Utah State University
DigitalCommons@USU
All Physics Faculty Publications
Winter 1-8-2015
Variation of the linear biconformal action
James Thomas Wheeler
Utah State University
Follow this and additional works at: http://digitalcommons.usu.edu/physics_facpub
Part of the Elementary Particles and Fields and String Theory Commons
Recommended Citation
Wheeler, James Thomas, "Variation of the linear biconformal action" (2015). All Physics Faculty Publications. Paper 2011.
http://digitalcommons.usu.edu/physics_facpub/2011
This Article is brought to you for free and open access by the Physics at
DigitalCommons@USU. It has been accepted for inclusion in All Physics
Faculty Publications by an authorized administrator of
DigitalCommons@USU. For more information, please contact
[email protected].
Physics
Variation of the linear biconformal action
James T. Wheeler
[email protected] USU Department of physics, 4415 Old Main Hill, Logan, UT
84322-4415
Abstract
We find the field equations of biconformal space in a basis adapted to Lagrangian submanifolds on
which the restriction of the Killing metric is non-degenerate.
1
Introduction
Starting with the conformal symmetries of Euclidean space, we have shown in [1] how construct a manifold
where time manifests as a part of the geometry. The result is based on a theorem by Spencer and Wheeler
[2] detailing when this is possible. In addition, though there is no matter present in the geometries studied
in [1], geometric terms analogous to dark energy and dark matter appear in the Einstein tensor.
Specifically, the quotient of the conformal group of Euclidean n-space by its Weyl subgroup results in a
biconformal geometry possessing many of the properties of relativistic phase space, including both a natural
symplectic form and non-degenerate Killing metric. The general solution for this homogeneous space posesses
orthogonal Lagrangian submanifolds, with the induced metric and the spin connection on the submanifolds
necessarily Lorentzian, despite the Euclidean starting point.
An action functional linear in the Cartan curvatures of the conformal group was developed in [8]. Though
an explicit gravitational theory is presented in [1], the main results of that work apply to the homogeneous
space. It is of great interest to extend this study to nontrivial gravitational solutions. In this work, we vary
the conformal connection fields to find the field equations. The variation is carried out in a basis adapted to
the Lagrangian submanifolds.
We begin in the next Section with a description of the biconformal gauging. Section 3 contains the
variation and the final Section a summary of the field equations.
2
Biconformal gravity
We review the construction of biconformal gravity theory.
2.1
General properties of the conformal group
We begin with the conformal group, C, of compactified Rn . The one-point compactification at infinity
allows a global definition of inversion, with translations of the point at infinity defining the special conformal
transformation. For pseudo-Euclidean spaces, Rp,q , (of dimension n = p + q and signature s = p − q; all
parts of this construction work for any (p, q)) the compactification requires a null cone at infinity in addition
at the point at infinity. Then C has a real linear representation in n + 2 dimensions, V n+2 (alternatively
[(n+2)/2]
we could choose the complex representation C2
for Spin (p + 1, q + 1)). The isotropy subgroup of Rn
is the subgroup of rotations, SO (p, q), together with dilatations. We call this subgroup the homogeneous
Weyl group, W and require our fibers to contain it, leaving only three allowed subgroups for a quotient: W
itself; the inhomogeneous Weyl group, IW, found by appending the translations; and W together with special
1
conformal transformations, isomorphic to IW. The quotient of the conformal group by either inhomogeneous
Weyl group, called the auxiliary gauging, leads naturally to Weyl gravity (see [4]). We concern ourselves with
the only other meaningful conformal quotient, the biconformal gauging, generalizing the principal W-bundle
formed by the quotient of the conformal group by its Weyl subgroup.
The Maurer-Cartan structure equations, follow immediately from a knowledge of the structure constants
generators M αβ of SO (p, q) and n translational generators
of the Lie algebra. In addition to the n(n−1)
2
Pα , there are n generators of translations of a point at infinity (special conformal transformations, or cotranslations) K α , and a single dilatational generator D. Dual to these, we have the connections ξ αβ , χα , π α ,δ,
respectively. Substituting the structure constants into the Maurer-Cartan dual form of the Lie algebra [5]
gives
dξ αβ
α
dχ
dπ α
β
=
(1)
α
(2)
∧ πβ − δ ∧ πα
(3)
= χ ∧
ξ βα
α
ξ αβ
+δ∧χ
= χ ∧ πα
(4)
1
tensors with respect to the original (p, q) metric,
≡ 21 δνα δβµ − δ αµ δνβ antisymmetrizes
1
dδ
where ∆αµ
νβ
ν
= ξ µβ ∧ ξ αµ + 2∆αµ
νβ π µ ∧ χ
δµν = diag (1, . . . , 1, −1, . . . , −1)
These equations, which are the same regardless of the gauging chosen, describe the Cartan connection on
the conformal group manifold. Before proceeding to the quotient, we note that the conformal group has a
nondegenerate Killing form,
 ac

∆db


0 δba

KAB ≡ tr (GA GB ) = cCAD cDBC = 
a


δb 0
1
(2n)
This provides a metric on the conformal Lie algebra. When restricted to M0 ≡ C/W, it remains nondegenerate.
