)
ea
21
qA
=m
0
:
This property will be referred as passive gauge invariance of the DHE. It
shows that the fact that even if writing of the Dirac-Hestenes is, of course,
frame dependent, this fact does not implies in the selection of any preferred
reference frame 10 .
The concept of active gauge invariance under local rotations of the DHE
will be studied in Chapter 10.
7.10 Amorphous Spinor Fields
Crumeyrolle [92] gives the name of amorphous spinor …elds to ideal sections of the Cli¤ord bundle C`(M; g). Thus an amorphous spinor …eld is a
section of C`(M; g) such that P = , where P = P2 is an idempotent section of C`(M; g). However, these …elds and also the so-called Dirac-Kähler
([154, 189]) …elds, which are also sections of C`(M; g), cannot be used in a
physical theory of fermion …elds since they do not have the correct transformation law under a Lorentz rotation of the local spin coframe. However,
1 0 The fundamental concept of reference frame in a Lorentzian spacetime has been
discussed in details in Chapter 6.
7.11 Bilinear Invariants
285
amorphous spinor …elds appears in many ‘Dirac like’ representations of
Maxwell equations and are often confounded with authentic spinor …elds
(i.e., sections of spinor bundles). Some of these ‘Dirac like’representations
of Maxwell equations will be discussed in Chapter 13.
7.11 Bilinear Invariants
7.11.1 Bilinear Invariants Associated to a DHSF
We are now in position to give a precise de…nition of the bilinear invariants
…elds of Dirac theory associated to a given DHSF. Recalling that
Vp
T M ,! C`(M; g), p = 0; 1; 2; 3; 4, and recalling Propositions 470 and
471, we have the
De…nition 495 Let
variants associated to
~ =
+
J=
e0 ~ 2 sec
=
5!
1
2 (1+e0 ),
^0
2 sec
^1
e12 ~ 2 sec
S=
where
2 sec C`lSpine1;3 (M; g) be a DHSF. The bilinear inV
are the following sections of T M ,! C`(M; g):
T M+
^4
T M; K =
^2
T M
;
e3 ~ 2 sec
T M;
~ 2 sec C`r e (M; g) and
Spin1;3
5
=
(7.100)
^1
T M
0123
2 sec
V4
T M.
Remark 496 Of course, since all bilinear invariants in Eq.(7.100) are
sections of C`(M; g) they have the right transformations properties under
arbitrary local Lorentz transformations, as required. As recalled in Chapter
3 these bilinear invariants and their Hodge duals satisfy a set of identities,
called the Fierz identities. They are crucial for the physical interpretation
of the Dirac equation (in …rst and second quantizations).
7.11.2 Bilinear Invariants Associated to a Representative of a
DHSF
We note that the bilinear invariants, when written in terms of
read (from proposition 476):
S=
~
=
~
J=
0
S=
1 2
~
+
5!
2 sec(
2 sec
^1
^0
T M+
T M; K =
^2
2 sec
T M;
3
^4
~
T M );
^1
2 sec
T M
:=
1r ,
286
7. Cli¤ord and Dirac-Hestenes Spinor Fields
where 5 = 0123 = 0 1 2 3 . These are all intrinsic quantities, as they
should be. For a discussion on the Fierz identities satis…ed by these bilinear
covariants, written with the representatives of Dirac-Hestenes spinor …elds,
see Section 3.7.3 and [219, 350].
7.11.3 Electromagnetic Gauge Invariance of the DHE
Proposition 497 The DHE is invariant under electromagnetic gauge transformations
7!
0
eq1
=
21
;
(7.101)
A 7! A + @ ,
!ea 7! !ea
0
+
(7.102)
(7.103)
V1
where
;
2 sec C` (M; g), A 2 sec
T M ,! sec C`(M; g) and where
V0
2 sec
T M ,! sec C`(M; g) is a gauge function.
Proof. It is a direct calculation.
But, what are the meaning of these transformations? Eq.(7.101) looks
similar to Eq.(7.99) de…ning the change of a representative of a DHSF once
we change spin coframe, but here we have an active transformation, since
we did not change the spin coframe. On the other hand Eq.(7.102) does
not correspond either to a passive (no transformation at all) or active local
Lorentz transformation for A. Nevertheless, writing = #=2 yields
e1
e
e
e
q
21
q
21
q
21
q
21
21
#=2
=
00
=
0
#=2 1 q
21
#=2
=
01
=
1
#=2 3 q
21
#=2
=
02
=
#=2 3 q
21
#=2
=
03
#=2 0 q
e
e
e
e
=
cos q# +
1
3
sin q +
:
2
sin q ;
2
cos q
2
;
(7.104)
0
We see that Eqs.(7.104) de…ne a spin coframe
to which corresponds, as
V1
00 01 02 03
we already know, a basis f ; ; ; g for
T M ,! C`(M; g). We
can then think of the electromagnetic gauge transformation as a rotation
0 , the repin the spin plane 21 by identifying 0 in Eq.(7.101) with
0
resentative of the DHSF in the spin coframe
and by supposing that
instead of transforming the spin connection !ea as in Eq.(7.50) it is taken
as …xed and instead of maintaining the electromagnetic potential A …xed
it is transformed as in Eq.(7.102).
We observe that, since in GRT and also in Riemann-Cartan theory (see
Chapter 9) the …eld !ea is associated with some aspects of that …eld, our
interpretation for the electromagnetic gauge transformation suggests a possible non trivial coupling between electromagnetism and gravitation, if the
Dirac-Hestenes equation is taken as a more fundamental representation of
fermionic matter than the usual Dirac equation. We will not have space
and time to explore that possibility in this book.
7.12 Commutator of Covariant Derivatives of Spinor Fields and Lichnerowicz Formula
7.12 Commutator of Covariant Derivatives of
Spinor Fields and Lichnerowicz Formula
In this section we complement the results of Section 4.5.10 and Section 4.6
and calculate the commutator of the covariant derivatives of spinor …elds
and the square of the spin-Dirac operator of a Riemann-Cartan connection
leading to the generalized Lichnerowicz formula. Let 2 sec C`(0) (M; g) be
a representative of a DHSF in a given spin frame
de…ning the orthonormal
V1
basis fea g for T M and let f a g, a 2 sec T M ,!2 sec C`(M; g) be the
corresponding dual basis. Let moreover f a g be the reciprocal basis of f a g.
We show that11
(s)
[r(s)
ea ; reb ] =
1
R(
2
a
^
(T cab
b)
! cab + ! cba )r(s)
ec
(s)
(7.105)
(s)
Let u = g(u; ); v = g(v; ) 2 sec T M . We calculate [ru ; rv ] . Taking into
(s)
account that, e.g., ru = ru + 21 !u , we have
(s)
r(s)
u rv
1
1
1
1
= ru rv + (rv )!u + (ru )!v + !v !u +
ru !v :
2
2
4
2
Then, using some of the results of Chapter 5, we have
(s)
[r(s)
u ; rv ] = [ru ; rv ] +
1
[R(u ^ v);
2
1
+
(ru !v
2
1
= [R(u ^ v);
2
1
+
(ru !v
2
1
= [R(u ^ v);
2
1
+
(ru !v
2
1
= [R(u ^ v);
2
1
= R(u ^ v)
2
=
1
(ru !v
2
rv !u
1
[!u ; !v ])
2
] + r[u;v]
1
[!u ; !v ])
2
1
(s)
] + r[u;v]
![u;v]
2
1
rv !u
[!u ; !v ])
2
rv !u
(s)
] + r[u;v]
1
[!u ; !v ] ![u;v] )
2
1
(s)
] + r[u;v] +
R(u ^ v)
2
rv !u
(s)
+ r[u;v] .
(7.106)
From Eq.(7.106) Eq.(7.105) follows trivially.
1 1 Compare Eq.(7.105) with Eq.(5.57). Also, compare Eq.(7.105) with Eq.(6.4.54) of
Rammond’s book [308].
287
288
7. Cli¤ord and Dirac-Hestenes Spinor Fields
7.12.1 The Generalized Lichnerowicz Formula
In this section we calculate the square of the spin-Dirac operator on a
Riemann-Cartan spacetime acting on a representative of the DHSF. We
have the
Proposition 498
@ (s)
2
ab
=
r(s)
ea
ac
1
! bac r(s)
+ R +J
eb
4
c
r(s)
ec
(s) b
Proof. Taking notice that since "b 2 sec C`(M; g), then rea
we have
@ (s)
and since rea
b
2
@ (s)
or
2
=
ab
! bac
=
or
h
(s)
2
2
b
,
(s)
b
re
reb
h a
i
(s)
(s) (s)
= a rea b reb + b rea reb
h
i
(s)
(s) (s)
= a y rea b reb + b rea reb
i
h
(s)
(s) (s)
+ a ^ re a b reb + b rea r eb
=
@ (s)
@ (s)
a
= rea
(7.107)
c
we get
h
= ay
! bac
h
! bac
+ a^
c
c
(s)
b
(s)
b
reb +
reb +
i
(s)
(s)
(s)
(s)
rea reb
rea reb
(s) (s)
(s)
! bac ac reb + ab rea reb
(s) (s)
b
a
c (s)
a
! ac ^ reb + ^ b rea reb
(s)
(s) (s)
= ! cab ab rec + ab rea reb
(s) (s)
c
a
b (s)
a
! ab ^ rec + ^ b rea reb
i ;
=
i
! cab re(s)
+
c
(s)
r(s)
e a re b
a
^
Now we can de…ne the operator
@ (s) @ (s) =
ab
h
(s)
r(s)
ea reb
called the generalized spin D’Alembertian and
h
@ (s) ^ @ (s) = a ^ b re(s)
re(s)
a
b
b
h
(s)
r(s)
ea reb
;
i
! cab r(s)
ec :
i
! cab r(s)
ec ;
i
! cab re(s)
;
c
(7.108)
(7.109)
which will be called twisted curvature operator. Then, we can write
@ (s)
2
= @ (s) @ (s) + @ (s) ^ @ (s) :
(7.110)
7.12 Commutator of Covariant Derivatives of Spinor Fields and Lichnerowicz Formula
On the other hand, we can write
@ (s) @ (s)
ab
=
h
(s) (s)
rea reb
(s)
(s)
= ab rea reb
(s) (s)
= hab rea reb
(s)
= ab rea
and
=
(s)
ab
ac
i
! cab rec
(s)
ac b
! iac reb
(7.111)
(s)
! bac reb
1
1 (s)
@ ^ @ (s) + @ (s) ^ @ (s)
2
2
i 1
h
b
a
(s)
+
r(s)
! cab re(s)
^
eb rea
c
2
@ (s) ^ @ (s) =
i
(s)
r(s)
! cba r(s)
ea reb
ec
i
h
1 a
(s)
(s)
=
^ b r(s)
(! cab ! cba ) r(s)
r(s)
ea reb
eb rea
ec
2
h
i
1 a
(s)
(s)
=
^ b r(s)
r(s)
(ccab + T cab )re(s)
:
(7.112)
ea reb
eb rea
c
2
1
2
a
^
b
h
(s)
! cab rec
Taking into account that T cab = ! cab ! cba ccab , we have from Eq.(7.111)
and Eq.(7.112) that
h
i
2
ac b
@ (s) = ab r(s)
! ac r(s)
ea
eb
h
i
1
(s)
(s)
r(s)
+ a ^ b r(s)
(ccab + T cab )r(s)
ea reb
eb rea
ec : (7.113)
2
On the other hand, from Eq.(7.105), we have
h
i
1
(s)
r(s)
;
r
= R ( a ^ b ) + ccab r(s)
ec
ea
eb
2
and then Eq.(7.113) becomes
h
i
2
1
ac b
@ (s)
= ab r(s)
! ac r(s)
+ (
ea
eb
4
1 a
b c
(s)
^ T ab rec
2
h
i
1
ac b
= ab r(s)
! ac r(s)
+ (
ea
eb
4
We need to compute (
(
a
^
b
)R (
a
^
b)
a
= h(
+ h(
b
^
a
a
^
^
)R (
b
b
^
a
)R (
a
)R (
a
b ).
^
^
a
^
b
)R (
a
^
b)
a
^
b
)R (
a
^
b)
c
From (Eq.(2.70)) we have
b )i0
b )i4
+ h(
a
b
^
)R (
a
Now, recalling Eq.(5.56), we can have
h(
a
^
b
)R (
a
^
b )i0
:= (
=
a
(
^
a
^
b
re(s)
:
c
)yR (
b
a
) R(
^
a
b)
^
b)
= R:
^
b )i2
289
290
7. Cli¤ord and Dirac-Hestenes Spinor Fields
Next, we recall the identity (Eq.(2.72))
a
h(
b
^
)R (
a
^
b )i2
a
=
b
^(
yR (
^
a
b ))
+
a
y(
b
^ R(
b
)R (
a
^
b ))
and also the identity Eq.(2.60)
a
y(
b
^ R(
a
^
a
b ))
b
^(
yR (
a
^
b ))
=(
a
a
^
b) :
Then, it follows that
h(
h
a
^
b
)R (
a
^
b )i2
It remains to calculate h
a
a
^
^
b
R(
a
^
b )i4
=
a
=(
a
^
b
b
b
)R (
R(
a
^ R(
a
^
a
^
^
b)
=
ab
R(
b )i4 .
We have
b)
1
Rabcd
2
=
a
a
^
^
1
(Rabcd abcd + Racdb acdb + Radbc
6
1
= (Rabcd + Racdb + Radbc ) abcd :
6
=
b
b)
=0
^
c
adbc
^
d
)
Now, using Eq.(4.198) we can write
Rabcd = Rabcd + Jab[cd]
where Rabcd are the components of the Riemann tensor of the Levi-Civita
connection of g and from Eq.(4.199)
J bacd = rc K bda
J ba[cd] = J bacd
K bck K kda + K kcd K bka ;
J badc ;
with K kcd given by Eq.(4.197), i.e.,
K kcd =
1
2
km
(
n
nc T md
+
n
nd T mc
n
nm T cd ):
Taking into account the well known …rst Bianchi identity Rabcd +Racdb +
Radbc = 0, we have
@ (s)
2
where
=
h
ab
r(s)
ea
ac
i
1
+ R +J
! bac r(s)
eb
4
1
(Jab[cd] + Jac[db] + Jad[bc] )
6
1
= (Jab[cd] + Jac[db] + Jad[bc] )
6
J=
and the proposition is proved.
c
r(s)
ec ; (7.114)
abcd
abcd
0123 g ;
(7.115)
7.12 Commutator of Covariant Derivatives of Spinor Fields and Lichnerowicz Formula
Remark 499 Eq.(7.114) may be called the generalized Lichnerowicz formula and (equivalent expressions) appears in the case of a totally skewsymmetric torsion in many di¤ erent contexts, like, e.g., in the geometry
of moduli spaces of a class of black holes, the geometry of NS-5 brane solutions of type II supergravity theories and BPS solitons in some string
theories ([93]) and many important topics of modern mathematics (see
[49, 141]). For a Levi-Civita connection we have that J = 0 and c = 0
and we obtain the famous Lichnerowicz formula [210].
Exercise 500 Describe the bundles of dotted and undotted algebraic spinor
…elds and their tensor product.12
Solution: We start by considering the bundle
C`(0) (M; g) = PSpine1;3 (M )
(0)
Ad
R1;3 ;
(7.116)
which may be called the bundle of Pauli …elds. Next we de…ne the spinor
bundles
S(M ) = PSpine1;3 (M )
l
I;
_
S(M
) = PSpine1;3 (M )
r
I_
(7.117)
0
(0)
(0)
(0)
where I = R1;3 e is the ideal in R1;3 de…ned in Eq.(3.129), I_ = R1;3 e. Also,
l stands for a left modular representation of Spine1;3 in R1;3 that mimics the
1
D( 2 ;0) representation of Sl(2; C) and r stands for a right modular represen1
tation of Spine1;3 in R1;3 that mimics the D(0; 2 ) representation of Sl(2; C).
We then have the obvious isomorphism
C`(0) (M; g) = PSpine1;3 (M )
= PSpine1;3 (M )
= S(M )
C
(0)
Ad
l r
_
S(M
):
R1;3
I
_
CI
(7.118)
1 2 For a detailed discussion of the theory of dotted and undotted algebraic spinor …elds
on arbitrary Lorentzian space times the reader may consult [282].
291
292
7. Cli¤ord and Dirac-Hestenes Spinor Fields
This is page 293
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8
A Cli¤ord Algebra Lagrangian
Formalism in Minkowski Spacetime
8.1 Some Preliminaries
Recall that in De…nition (276) Minkowski spacetime has been introduced as
the structure (M; ; D; ; ") where M ' R4 and R(D) = 0, T(D) = 0. As
we know from Chapter 6, Minkowski spacetime possess an in…nity of physically equivalent inertial reference frames. These are frames I 2 sec T M such
that DI = 0. Given an inertial frame I, a global coordinate chart with coordinate functions in the Einstein-Lorentz-Poincaré coordinate gauge fx g
@
for M is a (nacsjI) if I = @x
0 . We choose a global basis of the orthonormal
@
frame bundle
= fe g, with e0 = I and ei = @x
can be
i . The frame
used in order to de…ne an equivalence relation between tensors located at
di¤erent points e 2 M . This can be done as follows.
The dual basis of fe g will be denoted by f g, i.e.,
= dx 2 sec
V1
T M and
(e ) = . As previously introduced, the set f g, with
=
is called the reciprocal basis of the basis f g. Recall that
f e je g and f dx je g are the natural basis for the tangent vector space Te M
and the tangent covector space Te M . In what follows we write,
x (e) = x , x (e0 ) = x0 ;
(8.1)
294
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
and recall that
D
=
dx
@
= 0;
@x
@
@x
dx ;
where the matrix with entries
= (
(8.2)
@
@
;
);
@x @x
(8.3)
is the diagonal matrix diag(1; 1; 1; 1).
r
De…nition 501 Let T; T0 2 sec Tsr M . We say that Te 2 Tse
M is equivr
0
alent to Te0 2 Tse0 M (written Te = Te0 ) if and only if
T
1 ::: r
1 ::: s
(x (e)) = T0 11:::::: s r (x (e0 ))
(8.4)
As usual, the set of vectors equivalent to ve 2 Te M will be denoted by
[ve ]. The set of equivalent classes of tangent vectors over the tangent bundle
is a vector space over the reals, that we denote by
M = f[ve ] j for all ve 2 Te M g.
(8.5)
Note that f[ e je ]g is a natural basis for M. With the notations ~v =
[ve ] and ~e = [ e je ] we can write ~v = v ~e . The dual space of M will be
denoted by M = M . The dual basis of f~e g will be denoted by f g. A
scalar product in M , : M M ! R can be de…ned in a canonical way
using (the metric of the cotangent bundle) and the equivalence relation
for tensors at di¤erent spacetime points. Recall moreover that the Cli¤ord
algebra generated by the structure (M ; ) is C`(M ; ) ' R1;3 .
The pair (M; M ) has the structure of an a¢ ne space, and (M; M ; ) is
a metric a¢ ne space.
De…nition 502 Fix a point eo 2 M . Then for any e 2 M , we call x 2 M
such that
x := (e eo ) 2 M
the position covector of e relative to eo .
Given two di¤erent Einstein-Lorentz-Poincaré coordinate charts for M ,
with coordinate functions fx g and fy g, e 2 M is represented by the
coordinates
fx (e)g and fy (e)g;
(8.6)
which are related in general by Poincaré transformation (L;a) (L ;a ). Of
course, when we introduce coordinates using the Einstein-Lorentz-Poincaré
coordinate functions x , a given event, say eo has coordinates x (eo ) =
(0; 0; 0; 0) and since y (eo ) = L x (eo ) + a it follows that y (eo ) = a . So,
we can write
x=e
e0 = x (e)
=x
= y (e)
0
=y
0
,
(8.7)
8.1 Some Preliminaries
295
where1
0
=L
1
:
(8.8)
Remark 503 It is clear that once we …x a coordinate chart for M with coordinate functions in Einstein-Lorentz-Poincaré gauge fx g, with x (e0 ) =
0, there is a correspondence between the position covector x = (e e0 ) and
V1
the the object x (e) 2
T M ,! C`(M; ). However take notice that
x (e) is not a legitimate 1-form …eld because the x (e), = 0; 1; 2; 3 are
not the components of a vector …eld. However, if we restrict ourselves to
charts such that the coordinates are related by Poincaré transformations
then we can call x (e) the position 1-form of the event e 2 M relative to
event eo 2 M . When there is no chance of confusion we denote x (e) by
the same symbol x used to denote the position (co)vector (e e0 ) 2 M .
The introduction of the position covector and the position “1-form …eld”,
permit us to associate to each each Cli¤ord …eld C 2 sec C`(M; ) a multiform (valued) function C of the position covector x. Indeed, the representation of C in a Einstein-Lorentz-Poincaré chart with coordinate functions
fx g is
1
CJ (x ) J ;
(8.9)
C=
(J)
where CJ : R4 ! R are (smooth) functions. The multiform (valued) function associated to C is
C : M ! C`(M ; ); x 7! C(x);
(8.10)
such that in the Einstein-Lorentz-Poincaré chart with coordinate functions
fx g we have
1
C=
CJ (x) J ;
(8.11)
(J)
where the CJ : M ! R are (smooth) functions such that (recalling that
x = x (e)
=x
) we have
CJ (x ) = CJ (x):
(8.12)
Remark 504 Since in this chapter we shall use only coordinate charts
in Einstein-Lorentz-Poincaré gauge, taking into account the identi…cation
provided by Eq.(8.12), (e e0 ) $ x, C $ C every time we introduce a
Cli¤ ord …eld C 2 sec C`(M; ) we immediately identify it with the corresponding multiform (valued ) function C : M ! C`(M ; ) and when no
confusion arises we use the same symbol for both. The importance of the
identi…cation (e e0 ) $ x, C $ C is that it permits the use all identities
of the multiform functions of multiform variables, developed in Chapter 2
in an obvious way.
1 L 1 refers, of course, to the elements of the Lorentz transformation represented
by the matrix L 1 .
296
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
We recall now that a (p; q)-extensor (De…nition 45) is a linear mapping
t:
^p
M!
^q
(8.13)
M
Vp
Vq
and that the set of all (p; q)-extensors is denoted ext( M ;
M ).
De…nition 505 A smooth (p; q)-extensor …eld t on Minkowski spacetime
is a di¤ erentiable (p; q)-extensor valued function on M ,
^p
^q
M 3 e 7! te 2 ext(
Te M;
Te M ):
(8.14)
Remark 506 As it was the case for sections of C`(M; ), it is sometimes
convenient to associate (in an obvious way) to a smooth (p; q)-extensor
…eld t a (p; q)-extensor valued function t,
M 3 x 7! tx 2 ext(
^p
M;
^q
M ):
(8.15)
This again permit us to use the identities presented in the extensor calculus
developed in Chapter 2 in the computations that follows. Once again we
remark that since we are to use only charts in the Einstein-Lorentz-Poincaré
gauge we shall use (when no confusion arises) the same symbol for both t
and t.
8.2 Lagrangians and Lagrangian Densities for
Multiform Fields
Let Y be a Cli¤ord …eld, i.e., Y = X 2 sec C`(M; ), or a representative of
a Dirac-Hestenes spinor …eld, i.e., Y = 2 sec C`(0) (M; ).2 Since we restrict ourselves to Einstein-Lorentz-Poincaré coordinate charts and choose
a gauge where the connection coe¢ cients are null, the Dirac operator and
the representative of the spin-Dirac operator acting on sections of C`(M; )
has a very simple representation indeed, and in order to leave no chance
for confusion with the general case, we write, @ 7! @ x , @ (s) 7! @ (s)
x . Then,
with the above assumption
@ x = @ (s)
x =
@
:
@x
(8.16)
The operator @ x acts on the multiform (valued) function Y : M !
C`(M ; ) associated to Y 2 sec C`(M; ) as the vector derivative relative
2 To shorten the notation, when Y is a homogeneous section of the Cli¤ord bundle,
V
V
we write Y 2 sec p T M , instead Y 2 sec p T M ,! sec C`(M; ):
8.2 Lagrangians and Lagrangian Densities for Multiform Fields
297
to the position covector x also denoted @ x introduced in Section 2.11.2.
We write
(e e0 ) $ x;
@ x Y $ @ x Y(x);
and if
Y =
X
1
(J)
J
X
1
(J)
J
J
we have that
@xY =
J
YJ (x )
(8.17)
@
YJ (x ):
@x
(8.18)
Remark 507 In what follows we shall need to consider also the derivatives @ x ~ Y , where ~ is any one of the product of multiforms (Cli¤ ord,
y; x or ^) introduced in Section 2.12 and also derivatives of functionals of
multiform (valued ) functions.
De…nition 508 The Cli¤ ord jet bundle3 J~ (C`(M; )) of M is the (trivial ) vector bundle given by
S
J~ (C`(M; )) = x2M Jx1 (C`(M; ));
(8.19)
such that for X; Y 2 sec C`(M; ) such that X ~1x Y we have
@ x ~ Y (x) = @ x ~ X(x):
(8.20)
De…nition 509 We call a functional of the Cli¤ ord …eld Y 2 sec C`(M; )
any mapping F : J~ (C`(M; )) ! C`(M; ):
Remark 510 Given any …eld Y 2 sec C`(M; ) we abbreviate by F~ (Y )
(or F(x; Y; @ x ~ Y ) or yet4 F(Y (x); @ x ~ Y (x)) ) the mapping
F J~ (Y ):
(8.21)
Remark 511 To any functional of the Cli¤ ord …eld Y corresponds in an
obvious way a functional of the corresponding multiform (valued ) function
Y with values in C`(M ; ) and both objects are identi…ed (when no confusion arises) in order to to be possible to use in a standard way the theory
of multiform functions of multiform variables described in Chapter 2.
Remark 512 Clearly, given two functionals F, F 0 : J~ (C`(M; )) !
C`(M; ) any one of the products F ~ F 0 is well de…ned. We write
(F ~ F 0 )(Y ) = F (Y ) ~ F 0 (Y ):
3 For
(8.22)
the de…nition of jet bundles and notation employed see Appendix A.1.3.
this case we are using a common sloppy notation where the section Y of the
Cli¤ord bundle is written as Y (x). Note that Y(x) is also a sloppy notation for the
mutiform (valued) function Y:
4 In
298
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
V4
De…nition 513 In the case L : J~ (C`(M; )) !
T M , L is said a LaV0
grangian density for the …eld Y . In the case, L : J~ (C`(M; )) !
T M,
L is said a Lagrangian for the …eld Y .
For any Lagrangian density L(Y; @ x ~Y ) there corresponds a well de…ned
Lagrangian L(Y; @ x ~ Y ) and they are related by
L(Y; @ x ~ Y ) = L(Y; @ x ~ Y ) ;
where
(8.23)
is the volume element in M .
De…nition 514 To any Lagrangian density L~ (Y ) or Lagrangian L~ (Y )
the action for the multiform …eld Y 2 T M on U M is the real number
Z
Z
A(Y ) =
L~ (Y ) =
L~ (Y ) :
(8.24)
U
U
8.2.1 Variations
Vertical Variation
In this book, we restrict our investigation only to theories where the …elds
Y are Cli¤ord …elds (Y = X) or Dirac-Hestenes spinor …elds (Y = ).
As we learned in Chapter 7 these …elds carry di¤erent representations
of Spine1;3 ,which is the universal covering group of the homogeneous restrict and orthochronous Lorentz group L"0 . Let be u 2 Spine1;3 (M ) ,!
sec C`(M; ), i.e., for any x 2 M , u(x) 2 Spine1;3 ,! R1;3 . Then, if Y = X 2
sec C`(M; ) is a Cli¤ord …eld, an active local Lorentz transformation sends
X 7! X 0 2 sec C`(M; ), with
X 0 = uX u
~:
(8.25)
If Y = 2 sec C`(0) (M; ) is a representative of a Dirac-Hestenes spinor
0
…eld, then an active local transformation sends !
7
, with
0
=u :
(8.26)
From Section 3.3.4 we know that each u 2 Spine1;3 (M ) can be written as
V2
the exponential of a biform …eld F 2 sec
T M ,! sec C`(M; ). For
in…nitesimal transformations we must choose the + sign and write F = f ,
1, F 2 6= 0.
De…nition 515 Let Y = X be a Cli¤ ord …eld or Y = a representative
of a Dirac-Hestenes spinor …eld. The vertical variation of Y is the …eld
v Y (of the same nature of Y ) such that
vY
=Y0
Y:
(8.27)
Remark 516 The case where F is independent of x 2 M is said to be a
gauge transformation of the …rst kind, and the general case is said to be a
gauge transformation of the second kind.
8.2 Lagrangians and Lagrangian Densities for Multiform Fields
299
Horizontal Variation
Let t be a one-parameter group of di¤eomorphisms of M and let 2
sec T M be the vector …eld that generates t , i.e., in local coordinates in
Einstein-Lorentz-Poincaré gauge, we have
(x ) =
d
(x )
dt
t
:
(8.28)
t=0
De…nition 517 We call the horizontal variation of induced by a oneparameter group of di¤ eomorphisms of M the quantity
Y
tY
= $ Y:
(8.29)
h Y = lim
t!0
t
De…nition 518 We call total variation of a multiform …eld the quantity
Y =
vY
+
hY
=
vY
$ Y:
(8.30)
Remark 519 It is crucial to distinguish between the variations de…ned
above, something that sometimes is not done appropriately in textbooks. We
denote in what follows by a generic variation (horizontal or vertical). In
particular such a distinction is essential for the developments in Chapter
9.
Now, when we have a …eld theory formulated on Minkowski spacetime,
di¤eomorphisms associated with global Lorentz transformations have a very
important physical meaning, as we learned in Chapter 6. How to describe
such a di¤eomorphism using the Cli¤ord bundle formalism? Recall that a
di¤eomorphism related to a global Lorentz transformation is given by
` : e 7! e0 ; x = e
eo = x (e)
7! x0 = lx = x (e0 )
;
(8.31)
with
x (e0 ) = L x (e);
with L 2
S0e1;3
(8.32)
independent of e 2 M . We can then write
x0 = L x (e)
= x (e)u
1
u = x (e)
0
=u
1
xu
(8.33)
Spine1;3 (M )
for some constant section u 2
,! sec C`(M; ), i.e., independent of e 2 M such that the 2 1 homomorphism h : Spine1;3 ! S0e1;3
(Chapter 3) gives h(u 1 ) = l. Recalling now our discussion of Section 4.5.1
concerning the pullback mapping, we have: putting ` X = X 0 , `
= 0
that
X 0 (x) = X (x0 ) = uX(x)u
0
(x) =
0
(x ) = u (x):
1
;
(8.34)
Remark 520 Eq.(8.34) shows that the pullback mapping corresponding to
a global Lorentz transformation in the present formalism has the same form
as a local constant rotation, but care must be taken in using such formulas
in order to avoid misconceptions.
300
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
8.3 Stationary Action Principle and
Euler-Lagrange Equations
Let U be an open set of M , with boundary @U and consider an arbitrary
multiform …eld A 2 sec C`(M; ) such that Aj @U = 0. Given a Lagrangian
L~ (Y ) L(Y; @ x ~ Y ) we introduce the mapping5 LA : M I ! R
LA (x; ) := L(Y (x) + hA(x)iY ; @ x ~ Y (x) + h@ x ~ A(x)i@ x ~Y ); (8.35)
where 2 I and I is a subset of R containing zero. Note that LA (x; 0) =
L(Y (x); @ x ~ Y (x)). In view of this fact we call LA (x; ) the varied Lagrangian.
De…nition 521 Given a Lagrangian L~ (Y ) we de…ne its variation by
L~ (Y ) :=
d
:
L(Y (x) + hA(x)iY ; @ x ~ Y (x) + h@ x ~ A(x)i@ x ~Y )
d
=0
(8.36)
De…nition 522 The variation of the action is
Z
Z
A(Y ) =
L~ (Y ) =
L~ (Y ):
U
(8.37)
U
Axiom 523 In Lagrangian …eld theory the dynamics of a …eld Y is supposed to be derived from the Stationary Action Principle, hereafter denoted
SAP, i.e., we suppose that the laws of motion are to be deduced from
A(Y ) = 0 for all A such that Aj@U = 0 .
(8.38)
The SAP implies the so called Euler-Lagrange equations (ELE) for the
…eld Y .
Proposition 524 Given a …eld Y on U M and postulated a Lagrangian
L(Y; @ x ~ Y ) the SAP implies for the cases: (a) @ x ~ Y = @ x yY ; (b)
@ x ~ Y = @ x ^ Y ; (c) @ x ~ Y = @ x Y the following ELEs.
(a)
(b)
(c)
@Y L(Y; @ x yY ) @ x ^ @@ x yY L(Y; @ x yY ) = 0;
@Y L(Y; @ x ^ Y ) @ x y@@ x ^Y L(Y; @ x ^ Y ) = 0;
@Y L(Y; @ x Y ) @ x (@@ x Y L(Y; @ x Y )) = 0:
(8.39)
Proof. Here, we prove only case (c), leaving the proof of the other cases
as exercises for the reader. The Y -variation of L(Y; @ x Y ) gives immediately, using the de…nition of derivatives of multiform functions of multiform
variables (see Eq.(2.165)),
L(Y; @ x Y ) = A @Y L(Y; @ x Y ) + @ x A @@ x Y L(Y; @ x Y )
(8.40)
5 Recall that in Eq.(8.35) hAi
X means the projection of A in the grades of X. See
De…nition 4, Chapter 2.
8.4 Some Important Lagrangians
301
Now, recall the identity (c) in Eq.(2.219) which says that for any multiform
…elds Y and 1-form a = a
; with a constant functions, we have
(@ x Y ) X + Y (@ x X) = @ x [@a (aY ) X]
where @a =
@
@a
(8.41)
. Using this result we can write Eq.(8.40) as
L(Y; @ x Y ) = A [@Y L(Y; @ x Y )
@ x @Y L(Y (x); @ x Y )]
+ @ x [@a (aA) @@Y L(Y; @ x Y )]:
(8.42)
Then the SAP gives
Z
fA [@Y L @ x @Y L] + @ x [@a (aA) @@Y L]g
= 0;
(8.43)
U
for all A such that Aj@U = 0. Now, given any form …eld v we write Stokes
theorem for d(?v) in the form
Z
Z
(@ x v) =
vy :
(8.44)
U
@U
Then, the second term in Eq.(8.43) taking into account the boundary
condition can be written (in obvious notation)
Z
@ x [@a (aA) @@Y L(Y; @ x Y )]
U
Z
=
[@a (aA) @@Y L(Y; @ x Y )]( y )
Z@U
=
A ( @@Y L(Y; @ x Y )d3 S = 0:
(8.45)
@U
Using Eq.(8.45) in Eq.(8.43) we get
Z
A [@Y L(Y; @ x Y ) @ x @Y L(Y; @ x Y )]d4 x = 0,
(8.46)
U
for all A. By the arbitrariness of A we get
@Y L(Y; @ x Y )
@ x (@@Y L(Y; @ x Y )) = 0;
and the proposition is proved.
8.4 Some Important Lagrangians
8.4.1 Maxwell Lagrangian
V1
The Lagrangian associated with the Maxwell …eld A 2 sec
T M (i.e.,
V1
the electromagnetic potential ) generated by a current Je 2 sec
T M is
L(A; @ x ^ A) =
1
(@ x ^ A) (@ x ^ A)
2
A Je :
(8.47)
302
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
Using Eq.(8.39 b) we get immediately
@ x y(@ x ^ A) = Je :
(8.48)
V2
The electromagnetic …eld is (see Section 3.6) F = @ x ^ A 2 sec
T M.
Then taking into account that @ x ^ (@ x ^ A) = 0, we can write Eq.(8.48)
as
@ x F = Je ;
(8.49)
which is, of course, equivalent Eq.(4.264) of Chapter 4.
8.4.2 Dirac-Hestenes Lagrangian
We recall that once a global spin coframe 2 PSpine1;3 (M ) is …xed, a DiracHestenes spinor …eld can be represented by a even section of sec C`(M; ):
V0
V2
V4
Let then 2 sec( T M +
T M+
T M ) be a representative of a
Dirac-Hestenes spinor …eld 2 sec C`lSpine1;3 (M; ) relative to a global spin
coframe . A possible Lagrangian
for such a …eld in interaction with an
V1
electromagnetic …eld A 2 sec
T M is
L( ; @ x ) = (@ x i 3 )
q(A
0)
m
;
(8.50)
where the real parameters m 2 R+ and q 2 R are called the mass and
electric charge of the Dirac-Hestenes …eld and i := 5 . Lagrangian (8.50) is
of type (c) in Eq.(8.39). Then, the respective ELE is
@ L( ; @ x )
@ x (@@ L(x; Y; @ x )) = 0:
(8.51)
We have
@ L( ; @ x ) = h@ x i 3 i
= @x i
3
qhA
2Aq
0
0
+ A~ ~0 i
2m
2m :
(8.52)
Also,
@@ x L(Y; @ x ) =
Then, we have with 3 = 3 0
@x i
or, equivalently
3
@@ x (@ x
m
0
i 3) =
i 3:
(8.53)
qA = 0;
(8.54)
qA = 0;
(8.55)
3
@x
2 1
m
0
which we recognize as equivalent to the Dirac-Hestenes equation (Eq.(7.93))
introduced in Chapter 7, since the term containing the connection biform
in Eq.(7.93) is null in our case, because we are using only Einstein-LorentzPoincaré coordinate charts and using only exact global cotetrad …elds (as
introduced above), for which the connection coe¢ cients are all null.
8.5 Canonical Energy-Momentum Extensor Field
Exercise 525 Show that (A
0)
=A
0
303
~ (see Eq.(2.69)).
Exercise 526 Show that for the above Lagrangian
@ x (@@ x L(Y; @ x )) = @ (@@
L(Y; @ x )):
(8.56)
Solution:
We have from Eq.(8.53) that
@ x (@@ x L(Y; @ x )) =
@
i 3:
On the other hand we can write the term (@ x i 3 )
as
( @ i 3) :
in the Lagrangian
(8.57)
Then, using Eq.(2.68) and Eq.(2.69) we can write
(
@
i 3)
= (@
i 3)
=
@
(
i 3) =
(
i 3) @ :
(8.58)
Using now Eq.(2.192) we get
@@
( (
i 3) @
)=
i 3;
@@
( (
i 3) @
) =
@
and then
@
8.5
i 3:
Canonical Energy-Momentum Extensor Field
We now …nd the canonical energy-momentum extensor for each one of the
types of Lagrangians in Proposition 524.
We are interested here in di¤eomorphisms that are simply translations in
the Minkowski manifold. These are generated by very simple one-parameter
group of di¤eomorphisms. Indeed, under a translation in the constant ‘direction’ the position 1-form x (of a spacetime point e, with coordinates
x (e) = x ) goes in the position 1-form x0 (of a spacetime point e0 = t e,
with coordinates x0 (e) = x0 ) and we write
x0 = x + "
:
(8.59)
Vk
Given the rule for the pullbacks Y 0 = t Y for any Y 2 sec
T M ,!
sec C`(M; ), the equivalence relation (De…nition 501), and the convention
given by Eq.(8.1), we can write t Y (x) = Y (x0 ). Then, we have
Y 0 (x) = Y (x0 ):
(8.60)
304
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
Now, a Lagrangian L~ (Y ) is invariant under the action of a one-parameter
group of di¤eomorphisms generated by a vector …eld if the Lie derivative
$ L~ (Y ) = 0. This means that we must have
L(Y 0 (x); @ x ~ Y 0 (x)) = L(Y (x0 ); @ x0 ~ Y (x0 )):
(8.61)
To proceed we detail only the case where ~ refers to the Cli¤ord product. We derive both members of Eq.(8.61) in relation to the parameter "
obtaining,
@" Y 0 (x) @Y 0 L(Y 0 (x); @ x Y 0 (x)) + @ x @" Y 0 (x) @@ x Y 0 L(Y 0 (x); @ x Y 0 (x))
= @" x0 @ x0 L(Y (x0 ); @ x0 Y (x0 )):
(8.62)
In writing Eq.(8.62) we used the fact that both members can be considered (composite) functions of ", some chain rules for composition of multiform functions (Section 2.11) and that @ x @" = @" @ x . Now, we calculate
both members of Eq.(8.62) for " = 0. Noting that
Y (x0 )j"=0 = Y (x), @ x0 Y (x0 )j"=0 = @ x Y (x);
(8.63)
@ " x0 =
(8.64)
and at " = 0,
, @" Y 0 (x) = @ Y (x) =
@ x Y (x);
we can write Eq.(8.62) as
@ Y @Y L(Y (x); @ x Y (x)) + @ x @ Y @@ x Y L(Y (x); @x Y (x))
=
@ x L(Y (x); @ x Y (x)):
(8.65)
Now, the …rst member of Eq.(8.65) can be written as
@ Y @Y L(Y (x); @ x Y (x)) + @ x @ Y @@ x Y L(Y (x); @ x Y (x))
= @ Y @Y L(Y (x); @ x Y (x)) + @ x @ Y @@ x Y L(Y (x); @Y (x))
+ @ Y @ x @@ x Y L(Y (x); @ x Y (x)) @ Y @@ x Y @@ x Y L(Y (x); @x Y (x))
= @ Y [@Y 0 L(Y (x); @ x Y (x)) @ x @@ x Y L(Y (x); @ x Y (x)]
^
+@ x h@ Y (@@ x Y L(Y
(x); @Y (x))i1 :
(8.66)
~ =@ x A B + A @ x B,
In writing Eq.(8.66) we use the identity @ x hABi
valid for arbitrary A; B 2 sec C`(M; ). Moreover, to simplify the notation
we write in what follows
^
~
(@ @ x Y L(Y
(x); @ x Y (x)) = @
(8.67)
@ x Y L(Y (x); @ x Y (x)):
Observe also that the right side member of Eq.(8.65) can be written as
@ x L(Y (x); @ x Y (x))= @ L(Y (x); @ x Y (x))
@ L(Y (x); @ x Y (x))
=
[@
L(Y (x); @ x Y (x))]
= @x [
L(Y (x); @ x Y (x))]:
=
(8.68)
8.5 Canonical Energy-Momentum Extensor Field
305
Now, if we suppose that the …eld satisfy the Euler-Lagrange equations,
using Eqs.(8.66) (8.67), and (8.68), we can write Eq.(8.65) as
@ x [h(
@ x Y )@~@ x Y L(Y (x); @ x Y (x))i1
L(Y (x); @ x Y (x))] = 0: (8.69)
So, we conclude that associated to the Lagrangian L(Y; @ x Y ) there exists
V1
a (1; 1)-extensor …eld such that for any n 2 sec
T M ,! sec C`(M; ),
with constant coe¢ cients we have a (1; 1)- extensor …eld
^1
^1
T : sec C`(M; ) - sec
T M ! sec
T M ,! sec C`(M; )
T (n) = [h(n @ x Y )@~@ x Y L(Y; @ x Y )i1
nL(Y; @ x Y )];
(8.70)
such that
@x T (
Putting T = T (
) = 0:
(8.71)
), we have
@x T =
T =0
(8.72)
or
d ? T = 0:
(8.73)
De…nition 527 The extensor …eld
^1
^1
T : sec C`(M; ) - sec
T M ! sec
T M ,! sec C`(M; )
(8.74)
is called the (canonical )energy-momentum extensor …eld of the …eld Y 2
sec C`(M; ):
Taking into account the de…nition of T y (the adjoint of T ) satis…es
[@n @ x T y (n)]
= @x T (
);
(8.75)
we can write equivalently.
@n @T y (n) = 0:
(8.76)
V1
Remark 528 The T 2 sec
T M ,! sec C`(M; ) are called (canonV3
ical ) energy-momentum 1-forms and ?T 2 sec
T M ,! sec C`(M; )
the (canonical ) energy-momentum 3-forms. Note that some authors (e.g.,
[422]) call T y , instead of T the (canonical ) energy-momentum 1-forms.
Exercise 529 Show that the following conserved energy-momentum extensors may be derived from the Lagrangians L(Y; @ x yY ) and L(Y; @ x ^Y ).
T (n) = [h@~ @ x yY L(x; Y; @ x Y )y(n @ x Y )i1 nL(Y; @ x yY )];
T (n) = [(n @ x Ye )yh@ @ x Y L(Y; @ x ^Y )i1 nL(Y; @ x ^ Y )];
(8.77)
^
~
where @ @ x yY L(Y; @ x Y )
@@ x yY L(Y; @ x Y .
(a)
(b)
306
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
8.5.1 Canonical Energy-Momentum Extensor of the Free
Electromagnetic Field
The Lagrangian of the Free Electromagnetic …eld is
L(A; @ x ^A) =
1
(@ x ^A) (@ x ^A):
2
(8.78)
Using Eq.(8.77(b)) we get
1
TA (n) = (n @ x A)yF + nF F:
2
(8.79)
The adjoint of the energy-momentum extensor of the electromagnetic
…eld is given by
TAy (n) = @b n TA (b)
(8.80)
1
= @b n (b @ x A)yF + @b nF F:
2
Exercise 530 Show that
1
1
F nF~ = (nyF )yF + n(F F )
2
2
(8.81)
Solution:
1
1
1
(nyF )yF + n(F F ) = [(nyF )F F (nyF )] + n(F F )
2
2
2
1
1
= [nF F F nF F nF + F F n] + n(F F )
4
2
1
1
1
=
F nF + [ 2n(F F ) + n(F ^ F ) + (F ^ F )n] + n(F F )
2
4
2
1
1
1
1
=
F nF
n(F F ) + n ^ (F ^ F ) + n(F F )
2
2
2
2
1
1
~
=
F nF = F nF:
2
2
Exercise 531 Show that
TA (n) =
1
F nF~ + [d(n A)]yF:
2
(8.82)
Solution:
From Eq.(8.79) we can write
1
TA (n) = (n @ x A)yF + nF F
2
(n @ x A)yF
1
(nyF )yF + (nyF )yF + nF F:
2
(8.83)
8.5 Canonical Energy-Momentum Extensor Field
307
Now, recalling the identity given by Eq.(4.154) we have taking into account
that @ x n = 0 we have
@ x (n A) = (n @ x )A
= (n @ x )A
ny(@ x ^A)
nyF:
(8.84)
Then,
1
TA (n) = [@ x (n A)]yF + (nyF )yF + nF F
2
1
= [d(n A)]yF + (nyF )yF + nF F;
2
(8.85)
and taking into account Eq.(8.81), we can also write
TA (n) =
1
F nF~ + [d(n A)]yF;
2
which is what we wanted to show.
Exercise 532 Show that
TAy (n) =
1
F nF~
2
(nyF ) @ x A:
(8.86)
Exercise 533 Show that
?TA (n) =
1
1
(nyF ) ^ ?F + F ^ (ny ? F )
2
2
d(n A) ^ ?F:
(8.87)
Solution:
V1
First, observe that (for any n 2 sec
T M ,! sec C`(M; )),
d(n A) ^ ?F = d(n A) ^ F~ = d(n A) ^ F
1
=
[d(n A)F + F d(n A)]
2
1
=
[d(n A)F F d(n A)]
2
= f[d(n A)]yF g :
(8.88)
Also
(nyF ) ^ ?F = ny(F ^ ?F )
=n[(F F ) ]
F ^ (ny ? F )
F ^ (nyF~ )
(8.89)
308
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
and
(nyF ) ^ ?F =
=
=
=
(nyF ) ^ F
1
[(nyF )F + F (nyF )]
2
1
[(nyF )F F (nyF )]
2
[(nyF )yF ] :
(8.90)
Then, using Eq.(8.85) and the identities just derived above, we can write
1
?TA (n) = ? (nyF )yF + nF F + d(n A)]yF
2
1
= (nyF ) ^ ?F + n(F F ) + f[d(n A)]yF g
2
1
1
= (nyF ) ^ ?F + (nyF ) ^ ?F + F ^ (ny ? F ) + f[d(n A)]yF g
2
2
1
1
(nyF ) ^ ?F + F ^ (ny ? F ) + f[d(n A)]yF g
=
2
2
1
1
=
(nyF ) ^ ?F + F ^ (ny ? F ) d(n A) ^ ?F:
(8.91)
2
2
Remark 534 We easily verify that TA = TA (
but
TAB (n) =
)
is not symmetric,
1
1
~
(F F )n + (nyF )yF = F nF;
2
2
(8.92)
called the Belinfante energy-momentum extensor is symmetric. The non
symmetric part is also gauge dependent, but as can be easily veri…ed the
term d(nyA) ^ ?F in ?TA (n) can be written as an exact di¤ erential, since
taking into account that d ? F = 0 we have
d(nyA) ^ ?F = d(n A) ^ ?F
= d(n A ^ ?F ) + n A ^ d ? F
= d(n A ^ ?F ):
(8.93)
So, the gauge dependent term does not change the computation of the
energy-momentum of a free electromagnetic …eld, but as we are going to see,
it has a crucial role in the computation of the spin of the electromagnetic
…eld and consequently in the law of conservation of angular momentum.
This important issue is not discussed appropriately in textbooks.
8.6 Canonical Orbital Angular Momentum and Spin Extensors
309
8.5.2 Canonical Energy-Momentum Extensor of the Free
Dirac-Hestenes Field
The Lagrangian for the (representative) of a free Dirac-Hestenes spinor
V0
V2
V4
…eld 2 sec( T M +
T M+
T M ) is
L0 ( ; @ x ) = (@ x i 3 )
m
;
(8.94)
In this case, we get from Eq.(8.70) the energy-momentum extensor of the
Dirac-Hestenes …eld which, once we take into account that for any solution
of the ELE associated to L0 ( ; @ x ). We have,
T (n) = h(n @ x )@~@ x L0 ( (; @ x )i1
= h(n @ x )i 3 ~i1 :
nL0 ( ; @ x )
(8.95)
This extensor is called the free Tetrode energy-momentum extensor of the
Dirac-Hestenes …eld.
8.6 Canonical Orbital Angular Momentum and
Spin Extensors
In this section we show that the global rotational invariance of the Lagrangian of a Cli¤ord …eld Y = X 2 sec C`(M; ) or of a representative
of a Dirac-Hestenes spinor …eld, Y =
2 sec C`(M; ), implies the existence of a conserved (2; 1)-extensor …eld. From the possible Lagrangians
L0 ( ; @ x ~ ), we detail only the calculations for the case where ~ refers
to the Cli¤ord product and Y is a Cli¤ord …eld. For the other cases, we
give only the …nal results.
By rotational invariance here we mean that the Lagrangian is invariant
by di¤eomorphisms in Minkowski spacetime generated by one-parameter
group of di¤eomorphisms of M generated by the six vector …elds ( ) 2
sec T M such that
@
@
x
x
; ; = 0; 1; 2; 3;
(8.96)
( ) =
@x
@x
which close the Lie algebra of the homogeneous Lorentz group. As we already learned in Section 8.3, the representation of a di¤eomorphism in M
(generated a rotation by # in the …xed direction ( ) ) can be written as
x 7! x0 = e
#
2
^
xe
^
#
2
;
(8.97)
with x = x (e) and x0 = x (e0 )
x0
.
Recall that for any Cli¤ord …eld X 2 sec C`(M; ) or representative of a
Dirac-Hestenes …eld 2 sec C`(0) (M; ) their pullbacks X 0 2 sec C`(M; ),
0
2 sec C`(0) (M; ) under a di¤eomorphism generated by ( ) satisfy
X 0 (x) = X(x0 );
0
(x) = (x0 ):
(8.98)
310
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
and according to Eq.(8.34) we have,
(a)
(b)
#
#
2)
^
X 0 (x) = e( ^ 2 ) X(x)e(
0
( ^ #
)
0
2
(x) = e
(x ):
;
(8.99)
Then, we must have
L(Y 0 (x); @ x Y 0 (x)) = L(Y (x0 ); @ x0 Y (x0 ));
#
2)
^
for Y = X or Y = . Since x = e(
x0 e(
^
#
2)
(8.100)
, we can write
@# Y 0 (x) @Y 0 L(Y 0 (x); @ x Y 0 (x)) + @ x @# Y 0 (x) @@ x Y 0 L(Y 0 (x); @ x Y 0 (x))
(8.101)
= @# x0 @ x0 L(Y (x0 ); @ x0 Y (x0 )):
Next, we evaluate in details both members of Eq.(8.101) at # = 0 for the
case Y = X (Cli¤ ord …elds). The result for a Dirac-Hestenes spinor …eld
will be given in the next subsection. We need the results
@# x0 j#=0 = xy (
@# X 0 (x)j#=0 = [
^
1
2
=(
^
^
);
X(x) + xy(
)
^
X(x) + xy(
) @ x X(x)
^
) @ x X(x);
1
2
^
X(x)]
#=0
(8.102)
where we recall that ( ^ ) X = [( ^ ); X].
Now, the …rst member of Eq.(8.101) (Y = X) at # = 0 can be written
as a divergence, i.e.,
@# X(x) @X 0 L(X 0 (x); @ x X 0 (x)) + @ x @# X 0 (x) @@ x X 0 L(X 0 (x); @ x X 0 (x))
#=0
0
~
= @ x h @# X (x)j#=0 @ @ x X 0 L(X(x); @ x X(x)i1
= @ x h[( ^ ) X(x) + xy( ^ ) @ x X(x)]@~@ X 0 L(X(x); @X(x)i1 ;
(8.103)
and for the second member we have
@# x0 @ x0 L(X(x0 ); @ x0 X(x0 ))j#=0 = xy (
^
= @ x [xy (
) @X L(X(x); @ x X(x))
^
) L(X(x); @ x X(x))]:
(8.104)
Then, we get
@ x [h[(
^
)
@ x [xy (
^
X + xy(
^
) L(X; @ x X)]
) @ x X @~@ x Y L(X; @ x X)]i1 ] = 0:
(8.105)
8.6 Canonical Orbital Angular Momentum and Spin Extensors
311
Eq.(8.105) implies that there exists a conserved (2; 1)-extensor …eld,
JX : sec C`(M; )
- sec
^2
TM !
^1
T M ,! sec C`(M; );
(8.106)
JX (B) = xy (B) L h[B X + xyB @ x X @~@ x X L]i1 ;
(8.107)
V2
such that with B 2 sec
T M ,! sec C`(M; ) a constant biform
@ x JX (B) = 0:
(8.108)
De…nition 535 The (2; 1) extensor …eld JX given by Eq.(8.107) is called
the canonical angular momentum extensor of the …eld X 2 sec C`(M; ).
Exercise 536 Show that adjoint extensor of JX (B), i.e., the (1; 2)-extensor
V1
JyX (n), n 2 sec
T M with constant coe¢ cients is given by
JyX (n) = @B n JX (B)
y
= TX
(n) ^ x + hX
@~@ x X L(X; @ x X)ni2 :
(8.109)
Exercise 537 Show that
@n @ x JyX (n) = 0:
(8.110)
Remark 538 Sometimes, the (1; 2)-extensor
JyX : sec C`(M; )
JyX (n)
=
y
TX
(n)
- sec
^2
^ x + hX(x)
T M ! sec
^1
T M ,! sec C`(M; );
@~@ x X L(x; X; @ x X)ni2 :
(8.111)
instead of JX is called the canonical angular momentum extensor of a multiform …eld X 2 sec C`(M; ). We use that wording indi¤ erently since this
will generate no confusion.