Finally, we note that the conformal group is invariant under inversion. Within the Lie algebra, this
manifests itself as the interchange between the translations and special conformal transformations Pα ↔
δαβ K β along with the interchange of conformal weights, D → −D. The corresponding transformation of the
connection forms, χα ↔ δ αβ π β , δ → −δ, is easily seen to leave eqs.(1)-(4) invariant. This symmetry leads
to complex and Kähler structures.
2.2
The homogeneous quotient C/W
In the conformal group, translations and special conformal transformations are related by inversion. Indeed,
a special conformal tranformation is a translation centered at the point at infinity instead of the origin.
Because the biconformal gauging maintains the symmetry between translations and special conformal transformations, it is useful to name the corresponding connection forms and curvatures to reflect this. Therefore,
the biconformal basis will be described as the solder form and the co-solder form, and the corresponding
curvatures as the torsion and co-torsion. When we speak of “torsion-free biconformal space” we do not imply
that the co-torsion (Cartan curvature of the co-solder form) vanishes. In phase space interpretations, the
solder form is taken to span the cotangent spaces of the spacetime manifold, while the co-solder form is taken
to span the cotangent spaces of the momentum space. The opposite convention is equally valid.
Unlike other quotient manifolds arising in conformal gaugings, the biconformal quotient manifold posesses
natural invariant structures arising from the underlying groups. The first is the restriction of the Killing
2
metric, which is non-degenerate,
 ac
∆db



0
δba
δba
0


0

=
a

δ
b
1
M(2n)
This gives an inner product for the basis,
α β
χ , χβ πα , χ
hχα , π β i
hπ α , π β i
≡
0
δαβ
δba
0
δβα
0
,
2n×2n
(5)
This metric remains unchanged by the generalization to curved base manifolds, when we replace (χα , π α ) →
(ω α , ω α ) below.
The second natural invariant property is the generic presence of a symplectic form. The original fiber
bundle always has this, because the structure equation, eq.(4), shows that χα ∧ π α is exact hence closed,
d2 ω = 0, while it is clear that the two-form product is non-degenerate because (χα , π α ) together span
(2n)
M0 . Moreover, the symplectic form is canonical,
0
δαβ
[Ω]AB =
−δβα 0
so that χα and π α are canonically conjugate, in the sense that they form a canonical basis for the symplectic
form Ω. The symplectic form persists for the 2-form, ω α ∧ω α +Ω (see below), as long as it is non-degenerate,
so curved biconformal spaces are generically symplectic.
1
Next, we consider the effect of inversion symmetry. As a
tensor, the basis interchange takes the
1
form
µ αν
0 δ αν
χ
δ πν
I AB χB =
=
δβµ
0
πν
δβµ χµ
In order to interchange conformal weights, I AB must anticommute with the conformal weight operator, which
is given by
α
µ 0
δµ
χ
+χα
A B
W Bχ =
=
0 −δβν
πν
−π β
A
C
This is the case: we easily check that {I, W } B = I AC W CB + W A
C I B = 0. The commutator gives a new
object,
0
−δ αβ
A
J AB ≡ [I, W ] B =
δαβ
0
Squaring, J AC J CB = −δ AB , we see that J AB provides an almost complex structure. That the almost complex
structure is integrable follows immediately in this (global) basis by the obvious vanishing of the Nijenhuis
tensor,
A
D
A
A
D
D
N ABC = J D
C ∂D J B − J C ∂D J B − J D ∂C J B − ∂B J C = 0
Next, using the symplectic form to define the compatible metric
we find that in this basis g =
δαβ
0
δ
g (u, v) ≡ Ω (u, Jv)
0
, and we check the remaining compatibility conditions of the
αβ
triple (g, J, Ω),
ω (u, v)
J (u)
= g (Ju, v)
=
−1
(φg )
3
(φω (u))
where φω and φg are defined by
φω (u)
=
ω (u, ·)
φg (u)
=
g (u, ·)
(2n)
These are easily checked to be satisified, showing that that M0
is a Kähler manifold. Notice, however,
that the metric of the Kähler manifold is not the restricted Killing metric which we use in the following
considerations. Kähler manifolds play a role in string theory and geometric quantization, and we plan to
study this relationship further.
(2n)
Finally, a surprising result emerges if we require M0 to match our usual expectations for a relativistic
phase space. To make the connection to phase space clear, the precise requirements were studied in [2].
There it was required that the flat biconformal gauging of SO (p, q) in any dimension n = p + q have
Lagrangian submanifolds that are orthogonal with respect to the 2n-dim biconformal (Killing) metric and
have non-degenerate n-dim restrictions
metric. This is found to be possible only if the original space
of the
is Euclidean or signature zero p ∈ 0, n2 , n . The signature of the Lagrangian submanifolds is severely
limited (p → p ± 1), leading in the two Euclidean cases to Lorentzian configuration space, and hence the
origin of time. For the case of flat, 8-dim biconformal space we paraphrase the the following theorem from
[2]:
Flat n-dim biconformal space is a metric phase space with the following properties:
1. There exist Lagrangian submanifolds orthogonal with respect to the 2n-dim biconformal (Killing) metric
2. The restriction of the Killing metric to each Lagrangian submanifold is non-degenerate
if and only if the initial n-dim space we gauge is Euclidean or signature zero. In the Euclidean case the
resulting configuration sub-manifold is necessarily Lorentzian [2].
Thus, it is possible to impose the conditions necessary to make biconformal space a metric phase space
only in a restricted subclass of cases, and in 4-dim or Euclidean cases the configuration space metric must
be Lorentzian. In [1], it was shown that coordinates (wα , sβ ) exist such that the metric takes the form
2
(6)
kαβ = s2 δαβ − 2 sα sβ
s
where δαβ is the Euclidean metric and s2 = δ αβ sα sβ . The signature changing character is easily seen.
The Euclidean gauge theory necessarily posesses a special vector, v = ω − 21 ηab dη bc , built from the Weyl
vector and the conformal gauge of the metric [1]. It is preferable to write kαβ in terms of the components of
this vector, uα and vα defined above. In the (wα , sβ ) coordinates, vα = −kβsα and uα = βk αβ sβ , so we find
1 2
2
kαβ = 2 v δαβ − 2 vα vβ
β
v
2
with inverse k αβ = βv2 δ αβ − v22 δ αµ δ βν vµ vν . This shows that the vector vα determines the timelike directions. This vector gives the time direction on both the orthogonal Lagrangian submanifolds, making them
necessarily Lorentzian. The full manifold retains its original symmetry.
Thus, the structures of the conformal group give rise to a natural direction of time. The situation is
reminiscent of previous studies. In 1979, Stelle and West introduced a special vector field to choose the local
symmetry of the MacDowell-Mansouri theory. The vector breaks the de Sitter symmetry, eliminating the
need for the Wigner-Inönu contraction. Recently, Westman and Zlosnik[9] have looked in depth at both the
de Sitter and anti-de Sitter cases using a class of actions which extend that of Stelle and West by including
derivative terms for the vector field and therefore lead to dynamical symmetry breaking. In [10, 11] and
Einstein-Aether theory [12], there is also a special vector field introduced into the action by hand that can
make the Lorentzian metric Euclidean. These approaches are distinct from that of the biconformal approach,
where the vector necessary for specifying the timelike direction occurs naturally from the underlying group
structure.
4
2.3
Gauging
By gauge theory, we typically understand a theory (i.e., the specification of an action functional) which is
invariant under a local symmetry group – the gauge symmetry. Thus, there may be many gauge theories
having the same gauge group. However, gauge theories having the same gauge group share a common
structure: the underlying principal fiber bundle in which the base manifold is spacetime or some other
world manifold and the fibers are copies of the gauge group. Such a principal fiber bundle is most simply
constructed as the quotient of a larger group by the symmetry group. Constructed in this way, we have
immediate access to relevant tensor fields: any group invariant tensors, the curvatures of the bundle, and
the vectors of the group representation. Then any functional built invariantly from these tensors is a gauge
theory. For example, the quotient of the Poincaré group by its Lorentz subgroup may be generalized to a
principal fiber bundle with Lorentz group fibers and a general base manifold having arbitrary Riemannian
curvature. Identifying the curvature, solder form, Lorentz metric, and Levi-Civita tensor as tensors with
respect to this local Lorentz symmetry, it is clear that any functional built invariantly from them is a gauge
theory with local Lorentz symmetry. In addition, if we use a linear representation, SO(3, 1) or SL (2, C), of
the Lorentz group then the action functional may include vectors or spinors from that representation and
their covariant derivatives. For these reasons, we will define a gauging to be the fiber bundle of a specific
quotient, along with the identification of its associated tensors. A gauge theory remains the specification of
an action functional invariant on this bundle.