Eq.(8.109) and Eq.(8.110) suggests the
De…nition 539 The orbital angular momentum extensor and the spin extensor of the …eld X 2 sec C`(M; ) are respectively
y
LyX (n) = TX
(n) ^ x;
with x = x
(8.112)
and
SyX (n) = hX(x)
@~@ x X L(x; X; @ x X)ni2 :
(8.113)
Exercise 540 Show that the adjoint of SyX (n); i.e., the (2; 1)-extensor …eld
SX (B) such that SyX (n) = @B n SX (B) is given by
SX (B) =
h(B
X)@~@ x X L(x; X; @ x X)i1 :
(8.114)
312
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
Now, take B =
^
.
De…nition 541 The six SX ( ^ ) = S X are called spin 1-form …elds
of the …eld X and the six ?S X are called spin 3-form …elds (densities) of
the …eld X.
As promised, we give next the angular momentum and spin extensors for
Lagrangians of the types (a) and (b) in Proposition 524, i.e., L(x; Y; @ x yY )
and L(x; Y; @ x ^Y ), and also for the case of the Dirac-Hestenes spinor …eld
L(x; ; @ x ).
8.6.1 Canonical Orbital and Spin Density Extensors for the
Dirac-Hestenes Field
For the case of a Dirac-Hestenes …eld
2 sec C`(0) (M; ), repeating the
calculations done in the previous section, we …nd the following conserved
angular momentum extensor
Jy (n) = T y (n) ^ x + h
1 ~
@
L(x; ; @ x )ni2 :
2 @xY
(8.115)
The density of orbital angular momentum and spin extensors for the
Dirac-Hestenes …eld are respectively,
Ly (n) = T y (n) ^ x
(8.116)
and
1 ~
@
L(x; ; @ x )ni2
2 @xY
1
1
= h i 3 ~ni2 = i(s ^ n):
2
2
Sy (n) = h
(8.117)
De…nition 542 The bilinear invariant
s=
3
~
(8.118)
is called the spin covector of the Dirac-Hestenes …eld.
Exercise 543 Show that the adjoint of the canonical (1; 2)-extensor Sy (n)
of the Dirac-Hestenes …eld is the (2; 1)-extensor …eld S given by
1
S (B) = h B + (xyB)@~@ x L( ; @ x )ni1 :
2
(8.119)
8.6.2 Case L(X; @ x yX)
In this case, calculations analogous to the ones of the previous sections give
SX (B) =
e
h@
@ x yX L(X; @ x yX)y(B
X)i1
(8.120)
8.7 The Source of Spin
313
and
1
SyX (n) = hX
2
@e@ x yX L(x; X; @ x yX)n
X
n@e@ x yX L(X; @ x yX)i2 :
(8.121)
8.6.3 Case L(X; @ x ^X)
In this case we obtain,
SX (B) =
SyX (n) =
X)@~@ x ^X L(x; X; @ ^ X)i1 ;
h(B^
1
~
hX @
@ x ^X L(X; @ x ^ X)n + X
2
(8.122)
~
n@
@ x ^X L(X; @ x ^ Xi2 :
(8.123)
8.6.4 Spin Extensor of the Free Electromagnetic Field
In this case, using the Lagrangian given by Eq.(8.47) and Eq.(8.123) we
get immediately,
1
hA F~ n + A nF~ i2
2
= hA F xni2 = A ^ F xn
SyA (n) =
= (nyF ) ^ A
= ny(A ^ F )
(n A)F
(8.124)
This formula shows that the spin density of the electromagnetic …eld is
indeed gauge dependent and we think that it is just here that the reality of
A becomes apparent.6
Exercise 544 Show that
?SA (
^
)
?S
= ?[(Ay(
^
))yF ]:
(8.125)
8.7 The Source of Spin
We showed that for any Cli¤ord …eld X 2 sec C`(M; ) or representative
2 sec C`(0) (M; ) of a Dirac-Hestenes spinor …eld there exists a (1; 2)extensor …eld Jy (n) = T y (n)^x+Sy (n) such that @n @Jy (n) = 0. Recalling
6 We observe that this density of spin is analogous to the corresponding one used
in Relativistic Quantum Field Theory, which is also gauge dependent. See, e.g., the
discussion in [186].
314
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
De…nition 54 that says that bif (T ) = T (
)^
) we can write
@n @ x Jy (n) = @n @ x [T y (n) ^ x + Sy (n)]
@n [@ T y (n) + T y (n) ^ @ x] + @n @ x Sy (n)
=
= @n @ x T y (n) + bif (T y ) + @n @Sy (n) = 0:
(8.126)
Taking into account that @n @ x T y (n) = 0 we get the fundamental result
@n @ x Sy (n) =
bif (T y );
(8.127)
which says that Sy (n) alone is not conserved when the energy-momentum
extensor is not symmetric, its source being bif (T y ).
8.8 Coupled Maxwell and Dirac-Hestenes Fields
In this case, the Lagrangian for the coupled system of …elds is:
1
(@ x ^A) (@ x ^A):
2
(8.128)
Our Lagrangian depends then on two dynamic …elds, A and , which
is a combination of Lagrangians of types (a) and (c). A straightforward
computation gives as equations of motion for the dynamic …elds
L( ; @ x ; A; @ x A) = (@ x i 3 )
(a)
(b)
@x
A (q
2 1
0
~) m
m 0 eA = 0;
@ x F = Je :
(8.129)
This is a nonlinear system of partial di¤erential equations, since
Je = e
0
~:
(8.130)
Also, a straightforward computation gives for the energy-momentum extensor of the coupled …elds
~i1 + (n @ x A)yF + 1 nF F
2
= T (n) + TA (n);
T (n) = h(n @ x )i
3
(8.131)
which is the sum of the energy-momentum extensors of the free DiracHestenes and free electromagnetic …elds. We show below that there exists
an extensor …eld, called Tetrode extensor …eld TTy etr (n), with
TT etr (n) = T (n)
such that
(n Je )A;
@n @ x TTy etr (n) = F xJe :
(8.132)
(8.133)
8.8 Coupled Maxwell and Dirac-Hestenes Fields
315
Exercise 545 Show that
bif (TTy etr ) = 12 @ x yis = 21 i@ x ^s;
TT etr (n) TTy etr (n) = nybif (TTy etr );
@n @ x TTy etr (n) = @n @ x TT etr (n):
(i)
(ii)
(iii)
(8.134)
Solution:
We show (i). From the Dirac-Hestenes equation (Eq.(8.54)) we have
[@ x i 3 ] ~
m ~
~ = 0:
(8.135)
~ + i 3@ ~
(8.136)
eA
0
Now,
@ ( i
3
~) = @
i
3
and taking into account the notable identity valid in the spacetime algebra
which says that for any odd c 2 R1;3 , hci1 = 12 (c + c~) we have
h@
i
3
~i1 = 1 (@
2
3
~
i 3 @ ~)
(8.137)
and then
1
@ x ( i 3 ~) + h@ i 3 ~i1 = [@ x i 3 ] ~:
2
Using Eq.(8.136) in the second member of Eq.(8.138) yields
1
@ x (is) +
2
h@
i
3
~i1 = AJe + m ~;
(8.138)
(8.139)
~ is the
where Je = e 0 ~ is the electromagnetic current and s =
3
spin covector …eld introduced in Eq.(8.118). Note also that the hi2 part of
Eq.(8.139) yields
1
@ x y(is) +
2
^ h@
i
3
~i1 = A ^ Je :
(8.140)
Now,
bif (TTy etr ) =
=
bif (TT etr ) =
^ h@
i
3
~i1
^ TT etr (
)
AJe
~i1 A ^ Je
1
1
= @ x y(is) = i@ x ^s;
2
2
=
^ h@
i
3
(8.141)
and (i) in Eq.(8.134) is proved.
To prove Eq.(8.133) we start from the conservation of T (n),
@n @ x T (n) + @n @ x TA (n) = 0:
(8.142)
316
8. A Cli¤ord Algebra Lagrangian Formalism in Minkowski Spacetime
We have,
1
@n @ x TA (n) = @n @ x ( F nF~
2
Moreover,
(nyF ) @ x A):
(8.143)
1
1
@n [(@ F~ )nF + F~ (@ n)F + F~ n@ F ]
@n @ x F nF~ =
2
2
1
~
= (@g
xF F + F @xF )
2
1
= (Je F F Je ) = Je yF:
(8.144)
2
Also,
@n @ x (nyF ) @ x A)
=
=(
@n [(@ nyF ) @ x A + (ny@ F ) @ x A + (nyF ) @ @ x A]
y@ F ) @ x A + (
yF ) @ @ x A
= Je @ x A
1
= @ x Je A + Je @ x A
2
=
[(@ Je ) A + Je (@ A)]
@n [(@ n)Je A + n (@ Je ) A + n Je (@ A)
= @n @ x (n Je ) A:
(8.145)
Using Eqs.(8.144) and (8.145) in Eq.(8.142) gives
(n Je )A] = F xJe :
@n @ x [T (n)
(8.146)
Finally taking into account (iii) in Eq.(8.134) we get
(n Je )A] = F xJe ;
@n @ x [T (n)
(8.147)
which we recognize as Eq.(8.133).
Remark 546 This equation says that even if the Dirac-Hestenes …eld is
moving in a region where F = 0, but A 6= 0 the consideration of the coupling
term (n Je )A is necessary in order for energy-momentum conservation
to take place. Of course, this is related to the well known Bohm-Aharonov
e¤ ect.
Exercise 547 Show that Jy A (n) = T y (n) ^ x + S y (n)
@n @ x Jy A (n) = (F xJ e )^x:
Exercise 548 Show that J
d?J
A
=
A(
^
)=J
A
? f[(F xJ e )^x] (
(8.148)
satisfy
^
)g:
(8.149)
This is page 317
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9
Conservation Laws on
Riemann-Cartan and Lorentzian
Spacetimes
9.1 Preliminaries
9.1.1 Functional Derivatives on Jet Bundles
V
V
Let J 1 ( T M ) be the 1-jet bundle1 over T M ,! C`(M; g), i.e., the
vector bundle de…ned by
^
^
S
J 1 ( T M ) = x2M Jx1 ( T M ):
(9.1)
V
Then, with each local section 2 sec T M ,! sec C`(M; g)g we may asV
V1
sociate a local section j1 ( ) 2 sec J 1 ( T M ). Let f a g, a 2 sec
T M ,!
sec C`(M; g), a = 0; 1; 2; 3; be an orthonormal basis of T M dual to the baV1
sis fea g of T M and let ! ab 2 sec
T M ,! sec C`(M; g) be the connection
1-forms of the connection
r
such
that
rea eb = ! cab ec . We introduce also
V
the 1-jet bundle J 1 [( T M )n+2 ], i.e.,
^
^
^
^
S
J 1 [( T M )n+2 ] := x2M Jx1 ( T M
T M :::
T M ) (9.2)
V
over the con…guration space ( T M )n+2 ,! (C`(M; g))n+2 of V
a …eld theory describing n di¤erent …elds A belonging to sections of T p M ,!
C`(M; g) on a RCST, where for each di¤erent value
V of A we have in general a
di¤erent value of p. We denote sections of J 1 [( T M )n+2 ] by j1 ( a ; !ba ; )
or when no confusion arises simply by j1 ( ).
1 For
the de…nition of jet bundles and notations see Appendix A.3.1.
318
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
V
V
A functional for a …eld
2 sec TVM ,! sec C`(M; g) in J 1 ( T M ) is
V
a mapping F : sec J 1 ( T M ) ! sec T M , j1 ( ) 7! F(j1 ( )).
V
A Lagrangian density for a …eld theory described by …elds A 2 sec T M ,
A = 1; 2; :::; n over a Riemann-Cartan spacetime is a mapping
^
^4
Lm : sec J 1 [( T M )n+2 ] ! sec
T M;
(9.3)
j1 ( a ; !ba ; ) 7! Lm (j1 ( a ; !ba ; )):
(9.4)
Remark 549 When convenient and context is clear, we eventually use instead of Lm (j1 ( a ; !ba ; )) the sloppy notations Lm (x; a ; d a ; !ba ; d!ba ; ; d )
or Lm ( a ; d a ; !ba ; d!ba ; ; d ) when the Lagrangian density does not depend explicitly on x, or simply Lm [ ] and even only Lm . Moreover, in the
calculations done below Lm ( a ; d a ; !ba ; d!ba ; ; d ) will be considered as a
multiform functional of the …eld variables ( a ; d a ; !ba ; d!ba ; ; d ).
9.1.2 Algebraic Derivatives
Vp
De…nition 550 Let X 2 sec
T U . A multiform functional F of X (not
depending explicitly on x 2 M ) is a mapping
^p
^r
F : sec
T U ! sec
T U
Vp
Let w := X 2 sec
T VU . WriteVthe variation of F in the direction of
p
r
X as the functional F :
U!
U given by
F = lim
F (X +
X)
!0
F (X)
:
(9.5)
Remark 551 The algebraic derivative of F relative to X is given by
F := X ^
dF
:
dX
(9.6)
dF
Remark 552 We recall (see Remark 99) that dX
is related @X F de…ned
in Chapter 2, but take into account that these objects in general di¤ er. See
Remark 99
Remark 553 Sometimes, even if F depends only on X we write
dF
stead of dX
, i.e., we write Eq.(9.6) as
F := X ^
@F
:
@X
@F
@X
in-
(9.7)
When a functional depends on two independent variables, e.g., K(X; Y )
we de…ne the (partial) derivative @K
@X by
XK
= X^
@K
:
@X
(9.8)
9.1 Preliminaries
Moreover, for a composed functional F
chain rule::
319
G(X) = F (G(X)) we gave the
@
@G @F
F (G(X)) =
^
(9.9)
@X
@X @G
and for a functional Lp (X r ; Gk (X r )) where the superindices indicates here
the grade of the object, we have immediately
dLp
@Lp
@Gk
@Lp
=
+
^
:
(9.10)
r
r
r
dX
@X
@X
@Gk
Vp
Vp
Moreover,
let X 2 sec VT U . Given the
functionals F : sec
T U !
Vr
V
p
s
sec
T U ! and G : sec T U ! sec T U the variation satis…es
(F ^ G) = F ^ G + F ^ G;
(9.11)
and the algebraic derivative (as is trivial to verify) satis…es
@
@F
@G
(F ^ G) =
^ G + ( 1)rp F ^
:
(9.12)
@X
@X
@X
Another important property of is that it commutes with the exterior
derivative operator d, i.e., for any given functional F
d F = dF:
(9.13)
In general, we may have
Vpfunctionals
Vq depending on
Vrseveral di¤erent multiform …elds, say,
F
:
sec(
T
U
T
U
)
!
sec
T U , with (X; Y ) 7!
Vr
F (X; Y ) 2 sec T U . In this case, we have:
@F
@F
+ Y ^
:
(9.14)
@X
@Y
We are particularly interested
case where
Vpin the important
Vp+1
V4 the functional F
is such that F (X; dX) : sec( T U
T U ) ! sec T U , (X; dX) 7!
F (X; dX). Supposing that the variation X is chosen to be null on the
0
@F
= 0) and taking into account
boundary @U , U 0
U (or that @dX
@U 0
Stokes theorem, we can write de…ning F : j 1 (X) ! R by
Z
F(X) :=
F (X)
(9.15)
F = X^
U0
that
F(X) =
=
Z
U0
Z
U0
=
Z
U0
=
F (X) =
Z
U0
Z
U0
@F
@F
+ dX ^
@X
@dX
@F
@F
( 1)p d
+d X ^
@dX
@dX
Z
@F
@F
( 1)p d
+
X^
@dX
@dX
0
@U
@F
@X
@F
X^
@X
F(X)
X^
;
X
X^
X^
(9.16)
320
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
where
F(X)
@F
@F
=
( 1)p d
:
(9.17)
X
@X
@dX
Vp
Vp+1
V4 p
with F : sec( T U
T U) !
T U called the functional
X
2
derivative of F.
9.2 Euler-Lagrange Equations from Lagrangian
Densities
The principle of stationary action, is here formulated as the statement that
the variation of the action integral (see Eq.(8.24)) written in terms of a
Lagrangian density Lm (j1 ( a ; ! ab ; )) is null for arbitrary variations of
which vanish in the boundary @U of the open set U M ( i.e.,
j@U = 0)
Z
Z
A( ) =
Lm (j1 ( a ; ! ab ; )) =
Lm (j1 ( a ; ! ab ; )) = 0: (9.18)
U
U
Using Eq.(9.17) gives immediately
Z
A( ) =
U
^
A( )
(9.19)
where
? ( )=
A( )
:=
@Lm ( )
@
( 1)r d
@Lm ( )
@d
:
(9.20)
is known as the Euler-Lagrange functional for the …eld Since
trary in Eq.(9.19), the stationary action principle implies that
is arbi-
? ( ) = 0;
is the corresponding ELE for the …eld V
.
p
We recall also that if G(j1 ( )) 2 sec
T M is an arbitrary functional
and : M ! M a di¤eomorphism, then G(j1 ( )) is said to be invariant
under if and only if G(j1 ( )) = G(j1 ( )). Also, it is a well known result
that G(j1 ( )) is invariant under the action of a one parameter group of
di¤eomorphisms t if and only if
$ G(j1 ( )) = 0;
where
2 We
F
,
X
(9.21)
2 sec T M is the in…nitesimal generator of the group
observe that some authors (e.g.,[20, 98, 138]) denote
something we also did in the …rst edition of our book.
@F
@X
t.
( 1)p d
@F
@dX
by
9.3 Invariance of the Action Integral under the Action of a Di¤eomorphism
321
As an example, the Lagrangian
V1 density for the electromagnetic …eld generated by a current Je 2 sec
T M ,! sec C`(M; g) in a Riemann-Cartan
V1
spacetime where F = dA; A 2 sec
T M ,! sec C`(M; g) is
Lem (A) =
1
F ^ ?F
2
?Je ^ A:
(9.22)
The ELE (see Exercise 563) gives F = Je and since dF = 0 we have
Maxwell equations
dF = 0; F = Je :
(9.23)
9.3 Invariance of the Action Integral under the
Action of a Di¤eomorphism
Proposition 554 The action A( ) for any …eld theory formulated in terms
of …elds that are di¤ erential forms is invariant under the action of one parameters groups of di¤ eomorphisms if Lm (j1 ( a ; ! ab ; ))j@U = 0 on the
boundary of @U of a domain U M:
Proof. Let Lm (j1 ( a ; ! ab ; )) be the Lagrangian density of the theory. The
variation of the action which we are interested is the horizontal variation,
i.e.:
Z
A(
)
=
$ Lm (j1 ( a ; ! ab ; )):
(9.24)
h
U
Let
= g( ; ) 2 sec
^1
T M ,! sec C`(M; g):
(9.25)
Then we have (from a well known property of the Lie derivative) that
$ Lm (j1 ( a ; ! ab ; )) = d[ yLm (j1 ( a ; ! ab ; ))] +
y[dLm (j1 ( a ; ! ab ; ))]:
(9.26)
V4
But, since Lm (j1 ( a ; ! ab ; )) 2 sec
T M ,! sec C`(M; g) we have dLm =
0 and then $ Lm = d[ yLm ]. It follows, using Stokes theorem that
Z
Z
$ Lm (j1 ( a ; ! ab ; )) =
d[ yLm (j1 ( a ; ! ab ; ))]
U
U
Z
=
yLm (j1 ( a ; ! ab ; )) = 0;
(9.27)
@U
since Lm ( )j@U = 0.
Remark 555 It is important to emphasize that the action integral is always invariant under the action of a one parameter group of di¤ eomorphisms even if the corresponding Lagrangian density is not invariant (in
the sense of Eq.(9.21)) under the action of that one parameter group of
di¤ eomorphisms.
322
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
9.4 Covariant ‘Conservation’Laws
Let (M; g; r; g ; ") denotes a general Riemann-Cartan spacetime. As stated
above we suppose V
that the dynamic …elds A , A = 1; 2; :::; n, are r-forms,
r
A
i.e., each
2 sec
T M ,! sec C`(M; g), for some r = 0; 1; :::; 4.
Let fea g be an arbitrary global orthonormal basis for T M , and let f a g
V1
be the dual basis3 . We suppose that a 2 sec
T M ,! sec C`(M; g). Let
moreover f a g be the reciprocal basis to f a g. In Chapter 4 we learned how
to represent the gravitational …eld using f a g and how to write Einstein
equations for such objects.4
Here, we make the hypothesis that a Riemann-Cartan spacetime models a generalized gravitational …eld which must be described by f a ; ! ab g,
where ! ab are the connection 1-forms (in a given gauge).
VrThus, we suppose
that a dynamic theory for the matter …elds A 2 sec
T M is obtained
through
the
introduction
a
Lagrangian
density,
which
is
a functional on
V
J 1 [( T M )2+n ] as previously discussed.
Active local Lorentz transformations are represented by even sections
of the Cli¤ord bundle U 2 sec Spine1;3 (M ) ,! sec C`(0) (M; g), such that
~ = U
~ U = 1, i.e., U (x) 2 Spine1;3 ' Sl(2; C). Under a local Lorentz
UU
transformation the …elds transform as
a
7!
0a
7!
0A
=U
! ab 7! ! 0ab =
A
=U
a
U
1
=
a c
c ! d(
A
1
U
a b
;
b
1 d
)b + ac (d
1 c
)b ;
(9.28)
;
where ab (x) 2 SOe1;3 . In our formalism it is easy to see that Lm ( a ; ! ab ; )
is invariant under local Lorentz transformations. Indeed, since g := 5 =
V4
0 1 2 3
2 sec
T M ,! sec C`(M; g) commutes with even multiform
…elds, we have that a local Lorentz transformation produces no changes in
Lm ( a ; ! ab ; ), i.e.,
Lm ( a ; ! ab ; ) 7! U Lm ( a ; ! ab ; )U
1
= Lm ( a ; ! ab ; ):
(9.29)
However, this does not implies necessarily that the variation of the Lagrangian density Lm ( a ; !ba ; ) obtained by variation of the …elds ( a ; !ba ; )
is null, since v Lm = Lm ( a + v a ; !ba + v !ba ; + v ) Lm ( a ; !ba ; ) 6= 0,
unless it happens that for an in…nitesimal Lorentz transformation,
Lm (
a
+
= Lm (U
3 In
v
a
a
U
; ! ab +
a
v ! b;
1
; U ! ab U 1 ; U
+
v
U
1
)
) = U Lm U
1
= Lm :
(9.30)
this Chapter boldface latin indices, say a, take the values 0; 1; 2; 3:
Lagrangian density for the f a g for the case of GRT will be introduced in Section
8.5 and explored in details in the next Chapter.
4A
9.4 Covariant ‘Conservation’Laws
323
As we just showed above the action of any Lagrangian density is invariant under di¤eomorphisms. Let us calculate the total variation of the
Lagrangian density Lm , arising from a one-parameter group of di¤eomorphisms generated by a vector …eld 2 sec T M and by a local Lorentz
transformation, when we vary a ; ! ab ; A ; d A independently. We have (recalling Eq.(8.30)
Lm = v Lm $ Lm :
(9.31)
In what follows we suppose that the Lagrangian of the matter …eld is
invariant under local Lorentz transformations5 , i.e., v Lm = 0. Now, note
that Lm depends on the a due to the dependence of the …elds a on these
variables and on the ! ab because eventual covariant derivatives of the …elds
a
must appears in it. We suppose moreover that Lm does not depend
explicitly on d a and d! ab . Then
Lm =
=
a
$ Lm =
$
where ? A =
and we have:
a
^ ?Ta
A( )
A
^
@Lm
@Lm
+ ! ab ^
+
@ a
@! ab
$ ! ab ^ ?J ab
$
A
A
^?
^
A( )
A
A;
(9.32)
(9.33)
are the Euler-Lagrange functionals of the …elds
De…nition 556 The negative of the coe¢ cients of
?Ta =
@Lm
2 sec
@ a
? Ta :=
^3
a
=
$
T M
a
A
, i.e.
(9.34)
are called the
V1energy-momentum densities of the matter …elds, and the Ta =
T a 2 sec
T M are called the energy momentum 1-form …elds of the
matter …elds. The negative of coe¢ cients of ! ab = $ ! ab , i.e.,
?J ab =
? Jab =
^3
@Lm
2
sec
T M;
@! ab
(9.35)
are called the angular momentum densities of the matter …elds.
Taking into account that each one of the …elds A obey a Euler-Lagrange
equation, ? A = 0, we can write
Z
Z
Z
Lm =
$ Lm = ?Ta ^ $ a + ?Jab ^ $ ! ab :
(9.36)
Now, since all geometrical objects in the above formulas are sections of
the Cli¤ord bundle, recalling Eq.(4.61), we can write
$
a
=
yd
a
+ d( y a ):
(9.37)
5 We discuss further the issue of local Lorentz invariance and its hidden consequence
in [354, 326].
324
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
Moreover, recalling also the …rst Cartan’s structure equation,
a
d
+ ! ab ^
b
a
=
;
(9.38)
we get
$
a
=
y
a
=
y
a
y ! ab ^
! ab )
(
= D( y a ) +
y
+ d( y a )
b
b
+
b
(
! ab )
a
! ab + d( y a )
b
;
(9.39)
where D is the covariant exterior derivative of indexed p-form …elds introduced by (Eq.(4.118)). To continue we need the following
Proposition 557 Let ! be the 4 4 matrix whose entries are the connection 1-forms. For any x 2 M , y!ba 2 spine1;3 ' sl(2; C) = soe1;3 , the Lie
algebra of Spine1;3 (or of SOe1;3 ).
Proof. Recall that any in…nitesimal local Lorentz transformation (at x 2
M ) ab 2 SOe1;3 can be written as
a
b
=
ab
=
a
b
a
b,
+
ba :
j
a
bj
1;
(9.40)
Now,
! ab =
=
and we see that
(Lacb
c
c
)=(
d
d
) (Lacb
c
)
Lacb
(9.41)
!ab satisfy
!ab +
!ba =
c
(Lacb + Lbca ) = 0;
(9.42)
since in an orthonormal basis the connection coe¢ cients satisfy Lacb =
Lbca . We see then that we can identify if j c j << 1
a
b
! ab
:=
(9.43)
as the generator of an in…nitesimal Lorentz transformation, and the proposition is proved.
Now, the term (
! ab ) b has the form of a local vertical variation of
a
the
and thus we write
v
a
:=
! ab )
(
b
=
a b
b
(9.44)
Using Eq.(9.44) we can rewrite Eq.(9.39) as
$
a
= D(
a
)+
y
a
+
v
a
:
(9.45)
9.4 Covariant ‘Conservation’Laws
a
We see that $
=
a
v
325
only if we have the following constraint
a
D(
y
)+
a
= 0:
(9.46)
A necessary and su¢ cient condition for the validity of Eq.(9.46) is given
by Lemma 559 below.
Now, let us calculate $ ! ab . By de…nition,
$ ! ab =
=
y(d! ab ) + d(
y(Rab )
! ab )
! ac )! cb + (
(
! cb )! ac + d(
! ab );
(9.47)
where in writing the second line in Eq.(9.47) we used Cartan’s second
structure equation,
d! ab + ! ac ^ ! cb = Rab :
(9.48)
Under an in…nitesimal Lorentz transformation
we can write (in obvious matrix notation)
v!
=
d + !
= 1+ , recalling Eq.(9.28),
! ;
(9.49)
a
v ! b:
(9.50)
which using Eq.(9.43) gives for Eq.(9.47)
y(Rab ) +
$ ! ab =
Now, for a vertical variation we have:
Z
U
v Lm
:=
Z
v
a
U
^
@Lm
+
@ a
a
v! b
Then, if we recall that we assumed that
the …eld equations are satis…ed, i.e., ?
Z
Then
Z
=
Z
=
=
Z
Z
Z
Lm =
$ Lm
Z
[D(
a
$ Lm =
)+( y
[ y(Rab ) +
a
v ! b]
a
Z
)+
^
R
A
v Lm =
Lm
=
a
?Ta ^ $
v
a
@Lm
+
@! ab
A
v
A
^
Lm
A
:
(9.51)
0 and if we suppose that
= 0, Eq.(9.36) becomes,
+ ?Jab ^ $ ! ab :
(9.52)
] ^ ?Ta
^ ?Jab
?Ta ^ ( y
a
) + ?J ba ^ ( y(Rab ) + D[?T a (
a
?Ta ^ ( y
a
) + ?J ba ^ ( y(Rab )
a
(D?T a )(
)]
))
(D?T a )(
a
(9.53)
))
326
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
where we used that D[(
Z
d[(
U
a
a
)?T a ] = d[(
Z
a
)?T a ] =
(
)?T a ]; that ?T a j@U = 0 and
a
)?T a = 0:
(9.54)
@U
Now, writing
= a a = a a , and recalling that the action is invariant
under di¤eomorphisms we have (if as usual we suppose that Lm j@U = 0):
Z
Z
Lm =
$ Lm = ?T a ^ ( c y a ) + ?Jab ^ ( c yRab ) D?T c c = 0;
(9.55)
c
and since the
are arbitrary, we end with
D?T c + ?T a ^ ( c y
a
) + ?Jab ^ ( c yRab ) = 0:
Also, using the explicit expressions
for v a and
R
and Eq.(9.50)) in (Eq.(9.51) for
v Lm we get,
=
Z
Z
1
?T a ^
2
?T a ^
b
a b
b
?T b ^
+ ?J ba ^ (
a
a c
c! b
d ? Jab
! ac
(9.56)
a
v! b
given by Eq.(9.44)
c
b
d
! cb ^ ?Jcb
a
b)
?J ca ^ ! bc
a
b
= 0;
(9.57)
and since the coe¢ cients
a
b
D ? J ba +
are arbitrary we end with
1
?T b ^
2
a
?T a ^
b
= 0:
(9.58)
Eq.(9.56) and Eq.(9.58) are known as covariant conservation laws [38].
They are simply identities that follows from the hypothesis utilized, namely
that the theory is invariant under di¤eomorphisms and also invariant under the local action of the group Spine1;3 . Eq.(9.56) and Eq.(9.58) do not
encode genuine conservation laws and a memorable number of nonsense
have been generated along the years, by authors that use in a naive way
those equations. Some examples of the nonsense is discussed in the speci…c
case of Einstein’s theory in Section 8.7.
9.5 When Genuine Conservation Laws Do Exist?
We show now that when the Riemann-Cartan spacetime (M; g; r; g ; ")
admits symmetries, then Eq.(9.56) and Eq.(9.58) can be used for the construction of closed 3-forms, which then provides genuine conservation laws
for the matter …elds. We present that result in the form of the following
[38]
9.5 When Genuine Conservation Laws Do Exist?
327
Proposition 558 For each Killing vector …eld
2 sec T M , such that
a
$ g = 0 and $ = 0, where = ea
is the torsion tensor of r, and
a
the torsion 2-forms, we have
a
d [(
where L =
) ? Ta + (
a
L
b
) ? J ab ] = 0;
(9.59)
yD + Di is the so called Lie covariant derivative.
In order to prove the Proposition 558, we need some preliminary results,
which we recall in the form of lemmas.
Lemma 559 $
$ = 0.
a
=
v
a
and $ ! ab =
a
Proof. Let us show …rst that if $
$ g=
ab
a
($
=
b
)
+
a
v! b
v
a
if and only if $ g = 0 and
then $ g = 0. We have
ab
a
b
($
):
(9.60)
On the other since g is invariant under local Lorentz transformations, we
have
a
b
)
+ ab a ( v b ) = 0:
(9.61)
v g = ab ( v
Then, it follows from Eqs.(9.60) and (9.61) that if $ a = v a then $ g =
0.
Taking into account the de…nition of Lie derivative we can write
$ ea =
{ ba eb ;
{ ab
b
= [ea (
$
a
= { ab
b
;
m b
c am ]:
)
(9.62)
Now, if $ g = 0 we have from Eq.(9.60) that (
0, i.e., in this case we need to have
c
cb { a
{ab + {ba = 0;
+
c
a
ac { b )
b
=
(9.63)
and then it follows that for any x 2 M , {ab 2 spine1;3 . Using Proposition
557 and identifying { ab = ab =
! ab the vertical variation can be
a
a
written as v = $ .
The proof that if $ ! ab = v !ba v then $ = 0 is trivial. In the following
we prove the reciprocal, i.e., if $ = 0 then $ ! ab = v ! ab . We have,
$
Then, if $
a
= $ ea
+ ea
$
a
:
(9.64)
= 0 we conclude that
$
a
=
a
b
b
;
(9.65)
which is an in…nitesimal Lorentz transformation of the torsion 2-forms.
On the other hand, taking into account Cartan’s …rst structure equation,
328
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
a
Eq.(9.62) and the fact that $ d
$
a
=$ d
a
+ $ ! ab ^
a b
b
a
b) ^
=d
= d(
b
+ $ ! ab ^
b
), we can write
+ ! ab ^ $
b
a
b
bd
+
a
= d($
b
+ ! ab ^
b c
c
b
+ $ ! ab ^
b a
c! b
+
b
^
:
(9.66)
Also, using Eq.(9.65) we have
a
$
a
b
bd
=
a b
b! c
+
c
^
:
(9.67)
From Eqs.(9.66) and (9.67 it follows that
$ ! ab ^
b
a c
c! b
=
b
^
b a
c! b
b
^
a
b)
d(
^
b
;
(9.68)
or
$ ! ab =
a c
c! b
c a
b! c
a
b
d
Thus, recalling Eq.(9.49) we …nally have that $
Corollary 560 For any x 2 M ,
only if, $ g = 0.
a
Proof. The Lie covariant derivative of
a
L
=
yD
=
y d
a
a
+
=$
a
+(
=$
a
+(
a
=
where we put $
a
+ D(
! ab ^ b
! ab ) b
! ak ) k
a k
.
k
=
a
v ! b.
is given by
)
a
+ d(
b
(
=(
a
k
) + !ba
b
)! ab + ! ab
! ab )
+
k
b
;
Then,
a
L
b
(9.69)
! ab
is an element of spine1;3 , if and
a
L
b
:
a
b
=
! ab :
+
(9.70)
Now, we already showed above that for any x 2 M , the matrix of the
! ab
e
e
a
is an element of spin1;3 and then, b L
will be an element of spin1;3 if
and only if the matrix of the { ab is an element of spine1;3 . The corollary is
proved.
Lemma 561 If $ g = 0 and $
D(
L
b
= 0 then we have the identity
a
)+
yRab = 0:
(9.71)
Proof. Using the de…nitions of the exterior covariant derivative and the
Lie covariant derivative we have
D(
= df
b
b
L
[$
a
) = d(
a
+(
b
b
L
a
) + !bc (
! ac ) c ]g
d
$
+f
+(
d
L
c
a
[$
! dc )
a
c
)
!ca (
+(
! ad ;
b
L
c
)
! ac ) c ]g ! db
9.5 When Genuine Conservation Laws Do Exist?
329
i.e.,
D(
b
L
a
) = $ ! ab
+ d(
b
a
$
y (d! ab + ! ac ^ ! cb )
)+
! cb ( c
a
$
)
(9.72)
(
b
c
$
)! ac
:
If, $ g = 0, then for any x 2 M , b $ a 2 spine1;3 and the second
line of Eq.(9.72) is an in…nitesimal Lorentz transformation of the ! ab . If
besides that, also $ = 0 then $ ! ab = v ! ab and then the …rst term on
the second member of Eq.(9.72) cancels the term in the second line. Then,
taking into account Cartan’s second structure equation the proposition is
proved.
Proof. (Proposition 558).We are now in conditions of proving the Proposition 558. In order to do that we combine the results of Lemmas 559 and
561 with the identities given by Eqs.(9.56) and (9.58). We get,
a
d[(
a
) ? Ta ] = D[(
a
= D(
a
=L
) ? Ta ]
) ^ ?Ta + (
( y
^ ?Ta
a
a
)D ? Ta
a
) ^ ?Ta + (
)D ? Ta ;
i.e., using Eq.(9.56),
) ? Ta ] = L a ^ ?Ta ?Jba ^
yRba :
V1
Observe now that if A 2 sec
T M ,! sec C`(M; g) then, a ^ (
A. This permit us to write Eq.(9.73) as
a
d[(
a
d[(
b
) ? Ta ] =
(
a
L
b
) ^ ?Ta ^
b
yRba :
?Jba ^
(9.73)
a
A) =
(9.74)
If $ g = 0, we have by the Corollary of Lemma 559 that for any x 2 M ,
L a 2 spine1;3 . In that case, we can write Eq.(9.74) as
a
d[(
) ? Ta ] =
=
1
( b L a ) ^ [?Ta ^ b ?T b ^ a ] ?Jab ^
yRba
2
( b L a ) ^ D?Jab ?Jab ^
yRba :
(9.75)
On the other hand, if $
d[(
a
) ? Ta ] =
=
D(
D[ (
= 0, in view of Proposition 561 we can write
b
L
b
Finally, if $ g = 0 and $
d[(
a
) ? Ta + (
L
a
) ^ ?Jab
a
(
) ^ ?Jab ] =
b
L
d[ (
a
b
) ^ D?J ba
L
a
) ^ ?Jab ]:
(9.76)
= 0 we have
b
L
a
) ^ ?Jab ] = 0;
(7.40bis)
which is the result we wanted to prove.
The fact that the existence of symmetries implies in the existence of
closed 3-forms has been originally demonstrated by Trautman [428, 429,
430, 431]. See also [38]
330
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
9.6 Pseudopotentials in GRT
As we already said in Chapter 4, in Einstein’s gravitational theory, i.e.,
GRT each gravitational …eld is modelled by a Lorentzian spacetime M =
(M; g; D; g ; "). The ‘gravitational …eld’g is determined through Einstein’s
equation by the energy-momentum of the matter …elds A , A = 1; 2; :::; m,
living in M. As we already know from Chapter 4, Einstein’
Vs1 equation
(Eq.(4.276)) can be written in terms of the …elds a 2 sec
T M ,!
sec C`(M; g), where f a g is an orthonormal basis of T M as
(@ @)
a
a
+ @ ^ (@
) + @y(@ ^
a
1
)+ T
2
a
= T a;
(9.77)
where @ = a Dea ; T a = T a = Tba b are the energy-momentum 1-form
…elds and T = Taa . An explicit Lagrangian density giving that equation,
which di¤ers from the original Einstein-Hilbert Lagrangian LEH by an
exact di¤erential is (See exercise 565).
Lg =
1
d
2
a
^ ?d
a
+
1
2
a
^?
a
+
1
(d
4
a
^
a)
^? d
b
^
b
; (9.78)
with
1
? R = d( a ^ ?d a ) + Lg
(9.79)
2
The total Lagrangian density of the gravitational …eld and the matter
…elds can then be written as
LEH :
L = Lg + Lm ;
(9.80)
where Lm ( A ; d A ) = Lm ( A ; d A ) g = Lm ( A ; d A ) g is the matter Lagrangian. So, it depends on the the a but does not depends on the d a .
Now, variation of L with respect to the the …elds a yields
R
L = Ag + Am =
R
Ag R
=
+
a^
R
a
=
We de…ne
R
?ta :=
a
^
@Lg
;
@ a
@Lg
+d
@ a
? S c :=
@Lg
@d a
@Lg
;
@d a
R
Lg + Lm
Am
a^
a
+
R
a
^
? T a :=
@Lm
:
@ a
(9.81)
@Lm
@ a
(9.82)
m
where the T a = ? 1 @L
@ a are the matter energy momentum 1-form …elds.
We will show in a while that
R
R
a
Lg =
(9.83)
a ^ ?G ;
9.6 Pseudopotentials in GRT
331
V1
where G a = (Ra 12 R a ) 2 sec
T M ,! C` (T M; g) are the Einstein
V1
1-form …elds, with the Ra = Rba b 2 sec
T M ,! C` (T M; g) the Ricci
1-form …elds and R the scalar curvature.
V3
Moreover, a calculation (see Exercise 566) gives for ?tc 2 sec
T M ,!
V
2
c
C` (T M; g) and ?S 2 sec
T M ,! C` (T M; g) :
1
!ab ^ [! cd ^ ?(
2
@Lg
=
@ c
@Lg
=
?S c :=
@d c
?tc :=
1
!ab ^ ?(
2
a
^
b
a
^
b
^
c
^
d
! bd ^ ?(
)
a
^
d
^
):
c
)];
(9.84)
So, we have with
@Lg
+d
@ a
?G a :=
@Lg
@d a
? ta
=
d ? Sa =
? T a = ?T a (9.85)
We give now a proof that the second and third members of Eq.(9.85) are
equal starting from the Einstein-Hilbert Lagrangian density LEH instead
of using Lg .6 We have
LEH :=
1
R
2
g
Now,
? R = Rab ^ ?(
Indeed,
? [Rab y(
a
^
b
)] =
=
=
a
^
b
1
? R:
2
=
)=
1
? Rabcd [( c ^
2
1
? Rabcd [( c y(
2
1
? Rabcd ( cb da
2
(9.86)
? [Rab y(
d )y(
d y(
a
a
^
^
a b
c d)
b
=
a
^
b
)]
b
)]:
(9.87)
)]
? Rdccd =
? R:
(9.88)
Moreover
1
1
Rab ^ ?( a ^ b ) + Rab ^ [?( a ^ b )]
(9.89)
2
2
and taking into account Cartan’s second structure equation Rab = d!ab +
!ac ^ ! cd , we have
LEH =
Rab ^ ?(
a
^
= d !ab ^ ?(
6 For
b
a
)
^
b
) + [ !ac ^ ! cd ] ? (
a derivation using Lg see Exercise 520
a
^
b
) + [!ac ^ ! cd ] ? (
a
^ b ):
(9.90)
332
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
Now, taking into account Eq.(4.113) we have
a
d !ab ^ d ? (
= d [!ab ^ ?(
= d [!ab ^ ?(
a
b
^
)]
a
b
^
b
^
)]
a
)] + !ac ^ d[?(
!ac ^ f[! ac ^ ?(
c
b
^
b
^
)]
)] + ! bc ^ (
a
^
c
)g
and thus
a
Rab ^ ?(
^
b
a
) = d [!ab ^ ?(
b
^
)]:
Also, taking into account that
[?(
a
^
b
Rab ^ [?(
a
^
b
)] =
c
^ [ cy ? (
a
^
c
^ Rab ^ ?(
c
^
a
^
b
^ Rab ^ ?(
c
^
a
^
b
)]
it is
)] =
)
(9.91)
and so
LEH =
1
d[ !ab ? (
2
a
^
b
)] +
1
2
c
b
):
(9.92)
Then under the usual assumption that variations vanish at the boundary
we get
R
1R
c
LEH =
^ a ^ b ):
(9.93)
c ^ Rab ^ ?(
2
Now,
1
Rab ^ ?(
2
=
a
b
^
^
d
)=
1
Rabck ? [(
4
c
^
1
? [Rab y(
2
k
? (Rd
=
)y(
a
^
1
R
2
d
);
b
a
^
^
d
b
^
d
)]
)]
(9.94)
i.e.,
?G d =
and we get
R
LEH =
1
Rab ^ ?(
2
R
a
a
^
^ ?G a =
b
R
d
^
a
):
^ ?Ga :
(9.95)
(9.96)
9.7 Is There Any Energy-Momentum Conservation Law in GRT?
333
Next we use in Eq.(9.95) Cartan’s second structure equation. We get
2 ? Gd =
=
=
d!ab ^ ?(
a
d[!ab ^ ?(
a
d[!ab ^ ?(
a
!ac ^
!bc
^ ?(
b
^
^
b
^
b
a
^
d
^
^
d
^
d
b
^
!ac ^ ! cb ^ ?(
)
)] + !ab ^ d ? (
d
)
a
a
^
)] + !ab ^ ! ap ^ ?(
+ !ab ^ !pb ^ ?( a ^ p ^ d ) +
!ac ^ !bc ^ ?( a ^ b ^ d )
= !ab ^ [!pd ^ ?( a ^ b ^ p )
d[!ab ^ ?( a ^ b ^ d )]:
!ab ^
!pd
!pb ^ ?(
^ ?(
a
^
^
b
b
p
a
p
^
^
^
d
b
^
b
^
d
? td
d ? Sd:
)
)
d
^
)
p
^
)]
)]
Then,
taking into account Eq.(9.84) and that from Eq.(9.79) it is
R
Lg , the dual of Einstein 1-form …elds can be written as:
?G d =
d
(9.97)
R
LEH =
(9.98)
9.6.1 Pseudopotentials are not Uniquely De…ned
Now, we can write Einstein’s equation in a very interesting, but eventually
dangerous form, i.e.:
d ? S a = ?T a + ?ta :
(9.99)
In writing Einstein’s equations in that way, we have associated to the
gravitational …eld a set of 2-form …elds ?S a called pseudopotentials that
have as sources the currents (?T a + ?ta ). However, pseudopotentials are
not uniquely de…ned since, e.g., pseudopotentials (?S a + ? a ), with ? a
closed, i.e., d ? a = 0 give the same second member for Eq.(9.99).
9.7 Is There Any Energy-Momentum Conservation
Law in GRT?
Why did we say that Eq.(9.99) is a dangerous one?
The reason is that we may be led to think that we have discovered a
conservation law for the energy momentum of matter plus gravitational
…eld independent of the existence of appropriated Killing vector …elds7 ,
since from Eq.(9.99) it follows that
d(?T a + ?ta ) = 0:
(9.100)
7 Recall that from the previous section we learned that energy-momentum conservation law for the matter …elds alone exist only when appropriated Killing vector …elds
exist.
334
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
This thought however is only an example of wishful thinking, because in
Einstein theory the ?ta depends on the connection (see Eq.(9.84)) and thus
gauge dependent. They do not have the same tensor transformation law
as the ?T a (=
? T a ), i.e., there is no tensor …eld associated to the ta .
So, Stokes theorem cannot be used to derive from Eq.(9.100) conserved
quantities that are independent of the gauge, which is clear. However, and
this is less known, for this speci…c problem, Stokes theorem, also cannot
be used to derive conclusions that are independent of the local coordinate
chart used to perform calculations [54]. In fact, the currents ?ta are nothing
more than an old pseudo energy momentum tensor in a new dress. Non
recognition of this fact can lead to many misunderstandings. We present
some of them in what follows, in order to call our readers’ attention of
potential errors of inference that can be done when we use sophisticated
mathematical formalisms without a perfect domain of their contents.8
(i) First, it is easy to see that from Eq.(9.85) it follows that [247]
D?G = D ? T = 0;
where ?G = ea ?G a 2 sec T M
V3
sec T M sec
T M and where
D?G := ea
sec
V3
(9.101)
T M and ?T = ea
D ? Ga, D ? T =
ea
D?Ta
?T a 2
(9.102)
and D is the exterior covariant derivative of index valued forms (De…nition
257). Now, in [247] it is written (without proof) a ‘Stokes like theorem’
Z
D?T=
4-cub e
Z
?T:
(9.103)
3 b oundary
of this 4-cub e
We searched in the literature for a proof of Eq.(9.103) which appears also
in many other texts and scienti…c papers as, e.g., in [94, 434] but could, of
course, …nd none, which may be considered as valid9 . The reason is simply.
If expressed in details, e.g., the …rst member of Eq.(9.103) reads
Z
ea (d ? T a + ! ab ^ ?T b );
(9.104)
4-cub e
and it is necessary to explain what is the meaning (if any) of the integral.
Since the integrand is a sum of tensor …elds, this integral says that we
are adding tensors belonging to the tensor spaces of di¤erent spacetime
8 More
9 In
details on this issue may be found in [276].
particular, on this issue the reader should read page 108 of Parrot’s book [294].
9.7 Is There Any Energy-Momentum Conservation Law in GRT?
335
points. As, well known, this cannot be done in general, unless there is a
way for identi…cation of the tensor spaces at di¤erent spacetime points. This
requires, of course, the introduction of additional structure on the spacetime
representing a given gravitational …eld, and such extra structure is lacking
in Einstein theory. We unfortunately, must conclude that Eq.(9.103) do not
express any conservation law, for it lacks a precise mathematical meaning.
In Einstein theory possible pseudopotentials are, of course, the ?S a that
we identi…ed above (Eq.(9.84)), with
1
?Sc = [ !ab y(
2
a
^
b
^
c )]
5
:
(9.105)
Then, if we integrate Eq.(9.99) over a ‘certain …nite 3-dimensional volume’, say a ball B, and use Stokes theorem we have10
Z
Z
1
1
a
a
a
? (T + t ) =
? S a:
(9.106)
P :=
8
8
B
@B
In particular the energy or (inertial mass) of the gravitational …eld plus
matter generating the …eld is de…ned by11
Z
1
0
? S 0:
(9.107)
P = E = mI =
lim
8 R!1
@B
(ii) Now, a frequent misunderstanding is the following. Suppose that in
a given gravitational theory there exists an energy-momentum conservation law for matter plus the gravitational …eld expressed in the form of
Eq.(9.100), where T a are the energy-momentum 1-forms of matter and ta
are true 12 energy-momentum 1-forms of the gravitational …eld. This means
that the 3-forms (?T a + ?ta ) are closed, i.e., they satisfy Eq.(9.100). Is this
enough to warrant that the energy of a closed universe is zero? Well, that
would be the case if starting from Eq.(9.100) we could jump to an equation
like Eq.(9.99) and then to Eq.(9.107) (as done, e.g., in [420]). But that
sequence of inferences in general cannot be done, for indeed, as it is well
known, it is not the case that closed three forms are always exact. Take
a closed universe with topology, say R S 3 . In this case B = S 3 and we
have @B = @S 3 = ?. Now, as it is well known (see, e.g., [265]), the third
de Rham cohomology group of R S 3 is H 3 R S 3 = H 3 S 3 = R. Since
this group is non trivial it follows that in such manifold closed forms are not
exact. Then from Eq.(9.100) it did not follow the validity of an equation
analogous to Eq.(9.99). So, in that case an equation like Eq.(9.106) cannot
even be written.
1 0 The reason for the factor 8 in Eq.(9.106) is that we choose units where the numerical
value gravitational constant 8 G=c4 is 1, where G is Newton gravitational constant.
1 1 See the details of the calculation, e.g., in [276]
1 2 This means that the ta are no in this case pseudo 1-forms, as in Einstein’s theory.
336
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
Despite that commentary, keep in mind that in Einstein’s theory the
‘energy’of a closed universe13 supposed to be given by Eq.(9.107) is indeed
zero, since in that theory the 3-forms (?T a + ?ta ) are indeed exact (see
Eq.(9.99)). This means that accepting ta as the energy-momentum 1-form
…elds of the gravitational …eld, it follows that gravitational energy must be
negative in a closed universe.
(iii) But, is the above formalism a consistent one? Given a coordinate
chart fx g of the maximal atlas of M , with some algebra (left as exercise
to the reader) one can show that for a gravitational model represented by
a diagonal asymptotic ‡at metric14 , the inertial mass E = mI is given by
Z
xi @
1
(g11 g22 g33 g ij )r2 d ;
(9.108)
lim
mI =
16 r!1
r @xj
@B
where @B = S 2 (r) is a 2-sphere of radius r, gij xj = xi and d is the
element of solid angle. If we apply Eq.(9.108) to calculate, e.g., the energy
of the Schwarzschild space time15 generate by a gravitational mass m, we
expect to have one unique and unambiguous result, namely mI = m.
However, as showed in details, e.g., in [54] the calculation of E depends
on the spatial coordinate system naturally adapted to the reference frame
@
Z = q 1 2m @t
, even if these coordinates produce asymptotically ‡at
(1 r )
metrics. Then, even if in one given chart we may obtain mI = m there are
others where mI 6= m!16
Moreover, note also that, as showed above, for a closed universe, Einstein’s theory implies on general grounds (once we accept that the ta describes the energy-momentum distribution of the gravitational …eld) that
mI = 0. This result, it is important to quote, does not contradict the so
called “positive mass theorems”of, e.g., references [392, 393, 451], because
that theorems refers to the total energy of an isolated system. A system
of that kind is supposed to be modelled by a Lorentzian spacetime having
a spacelike, asymptotically Euclidean hypersurface.17 However, we want
to emphasize here, that although the energy results positive, its value is
1 3 Note that if we suppose that the universe contains spinor …elds, then it must be
a spin manifold, i.e., it is parallezible according to Geroch’s theorem [146, 147], as we
already know from Chapter 5.
1 4 A metric is said to be asymptotically ‡at in given coordinates, if g
= n (1 +
O r k ), with k = 2 or k = 1 depending on the author. See, eg., [392, 393, 442].
1
2m
1 5 For a Scharzschild spacetime we have g = 1
dt dt
1 2m
dr dr
r
r
2
2
r (d
d + sin d' d').
1 6 This observation is true even if we use the so called ADM formalism [24] to be
presented in Chapter 11. To be more precise, let us recall that we have a well de…ned
ADM energy only if the fall o¤ rate of the metric is in the interval 1=2 < k < 1. For
details, see [263].
1 7 The proof also uses as hypothesis the so called energy dominance condition [163].
9.7 Is There Any Energy-Momentum Conservation Law in GRT?
337
not unique, since depends on the asymptotically ‡at coordinates chosen to
perform the calculations, as it is clear from the elementary example of the
Schwarzschild …eld commented above and detailed in [54].
In a book written in 1970, Davis [98] said:
“Today, some 50 years after the development of Einstein’s generally covariant …eld
theory it appears that no general agreement regarding the proper formulation of the
conservation laws has been reached”.
Well, we hope that the reader has been convinced that the fact is: there
are in general no conservation laws of energy-momentum in GRT. Moreover, all discourses (based on Einstein’s equivalence principle)18 concerning
the use of pseudo-energy momentum tensors as reasonable descriptions of
energy and momentum of gravitational …elds in Einstein’s theory are not
convincing.
And, at this point it is better to quote page 98 of Sachs&Wu [367]:
“As mentioned in section 3.8, conservation laws have a great predictive power. It is
a shame to lose the special relativistic total energy conservation law (Section 3.10.2) in
general relativity. Many of the attempts to resurrect it are quite interesting; many are
simply garbage.”
In GRT, we already said, every gravitational …eld is modelled (module
di¤eomorphisms, according to present wisdom) by a Lorentzian spacetime.
In that particular case, when this spacetime structure admits a timelike
Killing vector …eld, we may formulate a law of energy conservation for
the matter …elds. Also, if the Lorentzian spacetime admits three linearly
independent spacelike Killing vectors, we have a law of conservation of
momentum for the matter …elds.
This follows at once from the theory developed in the previous section.
Indeed, in the particular case of GRT, the Lagrangian density is not supm
posed to be explicitly dependent on the ! ab . Then, the term @L
@! ab = 0 in
Eq.(9.59) is null. Then writing T ( ) = T Eq.(9.59) becomes d?T ( ) = 0
or and Eq.(9.59) becomes writing T ( ) = T ,
T ( ) = 0:
(9.109)
The crucial fact to have in mind here is that a general Lorentzian spacetime,
does not admits such Killing vector …elds in general as it is the case ,e.g.,
of the popular Friedmann-Robertson-Walker expanding universes models.