The biconformal gauging was first considered by Ivanov and Niederle [6]. They considered a curvaturequadratic action, with the extra four dimensions restricted to the minimum necessary for consistency with
conformal symmetry. The full 2n-dimensional geometry was first studied in [7], where tensorial constraints
on the full higher-dimensional curvatures were shown to reduce the geomety to general relativity. It was soon
realized that the dimensionless volume form permits an action linear in the curvatures [8], and the resulting
field equations were found to reduce to the system described in [7].
In the fiber bundle produced by the quotient C/W, the one-forms ξ αβ , δ span the W-fibers, with
(χα , π α ) spanning the remaining 2n independent directions of the homogeneous base manifold. We now
α
α
α
α
generalize the connection (and the manifold, if desired) by replacing ξ β , χ , π α , δ → ω β , e , ω α , ω in
the Maurer-Cartan equations, eqs.(1-4) to give the Cartan curvatures in terms of the new connection forms,
dω αβ
=
α
ν
ω µβ ∧ ω αµ + 2∆αµ
νβ ω µ ∧ ω + Ω β
(7)
deα
=
eβ ∧ ω αβ + ω ∧ eα + Tα
(8)
dω α
=
(9)
dω
=
ω βα
α
∧ ω β − ω ∧ ω α + Sα
ω ∧ ωα + Ω
(10)
Equations eq.(7-10) give the curvature two-forms in terms of the connection forms. We have therefore constructed a 2n-dim manifold based on the conformal group with local W symmetry. We require integrability
of these equations and horizontality of the curvatures.
Horizontality prevents the curvatures from depending on either the spin connection ω αβ or the Weyl
vector ω, but they now still depend on the 2n non-vertical forms, (ω α , ω α ). This means there are far more
components than for an n-dim Riemannian geometry. For example,
Ωαβ =
1
1 α
Ω
ω µ ∧ ω ν + Ωαβ µν ω µ ∧ ω ν + Ωαβ µν ω µ ∧ ω ν
2 βµν
2
The coefficients of the pure terms, Ωαβµν and Ωαβ µν each have the same number of degrees of freedom as the
Riemannian curvature of an n-dim Weyl geometry, while the cross-term coefficients Ωαβ µν have more, being
asymmetric on the final two indices.
Notice that the spin connection, ξ αβ , is antisymmetric with respect to the original (p, q) metric, δαβ , in
the sense that
ξ αβ = −δ αµ δβν ξ νµ
5
It is crucial to note that ω αβ retains this property, ω αβ = −δ αµ δβν ω νµ . This expresses metric compatibility
with the SO (p, q)-covariant derivative, since it implies
Dδαβ ≡ dδαβ − δµβ ω µα − δαµ ω µβ = 0
Therefore, the curved generalization has a connection which is compatible with a locally (p, q)-metric. This
relationship is general. If καβ is any metric, its compatible spin connection will satisfy ω αβ = −καµ κβν ω νµ .
Since we also have local scale symmetry, the full covariant derivative we use will also include a Weyl vector
term.
Integrability of the modified structure equations is guaranteed by a Bianchi identity for each of the
curvatures,
0
=
ν
ν
DΩαβ + 2∆αµ
νβ (Ωµ ∧ ω − ω µ ∧ Ω )
(11)
0
=
DTα − eβ ∧ Ωαβ + Ω ∧ eα
(12)
0
=
DSα + Ωαβ ∧ ω β − ω α ∧ Ω
(13)
0
=
α
α
DΩ + T ∧ ω α − ω ∧ Sα
(14)
where D is the Weyl covariant derivative,
DΩαβ
= dΩαβ + Ωµβ ∧ ω αµ − Ωαµ ∧ ω µβ
DTα
= dTα + Tβ ∧ ω αβ − ω ∧ Tα
DSα
= dSα − ω βα ∧ Sβ + Sα ∧ ω
DΩ
= dΩ
Notice that our development to this point was based solely on group quotients and generalization of
the resulting principal fiber bundle. We have arrived at the form of the curvatures in terms of the Cartan
connection, and Bianchi identities required for integrability, thereby describing certain classes of geometry
with local symmetry. Within the biconformal quotient, the demand for orthogonal Lagrangian submanifolds
with non-degenerate n-dim restrictions of the Killing metric leads to the selection of certain Lorentzian
submanifolds.
We are guided in the choice of action functional by the example of general relativity. In Riemannin
geometries we may write the Einstein-Hilbert action and proceed. More systematically, however, we may
write the most general, even-parity
action linear in the curvature and torsion. This turns out to be the
´
tetradic Palatini action, SP = Rab ∧ ec ∧ ed εabcd . A full variation of the connection, δeb , δω ab , implies
vanishing torsion in addition to the Einstein equation. The latter, more robust approach is what we follow
for conformal gravity theories.
It is generally of interest to build the simplest class of actions possible, and we use the following criteria:
1. The pure-gravity action should be built from the available curvature tensor(s) and other tensors which
occur in the geometric construction.
2. The action should be of lowest possible order ≥ 1 in the curvatures.
3. The action should be of even parity.
These are of sufficient generality not to bias our choice and sufficiently constraining to give a reasonable
class of theories.
Notice that if we perform an infinitesmal conformal transformation to the curvatures, Ωαβ , Ωα , Ωβ , Ω ,
they all mix with one another, since the conformal curvature is really a single Lie-algebra-valued two form.
However, the generalization to a curved manifold breaks the non-vertical symmetries, allowing these different
components to become independent tensors under the remaining Weyl group. Thus, to find the available
tensors, we apply an infinitesmal transformation of the fiber symmetry. Tensors are those objects which
6
transform linearly and homogeneously under these transformations. Note that this is not symmetry breaking,
but rather simply part of the construction of the manifold and local symmetry.
The biconformal gauging, based on C/W, has tensorial basis forms (ω α , ω α ). Moreover, each of the
component curvatures, Ωαβ , Ωα , Ωβ , Ω , becomes an independent tensor under the Weyl group. The volume
form eρσ...λαβ...ν ω α ∧ ω β ∧ . . . ∧ ω ν ∧ ω ρ ∧ ω σ ∧ . . . ∧ ω λ has zero conformal weight. Since both Ωαβ and
Ω also have zero conformal weight, there exists a curvature-linear action in any dimension [8]. The most
general linear case is
ˆ
S=
αΩαβ + βΩδβα + γω α ∧ ω β ∧ eβρ...σαµ...ν ω µ ∧ . . . ∧ ω ν ∧ ω ρ ∧ . . . ∧ ω σ
(15)
We now have three important properties of biconformal gravity that arise because of the doubled dimension:
(1) the non-degenerate conformal Killing metric induces a non-degenerate metric on the manifold, (2) the
dilatational structure equation generically gives a symplectic form, and (3) there exists a Weyl symmetric
action functional linear in the curvature, valid in any dimension.
There are a number of known results following from the linear action. In [8] torsion-constrained solutions
are found which are consistent with scale-invariant general relativity. Subsequent work along the same lines
shows that the torsion-free solutions are determined by the spacetime solder form, and reduce to describe
spaces conformal to Ricci-flat spacetimes on the corresponding spacetime submanifold. A supersymmetric
version is presented in [13], and studies of Hamiltonian dynamics [16, 15] and quantum dynamics [16] support
the idea that the models describe some type of relativistic phase space determined by the configuration space
solution.
In an orthonormal frame field, (ea , fa ), adapted to the Lagrangian submanifolds, two new tensor fields
emerge. The first, v = va ea + ua fa is a combination of the Weyl vector with the scale factor on the metric,
and determines the timelike directions on the submanifolds. The second, β ab = µab + ρab = µabc ec + ρab c fc ,
comes from the components of the spin connection, and is symmetric with respect to the new metric,
β ab = η ac ηbd β dc . Though this field is part of the original spin connection, it transforms homogeneously under
local Lorentz transformations and local dilatations. A complete discussion is given in [1].
We now proceed to write the action, eq.(15), in this adapted basis, then vary it to find the field gravitational field equations.
3
Variation
In the adapted basis, with the curvatures included, the Cartan structure equations become
d
f
e − hde fe + Ωab
dω ab = ω cb ω ac + ∆ac
db fc + hcf e
1
dea = ec ω ac + ωea + Dhac fc + hcd ed + Ta
2
1
b
dfa = ω a fb − ωfa − Dhab eb − hbc fc + Sa
2
dω = ea fa + Ω
and the action is immediately seen to be
ˆ
S = (αΩab + βδba Ω + γea fb ) εac...d εbe...f ec . . . ed fe . . . ff
We are able to separate the Levi-Civita tensor into two pieces because we demand that ea , fb span submanifolds.