At present, the authors know only one possibility of resurrecting a trustworthy conservation law for the energy-momentum of matter plus the gravitational …eld valid in all circumstances in a theory of the gravitational
…eld that resembles GRT (in the sense of keeping Einstein’s equation).19
1 8 Like, e.g., in [20, 296, 247] and many other textbooks. It is worth to quote here that,
at least, Anderson [20] explicitly said: “ In an interaction that involves the gravitational
…eld a system can loose energy without this energy being transmitted to the gravitational
…eld.”
1 9 On this issue, see also [356].
338
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
It consists in reinterpreting that theory as a …eld theory in ‡at Minkowski
spacetime. Theories using Minkowski spacetime have been proposed in the
past by, e.g., Feynman [133], Schwinger [389], Thirring [420] and Weinberg
[446, 447] among others and have been extensively studied by Logunov
and collaborators in a series of papers summarized in the monographs
[212, 213]. In the Chapter 11 we discuss the nature of the gravitational
…eld and give Cli¤ord bundle approach to the theory of the gravitational
…eld in Minkowski spacetime20 , following [341]. We also qualify a statement
in [99] that in the theory called teleparallel equivalent of GRT [227] there
is an energy-momentum conservation law.
Remark 562 As a …nal remark, we make the important observation that
even if a given Lorentziam spacetime modelling a gravitational …eld has
one timelike and three spacelike Killing vector …elds and so we can de…ned
four conserved quantities P a (see Eq.(9.106) we cannot in genral de…ne an
energy-momentum covector P (not a covetcotr …eld ) for the system as in
special relativistc …eld theories. For a thoughtful discussion of this issue see
[359].
9.8 Is there any Angular Momentum Conservation
law in the GRT
If the f a g and the f!ba g are varied independently in the sum of the
Einstein-Hilbert Lagrangian plus the matter Lagrangian then, as it is easy
to verify we get the additional …eld equation
D?
ab
= Jab =
? 2J ab
(9.110)
From this equation we get immediately
d?
a
b
= Jab
! cb ^ ?
a
c
+?
c
b
^ ! ac
(9.111)
and one is tempted to de…ne Sab = ( ! cb ^ ? ac + ? cb ^ ! ac ) as 2-form
densities of spin angular momentum of the gravitational …eld and to the
de…ne the orbital angular momentum of the system as
Z
ab
L :=
? ab :
(9.112)
S2
2 0 Another presentation the theory of the gravitational …eld in Minkowski spacetime
employing Cli¤ord algebra techniques has been given in [204]. However, that work, which
contains many interesting ideas, unfortunately contains also some equivocated statements that make (in our opinion) the theory, as originally presented by those authors
invalid. This has been discussed with details in [128].
9.9 Some Non Trivial Exercises
339
This de…nition, of course, has the same problems as the de…nition of
energy in the GRT because the 2-form …elds Sab are gauge dependent.
Moreover, the scalars Lab cannot be considered as components of any tensor
…eld in the spacetime manifold.
9.9 Some Non Trivial Exercises
Exercise 563 (a) Show that the energy-momentum densities ?Ta of the
Maxwell …eld are given by
1
?Ta = ? F
2
(b) Show also that Ta
b
= Tb
~
a F:
(9.113)
a.
Solution: (a) The Maxwell Lagrangian, here considered as the matter …eld
coupled to the background gravitational …eld must be taken (due to our
convention in the writing of Einstein equations and the de…nition of ?Ta )
as
1
Lm =
F ^ ?F;
(9.114)
2
V2
where F = 12 Fab a ^ b = 21 Fab ab 2 sec
T M ,! sec C`(M; g) is the
electromagnetic …eld. Now, recall that
?
ab
c
=
^ [ cy ?
ab
]
andVthat in general
and ? do not commute. Indeed, for any Ap 2
p
sec
T M ,! sec C`(M; g) we have
[ ; ?]Ap =
=
? Ap
a
? Ap
(9.115)
^ ( a y ? Ap )
?[
a
^ ( a yAp )] :
Multiplying both members of Eq.(9.115) with Ap = F on the right by F ^
we get
F^
a
?F =F ^? F +F ^f
Next we sum
^ ( ay ? F )
a
?[
^ ( a yF )]g:
F ^ ?F to both members of the above equation obtaining
(F ^ ?F ) = 2 F ^ ?F +
a
1
F ^ ?F
2
1
2
^ [F ^ ( a y ? F )
( a yF ) ^ ?F ]:
or,
=
F ^ ?F
a
^ [F ^ ( a y ? F )
( a yF ) ^ ?F ]:
340
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
It then follows from Eq.(9.33) that if a =
phism generated by the vector …eld that
?Ta =
1
@Lm
= [F ^ ( a y ? F )
a
@
2
$
a
for some di¤eomor-
( a yF ) ^ ?F ] :
Now,
( a yF ) ^ ?F =
? [( a yF )yF ] =
[( a yF )yF ]
g
and also using Eq.(2.60) we can write
( a yF ) ^ ?F =
a (F
F)
g
F ^ ( a y ? F ):
Using these results, we have
1
[F ^ ( a y ? F )
2
( a yF ) ^ ?F ]
1
f a (F F ) g ( a yF ) ^ ?F ( a yF ) ^ ?F g
2
1
= f a (F F ) g 2( a yF ) ^ ?F g
2
1
= f a (F F ) g + 2[( a yF )yF ] g g
2
1
1
=?
= ? (F a F~ );
a (F F ) + ( a yF )yF
2
2
=
where in writing the last line we used the identity given by Eq.(8.81).
(b) To prove that Ta b = Tb a we write:
Ta
=
b
=
1
hF
2
h(F x a )F
a F b i0
b i0
h(F x a )F
=
1
h( a F F
2
1
+ h a (F F )
2
b i0
=
b i0
1
+ h( a yF +
2
h(F x a )(F x
b)
^ F) F
+ (F x a )(F ^
1
h a (F ^ F ) b i0
2
1
= h(F x a )(F x b )i0 + h(F F )( a b )i0
2
1
= (F x b ) (F x a ) + (F F )( b a ) = Tb
2
b
a
b i0
b )i0
i0
a:
Note moreover that
Tab = Ta
a well known result.
b
=
cl
1
Fac Fbl + Fcd F cd
4
ab ;
(9.116)
9.9 Some Non Trivial Exercises
341
Exercise 564 Show that the connection 1-forms can be written as
!ab =
1
[ a yd
2
b yd a
b
or
!ab =
b yd a
a yd b
1
+ ( a y(
2
( a y(
b yd c )
b y(
^ d c )]
c
c
]
(9.117)
(9.118)
Solution: We prove Eq.(9.117). From Cartan’s …rst structure equation and
Eq.(2.60) we have
a yd b
=
=
c
c
a y(! b ^ c ) = ( a y! b )
( a y! cb ) c !ab :
^
! cb ^ ( a y c )
c
Moreover recalling Eq.(2.64) we have
a y( b yd c )
=(
a
^
b )yd c
^
b)
(! dc ^
d)
by d
d
a y! c )( b y d )
a y!bc
=
a
ay d
d
a y! c
d
b y! c
= det
=(
=(
( ay
d
d )( b y! c )
b y!ac :
Then
a yd b
= ( a y! cb )
c
!ab
(
b yd a
c
b y! a ) c
=
( a y(
+ !ba
b yd c )
c
( a y!bc )
c
+(
b y!ac )
c
2!ab :
and using the above results Eq.(9.117) is proved.
Exercise 565 Show that the Einstein Hilbert Lagrangian LEH := 21 R g
can be written as
1 a
1
1 a
d ^?d a +
^? a + (d a ^ a )^? d b ^ b
LEH = d( a ^?d a )
2
2
4
(9.119)
Solution: We already know from Eq.(9.79) that
1
1
R g = Rab ^ ?( a ^ b ):
2
2
We use now Cartan’s second structure equation to write
1
1
d!ab ^ ?( a ^ b ) + (!ac ^ ! cb ) ^ ?( a ^ b )
2
2
1
1
1
= d[!ab ^ ?( a ^ b )] + !ab ^ d ? ( a ^ b ) + !ac ^ ! cb ^ ?(
2
2
2
1
1
= d[!ab ^ ?( a ^ b )]
!ab ^ ! ac ^ ?( c ^ b ):
2
2
LEH =
a
^
b
)
342
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
Now, we have
!ab ^ ?(
a
b
^
? [!ab y(
)=
a
b
^
)] = 2 ? [(!ab y
b
) a]
Moreover, from Cartan’s …rst structure equation and Eq.(2.60) we have
immediately that
a
^ ?d a = ? [(!ab y b )] a ;
from where it follows that
1
d[!ab ^ ?(
2
a
^
b
b
)=
)] =
a
d[
^ ?d a ]:
(9.120)
Also,
=
?
!ab ^ ! ac ^ ?(
(!ab y[! ac y( c
^
b
c
^
? [(!ab ^ ! ac )y(
? [(!ab y
)] =
c
c
)(! ac y b )
^
b
)]
(!ab y
b
)(! ac y c )]:
But,
(!ab y c )(! ac y
b
) = !ab y[(! ac y c )
= !ab y[(
b
]
y(! ac ^ b ) + ! ab )
c
)y(! ac ^ b ) + !ab y! ab
c
= (!ab ^
and then
!ab ^ ! ac ^ ?(
=
(
b
c
!ab )(
b
^
c
)
^ ?! ac ) + (!ab ^
b
) ^ ?(! ac ^
Hence recalling that d a = ! ac ^ c and d ?
? 1 d ? a = !ab y b , we have
a =
(
b
!ab )(
c
^ ?! ac ) =
=
a
a
^d?
^??
1
a
c
) + !ab ^ ?! ab :
! ac ^ ?
=
c
and that
a
a
d?
=
a
^?
a
and
!ab ^ ! ac ^ ?(
c
^
b
)=
a
^?
a
d
a
^ ?d
a
+ !ab ^ ?! ab
Now, using Eq.(9.118) we can write
!ab ^ ?! ab = !ab ^ a ^ ?d b !ab ^ b ^ ?d a
1
+ !ab ^ ?[ a y( b y( c ^ d c )]
2
1
= d b ^ ?d b + d a ^ ?d a
d a ^ a ^ ?(d c ^ c )
2
9.9 Some Non Trivial Exercises
343
and get
!ab ^ ! ac ^ ?(
c
^
b
)=
a
+ 2d
a
a
^?
^ ?d
d a ^ ?d a
1
d a ^ a ? (d
2
a
c
^
c ):
(9.121)
Using Eqs.(9.120) and (9.121) we …nally have
LEH =
d( a ^?d a )
1 a
1
d ^?d a +
2
2
a
^?
1
(d
4
a+
a
^
a )^?
d
b
^
b
and Eq.(9.119) is proved.
@L
@L
Exercise 566 Find the algebraic derivatives @ dg and @d gd of EinsteinHilbert Lagrangian density LEH := 21 R g which can be written as
LEH =
+
1
2
a
^?
a
+
1
(d
4
a
a
d(
^
a)
1
d
2
^ ?d a )
b
^? d
^
a
^ ?d
=
b
a
a
d(
^ ?d a ) + Lg :
necessary to obtain Eq.(9.84).
Solution: We …rst show that Lg can be written as
Lg =
1
(d
2
a
^
b)
^? d
b
^
+
a
1
(d
4
a
^
a)
^? d
b
^
(9.122)
b
Indeed, using the identities in Eq.(2.77) we can write:
(d
a
^
b
) ^ ?(d
b
^
a)
=d
a
=d
a
=d
a
=d
a
=d
a
=d
a
=d
a
^[
b
^ ?[(
^ ?d
^ ?(d
b
+d
a
a
+ [(
b
a
+(
b
yd
b)
a
(
b
yd
b)
b
a
^ ?d
^ ?d
^ ?d
^ ?d
yd
b)
^
^
a
g
+ d a]
yd
b)
^?
yd
b)
^
a]
b
^ ?[
a
a
a )]
g
^(
a
^
a )]
^ ?d
a
^ ?d a )
^ ?( a yd a )
a:
(9.123)
from where Eq.(9.122) follows.
Now, we write
1
d a ^ [ b ^ ?(d
g
2
1
:= d a ^ ?S a :
2
Lg =
b
^
a
)
1
2
a
^? d
b
^
b
]
(9.124)
:
344
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
Then we have from Eq.(2.60)
k y(Lg )
1
a
k y(d a ^ ?S )
2
1
( k yd a ) ^ ?S a
2
=
=
1
d
2
^ ( ky ? S a)
a
(9.125)
and subtracting ( k yd a ) ^ ?S a on both sides of the last equation we get
k y(Lg )
1
[( k yd a ) ^ ?S a
2
( k yd a ) ^ ?S a =
1
d
2
a ^ ( ky ? S
a
)]: (9.126)
Now, we recall from the properties of the algebraic derivatives and of contraction operators (see Exercise 567) and taking also into account Eq.(9.10)
that for Lg it holds
k y(Lg )
@Lg
@d a
@Lg
dLg
=
+
^
d k
@ k
@ k
@d a
=
(9.127)
and we have immediately that
@d a @Lg
@Lg
^
= ( k yd a )^
= ( k yd a )^?S a : (9.128)
@ k @d a
@d a
@Lg
= ?tk : ,
@ k
Since
Lg =
k
^
Lg
+ d
k
k
^
@Lg
@d k
we …nally get
?tk =
k y(Lg )
( k yd a ) ^ ?
@Lg
=
@d a
k yLg
( k yd a ) ^ ?S a :
and we recall that according to Eq.(9.124)
a
?Sk =
^ ?(d
a
g
^
1
2
k)
k
^ ? (d
a
^
a) ;
(9.129)
which using the identities in Exercise 39 may be written in a suggestive
form (to be used in Chapter 11) as
?Sk =
?d
( ky ?
k
a
) ^ ?d ?
a
+
1
2
k
^ ? (d
a
^
a) :
(9.130)
Finally we recall that
Lg =
k
=
k
^
^
Lg
+d
k
dLg
+d
d k
k
^
@Lg
@d k
@Lg
@d k
+ d(
(9.131)
k
^
@Lg
):
@d k
9.9 Some Non Trivial Exercises
and for variations that are null on the boundary
Z
dLg
@Lg
Lg =
+d
k^
d k
@d k
345
(9.132)
and taking into account Eq.(9.96) we get immediately
@Lg
@d k
dLg
+d
d k
=
Gk :
Exercise 567 Verify that
d
(
d k
Since d a = 21 camn
and thus
d
(
d k
m
n
^
k y( a
^ d a) =
a
@
@
we have
=
a
k
ak
^ d a)
(9.133)
@d a
@ k
and
= cakn ^
^ d a ) @d a @( b ^ d
+
^
@ k
@ k
@d a
= d k + cakn n ^ a :
^ d a) =
a
@(
b
a
n
= cakn
)
On the other hand we have from21 Eq.(2.59)
k y( a
=d
k
a
^ d a) = ( ky a) ^ d
1 a
c
2 mn
ky
^
m
a
n
^
^ ( k yd a )
a
=d
k
+ cakn
n
^
a:
and Eq.(9.133) is proved.
Exercise 568 Prove that Eq.(9.129) and Eq.(9.130) are equivalent.
Solution: It is only necessary to verify that
a
Since ?d ?
a
^ ?(d
=
a
a
a
^
k)
=
^ ?(d
a
= ?d
k
+ ( ky ?
l
la
(where Dea
^
k)
= ?d
k
b
+
a
) ^ ?d ?
b c
ac
:=
a
( ky ?
a
a
)we have that
)
and
a
?(
k
k)
=
a
^
l
la
)=
?(
k
^
a
)
On the other hand
a
^ ?(d
= ?[d
a
k
^
a
= ?d
2 1 Recall
^ ?(
k
^ d a ) = ?[ a y(
k ^ ( yd a )] = ?d
a
k
that y is an antiderivation.
k
?
a
k y ? ( yd a )
k
^ d a)
a
k ^ ( yd a )
n
346
9. Conservation Laws on Riemann-Cartan and Lorentzian Spacetimes
So, it remains to prove that
ky
? ( a yd a ) =
But as easily veri…ed
a
yd
a
a
( ky ?
=
l
lc k y
?
c
a
)=
a
k
c
kc
and thus
=
l
lc
?(
k
^
?(
c
k
^
a
):
)
and the equivalence of Eq.(9.129) and Eq.(9.130) is proven.
Exercise 569 (a) Let ( a ; ma ) and ( a ; ma ) be two particles living in
Minkowski spacetime. Let pa = ma g( a ; ) 2 sec T a M and pb = mb g( b ; ) 2
sec T b M the momentum covectors of the particles. How you would de…ne
the total momentum of the two particles. If this object exists, where is the
space where it lives? (b) May you …nd a way to de…ne the total momentum
covector of the two particles if they live in a general Lorentzian manifold.?
(c) Has this question something to do which the absence of conservation
laws of energy-momentum in GRT?
This is page 347
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10
The DHE on a RCST and the
Meaning of Active Local Lorentz
Invariance
10.1 Formulation of the DHE on a RCST
Let (M; g; r; g ; ") be a general Riemann-Cartan spacetime. In this section
we investigate how to obtain a generalization in (M; g; r; g ; ") of the DiracHestenes equation (see Eq.(7.93)) for a representative
2 sec C`(M; g) of
a Dirac-Hestenes spinor …eld 2 sec C`lSpine1;3 (M; g): In order to do that we
…rst introduce a chart ('; U ) from the maximal atlas of M , with coordinates
fx g. The associate coordinate basis of T U is denoted by fe = @x@ = @ g
and we denoted by f = dx g its dual basis. Moreover, we suppose that
V1
the
2 sec
T M ,! sec C`(M; g) . Also, let fea g 2 sec PSOe1;3 (M ) an
orthonormal frame and f a g 2 sec PSOe1;3 (M ) the dual coframe. Note that,
V1
for each a = 0; 1; 2; 3, a 2 sec
T M ,! sec C`(M; g). In what follows
we are going to work in a …xed spin coframe 2 sec PSpine1;3 (M ) such that
s( ) = f a g and so, in order to simplify the notation we write simply
instead of
.
Recall now that the Lagrangian density for the free Dirac-Hestenes spinor
in Minkowski spacetime (M; ; D; ; ") (see Eq.(8.50)) can be written with
coordinates fx g in Einstein-Lorentz-Poincaré gauge as
L( ; @j ) = L(; ; @j )dx0 ^ dx1 ^ dx2 ^ dx3
= (@j i 3 )
m
dx0 ^ dx1 ^ dx2 ^ dx3 ;
(10.1)
348
10. The DHE on a RCST and the Meaning of Active Local Lorentz Invariance
where @j = dx D @ . The usual prescription of minimal coupling between
@x
a given spinor …eld and the (generalized) gravitational …eld,
a
dx 7!
, D
7! re a ;
@
@x
suggests that we take the following Lagrangian for a representative
Dirac-Hestenes spinor …eld in a Riemann-Cartan spacetime,
of a
L( ; @ (s) ) = L( ; @ (s) ) )dx0 ^ dx1 ^ dx2 ^ dx3
h
ip
= (@ (s) 0 2 1 )
m
jdet gjdx0 ^ dx1 ^ dx2 ^ dx3 ;
(10.2)
where (with the notations of Chapter 7)
@ (s)
Then
R
=
a
=
a
r(s)
ea
=
a
gea
1
+ !ea
2
:
1
+ !ea
2
ha @
(10.3)
(10.4)
L( ; @ (s) ) = 0 gives the Euler-Lagrange equation
@ L
@
L = 0:
@@
(10.5)
Using the identities given by Eqs. (2.68) and (2.69) we can write the
0 2 1
term a ha @
as
gea
0 2 1
= gea
=
0 2 1
a
gea
a
0 2 1
(10.6)
and
a
@gea L =
We now must express @ @@
To do that we …rst write
0 2 1
(10.7)
L in terms of the Pfa¤ derivatives gea .
a
= ha dx :
(10.8)
Then, putting det g := det[g(@ ; @ )] since
p
jdet gj =
1
det[ha ]
h
1
(10.9)
we have
Then,
h
L( ; @ (s) ) = (@ (s)
@@
L = (@@
0 2 1
)
m
i
h
1
:
(10.10)
gea )(@dea L) = ha @gea L
(10.11)
10.1 Formulation of the DHE on a RCST
349
and Eq.(10.5) becomes
@ L
(@ ha )@gea L gea @gea L = 0:
(10.12)
Now, we can verify that
h
1
@ h = ha @ ha ;
(10.13)
and taking into account that
[ea ; eb ] = (ha eb (hc )
hb ea (hc ))ec = ccab ec
(10.14)
we can write
cbab + h
@ ha =
1
ea (h);
(10.15)
and Eq.(10.12) becomes
@ L
cbab + h
[@ea
1
ea (h)] @gea L = 0:
(10.16)
Now, we have taking into account Eq.(2.194)
a
@
0 2 1
!ea
= h a !ea
0 2 1
+ !ea
a
0 2 1
i ;
and then
a
h@ L =
0 2 1
gea
gea @dea L =
1
2
a
!ea
@gea L =
h
h
+
1
0 2 1
1
a
ea (h)(@gea L)
1
+ !ea
2
0 2 1
h
1 a
a
0 2 1
2m
;
;
gea
0 2 1
;
(10.17)
a b
c ab
m =0
and Eq.(10.16) becomes
a
r(s)
ea
0 2 1
1
4
a
!ea
0 2 1
+
1
4
a
0 2 1
!ea
+
1
2
(10.18)
or, recalling that the components of the torsion tensor in an orthonormal
basis is given by
T cab = ! cab ! cba ccab ;
(10.19)
and that, in particular1 ! bab =
a
r(s)
ea
=
a
r(s)
ea
1
+ !ea
4
a
=
a
r(s)
ea
=
a
r(s)
ea
1 No
1
!e a 0
4 a
1 a
0 2 1
!ea
4
1 b a
0 2 1
c
2 ab
1 a
0 2 1
y!ea
2
1
0 2 1
+ T bab a
2
0 2 1
sum in b.
bb
bb ! a
2 1
= 0, we have
1 b
c
2 ab
a
0 2 1
m
0 2 1
0 2 1
0 2 1
0 2 1
m
1 b
c
2 ab
m = 0:
0 2 1
m
(10.20)
350
10. The DHE on a RCST and the Meaning of Active Local Lorentz Invariance
Finally we write the Dirac-Hestenes equation in a general RiemannCartan spacetime as [344]
@ (s)
2 1
1
+ T
2
0 2 1
0
m
= 0;
(10.21)
where
T = T bab
a
:
(10.22)
is (sometimes) called the torsion covector. Note that in a Lorentzian manifold T = 0 and we come back to the Dirac-Hestenes equation given by
Eq.(7.93). We observe moreover that the matrix representation of Eq.(10.21)
coincides with an equation …rst proposed by Hehl and Datta [170].
We observe yet that if we tried to get the equation of motion of a DiracHestenes spinor …eld on a Riemann-Cartan spacetime directly from that
equation on Minkowski spacetime by using the principle of minimal coupling, we would miss the term 21 T 2 1 appearing in Eq.(10.21). This
would be very bad indeed, because in a complete theory where the f a g
and the f!ea g are dynamical …elds we can easily show that spinor …elds
generate torsion (details in [170]).
10.2 Meaning of Active Lorentz Invariance of the
Dirac-Hestenes Lagrangian
In the proposed gauge theories of the gravitational …eld, it is said that the
Lagrangians and the corresponding equations of motion of physical …elds
must be invariant under arbitrary active local Lorentz rotations. In this
section we brie‡y investigate how to mathematically implement such an
hypothesis and what is its meaning for the case of a Dirac-Hestenes spinor
…eld on a Riemann-Cartan spacetime. The Lagrangian we shall investigate
is the one given by Eq.(10.10), i.e.,
h
L( ; @ (s) ) = ( a r(s)
ea
0 2 1
)
m
i
h
1
:
(Dirac-Hestenes)
Observe that the Dirac-Hestenes Lagrangian has been written in a …xed
gauge individualized by a spin coframe 2 sec PSpine1;3 (M ) and we already
know after Exercise 494, that it is invariant under passive gauge transformations 7! U 1 (U U 1 = 1, U 2 sec Spine1;3 (M ) sec C`(M; g)), once
the ‘connection’2-form !V transforms as given in Eq.(7.50), i.e.,
1
1
!V 7! U !V U
2
2
1
+ (rV U )U
1
:
(10.23)
10.2 Meaning of Active Lorentz Invariance of the Dirac-Hestenes Lagrangian
351
Under an active rotation (gauge) transformation the …elds transform in
new …elds given by
m
0
7!
=U ;
0m
7!
m
=U
em 7! e0m = (
U
1
m n
;
n
=
1 n
)m en :
(10.24)
Now, according to the mathematical ideas behind gauge theories brie‡y
outlined in Appendix A.4, we must search for a new connection r0s such
that the Lagrangian results invariant. This will be the case if connections rs
and r0s are generalized G-connections2 (De…nition 691), i.e.,
0(s)
re0m (U ) = U re(s)
;
m
or
r0(s)
en (U
m
(s)
n U re m
)=
:
(10.25)
Also, taking into account the structure of a representative of a spinor covariant derivative in the Cli¤ord bundle (see Eq.(7.55)) we must have for
the Pfa¤ derivative
gen 7! ge0a = m
(10.26)
n den ;
and for the connection
!e0 n =
m
n
1
U !em U
2gem (U )U
1
;
or
!e0 0m = U !em U
1
2gem (U )U
1
:
(10.27)
Write
1 0k l
!
2 m
1
= ! kml
2
!e0 n =
k
^
l
!en
k
^
l
U = eF , F =
1 0k l
!
2 m
1
= ! kml
2
=
1 rs
F
2
rs
kl
2 sec C`M; g);
kl
2 sec C`M; g);
2 sec C`M; g):
(10.28)
Recall that
! rns =
ra
!anb
! rnk = ! rns
sb
= ! rnb
sb
;
sk :
(10.29)
Then, from Eqs.(10.27), (10.28) and (10.29) we get
2 Note that rs and r0s are connection in the spin-Cli¤ord bundle of Dirac-Hestenes
spinor …elds whereas r(s) and r0(s) the e¤ective covariant derivative operators acting
on the representatives of Dirac-Hestenes spinor …elds in the Cli¤ord bundle.
352
10. The DHE on a RCST and the Meaning of Active Local Lorentz Invariance
b p
q ! mb
! 0rnk =
r
p
m
k
sk
m
k em
(F rs ) :
(10.30)
Now, we recall that the components of the torsion tensors,
and 0 of
n
the connections r and r0 in the orthonormal basis fer
^ k g are given
by
T rnk = ! rnk
! rkn
crnk ;
0r
T 0r
nk = ! nk
! 0rkn
crnk ;
(10.31)
where [en ; ek ] = crnk er .
Let us suppose that we start with a torsion free connection r. This
means that crnk = ! rnk ! rkn . Then ,
T 0r
nk =
b
n
and we see that
r p
p c mb
m
k
0
crnk
em (F rs ) [
sk
m
n
sn
m
k ];
(10.32)
= 0 only for very particular gauge transformations.
We then arrive at the conclusion that to suppose the Dirac-Hestenes Lagrangian is invariant under active rotational gauge transformations imply
in an equivalence between torsion free and non torsion free connections. It
is always emphasized that in a theory where besides , also the tetrad …elds
a
and the connection 1-forms ! ab are dynamical variables, the torsion is
not zero, because its source is the spin of the
…eld. Well, this is true
in particular gauges, because as showed above it seems that it is always
possible to …nd gauges where the torsion is null. The reader is invited to
re‡ect on this result, taking also into account a result proved in Chapter
6 that says that distinct LLRF and LLRF 0 that meet at p 2 M are not
physically equivalent.
Exercise 570 Show that whereas Maxwell Lagrangian density is invariant
under local Lorentz rotations, Maxwell equations (in general ) are not [326].
This is page 353
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11
On the Nature of the Gravitational
Field
11.1 Introduction
As well known, in GRT, each gravitational …eld generated by a given
energy-momentum tensor T is represented by an equivalence class1 of
Lorentzian spacetimes [(M; D; g; g ; ")], where we recall once again that a
Lorentzian spacetime is a structure (M; D; g; g ; ") where M is a non compact (locally compact) 4-dimensional Hausdor¤ manifold, g is a Lorentzian
metric on M and D is its Levi-Civita connection. Moreover M is supposed
to be oriented by the volume form g and the symbol " means that the
spacetime is time orientable. From the geometrical objects in the structure
(M; D; g; g ; ") we can calculate the Riemann curvature tensor R of D
and a nontrivial GRT model is one in which R 6= 0. In that way textbooks
often say that in GRT spacetime is curved. Unfortunately many people
mislead the curvature of a connection D on M with the fact that M can
eventually be a bent surface in a (pseudo)Euclidean space with a su¢ cient
number of dimensions2 . This confusion according to our view leads to all
sort of wishful thinking because many forget that GRT does not …x the
0
1 (M; D; g;
0
0 0
0
g ; ") is said to be equivalent to (M ; D ; g ; g ; " ) if there exists a di¤eomorphism h :M ! M 0 such that M =0 hM D 0 = h D, g 0 = h g, g0 = h g ; "0 = h "
and T 0 = h T .
2 Recall that bending is characterized by the shape operator introduzed in Chapter
5. Recall moreover that, e.g., the shape operator for a punctured sphere viewed as
a submanifold embedded in 3-dimensional Euclidean space is non null, but its Nunes
connection has zero curvature (Section 4.9.8).
354
11. On the Nature of the Gravitational Field
topology of M that often must be put “by hand”when solving a problem,
and thus think that they can bend spacetime or even change its topology if
they have an appropriate kind of some exotic matter. Worse, the insistence
in supposing that the gravitational …eld is geometry lead the majority of
physicists to relegate the search for the real physical nature of the gravitational …eld as not important at all (see a nice discussion of this issue in
[205]). As discussed with details in Chapter 9 what most textbooks with a
few exceptions (see, e.g., the excellent book by Sachs and Wu [367]) forget
to say and give a proof to their readers is that in the standard formulation
of GRT there are no genuine conservation laws of energy-momentum and
angular momentum unless spacetime has some additional structure which
is not present in a general Lorentzian spacetime [276]. Some textbooks e.g.,
[247] even claim that energy-momentum conservation for matter plus the
gravitational …elds is forbidden due the equivalence principle 3 because the
energy-momentum of the gravitational …eld must be non localizable. Only
a few people tried to develop consistent theories where the gravitational
…eld (at least from a classical point of view) is simple another …eld, which
like the electromagnetic …eld lives in Minkowski spacetime (see a list of
references in [127]). A …eld of that nature will be called, in what follows, a
…eld in Faraday’s sense.
Here we want to recall that: (i) the representation of gravitational …elds
by Lorentzian spacetimes is not a necessary one, for indeed, there are some
geometrical structures di¤erent from (M; D; g; g ; ") that can equivalently
represent such a …eld; (ii) That any realistic gravitational …eld can also be
nicely represented as a …eld living in a …xed background spacetime.
The preferred one which seems to describe all realistic situations is, of
course, Minkowski spacetime4 (M ' R4 ; D; ; ; ").
Concerning the possible alternative geometrical models, the particular
case where the connection is teleparallel (i.e., it is metric compatible, has
null Riemann curvature tensor and non null torsion tensor) will be brie‡y
addressed below (for other possibilities see [277]). What we will show, is
that starting with a thoughtful representation
V1 of the gravitational …eld in
terms of gravitational potentials ga 2 sec T M ,! sec C`(M; g), a =
0; 1; 2; 3 and postulating a convenient Lagrangian density for the gravitational potentials which does not use any connection there is a posteriori
di¤erent ways of geometrically representing the gravitational …eld, such
3 We
recall here that most presentations of the equivalence principle are according
to our view devoid from mathematical and physical sense. See, e.g., [415, 349] and our
discussion in Section 6.7.
4 Of course, the true background spacetime may be eventually a more complicated
one, since that manifold must represent the global topological structure of the universe,
something that is not known at the time of this writing [438]. We do not study this
possibility here, but the results we are going to present can be easily generalized for
more general spacetime backgrounds.
11.1 Introduction
355
that the …eld equations in each representation result equivalent in a precise
mathematical sense to Einstein’s …eld equations. Explicitly we mean by this
statement the following: any realistic model of a gravitational …eld in GRT
where that …eld is represented by a Lorentzian spacetime (with non null
Riemann curvature tensor and null torsion tensor ) which is also parallelizable, i.e., admits four global linearly independent vector …elds) is equivalent
to a teleparallel spacetime (i.e., a spacetime structure equipped with a metrical compatible teleparallel connection, which has null Riemann curvature
tensor and non null torsion tensor)5 . The teleparallel possibility follows almost directly from the results in Section 11.1 and a recent claim that it
can give a mathematical representation to “Einstein most happy though”
is discussed in Section 11.2. Comments about possible conservation laws in
the teleparallel equivalent of GRT is discussed in Section 11.6.
Equipped with the powerful Cli¤ord bundle formalism developed in previous chapters we give in Section 11.3 a …eld theory for the gravitational
…eld in Minkowski spacetime (with …eld equations equivalent to Einstein
equation in a precise mathematical sense). In our theory we are able to
identify in Subsection 11.4.1 a legitimate energy momentum tensor for the
gravitational …eld expressible in a very short and elegant formula.
Besides this important result we think that another important feature of
this chapter is that our representation of the gravitational …eld by the global
1-form …elds potentials fga g living on a manifold M and coupled among
themselves and with the matter …elds in a speci…c way (see below) shows
that we can dispense with the concept of a connection and a corresponding
geometrical description for that …eld. The simplest case is when M is part
of Minkowski spacetime structure, in which case the gravitational …eld is
(like the electromagnetic …eld) a …eld in Faraday’s sense6 . In section 11.4 we
present the Hamiltonian formalism for our theory and discuss the relation
of one possible energy concept7 naturally appearing in it and its relation to
the concept of ADM energy. In Section 11.5 we discuss the role of a possible
graviton mass in the the formulation of the …eld theory of gravitation.
Despite our present opinion that gravitation is a plastic distortion of the
5 There
are hundreds of papers (as e.g., [99]) on the subject, but none (to the best of
our knowledge) develope the theory from the point of view presented here and originally
in [356].
6 In Chapter 15 (see also [357]) we even show that when a Lorentzian spacetime
structure (M; D; g; g ; ") representing a gravitational …eld in GRT possess a Killing
vector …eld K, then there are Maxwell like equations with well determined source term
satis…ed for F = dA with A = g(K; ) encoding Einstein equation and more, there is a
Navier-Stokes equation encoding the Maxwell (like) and Einstein equations.
7 This other possibility does not de…ne in general a legitimate energy-momentum
tensor for the gravitational …eld in GRT, but it de…nes a legitimate energy-momentum
tensor in our theory in which the gravitational …eld is interpreted as a …eld in the sense
of Faraday living in Minkowski spacetime.
356
11. On the Nature of the Gravitational Field
Lorentz vacuum [127] and thus is to be described by a …eld in the sense of
Faraday living in Minkowski spacetime in Section 11.7 (titled On Cli¤ ord
Little Hills) using a nice result proved in Chapter 5 (namely, that the
Ricci operator is the negative square of the shape operator) we show a way
to represent as “marble” the “wood” part of Einstein equation, i.e., we
show how the phenomenological energy-momentum tensor can be explicitly
represented by a geometrical property of a 4-dimensional Lorentzian brane
embedded as a submanifold in a pseudo-Euclidean space of large dimension.
In Section 11.8 we present our conclusions.
11.2 Representation of the Gravitational Field
Suppose that a 4-dimensional M manifold is parallelizable, thus admitting
a set of four global linearly independent vector ea 2 sec T M , a = 0; 1; 2; 3
…elds such fea g is a basis for T M and let fga g; ga 2 sec T M be the
corresponding dual basis (ga (eb ) = ba ). Suppose also that not all the ga
are closed, i.e., dga 6= 0, for a least some a = 0; 1; 2; 3. This will be necessary
for the possible interpretations we have in mind for our theory. The 4-form
…eld g0 ^ g1 ^ g2 ^ g3 de…nes a (positive) orientation for M .
Now, the fga g can be used to de…ne a Lorentzian “metric”…eld in M by
de…ning g 2 sec T20 M by g:= ab ga gb , with the matrix with entries ab
being the diagonal matrix (1; 1; 1; 1). Then, according to g the fea g
are orthonormal, i.e., ea eb :=g(ea ; eb ) = ab .
g
Since the e0 is a global time like vector …eld it follows that it de…nes a
time orientation in M which we denote by ". Then, the 4-tuple (M;g; g ; ")
is part of a structure de…ning a Lorentzian spacetime and can eventually
serve as a substructure to model a gravitational …eld in GRT.
For future use we also introduce g 2 sec T02 M by g := ab ea eb , and
we write ga gb := g(ga ; gb ) = ab .
g
Due to the hypothesis that dga 6= 0 the commutator of vector …elds ea ,
a = 0; 1; 2; 3 will in general satisfy [ea ; eb ] = ckab ek ;where the ckab , the
structure coe¢ cients of the basis fea g, and we. easily show that dga =
1 a
k
l
2 c kl g ^ g .
Next, we introduce two di¤erent metric compatible connections on M ,
namely D (the Levi-Civita connection of g) and a teleparallel connection
r. Metric compatibility means that for both connections it is Dg = 0,
rg = 0. Now, we put
D ea eb = ! cab ec ; D ea gb =
rea eb = 0;
rea gb = 0:
! bac gc ;
(11.1)
As we know, the objects ! cab are called the connection coe¢ cients of
the connection D in the fea g basis and the objects ! ab 2 sec T M de…ned
11.2 Representation of the Gravitational Field
357
by ! ab := ! akb gk are called the connection 1-forms in the fea g basis. The
connection coe¢ cients $ b
ac of r and the connection 1-forms of r in the
basis fea g are null according to the second line of Eq.(11.1) and thus the
basis fea g is called teleparallel and the connection r de…nes an absolute
parallelism on M . Of course, as it is well known the Riemann curvature
tensor of the Levi-Civita connection D of g, is in general non null in all
points of M , but the torsion tensor of D is zero in all points of M . On the
other hand the Riemann curvature tensor of r is null in all points of M ,
whereas the torsion tensor of r is non null in all points of M .
We recall from Chapter 4 that for a general connection, say D on M (not
necessarily metric compatible) the torsion and curvature operators and the
torsion and curvature tensors are respectively the mappings:
: sec T M
TM
(u; v; w) = Du Dv w
: sec T M
(u;v) = Du v
T M ! sec T M;
Dv Du w
D[u;v ]w;
T M ! sec T M;
Dv u
[u; v]:
(11.2)
It is usual to write [77] (u; v; w) = (u; v)w and ( ; u;v) = ( (u; v))
and R(w; ; u;v) = ( (u;v)w), for every u;v;w 2 sec T M and
2
V1
sec
T M . In particular we write T abc := (ga ; eb ;ec ) and Rbacd :=
a
R(gb ; ea ; ec ; ed ), and de…ne the Ricci tensor by Ricci := RV
gc
ac g
b
with Rac := R acb = Rca . We take as in previous chapters T M =
L 4 Vr
T M ,! C`(M; g).
r=0
Given that we introduced two di¤erent connections D and r de…ned
in the manifold M we can write two di¤ erent pairs of Cartan’s structure
equations. Those pairs describe respectively the geometry of the structures
(M; D; g; g ; ") and (M; r;g; g ; ") called respectively a Lorentzian spacetime and a teleparallel spacetime. In the case (M; D; g; g ; ") we write
a
:= dga + ! ab ^ gb = 0; Rab := d! ab + ! ac ^ ! cb ;
V2
where the a 2 sec T M ,! sec C`(M; g), a = 0; 1; 2; 3 and the Rab 2
V2
sec T M ,! sec C`(M; g), a; b = 0; 1; 2; 3 are respectively the torsion
and the curvature 2-forms of D with
1
1
a
(11.3)
= T abc gb ^ gc ; Rab = Rabcd gc ^ gd .
2
2
In the case of (M; r;g; g ; ") since $ ab = 0 we have
F a := dga + $ ab ^ gb = dga ;
$
Rab := d$ ab + $ ac ^ $ cb = 0;
(11.4)
V2
V2
where the F a 2 sec T M , a = 0; 1; 2; 3 and the
2 sec T M ,
a; b = 0; 1; 2; 3 are respectively the torsion and the curvature 2-forms of
r given by formulas analogous to the ones in Eq.(11.3).
$
Rab
358
11. On the Nature of the Gravitational Field
We next postulate that the fga g are the basic variables representing the
gravitational …eld, and moreover postulate that the fga g interacts with the
matter …elds through the following Lagrangian density
L = Lg + Lm ;
(11.5)
where Lm is the matter Lagrangian density and
Lg =
1 a
1
1
dg ^ ?dga + ga ^ ? ga + (dga ^ ga ) ^ ? dgb ^ gb ; (11.6)
g
gg
g
2
2g
4
The form of this Lagrangian is notable, the …rst term is Yang-Mills like,
the second one is a kind of gauge …xing term and the third term is an
auto-interaction term describing the interaction of the “vorticities” of the
potentials (or if you prefer, the interaction between Chern-Simons terms
dga ^ ga ). Before proceeding we observe that this Lagrangian is not invariant under arbitrary point dependent Lorentz rotations of the basic
~ (where for each
cotetrad …elds. In fact, if ga 7! g0a = ab gb = Rga R
"
a
x 2 M , b (x) 2 L+ , the homogeneous and orthochronous Lorentz group
and R(x) 2 Spin1;3 R1;3 ) we get that
L0g =
1
1
1 0a
dg ^ ?dg0a + g0a ^ ? g0a + (dg0a ^ g0a ) ^ ? dg0b ^ g0b ; (11.7)
g
gg
g
2
2g
4
di¤ers from Lg by an exact di¤erential. So, the …eld equations derived by
the variational principle results invariant under a change of gauge and we
can always choose a gauge such that ga = 0.
g
Now, a derivation of the …eld equations directly from Eq.(11.6) using
constrained variations of the ga (i.e., variations induced by point dependent
Lorentz rotations) that do not change the metric …eld g has been given in
Chapter 9. The result is:
d ? Sd
g
? td = ?Td ;
g
g
(11.8)
with
@Lg
1
= [(gd ydga ) ^ ?dga dga ^ (gd y ? dga )]
d
g
g
g g
@g
2
1
1
1
a
a
+ d(gd y ? g ) ^ ?d ? ga + (gd y ? g ) ^ ?d ? ga + dgd ^ ? (dga ^ ga )
g g
g g
g g
g g
g
2
2
2
1
1 a
dg ^ ga ^ gd y ? (dgc ^ gc )
gd y (dgc ^ gc ) ^ ? (dga ^ ga ) ;
g g
g
g
4
4
(11.9)
?td :=
g
?Sd :=
g
@Lg
=
@dgd
? dgd
g
1
(gd y ? ga ) ^ ?d ? ga + gd ^ ? (dga ^ ga ) : (11.10)
gg
g g
g
2
11.2 Representation of the Gravitational Field
and the8
?Td :=
g
@Lm
=
@gd
? Td
g
359
(11.11)
with the ?Td the energy-momentum 3-forms of the matter …elds.
g
Recalling that from Eq.(11.4) it is F a := dga , it is, of course, dF a = 0
and the …eld equations (Eq.(11.8)) can be written (recalling Eq.(11.10)) as
d ? Fd =
? Td
g
g
?hd ;
(11.12)
1
gd ^ ? (F a ^ ga ) :
g
2
(11.13)
?td
g
g
where
hd = d (gd y ? ga ) ^ ?d ? ga
g g
g
g
RecallingVthe de…nition of the Hodge coderivative operator acting on
r
sections of
T M we can write Eq.(11.12) as
g
with the td 2 sec
V1
Fd =
(T d + td );
(11.14)
T M given by
td := td + hd ;
(11.15)
which are legitimate energy-momentum9 1-form …elds for the gravitational
…eld. Note that the total energy-momentum tensor of matter plus the gravitational …eld is trivially conserved in our theory, i.e.,
(?T d + td ) = 0:
g g
(11.16)
Remark 571 Recalling Eq.(11.9) and Eq.(11.13) the formula for the td
in Eq.(11.15) cannot be, of course, the nice and short formula we promised
to present in the introduction. However, it is equivalent to the nice formula
as shown in Section 11.3.
Exercise 572 Show that in a 2-dimensional spacetime the Einstein-Hilbert
Lagrangian density is an exact di¤ erential (see, e.g.,[327]).
Recall the similarity of the equations satis…ed by the gravitational …eld
to Maxwell equations. Indeed, in electromagnetic
V1 theory on a Lorentzian
spacetime we have only one potential A 2 sec T M ,! sec C`(M; g) and
the …eld equations are
dF = 0;
8 We
g
F =
J;
(11.17)
suppose that Lm does not depend explicitly on the dga .
will become evident after we present below the nice formula for the td .
9 This
360
11. On the Nature of the Gravitational Field
V2
where F 2 sec T M ,! sec C`(M; g) is the electromagnetic …eld and J 2
V1
sec T M ,! sec C`(M; g) is the electric current. As well known the two
equations in Eq.(11.17) can be written (if you do not mind in introducing
the connection D in the game as a mathematical tool to simplify some
formulas) as a single equation using the Cli¤ord bundle formalism, namely
@F = J:
where we can write @ = d
g
(11.18)
= ga D ea , where @ is the Dirac operator
(acting on sections of C`(M; g)).
Now, if you feel uncomfortable in needing four distinct potentials g a
for describing the gravitational …eld you can put them together de…ning a
vector valued di¤erential form10
^1
g = ga ea 2 sec
T M T M ,! sec C`(M; g) T M
(11.19)
and in this case the gravitational …eld equations are
dF = 0;
g
F=
(T + t);
(11.20)
where F = F a ea ; T = T a ea ; t = ta ea . Again, if you do not mind
in introducing the connection D in the game) by considering the bundle
C`(M; g) T M we can write the two equations in Eq.(11.20) as a single
equation, i.e.,
@F = T + t:
(11.21)
At this point you may be asking: which is the relation of the theory just
presented with Einstein’s GRT? The answer is that recalling (See Exercise
564) that the connection 1-forms ! cd of D are given by
! cd =
1 d
g ydgc
g
2
gc ydgd + gc y(gd ydga )ga
g
g
g
(11.22)
we already showed in Chapter 9 that the Lagrangian density Lg becomes
Lg = d(ga ^ ?dga ) + LEH ;
(11.23)
1
Rcd ^ ?(gc ^ gd ) =
g
2
(11.24)
g
where
LEH =
1
?R
2g
(with Rcd given by Eq.(11.3)) is the Einstein-Hilbert Lagrangian density.
This, as we know from Chapter 9 permits (with some algebra) to show that
Eqs.(11.8) are indeed equivalent to the usual Einstein equation.
1 0 Recall
that g = ga
ea is the identity operator in
^1
T M.
11.3 Comment on Einstein Most Happy Though
361
Before ending this section we recall from Chapter 9 that from Eq.(11.8)
we can also de…ne for our theory a meaningful energy-momentum for the
gravitational plus matter …elds Indeed, using Stokes theorem for a ‘certain
3-dimensional volume’, say a ball B we immediately get
P a :=
1 R
1 R
? (T a + ta ) =
? S a:
8 Bg
8 @B g
(11.25)
11.3 Comment on Einstein Most Happy Though
The exercises presented above indicate that a particular geometrical interpretation for the gravitational …eld is no more than an option among many
ones. Indeed, it is not necessary to introduce any connection D or r on M
to have a perfectly well de…ned theory for the gravitational …eld whose …eld
equations are (in a precise mathematical sense) equivalent to the Einstein
…eld equation. Note that we have not given until now details on the global
topology of the world manifold M , except that since we admitted that M
carries four global (not all closed) 1-form …elds ga which de…nes the object
g, it follows as we know from Chapter 7 that (M; D; g; g ; ") is a spin manifold [146, 147], i.e., it admits spinor …elds. This, of course, is necessary if the
theory is to be useful in the real world since fundamental matter …elds are
spinor …elds. The most simple spin manifold is clearly Minkowski spacetime
which is represented by a structure (M = R4 ; D; ; ; ") where D is the
Levi-Civita connection of the Minkowski metric . In that case it is possible to interpret the gravitational …eld as a (1; 1)-extensor …eld h which is a
…eld in the Faraday sense living in (M; D; ; ; "). The …eld h (as we know
from Eq.(2.121)) is a kind of square of g which has been called in [127] the
plastic distortion …eld of the Lorentz vacuum. In that theory the potentials
ga = h( a ) where a = a dx , with fx g being global naturally adapted
coordinates (in Einstein-Lorentz-Poincaré gauge) to the inertial reference
frame I = @=@x0 according to the structure (M = R4 ; D; ; ; "), i.e.,
DI = 0. In [127] we give the dynamics and coupling of h to the matter
…elds.
We want also to comment that, as well known, in Einstein’s GRT one
can easily distinguish in any real physical laboratory, i.e., not one modelled
by a time like worldline (despite some claims on the contrary) [280] a true
gravitational …eld from an acceleration …eld of a given reference frame in
Minkowski spacetime. This is because in GRT the mark of a real gravitational …eld is the non null Riemann curvature tensor of D, and the Riemann curvature tensor of the Levi-Civita connection of D (present in the
de…nition of Minkowski spacetime) is null. However if we interpret a gravitational …eld as the torsion 2-forms on the structure (M; r;g; g ; ") viewed
as a kind of deformation of Minkowski spacetime then one can also interpret an acceleration …eld of an accelerated reference frame in Minkowski
362
11. On the Nature of the Gravitational Field
e
spacetime as generating an e¤ective teleparallel spacetime (M; r; ; ; ").
This can be done as follows. Let Z 2 sec T U , U
M with (Z; Z) = 1
an accelerated reference frame on Minkowski spacetime. This means as we
know from Chapter 5 that a = D Z Z 6= 0. Put e0 = Z and de…ne an
accelerated reference frame as non trivial if #0 = (e0 ; ) is not an exact
di¤erential. Next recall that in U M there always exist [77] three other orthonormal vector …elds ei , i = 1; 2; 3 such that fea g is an -orthonormal
basis for T U , i.e., = ab #a #b , where f#a g is the dual basis11 of fea g.
We then have, D ea eb = ! cab ec ; D ea #b = ! bac #c :
What remains in order to be possible to interpret an acceleration …eld
as a kind of ‘gravitational …eld’ is to introduce on M a -metric come
patible connection r such that the fea g is teleparallel according to it,
e
e
i.e., rea eb = 0; rea #b = 0. Indeed, with this connection the structure
e
(M ' R4 ; r; ; ; ") has null Riemann curvature tensor but a non null
torsion tensor, whose components are related with the components of the
acceleration a and with the other coe¢ cients ! cab of the connection D,
which describe the motion on Minkowski spacetime of a grid represented
by the orthonormal frame fea g. Schücking [394] thinks that such a description of the gravitational …eld makes Einstein most happy though, i.e., the
equivalence principle (understood as equivalence between acceleration and
gravitational …eld) a legitimate mathematical idea. However, a true gravitational …eld must satisfy (at least with good approximation) Eq.(11.12),
whereas there is no single reason for an acceleration …eld to satisfy that
equation.
11.4 Field Theory for the Gravitational Field in
Minkowski Spacetime
Since the structure of the Lagrangian density for the gravitational …eld
and the resulting …eld equations do not use any connection we can assume the gravitational …eld represented by the potentials ga is a …eld in
Faraday sense, i.e., it leaves in the Minkowski spacetime structure (M =
R4 ; D; ; ; "). All geometrical like objects like g; D; r introduced above
are then to be understood as no more than auxiliary mathematical devices
to present formulas and to suggest possible geometrical interpretations for
the gravitational …eld. With this advise in mind we show next that we can
give a very nice and compact formula for the energy-momentum tensor of
the gravitational …eld.
1 1 In
general we will also have that d#i 6= 0, i = 1; 2; 3.
11.4 Field Theory for the Gravitational Field in Minkowski Spacetime
363
11.4.1 Legitimate Energy-Momentum Tensor of the
Gravitational Field. The Nice Formula
Taking into account that F d = dgd = @ ^ gd we return to Eq.(11.14) and
write it as
@ 2 gd = T d + td ;
(11.26)
with td = td d gd . Next we recall from Chapter 4 that the operator @ 2
(the Hodge D’
VAlembertian) has two distinct decompositions, namely, for
each M 2 sec T M ,! sec C`(M; g) we have
@2M =
(d + d)M
g
g
=@^@ M +@ @ M
(11.27)
where @ ^ @ is the Ricci operator and @ @ is the covariant D’Alembertian
operator. We have
@ ^ @ gd = Rd ;
(11.28)
V
1
where the Rd = Rad gd 2 sec T M ,! sec C`(M; g) (with Rad the components of the Ricci tensor) are the Ricci 1-form …elds. Then we can write
Eq.(11.26) as
@ ^ @ gd + @ @ gd = T d + td ;
(11.29)
or
Rd + @ @ gd = T d + td :
(11.30)
Now, we recall that Einstein equation in components form is
Rda
1
2
a
dR
Tda
(11.31)
T d = T d:
(11.32)
=
from where it follows immediately that
Rd
Then
1 d
Rg =
2
1
Rd + @ @ gd = T d + Rgd + @ @ gd ;
2
(11.33)
and comparing Eq.(11.29) with Eq.(11.33) we get
1 d
Rg + @ @ gd ;
2
(11.34)
1 d
Rg + @ @ gd + d gd
2
(11.35)
td =
and
td =
364
11. On the Nature of the Gravitational Field
the nice formula promised and that clearly demonstrates that the objects
tda = ac dl tc ygl are components of a legitimate gravitational energyg
momentum tensor tensor …eld t = tda gd ga 2 sec T02 M . We observe
moreover that even in the Lorentz gauge ( gd = 0)
tda
tad = 2@ @ gd yga ;
(11.36)
g
i.e., the energy-momentum tensor of the gravitational …eld in not symmetric. As shown in [127] this is important in order to have a total angular
momentum conservation law for the system consisting of the gravitational
plus the matter …elds. At least observe that td = td when the potentials
are chosen in the Lorenz gauge.
11.5 Hamilton Formalism
If we de…ne as usual the canonical momenta associated to the potentials
fga g by pa = @Lg =@dga = ?Sa and suppose that this equation can be solved
g
for the dga as function of the pa we can introduce a Legendre transformation
with respect to the …elds dga by
L : (g ; p ) 7! L(g ; p ) = dg ^ p
Lg (g ; dg (p ))
(11.37)
We write in what follows Lg (g ; p ) := Lg (g ; dg (p )) and observe that
de…ning12
Lg (g ; p )
:= dp
g
@L
;
@g
Lg (g ; p )
:= dg
p
@L
@p
we can obtain (see details in [127])
g ^
Lg (g ; dg )
= g ^
g
Lg (g ; p )
g
+
Lg (g ; p )
p
^ p :
(11.38)
Exercise 573 Prove Eq.(11.38).
To de…ne the Hamiltonian form, we need something to act the role of
time for our manifold, and we choose this ‘time’ to be given by the ‡ow
of an arbitrary timelike vector …eld Z 2 sec T M such that g(Z;Z) = 1.
V1
Moreover, we de…ne Z = g(Z; ) 2 sec
T M ,! C`(M; g). With this
1 2 We use only constrained variations of the ga , which as already recalled in Section
11.1 do not change the metric …eld g.
11.5 Hamilton Formalism
365
choice, the variation is generated by the Lie derivative $Z . Using Cartan’s
‘magical formula’, we have
Lg = $Z Lg = d(ZyLg ) + ZydLg = d(ZyLg ):
(11.39)
and after some algebra we get
d(ZyLg ) = d($Z g ^ p ) + $Z g ^
Lg
+ $Z p ^
g
and also
d($Z g ^ p
ZyLg ) =
L
p
Lg
:
g
$Z g ^
(11.40)
(11.41)
Now, we de…ne the Hamiltonian 3-form by
H(g ; p ) := $Z g ^ p
ZyLg :
(11.42)
We immediately have taking into account Eq.(11.41) that, when the …eld
equations for the free gravitational …eld are satis…ed (i.e., when the EulerLagrange functional is null, Lg = g = 0) that
dH = 0:
(11.43)
Thus H is a conserved Noether current. We next write
H = Z H + dB:
(11.44)
We can show (details in [127]) that H =
Lg = g and B = Z a pa and
now we investigate the meaning of the boundary term13 B. Consider an
arbitrary spacelike hypersuface . Then, we de…ne
Z
Z
Z
H = (Z H + dB) = Z H +
B:
@
If we recall that H =
Lg = g we see that the …rst term in the above
equation is null when the …eld equations (for the free gravitational …eld)
are satis…ed and we are thus left with
Z
E=
B;
(11.45)
@
which is called the quasi local energy [416].
Now, if fe g is the dual basis of fg g we have g0 (ei ) = 0; i =1; 2; 3 and
if we take Z = e0 orthogonal to the hypersurface , such that for each
p 2 , T p is generated by fei g we get recalling that p = ?S that
g
E=
Z
@
? S0 ;
g
(11.46)
1 3 More details on possible choices of the boundary term for di¤erent physical situations
may be found in [239].
366
11. On the Nature of the Gravitational Field
which we recognize (a constant factor apart) as being the same conserved
quantity as the one de…ned by Eq.(11.25).
The relation of the energy de…ned by Eq.(11.46) with the energy concept de…ned in ADM formalism [24] can be seen as follows [444]. Instead of
choosing an arbitrary unit timelike vector …eld Z, start with a global
timeV1
like vector …eld n 2 sec T M such that n = g(n; ) = N 2 dt 2 sec
T M ,!
C`(M; g), with N : R I ! R, a positive function called the lapse function
of M . Then n ^ dn = 0 and according to Frobenius theorem, n induces
a foliation of M , i.e., topologically it is M = I t ; where t is a spacelike hypersurface
with normal given by n. Now, we can decompose any
Vp
A 2 sec
T M ,! C`(M; g) into a tangent component A to t and an
orthogonal component ? A to t by
A=A+
?
A,
(11.47)
where
?
A := ny(dt ^ A),
A = dt ^ A? ,
A? := nyA:
(11.48)
Introduce also the parallel component d of the di¤erential operator d by:
dA := ny(dt ^ dA)
(11.49)
from where it follows (taking into account Cartan’s magical formula) that
dA = dt ^ ($n A
dA? ) + dA:
(11.50)
Call
m :=
g+n
n = gi
gi ;
(where n = n=N ) the …rst fundamental form on t and next introduce
the Hodge dual operator associated to m, acting on the (horizontal forms)
forms A by
n
? A := ?( ^ A):
(11.51)
m
g N
At this point, we come back to the Lagrangian density Eq.(11.42) and,
proceeding like above, but now leaving na to be non null, we eventually
arrive at the following Hamiltonian density
i
H(g ; pi ) = $n gi ^ ? pi
Kg ;
(11.52)
gi = dt ^ (nygi ) = ni dt;
(11.53)
m
where
gi
and where Kg depends on (n; dn; gi ; dgi ; $n gi ). We can show (after some
i
tedious but straightforward algebra that H(g ; pi ) can be put into the form
H = ni Hi + dB 0 ;
(11.54)
11.6 Mass of the Graviton
with as before Hi =
Lg = gi =
B0 =
367
Kg = ni and
N gi ^ ? dgi
(11.55)
m
Then, on shell, i.e., when the …eld equations are satis…ed we get
Z
E0 =
N gi ^ ? dgi
@
m
t
(11.56)
which is exactly the ADM energy, as can be seen if we take into account
that taking @ t as a two-sphere at in…nity, we have (using coordinates in
the Einstein-Lorentz-Poincaré gauge) gi = hij dxj and hij , N ! 1. Then
gi ^ ? dgi = hij (
m
@hik
) ? gk
@xj m
@hij
@xk
and under the above conditions we have the ADM formula
Z
@hik
@hik
? gk :
E0 =
i
k
m
@x
@x
@ t
(11.57)
(11.58)
which, as is well known , is positive de…nite14 . If we choose n = g0 it may
happen that g0 ^ dg0 6= 0 and thus it does not determine a spacelike
hypersurface t . However all algebraic calculations above up to Eq.(11.55)
are valid (and of course, gk = gk ). So, if we take a spacelike hypersurface
such that at spatial in…nity the ei (gk (ei ) = ik ) are tangent to , and
e0 ! @=@t is orthogonal to , then we have E = E 0 since in this case
N gi ^ ? dgi ! gi ^ ? (g0 ^ ? dgi ) which as can be easily veri…ed (see
m
m
m
Eq.(11.10)) is the asymptotic value of ?S 0 (taking into account that at
g
spatial in…nity dg0 ! 0).
11.6 Mass of the Graviton
In the Lagrangian given by Eq.(11.6) the mass of the graviton is supposed
to be zero. A non null mass m requires an extra term in the Lagrangian.
As an example, consider the Lagrangian density
1
1
1
1 a
dg ^?dga + ga ^? ga + (dga ^ ga )^? dgb ^ gb + m2 ga ^?ga
g
g
g
g
g
g
2
2
4
2
(11.59)
With the extra term the equations for the gravitational …eld, for the S a
result in
d ? S a = ?T a + ?ta + m2 ? ga ;
(11.60)
L0g =
g
1 4 See
a nice proof in [444].
g
g
g
368
11. On the Nature of the Gravitational Field
from where we get
g
(T a + ta ) =
m2 ga
g
(11.61)
If we impose the gauge ga = 0, which is analogous to the Lorenz gauge
g
in electrodynamics, Eq.(11.61) becomes
g
(T a + ta ) = 0;
(11.62)
which is the same equation valid in the case m = 0!
There are other possibilities of having a non null graviton mass, as, e.g.,
in Logunov’s theory [212, 213], which we do not discuss here15 .
11.7 Comment on the Teleparallel Equivalent of
GR
We observe that some people [99] think to have …nd a valid way of formulating a genuine energy-momentum conservation law in the teleparallel
equivalent to general relativity. In that theory, as we already know (see
also [227]), spacetime is teleparallel (a.k.a. Weintzenböck [268]), i.e., has a
metric compatible connection with non zero torsion and with null curvature16 . However, the claim of [99] must be quali…ed. Indeed, we have two
important comments (a) and (b) concerning this issue.
(a) First, it must be clear that the structure of the teleparallel equivalent
of GRT as formulated, e.g., by [227] or [99] consists in nothing more than
a trivial introduction of: (i) a bilinear form (a deformed metric tensor)
g = ab ga gb and (ii) a teleparallel connection in a manifold M ' R4
( part of the structure de…ning a Minkowski spacetime). Indeed, taking
advantage of the the discussion of the previous sections, we can present
that theory with a cosmological constant term as follows. Start with L0g
(Eq.(11.59)) and write it (after some algebraic manipulations) as
"
#
1
1 a
0
b
Lg =
dg ^ ? dga ga ^ (gb ydgb ) + ?( ga ^ ?(dg ^ gb ))
g
2
2 g
1
+ m2 ga ^ ?ga
g
2
1 a
=
dg ^ ?((1) dga
g
2
2(2) dga
1
2
(3)
1
dga ) + m2 ga ^ ?ga ;
g
2
(11.63)
1 5 We only observe that Lagrangian density of Logunov’s theory when written in terms
of di¤erential forms is not a very elegant expression.
1 6 In fact, formulation of teleparallel equivalence of GRT is a subject with a old history.
See, e.g., [167].
11.7 Comment on the Teleparallel Equivalent of GR
369
where
dga =(1) dga +(2) dga +(3) dga ;
(1)
dga = dga (2) dga (3) dga ;
1
(2)
dga = gb ^ (gb ydgb );
g
3
1
(3)
dga =
? (gb ^ ?(dgb ^ gb ):
g
3g
(11.64)
Next introduce a teleparallel connection by declaring that the cobasis fga g
…xes the parallelism, i.e., we de…ne the torsion 2-forms by
a
:= dga ;
a
1
2
(11.65)
and L0g becomes
L0g =
1
2
a
^?
(1)
g
a
2(2)
(3)
1
+ m2 ga ^ ?ga ;
g
2
a
(11.66)
where (1) a =(1) dga , (2) a =(31) dga and (3) a =(31) dga , called tractor
(four components), axitor (four components) and tentor (sixteen components) are the irreducible components of the tensor torsion under the action
of SOe1;3 .
(b) Recalling the results of Chapter 9 we now show that even if the
metric of a given teleparallel spacetime has some Killing vector …elds there
are genuine conservation laws involving only the energy-momentum and
angular momentum tensors of matter only if some additional condition is
satis…ed. Indeed, in the teleparallel basis where rea eb = 0 and [em ; en ] =
camn ea we have that the torsion 2-forms satisfy
a
= dga =
1 a
1
c
gm ^ gn = T amn gm ^ gn :
2 mn
2
(11.67)
Then, recalling once again that $ (dga ) = d($ ga ) = d({ ab gb ) and Eq.(9.62)
we can use Eq.(9.65) (which express the condition $ = 0) to write
d({ ab gb ) = { ab dgb ;
(11.68)
d{ ab ^ gb = 0:
(11.69)
which implies
Then, Eq.(11.69) is satis…ed only if the torsion tensor of the teleparallel
spacetime satisfy the following di¤erential equation:
a
Tm
bd em ( ) + ed (
m
T abm )
eb (
m
T adm ) = 0:
(11.70)
Of course, Eq.(11.70) is in general not satis…ed for a vector …eld that is
simply a Killing vector of g. This means that in the teleparallel equivalent
370
11. On the Nature of the Gravitational Field
of GRT even if there are Killing vector …elds, this in general do not warrant
that there are conservation laws as in Eq.(9.59) involving only the energy
and angular momentum tensors of matter.
Next, we remark that from L0g we get as …eld equations (in an arbitrary
basis, not necessarily the teleparallel one) satis…ed by the gravitational …eld
the Eq.(11.60), i.e.,
d ? S a = ?T a + ? ta ;
(11.71)
g
g
g
with
?ta = ?ta + m2 ? ga
g
g
(11.72)
g
and S a and ta given in Eq.(9.84) where it must also be taken into account
that in the teleparallel equivalent of GRT and using the teleparallel basis
a
the Levi-Civita connection 1-forms ! ab there must be substituted by
b,
with
cd
=
=
1 d
g ydgc
g
2
1 d c
g y
g
2
gc ydgd + gc y(gd ydga ) ga
g
gc y
g
g
d
g
+ gc y(gd y
g
g
a)
ga ;
(11.73)
where ab = K abc gc , with K abc the components of the so called contorsion
tensor17 . We have,
?tc =
1
2
ab
^[
c
d
^ ?(ga ^ gb ^ gc ) +
b
d
^ ?(ga ^ gb ^ gc )]:
(11.74)
Under a change of gauge, ga 7! g0a = U ga U = ab gb (U 2 sec Spine1;3 (M ) ,!
C`(M; g), ab (x) 2 SOe1;3 , 8 x 2 M ), we have that a 7! 0a = ab b . It
follows that the tab , which are the components of the energy-momentum
1-forms ta = tab gb de…nes a tensor …eld.
We then conclude that for each gravitational …eld modelled by a particular teleparallel spacetime, if the cosmological term is null or not there
is a conservation law of energy-momentum for the coupled system of the
matter …eld and the gravitational …eld which is represented by that particular teleparallel spacetime. Although the existence of such a conservation
law in the teleparallel spacetime is a satisfactory fact with respect of the
usual formulation of the gravitational theory where gravitational …elds are
modelled by Lorentzian spacetimes and where genuine conservation laws
(in general) do not exist because in that theory the components of ta de…nes only a pseudo-tensor, we cannot forget observation (a): the teleparallel
equivalent of GRT as formulated, e.g., by [227] or [99] consists in nothing
more than a trivial introduction of: (i) a bilinear form (a deformed metric
tensor) g = ab ga gb and (ii) a teleparallel connection in the manifold
1 7 See,
Eq.(4.197) with Q
= 0.
11.8 On Cli¤ord’s Little Hills
371
M ' R4 of Minkowski spacetime structure. The crucial ingredient is still
the old and good Einstein-Hilbert Lagrangian density.
Finally we must remark that if we insist in working with a teleparallel
spacetime we lose in general the other six genuine angular momentum conservation laws which always hold in Minkowski spacetime. Indeed, we do
not obtain in general even the chart dependent angular momentum ‘conservation’ law of GRT. The reason is that if we write the equivalent of
Eq.(11.60) in a chart (U; ') with coordinates fx g for U
M we did not
get in general that dx ^ ?t = dx ^ ?t , which as well known is necessary
in order to have a chart dependent angular momentum conservation law
[422].
11.8 On Cli¤ord’s Little Hills
Even if we leave clear that there are at present time not a single indication
that the topology of our spacetime is di¤erent from R4 we cannot leave out
the possibility that its topology is more complicated, in particular that it
is modelled by a Lorentzian brane (See Chapter 5) living in a large dimensional manifold. One interesting aspect of this possibility is to transform
the “wood” part of Einstein equation in “marble”. Let us see how to do
this applying the notable formula
@ ^ @ (v) = R(v) =
S2 (v)
(see Eq.(5.9)) of brane theory to General Relativity. As we will see this
permits to give a mathematical formalization to Cli¤ord’s intuition18 presented in [83], namely that:
(1) That small portions of space are in fact of a nature analogous
to little hills on a surface which is on the average ‡at; namely,
that the ordinary laws of geometry are not valid in them.
(2) That this property of being curved or distorted is continually
being passed on from one portion of space to another after the
manner of a wave.
(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of
matter, whether ponderable or ethereal.
(4) That in the physical world nothing else takes place but this
variation, subject (possibly) to the law of continuity.
1 8 Taking into account, of course, that di¤erently from Cli¤ord’s idea, instead of a
space theory of matter, we must talk about a spacetime theory of matter.
372
11. On the Nature of the Gravitational Field
To proceed. let (M; g; D; g ; ") be a model of a gravitational …eld generb
ated by an energy momentum tensor T := Tba a
describing all matter
of the universe according to General Relativity theory. As well already
know Einstein equation can be written as
@^@
a
=
1
Ta + T
2
a
;
(11.75)
where T a := Tba b and T = Taa , with Tba . If we suppose that the structure
(M; g) is a submanifold of (M ' Rn ; g) for n large enough as discussed in
the beginning of Section 5.3 we can write Eq.(11.75) taking into account
Eq.(5.146) as
1 a
S2 ( a ) = T a
T :
(11.76)
2
Thus, in a region where there is no matter S2 ( a ) = 0, despite the fact that
S( a ) = S( a ) may be non null. So, a being living in the hyperspace Rn
and looking at our brane world will see the little hills (i.e., “matter”) are
special shapes in M , places where the S2 ( a ) 6= 0 which act as sources for
P(dyS( a )) since P(dyS( a )) = S2 ( a ).
Remark 574 To properly appreciate the above argument one must take
in mind that the shape extensor depends for its de…nition on the metric g
and the Levi-Civita D connection of g used in M . So, a di¤ erent choice of
metric in M will imply in Cli¤ ord’s little hills to be represented by di¤ erent
shape extensors. Despite this fact, it seems to us that shape is most appealing than the curvature biform R19 or the Ricci 1-form …elds Ra = @^@ a
as indicator of the presence of matter as distortions in the world brane M .
Indeed, inner observers living in M in general may not have enough skills
and technology to discover the topology of M and so cannot know if their
brane world is a bended surface in the hyperspace (i.e., M ) or even if a open
set U
M is a part of an hyperplane or not. Moreover, those inner observers that have learned a little bit of di¤ erential geometry know that they
cannot say that their manifold is curved based on the fact that the curvature
biform is non null, for they know that the curvature biform is a property
of the connection (parallelism rule) that they decide to use by convention
in M and not an intrinsic property of M . They know that if they choose
a di¤ erent connection it may happen that its curvature biform may be null
and their connection (not their manifold) may have torsion and even a non
null nonmetricity tensor20 So, with their knowledge of di¤ erential geometry they infer that little hills (as seems for beings living in M ) can only be
associated to the shape extensor if they use Levi-Civita connection of g in
M.
1 9 Recall
that R is in general non null even in vacuum.
about these possibilities are discussed in [127] where a theory of the gravitational …eld on a brane di¤eomorphic to R4 is discussed.
2 0 Details
11.9 A Maxwell Like Equation for a Brane World with a Killing Vector Field
373
11.9 A Maxwell Like Equation for a Brane World
with a Killing Vector Field
When (M; g) admits a Killing vector …eld21 A 2 sec T M then it follows
V1
[355] that A = 0, where A = g(A; ) 2 sec T M ,! sec C`(M; g). In
this case we can show that the Ricci operator applied to A is equal to the
covariant D’Alembertian operator applied to A, i.e.,
@^@ A=@ @ A
(11.77)
Now, recalling Eq.(5.7) that the square of the Dirac operator @ 2 can be
decomposed in two ways, i.e.,
@ ^ @ A + @ @ A = @2A =
d A
dA
(11.78)
we have writing F = dA and taking into account that A = 0 that Einstein
equation can be written as
F = 2S2 (A)
(11.79)
and since dF = ddA = 0 we can write Einstein equation as:
@F =
2S2 (A):
(11.80)
Eq.(11.80) shows that in a Lorentzian brane M of dim 4 which contains a
Killing vector …eld A, Einstein equation is encoded in an “electromagnetic
like …eld” F having as source a current J = 2S2 (A) 2 sec C`(M; g).
Exercise 575 Prove Eq.(11.77).
11.10 Conclusions
This is a good point to end this chapter. Our intention in this book was only
to present some critical aspects of GRT theory (mainly using the Cli¤ord
bundle formalism when convenient) and to discuss matters of principle, in
particular to let the reader aware of some very controversial issues concerning the orthodox interpretation of Einstein’s theory, as, e.g., the case
of the energy momentum ‘conservation’. We showed that this particular
problem can be solved if we interpret the gravitational …eld as a physical …eld living in Minkowski spacetime. The gravitational …eld, when the
graviton mass is zero, creates an e¤ective Lorentzian geometry where probe
2 1 See
more details in Chapter 15.
374
11. On the Nature of the Gravitational Field
particles and probe …elds move. In such theory there are no exotic topologies, black holes22 , worm holes, no possibility for time-machines23 , etc.,
which according to our opinion are pure science …ction objects. Eventually,
many will not like the viewpoint just presented24 , but we feel that many
will become interested in exploiting new ideas, which may be more close to
the way Nature operates. We hope that our study clari…es the real di¤erence between mathematical models and physical reality and leads people
to think about the real physical nature of the gravitational …eld (and also
of the electromagnetic …eld25 ). We brie‡y discussed also an Hamiltonian
formalism for our theory and the concept of energy de…ned by Eq.(11.25)
and the one given by the ADM formalism, which are shown to coincide.
2 2 On
this issue recall Section 6.9 keeping in mind that there are articles criticizing the
notion that black holes are predicitons of GRT due mainly to some mathematical misunderstandings as, e.g.[1, 73, 411] and/or physical grounds. When thinking on this issue
take also into account the ‘pasticcio’ concerning the black hole information ‘paradox’
(see, [166, 164]) and its possible resolutions with the suggested existence of a “complementarity principle” [413] or existence of …rewalls [16] as an indication that the foundations of GRT and its relation to other theories of Physics are not well understood as some
people would like us to think. Recently adding stu¤ to the “pasticcio” Hawking [165] is
claiming that “The absence of event horizons mean that there are no black holes –in the
sense of regimes from which light can’t escape to in…nity”. But this statement seems to be
already an old idea. More information at http://asymptotia.com/2014/01/30/hawkingan-old-idea/ and http://www.physics.ohio-state.edu/~mathur/
2 3 The possibility for time machines arises when closed timelike curves exist in a
Lorentzian manifold. Such exotic con…gurations, it is said, already appears in Gödel’s
universe model. However, a recent thoughtful analysis by Cooperstock and Tieu (which
we endorse) shows that the old claim is wrong. Authors like, e.g, Davies [97] (which
are proposing to build time machines even at home), Gott [151] and Novikov [279] are
invited to read [88] and …nd a error in the argument of those authors.
2 4 For those people in that class we o¤er Chapter 12.
2 5 As suggested, e.g., by the works of Laughlin [205] and Volikov [440]. Of course„ it
may be necessary to explore also other ideas, like e.g., existence of Lorentzian branes in
string theory or generalizations.
This is page 375
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12
On the Many Faces of Einstein
Equations
12.1 Introduction
Even if we leave clear in the previous chapters our opinion that the gravitational …eld must be interpreted as a physical …eld in Minkowski spacetime,
we decided to present here how the orthodox Einstein theory can be formulated in a way that resembles the gauge theories of particle physics,
in particular a gauge theory with gauge group Sl(2; C). This exercise will
reveal yet another face of Einstein’s equations, besides the ones already discussed in previous chapters1 . For our presentation we introduce mathematical objects called Cli¤ord valued di¤erential forms (cliforms) and a new
operator D called the fake exterior covariant di¤ erential 2 (FECD) acting
on them. Moreover, with our formalism we show that Einstein’s equations
can be put in a form that apparently resembles Maxwell equations written
in coordinates and using the covariant of a Lorentziann spacetime structure
1 Of course, this will not exhaust the possible faces of Einstein’s equations, for …nding
new faces depends mainly on authors mathematical knowledge and imagination. Also,
the reader is here advised that this chapter do not intend to be a presentation of the
many developments that goes under the name of gauge theories of gravitation. The
interested reader on this issue may consult, e.g., [158, 240]
2 The name is due to the fact that D does not always satisfy a Leibniz type rule when
applied to the ^ product of arbitrary cli¤orms.
376
12. On the Many Faces of Einstein Equations
(M; g; D; g ; ")3 and which is very di¤erent from Eq.(11.80) of Chapter 11
which really encodes Einstein equation in a Maxwell like equation when
(M; g) admits a Killing vector …eld4 .
12.2 Preliminaries
Let (M; g; r; g ; ") be a Riemann-Cartan spacetime. We already introduced the Cli¤ord bundle of di¤erential forms C`(M; g). To present from
where ‘Sl(2; C) Gauge Theories of Gravitation’come from it will be useful
to introduce Cli¤ord valued di¤erential forms. For presenting these objects
we need to introduce besides the Cli¤ord bundle of multiforms C`(M; g),
also the Cli¤ord bundle of multivector …elds that will be denoted in what
follows by C`(M; g)
[
C`(M; g) =
C`(Tx M; g):
(12.1)
x2M
This bundle has, of course, an analogous structure as the bundle C`(M; g)
of multiform …elds and all products, contractions, etc..., are de…ned in analogous way, and we shall use the same symbols for them, since this procedure
produces no confusion. If fea g 2 PSOe1;3 (M ) is an orthonormal basis for
V1
TM =
T M ,! C`(M; g) we have
ea eb + eb ea = 2
ab :
(12.2)
Moreover, we introduce fea g as the reciprocal basis of fea g, i.e., ea eb = ab .
A general section m 2 sec C`(M; g), called a (nonhomogeneous) multivector
…eld has, e.g., the expansion
m = s + vi ei +
1
1
bij ei ej + tijk ei ej ek + pe5 ;
2!
3!
where e5 = e0 e1 e2 e3 is the pseudoscalar generator and
^0
s; vi ; bij ; tijk ; p 2 sec
T M ,! sec C`(M; g):
(12.3)
(12.4)
To motivate the introduction of Cli¤ord valued di¤erential forms we shall
need to recall some results from the general theory of connections (see
Appendix) adapted for the case of the spacetime structure (M; g; r; g ; ").
In the Appendix we learned that a connection (a gauge potential) is a
1-form in the cotangent space of a principal bundle, with values in the Lie
3 Such form of Einstein’s equations leaded some people equivocally [368, 369] to think
that they achieved an uni…ed theory of the gravitational and electromagnetic …eld. A
discussion of that issue may be found in [282, 352].
4 See also Chapter 15.
12.2 Preliminaries
377
algebra of a gauge group. For a gauge theory of the gravitational …eld it
seems natural (at least at …rst sight5 ) to take as gauge potential the linear
connection6
N
! 2 sec T PSOe1;3 (M )
sl(2; C);
(12.5)
which determines exterior covariant derivative operators acting on sections
of associated vector bundles to the principal bundle PSOe1;3 (M ). However,
N
a moment of re‡ection shows that ! cannot be the unique ingredient of our
theory, since, if our objective is to end with a geometrical spacetime structure (M; g; r; g ; ") modeling a gravitational …eld we needs to be able to
reproduce the well known results obtained with the usual covariant derivN
ative of tensor …elds in the base manifold. For that we need besides ! also
the soldering form
N
2 sec T PSOe (M )
1;3
Let be U
M and let & : U ! &(U )
N
N
R1;3 :
(12.6)
PSOe (M ). We are here interested
1;3
once we give a local trivialization of the
in the pullbacks & ! and &
respective bundles. As it is well known [77, 195], in local coordinates fx g
covering U and with fe = @ ,g a basis for T U , &
the tensor …eld7
=e
dx
e dx 2 sec T M
N
^1
uniquely determines
T M:
(12.7)
Remark 576 In Eq.(12.7) and other formulas involving Cli¤ ord valued
di¤ erential forms we will omit the symbol when no confusion arises.
Now, if we give: (i) the Cli¤ord algebra structure to the tangent bundle,
thus generating the Cli¤ord bundle C`(M; g) and; (ii) recall moreover (see
Section 3.3.4) that for each x 2 U
M the bivectors of C`(Tx M; gx )
generate under the product de…ned by the commutator, the Lie algebra
[217] spine1;3 ' sl(2; C), we are naturally lead to de…ne the representatives
V
N
in C`(M; g)
T M for and for the pullback ! = & ! of the connection
5 We are not going to discuss this issue here. Such a de…ciency may be supplied by a
careful and lucid analysis of the problem of formulating GRT as a possible gauge theory
with gauge groups Sl(2; C) or T(4) as done by Wallner [443].
N
6 In words, ! is a 1-form in the cotangent space of the bundle of orthonormal frames
with values in the Lie algebra soe1;3 ' sl(2; C) of the group SO e1;3 .
7 Note that this tensor is the identity tensor acting on the space of vector …elds on
U
M . We denoted the identity tensor by g in Chapter 11.
378
12. On the Many Faces of Einstein Equations
N
! in a given gauge.
= e dx = ea
a
2 sec
1
! = ! ba c eb ec a
2
1 bc
= (! a eb ^ ec )
2
^1
a
TM
2 sec
^1
^2
T M ,! C`(M; g)
TM
^1
^1
T M;
T M ,! C`(M; g)
^1
T M:
(12.8)
From Appendix A.4 we recall that whereas is a true tensor, ! is not
a true tensor, since its ‘components’ do not have the appropriate tensor
transformation properties. Indeed, the ! ba c are the ‘components’ of the
connection relative to the basis fea g. They are de…ned by
rea eb =
! bac ec ;
!abc =
!cba =
d
ad ! bc ;
(12.9)
where rea is a metric compatible covariant derivative operator de…ned on
the tensor bundle and that acts naturally on C`(M; g) as we already learned
in previous chapters. Objects like and ! will be called Cli¤ord valued
di¤erential forms (or Cli¤ord valued forms, or yet cliforms, for short), and
below we give a detailed account of the algebra and calculus of that objects. Let us now recall some additional concepts of the theory of linear
connections in the form appropriated for our enterprise.
12.2.1 Exterior Covariant Di¤erential
To achieve our objectives of presenting a ‘gauge like’ formulation of the
theory of the gravitational …eld we need to …nd an operator which acts
naturally on sections of cliforms and which mimics the action of the pullback of the exterior covariant derivative operator acting on sections of
a vector bundle associated with the principal bundle PSOe1;3 (M ), once a
linear metric compatible connection is given. This operator is introduced
below and called fake exterior di¤erential operator (FECD). It is used in
the calculations of curvature bivectors, Bianchi identities, etc. With the
FECD D and its associated operator Der we can formulate Einstein’s theory in such a way that the resulting equations looks like the equations
for a gauge theory with Sl(2; C) as the gauge group. Before introducing
D we …rst recall the concept of the exterior covariant derivative (di¤erential) acting on arbitrary sections of a vector bundle E(M ) associated with
PSOe1;3 (M ) (having as typical …ber a l-dimensional real vector space) and
on EndE(M ) = E(M ) E (M ), the bundle of endomorphisms of E(M )
introduced in Appendix A.5 and next the concept of absolute di¤erential
Vl
acting on sections of the tensor bundle, for the particular case of
TM.
In what follows we use a simpli…ed notation in relation to the general one
in Appendix A.5
12.2 Preliminaries
379
Now, from Appendix A.5.4 we know that the exterior covariant di¤erential operator dE acting on sections of E (M ) and EndE(M ) is the mapping
^1
dE : sec E (M ) ! sec E (M )
T M;
(12.10)
such that for any di¤erentiable
function f : M ! R, AV2 sec E (M ) and
Vp
q
any F 2 sec EndE(M )
T M ); G 2 sec EndE(M )
T M we have8 :
dE (f A) = df
dE (F
dE (F
^
^
A + f dE A;
A) = dE F
G) = dE F
^
^
A + ( 1)p F
^
G + ( 1)p F
^
dE A;
dE G:
(p)
(12.11)
(q)
In Eq.(12.11), writing F = F a fa , G = Gb gb where F a , Gb 2
Vp
Vq
(p)
(q)
sec EndE(M ), fa 2 sec T M and gb 2 sec T M we have
F
F
^
^
A = Fa
fa(p)
G = Fa
fa(p)
^
^
A;
Gb
(q)
gb ;
(12.12)
with the product ^ given through De…nition 693 in Appendix A.5.4. In
order to simplify even more the notation we eventually use when there is
no possibility of confusion, the simpli…ed (sloppy) notation
(F a A)
Fa
fa(p)
^
Gb
fa(p)
(q)
gb
(F a A) fa(p) ;
(q)
= F a Gb fa(p) ^ gb ;
(12.13)
where F a A 2 sec E (M ) and F a Gb means the composition of the respective
endomorphisms.
Let U
M be an open subset of M , fx g coordinate functions of a
maximal atlas of M , fe g a coordinate basis of T U
T M and fsK g;
K = 1; 2; :::l a basis for any sec E (U ) sec E (M ). Then, a basis for any
V1
section of E (M )
T M is given by fsK dx g.
Recall also, that the covariant derivative operator re : sec E (M ) !
sec E (M ) is given by
dE A := re A
where, writing A = AK
that the product
(12.14)
sK we have
r e A = @ AK
8 Recall
dx ;
^ is
sK + AK
re sK :
given in De…nition 693.
(12.15)
380
12. On the Many Faces of Einstein Equations
V1
Now, let examine the case where E (M ) = T M
T M ,! C`(M; g).
Let fej g; be an orthonormal basis of T M . Then, using Eq.(12.14) and
Eq.(12.9)
dE ej = (rek ej )
! kj = ! krj
r
k
ek
! kj ;
;
(12.16)
V1
where the ! kj 2 sec T M are the connection 1-forms.
Also, for v = v i ei 2 sec T M , we have
dE v = rei v
i
dE v i ;
= ei
dE v i = dv i + ! ik v k :
(12.17)
12.2.2 Absolute Di¤erential
Vl
Now, let E (M ) = T M ,! C`(M; g): In this case, dE is the usual absolute
Vl
di¤ erential r of A 2 sec T M ,! sec C` (M; g), i.e., it is a mapping (see,
e.g., [77])
^l
^l
^1
r: sec
T M ! sec
TM
T M;
(12.18)
such that for any di¤erentiable A 2 sec
Vl
T M we have
rA = (rei A)
i
;
(12.19)
Vl
where rei A is the standard covariant derivative of A 2 sec T M ,!
sec C` (M; g). Also, for any di¤erentiable function f : M ! R, and difVl
ferentiable A 2 sec T M we have
r(f A) = df
A + f rA:
(12.20)
Now, if we suppose that the orthonormal basis fej g of T M is such that
V1
each ej 2 sec T M ,! sec C` (M; g) ; we can easily …nd using the Cli¤ord
algebra structure of the space of multivectors that we can write:
rej = (rek ej )
!=
k
1 ab
! ea ^ eb
2 k
1 ab
! ea eb
2 k
=
1
[!; ej ] =
2
ej y!
k
k
2 sec
^2
TM
^1
T M ,! sec C`(M; g)
where ! is the representative of the connection in the given gauge.
The general case is given by the following proposition:
^1
T M;
(12.21)
12.3 Cli¤ord Valued Di¤erential Forms
Proposition 577 For A 2 sec
Vl
381
T M ,! sec C` (M; g) we have
1
rA = dA + [!; A]:
2
(12.22)
Vl
Proof. Left to the reader. Note that A is to be considered as a
TMvalued 0-form.
Eq.(12.22) can now be extended by linearity for an arbitrary nonhomogeneous multivector A 2 sec C` (T M; g).
We proceed now to …nd an appropriate exterior covariant di¤erential
which acts naturally
V on Cli¤ord valued di¤erential forms,
V i.e., objects that
are sections of T M
C`(M; g) ( C`(M; g)
T M ) (see next
section). Note that we cannot simply use the above de…nition putting
E (M ) = C`(M;
V g) and EndE(M ) = EndC`(M; g), because EndC`(M; g) 6=
C`(M; g)
T M . Instead, we must …nd inspiration in the general theory
in order to …nd an appropriate de…nition. Let us see how this can be done.
12.3 Cli¤ord Valued Di¤erential Forms
De…nition 578 A homogeneous
multivector
valued di¤ erential
Vl
Vp
V form of
type (l; p) is a section of
TM
T M ,!
C`(T
M;
g)
T M , for
V
0 l 4, 0 p 4. A section of C`(M; g)
T M such that the multivector part is non homogeneous is called a Cli¤ ord valued di¤ erential form
(cliform).
We recall, that any A 2 sec
can always be written as
A = m(l)
Vl
(p)
TM
Vp
T M ,! sec C`(M; g)
1 i1 :::il
m
ei1
l! (l)
1
(p)
j1
m(l)
^
j1 :::jp
p!
1 i1 :::il
=
m
ei1 :::eil
l!p! (l)
1 i1 :::il
=
A
ei :::eil
l!p! j1 :::jp 1
^
=
^
T M
(p)
eil
jp
(p)
j1
j1 ::::jp
j1
Vp
^
^
^
jp
jp
(12.23)
:
m
Moreover, recall that (See De…nition 693) the ^ product of A = A
m
Vp
Vq
(p)
(q)
2 sec C`(M; g)
T M and B = B
2 sec C`(M; g)
T M
382
12. On the Many Faces of Einstein Equations
is the mapping:9
^
: sec C`(M; g)
! sec C`(M; g)
mm
A
^
(p)
B = AB
^
^q
^p
T M sec C`(M; g)
^p+q
T M;
(q)
T M
:
(12.24)
Vl
Vp
De…nition 579V The commutator V
[A; B] of AV2 sec T M
T M ,!
p
m
q
sec
C`(M;
g)
T
M
and
B
2
T
M
T
M
,!
sec
C`(M;
g)
Vq
T M is the mapping:
[
;
] : sec
^l
TM
1
2 [l+m
! sec
[A; B] = A
Xjl
^p
mj]
k=0
^
B
T M
^jl
pq
( 1)
B
sec
mj+2k
^
^m
TM
TM
A
^q
^p+q
1
B i1 :::im ei1 :::eim
Writing A = l!1 Aj1 :::jl ej1 ::: ejl (p) , B = m!
Vp
Vq
(q)
sec T M and
2 sec T M , we have
[A; B] =
1 j1 :::jl i1 :::im
A
B
[ej1
l!m!
ejl ;ei1
eim ]
(p)
T M
T M;
^
(12.25)
(q)
, with
(q)
:
(p)
2
(12.26)
The de…nition of the
V commutator is extended by linearity to arbitrary sections of C`(M; g)
T M.
Now, we have the following proposition:
Vp
Proposition
580 Let A 2 sec C`(M;
T M , B 2 sec C`(M; g)
Vq
Vr g)
T M , C 2 A 2 sec C`(M; g)
T M . Then,
[A; B] = ( 1)1+pq [B; A];
(12.27)
and
( 1)pr [[A; B] ; C] + ( 1)qp [[B; C] ; A] + ( 1)rq [[C; A] ; B] = 0:
(12.28)
Proof. It follows directly from a simple calculation, left to the reader.
Eq.(12.28) which is analogous to Eq.(A.45) may be called the graded
Jacobi identity [51].
9 Note
m
m
that A and B are general non-homogeneous multivector …elds.
12.3 Cli¤ord Valued Di¤erential Forms
383
V2
Vp
Vr
Corollary
581 Let A(2) 2 sec T M
T M and B 2 sec T M
Vq
T M . Then,
[A(2) ; B] = C;
(12.29)
Vr
Vp+q
where C 2 sec T M
T M.
Proof. It follows from a direct calculation, left to the reader.
V2
V1
Vl
Vp
Proposition
! 2 sec T M
T M , A 2 sec T M
T M,
Vm 582 VLet
q
B 2 sec
TM
T M . Then, we have
[!; A
^
B] = [!; A]
B + ( 1)p A
^
^
[!; B]:
(12.30)
Proof. Using the de…nition of the commutator we can write
[!; A]
^
B = (!
= (!
+ ( 1)
^
^A
p+q
= [!; A
( 1)p A
A
^
A
^ B]
B
^
^ !)
p+q
( 1)
B
^
^
A
B
^
p
B
^
( 1) A
!
( 1)p A
^
^ [!; B];
!)
!
^
B
from where the desired result follows.
From Eq.(12.30) we have also10
(p + q)[!; A
= p[!; A]
+ q[!; A]
B]
^
^
B + ( 1)p qA
p
^ B + ( 1) pA
^
[!; B]
^ [!; B]:
De…nition 583 The action of the di¤ erential operator d acting on
^l
^p
^p
A 2 sec
TM
T M ,! sec C`(M; g)
T M;
is given by:
dA := ej1
ejl
= ej1
e jl
dAj1 :::jl
1
:::jl
d Aji11:::i
p
p!
(12.31)
i1
^
^
ip
:
We have the important
Vp
Proposition
584 Let A 2 sec C`(M; g)
T M and B 2 sec C`(M; g)
Vq
T M . Then,
d[A; B] = [dA; B] + ( 1)p [A; dB]:
(12.32)
Proof. The proof is a simple calculation, left to the reader.
We now de…ne the fake exterior covariant di¤erential operator (FECD)
Vl
D
Der acting on a cliform A 2 sec T M
Vpand also an associated operator
Vp
T M ,! sec C`(M; g)
T M , (l; p 1) as follows.
1 0 The
result printed in [352] is wrong.
384
12. On the Many Faces of Einstein Equations
12.4 Fake Exterior Covariant Di¤erential of
Cliforms
De…nition 585 The fake exterior covariant di¤ erential (FECD) of A is
the mapping :
^l
^p
^l
^p
T M ! sec
TM
T M
^p+1
T M;
^l
^p
p
TM
T M; l
DA = dA + [!; A]; if A 2 sec
2
D: sec
TM
^l
sec
TM
^
^1
T M
0; p
1:
(12.33)
where ! is given by Eq.(12.21).
Vl
V0
Vl
Remark 586 For p = 0 we identify T M
T M = T M and thus
in order to have an agreement with Eq.(12.22) we put DA = dA + 21 [!; A].
When l = 0 we have DA = dA.
Vl
Vp
Vp
Proposition
Let A 2 sec T M
T MV,! sec C` (M; g)
T M,
Vm 587 V
q
q
B 2 sec
TM
T M ,! sec C`(M; g)
T M . Then, the FECD
satis…es
D(A
^
B) = DA
^
+ q[!; A]
B+( 1)p A
^
p
^
DB
B+( 1) pA
^
[!; B]:
(12.34)
Proof. It follows directly from the de…nition if we take into account the
properties of the product ^ and Eq.(12.30).
Remark 588 According to the above proposition D does not satisfy Leibniz’s rule when applied to the product A ^ B . This may seems strange, but
we recall that there are some important derivatives operators as, e.g., the
Dirac operator acting on sections of a Cli¤ ord bundle of multiforms that
also does not satisfy Leibniz’s rule.
12.4.1 The Operator Der
De…nition 589 The operator Der is the mapping
Der : sec
^l
TM
^p
T M ! sec
(l)
such
P (q) 2 sec
Vp that for any A = L
T M , l; p 1, we have
(Der A)
r
^
Vl
^l
TM
= DA:
TM
Vp
^p
T M;
T M ,! sec C`(M; g)
(12.35)
12.4 Fake Exterior Covariant Di¤erential of Cliforms
385
We have immediately
p
Der A = @er A + [! r ; A];
2
1
i1 :::ip
pj er (Pi1 :::ip )
where @er A : =L(l)
general11
(12.36)
! r = !(er ) and, of course, in
Der A =
6 rer A:
(12.37)
Let us examine some important cases which will appear latter.
Case p = 1
Let A 2 sec
Vl
V1
TM
or
T M ,! sec C`(M; g)
1
DA = dA + [!; A];
2
V1
T M . Then,
(12.38)
1
Dek A = @er A + [! k ; A]:
2
(12.39)
Case p = 2
Let F 2 sec
Vl
V2
TM
T M ,! sec C`(M; g)
DF = dF + [!; F];
V2
T M . Then,
(12.40)
Der F = @er F + [! r ; F]:
(12.41)
12.4.2 Cartan Exterior Di¤erential of Vector Valued Forms
Recall that [138] Cartan de…ned the exterior covariant di¤erential of C =
V1
Vp
ei Ci 2sec T M
T M as a mapping
rc :
c
^1
TM
c
r C = r (ei
c
r ej = (rek ej )
^p
i
T M !
C ) = ei
k
^1
i
TM
c
^p+1
T M;
i
dC + r ei ^ C ;
(12.42)
which in view of Eq.(12.31) and Eq.(12.21) can be written as
rc C = rc (ei
1 1 For
1
Ci ) = dC + [!; C]:
2
a Cli¤ord algebra formula for the calculation of rer A, A 2 sec
Eq.(7.44).
(12.43)
Vp
T M recall
386
12. On the Many Faces of Einstein Equations
So, we have, for p > 1, the following relation between the FECD D and
Cartan’s exterior di¤erential (p > 1)
DC = rc C +
Note moreover that when C(1) = ei
p
1
2
[!; C]:
Ci 2 sec
DC(1) = rc C(1) :
V1
(12.44)
TM
V1
T M , we have
(12.45)
We end this section with two observations:
(i) We emphasize that to the concept of the fake exterior covariant di¤erential just introduced is di¤erent from the usual de…nition of the V
exterior
p
covariant di¤erential acting on sections
of
a
vector
bundle
E(M
)
T M
Vp
and also in sections of EndE(M )
T M , as discussed in the Appendix
and as appear in well know texts, e.g., [30, 39, 138, 150, 267, 290, 409]. Our
intention in de…ning the fake exterior covariant di¤erential of Cli¤ord valued forms was to …nd an object that could mimic coherently the pullback
under a local section of the covariant di¤erential acting on sections of vector
bundles associated with a given principal bundle as used in gauge theories.
The consistence of our de…nition will be checked in several situations below.
(ii) We note also that the fake exterior covariant di¤erential is di¤erent
from the concept of exterior covariant derivative of indexed p-forms12 which
are objects like the curvature 2-forms (see below) or the connection 1-forms
introduced above.
12.4.3 Torsion and Curvature Once Again
V1
V1
V1
Let
= e dx = ea a 2 sec T M
T M ,! C`(M; g)
T M
V
V
2
1
1 bc
a
a
!
e
e
2
sec
T
M
T
M
,!
and ! = 21 ! ba c eb ^ ec
b c
2 a
V1
C`(T M ; g)
T M be respectively the representatives of a soldering
form and a connection (in a …xed gauge) on the basis manifold. Then,
following the standard procedure [195], the torsion of the connection and
the curvature of the connection on the basis manifold are de…ned by
= D 2 sec
^1
R = D! 2 sec
^2
and
We now calculate
1 2 These
TM
TM
^2
T M ,! C`(M; g)
^2
T M ,! C`(M; g)
and then DR. We have,
objects have been introduced in De…nition 257.
^2
T M;
^2
T M:
(12.46)
(12.47)
12.4 Fake Exterior Covariant Di¤erential of Cliforms
a
D = D(ea
a
1
+ [! a ; eb ]
2
a
^
d
(12.48)
ed y! a = ! cad ec we have
and since 21 [! a ; ed ] =
D(ea
) = ea d
387
a
) = ea [d
a
+ ! abd
b
^
d
a
+ ! abd
b
^
d
] = ea
a
;
(12.49)
and we recognize
a
=d
;
(12.50)
as Cartan’s …rst structure equation.
12.4.4 FECD and Levi-Civita Connections
For a torsion free connection, the torsion 2-forms a = 0, and it follows that
= 0. We recall that a metric compatible connection ! (for which Dea g =
0; a = 0; 1; 2; 3) satisfying a = 0 is called a Levi-Civita connection. In the
remaining of this Chapter we restrict ourself to that case.
To start recall that from Eq.(12.33) we have,
DR = dR + [!; R]:
(12.51)
Then, taking into account that
1
R = d! + [!; !];
2
(12.52)
and that from Eqs.(12.27), (12.28) and (12.32) it follows that
d[!; !] = [d!; !]
[d!; !] =
[!; d!];
[!; d!];
[[!; !]; !] = 0;
(12.53)
and it results that
DR = dR + [!; R] = 0:
(12.54)
which is known as the Bianchi identity.
Note that, since fea g is an orthonormal frame we can write (Ra
ab
R )
1 ab
R ea ^ eb (dx ^ dx )
4
1 ab
1
c
R ea eb
^ d= R e e
4 cd
4
1
= R
e e
dx ^ dx ;
4
b
R=
dx ^ dx
(12.55)
388
12. On the Many Faces of Einstein Equations
where R
are the components of the curvature tensor. We recall from
Chapter 5 the well known symmetries
R
=
R
;
R
=
R
;
R
=R
:
(12.56)
We can also write Eq.(12.55) as
1 ab
R ea eb
4 cd
1
= Rab ea eb ;
2
R=
(
c
^
d
)=
1
R dx ^ dx
2
(12.57)
with
^2
1
1 ab
R ea eb = Rab ea ^ eb 2 sec
T M ,! C`(M; g);
2
2
^2
1
Rab = Rab dx ^ dx 2 sec
T M;
(12.58)
2
where R is called curvature bivectors and we recognize from Chapter
4 the Rab as the Cartan curvature 2-forms, which satisfy Cartan’s second
structure equation
Rab = d! ab + ! ac ^ ! cd ;
(12.59)
R
=
a result that follows immediately calculating dR from Eq.(12.52). Now, we
can also write,
DR = dR + [!; R]
1
1
= fd( Rab ea eb dx ^ dx ) + [! ; R ]dx ^ dx ^ dx g
2
2
1
(12.60)
= f@ R + [! ; R ]gdx ^ dx ^ dx
2
1
= De R dx ^ dx ^ dx
2
1
=
De R + De R + De R
dx ^ dx ^ dx = 0;
3!
from where it follows that
De R
+ De R
+ De R
= 0:
(12.61)
Remark 590 Eq.(12.61) is called in Physics textbooks on gauge theories
(see, e.g., [267, 365]) Bianchi identities. Physicists call the operator
De
D = @ + [! ; ];
(12.62)
acting on the curvature bivectors as the ‘covariant derivative’. Note however
that, as detailed above, this operator is not the usual covariant derivative
operator Dea acting on sections of the tensor bundle, and not realizing this
fact may give rise to a real confusion.
12.5 General Relativity as a Sl(2; C) Gauge Theory
389
We now …nd the explicit expression for the curvature bivectors R in
terms of the connections bivectors ! = !(e ), which will be used latter.
First recall that by de…nition13
R
=R(
^
)=
R(
^
)=
R :
(12.63)
Now, observe that using Eqs.(12.27), (12.28) and (12.32) we can easily
show that
[!; !] (
^
)=!
^
!(
= 2[! ; ! ]:
^
) = 2[!(e ); !(e )]
(12.64)
Using Eqs. (12.52), (12.63) and (12.64) we get
R
=@ !
@ ! + [! ; ! ]:
(12.65)
which is to be compared with Eq.(5.44) of Chapter 5.
Vp
Exercise 591 Let A 2 sec T M ,! sec C`(M; g) and R the curvature of
the connection as de…ned in Eq.(12.47). Show that 14 ,
D2 A =
1
1
[R; A] + [d!; A]:
4
4
(12.66)
12.5 General Relativity as a Sl(2; C) Gauge Theory
12.5.1 The Non-homogeneous Field Equations
V2
The analogy of the …elds R = 12 Rab ea eb = 12 Rab ea ^eb 2 sec T M ,!
C`(M; g) with the gauge …elds of particle …elds is so appealing that it is
irresistible to propose some kind of a Sl(2; C) formulation for the gravitational …eld. And indeed this has already been done, and the interested
reader may consult, e.g., [68, 240] for details. Here, we observe that despite
the similarities, the gauge theories of particle physics are in general formulated in ‡at Minkowski spacetime and the theory here must be for a …eld on
a general Lorentzian spacetime. This introduces additional complications,
but it is not our purpose to discuss that issue with all attention it deserves
here. We only wants to discuss some issues related to the formalism just
introduced above.
To start, recall that in gauge theories besides the homogeneous …eld
equations given by Bianchi identities, we also have a nonhomogeneous …eld
13 f
g is the dual basis of fe g and the scalar product in Eqs.(12.63) and (12.64)
refers to the scalar product of the form factors of R.
1 4 The value obtained for D 2 A in [352] is wrong.
390
12. On the Many Faces of Einstein Equations
equation. This equation which is to be the analog of the nonhomogeneous
equation for the electromagnetic …eld is written here as
D?R = d?R + [!; ?R] = ? J ;
(12.67)
V2
V1
V1
where the J 2 sec T M
T M ,! C`(M; g)
T M is a ‘current’,
which, if the theory is to be one equivalent to GRT, must be in some
way related with the energy momentum tensor in Einstein theory. In order
to write from this equation Van equation for the curvature bivectors, it is
2
very useful to imagine that
T M ,! C`(T M; g), the Cli¤ord bundle of
di¤erential forms, for in that case the powerful calculus used in previous
chapters can be used. So, we write:
^2
^1
T M
^1
,! C`(T M ; g)
T M ,! C`(M; g) C`(T M; g);
^2
^2
R = D! 2 sec
TM
T M
^2
,! C`(M; g)
T M ,! C`(M; g) C`(T M; g)
^2
^1
J =J
J
2 sec
TM
T M
(12.68)
! 2 sec
TM
,! C`(M; g)
C`(T M; g):
Now, using De…nition 27 of the Hodge star operator and recalling that
for a torsionless connection it is d = @^ and = @y we can write
d?R=
5
( @yR) =
? (@yR) =
? ((@ R ) ):
(12.69)
Also,
[!; ?R] = [! ; R
=
]
[! ; R ]
1
[! ; R ]
2
=
^ ?(
^
5
(
5
f
^
(
)
^
)
^
)+ 5 (
^
)
)
g
g
5
=
=
2
5
=
[! ; R
]
f
(
^
)
[! ; R
]
f
y(
^
)
2 ? ([! ; R ]
:
(
^
(12.70)
Using Eqs.(12.67-12.70) we get15
@ R + 2[! ; R ] = De R
1 5 Recall
that J 2 sec
V2
T M ,! sec C`(M; g).
=J :
(12.71)
12.6 Another Set of Maxwell-Like Equations for Einstein Theory
391
So, the gauge theory of gravitation has as …eld equations the Eq.(12.71),
the nonhomogeneous …eld equations, and Eq.(12.61) the homogeneous …eld
equations (which is Bianchi identities). We summarize that equations, as
De R
=J ,
De R
+ De R
+ De R
= 0:
(12.72)
Eqs.(12.72) which looks like Maxwell equations, must, of course, be compatible with Einstein’s equations, which may be eventually used to determine R ; ! and J , an exercise left to the interested reader.
12.6 Another Set of Maxwell-Like Equations for
Einstein Theory
We now show, e.g., how a special combination of the Rab are directly related
with a combination of products of the energy-momentum 1-vectors Ta and
the tetrad …elds ea (see Eq.(12.75) below) in Einstein theory. In order to
do that, we recall from previous chapters that Einstein’s equations can be
written in components in an orthonormal basis as
Rab
1
2
ab R
=
Tab ;
(12.73)
where Rab = Rba are the components of the Ricci tensor (Rab = Rcabc ),
Tab are the components of the energy-momentum tensor of matter …elds
and R = ab Rab is the curvature scalar. We next introduce as (in analogy
to what has been done in Chapter 9 for form …elds) the Ricci 1-vectors
and the energy-momentum 1-vectors by
We have that
^1
Ra = Rab eb 2 sec
T M ,! C`(M; g);
^1
Ta = Tab eb 2 sec
T M ,! C`(M; g):
Ra =
eb yRab :
(12.74)
(12.75)
(12.76)
b
Now, multiplying Eq.(12.73) on the right by e we get
Ra
1
Rea =
2
Ta :
(12.77)
Multiplying Eq.(12.77) …rst on the right by eb and then on the left by
eb and making the di¤erence of the resulting equations we get
( ec yRac ) eb
eb ( ec yRac )
1
R(ea eb
2
eb ea ) = (eb Ta
Ta eb ):
(12.78)
392
12. On the Many Faces of Einstein Equations
De…ning
Fab = ( ec yRac ) eb
=
eb ( ec yRac )
1
(Rac ec eb + eb ec Rac
2
1
R(ea eb
2
ec Rac eb
eb ea )
eb Rac ec )
1
R(ea eb
2
eb ea )
(12.79)
and
Jb = Dea (eb Ta
Ta eb );
(12.80)
we have16
Dea Fba = Jb :
(12.81)
It is quite obvious that with coordinates fx g covering an open set U
M we can write
De F = J ;
(12.82)
with F = g
F
F
and
= ( e yR
J = De (e T
)e
e ( e yR
)
1
R(e e
2
T e ):
e e );
(12.83)
(12.84)
Remark 592 Eq.(12.81) (or Eq.(12.82)) is a set of Maxwell-like nonhomogeneous equations. It looks like the nonhomogeneous classical Maxwell
equations when that equations are written in components, but keep in mind
that Eq.(12.82) is only a new way of writing the equation of the nonhomogeneous …eld equations in the Sl(2; C) like gauge theory version of Einstein’s
theory, discussed inVthe previous section. In particular, recall that any one
2
of the six F 2 sec T M ,! C`(M; g). Or, in words, each one of the F
is a bivector …eld, not a set of scalars which are components of a 2-form,
as is the case in Maxwell theory. Also, recall that according to Eq.(12.84)
V2
each one of the four J 2 sec
T M ,! C`(M; g).