3.1
The variation of the Weyl vector
We begin with the variation of the Weyl vector, ω,
0
=
δS
7
ˆ
=
β
=
β
d (δω) εac...d εae...f ec . . . ed fe . . . ff
ˆ
εac...d εae...f d δωec . . . ed fe . . . ff + δωd ec . . . ed fe . . . ff
Dropping the surface term and using the structure equations,
ˆ
h
i
n−1
0 = β δω εacd...e εaf ...g d ec ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed d (fe ff . . . fg )
ˆ
h
i
n−1
= β δω (n − 1) εacd...e εaf ...g (dec ) ed . . . ee ff . . . fg + (−1)
(n − 1) εac...d εaef ...g ec . . . ed (dfe ) ff . . . fg
ˆ
1
ch
i
c
af ...g
h c
c
fh + hhi e + T ed . . . ee ff . . . fg
= β δω (n − 1) εacd...e ε
e ω h + ωe + Dh
2
ˆ
1
n−1
aef ...g c
d
h
h
hi
+β δω (−1)
(n − 1) εac...d ε
e . . . e ω e fh − ωfe − Dheh e − h fi + Se ff . . . fg
2
Define
T̃c
S̃a
1
= Tc + Dhch fh + hhi ei
2
1
= Sa − Dhab eb − hbc fc
2
and collect like terms,
ˆ
n−1
0 = (n − 1) β δω εacd...e εaf ...g eh ω ch ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed ω he fh ff . . . fg
ˆ
n
+ (n − 1) β δω εacd...e εaf ...g ωec ed . . . ee ff . . . fg + (−1) εac...d εaef ...g ec . . . ed ωfe ff . . . fg
ˆ
n−1
+ (n − 1) β δω εacd...e εaf ...g T̃c ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed S̃e ff . . . fg
For the Weyl vector terms we have
ˆ
n
δ1 = (n − 1) β δω εacd...e εaf ...g ωec ed . . . ee ff . . . fg + (−1) εac...d εaef ...g ec . . . ed ωfe ff . . . fg
ˆ
q
n
n−1
q
f ...g
ef ...g c
= (n − 1) β δω (−1) (n − 1)!δcd...e
ωec ed . . . ee ff . . . fg + (−1) (−1)
ω (−1) (n − 1)!δc...d
e . . . ed fe ff . . . fg
ˆ
q
ef ...g c
= (n − 1) β δω (ω − ω) (−1) (n − 1)!δc...d
e . . . e d fe ff . . . fg
=
0
c i
Now check the spin connection terms. Expand the spin connection as ω ch = ωhi
e + ωhci fi , and the variation
j
j
as δω = Aj e + B fj so that
ˆ
n−1
δ2 = (n − 1) β δω εacd...e εaf ...g eh ω ch ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed ω he fh ff . . . fg
ˆ
n−1
c i d
h i
= (n − 1) β δω εacd...e εaf ...g eh ωhi
e e . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed ωei
e fh ff . . . fg
ˆ
n−1
+ (n − 1) β δω εacd...e εaf ...g eh ωhci fi ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed ωehi fi fh ff . . . fg
ˆ
n−1
h
= (n − 1) β Aj ωki
εahd...e εaf ...g ej ek ei ed . . . ee ff . . . fg + (−1)
εac...d εakf ...g ej ec . . . ed ei fh ff . . . fg
8
=
ˆ
n
h
(−1) (n − 1) β B j ωki
εahd...e ek ei ed . . . ee εaf ...g fj ff . . . fg + εac...d ei ec . . . ed εakf ...g fj fh ff . . . fg
ˆ
n−1
εac...d εakf ...g ej ec . . . ed fi fh ff . . . fg
+ (n − 1) β Aj ωkhi εahd...e εaf ...g ej ek fi ed . . . ee ff . . . fg + (−1)
Define the volume forms,
ea . . . eb
= εa...b Φn
fa . . . fb
= εa...b Φn
Φnn
Then
δ2
= Φn Φn
ˆ
n
h
B j ωki
εahd...e ek ei ed . . . ee εaf ...g fj ff . . . fg + εac...d ei ec . . . ed εakf ...g fj fh ff . . . fg
ˆ
n
+ (−1) (n − 1) β Aj ωkhi εahd...e εaf ...g ej ek ed . . . ee fi ff . . . fg − εac...d εakf ...g ej ec . . . ed fi fh ff . . . fg
ˆ
n
h
= (−1) (n − 1) β B j ωki
εahd...e εkid...e Φn εaf ...g εjf ...g Φn + εac...d εic...d Φn εakf ...g εjhf ...g Φn
ˆ
n
+ (−1) (n − 1) β Aj ωkhi εahd...e εaf ...g εjkd...e Φn εif ...g Φn − εac...d εakf ...g εjc...d Φn εihf ...g Φn
ˆ
n
n
h
ki
ak
= (−1) (n − 1) β B j ωki
2 (n − 2)!δah
(n − 1)!δja + (n − 1)!δai 2 (n − 2)!δjh
Φn
ˆ
n
jk
ak
+ (−1) (n − 1) β Aj ωkhi 2 (n − 2)!δah
(n − 1)!δia − 2 (n − 2)!δih
(n − 1)!δaj Φnn
ˆ
n
n
i
k
k
h
= (−1) (n − 1) (n − 1)! (n − 2)!β B j ωji
− ωkj
+ ωkj
− ωjh
Φn
ˆ
n
+ (−1) (n − 1) (n − 1)! (n − 2)!β Aj ωkkj − ωkjk − ωkkj + ωkjk Φnn
=
(−1) (n − 1) β
=
0
This leaves the field equation in terms of the torsion and co-torsion, as expected.