From Eq.(12.81) it is not obvious that in vacuum we must have Fab = 0.
However that is exactly what happens if we take into account Eq.(12.79)
which de…nes that object. Moreover, Fab = 0 does not imply that the
curvature bivectors Rab are null in vacuum. Indeed, in that case, Eq.(12.79)
implies only the identity (valid only in vacuum)
(ec yRac ) eb = (ec yRbc ) ea :
(12.85)
1 6 Note that we could also produce another Maxwell-like equation, by using the usual
Levi-Civita covariant derivative operator D in the de…nition of the current, i.e., we can
put Jb = Dea (Ta eb eb Ta ), and in that case we obtain Dea Fba = Jb . An equation of
this form appears in [368, 369].
12.6 Another Set of Maxwell-Like Equations for Einstein Theory
393
Moreover, recalling de…nition (Eq.(12.58)) we have
Rab =
1
Rabcd ec ed ;
2
(12.86)
and we see that the Rab are zero only if the Riemann tensor is null which
is not the case in any non trivial general relativistic model.
The important fact that we want to emphasize here is that although
eventually interesting, Eq.(12.81) does not seem (according to our opinion)
to contain anything new in it. More precisely, all information given by that
equation is already contained in the original Einstein’s equation, for indeed
it has been obtained from it by simple algebraic manipulations. We state
again: According to our view terms like
1
(Rac ec eb + eb ec Rac ec Rac eb eb Rac ec )
2
1
= (eb Ta Ta eb )
R(ea eb eb ea );
2
1
= R(ea eb eb ea );
2
Fab =
Rab
Fab
1
R(ea eb
2
eb ea );
(12.87)
are pure gravitational object and there is no relationship of any one of these
objects with the ones appearing in Maxwell theory. Of course, these objects
may eventually be used to formulate ‘interesting’equations, like Eq.(12.81)
which are equivalent to Einstein’s …eld equations, but this fact does not
seem to us to point to any new Physics. Even more, from the mathematical
point of view, to …nd solutions to the new Eq.(12.81) is certainly as hard
as to …nd solutions to the original Einstein equations.
In [368, 369] there appear equations resembling the above ones using paravector …elds q a = ea e0 2 sec C`(0) (M; g). However, in those papers, the
author mislead the equations that he obtained with the Maxwell equations
describing the electromagnetic …eld, and claimed to have obtained an uni…ed …eld theory for the gravitational plus electromagnetic …eld. According
to our discussion, such claims are unfortunately equivocated. More details
on this issue may be found in [282, 352].
394
12. On the Many Faces of Einstein Equations
This is page 395
Printer: Opaque this
13
Maxwell, Dirac and Seiberg-Witten
Equations
13.1 Dirac-Hestenes Equation and i =
p
1
We already learned in Chapter 7 that we can write Dirac equation in interaction with the electromagnetic …eld in the spin-Cli¤ord bundle C`lSpine1;3 (M; ),
as
@ s e21 m e0 qA = 0
(13.1)
2 sec C`lSpine1;3 (M; ) is a DHSF and the ea 2 R1;3 are such that
V1
ea eb + eb ea = 2 ab and A 2 sec T M ,! sec C`(M; ). Eq.(13.1) that
we named DEC`l does not involve complex numbers. We also learned that
once we …x a spin frame the DHC`l has a representative in the Cli¤ord
bundle that we called the Dirac-Hestenes equation,1
where
@ (s)
21
+m
0
qA
= 0;
(13.2)
where
2 sec C`(M; ) is the representative of
2 sec C`lSpine1;3 (M; )
in the spin coframe 2 PSpine1;3 (M ). Again this equation does not contain
any complex number in its formulation, and we already know from previous
Chapters (3 and 7) the important geometrical meaning of representatives
of DHSF.
(s)
1 Recall that @ (s) = "a r
ea is the representative of the spin-Dirac operator in the
Cli¤ord bundle.
396
13. Maxwell, Dirac and Seiberg-Witten Equations
p
Now, we recall that i =
1 enters Dirac theory once we multiply
Eq.(7.80) on the right by the idempotent f = 21 (1 + e0 ) 21 (1 + ie2 e1 ) 2
C R1;3 . We get after some simple algebraic manipulations the following equation for the (complex) ideal left spin-Cli¤ord …eld f =
2
sec C`lSpine1;3 (M; g),
i@ s
m
qA = 0:
(13.3)
From that equation we get using the standard matrix representation
of the f g the standard Dirac equation for column spinor …elds (c.r.,
Eq.(7.77)).
We would like to emphasize here that this introduction of i
p
=
1 in (…rst quantization) Dirac theory is super‡uous from any fundamental point of view, although it may be useful in calculations. However
it must be clear to the reader that Eq.(13.3) hides the crucial geometrical
information that 21 refers to the spin plane.
13.2 How i =
p
1 enters Maxwell Theory.
The plane wave solutions (PWS) of ME are usually obtained by looking
for solutions to these equations such that the potential A = A
is in the
Coulomb (or radiation) gauge, i.e.,
~=0
A0 = 0; @i Ai := r A
(13.4)
Eq.(13.4), of course, implies that @ A =
A = 0, i.e., the Lorenz
gauge condition is also satis…ed. Now, in the absence of sources, A and
F = @ ^ A = dA satisfy, respectively the homogeneous wave equation
(HWE) and the free ME
A = 0;
@F = 0
(13.5)
(13.6)
The PWS can be obtained directly from the free ME (@F = 0) once
we suppose that F is a null …eld, i.e., F 2 = 0 (see Exercise 593). However, for the purposes we have in mind we think more interesting to …nd
these solutions by solving A = 0 with the subsidiary condition given by
Eq.(13.4).
In order to do that we introduce besides fx g another set of coordinate fx0 g also in the Einstein-Lorentz-Poincaré gauge which are also a
(nacsj@=@x0 ), such that , x00 = x0 , x0i = Rji xj . Putting " = dx0 we then
have
~ "0 = 0 , "i 6= i ,
" = R R;
(13.7)
where the (constant) R 2 sec Spin3 (M ) generates a global rotation of the
space axes. We also write
"i "j
"ij ;
"i e0 = ~ei ; ;
~ei~ej
eij =
i; j = 1; 2; 3 and i 6= j:
"ij ;
i = ~e1~e2~e3 ;
(13.8)
13.2 How i =
p
1 enters Maxwell Theory.
397
We consider next the following two linearly independent monochromatic
plane solutions A(i) , i = 1; 2, of Eq.(13.5) satisfying the subsidiary condition giving by Eq.(13.4) and moving in the ~e3 direction,
A(i) = exp
i
( 1)i+1 e21
2
: M ! R, x 7!
i
( 1)i+1 e21
2
e1 exp
i (x)
i
= exp e21 ( 1)i+1
i
"1 ;
~k 0 ~x0 + 'i , ! = j~k 0 j;
= k 0 x0 + 'i = !t
~k 0 = !~e3 :
(13.9)
(i)
where the 'i are real constants, called the initial phase. Since A0 = 0 we
write,
~ (i) = A(i) "0 = exp
A
( 1)i+1 e21
i
~e1 = ~e1 exp ( 1)i+1 e21
i
; (13.10)
Now,
F (i) = @ ^ A(i) = @ ^ A(i) = @"0 "0 A(i)
~ (i) ):
= (@t r)( A
(13.11)
13.2.1 From Spatial Rotation to Duality Rotation
We calculate in details F (1) in order for the reader to see explicitly how
i = ~e1~e2~e3 enters in the classical formulation of the electromagnetic …eld.
We have,
F (1) =
! ~e1~e2~e1 exp(e21
= ! [~e2
i~e1 ] exp( e21
= ! [~e1 + i~e2 ] exp( i
1
= !t
1)
~0
k
0
~e3~e1~e2~e1 exp(e21
1)
1)
1)
~x + '1 ; '1 = '1
2
(13.12)
This formula shows how a spatial rotation turns up into a duality rotation, a really non trivial result.
13.2.2 Polarization and Stokes Parameters
Now, a monochromatic plane wave can be written as a linear combination
of F (1) and F (2) , which have, of course, opposite helicities 2 , denoted from
now on respectively by + and . We write
n
o
F
~0 0
~0 0
= (i~e1 ~e2 ) a+ e)[ i(!t k ~x +'+ ] a e[i(!t k ~x +' )]
(13.13)
!
2 For the moment di¤erent helicities means that the vectors A
~ (i) have opposite sense
of rotation. We will be more precise later.
398
13. Maxwell, Dirac and Seiberg-Witten Equations
where a are real constants. Now, observe that F can be written as the
product of two (di¤erent) 2-dimensional matrix …elds with values in the
Cli¤ord bundle I(M; ) = C`0 (M; )e;where
e=
(1
~e3 )
2
2 sec C`0 (M; )
(13.14)
is an idempotent …eld3 . We write
F =
p
h
2i~e1 ee
i(!t ~
k0 ~
x0 )
~
k0 ~
x0 )
; eei(!t
i
p
2!a+ e i'+ e
p
2!a e+i' e
(13.15)
This means that each line of , e.g., the 2-dimensional matrix …eld
p
2!a+ e i'+ e
p
=
(13.16)
2!a e+i' e
is an amorphous spinor …eld (recall Section 7.10).
Writing
~ + iB,
~ F =E
~ iB;
~
F =E
we recall that the polarization of a given wave is in general speci…ed by
~ = (F +F )=2, the electric …eld. Now, for every formally complex number
E
or formally complex Euclidian vector …eld we de…ne in an obvious way the
operators Re and Im, such that, e.g.,
h
i
~ i(!t ~k0 ~x0 ) ;
~ = Re F = Re F = Re Ee
E
(13.17)
p
i'+
~ = pi [~e1 + i~e2 ; ~e1 i~e2 ] ! p2a+ e
E
:
(13.18)
i'
! 2a e
2
The Cli¤ord valued two dimensional complex vector …eld
p
! p2a+ e i'+
E=
! 2a ei'
(13.19)
plays an important role since it enters in the de…nition of the coherence
density matrix.
~ looks like the complex electric …eld used in electroWe observe that E
p
dynamic text books, (with the substitution i 7! i =
1) as, e.g., in the
books
by
[184,
203]
.
However,
it
is
necessary
to
emphasize
once more that
p
1 has no meaning in classical electromagnetic …eld and its use in general
occults the fundamental geometrical meaning of i.
To continue we put
a1 = a+ e
i'+
;
a2 = a ei' ;
(13.20)
3 Recall that I(M; ) is a bundle of amorphous spinor …elds and it is not to be confused
with the bundle I(M; ) (De…nition 453) of algebraic spinor …elds.
13.2 How i =
p
1 enters Maxwell Theory.
399
which are formally complex numbers. Now,
~ = 2F F
u+S
~
k0 ~
x0 ) +
a1 a2
= 4! 2 (ja1 j2 + e2i(!t
+e
2i(!t ~
k0 ~
x0 )
2
a1 a+
e3 );
2 + ja2 j )(1 + ~
(13.21)
~ are respectively the energy density and the Poynting vector.
where u and S
The mean value of this equation over the rapid oscillations of the …eld gives
~ = 4! 2 (ja1 j2 + ja2 j2 )(1 + ~e3 ):
(u + S)
(13.22)
Now we calculate the product EE + . This gives,
ja1 j2
a+
1 a2
1
EE + = e
2! 2
0
=e
In Eq.(13.23) the
i,
+
1
a1 a+
2
ja2 j2
1
+
2
2
+
3
3
:
(13.23)
i = 1; 2; 3 are the Pauli spin matrices and the matrix
,
=
0
+
1 1
+
2
0
=
2 2
+
2
ja1 j + ja2 j
;
2
1
3 3;
= Re a+
1 a2 ;
2
= Ima+
1 a2 ,
3
=
ja1 j2
2
ja2 j2
:
(13.24)
is called the Cartesian representation of the coherence density . The i
are called the Stokes parameters. The expression for EE + suggest to us
to de…ne the fake ‘spinor’ …eld
=
e 2 sec C`0 (M; ) I(M; )
sec C`(M; ) C`(M; ), where (the coherence density matrix ) is given
by,
= 0 + 1~e1 + 2~e2 + 3~e3 = 0 + ~:
(13.25)
Now, consider the action of4 Spin3 (M ) Spin3 (M ) on the sections of
sec Cl0 (M; ) I(M; ) sec Cl(M; ) C`(M; ) de…ned by
7!
0
=U U
1
U
1
e; U 2 sec Spin3 (M ):
(13.26)
Since the information contained in
and 0 are the same we see that
1
U U has also the same information carried by : The matrix representation of U U 1 is given by
U U 1
(13.27)
4 Spin
3
' SU(2).
400
13. Maxwell, Dirac and Seiberg-Witten Equations
where for any x 2 M , U(x) 2 SU (2) is the matrix representation of U 2
sec Spin3 (M ) and the matrix representation of .. For example, for
U=
1
2
i
i
1
1
(13.28)
we have the so called Jones representation. Writing
=
0
e+~
e
(13.29)
we see that the coherence matrix, de…nes an unitary5 vector ~ e in an
internal space. This means that any tentative of associating ~ or better,
0
i
with a spacetime vector …eld will produce nonsense6 .
0 = 0
i
Recall the ratios
s
2
2
1 + 3
2
; C=
(13.30)
L =
2
0
0
which are called respectively the degree of linear polarization and the degree
of circular polarization. We can write recalling Eq.(13.24) and Eq.(13.20)
C
=
ja+ j2 ja j2
ja+ j2 + ja j2
= i~e3
F F+
:
F F+
(13.31)
This last formula for C appears in electrodynamics and optics books
written in terms of a complex Euclidean vector …eld E~ which has the same
form
as the biform …eld F 2 sec C`0 (M; ) (with the substitution i 7! i =
p
1). Explicitly, we …nd, e.g., in Silverman’s book [391],
C
= i~e3
+
E~ E~
+ ;
E~ E~
i=
p
1:
(13.32)
Now, in [117, 118, 119, 120, 121, 122, 123] it is de…ned
~ (3) =
B
ie ~ ~+
E E
!
(13.33)
~ (3) is a fundamental longitudinal
and it is claimed that this phaseless …eld B
magnetic …eld with is an integral part of the plane wave …eld con…gurations.
Obviously, this is sheer nonsense, and Silverman’s in his wonderful book
[391] writes in this respect:
5 Recall
that ~ is an unitary vector.
~ (3) theory presented in a series of
is just what happened with the misleading B
books and inumerous articles, see e.g., [117, 118, 119, 120, 121, 122, 123].
6 This
13.2 How i =
p
1 enters Maxwell Theory.
401
“Expression (13.32)7 is specially interesting, for it is not, in my experience, a
particularly well-known relation. Indeed, it is su¢ ciently obscure that in recent
years an extensive scienti…c literature has developed examining in minute detail
the far reaching electrodynamic, quantum, and cosmological implications of a
“new” nonlinear light interaction proportional to E~ E~
+
(deduced by analogy to
+
~ _ E~ H
~ ) and interpreted as a “longitudinal magnetic
the Poynting vector S
…eld” carried by the photon. Several books have been written on the subject.
Were any of this true, such a radical revision of Maxwellian electrodynamics
would of course be highly exciting, but it is regrettably the chimerical product
of self-delusion— just like the “discovery” of N-Rays in the early 1900s.(During
the period 1903-1906 some 120 trained scientists published almost 300 papers on
the origins and characteristics of a totally spurious radiation …rst purported by
a french scientist, René Blondlot).”8
Of course, the real meaning of the right hand side of Eq.(13.31) (or
Eq.(13.32)) is that it is a generalization of the concept of helicity which is
de…ned for a single photon in quantum theory (see, e.g., [186, 391]). We
shall not give the details here, we only quote that, e.g., for a right circularly
polarized plane wave (helicity 1), a+ = 0; a = 1 and C = 1.
Exercise 593 Find PWS of the free Maxwell equations @F = 0 directly.
Solution: As we know ME in vacuum can be written as
@F = 0;
(13.34)
V2
where F 2 sec
T M ,! sec C`(M; ). The well known PWS of Eq.(13.34)
are obtained as follows. In a given Lorentzian chart fx g of the maximal
atlas of M naturally adapted to the inertial reference frame @=@x0 a PWS
moving in the z-direction is written as
F = fe
k=k
1
5k
2
x
;
; k = k = 0; x = x
(13.35)
;
(13.36)
V1
where k; x 2 sec
T M ,! sec C`(M; ) and where f is a constant 2-form.
From Eqs.(13.34) and (13.35) it follows that
kF = 0 :
(13.37)
Multiplying Eq.(13.37) by k we get
k2 F = 0
(13.38)
7 In Silverman’s book his Eq.(34), pp.167 is the one that corresponds to our
Eq.(13.32).
8 Of course, Silverman is refering to papers like, e.g., [117, 118, 119, 120, 121, 122, 123]
and hundereed of others by one author and his many associates. Some of the absurdities
of those papers are discussed in [70].
402
13. Maxwell, Dirac and Seiberg-Witten Equations
and since k 2 sec
V1
T M ,! sec C`(M; ) then
j~kj = k 3 ;
k 2 = 0 $ k0 =
(13.39)
i.e., the propagation vector is light-like. Also
F2 =
F F +F ^F =0
(13.40)
as can be easily seen by multiplying both members of Eq.(13.37) by F and
taking into account that k 6= 0. Eq.(13.40) says that the …eld invariants are
null for PWS. Such …elds are known as null …elds.
As in the previous section we emphasize again the fundamental role of
the volume element 5 (duality operator) in electromagnetic theory. In
particular since e 5 k x = cos k x + 5 sin k x, we see that
F = f cos k x +
5f
sin k x:
(13.41)
~ + iB,
~ with i
Writing F = E
e1 + i~e2 , ~e1 ~e2 = 0, ~e1 ,
5 and choosing f = ~
~e2 constant vectors in the Pauli subbundle sense, Eq.(13.35) becomes
~ + iB)
~ = ~e1 cos kx
(E
~e2 sin k x + i(~e1 sin k x + ~e2 cos k x):
(13.42)
This equation is important
because it shows once again that we must take
p
1 that appears in usual formulations
care with the i =
p of Maxwell
theory using complex electric and magnetic …elds. The i =
1 in many
cases unfolds a secret that can only be known through Eq.(13.42). From
~ = ~k B
~ = 0, i.e., PWS of
Eq.(13.37) we can also easily show that ~k E
ME are transverse waves. However, contrary to common belief, the free
Maxwell equations possess also solutions that are not transverse waves and
for which F 2 6= 0. Those (extraordinary) solutions [335, 334, 346] and their
properties will be fully investigated in a forthcoming book [283].
We can rewrite Eq.(13.37) as
k
and since k
0
= k0 + ~k;
0F 0
0 0F 0
=
=0
(13.43)
~ + iB
~ we have
E
~kf = k0 f:
(13.44)
Now, we recall that in C`0 (M; ) (where the typical …ber is isomorphic to
the Pauli algebra R3;0 ) we can introduce the operator of space conjugation [172] denoted by such that writing f = ~e + i~b we have
f =
~e + i~b ; k0 = k0 ; ~k =
~k:
(13.45)
We can now interpret the two solutions of k 2 = 0, i.e., k0 = j~kj and
k0 = j~kj as corresponding to the solutions k0 f = ~kf and k0 f = ~kf ; f
13.3 ‘Dirac like’Representations of ME
403
and f correspond in quantum theory to “photons” which are of positive
or negative helicities. We can interpret k0 = j~kj as a particle and k0 = j~kj
as an antiparticle.
Summarizing we have the following important facts concerning PWS of
ME: (i) the propagation vector is light-like, k 2 = 0; (ii) the …eld invariants
~ = ~k B
~ = 0.
are null, F 2 = 0; (iii) the PWS are transverse waves, i.e., ~k E
13.3 ‘Dirac like’Representations of ME
We exhibit in this section some faces (i.e., di¤erent mathematical representations) of ME where in any case the Maxwell …eld is represented by an
amorphous spinor …eld (see Section 7.8.) satisfying a ‘Dirac like’equation.
In the next section we show how, given a spin coframe, we can associate
a representative
2 sec C`(M; ) of a Dirac-Hestenes spinor …eld to the
V2
Faraday …eld F 2 sec T M ,! sec C`(M; ). The equation satis…ed by
gives a mathematical Maxwell-Dirac equivalence of the …rst kind.
We start from the Maxwell equation written in the Cli¤ord bundle, i.e.,
@F = J;
(13.46)
V2
V1
where F = 21 F
2 sec T M ,! sec C`(M; ) and J 2 sec T M ,!
sec C`(M; ). Moreover, we recall that
2
3
0
E1
E2
E3
6 E1
0
B3 B2 7
7 ;
[F ] = 6
(13.47)
4 E 2 B3
0
B1 5
E3
B2 B1
0
and that we can write
~ + iB
~
F =E
~ = E i~ i , B
~ = B i~ i , ~ i =
with E
I 0
(13.48)
as discussed in the previous section.
13.3.1 ‘Dirac like’Representation of ME on I 0 (M; )
Now, consider the bundle of amorphous spinor …elds I 0 (M; ) = C`0 (M; )e,
where
1
1
3 0
) = (1 + ~ 3 ) 2 sec C`0 (M; )
(13.49)
e = (1
2
2
is a primitive idempotent …eld.
We can perform the following algebraic manipulations in ME.
@F = J ) @
0 0
F
0
=J
0
:
(13.50)
404
13. Maxwell, Dirac and Seiberg-Witten Equations
Also,
0
~
Writing F + = E
@
0
F
0
= ~ @ ; ~ 0 1; ~ i = ~ i ;
~ + iB;
~ J 0=J ~ :
= E
~ , ' = F + e;
iB
=
(13.51)
J ~ e 2 sec I 0 (M; ), we have,
~ @ '= :
(13.52)
As the reader can easily verify recalling the results of Chapter 3, Eq.(13.52)
has a two dimensional matrix representation, namely
@ '=
(13.53)
with
'=
E3 + iB3
E1 + iE2 + iB1
0
0
B2
;
=
J0 + J3
J1 + iJ2
0
0
: (13.54)
13.3.2 Sachs ‘Dirac like’Representation of ME
Consider the equation @ = {, = F e00 and { = Je00 , where the idempotent …eld e00 = 21 (1 + 3 0 ) 2 sec C`0 (M; ) generates the bundle of
amorphous (Weyl) spinor …elds I 00 (M; ) = C`0 (M; )e00 . The objects living in I 00 (M; ) have a 4 4 matrix representation, as the reader may easily
verify using the results of Chapter 3. We have,
2
3
E3 + iB3
0 0 0
7
6 E1 iE2 + iB1 B2 0 0 0
7;
=6
4 0
0 0 E1 iE2 + iB1 + B2 5
0
0 0
E3 iB3
2
3
0
0 0
J1 + iJ2
6 0
0
0
J0 + J3 7
7:
(13.55)
=6
4 J0 + J3 0 0 0
5
J1 + iJ2
0 0 0
Putting now,
'
^1 =
'
^2 =
E3 + iB3
E1 iE2 + iB1
;
B2
E1 iE2 + iB1 + B2
E3 iB3
;
J0 + J3
J1 + iJ2
{
^1 =
{
^2 =
J1 + iJ2
J0 + J3
;
(13.56)
;:
the matrix representation of the equation @F e00 = e00 decouples in two
di¤erential equations for matrix representation of the amorphous (Weyl)
spinor …elds namely
^ @ b a'
^ a = ba ;
a = 1; 2:
(13.57)
13.3 ‘Dirac like’Representations of ME
405
Eq.(13.57) has been …rst obtained by Sachs [368] in an ad hoc way starting from a non covariant equation. Of course, both equations in Eq.(13.57)
carries the same information, exactly the same information carried by
Eq.(13.53)9 .
13.3.3 Sallhöfer ‘Dirac like’Representation of ME
Consider now the idempotent e = 21 (1 + 0 ) 2 sec C`(M; ). The 4 4
matrix representations of the amorphous spinor …elds = F e, = Je 2
sec C`(M; )e are easily found as
2
3
iB3
iB1 + B2 0 0
6 iB1 + B2
iB3
0 0 7
7;
(13.58)
=6
4 E3
E1 iE2 0 0 5
E1 + iE2
E3
0 0
3
2
J0
0
0 0
6 0
J0
0 0 7
7:
(13.59)
=6
4 J3
J1 iJ2 0 0 5
J1 + iJ2
J3
0 0
We can easily verify that each one of the non null columns of satis…es
a ‘Dirac-like’equation. In particular, taking into account that the ME are
~ 7! E;
~ E
~ 7! B
~ and considering a
invariant under the substitutions B
medium with dielectric susceptibility and magnetic permeability (both
~ 7! H
~ in
possibly, spacetime functions) and making the substitution B
10
Eq.(13.58), writing moreover c for the speed of light in vacuum , we get
that the free electromagnetic …elds in the medium satis…es
~ r
I2
I2
1 @
c @t
= 0;
(13.60)
which is an equation originally obtained by Sallhöfer [372, 373, 374, 375] in
an ad hoc way. Recently Smulik [396] obtained several interesting solutions
for ME from known solutions of Eq.(13.60) for some and functions.
13.3.4 A Three Dimensional Representation of the Free ME
Here we derive a three dimensional representation of the free ME, …rst
presented by Majorana [226] (see also [202]), but obtained in a completely
di¤erent way from the one given below. Our starting point is Eq.(13.50)
(obviously equivalent to ME) which since it is also satis…ed by 5 F can
9 The pair of Eq.(13.57) suggest the existence of new invariants for the electromagnetic
…elds, and indeed Sachs made interesting use of them in [368] .
1 0 We are using s system of units such that c = 1:
406
13. Maxwell, Dirac and Seiberg-Witten Equations
be rewritten (in the case J = 0) in the following equivalent ways in
C`0 (M; ) C`(M; ).
@( 5 F ) = 0;
i~ @ F = 0;
(13.61)
0
(i
~ @
) F =
2 @t
i
i
~
@i F:
2
k
i
Now, we recall that ~ i =2; ~ j =2 = i"ij
k ~ =2, i.e., the set {~ =2} is a
basis for any x 2 M of the Lie algebra su(2) of SU(2), the universal covering group of SO3 , the special rotation group in three dimensions. A three
dimensional representation of su(2) is given by the Hermitian matrices
2
3
0
i p3 i p2
0
i p1 5
Kp = 4 i p3
(13.62)
p2
p1
i
i
0
and
[Kp ; Kq ] = i
pq r
r K :
(13.63)
0
Writing moreover K = I3 for the three dimensional unitary matrix and
de…ning
2
3
E1 + iB1
jFi = 4 E2 + iB2 5 ;
(13.64)
E3 + iB3
we can obtain ME in three dimensional form with the substitutions
1
~ 7! K ; i 7! i; F 7! jFi:
(13.65)
2
in Eq.(13.61). We get]
i
@
jFi =
@t
iK rjF i
(13.66)
Note the doubling of the representative of the unity element of C`0 (M; )
when going to the three dimensional representation. This corresponds to the
fact that in relativistic quantum …eld theory, the 2 2 matrix representation
of Eq.(13.61) (projected in the idempotent 21 (1 + ~ 3 )) represents11 the
wave equation for a single quantum of a massless spin 1/2 …eld, whereas
Eq.(13.66) represents the wave equation for a single quantum of a massless
spin 1 …eld12 .
1 1 Of course, it is necessary for the quantum mechanical interpretation to multiply
both sides of eq.(13.66) by ~, the Planck constant.
1 2 Indeed in quantum mechanics the Pauli matrices
i and the matrices Ki are the
quantum mechanical spin operators and
3
X
i=1
(
i)
2
=
3
1
1
3 X
(1 + ) = ;
(Ki )2 = 1:(1 + 1) = 2:
2
2
4 i=1
13.4 Mathematical Maxwell-Dirac Equivalence
407
13.4 Mathematical Maxwell-Dirac Equivalence
In ([63, 64, 65, 66, 67]) using standard covariant spinor …elds Campolattaro
proposed that Maxwell equations are equivalent to a non linear Dirac like
equation. The subject has been further developed in ([347, 351, 435, 436])
using the Cli¤ord bundle formalism. The crucial point in proving the mentioned equivalence (abbreviated as MDE in what follows, when no confusion arises), starts once we observe that to any given representative
V0
V2
V4
2 sec( T M +
T M+
T M ) ,! sec C`(M; ) of a Dirac-Hestenes
spinor …eld in V
a given spin coframe there is associated an electromagnetic
2
…eld F 2 sec
T M ,! sec C`(M; ), (F 2 6= 0) through the RainichMisner theorem, i.e., we have ([307, 347, 351, 435, 436])
F =
21
~:
(13.67)
Before proceeding we recall that for null …elds, i.e., F 2 = 0, the spinor
associated with F through Eq.(13.67) must be a Majorana spinor …eld (Section 3.3). Now, since an electromagnetic …eld F , with F 2 6= 0 satisfying
Maxwell equation13 has six degrees of freedom and a Dirac-Hestenes spinor
…eld has eight (real) degrees of freedom some authors felt uncomfortable
with the approach used in ([347, 351, 435, 436]) where some gauge conditions have been imposed on a nonlinear equation (equivalent to Maxwell
equation), thereby transforming it into an usual linear Dirac equation (the
Dirac-Hestenes equation in the Cli¤ord bundle formalism). The claim, e.g.,
in [159] is that the MDE found in ([347, 351, 435, 436]) cannot be general.
The argument there is that the imposition of gauge conditions implies that
a satisfying Eq.(13.67) can have only six (real) degrees of freedom, and
this implies that the Dirac-Hestenes equation corresponding to Maxwell
equation can be only satis…ed by a restricted class of Dirac-Hestenes spinor
…elds, namely the ones that have six degrees of freedom.
Incidentally, in [159] it is also claimed that the generalized Maxwell equation
@F = Je + 5 Jm
(13.68)
V1
(where Je ; Jm 2 sec
T M ) describing the electromagnetic …eld generated by charges and monopoles [222] cannot hold in the Cli¤ord bundle
formalism, because according to that author the formalism implies that
Jm = 0.
In what follows we analyze those claims of [159] and prove that they are
equivocated. The reason for our enterprise is that as will become clear in
what follows, understanding of Eqs.(13.67) and (13.68) together with some
reasonable hypothesis permit a derivation and eventually even a possible
physical interpretation of the famous Seiberg-Witten monopole equations
1 3 Such solutions exist [335, 334, 346] and are investigated in details in a forthcoming
book [283].
408
13. Maxwell, Dirac and Seiberg-Witten Equations
([264, 267, 395]). So, our plan is the following: …rst we prove that given
F in Eq.(13.67) we can solve that equation for , and we …nd that has
eight degrees of freedom, two of them being undetermined, the indetermination being related to the elements of the stability group of the spin plane
21 . This is a non trivial and beautiful result which can be called inversion
formula. Next, we introduce a generalized Maxwell equation and the generalized Hertz equation. After that we prove a mathematical Dirac-Maxwell
equivalence of the …rst kind ([347, 351, 435, 436]), thereby deriving a DiracHestenes equation from the free Maxwell equations. Moreover, we introduce
a new form of a mathematical Maxwell-Dirac equivalence, called MDE of
the second kind. This new MDE of the second kind suggests that the electron is a ‘composite’ system. To prove the Maxwell-Dirac equivalence of
the second kind we decompose a Dirac-Hestenes spinor …eld satisfying a
Dirac-Hestenes equation in such a way that it results in a nonlinear generalized Maxwell (like) equation satis…ed by a certain Hertz potential …eld,
mathematically represented by an objectVof the same mathematical nature
2
as an electromagnetic …eld, i.e., 2 sec
T M ,! sec C`(M; ). This new
equivalence is very suggestive in view of the fact that there are recent (wild)
speculations that the electron can be splitted in two components [228] (see
also [80]). If this fantastic claim announced by Maris [228] is true, it is necessary to understand what is going on. The new Maxwell-Dirac equivalence
of the second kind may eventually be useful to understand the mechanism
behind the “electron splitting” into electrinos, but we are not going to
discuss these ideas here. Instead, we concentrate our attention in showing
that (the analogous on Minkowski spacetime) of the famous Seiberg-Witten
monopole equations arises naturally from the MDE of the …rst kind once
a reasonable hypothesis is imposed. We also present a possible coherent
interpretation of that equations. Indeed, we prove that when the DiracHestenes spinor …eld satisfying the …rst of Seiberg-Witten equations is an
eigenvector of the parity operator, then that equation describe a pair of
massless ‘monopoles’of opposite ‘magnetic’like charges, coupled together
by its interaction electromagnetic …eld.
13.4.1 Solution of F =
21
~
We now want solve Eq.(13.67) for . Before proceeding we observe that on
Euclidian spacetime this equation has been solved using Cli¤ord algebra
methods in [437]. Also, on Minkowski spacetime a particular solution of an
equivalent equation (written in terms of biquaternions) appear in [161]. We
are going to show that contrary to the claims of [159] a general solution for
has indeed eight degrees of freedom, although two of them are arbitrary,
i.e., not …xed by F alone. Once we give a solution of Eq.(13.67) for ,
the reason for the indetermination of two of the degrees of freedom will
become clear. This involves the Fierz identities, the boomerangs and the
13.4 Mathematical Maxwell-Dirac Equivalence
409
general theorem permitting the reconstruction of spinors from their bilinear
invariants discussed in Chapter 3.
We start by observing that from Eq.(13.67) and from Eq.(3.92) valid for
invertible representatives of DHSF we can write
F = e
Then, de…ning f = F= e
5
5
R
~
21 R:
(13.69)
it follows that
f =R
2
f =
~
21 R;
(13.70)
1:
(13.71)
Now, since all objects in Eq.(13.69) and Eq.(13.70) are even we can take
advantage of the isomorphism R3;0
R01;3 and making the calculations
when convenient in the Pauli algebra. To this end we recall Eq.(13.47) for
the components of F , where (E 1 ; E 2 ; E 3 ) and (B 1 ; B 2 ; B 3 ) are respectively
the Cartesian components of the electric and magnetic …elds.
We now write as already done above F in C`0 (M; ), the even subbundle
of C`(M; ).
~ + iB;
~
F =E
(13.72)
i
j
~
~
with E = E ~ i , B = B ~ j , i; j = 1; 2; 3. We can write an analogous
equation for f;
f = ~e + i~b:
(13.73)
Now, since F 2 6= 0 and
F 2 = F yF + F ^ F
~2 B
~ 2 ) + 2i(E
~ B)
~
= (E
(13.74)
~2
the above equations give (in the more general case where both I1 = (E
2
~
~
~
B ) 6= 0 and I2 = (E B) 6= 0):
p
!
~2 B
~2
~ B)
~
1
E
2(E
;
= arctan
:
(13.75)
=
~2 B
~2
cos[arctg2 ]
2
E
Also,
~e =
1 ~
[(E cos
~b = 1 [(B
~ cos
~ sin )];
+B
~ sin )]:
E
(13.76)
13.4.2 A Particular Solution
Now, we can verify that
L= p
21
2(1
I = f 03
~ 3 if~
= q
;
5 I)
i 2(1 i(f~ ~ 3 )
+f
5f
12
f~ ~ 3 :
(13.77)
(13.78)
410
13. Maxwell, Dirac and Seiberg-Witten Equations
~ = LL
~ = 1. Moreover, L is a particular
is a Lorentz transformation, i.e., LL
solution of Eq.(13.70). Indeed,
p 21
2(1
+f
5 I)
21 p
12
2(1
f
5 I)
=
f [2(1
2(1
5 I)]
5 I)
= f:
(13.79)
Of course, since f 2 = 1, ~e2 = ~b2 1and ~e ~b = 0; there are only four
real degrees of freedom in the Lorentz transformation L. From this result
in [159] it is concluded that the solution of the Eq.(13.67) is the DiracHestenes spinor …eld
p 5
e L;
(13.80)
=
which has only six degrees of freedom and thus is not equivalent to a general
Dirac-Hestenes spinor …eld (the spinor …eld that must appears in the DiracHestenes equation), which has eight degrees of freedom. In this way it is
stated in [159] that a the MDE of …rst kind proposed in ([347, 351, 435,
436]) cannot hold. Well, although it is true that Eq.(13.80) is a solution of
Eq.(13.67) it is not a general solution, but only a particular solution.
13.4.3 The General Solution
The general solution R of Eq.(13.67) is trivially found. It is
R = LS;
(13.81)
where L is the particular solution just found and S is any member of the
stability group of 21 , i.e.,
S
~=
21 S
21 ;
~ = 1:
S S~ = SS
(13.82)
It is trivial to …nd that we can parametrize the elements of the stability
group as
S = exp( 03 ) exp( 21 ');
(13.83)
with 0
< 1 and 0
' < 1. This shows that the most general
Dirac-Hestenes spinor …eld that solves Eq.(13.67) has indeed eight degrees
of freedom (as it must be the case, if the claims of ([347, 351, 435, 436]) are
to make sense), although two degrees of freedom are arbitrary, i.e., they
are like hidden variables!
Now, the reason for the indetermination of two degrees of freedom has
to do with a fundamental mathematical result: the fact that a spinor can
only be reconstruct through the knowledge of its bilinear invariants and
the Fierz identities as discussed in Chapter 3 and 7.
13.4.4 The Generalized Maxwell Equation
To comment on the basic error in [159] concerning the Cli¤ord bundle
formulation of the generalized Maxwell equation we recall the following.
13.4 Mathematical Maxwell-Dirac Equivalence
411
The generalized Maxwell equation ([222]) which describes the electromagnetic …eld generated by charges and monopoles, can be written in the
Cartan bundle (of the oriented manifold M ) as
dF = Km ;
dG = Ke
(13.84)
V3
where F; G 2
T M and Km ; Ke 2
T M , i.e., all these objects are
taken to be pair forms (see Section 4.9.5)
These equations are independent of any metric structure de…ned on the
world manifold. When a metric is given and the Hodge dual operator ? is
introduced it is supposed that in vacuum we have G = V
?F . In this case
1
putting Ke = ? Je and Km = ?Jm , with Je ; Jm 2 sec
T M , we can
write the following equivalent set of equations
V2
dF =
? Jm , d ? F =
? F = Jm , F =
dF =
Now, supposing that any sec
get from the above equations
(d
(13.85)
Je
? Jm , F =
Vj
? Je ;
(13.86)
Je :
(13.87)
T M ,! sec C`(M; ) (j = 0; 1; 2; 3; 4) we
)F = Je + Km or
(d
)?F =
Jm + Ke ;
(13.88)
or equivalently
@F = Je +
5 Jm
or
@(
5F )
=
Jm +
1 ?
( F
2
)
5 Je :
(13.89)
Now, writing
F =
1
F
2
, ?F =
;
(13.90)
then generalized Maxwell equations in the form given by Eq.(13.86) can be
written in components ( in a Lorentz coordinate chart) as
@ F
= Je , @ (? F
)=
Jm :
(13.91)
~ and taking into account
Now, assuming as in Eq.(13.67) that F =
21
the relation between and the representation of the standard Dirac spinor
…eld D we can write Eq.(13.91) as
@
D
[^ ; ^ ]
F
D
= 2Je ,
@ D ^5 [^ ; ^ ] D = 2Jm ;
1
1
?
)=
=
D [^ ; ^ ] , ( F
D ^5 [^ ; ^ ]
2
2
D:
(13.92)
The reverse of the …rst formula in Eq.(13.89) reads
@F = Je
Km :
(13.93)
412
13. Maxwell, Dirac and Seiberg-Witten Equations
First summing, and then subtracting Eq.(13.68) with Eq.(13.84) we get
the following equations for F = 21 ~:
@
21
~ + (@
21
~) = 2Je ;
@
21
~
(@
21
~) = 2Km ;
(13.94)
which is equivalent to Eq.(3.76) in [159] (where G is used for the three form
of monopolar current). There, it is observed that Je is even under reversion
and Km is odd. Then, it is claimed that “since reversion is a purely algebraic
operation without any particular physical meaning, the monopolar current
Km is necessarily zero if the Cli¤ord formalism is assumed to provide a
representation of Maxwell’s equation where the source currents Je and Km
correspond to fundamental physical …elds.”It is also stated that Eq.(13.92)
and Eq.(13.94) imposes di¤erent constrains on the monopolar currents Je
and Km .
It now is clear that those arguments of [159] are fallacious. Indeed, it is
obvious that if any comparison is to be made, it must be done between
Je and Jm or between Ke and Km . In this case, it is obvious that both
pairs of currents have the same behavior under reversion. This kind of
confusion is widespread in the literature, mainly by people that works with
the generalized Maxwell equation(s) in component form (Eqs.(13.91)).
It seems that experimentally Jm = 0 and the following question suggests
itself: is there any real physical …eld governed by a equation of the type of
the generalized Maxwell equation (Eq.(13.68))? The answer is yes.
13.4.5 The Generalized Hertz Potential Equation
In what follows we accept that Jm
V2= 0 and take Maxwell equations for
the electromagnetic …eld F 2 sec
T M ,! sec C`(M; ) and a current
V1
Je 2 sec
T M ,! sec C`(M; ) as
@F = Je :
(13.95)
V
2
= ~ e + i ~ m 2 sec
T M ,! sec C`(M; ) be the so
Let = 12
called Hertz potential ([70, 346]). We write
3
2
2
3
1
0
e
e
e
3
2
7
6 1e
0
m
m 7
(13.96)
[
]=6
2
3
1 5
4 e
0
m
m
3
2
1
0
e
m
m
and de…ne the electromagnetic potential by
A=
Since
2
2 sec
1
T ? M ,! sec C`(M; );
(13.97)
= 0 it is clear that A satis…es the Lorenz gauge condition, i.e.,
A = 0:
(13.98)
13.4 Mathematical Maxwell-Dirac Equivalence
Also, let
5
S=d
2 sec
^
3
T M ,! sec C`(M; );
413
(13.99)
and call S, the Stratton potential 14 . It follows also that
5
d
But d(
5
S) =
5
S = d2
= 0:
(13.100)
S from which we get, taking into account Eq.(13.100),
S = 0:
(13.101)
We can put Eq.(13.97) and Eq.(13.99) in the form of a single generalized
Maxwell like equation, i.e.,
@
= (d
)
=A+
5
S = A:
(13.102)
Eq.(13.102) is the equation we were looking for. It is a legitimate physical
equation. We also have,
)2
= (d
= dA +
5 dS:
(13.103)
Next, we de…ne the electromagnetic …eld by
F = @A =
= dA +
5 dS
= Fe +
5 Fm :
(13.104)
We observe that,
= 0 ) Fe =
5 Fm :
(13.105)
Now, let us calculate @F . We have,
@F
=
(d
)F
= d2 A + d(
5
dS)
(dA)
(
5
(13.106)
dS):
The …rst and last terms in the second line of Eq.(13.106) are obviously null.
Writing,
Je =
dA; and 5 Jm = d( 5 dS);
(13.107)
we get Maxwell equation
@F = (d
if and only if the magnetic current
)F = Je ;
5
Jm = 0, i.e.,
dS = 0:
1 4 This
(13.108)
object …rst appears for the best of our knowledge in [410].
(13.109)
414
13. Maxwell, Dirac and Seiberg-Witten Equations
a condition that we suppose to be satis…ed in what follows. Then,
A = Je =
dA;
S = 0:
(13.110)
Now, we de…ne,
~ e + iB
~ e;
Fe = dA = E
(13.111)
~ m + iE
~ m:
Fm = dS = B
(13.112)
and also
F = Fe +
5 Fm
~ + iB
~ = (E
~e
=E
~ m ) + i(B
~e + B
~ m ):
E
(13.113)
~ m = B:
~
(13.114)
Then, we get
~ e = E;
~
It is important to keep in mind that:
~ = 0, and B
~ = 0:
=0)E
(13.115)
Nevertheless, despite this result we have,
Proposition 594 (Hertz Theorem)
= 0 =) @Fe = 0:
(13.116)
Proof. We have immediately from the above equations that
@Fe =
d( 5 dS) + ( 5 dS) =
5d
2
S
5
dS = 0;
which proves the proposition.
We remark that Eq.(13.116) has been called the Hertz theorem in ([70,
346]). Hertz theorem has been used to …nd nontrivial subluminal and superluminal solutions of the free Maxwell equation in [335, 334]. See also
our forthcoming book [283].
13.4.6 Maxwell Dirac Equivalence of First Kind
Let us consider a generalized Maxwell equation
@F = J ;
(13.117)
where J is the generalized electromagnetic current (an electric current Je
V1
plus a magnetic monopole current
T M ,!
5 Jm , where Je , Jm 2 sec
C`(M; )). We proved in a previous section that if F 2 6= 0, then we can
write
F = 21 ~ ;
(13.118)
13.4 Mathematical Maxwell-Dirac Equivalence
415
where 2 sec C`0 (M; ) is a representative of a Dirac-Hestenes …eld. If we
use Eq.(13.118) in Eq.(13.117) we get
@(
21
~) =
@ (
21
~) =
(@
21
~+
~) = J :
21 @
(13.119)
from where it follows that
2
h@
21
~i2 = J ;
(13.120)
Consider the identity
h@
21
~i2 = @
~
21
h@
21
~i0
h@
21
~i4 ;
(13.121)
and de…ne moreover the covector …elds
j=
h@
g=
h@
21
~i0 ;
5 21
(13.122)
~i0 :
(13.123)
Taking into account Eqs.(13.120)-(13.123), we can rewrite Eq.(13.119) as
@
21
~ = 1 J + (j +
2
5 g)
:
(13.124)
Eq.(13.124) yields in the case where is non-singular (which corresponds
to non-null electromagnetic …elds) a representation of Maxwell equation
satis…ed by a representative of a DHSF. Indeed, we have
@
21
=
e
5
1
J + (j +
2
5 g)
:
(13.125)
The Eq.(13.125) representing Maxwell equation, written in that form,
does not appear to have any relationship with the Dirac-Hestenes equation
(Eq.(7.97)). However, we shall make some algebraic modi…cations on it in
such a way as to put it in a form that suggests a very interesting and intriguing relationship between them, and eventually a possible hidden connection
between electromagnetism and quantum mechanics.
Since is supposed to be non-singular (F 2 6= 0) we can use the canonical
V0
T M ,!
decomposition of and write = 1=2 e 5 =2 R, with , 2 sec
sec C`(M; ) and R 2 secSpine1;3 (M ). Then
@
=
1
(@ ln +
2
5@
+
) ;
(13.126)
where we de…ne the 2-form
~
= 2(@ R)R:
(13.127)
416
13. Maxwell, Dirac and Seiberg-Witten Equations
Using this expression for @
into the de…nitions of the covectors j and
g (Eqs.(13.122,13.123)) we obtain that
j=
(
S) cos
g=[
+
[
( 5 S)] cos
( 5 S)] sin ;
(13.128)
S) sin ;
(13.129)
(
where we de…ne the spin 2-form S by
S=
1
2
1
21
=
1
R
2
~
21 R:
(13.130)
We de…ne moreover
J=
0
~= v= R
0
1
R
;
(13.131)
where v is a ‘velocity’ …eld for the system. To continue, we de…ne also the
2-form = v
and the scalars and K by
=
S;
(13.132)
( 5 S):
(13.133)
S= v ;
(13.134)
( 5 S) = Kv ;
(13.135)
K=
Using these de…nitions we have that
and the vectors j and g can be written as
j = v cos
+ Kv sin
=
v;
(13.136)
g = Kv cos
v sin
=
v;
(13.137)
where we putted
=
cos
+ K sin ;
(13.138)
= K cos
sin :
(13.139)
The spinorial representation of Maxwell equation is written now as
@
21
=
e 5
J
2
+
0
+
5
0:
(13.140)
As we already mentioned [335, 334, 346] there are in…nite families of
non trivial solutions of Maxwell equations such that F 2 6= 0 (which correspond to subluminal and superluminal free boundary solutions of Maxwell
equation). Then, it is opportune to consider the case J = 0. We have,
@
21
=
0
+
5
0;
which is very similar to the Dirac-Hestenes equation.
(13.141)
13.4 Mathematical Maxwell-Dirac Equivalence
417
Constraining the Degrees of Freedom of
In order to go a step further into the relationship between those equations,
we remember that the electromagnetic …eld has six degrees of freedom,
while a Dirac-Hestenes spinor …eld has eight degrees of freedom and as
proved above two of those degrees of freedom are hidden variables. We are
free therefore to impose two constraints on if it is to represent an electromagnetic …eld. We choose as constraints the following equations saying
that the “currents” j and g are conserved
@ j = 0 and @ g = 0:
(13.142)
Using Eqs.(13.136,13.137) these two constraints become
@ j = _ + @ J = 0;
(13.143)
@ g = _ + k@ J = 0;
(13.144)
_
_
where J = v and = (v @) ; k = (v @)k. These conditions imply that
=
(13.145)
which gives ( 6= 0):
= constant =
tan
0;
(13.146)
or from Eqs.(13.138,13.139):
K
= tan(
0 ):
(13.147)
Now we observe that is the angle of the duality rotation from F to
F 0 = e 5 F: If we perform another duality rotation by 0 we have F 7!
e 5 ( + 0 ) F; and for the Takabayasi angle 7! + 0 : If we work therefore
with an electromagnetic …eld duality rotated by an additional angle 0 , the
above relationship becomes
K
= tan :
(13.148)
This is, of course, just a way to say that we can choose the constant
Eq.(13.146) to be zero. Now, this expression gives
=
cos
=
+
tan sin
tan cos
sin
=
cos
= 0;
;
0
in
(13.149)
(13.150)
and the spinorial representation of the Maxwell equation (Eq.(13.141)) becomes
@ 21
(13.151)
0 =0
418
13. Maxwell, Dirac and Seiberg-Witten Equations
Note that
is such that
_ =
The current J =
suppose also that
0
@ J:
(13.152)
~ is not conserved unless
is constant. So, if we
@ J =0
(13.153)
we must have
= constant.
Now, throughout these calculations we have assumed ~ = c = 1. We observe that in Eq.(13.151) has the units of (length) 1 , and if we introduce
the constants ~ and c we have to introduce another constant with unit of
mass. If we denote this constant by m such that
=
mc
;
~
(13.154)
then Eq.(13.151) assumes a form which is identical to Dirac-Hestenes equation:
mc
@ 21
(13.155)
0 = 0:
~
Remark 595 It is true that we did not prove that Eq.(13.155) is really
the Dirac-Hestenes equation since the constant m has to be identi…ed in
this case with the electron’s mass, and we do not have any good physical
argument to make that identi…cation, until now. In resume, Eq.(13.155) has
been obtained from Maxwell equation by imposing some gauge conditions
allowed by the hidden parameters in the solution of Eq.(13.67) for
in
terms of F . In view of that, it is certainly more appropriate instead of
using the term mathematical Maxwell-Dirac equivalence of …rst kind to talk
about a correspondence between that equations under which two degrees of
freedom of the Dirac-Hestenes spinor …eld are treated as hidden variables.
We end this section with the observation that it is to earlier to know if the
above results are of some physical value or only a mathematical curiosity.
Let us wait...
13.4.7 Maxwell-Dirac Equivalence of Second Kind
We now look for a Hertz potential …eld
following equation
@
= (@G+mP 3 +mh
2 sec
012 i1 )+ 5 (@P+mG 3
V2
T M satisfying the
5 hm
012 i3 )
(13.156)
V0
where G; P 2 sec
T M , and m is a constant. According to previous
results the electromagnetic and Stratton potentials are
A = @G + mP
3
+ mh
012 i1;
(13.157)
13.5 Seiberg-Witten Equations
5S
=
5 (@P
+ mG
5 hm
3
012 i3 );
419
(13.158)
and must satisfy the following subsidiary conditions,
(@G + mP
( 5 (@P + mG
3
+ mh
3
G + m@ h
P
m@ ( 5 h
012 i1 )
5 hm
= Je ;
012 i3 ))
012 i1
= 0;
012 i3 )
= 0:
= 0;
(13.159)
(13.160)
(13.161)
(13.162)
Now, in the Cli¤ord bundle formalism, as we already explained above,
the following sum is a legitimate operation
=
G+
+
5P
(13.163)
and according to previous results Eq.(13.163) de…nes as a(representative
of some Dirac-Hestenes spinor …eld. Now, we can verify that
satis…es
the equation
@ 21 m 0 = 0
(13.164)
which is as we already know a representative of the standard Dirac equation
(for a free electron) in the Cli¤ord bundle.
The above developments suggest (consistently with the spirit of the generalized Hertz potential theory developed above) the following interpretation. The Hertz potential …eld
generates the real electromagnetic …eld
of the electron15 . Moreover, the above developments suggest that the electron is “composed” of two “fundamental” currents, one of electric type
and the other of magnetic type circulating at the ultra microscopic level,
which generate the observed electric charge and magnetic moment of the
electron. Then, it may be the case, as speculated by Maris [345], that the
electromagnetic …eld of the electron can be spliced into two parts, each
corresponding to a new kind of subelectron type particle, the electrino.
Of course, the above developments leaves open the possibility to generate
electrinos of fractional charges. Well, it is time to stop speculations on this
issue.
13.5 Seiberg-Witten Equations
As it is well known, the original Seiberg-Witten (monopole) equations have
been written in Euclidean “spacetime” and for the self dual part of the
…eld F . However, on Minkowski spacetime, of course, there are no self dual
electromagnetic …elds. Indeed, the equation ?F = F implies that the unique
1 5 The question of the physical dimensions of the Dirac-Hestenes and Maxwell …elds is
discussed in [347].
420
13. Maxwell, Dirac and Seiberg-Witten Equations
solution (on Minkowski spacetime) is F = 0. This is the main reason for the
di¢ culties in interpreting that equations in this case, and indeed in [437]
it was attempted an interpretation of that equations only for the case of
Euclidean manifolds. Here we derive and give a possible interpretation to
that equations on Minkowski spacetime based on a reasonable assumption.
To start we recall that the analogous of Seiberg-Witten monopole equations read in the Cli¤ord bundle formalism and on Minkowski spacetime
as
8
< @ 21 A = 0;
(13.165)
F = 21 21 ~;
:
F = dA;
where
2 sec C`0 (M; ) is a representative of a Dirac-Hestenes spinor
V1
…eld in a given spin coframe, A 2 sec
T M ,! sec C`(M; ) is an elecV2
tromagnetic vector potential and F 2 sec
T M ,! sec C`(M; ) is an
electromagnetic …eld.
Our intention in this section is:
(a) To use the Maxwell Dirac-Equivalence of the …rst kind (proved above)
and an additional hypothesis to be discussed below to derive the SeibergWitten equations on Minkowski spacetime.
(b) to give a (possible) physical interpretation for that equations.
13.5.1 Derivation of Seiberg-Witten Equations
Step 1. Assume that the electromagnetic …eld F appearing in the
second of the Seiberg-Witten equations satisfy the free Maxwell equation, i.e., @F = 0:
Step 2. Use the Maxwell-Dirac equivalence of the …rst kind proved
above to obtain Eq.(13.151),
@
21
0
= 0:
(13.166)
1
(13.167)
Step 3. Introduce the ansatz
A=
0
:
This means that the electromagnetic potential (in our geometrical units)
is identi…ed with a multiply of the velocity …eld de…ned through Eq.(13.131).
Under this condition Eq.(13.166) becomes
@
21
A = 0;
which is the …rst Seiberg-Witten equation!
(13.168)
13.5 Seiberg-Witten Equations
421
13.5.2 A Possible Interpretation of the Seiberg-Witten
Equations
Well, it is time to …nd an interpretation for Eq.(13.168). In order to do
that we recall that (as it is well known) if
are Weyl spinor …elds (recall
Eq.(3.103)), then
satisfy a Weyl equation, i.e.,
= 0:
@
(13.169)
Consider now,
V1 the equation for + coupled with an electromagnetic …eld
A = gB 2 sec
T M ,! sec C`(M; ), i.e.,
@
+ 21
+ gB
+
= 0:
(13.170)
This equation is invariant under the gauge transformations
+
7!
Also, the equation for
+e
g
5
; B 7! B + @ :
(13.171)
coupled with an electromagnetic …eld gB is
@
21
+ gB
= 0:
(13.172)
which is invariant under the gauge transformations
eg
7!
5
; B 7! B
@ :
(13.173)
showing clearly that the …elds + and
carry opposite ‘charges’. Consider
now the Dirac-Hestenes spinor …elds " ; # (recall Eq.(3.107)) which are
eigenvectors of the parity operator (recall Eq.(3.106)) and look for solutions
of Eq.(13.168) such that = " : We have,
@
"
21
+ gB
"
=0
(13.174)
which separates in two equations,
@
"
+
+g
5B
"
+
= 0;
@
"
g
5B
"
= 0:
(13.175)
These results show that when a Dirac-Hestenes spinor …eld associated
with the …rst of the Seiberg-Witten equations is in an eigenstate of the
parity operator, that spinor …eld describes a pair of particles with opposite
‘charges’. We interpret these particles (following Lochak16 [211]) as massless ‘monopoles’in auto-interaction. Observe that our proposed interaction
is also consistent with the third of Seiberg-Witten equations, for F = dA
implies a null magnetic current.
1 6 Lochak suggested that an equation equivalent to Eq.(13.174) describe massless
monopoles of opposite ‘charges’.
422
13. Maxwell, Dirac and Seiberg-Witten Equations
It is now well known that Seiberg-Witten equations have non trivial
solutions on Minkowski manifolds (see [266]). From the above results, in
particular, taking into account the inversion formula (Eq.(13.79)) it seems
to be possible to …nd whole family of solutions for the Seiberg-Witten
equations, which has been here derived from a Maxwell-Dirac equivalence
of …rst kind with the additional hypothesis that electromagnetic potential
A is parallel to the velocity …eld v (Eq.(13.131)) of the system described by
Eq.(13.172). We conclude that a consistent set of Seiberg-Witten equations
on Minkowski spacetime must be
8
@ 21 A = 0;
>
>
<
F = 12 21 ~;
(13.176)
F = dA;
>
>
:
1
A=
:
0
We end this long chapter recalling that we exhibit two di¤erent kinds
of possible Maxwell-Dirac equivalences. Many will …nd the ideas presented
above speculative from the physical point of view, but we think that all
will agree that the mathematical coincidences found deserves more careful
investigation. We think it is really provocative that the MDE of the second
kind reveal an unsuspected possible interpretation of the Dirac equation,
namely that the electron seems to be a composed system build up from the
self interaction of two currents of the ‘electrical’and ‘magnetic’types. Of
course, it is to earlier to say if this discovery has any physical signi…cance.
We showed also, that by using the MDE of the …rst kind together with a
reasonable hypothesis we can shed some light on the meaning of SeibergWitten monopole equations on Minkowski spacetime. We hope that the
results just described may be an indication that Seiberg-Witten equations
(which are a fundamental key in the study of the topology of four manifolds
equipped with an euclidean metric tensor), may play an important role in
Physics, whose arena where phenomena occur is a Lorentzian manifold.
This is page 423
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14
Superparticles and Super…elds
14.1 Spinor Fields and Classical Spinning Particles
We supposed in what follows that spacetime (M; g; D; g ; ") is a spin
manifold, which implies, as we learned in Chapter 7 the existence
of a
V1
global tetrad frame fea g 2 sec PSOe1;3 (M ). Let f a g; a 2 sec
T M ,!
C`(M; g) be the dual frame of fea g. Also, let f a g be the reciprocal frame
of f a g, i.e., a b = ba . Suppose moreover that the reference frame (De…nition 378) de…ned by e0 is in free fall, i.e., De0 e0 = 0 and that the spatial
axis along each one of the integral lines of e0 have been constructed by
Fermi transport of spinning gyroscopes. This is translated by the requirement that De0 ei = 0, i =1; 2; 3 , and we have, equivalently De0 a = 0.
We introduce a spin coframe 2 PSpine1;3 (M ) by the method described in
Chapter 7 such that s( ) = f a g. Now, let be the representative of an
invertible Dirac-Hestenes spinor …eld over
(the world line of a spinning
particle) in the spin coframe . Let moreover ffa g be a Frenet frame over
such that f0 = 0 j and g(f0 ; ) = e0 j . Then, as we know from Chapter
6 we can write
1
fa =
:
(14.1)
a
Now, the general form of a representative of an invertible Dirac-Hestenes
1
5
over is = 2 e 2 R, where , : (I) ! R and R : (I) ! Spine1;3
R01;3 . Recalling Eq.(6.21) we get using Eq.(14.1) that
De0 R =
1
2
D R;
(14.2)
424
14. Superparticles and Super…elds
which may be called the spinor equation of motion of a classical spinning
particle.
14.1.1 Spinor Equation of Motion for a Classical Particle on
Minkowski Spacetime
Now, let us analyze the spinor equation of motion of a free particle in
Minkowski spacetime. In that case, recalling Eq.(6.24) we have D = S .
We can trivially rede…ne the Frenet frame in such a way as to have 3 = 0
(recall Section 6.2.4). Indeed, this can be done by rotating the original frame
3
. So, in what follows we
with U = ef3 f1 2 and choosing = arctan
1
suppose that this choice has already been done. We are interested in the
case where 2 is a real constant. Then, Eq.(14.2) becomes
1
2
D e0 R =
2f
2 1
f R:
(14.3)
The solution of Eq.(14.3) is
R = exp(
2
2 1
2
t):
(14.4)
V1
Suppose next the existence of a covector …eld V 2 sec
T M ,! C`(M; g)
and of an unitary Dirac-Hestenes spinor …eld with representative
2
C`(M; g) ( ~ = 1) in the spin coframe such that V j = v and j = R.
Without any loss of generality we can choose a global tetrad …eld such
that a 7!
= dx (where fx g are coordinates in Einstein-LorentzPoincaré gauge for the Minkowski spacetime). Then, under all these these
conditions we can rewrite Eq.(14.3) (taking into account the de…nition of
the spin-Dirac operator and its action on representatives of Dirac-Hestenes
spinor …elds) as:
1
2 1
D e0 R = 0 @ =
:
(14.5)
2
2
De…ning
p=
2
0
2
;x = x
;
(14.6)
we have that
= ep x
2
1
:
(14.7)
Now,
0
@ =
0
0
@ =(
1
)@ ;
(14.8)
and substituting this result in Eq.(14.5) we get
@
2 1
+
2
2
0
= 0:
(14.9)
14.1 Spinor Fields and Classical Spinning Particles
Put m =
2
2
425
and end with
@
2 1
m
0
= 0:
(14.10)
Eq.(14.10) is formally identical to the Dirac-Hestenes equation for a unitary spinor. Note that the signal of 2 merely de…nes the sense of rotation
in the e2 ^ e1 plane.
Recalling the various spin operators
introduced in Chapter 7 we can call
V2
the bilinear invariant S 2 sec
T M ,! C`(M; g)
S
=k
2 1~
(14.11)
the Vspin biform. Note that for our example S = ? S xv = k
1
sec
T M ,! C`(M; g) and may be called the spin covector.
3~
2
Remark 596 The results just obtained shows that a natural interpretation
suggests itself for the plane ‘wave function’ in the theory just presented.
It describes a ‘spinning ‡uid’ (i.e., determines a velocity …eld V ) such
that the particle follows one of its streamlines. It is not a physical …eld
in any sense 1 . This interpretation is reinforced by the derivation in the
next section of a classical non linear Dirac-Hestenes equation for a charged
spinning particle moving under the action of an electromagnetic …eld.
14.1.2 Classical (Nonlinear) Dirac-Hestenes Equation
We start from the classical Lorentz force law, which as well known, describes the motion of a classical particle of mass m and charge e following
a worldline : R I ! M under the action of an electromagnetic …eld.
Let
be the velocity of the particle and let v = g( ; ) be the physically
equivalent 1-form. From now one we suppose, asVin the previous section
2
that v 2 sec T M ,! sec C`(M; g). Let F 0 2 sec
T M ,! sec C`(M; g)
be the total electromagnetic …eld acting on the charged particle, i.e., the
0
sum of external
V1 electromagnetic …eld F and the self …eld Fs . Let A =
A + As 2 sec
T M ,! sec C`(M; g) be a potential generating F . Then
we have
mv_ = evyF 0 = evy(dA0 ):
(14.12)
V1
To continue, we suppose the existence of a 1-form …eld V 2 sec
T M ,!
sec C`(M; g) such that its restriction over is v, i.e., Vj = v. Also we impose that V 2 = 1. Next we observe that
1
v_ = V y@V = V y(@ ^ V ) + @V 2 = V y(@ ^ V ) = V y(dV ) =
2
1 Of course, the question of renormalization of the wave function
the same way as in standard quantum mechanics.
(dV )xV;
(14.13)
is to be deal in
426
14. Superparticles and Super…elds
Using this result in the …rst member of Eq.(14.12) we get
[d(mV
eA
eAs ]yV = 0:
(14.14)
A su¢ cient condition for Eq.(14.14) to hold is the existence of a 0-form
…eld such that
mV eA eAs = d = @ :
(14.15)
Eq.(14.15) is more conveniently written as
d + e(A + As ) = mV:
(14.16)
Squaring this equation we get putting A0 = A + As
(d + eA0 )2 = m2
(14.17)
which we recognize as the classical relativistic Hamilton-Jacobi equation,
if we ignore the self …eld As :
5
1
As in the previous section, let = 2 e 2 R 2 sec C`0 (M; g), be the representative (in the spin coframe ) of a particular invertible Dirac-Hestenes
spinor …eld such that
~ 6= 0
(14.18)
and since
1
2
=
e2
5
R we have
0
1
5
R 0R 1:
(14.19)
V1
Remark 597 Observe that since V 2 sec
T M we must necessarily
have that = 0 or = . These values correspond to charges of opposite
signs.
V =
=e
Substituting Eq.(14.19) in Eq.(14.16) after multiplying both members by
and taking = 0 we get
(@ + eA0 ) = m
0
=
2 1
:
(14.20)
We now write
0e
21
;
21
=
(14.21)
where 0 is also a representative of some invertible Dirac-Hestenes spinor
…eld that determines the same current J = 0 0 ~0 as the one determined
by .
We observe moreover that
@
=@
21
+@
0e
12 :
21
(14.22)
Using this result in Eq.(14.20) and observing that e
arrive at the non linear equation
@
21
+ eA + eAs
m
0
@(ln
0)
21
21
= 0:
=
1
0
we
(14.23)
14.1 Spinor Fields and Classical Spinning Particles
427
This result is to be compared with the quantum Dirac-Hestenes equation
(presented in Chapter 76) for an electron interacting with a electromagnetic
…eld A which is
@ 21 + eA
m 0 = 0;
(14.24)
5
where 2 sec C`0 (M; g), but with ~ = e
with = 0 or = .
Comparison of (14.23) and (14.24) shows that besides the di¤erence in
normalization there is a nonlinear term in Eq.(14.23) namely @(ln 0 ) 21
and that the quantum Dirac-Hestenes equation does not include the self
interaction term eAs . The term @(ln 0 ) 21 is identical to Bohm’s quantum potential (see, e.g., [155]).
Our exercise, may eventually serve as a prelude for a interpretation of
quantum theory, since it is clear that must be thought as a kind of probability amplitude de…ning a probability current distribution through the
bilinear invariant J = e 0 ~ (and a probability spin distribution biform
S = ~2 21 ~). In this way, as it was the case with the discussion on the
previous section, the ‘wave function’ is not to be interpreted as a real
…eld in any sense, at least in the theory here presented.
14.1.3 Digression
Despite this fact, a ‘fanatic’ …eld theorist may argue that ‘the …eld ate
the particle’, i.e., the electron is indeed a …eld, not a particle and that
satisfying Eq.(14.23) is a real …eld living in Minkowski spacetime. If that
is the case, it seems tempting to that …eld theorist to present the following
Conjecture 598
eAs =
~
@(ln
2
0)
1
21
:
(14.25)
Indeed, if Eq.(14.25) holds, then Eq.(14.23) becomes
@
21
+ eA
m
0
= 0:
(14.26)
5
0
1
If we left V = e
and do not use Remark 597 then under the
same hypothesis as above we would arrive at
@
21
+ eA
m
0e
5
= 0:
(14.27)
Well, we stop just here to conjecture and end this section observing that
Eq.(14.27), which will be called hereafter the non linear Dirac-Hestenes
equation (NLDHE), has been extensively studied by Daviau [95, 96]. He
showed that it gives very good results for the hydrogen spectrum. Also,
Eq.(14.27) equation possess very nice properties not possessed by the DHE
and it is surprisingly connected in an intriguing way with the free Maxwell
5
equation as discussed in [351]. Moreover, the nonlinear term e
causes no
di¢ culty concerning the superposition principle, since it has been shown
428
14. Superparticles and Super…elds
in [347] that we can superpose only ‘wave functions’having the same …xed
value of the Takabaysi angle .
Remark 599 It is worth to observe that Eq.(14.24) can be derived heuristically by a very simple argument. Indeed, in Hamiltonian mechanics the
canonical momentum of particle in interaction with an electromagnetic
A potential and following a world line such that v =
satis…es
=p
eA
Then if is a Dirac-Hestenes spinor …eld such that
write Eq.(14.28) as
+ eA = m 0
(14.28)
0
1
= v we can
(14.29)
and thus heuristically postulating that in relativistic quantum mechanics
7! @ 2 1 we get Eq.(14.24).2
14.2 The Superparticle
The results of the previous sections showed that the Frenet equations and
a Dirac-Hestenes like equation are (when properly interpreted) appropriate equations describing important aspects of the motion of classical spinning particle. In what follows we propose to develop a Lagrangian and
Hamiltonian formalism for Frenet equations using the multiform calculus
developed in Chapter 2. For reasons that will become clear in a while
we call the classical spinning particle the superparticle. For great generality we consider in what follows a n-dimensional ‡at Minkowski spacetime
(M; ; D; ; "), where (Tx M; x ) = R1;n 1 . The Cli¤ord bundle of multiforms is V
C`(M; ) and we choose as before a basis generated by the f g,
2 sec T M ,! sec C`(M; ); such that
+
=2
;
= diag(1; 1; :::; 1);
= 0; 1; 2; :::; n:
(14.30)
We have
De…nition 600 A superparticle is a pair ( ; X), where : R I ! M is
a time like curve and X : R I ! sec C`(M; ) is a Cli¤ ord-…eld over
(or a set of Cli¤ ord …elds over ).
De…nition 601 A multiform Lagrangian is a mapping
_
_
L : (X(s); X(s))
7! L(X(s); X(s))
2 sec C`(M; )
(14.31)
where X 2 sec C`(M; ) and s is an invariant time parameter on .
2 This simple heuristic argument has been generalized in order to obtain the wave
equation for a spin 1=2 particle in [360]
14.2 The Superparticle
429
Recalling Chapter 7, we see that L is a multiform functional, i.e., given
a trivialization of C`(M; ) it has values in the Cli¤ord algebra R1;n 1 for
each .
The most general L can then be written as
X
X
L=
hLik
Lk :
k
(14.32)
k
To gain con…dence in the sophisticated multiform derivative calculations
that we shall need to do,
Vrwe start by studying a simple case, namely, we
choose X = Xr 2 sec T M ,! sec C`(M; ) and L = hLi0 , a scalar
multiform functional.
We postulate next that the action for the superparticle is
Z 2
A(Xr ) =
ds L(Xr (s); X_ r (s))
(14.33)
1
and that the equations of motion can be derived from the principle of
stationary action (recall Chapter 7), that reads
d
A(Xr + tAr )jt=0 = Ar @Xr A(Xr ) = 0;
dt
Vr
where Ar 2 sec T M ,! sec C`(M; ) is a Cli¤ord …eld over
Ar (s1 ) = Ar (s2 ) = 0:
From Eq.(14.34) we get
Z s2
ds [(Ar @Xr )L + A_ r @X_ r L] = 0;
(14.34)
such that
(14.35)
(14.36)
s1
Since L = hLi0 and Xr , Ar 2 sec
Vr
T M ,! sec C`(M; ) we have
Ar @Xr hLi0 = hAr (@Xr L)i0 = A~r (@Xr L)r ;
(14.37)
e_ (@ L) :
A_ r @X_ r hLi0 = hAr (@X_ r L)i0 = A
r
r
X_ r
Using Eq.(14.37) into Eq.(14.36) results
Z s2
d
(@X_ r L)r ] = 0;
ds [(A~r (@Xr L)r A~r
ds
s1
(14.38)
i.e.,
A~r h@Xr L
d
(@ _ L)ir = 0:
ds Xr
(14.39)
430
14. Superparticles and Super…elds
Since for L = hLi0 , we have @Xr L = h@Xr Lir and @X_ r L = h@X_ r Li and
since Ar is arbitrary then Eq.(14.39) implies
d
(@ _ L)ir = 0;
ds Xr
h@Xr L
(14.40)
or
d
(14.41)
(@ _ L) = 0;
ds Xr
that is the Euler-Lagrange equation. It is quite clear that the Eq.(14.41)
holds for L being a functional of a general Cli¤ord …eld X over .
@Xr L
Next we study the general case where L is given by Eq.(14.32). However,
we restrict
ourselves without loss of generality to the case where X = Xr 2
Vr
sec T M ,! sec C`(M; )
We de…ne
X
X
=
h ik =
(14.42)
k;
Vr
where the k 2 sec T M ,! sec C`(M; ) are constant multiforms.
Then,
X
X
hL~i0 = L
=
Lk k =
Lk k :
(14.43)
k
k
In this way hL~i0 has the role of a scalar valued Lagrangian and we
de…ne the action by
Z s2
_ i0 :
A(X) =
dshL(X; X)~
(14.44)
s1
The principle of stationary action then gives
Z s2
ds[(Ar @X )hL ~i0 + (A_ r @X_ )hL~i0 ]
s1
X Z s2
=
ds[(Ar @X )(Lk k ) + (A_ r @X_ )(Lk
k
k )]
= 0:
(14.45)
k:
(14.46)
s1
Since (Ar @X )
(Ar @X ) (Lk
k
= 0, we have (recalling Eq.(2.177))
k)
= hAr @X Lk i
k
= hAr @X Lk ik k = hAr (@X Lk )ik
Vr
But, since X = Xr 2 sec T M ,! sec C`(M; ), then
@X Lk = h@X Lk ijr
kj
+ h@X Lk ijr
kj+2
+
h@X Lk ir+k ;
(14.47)
and we can write,
(Ar @X ) (Lk
k)
= hAr (@X Lk )jr
+
1
2 (r+k
=
k ik
k
+ hAr (@X L)jr
+ hAr (@X Lk )r+k ik
Xjr
`=0
kj+2 ik
k
k
kj)
hAr (@X Lk )jr
kj+2` i
k:
(14.48)
14.2 The Superparticle
431
Also,
(A_ r @X_ ) (Lk
k)
= hA_ r (@X_ Lk )ik
1
2 [r+k
Xjr
=
`=0
1
2 [r+k
Xjr
=
`=0
hAr
k
kj]
hA_ r (@X_ Lk )jr
kj]
kj+2` ik
d
hAr (@X_ Lk )jr
ds
d
(@ _ Lk )jr
ds X
kj+2` ik
k
kj+2` ik
k:
(14.49)
Using Eq.(14.48) and Eq.(14.49) into Eq.(14.45) and taking into account
that Ar (s1 ) = Ar (s2 ) = 0 we get
X Z
k
1
2 [r+k
s2
s1
ds
Xjr
kj]
hAr [(@X Lk )jr
`=0
kj+2`
d
(@ _ Lk )jr
ds X
kj+2` ik
k
= 0:
(14.50)
Vr
Now, the k are constant sections of sec T M ,! sec C`(M; ). Then
if p = nk , it follows that k is of the form ( k ) 1 ::: p
1 :::
p where
( k ) 1 ::: p are arbitrary real constants. Also the term in the brackets in
Eq.(14.50) is of the form h ik = (h ik ) 1 ::: p 1 : : : p and Eq.(14.50)
results in a sum of terms of the form ( ) 1 ::: p (h ik ) 1 : : : p . Since the
( k ) 1 ::: p are arbitrary constants, Eq.(14.50) implies that for each k we
must have
hAr [@X Lk
d
(@ _ Lk )]jr
ds X
kj + : : : [@X Lk
d
(@X Lk
ds
d
(@ _ Lk )]r+k ik = 0;
ds X
(14.51)
or
d
(@ _ Lk )ik = 0:
(14.52)
ds X
Next we show that Eq.(14.52) implies the multiform Euler-Lagrange
equation
d
@X Lk
(@ _ Lk ) = 0;
(14.53)
ds X
which means
hAr [@X Lk
[@X Lk
[@X Lk
[@X Lk
d
(@ _ Lk )]jr kj = 0;
ds X
d
(@ _ Lk )]jr kj+2 = 0;
ds X
..
.
d
(@ _ Lk )]r+k = 0:
ds X
(14.54)
432
14. Superparticles and Super…elds
Observe that if k = 0, Eq.(14.53) implies nk = n0 = 1 equation,
whereas the variation of Ar implies nr arbitrary variations. Then nr 1 =
n
n
namely,
r and we conclude the existence of r Euler-Lagrange equations,
Vr
n
d
one for each of the r components of [@Xr L0 ds (@X_ r L0 )] 2 sec T M ,!
sec C`(M; ). The same happens with k 6= 0 and r = 0 since in this case we
d
(@X_ r Lk )] 2
have nk
1 = nk Euler-Lagrange equations for [@Xr Lk ds
Vr
n
sec T M ,! sec C`(M; ) which has k components.
We now must extend the above reasoning for k 6= 0; r 6= 0. Observe that
in this general case we need
1
2 [r+k
Xjr
`=0
kj]
jr
n
kj + 2`
=
n
r
n
jr
n
k
kj
+
; n; k
jr
n
n
+ ::: +
kj + 2
r+k
n ; r + k < n;
(14.55)
in order to deduce from Eq.(14.52) the validity of Eq.(14.53). This happens
P 21 [r+k jr kj]
because Eq.(14.53) is equivalent to Eq.(14.54) which is a set of `=0
Euler-Lagrange like equations, and from Eq.(14.52) we can deduce only
n
n
r
k equations of the Euler-Lagrange type.
Now, Eq.(14.55) has been tested in a computer program to be true, and
we conclude for the validity of Eq.(14.53), the multiform Euler-Lagrange
equation. Indeed, Eq.(14.53) is valid also if X is a general multivector
…eld P
over , and we conclude that the principle of stationary action with
L = k Lk produces the general multivector Euler-Lagrange equation
@X L
14.3
d
(@ _ L) = 0:
ds X
(14.56)
Superparticle in Minkowski Spacetime
We now postulate the following biform valued Lagrangian for a superparticle whose world line in 4-dimensional Minkowski spacetime is
LS =
1
e_ ^ e
2
1
!
2
e ^e ;
(14.57)
where fe g; = 0; 1; 2; 3 a comoving coframe for in Minkowski spacetime
and where the ! are functions over . From Eq.(14.57) we have four
multiform Euler-Lagrange equations
@e LS
d
(@e_ LS ) = 0:
ds
(14.58)
14.4 Super…elds
433
We have taking into account the results of Exercise 108 (Chapter 2):
1
e_ ^ e
2
@e
=
@e_
De…ning
D
3
e_ ; @e
2
1
e_ ^ e
2
1
!
2
=
e ^e
= 3!
e ;
3
e_ :
2
= 12 ! e ^ e we arrive at
e_ =
D xe
(14.59)
which is similar to the equations of motion of a Frenet tetrad (see Eq.(6.21)).
And, indeed, for particular values of ! Eq.(14.59) may be identi…ed with
Frenet equations.
14.4
Super…elds
Here we show the connection of the Cli¤ord bundle formalism and the
concept of multivector derivatives with the Berezin supercalculus.
In 1977 Berezin [43] introduced the following calculus now known today
as supercalculus [101, 433].
Let i ; i = 1; : : : ; n be the generations of the Grassmann algebra Gn .
Then,
f i; j g = 0
(14.60)
where f g is the anticommutator.
Obviously, we have the following isomorphism
i
$ ei ;
i j
$ ei ^ ej
(14.61)
where ei are orthonormal covectors generating the Cli¤ord algebra Rp;q ,
p + q = n. (We leave p; q unspeci…ed at this moment).
With that identi…cation any combination of Grassmann variables is isomorphic to a certain multiform.
Berezin introduced the di¤erentiation of functions of Grassmann variables by the rules :
@ j
@
= ij ; j
= ij :
@ i
@ i
Introducing the reciprocal basis fei g of Rp;q ; ei ej =
(14.62)
i
j
we have
@
( 7! ei y ,
@ i
(14.63)
@
7! xej .
@ j
(14.64)
)
434
14. Superparticles and Super…elds
We immediately verify with the above identi…cations that di¤erentiation
in the Berezin calculus satis…es the so called graded Leibniz’s rule [101, 433].
Then if f ( ) = f ( 1 ; : : : n ) is a general Grassmann function a Taylor
expansion yields
f ( ) = f0 + fi
i
1
+ fij
2
i
1 + :::
1
fi :::i
n! n n
in
:::
in
:
(14.65)
Berezin de…ned integration by the rules
Z
Z
1d i = 0 ;
i d i = 1; 8i;
Z
(14.66)
(
(
f(
1
:::
n )d n
d
n 1
:::d
1
= f( )
@
@
@ n @ n
(
:::
1
@
:
@ 1
It is obvious that f ( ) is clearly isomorphic to a multiform F 2 Rpq with
the same coe¢ cients as in Eq.(14.65) and Eq.(14.65) is equivalent to
(: : : ((F xen )xen
1
n
n
) : : :) = hF E n )0 ;
E =e ^e
n 1
(14.67)
1
::: ^ e :
(14.68)
With this identi…cation almost all supercalculus as presented, e.g., in
deWitt [101] reduces to elementary algebraic identities for multiform functions. This also shows that super…elds, …rst introduced by Salam and
Strathdee [371] are isomorphic to sections of the Cli¤ord bundle. Indeed to
the super…eld A : M Gn ! Gn , which has the obvious Taylor expansion,
A(x; ) = A0 (x) + (A1 (x))i
+ :::
1
An (x)
n!
i
+
1
A2 (x)
2
1 :::
i j
(14.69)
ij
n
(14.70)
1 ::: n
it corresponds C(x) 2 sec C`(M; ) given by
C(x) = A(x) + (A1 (x))i ei +
::: +
1 (x)
A (x)
n! n
1
A2 (x)
2
e
1
:::e
ei ej +
ij
n
:
(14.71)
1 ::: n
Remark 602 The amazing fact that we would like to emphasize here is
that, as we learned in Chapter 7, any representative of a Dirac-Hestenes
spinor …eld is an even section of the Cli¤ ord bundle and thus has the same
structure of super…eld. Moreover the generalized potential A = A + 5 S,
where A is the usual electromagnetic potential and S is the Stratton potential introduced in Section 13.4.5 is also a super…eld.
14.4 Super…elds
435
To end this section we write a Berezin-Marinov’s like Lagrangian [43] for
a spinning particle in Minkowski spacetime as
LBM =
1_
2
1
!
2
(14.72)
where
: t 7! G4 ; = 0; 1; 2; 3 are elementary Grassmann …elds over ,
and ! are functions over , which in the original Berezin-Marinov model
are constant functions.
With the isomorphism de…ned by Eq.(14.61), namely 7! e where fe g
is an orthonormal coframe over (introduce above) we get the isomorphism
LBM ' LS
(14.73)
where LS is the biform Lagrangian de…ned by Eq.(14.57).
From the identi…cation LBM ' LS it becomes clear that in BerezinMarinov Lagrangian in a 4-dimensional Minkowski spacetime can produce
a (classical) Dirac-Hestenes equation as discussed in Section 14.1.3
In conclusion, we showed that Frenet equations are naturally appropriate
equations of motion a classical spinning particle, and from the spinor form
of Frenet equations we even obtain a ‘classical’ Dirac-Hestenes equation
for a unitary representative of a Dirac-Hestenes spinor …eld and also a nonlinear Dirac-like equation for a general representative of a Dirac-Hestenes
spinor …eld.
We succeeded also in giving a multiform Lagrangian formalism for Frenet
equations and showed that it is isomorphic to a generalization in a 4dimensional Minkowski spacetime of the Lagrangian of the famous BerezinMarinov [43] model, thus eventually providing a satisfactory geometrical
interpretation of that model a physical system living in usual Minkowski
spacetime. Moreover, our developments also indicates strongly that eventually super…elds may also have geometrical interpretation as Cli¤ord …elds4 .
This statement is based on the observation that any representative of a
Dirac-Hestenes spinor …eld in a spin frame may be identi…ed with a super…eld!
3 Compare this with the original Berezin-Marinov model [43], where it is necessary to
use a pentadimensional Grassmann algebra in order to obtain the Dirac equation after
quantization.
4 In [343] we showed that super…elds as de…ned by Witten [452] may be identi…ed
with ideal sections of an hyperbolic Cli¤ord bundle over Minkowski spacetime.
436
14. Superparticles and Super…elds
This is page 437
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15
Maxwell, Einstein, Dirac and
Navier-Stokes Equations
15.1 The Conserved Komar Current
Let K 2 sec T M be the generator of a one parameter group of di¤eomorphisms of M in the spacetime structure (M; g; D; g ; ") which is a model of
a gravitational …eld generated by T 2 sec T20 M (the matter plus non gravitational …elds energy-momentum momentum tensor) in Einstein GRT. It
V1
is quite obvious that if we put F := dK,
V where K = g(K;) 2 sec T M
then we can de…ne a current JK 2 sec T M by
JK =
F
(15.1)
JK = 0:
(15.2)
g
which, of course, is conserved, i.e.,
g
Surprisingly such a trivial mathematical result seems to be very important
by people working in GRT who call JK the Komar current1 [197]. Komar
called2
1 R
1 R
E:=
? JK =
?F
(15.3)
8 V g
8 @V g
1 Komar
called a related quantity the generalized ‡ux.
denotes a spacelike hypersurface and S = @V its boundary. Usually the integral
E is calculated at a constant x0 time hypersurface and the limit is taken for S being the
boundary at in…nity.
2V
438
15. Maxwell, Einstein, Dirac and Navier-Stokes Equations
the generalized energy.
To understand why JK is considered important write the action for the
gravitational plus matter and non gravitational …elds as
R
R
A = LEH + Lm =
1R
R
2
g
R
+ Lm :
(15.4)
Now, under the (in…nitesimal) di¤eomorphism h : M ! M generated
by K we have that g 7! g 0 = h g = g + g where the variation g =
$K g and taking into account Cartan’s magical formula ($K P = KydP +
g
V
d(KyP ), for any P 2 sec T M ) we have
g
A=
=
R
=
R
LEH + Lm
R
R
$K LEH
$K Lm
R
R
d(KyLEH )
d(KyLm )
R
:= d(?C)
(15.5)
g
Of course, if ?C is null on the boundary of the integration region we have
g
A = 0. On the other hand, recall that [203]
A=
p
R
1R p
R
det gdx0 dx1 dx2 dx3 + Lm
det gdx0 dx1 dx2 dx3 : (15.6)
2
Variation of A with respect to g gives as we know Einstein equation G =
1
R
R = T with R and T the components of the Ricci and
2
the energy-momentum tensor and R the scalar curvature. Now, recall that
1
G := R
= 0 =
2 Rg are the Einstein 1-form …eld and that D G
D T . Put T := T g and
E := G + T ; E := E
g = (G
+T
)g
(15.7)
to get
A=
=
=
=
=
=
R
p
1R
E (LK g)
det gdx0 dx1 dx2 dx3
2
p
R
E D K
det gdx0 dx1 dx2 dx3
p
R
D (E K )
det gdx0 dx1 dx2 dx3
R
(@yE K ) g
g
? (E K )
g g
R
d ? (E K ):
(15.8)
g
From Eqs.(15.5) and (15.6) we have immediately that
R
d(?E K ) + d(?C) = 0;
g
g
(15.9)
15.1 The Conserved Komar Current
439
and thus
g
(E K ) + C = 0:
(15.10)
g
V1
It follows that the current C 2 sec T M is conserved if the …eld equations E = 0 are satis…ed. An equation (in component form) equivalent to
Eq.(15.10) already appears in [197] (and also previously in [44] ) who took
C = E K + N where N = 0.
g
R Here, to continue we prefer to write an identity involving only Ag =
Lg . Proceeding exactly as before we get putting G(K) = G K that
V1
there exists P 2 sec T M such that
@yG(K)+@yP = 0:
(15.11)
and we see that we can identify
P :=
G K +L
(15.12)
where L = 0. Now, we claim that we can …nd L 2 sec
g
P=
G K + L = dK:
V1
T M such that
(15.13)
g
Let us …nd such a L and investigate
if we can give some nontrivial physical
V1
meaning to such P 2 sec T M .
In order to prove our claim we observe that we can write (taking into account the identities of Chapter 4 involving the Ricci, covariant D’Alembertian
operators and the square of the Dirac operator)
1
RK
2
1
= @ ^ @K
RK
2
G K =R K
= @ ^ @K + @ @K
1
RK
2
= @2K
=
g
dK
d K
g
1
RK
2
@ @K
@ @K
1
RK
2
(15.14)
@ @K
Then we take
P :=
G K
d K
g
and thus3
L=
3 Note
d K
g
1
RK
2
1
RK
2
@ @K = dK
(15.15)
g
(15.16)
@ @K:
that since (G K ) = 0 it follows from Eq.(15.16) that indeed L = 0.
g
g
440
15. Maxwell, Einstein, Dirac and Navier-Stokes Equations
Next, we recall the action of the extensor …eld T := T dx
T (K) = T K :
@ on K is
(15.17)
Then, we can write Eq.(15.15) as
g
dK = T (K)
d K
g
1
RK
2
@ @K:
(15.18)
We can write Eq.(15.18) taking into account that R = T := T and putting
F := dK as
F = JK ;
(15.19)
g
with
JK =
1
T (K) + T K+ d K + @ @K:
g
2
(15.20)
Eq.(15.20) gives the explicit form for the Komar current4 . Moreover, since
F = ?d ? F , we have:
g
g
g
d?F =?
g
g
1
1
TK
2
T (K)
1
TK
2
= ? T (K)
g
d K
g
d K
g
@ @K
@ @K
and thus taking into account Stokes theorem
R
R
d ? F = @V ? F
V
g
(15.22)
g
we arrive at the conclusion that the quantity
Z
1
E :=
?F
8 S=@V g
1 R
1
=
? (T (K)
TK d K
g
8 V
2
(15.21)
@ @K
(15.23)
is a conserved one.
Remark 603 As already remarked an equation equivalent to Eq.(11.36)
has already been obtained in [197] who called that quantity the conserved
generalized energy. But according to our best knowledge Eq.(15.23) …rst
appeared in [357] and it shows explicitly that all terms in the integrand
are legitimate 3-form …elds and thus the value of the integral is, of course,
independent of the coordinate system used to calculate it.
However, considering that for each K 2 sec T M that generates a one
parameter group of di¤ eomorphisms of M we have a conserved quantity it
4 Something
that is not given in [197].
15.1 The Conserved Komar Current
441
is not in our opinion appropriate to think in this quantity as a generalized
energy. Indeed, why should the energy depends on terms like d K and @ @K
g
if K is not a dynamical …eld?
We show in the next section that when K = A is a Killing vector …eld,
i.e., $A g = 0, we can write Eq.(15.23) as
E=
1 R
? (T (A)
4 V g
1
T A)
2
(15.24)
which is a conserved quantity 5 which relativistic physicists [367] think when
A is a timelike Killing vector …eld to be more directly associated with
the energy-momentum tensor of the matter plus non gravitational …elds 6 .
For a Schwarzschild spacetime, as well known, A = @=@t is a timelike
Killing vector …eld and in his case since the components of T are T =
p 8
(r)v v and v i vj = 0 (since v = pg100 (1; 0; 0; 0)) we get E = m.
det g
Remark 604 Originally Komar obtained the same result directly from
Eq.(15.23) supposing that the generator of the one parameter group of
di¤eomorphism was A = @=@t, so he got E = m by pure chance. Had he
picked another vector …eld generator of a one parameter group of di¤ eomorphisms A6= @=@t, he of course, would not obtained that result.
Remark 605 The previous remark shows clearly that the above approach
does not to solve the energy-momentum conservation problem for a system
consisting of the matter and non gravitational …elds plus the gravitational
…eld. It only gives a conserved energy for the matter plus non gravitational
…elds if the spacetime structure possess a timelike Killing vector …eld. To
claim that a solution for total energy-momentum of the total system 7 problem exist it is necessary to …nd a way to de…ne a total energy-momentum
1-form for the total system. This can only be done if the spacetime structure modeling a gravitational …eld (generated by the matter plus non gravitational …elds energy-momentum tensor T) possess additional structure, or
if we interpret the gravitational …eld as a …eld in the Faraday sense living
in Minkowski spacetime as did in Chapter 11. See also [127, 356].
R
R
that when A is a Killing vector …eld the quantities V ?T (A) and V 21 ?T A
are separately conserved as it is easily veri…ed.
6 An equivalent formula appears, e.g., as Eq.(11.2.10) in [442]. However, it is to be
emphasized here the simplicity and transparency of our approach concerning traditional
ones based on classical tensor calculus.
7 The total system is the system consisting of the gravitational plus matter and non
gravitational …elds.
5 Observe
442
15. Maxwell, Einstein, Dirac and Navier-Stokes Equations
15.2 A Maxwell like Equation Encoding Einstein
Equation
We have seen above that if (M; g; D; g ; ") is a model of a gravitational
…eld generated by T 2 sec T20 M (the matter plus non gravitational …elds
energy-momentum momentum tensor) in Einstein GRT and K 2 sec T M
is a vector …eld generating a one parameter group of di¤eomorphisms of M
the we can encode Einstein equations in Maxwell like equations for F = dK
where K = g(K; ). Indeed, the encoding is given by
dF = 0;
g
F =
JK ;
1
T (K) + T K+ d K + @ @K
g
2
JK =
(15.25)
(15.26)
Moreover, taking into account that the Dirac operator acting on sections
of the Cli¤ord bundle is @ = d
we have the remarkable result that
g
we can write a single Maxwell like equation encoding Einstein equations,
namely
@F = JK :
(15.27)
15.3 The Case When K = A is a Killing Vector
Field
V1
To proceed we suppose that A = g(A; ) 2 sec T M , where A is a
nontrivial Killing vector …eld in a LSTS (M; g; D; g ; ") which represents
a gravitational …eld generated by a given energy-momentum distribution
T 2 sec T20 M according to GRT. We will need two results that are presented
in the form of exercises whose solutions are left to the reader.8
Exercise 606 Show that if A2 sec T M is a Killing vector …eld in the LSTS
(M; g; D; g ; "), then
A = 0;
(15.28)
g
a
a
where A = g(A; ) = Aa g = A ga .
Remark 607 We recall now Exercise 575 (Chapter 11) that says that if
A2 sec T M is a Killing vector …eld in the LSTS (M; g; D; g ; "), then
@ ^ @A =
A = R a Aa ;
(15.29)
where @ = ga Dea is the Dirac operator acting on the sections of the Cli¤ ord
V1
bundle C`(M; g) and @ ^ @ is the Ricci operator acting on sec T M ,!
8 If
you need help for the solution of the exercises, see, [355, 357].
15.4 From Maxwell Equation to a Navier-Stokes Equation
443
V1
sec C`(M; g). Finally Ra 2 sec T M ,! sec C`(M; g) are the Ricci 1form …elds, with Ra = Rba gb , where Rba are the components of the Ricci
tensor. Also @ @A = A is the covariant D’Alembertian.
Using Eq.(14.4) we can immediately show taking into account the extensorial property of the Ricci operator @ ^ @ that Einstein equations
Ra 21 Rg a = T a after multiplying both members by Aa can be written as
@ ^ @A
1
RA = @ @A
2
1
RA =
2
T (A);
(15.30)
from where the current in Eq.(15.26) now denoted JA is given by
JA = RA
2T (A)
(15.31)
thus justifying Eq.(15.24) as a “generalized energy”in the case in which A
is a Killing vector …eld.
In this case the Maxwell like equation written in terms of the Dirac
operator or the …eld F associated to the Killing form A is simply
@F = RA
2T (A).
(15.32)
Remark 608 In the theory presented in [355] the …eld A is supposed to
be (up to a dimensional constant) the electromagnetic potential of a genuine electromagnetic …eld created by a given superconducting current and
interacting with the gravitational …eld. Then, clearly, the …eld F = dA in
[355] is supposed to automatically satisfy Maxwell equations. In what follows we only suppose that A = g(A; ), where A is a Killing vector …eld in
the structure (M; g; D; g ; "). In this chapter, of course, it is not supposed
that A is the electromagnetic potential of any electromagnetic …eld.
15.4 From Maxwell Equation to a Navier-Stokes
Equation
In this Section we obtain a Navier-Stokes equation following from the
Maxwell like equation (Eq.(15.32)) that encodes Einstein equation once we
identify appropriately the magnetic and electric like components of F = dA
with variables appearing in theory of the Navier-Stokes equation.
To appreciate what follows we recall that the original Navier-Stokes equation describes the non relativistic motion of a general ‡uid in Newtonian
spacetime. It is not thus adequate to use — at least in principle — a general
LSTS (M; g; D; g ; ") to describe a ‡uid motion. In fact we want to describe
a ‡uid motion in a background spacetime such that the ‡uid medium, together with its dynamics, is equivalent to a LSTS governed by Einstein
equation in the sense described below.
444
15. Maxwell, Einstein, Dirac and Navier-Stokes Equations
Next we recall that the theory of the gravitational …eld in Minkowski
spacetime presented in Chapter 11 interprets gravitation as a plastic distortion of the Lorentz vacuum9 . In that theory
…eld is repV1 the gravitational
V1
resented by a (1; 1)-extensor …eld h : sec T M ! sec T M living in
Minkowski spacetime structure. The …eld h — generated by a given energymomentum distribution in some region U of Minkowski spacetime — distorts the Lorentz vacuum described by the global cobasis10 f
= dx g,
dual with respect to the basis he = @=@x i of T M , thus generating the
gravitational potentials ga = h( a ).
Now, in the inertial reference frame e0 (according to the Minkowski
spacetime structure (M = R4 ; g; D; g ; ")), we write using the global coordinate functions fx g for M ' R4 ,11
A = A e :=
p
1
1
v2
+ V0 + q e0 v i ei = A e = e0 vi ei ; (15.33)
where the vector function v = (v1; v2 ; v3 ) is to be identi…ed with the 3velocity of a Navier-Stokes ‡uid — in the inertial frame e0 according to the
conditions disclosed below. Also, V0 denotes a scalar function representing
an external potential acting on the ‡uid, and
Z (t;x)
dp
q=
;
(15.34)
0
where the functions p and are identi…ed respectively with the pressure
and density of the ‡P
uid and supposed functionally related, i.e., dp ^ dq = 0.
3
Furthermore v 2 := i=1 (vi )2 .
Observe that looks like the relativistic energy per unit mass of the
‡uid. Then we will write A as
A=
1 2
v + V + q e0
2
v i ei = e 0
v i ei ;
(15.35)
where the new potential function V is the sum of V0 with the sum of the
Taylor expansion terms of [(1 v 2 ) 1=2 21 v 2 ]: Hence we have
A = g(A;) = A
=
A
=A
;
F = dA =
1
F
2
^
;
1
F
2
^
:
(15.36)
and
A = g(A;) = A
9 More
=g A
=A
;
F = dA =
(15.37)
details may be found in [127].
fx g are global coordinate functions in Einstein-Lorentz Poincaré gauge for
the Minkowski spacetime that are naturally adapted to an inertial reference frame e0 =
@=@x0 ; De0 = 0.
1 1 The basis fe g is the reciprocal basis of the basis fe g, i.e., g(e ; e ) =
:
1 0 The
15.4 From Maxwell Equation to a Navier-Stokes Equation
445
Identi…cation Postulate
We proceed by identifying the magnetic like and the electric like components of the …eld F . It seems natural to identify the magnetic like components of F with the vorticity …eld, but we cannot identify electric like
components of F with the Lamb vector …eld. The correct identi…cation is
given as follows. Write F = (dA) as
0
1
0
l1 d1 l2 d2 l3 d3
B l1 + d1
0
w3
w2 C
C
(15.38)
F =B
@ l2 + d2
w3
0
w1 A
l3 + d3
w2
w1
0
were w is the vorticity of the velocity …eld
w := r
v;
(15.39)
l := w
v;
(15.40)
and
is the so called Lamb vector and moreover
d=
where
r ;
(15.41)
is a smooth function.
Remark 609 We emphasize that the identi…cation of the components of
F has been done in an arbitrary but …xed inertial frame e0 = @=@x0 as
introduced above.
Next we recall that the non relativistic Navier-Stokes equation for an
inviscid ‡uid is given by [79, 136]
@v
+ (v r)v =
@t
r(V + q);
(15.42)
or using a well known vector identity,
@v
=
@t
w
v
r V +
p
1
+ v2 :
2
(15.43)
By these identi…cations12 , we get a Navier-Stokes like equation from the
straightforward identi…cation of l d = (F01 ; F02 ; F03 ) and w = (F32 ; F13 ; F21 ).
Indeed, we have
F0i = (w v)i di =
P3
Fjk =
i=1 ijk wi ;
@vi
@t
@
;
@xi
(15.44)
(15.45)
1 2 Other identi…cations of Navier-Stokes equation with Maxwell equations may be
found in [404, 405].
446
where
15. Maxwell, Einstein, Dirac and Navier-Stokes Equations
ijk
is the 3-dimensional Kronecker symbol. Eq.(15.44) becomes
@v
+w
@t
and since d =
written as
r
v+r
1 2
v
2
=
r V +
p
for some smooth function
@v
+w
@t
v+r
1 2
v
2
=
+ di ;
(15.46)
then Eq.(15.46) can be
r V +
+
p
(15.47)
which is now a Navier-Stokes like equation for a ‡uid moving in an external
potential V 0 = V + : Moreover, the homogeneous Maxwell equation dF = 0
is equivalent to
r
@w
= 0;
@t
r w = 0;
l+
(15.48)
which express Helmholtz equation for conservation of vorticity.
Remark 610 Note that since A = g(A; ) = A
write 13
A = g(A);
= A g
, we can
(15.49)
where the extensor …eld g is de…ned by g( ) = g
. Thus, in general,
we can write since d(F F ) = 0 , F = F + G where G is a closed 2-form
…eld. So, dF = dF = 0 express the same content, namely Eq.(15.48), the
Helmholtz equation for conservation of vorticity. Even more, taking into
account that the Minkowski manifold is star shape we have that G is exact.
Thus F = F + dP , for some smooth 1-form …eld P .
15.4.1 A Realization of Eqs.(15.40) and (15.41)
The attentive reader will realize that what has been done until now will
be not an empty exercise only in case there exist nontrivial solutions of
Eq.(15.40) and Eq.(15.41), i.e., F0i = (w v)i + di for at least a model
of GRT given by the LSTS (M; g; D; g ; ") modelling a gravitational …eld.
That this is the case, is easily seem if we take, e.g., the Schwarzschild
spacetime structure. As well known [247] the vector …eld
A = @' =
x2 @x1 + x1 @x2
1 3 We have (details in [127]) g = hy h and g(g(
g(g ; g ).
);
)=g
(15.50)
= g(h(
); h(
)) =
15.4 From Maxwell Equation to a Navier-Stokes Equation
447
is a Killing vector …eld for the Schwarzschild metric14 . The 1-form …eld
corresponding to it and living in Minkowski spacetime is A = x2 dx1 x1 dx2 .
Thus = 0 and v = (x2 ; x1 ; 0). This gives 0 = F0i = di + (w v)i ; i.e.,
d=
(r
v
v) = r[(x1 )2 + (x2 )2 ];
(15.51)
and Eq.(15.47) holds. The reader is invited to …nd other examples for other
solutions of Einstein equations.
Remark 611 For the example just given above we have simply A = f A
where f = r2 cos2 . Thus in this case, we have the simple expression
F = df ^ A + f F:
(15.52)
There are many examples of Killing vector …elds for which A = f A and for
such …elds that developments given below in terms of A are easily translated
in terms of A. In particular, when A = f A we have the following identi…cation of the components of the Lamb and vorticity vector …elds with the
components of F ,
li = (w
wi =
1
F0i (d ln f ^ A)0i ;
f
1
Fjk (d ln f ^ A)jk :
ijk
f
v)i =
1 P3
2 i=1
(15.53)
15.4.2 Constraints Imposed by the Nonhomogeneous Maxwell
Like Equation
To continue, we recall that in order for the Navier-Stokes equation just
obtained to be compatible with Einstein equation it is necessary yet to take
into account the equation F = JA written in the LSTS (M; g; D; g ; ")
g
since that equation produce as we are going to see constraints among the
several …elds involved. We want now to express the constraints implicit
in F = JA in terms of the objects de…ning the Minkowski spacetime
g
structure (M = R4 ; g; D; g ; ").
Now, taking into account that Dg = 0, we have
V1
Dg = A 2 sec T02 M
T M;
(15.54)
V1
where A 2 sec T02 M
T M is the nonmetricity tensor of D with respect
to g. In the coordinates fx g introduced above it follows that
A=Q
1 4 The
spherical coordinate functions are (r, ; ').
:
(15.55)
448
15. Maxwell, Einstein, Dirac and Navier-Stokes Equations
Then, as we recall from Chapter 4 the relation between
and
associated to the connections D and D (De
De g =
g ) in an arbitrary coordinate vector f @x@
h# = dx i bases — associated to arbitrary coordinate
covering U M — are given by15
1
+ S
2
=
the coe¢ cients
g =
g ,
g and covector
functions fx g
;
(15.56)
where
S
= g (Q
+Q
Q
)
(15.57)
are the components of the so called strain tensor of the connection.
In the coordinate bases f@=@x g and f
= dx g, associated to the
coordinate functions fx g, it follows that
= 0 and in addition the
following relation for the Ricci tensor of D holds:
R
Denoting K
J
=
1
2S
= J(
, the J(
=D K
)
D K
):
is the symmetric part of
+K
K
K
K
:
Now, we introduce the Dirac operator associated to the Levi-Civita connection D of g in our game,
@j := # D@=@x =
(15.58)
D@=@x
and recall Eq.(4.248) of Chapter 4, i.e.,
@ ^ @A = (@j ^ @j )A + L
where A = A
, A :=
g
A , and L =
g
A;
J
(15.59)
and the symbol
g
denotes the scalar product accomplished with g. Since @j ^ @j A = R A =
R A
= 0, Eq.(15.59) reads
@ ^ @A =
J
A=
J
g A
:
(15.60)
Recalling now Eq.(15.29) and Eq.(15.30), Eq.(15.60) can be written as an
algebraic constraint16 , relating the components A to the components of
1 5 We use that g = g
#
# = g #
# , where f# g is the reciprocal basis
of f# g, namely # = g # and g g
= . In the bases associated to fx g it is
g=
=
.
1 6 Of course, it is a partial di¤erential equation that needs to be satis…ed by the
components of the stress tensor of the connection.
15.5 Conclusions
449
the energy-momentum tensor of matter and the components of the g …eld
that is part of the original LSTS. We have,
1
J
g A = g J( ) A
T A :
(15.61)
2
Eq.(15.61) are the constraints need to be satis…ed by the variables of our
theory in order for the Navier-Stokes equation to be compatible with the
contents of Einstein equation.
15.5 Conclusions
Besides having determined the precise form of the so called Komar current
we showed that for each LSTS (M; g; D; g ; ") representing a gravitational
…eld in GRT which contains an arbitrary Killing vector …eld A, the …eld
F = dA (where A = g(A; )) satis…es Maxwell like equations dF = 0;
F = JA with JA given by Eq.(15.31). Moreover we showed that for
g
Killing 1-form …eld A = g(A; ), when some identi…cations of the components of A and the variables entering the Navier-Stokes equation are
accomplished and in particular when the postulated nontrivial conditions
(Eq.(15.38)— F0i = (dA)0i = li di — is satis…ed, the Maxwell like equations for F and thus the ones for F can be written as a Navier-Stokes equation representing an inviscid ‡uid. Thus, the Maxwell and Navier-Stokes
like equations presented above17 are almost directly obtained from Einstein equation through thoughtful identi…cation of …elds. All …elds in our
approach live in a 4-dimensional background spacetime, namely Minkowski
spacetime and a LSTS (M; g; D; g ; ") is considered only a (sometimes useful) description of a gravitational …eld, as discussed in detail in Chapter 11.
We observe moreover that the results just presented are in contrast with
the very interesting and important studies in, e.g., [56, 180, 312] where it is
shown through some identi…cations that every solution for an incompressible Navier-Stokes equation in a (p + 1)-dimensional spacetime gives rise
to a solution of Einstein equation in (p + 2)-dimensional spacetime.18 It
is worth also to quote [291] where it is also suggested an interesting relation between Einstein equations and the Navier-Stokes equation. Finally
1 7 In
[387] a ‡uid satisfying a particular Navier-Stokes equation is also shown to be
approximately equivalent to Einstein equation. The approach here which follows the one
in [357] is completely di¤erent from the one in [387].