ˆ
n−1
0 = (n − 1) β δω εacd...e εaf ...g T̃c ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed S̃e ff . . . fg
Expanding the variation gives two equations,
n−1
0 = (n − 1) βej εacd...e εaf ...g T̃c ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed S̃e ff . . . fg
n−1
0 = (n − 1) βfj εacd...e εaf ...g T̃c ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed S̃e ff . . . fg
For the first,
0
n−1
(n − 1) βej εacd...e εaf ...g T̃c ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed S̃e ff . . . fg
n−1
= (n − 1) β εacd...e εaf ...g T̃c ej ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ej ec . . . ed S̃e ff . . . fg
1
n−1
= (n − 1) β εacd...e εaf ...g T̃ ch k fh ek ej ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ej ec . . . ed S̃e hk fh fk ff . . . fg
2
1
n
n−1
= (n − 1) β (−1) εacd...e εaf ...g T̃ ch k εkjd...e εhf ...g + (−1)
S̃e hk εac...d εaef ...g εjc...d εhkf ...g Φnn
2
=
9
1
n
kj ch
2 (n − 2)! (n − 1)! (n − 1) (−1) β δha δac
T̃ k − S̃e
2
1
n
= (n − 2)! (n − 1)! (n − 1) (−1) β T̃ jk k − T̃ kj k − S̃e
2
=
so that
β S̃k
jk
hk j ae
δa δhk
je
1
+ S̃e
2
Φnn
ej
Φnn
= β T̃ jk k − T̃ kj k
and similarly, for the second,
n−1
0 = (n − 1) βfj εacd...e εaf ...g T̃c ed . . . ee ff . . . fg + (−1)
εac...d εaef ...g ec . . . ed S̃e ff . . . fg
n
= (n − 1) β (−1) εacd...e εaf ...g T̃c ed . . . ee fj ff . . . fg + εac...d εaef ...g ec . . . ed S̃e fj ff . . . fg
n 1
af ...g c
h k d
e
aef ...g c
d
h
k
T̃ hk e e e . . . e fj ff . . . fg + εac...d ε
e . . . e S̃e k fh e fj ff . . . fg
= (n − 1) β (−1) εacd...e ε
2
n 1
af ...g c
h k d
e
aef ...g
h
c
d k
= (n − 1) β (−1) εacd...e ε
T̃ hk e e e . . . e fj ff . . . fg − εac...d ε
S̃e k e . . . e e fh fj ff . . . fg
2
n 1
af ...g 1 c
hkd...e
aef ...g
h
c...dk
T̃ hk ε
εjf ...g − εac...d ε
S̃e k ε
εhjf ...g Φnn
= 2 (n − 2)! (n − 1)! (n − 1) β (−1) εacd...e ε
2
2
a1 c
n−1 k
n
a e
e a
h
h k
h k
δa S̃e k Φnn
= (n − 2)! (n − 1)! (n − 1) β (−1) δa δc − δc δa δj T̃ hk − δh δj − δh δj (−1)
2
n
= (n − 2)! (n − 1)! (n − 1) (−1) β T̃ c jc + S̃j k k − S̃h h j Φnn
so that
β T̃ c
3.1.1
jc
= β S̃h h
j
− S̃j k
k
Check
Assume the tensorial character, so from the action
ˆ
S = (αΩab + βδba Ω + γea fb ) εac...d εbe...f ec . . . ed fe . . . ff
we have
0
=
δSˆ
= β d (δω) εac...d εae...f ec . . . ed fe . . . ff
ˆ
= β εac...d εae...f d δωec . . . ed fe . . . ff + δωd ec . . . ed fe . . . ff
ˆ
= β εac...d εae...f δωd ec . . . ed fe . . . ff
ˆ
n−1
= (n − 1) β δω εacg...d εae...f T̃c eg . . . ed fe . . . ff + (−1)
εac...d εaeg...f ec . . . ed S̃e fg . . . ff
so with δω = Ak ek + B k fk
ˆ
n−1
0 = (n − 1) β δω εacg...d εae...f T̃c eg . . . ed fe . . . ff + (−1)
εac...d εaeg...f ec . . . ed S̃e fg . . . ff
ˆ
n−1
= (n − 1) β Ak εacg...d εae...f T̃c ek eg . . . ed fe . . . ff + (−1)
εac...d εaeg...f ek ec . . . ed S̃e fg . . . ff
10
ˆ
n