1 8 For other papers relating Einstein equations and Navier-Stokes equations we quote
also here that the authors in [46] show that cosmic censorship might be associated to
global existence for Navier-Stokes or the scale separation characterizing turbulent ‡ows,
and in the context of black branes in AdS 5 , Einstein equations are shown to be led to
the nonlinear equations of boundary ‡uid dynamics [47]. In addition, gravity variables
can provide a geometrical framework for investigating ‡uid dynamics, in a sense of a
geometrization of turbulence [112].
450
15. Maxwell, Einstein, Dirac and Navier-Stokes Equations
we remark that it is clear that we can …nd examples [376] of Lorentzian
spacetimes that do not have any nontrivial Killing vector …eld19 and of
course, for such cases, it is not possible to …nd a Navier-Stokes equation
encoding Einstein equation.
As a last remark, we observe that the Killing vector …eld of the Schwarzschild spacetime given by Eq.(15.50) is such that F = dA is given by
F = 2 21 and thus F 2 6= 0. Thus, for this case taking into account the
MDE of …rst kind discussed in Chapter 13 we can also encode the contents
of Einstein equation in Maxwell like, Navier-Stokes like and Dirac-Hestenes
like equations. All the …elds entering the original named equations are, of
course, of very di¤erent nature, but it seems to the authors a noteworthy fact that for the case just studied the very di¤erent equations may
have their contents encoded by equations resembling the most important
equations of XX-th century Physics.
1 9 Although, as asserted in Weinberg [446], all Lorentzian spacetimes that represent
gravitational …elds of physical interest possess some Killing vector …elds.
This is page 451
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16
Magnetic Like Particles and Elko
Spinor Fields
16.1 Introduction
Modern Cosmology based on GRT implies that our universe is permeated
by dark matter and dark energy. Elko spinor …elds have been introduced in
[6, 7] to supposedly describe dark matter. These objects are dual helicity
eigenspinors of the charge conjugation operator satisfying Klein-Gordon
equation and carrying according to the authors of [6, 7] mass dimension 1
instead of mass dimension 3=2 carried by Dirac spinor …elds. A considerable
number of interesting papers have been published in the literature on these
intriguing objects in the past few years. In particular, according to the
theory in [6, 7] the anticommutator of a second quantized elko spinor …eld
with its conjugate momentum is nonlocal and thus it is claimed that the
theory possess an axis of locality which implies also that the theory of
elko spinor …elds break Lorentz invariance. We discuss this issue below
(Section 16.7) which according our view is an odd and acceptable feature
of the theory in [6, 7]. We recall in section 16.2 that di¤erently from the
theory described in [6, 7] where a second quantized elko spinor …eld satis…es
a Klein-Gordon equation (instead of a Dirac equation) the classical elko
spinor …elds of and types satisfy by their construction a csfopde that
is Lorentz invariant. The csfopde once interacted leads to Klein-Gordon
equations for the and type …elds. However, since the csfopde is the basic
one and since the Klein-Gordon equations for and possess solutions that
are not solutions of the csfopde for and we think that it is not necessary
to get the …eld equations for and from a Lagrangian where those …elds
452
16. Magnetic Like Particles and Elko Spinor Fields
have mass dimension 1 as in [6, 7]. Indeed, we claim that we can attribute
mass dimension of 3=2 for these …elds as it is the case for Dirac spinor
…elds. A proof of this fact is o¤ered by deriving in Section 16.3 the csfopde
for
and from a Lagrangian where these …elds have mass dimension
3=2. This, fact is to be contrasted with the quantum theory of these …elds
as presented in [6, 7, 10, 11, 329, 330, 390] (and references therein), namely
that elko …elds have mass dimension 1.
Taking seriously the view that elko spinor …elds due to the special properties given by their bilinear invariants may be the description of some kind
of particles in the real world a question then arises: what is the physical
meaning of these …elds?
In what follows we propose on Section 16.4 that the …elds and (the
representatives in the Cli¤ord bundle C`(M ; ) of the covariant spinor …elds
and ) serve the purpose of building Cli¤ord valued multiform …elds,
i.e., K 2 C`0 (M ; ) R01;3 and M 2 sec C`0 (M ; ) R01;3 (see Eq.(16.40)).
These …elds are electrically neutral but carry magnetic like charges which
permit that they couple to a su(2) ' spin3;0 R01;3 valued potential A 2
V1
sec T M spin3;0 . If the …eld A is of short range the particles described
by the K and M may be interacting and forming a system of spin zero
particles with zero magnetic like charge and eventually form condensates
something analogous to dark matter, in the sense that they do not couple
with the electromagnetic …eld and are thus invisible.
In Section 16.5 we investigate the similarities and main di¤erence between Majorana and elko spinor …elds. We observe that elko and Majorana
…elds are in class 5 of Lounesto classi…cation [217] and although an elko
spinor …eld does not satisfy the Dirac equation as correctly claimed in [6, 7],
a Majorana spinor …eld M : M ! C4 which is a dual helicity object according to some authors (see e.g., [224]) does satisfy the Dirac equation.
However this statement is not correct. An operator (quantum) Majorana
…eld M can satisfy Dirac equation if it is not a dual helicity object (see
Section 16.5.3). Also, even at a “classical level” a Majorana spinor …eld
satis…es Dirac equation if for any x 2 M their components take values in
a Grassmann algebra. Also, di¤erently from the case of elko spinor …elds
some authors claim that Majorana …elds are not dual helicities objects
[6], a statement that is correct only for Majorana quantum …elds as constructed in Section 16.5.3. For a Majorana …eld (even at “classical level”)
whose components take values in a Grassmann algebra the statement is not
correct.
Finally, since according to our …ndings the elko spinor …elds as well as
the …elds K and M are of mass dimension 3=2 we show in Section 16.6 how
to calculate the correct propagators for K and M. We also show that the
causal propagator for the covariant and …elds (of mass dimension 3=2)
is simply the standard Feynman propagator of Dirac theory.
16.2 Dictionary Between Covariant and Dirac-Hestenes Spinor Fields Formalisms
In presenting the above results we use the representation of spinor …elds
in the Cli¤ord bundle formalism (CBF) as developed in Chapter 8. For
for the convenience of the reader the necessary results are summarized in
Section 16.2 where a useful translation for the standard matrix formalism
to the CBF is given. The CBF makes all calculations easy and transparent
and in particular permits to infer [325] in a while that elko spinor …elds are
class 5 spinor …elds in Lounesto classi…cation [217, 325]. In Section 16.7 we
present a note on the calculation of the anticommutator of elko spinor …elds
of mass dimension 1 as introduced originally in [6] which implies nonlocality
and worse,we show that it leads to an odd (and unacceptable) inference,
namely breakdown of rotation invariance and Lorentz by a simple choice by
an observer of the (x; y; z) of his laboratory! For completeness of our study
on Dirac, Majorana and elko spinor …elds we present also in Section 16.8 a
new representation for the parity operator acting on Dirac-Hestenes spinor
…elds which although not well known is really noteworthy. In Section 16.9
we present our conclusions.
Remark 612 The contents of this Chapter has been …rst published in
[285].
16.2 Dictionary Between Covariant and
Dirac-Hestenes Spinor Fields Formalisms
Let (M ' R4 ; ; D; ) be the Minkowski spacetime structure where 2
sec T20 M is Minkowski metric and D is the Levi-Civita connection of .
V4
Also,
2 sec T M de…nes an orientation. We denote by 2 sec T02 M
the metric of the cotangent bundle. It is de…ned as follows. Let fx g be
coordinates for M in the Einstein-Lorentz-Poincaré gauge. Let fe =
@=@x g a basis for T M and f = dx g the corresponding dual basis for
T M , i.e.,
(e ) = . Then, if =
then =
e
e ,
where the matrix with entries
and the one with entries
are the equal
V1
to the diagonal matrix diag(1; 1; 1; 1). If a; b 2 sec T M we write
a b = (a; b). We also denote by f g the reciprocal basis of f = dx g,
which satis…es
= .
We denote the Cli¤ord bundle of di¤erential forms on Minkowski spacetime by C`(M; ) and recall once more the fundamental relation
+
=2
:
(16.1)
As we know from Chapter 3 (covariant) spinor …elds carrying a (1=2; 0)
(0; 1=2) representation of Spin01;3 ' Sl(2; C) belongs to one of the six
Lounesto classes [217]. Moreover, we recall that a (1=2; 0) (0; 1=2) spinor
…eld in Minkowski spacetime is an equivalence class of triplets ( ; ; )
where for each x 2 M; (x) 2 C4 ,
is an orthonormal coframe and
453
454
16. Magnetic Like Particles and Elko Spinor Fields
= u 2 Spin01;3 (M; ) C`(M; ) is a spinorial frame. If we …x a …ducial
global coframe 0 = f g and take, e.g., 0 = u0 = 1 2 Spin01;3 (M; )
C`(M; ) the triplet ( 0 ; 0 ; 0 ) is equivalent to ( ; ; ) if
=
=
( u) ( u 1 ) and (x) = S(u) 0 (x) where S(u) is the standard (1=2; 0)
(0; 1=2) matrix representation of Sl(2; C). Dirac gamma matrices in standard and Weyl representations will be denoted in this
and
V1Chapter by
0
and are not to be confused with the
2 sec T M ,! C`(M; ).
As well known the gamma matrices also satisfy
+
=2
and
0
0
+ 0 0 =2
. The relation between the
and the 0 is given
by
0
=S S 1
(16.2)
where1
1
1 1
:
(16.3)
S=p
1
1
2
We recall also that a representation of a (1=2; 0) (0; 1=2) spinor …eld
in the Cli¤ord bundle is an equivalence class of triplets ( ; ; ) where
2 sec C`0 (M; ) (the even subbundle of sec C`(M; )), is an orthonormal
C`(M; ) is a spinorial frame. If
coframe and u = u 2 Spin01;3 (M; )
we …x a …ducial global coframe 0 = f g and take u0 = u0 = 1 2
sec C`(M; ) the triplet ( 0 ; 0 ; 0 ) is equivalent to2
sec Spin01;3 (M; )
= (u) (u 1 ) and = 0 u 1 . Field is called an
( ; ; u ) if
=
operator spinor …eld and the operator spinor …elds belonging to Lounesto
classes 1; 2; 3 are also known as Dirac-Hestenes spinor …elds.
If
, = 0; 1; 2; 3 are the Dirac gamma matrices in the standard representation and f g are as introduced above, we de…ne
V2
T M ,! sec C`0 (M; ), k = 1; 2; 3;
(16.4)
k := k 0 2 sec
V4
i = 5 := 0 1 2 3 2 sec T M ,! sec C`(M; );
(16.5)
5
:=
0
1
2
3
2 Mat(4; C):
(16.6)
Then, to the covariant spinor
:pM ! C4 (in standard representation
of the gamma matrices) where (i =
1, ; & : M ! C2 )
0
1
m0 + im3
B
C
m2 + im1
C;
=
=B
(16.7)
@
A
&
n0 + in3
n2 + in1
there corresponds the operator spinor …eld
=
+&
3
= (m0 + mk i
k)
2 sec C`0 (M; ) given by
+ (n0 + nk i
k ) 3:
(16.8)
1 We will supress the writing of the 4
4 and the 2 2 unity matrices when no
confusion arises.
2 Take notice that ( ; ;
(u 1 ) =
u ) is not equivalent to ( ; ;
u ) even if (u)
( u) ( u 1 ).
16.2 Dictionary Between Covariant and Dirac-Hestenes Spinor Fields Formalisms
We then have the useful formulas in Eq.(16.9) below that one can use
to immediately translate results of the standard matrix formalism in the
language of the Cli¤ord bundle formalism and vice-versa
$
i
i
0;
$
$
5
y
21
3
= i
=
3 0;
$ ~;
=
y
0
$
0
~ 0;
$
3;
2
2:
(16.9)
Remark 613 Note that
; i14 and the operations and y are for each x 2
M mappings C4 ! C4 . Then they are represented in the Cli¤ ord bundles
formalism by extensor …elds which maps C`0 (M; ) ! C`0 (M; ). Thus, to
the operator
there corresponds an extensor …eld, call it
: C`0 (M; )
0
! C` (M; ) such that
=
0.
Using the above dictionary the standard Dirac equation for a Dirac spinor
…eld : M ! C4
i @
m =0
(16.10)
translates immediately in the so-called Dirac-Hestenes equation, i.e.,
@
21
m
0
= 0;
(16.11)
where @ is the Dirac operator acting on C 2 sec C`(M; ), which using the
basis introduced above is simply
@C : =
y(@ C) +
^ (@ C)
(16.12)
Remark 614 It is sometimes useful, in particular when studying in Section 16.6 solutions for the Dirac-Hestenes equation to consider besides the
Cli¤ ord bundle of di¤ erential forms C`(M; ) also the Cli¤ ord bundle of
multivector …elds C`(M; ). We will write 2 sec C`(M; ) for the sections
of the C`(M; ) bundle. The Dirac-Hestenes equation in C`(M; ) is.
@ e21
m e0 = 0:
where e e + e e = 2
and @ := e @ with e :=
the basis introduced above)
@ C := e y(@ C)+e ^ (@ C);
(16.13)
and (when using
(16.14)
for C 2 sec C`(M; ). Keep in mind that in de…nition of @ the e are
not supposed to act as a derivatives operators, i.e., e y(@ C) (respectively
e ^ (@ C)) is the left contraction of e with @ C (respectively, the exterior
product of e with @ C).
455
456
16. Magnetic Like Particles and Elko Spinor Fields
The basic positive and negative energy solutions of Eq.(16.10) which are
eigenspinors of the helicity operator are [385]
u(1) (p)e
ip x
;
u(2) (p)e
ip x
v(1) (p)eip
;
x
;
v(2) (p)eip
x
:
(16.15)
The u( ) (p) and v( ) (p) ( = 1; 2) are eigenspinors of the parity operator3
P, i.e.,
Pu( ) (p) = u( ) (p); Pv( ) (p) = v( ) (p);
(16.16)
which makes Dirac equation invariant under a parity transformation. This
will be discussed below. These …elds are represented in the Cli¤ord bundle
formalism by the following operator spinor …elds,
u(r) (p) =L(p){ (r) ;
where { (r) = f1; i
2g
v (r) (p) =L(p){ (r)
3;
(16.17)
and L(p) is the following boost operator4
~
satisfying L(p)L(p)
= 1:
L(p) = p
0
p
+m
2m(E + m)
;
(16.18)
Remark 615 Recall that Dirac-Hestenes
spinor …elds couple to the elecV1
tromagnetic potential A 2 sec T M ,! sec C`(M; ) as
@
21
m
0
+ eA = 0:
(16.19)
As it is well known this equation is invariant under a parity transformation
of the …elds A and .
In [6] the following (covariant) self and anti-self dual elko spinor …elds
0s;a
0s;a
0s;a
4
f +g f+ g ; f +g : M ! C which are eigenspinors of the charge
5
conjugation operator (C) are de…ned using the Weyl (chiral) representation of the gamma matrices by
0s;a
f+ g ;
0s
f
g (p)
=
2 [ L (p)]
L
(p)
0a
f
;
g (p)
=
2 [ L (p)]
L
(p)
;
(16.20)
0s
f
g (p)
=
(p)
2 [ R (p)]
R
;
0a
f
g (p)
=
(p)
2 [ R (p)]
R
;
(16.21)
3 The parity operator acting on covariant spinor …elds is de…ned as in [6], i.e., P =
i 0 R, where R changes p 7! p and changes the eingenvalues of the helicity operator.
For other possibilities for the parity operator, see e.g., page 50 of [41].
4 Recall that p 0 = p
0 = E + p.
2
5 The conjugation operator used in [6] is C
=
. Using the dictionary given
by Eq.(3.68) we …nd that in the Cli¤ord bundle formalism we have C =
20 .
16.2 Dictionary Between Covariant and Dirac-Hestenes Spinor Fields Formalisms
0a
where the C 0s = + 0s , C 0a =
and the indices f+ g; f +g refers
to the helicities of the upper and down components of the elko spinor …elds,
and where as in [6] we introduce the following helicity eigenstates6 , +
L (0)
p
and L (0) and +
we
have
(0)
and
(0)
such
that
with
p
^
R
R
jpj
p
jpj
p
jpj
L (0)
:=
L (0);
R (0)
:=
R (0);
p
[
jpj
p
[
jpj
2 ( L (0))
]=
2 ( R (0))
[
]=
2 ( L (0))
[
2 ( R (0))
K 0;1=2 = e 2
e
D0;1=2 rep(16.23)
2
we have, e.g., taking
0s
f
=p
r
E+m
+g (p) =
m
]:
(16.22)
Also recall that being a general boost operator in the D1=2;0
resentation of Sl(2; C)
K = K 1=2;0
];
1
jpj
E+m
0s
f +g (0);
(16.24)
More details, if necessary, may be found in [6].
Remark 616 By dual helicity …eld we simply mean here that the formulas
in Eq.(16.22) are satis…ed. Note that the helicity operator (in both Weyl
and standard representation of the gamma matrices) is
!
p
0
p
jpj
=
:
(16.25)
p
0
jpj
jpj
C4 -valued spinor …elds depends for its de…nition of a choice of an inertial frame where the momentum of the particle is (p0 ; p). The operator
p
(K 1=2;0 K 0;1=2 ) commutes with
is proportional to
jpj only if
p
jpj . So, the statement in [6] that the helicity operator commutes with the
boost operator must be quali…ed. However, it remains true that 2 [ +
l (p)]
and +
(p)
have
opposite
helicities
for
any
p.
l
Remark 617 Recall that a C4 -valued spinor …eld 0s
f +g (p) given in the
Weyl representation of the gamma matrices is represented by sf +g (p) in
the standard representation of the gamma matrices. We have
s
f
+g (p)
=S
0s
f
1
=p
2
1
1 1
=p
1
1
2
+
+
2 [ L (p)] + L (p)
+
+
2 [ L (p)]
L (p)
+g (p)
+
2 [ L (p)]
+
L (p)
(16.26)
6 The indices L and R in
L (p) and L (p) refer to the fact that these spinors …elds
transforms according to the basic non equivalent two dimensional representation of
Sl(2; C).
457
458
16. Magnetic Like Particles and Elko Spinor Fields
and then
+
2 [ L (p)]
+
2 [ L (p)]
p 1
p
jpj 2
0
+
+
L (p)
+
L (p)
+
2 [ L (p)]
+
2 [ L (p)]
1
=p
2
+
+
L (p)
+
L (p)
:
(16.27)
Eq.(16.26) and Eq.(16.27) show that the labels f +g (and also f+ g)
as de…ning the helicities of the upper and down C2 -valued components of a
type spinor …eld in the standard representation of the gamma matrices
have no meaning at all.
Also, in [6] the following identi…cations are made:
s
f+ g (p)
a
f+ g (p)
= +i
=
i
a
f+ g (p);
s
f+ g (p);
s
f +g (p)
a
f +g (p)
=
a
f +g (p);
s
f +g (p):
i
= +i
(16.28)
Moreover, we recall that the elko spinor …elds are not eigenspinors of the
parity operator and indeed (see Eq.(4.14) and Eq.(4.15) in [6]),
s
f +g (p)
s
f+ g (p)
a
f +g (p)
a
f+ g (p)
P
P
P
P
a
f+ g (p)
a
f +g (p)
s
f+ g (p)
s
f +g (p)
= +i
=
i
=
i
= +i
=
=
=
=
s
f+ g (p);
s
f +g (p);
a
f+ g (p);
a
f +g (p):
(16.29)
Then if s;a (x) := s;a (p) exp( s;a ip x ), with s = 1 and a = +1 we
have due to their construction that the elko spinor …elds must satisfy the
following csfopde:
i
@
i
@
i
@
i
@
s
f +g
a
f +g
s
f+ g
a
f+ g
+m
m
m
+m
a
f+ g
s
f+ g
a
f +g
s
f +g
s;a
s;a
If s;a
f+ g ; f +g ; f+ g ;
the covariant spinors s;a
f+
satisfy the csfopde:
@
@
@
@
s
f +g
a
f +g
s
f+ g
a
f+ g
21
+m
21
m
21
m
21
+m
a
f+ g
s
f+ g
a
f +g
s
f +g
= 0;
i
@
= 0;
i
@
= 0;
i
@
= 0;
i
@
a
f +g
s
f +g
a
f+ g
s
f+ g
+m
m
m
+m
s;a
0
f +g 2 sec C` (M; ) are
s;a
s;a
s;a
g,
f +g , f+ g , f +g
0
= 0;
@
0
= 0;
@
0
= 0;
@
0
= 0;
@
a
f +g
s
f +g
a
f+ g
s
f+ g
21
+m
21
m
21
m
21
+m
s
f+ g
a
f+ g
s
f +g
a
f +g
= 0;
= 0;
= 0;
= 0:
(16.30)
the representatives of
: M ! C4 then they
s
f+ g
a
f+ g
s
f +g
a
f +g
0
= 0;
0
= 0;
0
= 0;
0
= 0:
(16.31)
16.3 The CSFOPDE Lagrangian for Elko Spinor Fields
459
Remark 618 From Eq.(16.31) it follows trivially that the operator spinor
s;a
s;a
s;a
0
…elds s;a
f+ g ; f +g ; f+ g ; f +g 2 sec C` (M; ) satisfy Klein-Gordon equations. However, e.g., the Klein-Gordon equations
+ m2
s
f +g
s
f +g
+ m2
a
f+ g
= 0;
a
f+ g
= 0;
(16.32)
possess (as it is trivial to verify) solutions that are not solutions of the
csfopde satis…ed sf +g and af+ g . An immediate consequence of this observation is that attribution of mass dimension 1 to elko spinor …elds
seems equivocated. elko spinor …elds as Dirac spinor …elds have mass dimension 3=2, and the equation of motion for the elkos can be obtained from
a Lagrangian (where the mass dimension of the …elds are obvious) as we
recall next.
16.3 The CSFOPDE Lagrangian for Elko Spinor
Fields
A (multiform) Lagrangian that gives the Eqs.(16.31) for the operator elko
spinor …elds sf +g ; af +g ; af+ g ; sf +g 2 sec C`0 (M; ) having mass dimension 3=2 is:
1
L=
2
(
(@ sf+
+(@
gi 3)
s
f +g
s
a
f+ g + (@ f +g i 3 )
s
2m sf+ g
3)
f +g
i
a
a
a
f +g + (@ f+ g i 3 )
f+ g
a
a
s
+
2m
f +g
f +g
f+ g
(16.33)
We know from Chapter 8 that the Euler-Lagrange equation obtained,
from the variation ,e.g., of the …eld Sf+ g is:
@
S
f+
g
L
@ @@
s
f+
g
L = 0:
(16.34)
We have immediately7
@
@@
@ @@
1 s
@
i 3 m
2 f+ g
1
@@ sf+ g @ sf
L=
+g
2
1
L = + @ sf+ g i 3 :
g
2
s
f+
s
f
s
f+
Recalling that i
g
L=
3
=
0 1 2
@
a
f +g ;
+g )
s
f+ g i 3
=
1
2
s
f+ g i 3 ;
(16.35)
the resulting Euler-Lagrange equation is
s
f+ g 21
m
a
f +g 0
= 0:
7 In the second line of Eq.(16.35) we used the identity (KL) M = K (M L)
~ for all
K; L; M 2 sec C`(M; ):
)
460
16. Magnetic Like Particles and Elko Spinor Fields
Remark 619 With this result we must say that the main claim concerning
the attributes of elko spinor …elds appearing in recent literature , i.e., that
these objects are of mass dimension 1, seems to us not necessary if not
equivocated and the question arises: which kind of particles are described
by these …elds and to which gauge …eld do they couple? This question is
answered in the next section.
16.4 Coupling of the Elko Spinor Fields a
su(2) ' spin3;0 valued Potential A
We start by introducing Cli¤ord valued di¤erential multiforms …elds, i.e.,
the objects
K=
M=
s
f +g
s
f +g
a
f+ g
1
a
f+ g
1
i
2
i
2
2 sec C`0 (M; )
R01;3
2 sec C`0 (M; )
R01;3
(16.36)
where 1; 2; ; 3 are the generators of the Pauli algebra R3;0 ' R01;3 and
i : = 1 2 3 . So, we have i := i 0 where the
are the generators of
R1;3 , i.e.,
+
= 2 . Also, i : = 1 2 3 = 0 1 2 3 =: 5 .
We de…ne the reverse a general Cli¤ord valued di¤erential multiforms
…eld
1 ikj
1 k
N
R01;3 ;
N = N0 1+Nk
i j+
i k j 2 sec C`(M; )
k; + N
2
3!
(16.37)
where N 0 ; N k ; N k ; N ikj 2 sec C`(M; ) by
e =N
e0
N
ek
1+N
k;
1 e ij
+ N
2
j i
+
1 e ijk
N
3!
k j i
(16.38)
Since, as well known the 1; 2; ; 3 have a matrix representation in C(2),
namely 1; 2; ; 3 , a set of Pauli matrices, we have the correspondences
!
!
K$
s
f +g
a
f+ g
a
f+ g
s
f +g
;
We observe moreover that
1
K = K (1 +
2
3 )$
s
f +g
a
f+ g
s
f+ g
a
f +g
M$
!
0
0
1 1
; M = K (1 +
2 2
a
f +g
s
f+ g
3 )$
(16.39)
s
f+ g
a
f +g
0
0
!
(16.40)
Then, from Eqs.(16.31) we can show that the K and M …elds satisfy the
following linear partial di¤erential equations
@K
@M
mKi
2 0
= 0;
(16.41)
+ mMi
2 0
= 0:
(16.42)
21
21
16.4 Coupling of the Elko Spinor Fields a su(2) ' spin3;0 valued Potential A
Indeed, Ki
i 2
s
f +g
a
= i 2 K = sf +g i 2
1, Mi
f+ g
1 and we have the correspondences:
2
a
f +g
s
f +g
a
f+ g
K$
a
f+ g
s
f +g
Ki 2 $
!
a
f+ g
s
f +g
!
s
f +g
a
f+ g
; M$
; Mi 2 $
2
s
f+ g
a
f +g
a
f +g
s
f+ g
a
f +g
s
f+ g
s
f+ g
a
f +g
= iM
!
!
461
2
=
;
: (16.43)
Then, from Eqs.(16.41) and (16.42) we see that K and M satisfy the
following linear partial di¤erential equations
@K
21
+ im 2 K
0
= 0;
(16.44)
@M
21
im 2 M
0
= 0;
(16.45)
which, on taking the corresponding matrix representation gives the coupled equations for the pairs ( sf +g ; af+ g ) and ( sf+ g ; a +g ) appearing
in Eqs.(16.31).
Before proceeding we observe that the currents
e 2 sec V1 T M spin3;0 ,! sec C`(M; ) R0 ;
J K = K 1 0K
(16.46)
1;3
V1
0
f
J M = M 1 0 M 2 sec T M spin3;0 ,! sec C`(M; ) R1;3 ; (16.47)
are conserved, i.e.,
@yJ K = 0, @yJ M = 0:
(16.48)
Indeed, let us show that @yJ K = 0. We have
@yJ K =
1
@K
2
From Eq.(16.41) we have
@K = imK
2 012 ;
Then,
e@ = @ K
e
K
1
(imK
2
im
=
(K(
2
@yJ K =
e
e
1 0 K+K 1 0 K @
= im
e + imK
2 012 1 0 K
2 1
+
(16.49)
e
012 2 K:
(16.50)
e
12 1 2 K)
e = 0:
1 2 ) 12 K
The …elds K and M are electrically neutral, but they can couple with
an su(2) ' spin3;0 R3;0 valued potential
A = Ai
i
2 sec
V1
T M
spin3;0 ,! sec C`(M; )
R1;3 :
(16.51)
462
16. Magnetic Like Particles and Elko Spinor Fields
Indeed, we have taking into account that i =
is
@K
@M
mK
21
21
+ mM
5
5
20 0
20 0
5; i
+q
=
5 AK
+q
5 AM
that the coupling
i0
= 0;
(16.52)
= 0:
(16.53)
Equations (16.52) and (16.53) are invariant under the following transformation of the …elds and change of the basis of the spin3;0 R00
1;3 algebra:
0
K 7! K0 = e
A 7! A = e
5q
i
i0
5q
Ae
i
i0
5q
K; M 7! M0 = e
i
i0
;
i
7!
0
i
=e
5q
5q
i
i0
i
M;
i0
ie
5q
i
:
(16.54)
i0
With the above result we propose that elko spinor …elds of the and
types,are the crucial ingredients permitting the existence of the K and M
…elds which do not carry electric charges but possess magnetic 8 like charges
that couple to an spin3;0 R00
1;3 valued potential A.
16.5 Di¤erence Between Elko and Majorana Spinor
Fields
Here we recall that a Majorana …eld (also in class …ve in Lounesto classi…cation9 and supposedly describing a Majorana neutrino) di¤erently from
an elko spinor …eld is supposed in some textbooks to satisfy the Dirac
equation (see, e.g., [224]), even if that equation cannot be derived from a
Lagrangian (unless, as it is well known the components of Majorana …elds
for each x 2 M are Grassmann ‘numbers’). The “proof” in [224] for the
statement that a Majorana …eld 0M : M ! C4 satis…es the Dirac equation
is as follows. That author writes that r : M ! C2 and l : M ! C2
belonging respectively to the carrier spaces of the representations D0;1=2
and D1=2;0 of Sl(2; C) satisfy
i@
i@
r
= m l;
(16.55)
l
=m
(16.56)
r;
8 The use of the term magnetic like charge here comes from the analogy to the possible
coupling of Weyl …elds
describing massless magnetic monoples with the electromagnetic
V
potential A 2 sec 1 T M . See Chapter 13.
9 We mention Dirac spinor …elds are the real type fermion …elds and that Majorana
and Elko spinor …elds are the imaginary type fermion …elds according to Yang and
Tiomno [453] classi…cation of spinor …elds according to their transformation laws under
parity .
16.5 Di¤erence Between Elko and Majorana Spinor Fields
i
with
= (1; i ) and
= (1;
) where
matrices. From this we can see that we can write:
0
i
0
r
@
0
i
(=
r
=m
l
i)
463
are the Pauli
:
(16.57)
l
0
is ( as well known) a represen0
tation of Dirac matrices in Weyl representation,. It follows that 0 satisfy
the Dirac equation, i.e.,
The set of matrices
:=
0
i
0
@
0
m
= 0:
(16.58)
Have saying that, [224] de…nes a Majorana …eld (in Weyl representation)
by
0
M
=
i
0
l
=
l
2
i
r
;
(16.59)
l
and write
0
M
@
0
M
m
= 0;
(16.60)
concluding his “proof”.
Now, let us investigate more deeply that “proof”. First recall that writing
r (x)
=
r (p)e
ip x
;
l (x)
=
l (p)e
ip x
;
(16.61)
we have from Eq.(16.55) and Eq.(16.56) that
(p0
p)
r (p)
=
(p0 +
p) l (p) =
m l (p);
(16.62)
m
(16.63)
r (p):
However, if l (0) and r (0) are the zero momentum
p …elds we have (with
(
1)=2) with
=
{p
being the boost parameter, i.e., sinh {=2 =
1= 1 v 2 and n the direction of motion) by de…nition:
r (p)
l (p)
1
:= e 2 { r (0) = (cosh {=2 +
p0 + m + p
=
r (0);
[2m(p0 + m)]1=2
1
:= e 2 { l (0) = (cosh {=2
p0 + m
p
=
l (0):
1=2
[2m(p0 + m)]
n sinh {=2) l (0)
(16.64)
n sinh {=2) l (0)
(16.65)
We can now verify that Eq.(16.64) and Eq.(16.65) only imply Eq.(16.62)
and Eq.(16.63) if10
(16.66)
l (0) =
r (0):
1 0 That
: M ! C4 to satisfy
l (0) =
r (0) is a necessary condition for a spinor …eld
Dirac equation can bee seem, e.g., from Eq.(2.85) and Eq.(2.86) in Ryder’s book [365].
However, Ryder misses the possible solution l (0) =
r (0). This has been pointed by
Ahluwalia [8] in his review of Ryder’s book.
464
16. Magnetic Like Particles and Elko Spinor Fields
But this condition cannot be satis…ed by a Majorana …eld 0M : M ! C4
as de…ned by [224] where r (0) = i 2 l (0). Indeed, writing tl (0) = ( ; w)
with v; w 2 C we see that to have l (0) =
= ! and
r (0) we need
!=
, i.e., = ! = 0. We conclude that a Majorana …eld 0M : M ! C4
cannot satisfy the Dirac equation.
16.5.1 Some Majorana Fields are Dual Helicities Objects
Before continuing we recall also that it is a well known fact (see, e.g.,[145])
that the Dirac Hamiltonian commutes with the operator
p
^ given by
Eq.(16.25). Thus any
: M ! C4 satisfying Dirac equation which is an
eigenspinor of the Dirac Hamiltonian may be constructed such that l and
0
4
r have the same helicity. Since a Majorana spinor …eld
M : M ! C
as de…ned by [224] does not satisfy Dirac equation we may suspect that it
is not an eigenspinor of the of the operator
p
^ . And indeed this is the
case, for we now show l (0) and r (0) in a Majorana …eld 0M : M ! C4
are not equal. Taking the momentum (without loss of generality) in the
direction of the z-axis (of an inertial frame) and tl (0) = (1; 0) we have
p
^ l (0) =
l (0),
p
^ (i
2
l (0))
=
i
2
l (0),
(16.67)
and as the elko spinor …elds they are also dual helicities objects.
Remark 620 Keep also in mind that as well known even if a Majorana
…eld is described by a …eld [235] ' : M ! C2 carrying the D1=2;0 (or D0;1=2 )
representation of Sl(2; C) the value of the helicity obviously depends on the
inertial reference frame where the measurement is done [41, 289] because
the helicity is invariant only under those Lorentz transformations which did
not alter the direction of p along which the angular momentum component
is taken.
16.5.2 The Majorana Currents J M and J 5M
We observe moreover that if a Majorana …eld 0M : M ! C4 should satisfy
the Dirac equation then Eq.(16.60) should translate in the Cli¤ord bundle
formalism as
@ M 21 m M 0 = 0;
(16.68)
V
1
where M 2 sec C`0 (M; ). Then, current J M = M 0 ~M 2 sec T M ,!
sec C`(M; ) is conserved as it is trivial to verify. Moreover, it is lightlike (since for a class …ve spinor …eld ~M M = 0 and thus J M J M =
~
~
M 0 ( M M ) 0 M = 0) but it is a non null covector …eld if the components of the spinor …eld 0M : M ! C4 have values in C2 . Indeed writing
J M = M 0 ~M = JM we see immediately that
0
JM
=
0
M
00
0
M
6= 0:
(16.69)
16.5 Di¤erence Between Elko and Majorana Spinor Fields
465
Also, the current
J 5M :=
M 3
~M = (
0
M
05
0
0
M)
(16.70)
is non null as it is easy to verify, and is also lightlike. If the Majorana
spinor …eld was to satisfy the Dirac equation the current J 5M would be also
conserved, i.e., @y J 5M = 0. In that case we would have a subtle question
to answer: how can a massive particle have associated to it currents J M
and J 5M that are ligthlike? What is the meaning of these currents?
Remark 621 Of course, the answer to the above question from the point
of view of a …rst quantized theory is that a Majorana …eld cannot carry any
electric or magnetic charge, i.e., the physical currents eM J M and qM J 5M
are null because eM = qM = 0.
16.5.3 Making of Majorana Fields that Satisfy the Dirac
Equation
Is it possible to construct a Majorana spinor …eld that satis…es the Dirac
equation?
There are two possibilities of answering yes for the above question.
First possibility: As, e.g., in [209] and [59] we consider ab initio a
Majorana …eld as a quantum …eld and which is not a dual helicity object.
Indeed, de…ne a Majorana quantum …eld as 0M as an operator valued …eld
0
02 0?
satisfying Majorana condition 0c
M =
M . This condition can
M :=
11
be satis…ed if we de…ne
Z
d3 p X
0
(u(p;s)a(p;s)eip x + v(p;s)ay (p;s)e ip x );
M (x) :=
(2 )3=2 s
(16.71)
with
fa(p;s); ay (p0 ;s0 )g =
ss0
(p
p0 );
fa(p;s); a(p0 ;s0 )g = 0
(16.72)
and where the zero momentum spinors u(0;s) and v(0;s) are
0
1
0 1
1
0
1 B
1 B
0 C
1 C
B
B
C
u(0;1=2) = p @
; u(0; 1=2) = p @ C
;
A
1
0 A
2
2
0
1
0
1
0
1
0
1
B
C
1 B 1 C
C ; v(0; 1=2) = p1 B 0 C ;
v(0;1=2) = p B
@
A
@
A
0
1
2
2
1
0
1 1 Here
0?
M
:=
R
d3 p
(2 )3=2
P
s (u
(p;s)ay (p;s)e
ip x
+ v (p;s)a(p;s)eip
(16.73)
x
)
466
16. Magnetic Like Particles and Elko Spinor Fields
satisfying
0
0 u(0;s)
0
0 v(0;s)
= u(0;s);
=
v(0;s)
(16.74)
Indeed, we can verify by explicit calculation that
m p 0 00
u(p;s) = p
v(0;s) (16.75)
2p0 (p0 + m)
m + p 0 00
u(0;s);
u(p;s) = p
2p0 (p0 + m)
and taking into account Eq.(16.74) we see that
(p
0
m)u(p;s) = 0;
(p
0
+ m)v(p;s) = 0:
(16.76)
With this results we can immediately verify that the quantum Majorana
…eld 0M (x) satisfy the Dirac equation,
0
(i
@
m)
0
M (x)
= 0:
(16.77)
For that Majorana …eld that is not a dual helicity object we can construct
in the canonical way the causal propagator that is nothing more than the
standard Feynman propagator for the Dirac equation (see, e.g.,[209]).
Remark 622 In a second quantized theory the currents J M and J 5M are
given by the normal product of the …eld operators and in this case the
0
0
current: M 00 0M : as well known is null, but : M 05 0 0M : is non
null. For a proof of these statements, see, e.g., [209] where in particular
a consistent quantum …eld theory for Majorana …elds satisfying the Dirac
equation is presented, showing in particular that the causal propagator for
that …eld is the standard Feynman propagator of Dirac theory.
Second possibility: In several treatises, e.g., [308, 423] even at the
“classical level” it is supposed that any Fermi …eld must be a Grassmann
valued spinor …eld, i.e., an object where tL = ( !) and v; ! : M ! G,
with G a Grassmann algebra [42, 101], i.e., v(x) and w(x) are Grassmann
elements of a Grassmann algebra for all x 2 M .
In this case it is possible to show that the Majorana …eld de…ned, e.g.,
in [308] by
0M
L
=
(16.78)
2 L
does satisfy the Dirac equation.
To prove that statement write tL = ( !) where for any x 2 M , v(x)
and !(x) take values in a Grassmann algebra. If 0M does satisfy Dirac
equation we must have for 0M (p) at p = 0 that
2 L
m
L
m
L
= 0:
(16.79)
2 L
Then we need simultaneously to satisfy the equations
L
=
2 L
and
L
=
2 L;
(16.80)
16.5 Di¤erence Between Elko and Majorana Spinor Fields
467
which at …rst sight seems to be incompatible, but are not. Indeed, from
L =
2 L we obtain
= i! and ! =
and from
L
=
2 L
i
(16.81)
we obtain
=
i! and ! = i
:
(16.82)
So, if we understand the symbol as denoting the involution de…ned by
Berizin 12 Eq.(16.81) is consistent if we take
=
and ! = ! :
(16.83)
But since for any c 2 C and ' 2 G it is (c') = c ' the equation
implies
v = (i! ) =
i!
=
i!
= i!
(16.84)
and since =
and ! = ! Eq.(16.80) implies = i! . Thus, surprisingly as it may be at …rst sight Eq.(16.81) is compatible with Eq.(16.82).
Claim 623 We may then claim that a Majorana …eld whose components
take values in a Grassmann algebra satis…es the Dirac equation. This is
consistent with the fact that Dirac equation under these conditions may be
derived from a Lagrangian [308, 423]. We can also verify that for such a
Majorana …eld the current JM = 0.
Remark 624 In resume, from the algebraic point of view there is no difference between elko spinor …elds ; : M ! C4 and Majorana spinor …elds
0
4
M : M ! C . However have in mind that the Majorana …eld de…ned in
[308] (Eq.(16.78) above) looks like an elko spinor …eld, but, of course, is not
the same object, since the components of an elko spinor …elds are for any
x 2 M complex numbers but the components of 0M in [308] take values
in a Grassmann algebra.
Of course, if we recall that in building a quantum …eld theory for elkos
make automatically the components of elko spinor …elds objects taking values in a Grassmann algebra we cannot see any good reason for the building
of a theory like in [6]. Instead we think that elko spinor …elds are worth
objects of study because they permit the construction of the K and M …elds
introduced above which may describe possible of “magnetic like” particles.
1 2 See
pages 66 and of [42]
468
16. Magnetic Like Particles and Elko Spinor Fields
16.6 The Causal Propagator for the K and M
Fields
We now calculate the causal propagator SF (x x0 ) for, e.g., the K 2 sec C`0 (M; )
R01;3 …eld. Recall from Remark 614 that the K …eld must satisfy
@ Ke21
mK
5 20 e0
+
5 q AK
= 0:
(16.85)
If Ki (x) is a solution of the homogeneous equation
@ Ki e21
mK i
5 20 e0
= 0;
we can rewrite Eq.(16.85)as an integral equation
Z
K(x) = Ki (x) + q d4 ySF (x; y)A(y)K(y)
5 20 5 :
(16.86)
Putting Eq.(16.86) in Eq.(16.85) we see that SF (x; y) must satisfy for an
arbitrary P 2 sec C`(M; ) R01;3
@SF (x
y)P(y)e21
mSF (x
y)P(y)e0 =
4
(x
y)P(y)
(16.87)
whose solution as it is easy to verify is [160]
SF (x
y)P(y) =
1 R 4 pP(y) + mP(y)e0
d p
e
(2 )4
p2 m2
ip (x
For the causal Feynman propagator we get with E = p0 =
SF (x
y)K(x) =
1
(t
2(2 )3
+
1
(t
2(2 )3
y )
p
:
(16.88)
p2 + m2
R
(pK(y) + mK(y)e0 )e21
t0 ) d3 p
e
E
R
(pK(y) mK(y)e0 )e21
t0 ) d3 p
e
E
ip (x
y )
ip (x
y )
:
(16.89)
For a scattering problem de…ning Ks = K Ki with Ki an asymptotic
in-state we get when t ! 1
Z
R
(pA(y)K(y) + mA(y)K(y)e0 )e21 ip (x y )
Ks (x) = q d4 y d3 p
e
2E
(16.90)
This permits to de…ne a set of …nal states Kf given by
Z
(pf A(y)K(y) + mA(y)K(y)e0 )e21 ip (x y )
Kf (x) = q d4 y
e
(16.91)
2Ef
which are plane waves solutions to the free …eld Dirac-Hestenes equation
with momentum pf . Equipped with the Ki (x) and Kf (x) we can proceed to
16.7 A Note on the Anticommutator of Mass Dimension 1 Elkos According to [6]
calculate the scattering matrix elements, Feynman rules and all that (see
details if necessary in [160]).
For the covariant and …elds the causal propagator is the standard
Dirac propagator SF (x x0 ). Indeed, it can be used to solve, e.g., the
csfopde
i
s
f +g 21
@
a
f+ g
+m
= 0;
i
@
a
f+ g
m
s
f +g
=0
(16.92)
once appropriate initial conditions are given. To see this it is only necessary
to rewrite the formulas in Eq.(16.92) as
i
i
@
@
s
f +g
a
f+ g
m
m
s
f +g
a
f+ g
=
m(
= m(
s
f +g
s
f +g
+
a
f+ g )
a
f+ g )
+
Eqs.(16.93) and (16.94) have solutions
Z
s
(x)
=
d4 ySF (x
f +g
Z
a
(x)
=
d4 ySF (x
f+ g
= ;
={
(16.93)
(16.94)
y) ;
(16.95)
y){
(16.96)
once we recall that
(i
@
m)SF (x
y) =
4
(x
y):
(16.97)
16.7 A Note on the Anticommutator of Mass
Dimension 1 Elkos According to [6]
According to the theory of elko spinor …elds as originally developed in [6]
(see also [7, 8, 9, 10, 11] the evaluation of the anticommutator of an elko
spinor …eld with its canonical momentum gives
Z
d3 p ip (x x0 )
e
G(p);
(16.98)
f (x;t); (x; tg = i (x x0 )I + i
(2 )3
with
G(p) :=
5
n (p);
(16.99)
where the spacelike p-dependent …eld n = n (p)e is
(n0 (p); n1 (p); n2 (p); n3 (p)) := (0; n(p));
p = (p cos ; p sin cos '; p sin sin ')
n(p) :=
=
(1 +
2
)
1 @
sin @'
1=2
; (1 +
p
jpj
2
)
(16.100)
= ( sin '; cos '; 0)
1=2
;0 ;
= py =px :
(16.101)
469
470
16. Magnetic Like Particles and Elko Spinor Fields
Putting
x0 it is13
=x
^ )=
G(
=
Z
d3 p ip (x
e
(2 )3
5 1
P( ) +
x0 )
G(p)
5 2
Q( ):
(16.102)
with
P( ) =
i
(
2
z)
y
(
2
x
+
2 ) 23
y
;
Q( ) =
i
(
2
z)
x
(
2
x
+
2 ) 32
y
(16.103)
Remark 625 In [284] the integral in Eq.(16.102) has been evaluated for
the case when lies in the direction of one of the spatial axes ei = @=@xi of
an arbitrary inertial reference frame e0 = @=@x0 (where x ; = 0; 1; 2; 3)
are coordinates in Einstein-Lorentz-Poincaré gauge naturally adapted to
e0 ).We note that the evaluation of each one of the integrals in [284] is cor^
rect, but they do not express the values of the Fourier transform G(x
x0 )
for the particular values of
used in the calculations of those integrals. It
is not licit to …x a priori Rtwo of the components of
as being null to cal0
culate the integral (2 ) 3 d3 peip (x x ) G(p) for this procedure excludes the
singular behavior in the sense of distributions of the Fourier integral. So,
it is wrong the statement in [284] that elko theory as constructed originally
in [6] is local. However, let us show now that this nonlocality is really odd.
16.7.1 Plane of Nonlocality and Breakdown of Lorentz
Invariance
^
When z 6= 0, G(x
x0 ) is null the anticommutator is local and thus there
exists in the elko theory as constructed in [6, 10] an in…nity number of
^
“ locality directions”. On the other hand G(x
x0 ) is a distribution with
support in z = 0. So;the directions
= ( x ; y ; 0) are nonlocal in each
^
arbitrary inertial reference frame e0 chosen to evaluate G(x
x0 ). Recall
that given an inertial (coordinate) reference frame frame e0 = @=@x0 in
Minkowski spacetime there exists [77] and in…nity of triples of vector …elds
(k)
(k)
(k)
fe1 = @=@x1(k) ; e1 = @=@x2(k) ; e1 = @=@x3(k) g (with x0 ; x1(k) ; x2(k) ; x3(k) ),
coordinates in Einstein-Lorentz-Poincaré gauge naturally adapted to e0 differing by a spatial rotation) which constitutes a global section of the frame
bundle. So, the labels x,y and z directions in inertial reference frame e0 are
no more than a mere convention and thus without any physical signi…cance.
1 3 This result has been also found by Ahluwalia and Grumiller. We …nd the above
result without knowing their calculations. See erratum to [284] at Phys. Rev. D 88,
129901 (2013).
16.8 A New Representation of the Parity Operator Acting on Dirac Spinor Fields
This means that the theory as constructed in [6] breaks in each inertial reference frame rotational invariance by a subjective choice of an observer
and since in di¤erent inertial references frames there are di¤erent (x; y)
planes the theory breaks also Lorentz invariance. This physically unacceptable feature of the theory of elko spinor …elds as constructed originally in
[6] was eventually the main reason that lead us to present investigation.
16.8 A New Representation of the Parity Operator
Acting on Dirac Spinor Fields
Let as before fe = @x@ g and fe = @x@ g be two arbitrary orthonormal frames for T M and let 0 = f
= dx g and
= f
= dx g
be the respective dual frames. Of course, e0 and e0 are inertial reference
frames and we suppose now that e0 is moving relative to e0 with 3-velocity
1
v = (v ; v 2 ; v 3 ), i.e.,
e0 = p
1
1
v2
e0
3
P
i=1
p
vi
v2
1
(16.104)
ei
Let u0 and u be the spinorial frames associated with 0 and . Consider a Dirac particle at rest in the inertial frame e0 (take as a …ducial
frame). The triplet ( 0 ; 0 ; 0 ) is the representative of the wave function
of our particle in ( 0 ; 0 ) and of course, its representative in ( ; ) is
( ; ; ). Now,
=u 0
(16.105)
where u describes in the spinor space the boost sending
to , i.e., = u
. Now, the representative of the parity operator in ( 0 ; 0 )
u 1=
is Pu0 and in ( ; ) is Pu ; We have according to our dictionary (Eq.(3.68))
that
Pu = 0 0 ; Pu0 0 = 0 0 0 ;
(16.106)
or
Pu
where
and
0
=
0
R ;
Pu0
0
=
0
R
0;
(16.107)
are Dirac ideal real spinor …elds. From Chapter 3 we have
=
1
(1 +
2
0
);
0
=
0
1
(1 +
2
0
);
(16.108)
and if the momentum of our particle is the covector …eld p = p
=p
with (p0 ; p1 ; p2 ; p3 ) := (m; 0) and (p0 ; p1 ; pp2 ; p3 ) := (E; p) (and of course
p =
p = 0 p0 ) R an the operator such that if = (p)eipx then
R = (p)e
ip x
= (p)e
i(p0 x0 pi xi )
:
(16.109)
471
472
16. Magnetic Like Particles and Elko Spinor Fields
Also uR = Ru and clearly R
uPu0 u
1
u
0
0
=u
R
0
=
0.
0
=u
Now,
0
u
1
Ru
0
=
0
R ;
(16.110)
from where it follows that
Pu = uPu0 u
Now we rewrite Pu
Pu
0
=
R
1
:
(16.111)
as
p0 0
u R
m
p0 0
=
m
=
p0
u
m
1
= p
m
0
=
0
1
u
u
:
0;
(16.112)
We conclude that the parity operator in an arbitrary orthonormal and
spin frames ( ; ) acting on a Dirac ideal spinor …eld is
P = Pu =
1
p
m
:
Of course, when applied to covariant spinor …elds
operator P is represented by
P =
1
p
m
:
(16.113)
: M ! C4 the
(16.114)
A derivation of this result using covariant spinor …elds (and which can
be easy generalized for arbitrary higher spin …elds) has been obtained in
[401].
16.9 Conclusions
In Chapter 13 (see also [351]) it was shown that the massless Dirac-Hestenes
equation decouples in a pair of operator Weyl spinor …elds, each one carrying opposite magnetic
like charges that couple to the electromagnetic
V1
potential A 2 sec T M in a non standard way14 Here we proposed
that the …elds and serves the purpose of building the …elds K;M 2
sec C`(M; ) R01;3 . These …elds are electrically neutral but carry magnetic
like charges which permit them to couple to a spin3;0 valued potential
V1
A 2 sec T M spin3;0 . If the …eld A is of short range the particles
described by the K and M may interact forming something analogous to
dark matter, in the sense that they may form a condensate of spin zero
1 4 In [211] it is proposed that the massless Dirac equation describe (massless) neutrinos
which carry pair of opposite magnetic charges.
16.9 Conclusions
473
particles with zero total magnetic like charges that do not couple with the
electromagnetic …eld and are thus invisible.
We obtained also the causal propagators for the K and M …elds, which
can be used to calculate scattering matrix elements, Feynman rules, etc.15
Before closing this last chapter of our book we observe yet that elko
spinor …elds already appeared in the literature before the publication of [6].
A history about these objects may be found in [102, 103]. In those papers a
Lagrangian equivalent to Eq.(3.79) written for the covariant spinor …elds
and is given. However, the author of those papers did not comment that
since the basic csfopde satis…ed by the elko spinor …elds is by construction
the ones given in Eq.(3.74) and as a consequence these …elds, contrary to
the claim of[6], must have mass dimension 3=2 and not 1:
We recalled also that as claimed in [6] an elko spinor …eld (of class …ve
in Lounesto classi…cation) does not satisfy the Dirac equation. According
to some claims in the literature (see, e.g., [224]) a Majorana spinor …eld
0
4
M : M ! C and which is a dual helicity object (that also belongs
to class …ve in Lounesto classi…cation) does satisfy the Dirac equation.
However we showed that this claim is equivocated. At “classical level” a
Majorana spinor …eld can satisfy Dirac equation only if its components for
any x 2 M take values in a Grassmann algebra.
It is important to emphasize in order to avoid misunderstandings that
the theory presented in this paper is an alternative theory to the one originally built in [6] and developed in a series of interesting and challenging
papers (see references). It di¤ers drastically from that theory. The main
di¤erences are that the equations satis…ed by our elko spinor …elds of mass
dimension 3=2 (see Eq.(16.31)) and their solutions are trivially Lorentz invariant. In the theory in [6] the elko spinor …elds are of mass dimension 1
and that theory breaks Lorentz invariance as shown in Section 16.7. Also
our theory gives a prediction of a new type of particle that is electrically
and magnetically neutral but has a magnetic like charge which can couple
with an spin3;0 valued gauge …eld. The other theory (for the best of our
understanding) does not …x the nature of the …eld that intermediates the
interaction of the particles described by their elko spinor …elds of mass
dimension 1.
1 5 At least, we can say that now we have all the ingredients to formulate a quantum
…eld theory for the K and M objects if one wish to do so.
474
16. Magnetic Like Particles and Elko Spinor Fields
This is page 475
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Appendix A
Principal Bundles, Vector Bundles
and Connections
A.1 Fiber Bundles
De…nition 626 A …ber bundle over M with Lie group G will be denoted
by E = (E; M; ; G; F ). E is a topological space called the total space of
the bundle, : E ! M is a continuous surjective map, called the canonical projection and F is the typical …ber. The following conditions must be
satis…ed:
1
(a)
(x), the …ber over x, is homeomorphic to F .
(b) Let fUi ; i 2 Ig, where I is an index set, be a covering of M , such
that:
Locally a …ber bundle E is trivial, i.e., it is di¤ eomorphic to a product
1
bundle, i.e.,
(Ui ) ' Ui F for all i 2 I.
The di¤ eomorphisms
i
i (p)
ij
The collection f(Ui ;
alizations for E.
1
:
(Ui ) ! Ui
= ( (p);
1 (x)
i )g,
i;x
i;x (p));
1
:
F have the form
(x) ! F is onto.
(A.1)
(A.2)
i 2 I, are said to be a family of local trivi-
The group G acts on the typical …ber. Let x 2 Ui \ Uj . Then,
j;x
1
i;x
:F !F
(A.3)
476
Appendix A. Principal Bundles, Vector Bundles and Connections
FIGURE A.1. Transition functions on a …ber bundle
must coincide with the action of an element of G for all x 2 Ui \ Uj
and i; j 2 I.
We call transition functions of the bundle the continuous induced
mappings
gij : Ui \ Uj ! G, where gij (x) =
i;x
1
j;x :
(A.4)
For consistence of the theory the transition functions must satisfy the
cocycle condition
gij (x)gjk (x) = gik (x):
(A.5)
Transition functions on a …ber bundle
De…nition 627 (P; M; ; G; F
G)
(P; M; ; G) is called a principal
…ber bundle (PFB) if all conditions in De…nition 626 are ful…lled and moreover, if there is a right action of G on elements p 2 P , such that:
(a) the mapping (de…ning the right action) P G 3 (p; g) 7! pg 2 P is
continuous.
(b) given g; g 0 2 G and 8p 2 P , (pg)g 0 = p(gg 0 ):
(c) 8x 2 M; 1 (x) is invariant under the action of G, i.e., each element
1
1
of p 2
(x) is mapped into pg 2
(x), i.e., it is mapped into an element
of the same …ber.
1
(x), which means that
(d) G acts free and transitively on each …ber
1
all elements within
(x) are obtained by the action of all the elements
1
of G on any given element of the …ber
(x). This condition is, of course
necessary for the identi…cation of the typical …ber with G.
De…nition 628 A bundle (E; M; 1 ; G = Gl(m; F); F = V ), where F = R
or C (respectively the real and complex …elds), Gl(m; F) is the linear group,
and V is an m-dimensional vector space over F is called a vector bundle.
A.1 Fiber Bundles
477
De…nition 629 A vector bundle (E; M; ; G; F ) denoted E = P
F is
said to be associated to a PFB bundle (P; M; ; G) by the linear representation of G in F = V (a linear space of …nite dimension over an appropriate
…eld, which is called the carrier space of the representation) if its transition
functions are the images under of the corresponding transition functions
of the PFB (P; M; ; G). This means the following: consider the following
local trivializations of P and E respectively
i
i
i (q)
ij
1
1
(x)
:
1
:
1
1
=(
(Ui ) ! Ui
G;
(A.6)
(Ui ) ! Ui
F;
(A.7)
1 (q);
i;x
:
i (q))
1
1
= (x;
i (q));
(x) ! F;
(A.8)
(A.9)
where 1 : P
F ! M is the projection of the bundle associated to
(P; M; ; G). Then, for all x 2 Ui \ Uj , i; j 2 I, we have
j;x
1
In addition, the …bers
tation space V .
1
i;x
= (
j;x
1
i;x ):
(A.10)
(x) are vector spaces isomorphic to the represen-
De…nition 630 Let E = (E; M; ; G; F ) be a …ber bundle and U M an
open set. A local (cross) section of the …ber bundle (E; M; ; G; F ) on U is
a mapping
s : U ! E such that
s = IdU :
(A.11)
A global section s is one for which U = M . Not all bundles admit global
sections (see below ).
Notation 631 Let U M . We will say that 2 sec EjU (or 2 sec EjU )
if there exists a local section sj : U ! E, x 7! (x; (x)), with : U ! F .
Remark 632 There is a relation between sections and local trivializations
for principal bundles. Indeed, each local section s; (on Ui M ) for a prin1
cipal bundle (P; M; ; G) determines a local trivialization i :
(U ) !
U G; of P by setting
1
i
Conversely,
i
(x; g) = s(x)g = pg = Rg p:
(A.12)
determines s since
s(x) =
1
i
(x; e).
(A.13)
Proposition 633 A principal bundle is trivial, if and only if, it has a
global cross section.
Proposition 634 A vector bundle is trivial, if and only if, its associated
principal bundle is trivial.
478
Appendix A. Principal Bundles, Vector Bundles and Connections
Proposition 635 Any …ber bundle (E; M; ; G; F ) such that M is contractible to a point is trivial.
Proposition 636 Any …ber bundle (E; M; ; G; F ) such that M is paracompact and the …ber F is a vector space admits a local section.
Remark 637 Take notice that any vector bundle admits a global section,
namely the zero section. However, it admits a global nonvanishing global
section if its Euler class is zero.
De…nition 638 The structure group G of a …ber bundle (E; M; ; G; F )
is said to be reducible to G0 if the bundle admits an equivalent structure
de…ned with a subgroup G0 of the structure group G. More precisely, this
means that the …ber bundle admits a family of local trivializations such that
the transition functions takes values in G0 , i.e., gij : Ui \ Uj ! G0 .
A.1.1 Frame Bundle
The tangent bundle T M to a di¤erentiable n-dimensional manifold M
is an associated
bundle to a principal bundle called the frame bundle
S
F (M ) = x2M Fx M , where Fx M is the set of frames at x 2 M . Let fxi g
be coordinates associated to a local chart (Ui ; 'i ) of the maximal atlas of
@
g on Ui M .
M . Then, Tx M has a natural basis f @x
i
x
De…nition 639 A frame at Tx M is a set
independent vectors such that
ei jx = Fij
x
@
@xj
= f e1 jx ; :::; en jx g of linearly
;
(A.14)
x
and where the matrix (Fij ) with entries Aji 2 R, belongs to the the real
general linear group in n dimensions Gl(n; R). We write (Fij ) 2 Gl(n; R).
A local trivialization
i
1
:
i (f )
(Ui )!Ui
= (x;
x ),
Gl(n; R) is de…ned by
(f ) = x:
(A.15)
The action of a = (aji ) 2 Gl(n; R) on a frame f 2 F (U ) is given by
(f; a) ! f a, where the new frame f a 2 F (U ) is de…ned by i (f a) =
(x; 0x ), (f ) = x, and
0
x
= f e01 jx ; :::; e0n jx g;
e0i jx = ej jx aji :
(A.16)
Conversely, given frames x and 0x there exists a = (aji ) 2 Gl(n; R)
such that Eq.(A.16) is satis…ed, which means that Gl(n; R) acts on F (M )
actively.
A.1 Fiber Bundles
479
Let fxi g and fxi gbe the coordinates associated to the local chart (Ui ; 'i )
(Ui0 ; '0i ) and of the maximal atlas of M . If x 2 Ui \ Uj we have
ei jx = Fij
@
@xj
= Fij
x
@
@ xj
;
x
(Fij ); (Fij ) 2 Gl(n; R):
Since Fij = Fkj
@xk
@ xi
x
(A.17)
we have that the transition functions are
@xk
@ xi
gik (x) =
x
2 Gl(n; R):
(A.18)
Remark 640 Given U
M we shall denote by
2 sec F (U ) a section of F (U )
F (M ). This means that given a local trivialization
:
1
(U )!U Gl(n; R), ( ) = (x; x ), (f ) = x. Sometimes, we also
use the sloppy notation fei g 2 sec F (U ) or even fei g 2 sec F (M ) when the
context is clear.
A.1.2 Orthonormal Frame Bundle
Suppose that the manifold M is equipped with a metric …eld g 2 sec T20 M
of signature (p; q), p + q = n. Then, we can introduce orthonormal frames
in each Tx U . In this case we denote an orthonormal frame by x =
f e1 jx ; :::; en jx g and
ei jx = hji
g( ei jx ; ej jx )
x
@
@xj
;
(A.19)
x
= diag(1; 1; :::1; 1; :::; 1)
(A.20)
with (hji ) 2 Op;q , the real orthogonal group in n dimensions. In this case
we say that the frame bundle has been reduced to the orthonormal frame
bundle, which will be denoted by POn (M ). A section
2 secPOn (U ) is
called a vierbein.
Remark 641 The principal bundle of orthonormal frames PSOe1;3 (M ) over
a Lorentzian manifold modelling spacetime and its covering bundle called
spin bundle PSpine1;3 (M ) discussed in Chapter 6 play an important role in
this book. Also, vector bundles associated these bundles are very important. Associated to PSOe1;3 (M ) we have the tensor bundle, the exterior
bundle and the Cli¤ ord bundle. Associated to PSpine1;3 (M ) we have several
spinor bundles, in particular the spin-Cli¤ ord bundle, whose sections are
the Dirac-Hestenes spinor …elds. All those bundles and their relationship
are studied in Chapter 7.
480
Appendix A. Principal Bundles, Vector Bundles and Connections
Remark 642 In complete analogy to the construction of orthonormal frame
bundle we may de…ne an orthonormal coframe bundle that may be denoted
by POn (M ). Since to each given frame 2 sec POn (M ) there is a natural
coframe …eld 2 sec POn (M ), the one where the covectors are the duals of
the vectors of the frame. It follows that POn (M ) ' POn (M ). In particular
PSOe1;3 (M ) ' PSOe1;3 (M ).
A.1.3 Jet Bundles
Let E = (E; M; ; G; F ) be a …ber bundle. Let x 2 M and let Ux M be a
neighborhood of x. Let sec EjUX be the set of all local sections of E de…ned
in Ux . Let r 2 Z+ , r 1. Next de…ne in sec EjUX and equivalence relation
~rx such that if sj ; sj : Ux ! E we say that
sj ~rx sj
(A.21)
if for any smooth function f : E ! R and any smooth curve : R !M ,
t 7! (t), with (0) = x we have that the mappings f
and f
have the same r-order Taylor expansion at t = 0.
If we introduce local coordinates for EjUX we immediately see that the
equivalence relation ~rx reduces to the requirement that the local expressions
of and have the same r-order Taylor expansion at x. We denote by jxr
the equivalence class identi…ed by the representative 2 sec EjUX :
De…nition 643 Let j rx (E) be the quotient space sec EjUX =~rx . We call the
r
r
the
S r-jetr bundle of E the disjoint union of all j xr(E), i.e.,r the bundle J (E) =
and 0 such that
x2M j x (E). We moreover de…ne the maps
r
r
0
: J r (E) ! M;
: (J r (E)) ! E;
r
(jxr ) = x;
r
r
0 ((jx
)) = sj :
More details on jet bundles, a natural setup for rigorous formulation of
…eld theories may be found, e.g., in [124].
A.2 Product Bundles and Whitney Sum
Given two vector bundles (E; M; ; G; V) and (E 0 ; M 0 ;
0
; G0 ; V0 ) we have
De…nition 644 The product bundle E E 0 is a …ber bundle whose basis
space is M M 0 , the typical …ber is V V0 , the structural group of E
E 0 acts separately as G and G0 in each one of the components of V V0
and the projection
0
is such that E
0
E0 ! M
M 0.
De…nition 645 Let (E; M; ; G; V) and (E 0 ; M; 0 ; G0 ; V0 ) be vector bundles over the same basis space. The Whitney sum bundle E E 0 is the
pullback of E E 0 by h : M ! M M , h(p) = (p; p):
A.3 Connections
481
De…nition 646 Let (E; M; ; G; V) and (E 0 ; M; 0 ; G0 ; V0 ) be vector bundles over the same basis space. The tensor product bundle E E 0 is the
bundle obtained from E and E 0 by assigning the tensor product of …bers
1
0 1
for all x 2 M:
x
x
Remark 647 With the above de…nitions we can easily show that given
three vector bundles, say, E; E 0 ; E 00 we have
E
(E 0
E 00 ) = (E
E0)
(E
E 00 ):
(A.22)
A.3 Connections
A.3.1 Equivalent De…nitions of a Connection in Principal
Bundles
To de…ne the concept of a connection on a PFB (P; M; ; G), we recall that
since dim(M ) = m, if dim(G) = n, then dim(P ) = n + m. Obviously, for all
1
x 2 M,
(x) is an n-dimensional submanifold of P di¤eomorphic to the
1
structure group G and is a submersion,
(x) is a closed submanifold
of P for all x 2 M .
1
The tangent space Tp P , p 2
(x), is an (n + m)-dimensional vector
space and the tangent space Vp P
Tp ( 1 (x)) to the …ber over x at the
1
same point p 2
(x) is an n-dimensional linear subspace of Tp P called
the vertical subspace of Tp P 1 .
Now, roughly speaking a connection on P is a rule that makes possible a
correspondence between any two …bers along a curve : R I ! M; t 7!
(t). If p0 belongs to the …ber over the point (t0 ) 2 , we say that p0 is
parallel translated along by means of this correspondence.
De…nition 648 A horizontal lift of
by the parallel transport of p).
is a curve ^ : R
I ! P (described
It is intuitive that such a transport takes place in P along directions
speci…ed by vectors in Tp P , which do not lie within the vertical space Vp P .
Since the tangent vectors to the paths of the basic manifold passing through
a given x 2 M span the entire tangent space Tx M , the corresponding vectors Y p 2 Tp P (in whose direction parallel transport can generally take
place in P ) span a n-dimensional linear subspace of Tp P called the horizontal space of Tp P and denoted by Hp P . Now, the mathematical concept
1 Here we may be tempted to realize that as it is possible to construct the vertical
space for all p 2 P then we can de…ne a horizontal space as the complement of this space
in respect to Tp P . Unfortunately this is not so, because we need a smoothly association
of a horizontal space in every point. This is possible only by means of a connection.
482
Appendix A. Principal Bundles, Vector Bundles and Connections
of a connection can be presented. This is done through three equivalent de…nitions given below which encode rigorously the intuitive discussion given
above. We have,
De…nition 649 A connection on a PFB (P; M; ; G) is an assignment
to each p 2 P of a subspace Hp P Tp P , called the horizontal subspace for
that connection, such that Hp P depends smoothly on p and the following
conditions hold:
(i)
: Hp P ! Tx M , x = (p); is an isomorphism.
(ii) Hp P depends smoothly on p.
(iii) (Rg ) Hp P = Hpg P; 8g 2 G; 8p 2 P .
Here we denote by
the di¤ erential of the mapping and by (Rg )
the di¤erential of the mapping Rg : P ! P (the right action) de…ned by
Rg (p) = pg.
1
Since x = (^ (t)) for any curve in P such that ^ (t) 2
(x) and
^ (0) = p0 , we conclude that
maps all vertical vectors in the zero vector
in Tx M , i.e., (Vp P ) = 0 and we have2 ,
Tp P = Hp P
Vp P:
(A.23)
Then every Y p 2 Tp P can be written as
Yp = Yph + Ypv ;
Yph 2 Hp P;
Ypv 2 Vp P:
(A.24)
Therefore, given a vector …eld Y over M it is possible to lift it to a horizontal vector …eld over P , i.e., (Y p ) = (Y hp ) = Y x 2 Tx M for all p 2 P
with (p) = x. In this case, we call Y hp the horizontal lift of Y x . We say
moreover that Y is a horizontal vector …eld over P if Y h =Y .
De…nition 650 A connection on a PFB (P; M; ; G) is a mapping p :
Tx M ! Tp P , such that 8p 2 P and x = (p) the following conditions hold:
(i) p is linear.
(ii)
p = IdTx M :
(iii) the mapping p 7! p is di¤ erentiable.
(iv) Rg p = (Rg ) p , for all g 2 G.
We need also the concept of parallel transport. It is given by,
De…nition 651 Let : R I ! M; t 7! (t) with x0 = (0) 2 M , be a
curve in M and let p0 2 P such that (p0 ) = x0 . The parallel transport of
p0 along is given by the curve ^ : R I ! P; t 7! ^ (t) de…ned by
d
^ (t) =
dt
2 We
also write T P = HP
V P:
p(
d
(t));
dt
(A.25)
A.3 Connections
483
with p0 = ^ (0) and ^ (t) = pkt , (pkt ) = x:
In order to present yet a third de…nition of a connection we need to know
more about the nature of the vertical space Vp P . For this, let Y 2 Te G = G
be an element of the Lie algebra G of G. The vector Y is the tangent to
the curve produced by the exponential map
Y=
d
(exp(tY))
dt
:
(A.26)
t=0
Then, for every p 2 P we can attach to each Y 2 Te G = G a unique
element Y vp 2 Vp P as follows: let f : ( "; ") ! P , t 7! p exp tY be a curve
on P . Observe that it is obtained by right translation and then (p) =
1
(p exp tY) = x and so the curve lies in
(x), the …ber over x 2 M . Next
let F : P ! R be a smooth function. Then we de…ne
Ypv F(p) =
d
F f (t)
dt
=
t=0
d
F (p exp(tY))
dt
:
(A.27)
t=0
By this construction we attach to each Y 2 Te G = G a unique vector
…eld over P , called the fundamental …eld corresponding to this element.
We then have the canonical isomorphism
Ypv $ Y;
Ypv 2 Vp P;
Y 2 Te G = G
(A.28)
from which we get
Vp P ' G:
(A.29)
De…nition 652 A connection on a PFB (P; M; ; G) is a 1-form …eld !
on P with values in the Lie algebra G = Te G such that 8p 2 P we have,
(i) ! p (Y vp ) = Y and Y vp $ Y, where Y vp 2 Vp P and Y 2 Te G = G.
(ii) ! p depends smoothly on p.
(iii) ! p [(Rg ) Y p ] = (Adg 1 ! p )(Y p ), where Ad! p = g 1 ! p g.
It follows that if fGa g is a basis of G and f i g is a basis for T P then
! p = !pa
Ga = ! ai (p)
i
p
Ga ;
(A.30)
where ! a are 1-forms on P .
Then the horizontal spaces can be de…ned by
Hp P = ker(! p );
which shows the equivalence between the de…nitions.
(A.31)
484
Appendix A. Principal Bundles, Vector Bundles and Connections
A.3.2 The Connection on the Base Manifold
De…nition 653 Let U
M and
s:U !
1
(U )
P;
s = IdU ;
(A.32)
be a local section of the PFB (P; M; ; G).
De…nition 654 Let ! be a connection on P . The 1-form s ! (the pullback
of ! under s) given by
(s !)x (Yx ) = ! s(x) (s Yx ); Yx 2 Tx U; s Yx 2 Tp P;
p = s(x);
(A.33)
is called the local gauge potential.
It is quite clear that s ! 2 sec T U G. This object di¤ers from the gauge
…eld used by physicists by numerical constants (with units). Conversely we
have the following
Proposition 655 Given ! 2 sec T U G and a di¤ erentiable section of
1
(U )
P, U
M , there exists one and only one connection ! on
1
(U ) such that s ! = !.
Consider now
!2T U
!0 2 T U 0
G;
G;
!=(
1
!0 = (
0 1
(x; e)) ! = s !;
s(x) =
(x; e)) ! = s0 !;
s0 (x) =
1
(x; e);
0 1
(x; e):
(A.34)
Then we can write, for each p 2 P ( (p) = x), parameterized by the local
trivializations and 0 respectively as (x; g) and (x; g 0 ) with x 2 U \ U 0 ,
that
!p = g
1
dg + g
1
!x g = g 0
1
dg 0 + g 0
1
!x0 g 0 :
(A.35)
Now, if
g 0 = hg;
(A.36)
we immediately get from Eq.(A.35) that
! 0x = hdh
1
+ h! x h
1
;
which can be called the transformation law for the gauge …elds.
(A.37)
A.4 Exterior Covariant Derivatives.
485
A.4 Exterior Covariant Derivatives.
Vk
Vk
Let
(P; G) =
T P G; 0 k n, be the set of all k-form …elds over
P with values in the Lie algebra G of the gauge group G (and, of course,
V1
the connection ! 2 sec (P; G)):
Vk
De…nition 656 For each ' 2 sec (P; G) we de…ne the so called horiV
k
zontal form 'h 2 sec (P; G) by
'hp (X1 ; X2 ; :::; Xk ) = '(Xh1 ; Xh2 ; :::; Xhk );
(A.38)
where Xi 2 Tp P , i = 1; 2; ::; k.
Notice that 'hp (X1 ; X2 ; :::; Xk ) = 0 if one (or more) of the Xi 2 Tp P are
vertical.
Vk
De…nition 657 ' 2 sec T P V (where V is a vector space) is said
to be horizontal if 'p (X1 ; X2 ; :::; Xk ) = 0, implies that at least one of the
Xi 2 Tp P , i = 1; 2; :::; n is vertical.
Vk
De…nition 658 ' 2 sec T P V is said to be of type ( ; V) if 8g 2 G
we have
Rg ' = (g 1 )':
(A.39)
De…nition 659 Let ' 2 sec
be tensorial of type ( ; V).
Vk
T P
V be horizontal. Then, ' is said to
De…nition 660 The exterior covariant derivative of ' 2 sec
relation to the connection ! is
Vk
D! ' = (d')h 2 sec (P; G);
Vk
(P; G) in
(A.40)
where D! 'p (X1 ; X2 ; :::; Xk ; Xk+1 ) = d'p (Xh1 ; Xh2 ; :::; Xhk ; Xhk+1 ). Notice
Vk
that d' = d'a Ga where 'a 2 sec T P , a = 1; 2; :::; n.
Vi
Vj
De…nition 661 The commutator of ' 2 sec (P; G) and 2 sec (P; G),
Vi+j
0 i; j n, denoted by ['; ] 2 sec
(P; G) such that if X1 ; :::; Xi+j 2
sec T P , then
['; ](X1 ; :::; Xi+j )
(A.41)
X
1
( 1) ['(X (1) ; :::; X (i) ); (X (i+1) ; :::; X (i+j) )];
=
i!j!
2Sn
where Sn is the permutation group of n elements and ( 1) = 1 is the sign
of the permutation. The brackets [ ; ] in the second member of Eq.(A.41)
are the Lie brackets in G.
486
Appendix A. Principal Bundles, Vector Bundles and Connections
Writing
' = 'a
Ga ;
=
a
Ga ;
we have
['; ]
=
=
'a 2 sec
'a ^
^i
b
f cab ('a ^
(T P );
a
2 sec
[Ga ; Gb ]
b
)
^j
(T P );
(A.42)
(A.43)
Gc
where f cab are the structure constants of the Lie algebra.
Remark 662 In this section we are using (in order to get formulas as the
ones printed in main textbooks) the exterior product as given in Remark
_ We hope this will cause no misunder24 of Chapter 2 (there denoted ^).
standing.
With Eq.(A.43) we can prove easily the following important properties
involving commutators:
['; ] = ( 1)1+ij [ ; '];
(A.44)
( 1)ik [['; ]; ] + ( 1)ji [[ ; ]; '] + ( 1)kj [[ ; ']; ] = 0;
i
d['; ] = [d'; ] + ( 1) ['; d ];
Vj
Vk
for ' 2 sec (P; G), 2 sec (P; G), 2 sec (P; G).
We shall also need the following identity
(A.45)
(A.46)
Vi
[!; !](X1 ; X2 ) =2[!(X1 ); !(X2 )]:
The proof of Eq.(A.47) is as follows:
(i) Recall that
[!;!] = (! a ^ ! b )
(A.47)
[Ga ; Gb ]:
(A.48)
(ii) Let X1 ; X2 2 sec T P (i.e., X1 and X2 are vector …elds on P ). Then,
[!; !](X1 ; X2 )
=
(! a (X1 )! b (X2 )
=
2[!(X1 ); !(X2 )]:
! b (X2 )! a (X1 ))[Ga ; Gb ]
(A.49)
De…nition 663 The curvature of the connection ! 2 sec
V2
sec (P; G) de…ned by
!
= D! !:
De…nition 664 The connection ! is said to be ‡at if
V1
!
(P; G) is
!
2
(A.50)
= 0.
Proposition 665
D! !(X1 ; X2 ) = d!(X1 ; X2 ) + [!(X1 ); !(X2 )]:
(A.51)
A.4 Exterior Covariant Derivatives.
487
Eq.(A.51) can be written using Eq.(A.49) (and recalling that !(X) =
! a (X)Ga ) as
1
!
= D! ! = d! + [!; !]:
(A.52)
2
Proof. See, e.g., [77].
Proposition 666 (Bianchi identity):
D
Proof. (i) Let us calculate d
d
!
!
!
= 0:
(A.53)
. We have,
1
= d d! + [!; !] :
2
(A.54)
We now take into account that d2 ! = 0 and that from the properties of
the commutators given by Eqs.(A.44), (A.45), (A.46) above, we have
d[!; !)] = [d!; !]
[d!; !] =
[!; d!];
[!; d!];
[[!; !]; !] = 0:
(A.55)
By using Eq.(A.55) in Eq.(A.54) gives
d
!
= [d!; !]:
(A.56)
(ii) In Eq.(A.56) use Eq.(A.52) and the last equation in Eq.(A.55) to
obtain
d ! = [ ! ; !]:
(A.57)
(iii) Use now the de…nition of the exterior covariant derivative [Eq.(A.40)]
together with the fact that !(Xh ) = 0, for all X 2 Tp P to obtain
D!
!
= 0;
which is the result we wanted to prove.
We can then write the very important formula (known as the Bianchi
identity),
D! ! = d ! + [!; ! ] = 0:
(A.58)
A.4.1
Local Curvature in the Base Manifold M
1
Let (U; ) be a local trivialization of
(U ) and s the associated cross
section as de…ned above. Then, s ! := ! (the pull back of ! ) is a
well de…ned 2-form …eld on U which takes values in the Lie algebra G:
488
Appendix A. Principal Bundles, Vector Bundles and Connections
Let ! = s ! (see Eq.(A.34)). If we recall that the di¤erential operator d
commutes with the pullback, we immediately get
!
= s D! ! = d! +
1
[!;!] :
2
(A.59)
It is convenient to de…ne the symbols
D! := s D! !;
!
D
and to write
:= s D
D
!
=
0;
D
!
=
d
!
!
+ [!;
(A.60)
!
!
(A.61)
] = 0:
(A.62)
Eq.(A.62) is also known as Bianchi identity.
Remark 667 In gauge theories (Yang-Mills theories) ! is (except for
numerical factors with physical units) called a …eld strength in the gauge
.
Remark 668 When G is a matrix group, as is the case in the presentation
of gauge theories by physicists, De…nition 661 of the commutator ['; ] 2
Vi+j
Vi
Vj
sec
(P; G) (' 2 sec (P; G), 2 sec (P; G)) gives
( 1)ij
['; ] = ' ^
^ ';
(A.63)
where ' and are considered as matrices of forms with values in G and '^
stands for the usual matrix multiplication where the entries are combined
via the exterior product. Then, when G is a matrix group, we can write
Eq.(A.52) and Eq.(A.59) as
!
!
= D! ! = d! + ! ^ !;
:= D! = d! + !^!:
(A.64)
(A.65)
A.4.2 Transformation of the Field Strengths Under a Change
of Gauge
Consider two local trivializations (U; ) and (U 0 ; 0 ) of P such that p 2
1
(U \U 0 ) has (x; g) and (x; g 0 ) as images in (U \U 0 ) G, where x 2 U \U 0 .
Let s; s0 be the associated cross sections to and 0 respectively. By writing
s0 ! = !0 , we have the following relation for the local curvature in the
two di¤erent gauges such that g 0 = hg
!0
=h
!
h
1
;
8x 2 U \ U 0 :
(A.66)
A.4 Exterior Covariant Derivatives.
489
We now give the coordinate expressions for the potential and …eld strengths
in the trivialization . Let fx g be coordinates for a local chart for U M
and let @ = @x@
and fdx g, = 0; 1; 2; 3, be (dual) bases of T U and
T U respectively. Then,
! = !a
!
where ! a ,
a
Ga = ! a dx
1
= ( ! )a G a =
2
:M
Ga ;
a
(A.67)
dx ^ dx
Ga :
(A.68)
U ! R (or C) and we get
a
= @ !a
@ ! a + f abc ! b ! c ;
(A.69)
with f cab the structure constants of G, i.e., [Ga ; Gb ] = f cab Gc .
The following objects appear frequently in the presentation of gauge
theories by physicists.
(
! a
) =
!
1
2
a
a
=
a
1
dx ^ dx = d! a + f abc ! b ^ ! c ;
2
Ga = @ !
@ ! + [! ; ! ];
! = ! Ga :
(A.70)
(A.71)
(A.72)
Next we give the local expression of Bianchi identity. Using Eq.(A.62)
and Eq.(A.70) we have
D
!
:=
1
(D
2
Putting
(D
we have
!
D
and
D
!
!
!
)
dx ^ dx ^ dx = 0:
)
:= D
=@
+D
!
!
!
+D
(A.74)
!
+ [! ;
!
(A.73)
];
(A.75)
= 0:
(A.76)
Physicists call the operator
D := @ + [! ; ]:
(A.77)
the covariant derivative. The reason for this name will be given below.
A.4.3 Induced Connections
Let (P1 ; M1 ; 1 ; G1 ) and (P2 ; M2 ; 2 ; G2 ) be two principal bundles and let
F : P1 ! P2 be a bundle homomorphism, i.e., F is …ber preserving, it
induces a di¤eomorphism f : M1 ! M2 and there exists a homomorphism
: G1 ! G2 such that for g1 2 G1 , p1 2 P1 we have
F(p1 g1 ) = R
(g1 ) F(p1 ):
(A.78)
490
Appendix A. Principal Bundles, Vector Bundles and Connections
Proposition 669 F : P1 ! P2 be a bundle homomorphism. Then a connection ! 1 on P1 determines a unique connection om P2 :
Remark 670 Let (P; M; 0 ; Op;q ) = POp;q (M ) be the orthonormal frame
bundle, which is as explained above reduction of the frame bundle F (M ).
Then, a connection on POp;q (M ) determines a unique connection on F (M ).
This is a very important result that has been used implicitly in Section 4.9.9
and the solution of Exercise 318.
Proposition 671 Let F (M ) be the frame bundle of a paracompact manifold M . Then, F (M ) can be reduced to a principal bundle with structure
group Op;q , and to each reduction there corresponds a Riemannian metric
…eld on M:
Remark 672 If M has dimension 4, and we substitute Op;q 7! SOe1;3 then
to to each reduction of F (M ) there corresponds a Lorentzian metric …eld
on M:
A.4.4 Linear Connections on a Manifold M
De…nition 673 A linear connection on a smooth manifold M is a connection ! 2 sec T F (M ) gl(n; R):
Remark 674 Given a Riemannian (Lorentzian) manifold (M; g) a connection on F (M ) which is determined by a connection on the orthonormal
frame bundle POp;q (M ) (PSOe1;3 (M )) is called a metric connection. After
introducing the concept of covariant derivatives on vector bundles, we can
show that the covariant derivative of the metric tensor with respect to a
metric connection is null.
Consider the mapping f jp : Tx (M ) ! Rn (with p = (x; x ) in a given
trivialization) which sends v 2 Tx M into its components relative to the
frame x = f e1 jx ; :::; en jx g. Let f j x g be the dual basis of f ei jx g. We
write
f jp (v) = ( j x (v)):
(A.79)
De…nition 675 The canonical soldering form of M is the 1-form
2
sec T F (M ) Rn such that for any v 2 sec Tp F (M ) such that v = v we
have
( (v)): =
=
a
a
jp (v)ea
jx (v)ea ;
where fea g is the canonical basis of Rn and f
a
with a =
, a jp (v) = a jx (v).
(A.80)
a
g is a basis of T F (M ),
De…nition 676 The torsion of a linear connection ! 2 sec T F (M )
V2
gl(n; R) is the 2-form D = 2 sec
T F (M ) Rn .
A.4 Exterior Covariant Derivatives.
491
As it is easy to verify, the soldering form and the torsion 2-form are
tensorial of type ( ; Rn ), where (u) = u, u 2 Gl(n; R).
Using the same techniques employed in the calculation of D! !(X1 ; X2 )
(Eq.(A.51)) it can be shown that
= d + [!; ];
(A.81)
where [ , ] is the commutator product in the Lie algebra of the a¢ ne group
A(n; R) = Gl(n; R) Rn , where means the semi-direct product. Suppose
that (eba ; ec ) is the canonical basis of a(n; R), the Lie algebra of A(n; R).
Recalling that
!(v) = ! ab (v)eba ;
a
(v) =
(A.82)
(v)ea ;
(A.83)
we can show without di¢ culties that
D!
= [ ; ]:
(A.84)
A.4.5 Torsion and Curvature on M
Let fxi g be the coordinates associated to a local chart (U; ') of the maximal
@
= a ea . Take
atlas of M . Let
2 sec F (U ) with ei = Fij @x
j and
v = v. Then
(
p (v))
= f jp (v) = f jp (dxj (v)@ j ) = f jp (dxj (v)(Fjk )
= ((Fjk )
1
dxj (
v)):
With this result it is quite obvious that given any w 2 Rn ,
a horizontal …eld vw 2 secT F (M ) by
( (vw (p))) = w:
1
ek )
(A.85)
determines
(A.86)
With these preliminaries we have the
Proposition 677 There is a bijective correspondence between sections of
T M Tsr M and sections of T F (M ) Rnq , the space of tensorial forms
of the type ( ; Rnq ) in F (M ), with and q being determined by Tsr M .
Using the above proposition and recalling that the soldering form is tensorial of type ( (u); Rn ), (u) = u; we see that it determines on M a vector
V1
a
valued di¤erential 1-form3
= ea
2 sec T M
T M . Also, the
torsion
is tensorial of type ( (u); Rn ), (u) = u and thus de…ne a vector
3
is clearly the identity operator in the space of vector …elds.
492
Appendix A. Principal Bundles, Vector Bundles and Connections
a
valued 2-form on M , = ea
2 sec T M
Eq.(A.81) that given u; w 2 Tp F (M );
a
(
w) = d a (
u;
u;
w) + !ba (
V2
u) b (
T M . We can show from
w)
!ba (
w) b (
u):
(A.87)
On the basis manifold this equation is often written:
= D = ea
= ea
(d
a
(D a )
+ ! ab ^
b
);
(A.88)
where we recognize D a as the exterior covariant derivative of index forms
introduced in Section 3.3.4.4
2
Remark 678 Also, the curvature ! is tensorial of type (Ad; Rn ). It then
V2
b
de…nes = ea
Rab 2 sec T11 M
T M which we easily …nd to be
given by
= ea
b
= ea
b
Rab
(d! ab + ! ac ^ ! cb );
(A.89)
V2
where the Rab 2 sec
T M are the curvature 2-forms introduced in Chapter 4, explicitly given by
Rab = d! ab + ! ac ^ ! cb :
(A.90)
Note that sometimes the symbol D! ab such that Rab := D!ba is introduced
in some texts. Of course, the symbol D cannot be interpreted in this case
as the exterior
covariant derivative of index forms 5 . This is expected since
V1
! 2 sec
T P gl(n; R) is not tensorial.
A.5 Covariant Derivatives on Vector Bundles
Consider a vector bundle (E; M; 1 ; G; V)6 associated to a PFB bundle
(P; M; ; G) by the linear representation of G in the vector space V over
the …eld F = R or C. Also, let dimF V = m. Consider again the trivializations of P and E given by Eqs.(A.7)-(A.9). Then, we have the
4 Rigorosly
speaking, if Eq.(A.88) is to agree with Eq.(A.87) we must have ! a
b ^
b
_
=
!a
,
i.e,
as
already
observed
^
must
be
understood
as
^
given
b
in Remark 24. However, this cause no troubles in the calculations we done using the
Cli¤ord bundle formalism.
5 See Section 3.
6 Also denoted E = P
V:
b
!a
b
b
A.5 Covariant Derivatives on Vector Bundles
493
De…nition 679 The parallel transport of 0 2 E, 1 ( 0 ) = x0 , along
the curve : R 3 I ! M , t 7! (t) from x0 = (0) 2 M to x = (t) is the
element kt 2 E such that:
(i) 1 ( kt ) = x,
(ii) i ( kt ) = ('i (pkt ) 'i 1 (p0 )) i ( 0 ).
(iii) pkt 2 P is the parallel transport of p0 2 P along from x0 to x as
de…ned in Eq.(A.25) above.
De…nition 680 Let Y be a vector at x0 tangent to the curve (as de…ned above). The covariant derivative of 2 sec E in the direction of Y is
E
denoted (DY
)x0 2 sec E and
1
E
(DY
)x0 = lim (
t!0 t
E
(DY
)(x0 )
0
kt
0 );
(A.91)
where 0kt is the “vector” t
( (t)) of a section
2 sec E parallel
transported along from (t) to x0 , the only requirement on being
d
(t)
dt
= Y:
(A.92)
t=0
In the local trivialization (Ui ; i ) of E (see Eqs.(A.7)-(A.9)) if
element in V representing t , we have
i(
0
kt )
= (g0 gt 1 )
ij (t) (
t ):
t
is the
(A.93)
By choosing p0 such that g0 = e we can compute Eq.(A.91):
d
(g 1 (t)
dt
d t
=
dt t=0
E
(DY
)x0 =
t)
t=0
0
(e)
dg(t)
dt
(
0 ):
(A.94)
t=0
This formula is trivially generalized for the covariant derivative in the
direction of an arbitrary vector …eld Y 2 sec T M:
With the aid of Eq.(A.94) we can calculate, e.g., the covariant derivative
of
2 sec E in the direction of the vector …eld Y = @x@
@ . This
covariant derivative is denoted D@E .
We need now to calculate
dg(t)
dt
t=0
. In order to do that, recall that if
d
dt
is
d
a tangent to the curve in M , then s dt
is a tangent to ^ , the horizontal
d
lift of , i.e., s dt 2 HP
T P: As de…ned before s = i 1 (x; e) is the
cross section associated to the trivialization i of P (see Eq.(A.6). Then,
as g is a mapping U ! G we can write
s (
d
d
) (g) = (g
dt
dt
):
(A.95)
494
Appendix A. Principal Bundles, Vector Bundles and Connections
1
To simplify the notation, introduce local coordinates fx ; gg in
write (t) = (x (t)) and ^ (t) = (x (t); g(t)). Then,
d
dt
s
= x_ (t)
@
@
+ g(t)
_
;
@x
@g
(U ) and
(A.96)
in the local coordinate basis of T ( 1 (U )). An expression like the second
member of Eq.(A.96) de…nes in general a vector tangent to P but, according
d
to its de…nition, s dt
is in fact horizontal. We must then impose that
s
d
dt
= x_ (t)
@
@
+ g(t)
_
=
@x
@g
@
@
+ ! a Ga g
@x
@g
;
(A.97)
for some
.
@
We used the fact that @x@ + ! a Ga g @g
is a basis for HP , as can easily be
h
veri…ed from the condition that !(Y ) = 0, for all Y 2 HP . We immediately
get that
= x_ (t);
(A.98)
and
dg(t)
= g(t)
_
= x_ (t)! a Ga g;
dt
dg(t)
= x_ (0)! a Ga :
dt t=0
(A.99)
(A.100)
With this result we can rewrite Eq.(A.94) as
E
(DY
)x0 =
d t
dt
0
(e)!(Y)(
0 );
Y=
t=0
d
dt
:
(A.101)
t=0
which generalizes trivially for the covariant derivative along a vector …eld
Y 2 sec T M:
Remark 681 Many texts introduce the covariant derivative operator DYE
acting on sections of the vector bundle E as follows.
De…nition 682 A connection DE on M is a mapping
DE : sec T M
(X;
E
)7!DX
:
sec E ! sec E;
(A.102)
E
: sec E ! sec E satis…es the following properties:
such that DX
E
E
(i)
DX
(a ) = aDX
;
E
E
E
;
(ii) DX ( + ) = DX +DX
E
E
(iii) DX (f ) = X(f ) + f DX ;
E
E
E
(iv)
DX+Y
= DX
+DY
;
E
E
(v)
Df X = f DX ;
(A.103)
A.5 Covariant Derivatives on Vector Bundles
495
for all X; Y 2 sec T M , ; 2 sec E, 8a 2 F = R or C (the …eld of scalars
entering the de…nition of the vector space V of the bundle) 8f 2 C 1 (M ),
where C 1 (M ) is the set of smooth functions with values in F.
Of course, all properties in Eq.(A.103) follows directly from Eq.(A.101).
However, the point of view encoded in De…nition 682 may be appealing to
physicists. To see, recall that in general in physical theories E = P
V
where % stands for the representation of G in the vector space V.
De…nition 683 The dual bundle of E is the bundle E = P
V , where
V is the dual space of V and % is the representation of G in the vector
space V .
Example 684 As examples of bundles of the kind E = P V we have the
tangent bundle which is T M = F (M )
Rn where : Gl(n; R) !Gl(n; R)
denotes the standard representation. and T M = F (M )
(Rn ) where the
1 t
dual representation
satis…es
(g) = (g ) . Other important examples are the tensor bundle of tensors of type (r; s), the bundle of homogenous
k-vectors and the bundle of homogeneous k-forms, respectively:
Or
Or
Tsr M =
T M = F (M ) Nrs (
Rn );
s
s
^k
^k
T M = F (M ) Vk
Rn ;
^k
^k
T M = F (M ) Vk
Rn ;
(A.104)
N r Vk
Vk
where s ,
and
are the induced tensor product and exterior powers
representations.
De…nition 685 The bundle E E is called the bundle of endomorphisms
of E and will be denoted by End(E).
De…nition 686 A connection DE acting on E is de…ned by
E
(DX
for 8
2 sec E , 8
)( ) = X (
( ))
E
(DX
);
2 sec E and 8X 2 sec T M .
De…nition 687 A connection DE E acting on sections of E
2 sec E , 8 2 sec E and 8X 2 sec T M by
…ned for 8
E
DX
E
(A.105)
E
= DX
+
E
DX
:
E is de-
(A.106)
We shall abbreviate DE E by DEndE . Eq.(A.106) may be generalized in
an obvious way in order to de…ne a connection on arbitrary tensor products
of bundles E E 0 :::E 0:::0 . Finally, we recall for completeness that given
0
two bundles, say E and E 0 and given connections DE and DE there is an
496
Appendix A. Principal Bundles, Vector Bundles and Connections
0
obvious connection DE E de…ned in the Whitney bundle E
de…nition 645) It is given by
E0
E
DX
for 8
2 sec E, 8
0
0
(
E
) = DX
0
E
DX
0
;
E 0 (recall
(A.107)
2 sec E 0 and 8X 2 sec T M .
A.5.1 Connections on E over a Lorentzian Manifold
In what follows we suppose that (M; g) is a Lorentzian manifold (De…nition
241). We recall that the manifold M in a Lorentzian structure is supposed
paracompact. Then, according to Proposition 635 the bundles E; E , Tsr M
and EndE admit cross sections.
We then write for the covariant derivative of 2 sec E and X 2 sec T M;
E
DX
0E
= DX
+ W(X) ;
(A.108)
where W 2 sec EndE T M will be called connection 1-form (or potential )
E
0E
for DX
and DX
is a well de…ned connection on E, that we are going to
determine.
Consider then a open set U M and a trivialization of E in U . Such a
trivialization is said to be a choice of a gauge.
Let fei g be the canonical basis of V. Let
jU be a section of the
1
bundle E. Consider the trivialization
:
(U ) ! U
V, ( ) =
( ( ); ( )) = (x; ( )). In this trivialization we write
jU : = (x; (x));
(A.109)
(x) 2 V, 8x 2 U , with
: U ! V a smooth function. Let fsi g 2
1
sec EjU , si =
(ei ) i = 1; 2; :::; m be a basis of sections of EjU and
fe g 2 sec F (U ); = 0; 1; 2; 3 a basis for T U . Let also f" g, " 2 sec T U ,
be the dual basis of fe g and fs i g 2 sec E jU , be a basis of sections of
E jU dual to the basis fsi g.
We de…ne the connection coe¢ cients in the chosen gauge by
DeE si = W j i sj :
Then, if
=
i
(A.110)
si and X = X e
E
DX
= X DeE (
i
=X
i
e (
si )
) + Wi
j
j
si :
(A.111)
Now, let us concentrate on the term X W i j j si . It is, of course a new
section z := (x; X W i j j si ) of EjU and X W i j j si is linear in both X
and :
A.5 Covariant Derivatives on Vector Bundles
This observation shows that W U 2 sec(End EjU
the trivialization introduced above is given by
W U = W i j si
s
j
497
T U ), such that in
"
(A.112)
is the representative of W in the chosen gauge.
Note that if X 2 sec T U and
:= (x; (x)) a section of EjU we have
U
! U (X) := !X
= X W i j si
U
!X
( ) = X Wi
j
j
s j;
si :
(A.113)
We can then write
E
DX
U
= X( ) + !X
( );
0E
DX
(A.114)
0E
DX
thereby identifying
= X( ). In this case
is called the standard
‡at connection.
Now, we can state a very important result which has been used in Chapter 2 to write the di¤erent decompositions of Riemann-Cartan connections.
Proposition 688 Let D0E and DE be arbitrary connections on E then
there exists W 2 sec EndE T M such that for any
2 sec E and X 2
sec T M;
E
0E
DX
= DX
+W(X) :
(A.115)
A.5.2 Gauge Covariant Connections
De…nition 689 A connection DE on E is said to be a G-connection if for
any u 2 G and any 2 sec E there exists a connection D0E on E such that
for any X 2 sec T M
0E
E
DX
( (u) ) = (u)DX
:
E
0E
Proposition 690 If DX
= DX
+W(X)
0
0E
0E
sec T M , then DX = DX +W (X) with
0
W (X) = uW(X)u
1
(d11)
for
2 sec E and X 2
1
(A.116)
+ udu
:
Suppose that the vector bundle E has the same structural group as the
orthonormal frame bundle PSOe1;3 (M ), which as we know is a reduction of
the frame bundle F (M ). In this case we give the
De…nition 691 A connection DE on E is said to be a generalized Gconnection if for any u 2 G and any
2 sec E there exists a connection
D0E on E such that for any X 2 sec T M , T M = PSOe1;3 (M ) T M R4
0E
E
DX
) = (u)DX
;
0 ( (u)
where X 0 =
TM
X 2 sec T M:
(A.117)
498
Appendix A. Principal Bundles, Vector Bundles and Connections
A.5.3 Curvature Again
E
De…nition
on E. The curvature operator
V2 692 Let D be a G-connection
RE 2 sec
T M EndE of DE is the mapping
RE : sec T M
E
R (X; Y )
=
TM
E
E
DX
DY
E
E
RE (X; Y ) = DX
DY
for any
E ! E;
E E
DY
DX
E E
DY
DX
(A.118)
E
D[X;Y
]
E
D[X;Y
];
(d14)
2 sec E and X; Y 2 sec T M .
If X = @ ; Y = @ 2 sec T U are coordinate basis vectors associated to
the coordinate functions fx g we have
h
i
RE (@ ; @ ) := RE = D@E ; D@E :
(A.119)
s j g of EndE we have under the local trivialization
In a local basis {si
used above
RE = Ra b sa
Ra
b
= @ Wa b
s b;
@ W a b + W a cW c b
W a cW c b:
(A.120)
Eq.(A.120) can also be written
RE = @ W
@ W + [W ; W ] :
(A.121)
A.5.4 Exterior Covariant Derivative Again
De…nition 693 Consider
Ar 2 sec E
We de…ne (
Ar ) ^ Bs by
(
De…nition
694 Let
Vs
T M: We de…ne (
(
Bs )
Ar )
^
Bs =
Vr
T M and Bs 2
(Ar ^ Bs ):
Vr
Ar 2 sec E
T M and
Bs ) ^ (
Ar ) by
^
(
Ar ) =
Vs
T M .
(A.122)
Bs 2 sec EndE
( ) (Bs ^ Ar ):
(A.123)
De…nition 695 Given a connection DE acting on E; the exterior coVr
E
variant derivative dD acting on sections of E
T M and the exteVs
EndE
D
rior covariant derivative d
acting on sections of EndE
T M
(r; s = 0; 1; 2; 3; 4) is given by
(i) if
2 sec E then for any X 2 sec T M
dD
E
(X) = DE ;
(A.124)
A.5 Covariant Derivatives on Vector Bundles
(ii) For any
Vr
Ar 2 sec E
E
dD (
(iii) For any
T M
Ar ) = dD
E
Vs
Bs 2 sec EndE
dD
EndE
Bs ) = dD
(
499
^Ar +
dAr ;
(A.125)
T M
EndE
^ Bs
+
dBs ;
(A.126)
Vs
Proposition
696 Consider the bundle product EV= (EndE
T M) ^
Vr
s
(E
T M ). Let
=
Bs 2 sec EndE
T M and
=
Ar
Vr
E
2 sec E
T M . Then the exterior covariant derivative dD acting on
sections of E satis…es
E
dD (
^
) = (dD
EndE
)
s
+(
^
1)
DE
^d
:
(A.127)
Exercise 697 The reader can now show several interesting results, which
make contact with results obtained earlier when we analyzed the connections
and curvatures on principal bundles and which allowed us sometimes the
use of sloppy notations in the main text:
0E
D 0E
(i) Suppose that the bundle
=
Vr admits a ‡at connection D : We put d
d. Then, if
2 sec E
T M we have
dD
(ii) If
(iii) If
2 sec E
2 sec E
Vr
0E
=d +W
:
^
T M we have
E
(dD )2 = RE
Vr
^
:
(A.128)
T M we have
E
DE
^d
(dD )3 = RE
:
(A.129)
(iii) Suppose that the bundle admits a ‡at connection D0E . We put
0E
dD = d. Then, if
Vs
(iv)
2 sec EndE
T M
dD
EndE
=d
+ [W;
]:
(A.130)
(v)
dD
EndE
RE = 0:
(A.131)
(vi)
RE = dW + W
Remark 698 Note that RE 6= dD
EndE
^ W:
(A.132)
W:
We end here this long Appendix, hopping that the material presented be
enough to permit our reader to follow the more di¢ cult parts of the text
and in particular to see the reason for our use of many eventual sloppy
notations.
500
Appendix A. Principal Bundles, Vector Bundles and Connections
This is page 501
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Acronyms and Abbreviations
ADM
CBF
GRT
GSS
MCGSS
MCPS
DHSF
MECM
i¤
i.e.
e.g.
Arnowitt-Deser-Misner
Cli¤ord Bundle Formalism
General Relativity Theory
Geometrical Space Structure
Metrical Compatible Geometrical Space Structure
Metrical Compatible Parallelism Structure
Dirac-Hestenes Spinor Field
Multiform and Extensor Calculus on Manifolds
if and only if
Id Est
Exempli Gratia
502
Appendix A. Principal Bundles, Vector Bundles and Connections
This is page 503
Printer: Opaque this
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Index
(p; q)-extensor, 36
3-acceleration, 207, 209
3-velocity, 207, 209
adjoint operator, 36
ADM formalism, 366
algebra
homomorphism, 65
Lie algebra of Spine1;3 , 74
Pauli, 89
quaternin, 90
algebraic spin frames, 78
antiderivation, 112
associative algebra, 63
atlas, 98
Bianchi identity, 488
bilinear covariants, 85
bilinear covariants for DHSF, 284
blackhole, 252
boomerang, 86
bradyons, 201
branes, 169
bundle
…ber bundle, 475
frame bundle, 478
left and right spin-Cli¤ord bundles, 263
local section, 477
orthonormal frame bundle, 479
principal bundle, 476
product bundle, 480
spinor bundles, 262
trivialization, 475
vector bundle, 476
Whitney sum, 480
Cartan structure equations, 124
chain rules, 55
characteristic biform, 40
chart, 98
classical Dirac-Hestenes equation,
425
Cli¤ord algebra
Cli¤ord product, 31
de…nition, 30
even subalgebra, 31
real and complex, 67
universality, 32
Cli¤ord bundle, 130, 257
534
Index
Cli¤ord …elds, 130, 259
Cli¤ord numbers, 33
cliforms, 375
conjugation, 29
conjugation operator, 24
connection
connection on a principal bundle, 481
dilation of a connection, 143
Fermi-Walker, 204
gauge covariant connections,
497
generalized G-connection, 497
induced conection, 489
Levi-Civita, 128
linear connection, 490
Nunes, 161
shear of a connection, 143
strain of a connection, 143
torsion of a connection, 143
connection
on the base manifold, 484
connection 1-forms, 124
connection coe¢ cients, 124
conservation laws
Riemann-Cartan spacetimes
Lorentz spacetimes, 317
contorsion tensor, 145
contractions, 29
coordinate functions, 98
cotangent space, 100
covariant ’conservation’laws, 322
covariant D’Alembertian, 150
covariant derivative
of Cli¤ord …elds, 273
of spinor …elds, 275
on a vector bundle, 492
curvature 2-forms, 124
curvature and bending, 161
curvature biform, 177
curvature extensor, 175
curvature operator, 175
curve
lightlike, 200
spacelike, 200
timelike, 200
de Rham periods, 119
derivative
algebraic, 318
determinant, 44
dett, 39
DHSF
representative in C`(M; g), 272
di¤eomeorphism
Poincaré, 225
di¤eomorphism
di¤eomorphism invariance of
GRT, 227
induced connection, 127
invariance, 222
Lorentz, 299
di¤erential
absolute, 380
di¤erential form
Cli¤ord valued, 381
dilation, 46
Dirac algebra, 69
Dirac anticommutator, 135
Dirac commutator, 135
Dirac equation
the many faces, 279
Dirac like representations of ME,
403
Dirac operator, 133
associated, 138
standard, 133
Dirac-Hestenes equation
DHE, 281
dual basis, 24
Einstein equations, 161
the many faces, 375
Einstein-Lorentz Poincaré gauge
de…nition, 233
Einstein-Lorentz-Poincaré coodiante gauge, 2
energy-momentum
1-forms
densities, 323
Index
energy-momentum tensor, 161
equivalence principle, 244
Euler-Lagrange Equation, 300
Euler-Lagrange equations, 320
extensor, 35
(1; 1)-extensor, 37
(p; q)- extensor …eld, 296
Belinfante
energy-momentum, 308
canonical angular momentum
extesor …eld, 309
canonical energy-momentum
extensor …eld, 305
canonical spin extensor …eld,
311
exterior power extenion, 37
spin density extensor for DiracHestenes …eld, 312
spin extensor of the free electromagnetic …eld, 313
symmetric and antisymmetric parts, 37
extensor
canonical energy-momentum
extensor …eld, 303
exterior covariant derivative, 485
indexed form …elds, 125
exterior covariant di¤erential, 378
exterior power extension, 32
fake exterior covariant di¤erential,
375
Fermi transport, 203
physical meaning, 206
…ducial section, 268
Fierz identities, 85
frame
algebraic spin frame, 77
Frenet, 206
inertial reference frame, 203
local Lorentz reference frame,
243
locally synchronizable reference frame, 214
moving frame, 204
535
physically equivalent reference
frames, 232
pseudo inertial reference frame,
242
reference frame, 203
spin coframe bundle, 260
spin frame bundle, 260
spin frame …eld, 260
synchronizable reference frame,
215
function
extensor valued, 296
functional
algebraic derivative, 319
functional
derivative, 320
functional deriviatives
on jet bundles, 317
gauge metric extensor, 42
generalization operator, 40
generalized Hertz potentials
generalized Hertz potentials,
412
generalized Lichnerowicz formula,
287
generalized Maxwell equation, 411
gravitational …eld
representation of gravitational
…eld, 356
graviton, 367
group
Cli¤ord-Lipschitz, 71
Lorentz, 50
non standard realization of Lorentz
group, 237
pinor, 72
Poincaré, 50
spinor, 72
GRT
angular momentum conservation law in the GRT?,
338
Droste and Hilbert solution,
254
536
Index
energy-momentum momentum
in GRT?, 333
Kruskal solution, 254
Schwarzschild original solution,
252
teleparallel equivalent, 368
Hamilton-Jacobi equation, 426
Hamiltonian formalism, 364
Hodge bundle, 120
Hodge codi¤erential, 121
Hodge star operator
Hodge dual, 28
ide