+ (n − 1) β B k (−1) εacg...d εae...f T̃c eg . . . ed fk fe . . . ff + εac...d εaeg...f ec . . . ed S̃e fk fg . . . ff
ˆ
1
n
n−1
= (n − 1) β Ak (−1) εacg...d εae...f T̃ cmn en ek eg . . . ed fm fe . . . ff + (−1)
εac...d εaeg...f ek ec . . . ed S̃e mn fm fn fg . . . ff
2
ˆ
1
n
n
+ (n − 1) β B k (−1) εacg...d εae...f T̃ c mn em en eg . . . ed fk fe . . . ff + (−1) εac...d εaeg...f en ec . . . ed S̃e m n fm fk fg . . . ff
2
Therefore, the field equations are
1
n
n−1
0 = (n − 1) β (−1) εacg...d εae...f T̃ cmn εnkg...d εme...f + (−1)
εac...d εaeg...f εkc...d S̃e
2
n
a nk cm
ae 1
= 2 (n − 1)! (n − 1)! (−1) β δm
δac T̃ n − δak δmn
S̃ mn
2 e
n
= (n − 1)! (n − 1)! (−1) β T̃ kmm − T̃ nkn − S̃e ke
n
n ae
m
a mn 1 c
0 = 2 (n − 1)! (n − 1)! (−1) β δk δac T̃ mn + δa δmk S̃e n
2
n
= (n − 1)! (n − 1)! (−1) β T̃ c kc + S̃k m m − S̃mm k
mn
εmng...f
and therefore,
β T̃ kmm − T̃ nkn
β S̃k m m − S̃mm k
= β S̃e
ke
= −β T̃ c
kc
These check.
3.1.2
Check
Check again:
ˆ
S =
(αΩab + βδba Ω + γea fb ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
0 = δ (αΩab + βδba Ω + γea fb ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
=
βδba d (δω) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
=
βεac...d εae...f d δωec . . . ed fe . . . ff + δωd ec . . . ed fe . . . ff
ˆ
n−1
=
βδω (n − 1) εach...d εae...f dec eh . . . ed fe . . . ff + (−1)
(n − 1) εac...d εaeh...f ec . . . ed dfe fh . . . ff
ˆ
n−1
=
β Ak ek + B k fk (n − 1) εach...d εae...f Tc eh . . . ed fe . . . ff + (−1)
(n − 1) εac...d εaeh...f ec . . . ed Se fh . . . ff
ˆ
n−1
=
βAk (n − 1) εach...d εae...f Tc ek eh . . . ed fe . . . ff + (−1)
(n − 1) εac...d εaeh...f ek ec . . . ed Se fh . . . ff
ˆ
n
+ βB k (−1) (n − 1) εach...d εae...f Tc eh . . . ed fk fe . . . ff + (n − 1) εac...d εaeh...f Se ec . . . ed fk fh . . . ff
Then
n
0
=
(−1) (n − 1) βεach...d εae...f T cmn en ek eh . . . ed fm fe . . . ff + (−1)
0
=
(−1)
n
1
(n − 1) βεach...d εae...f T c
2
m n h
mn e e e
n−1
1
(n − 1) βεac...d εaeh...f Se
2
n
. . . ed fk fe . . . ff + (−1) (n − 1) βεac...d εaeh...f Se
11
mn
m
ek ec . . . ed fm fn fh . .
n c
ne e
. . . ed fm fk fh . . .
For the first,
0
n
n−1
(−1) (n − 1) βεach...d εae...f T cmn εnkh...d εme...f + (−1)
=
1
(n − 1) βεac...d εaeh...f Se
2
mn kc...d
ε
n
n−1 1
ae
a nk cm
2 (n − 1)! (n − 1) (n − 2)!βδak δmn
Se
(−1) 2 (n − 1)! (n − 1) (n − 2)!βδm
δac T n + (−1)
2
n
n−1
n k
k
= (−1) (n − 1)! (n − 1) (n − 2)!β δm
δc − δcn δm
T cmn + (−1)
(n − 1)! (n − 1)!βSe ke
n
= (−1) (n − 1)! (n − 1)!β T kmm − T nkn − Se ke
=
εmnh...f
mn
so that
β T kmm − T nkn − Se
ke
=0
For the second,
0
1
n
(n − 1) βεach...d εae...f T c mn εmnh...d εke...f + (−1) (n − 1) βεac...d εaeh...f Se
2
n
n
mn c
ae
= (−1) (n − 1)! (n − 1)!βδkc
T mn + (−1) 2 (n − 1)! (n − 1)!βδan δmk
Se m n
n
=
(−1)
=
(−1) (n − 1)! (n − 1)!β (T c
n
kc
+ Sk
m
m
− Sm
m
m
− Sm
m
nε
nc...d
εmkh...f
k)
so that
β (T c
3.2
kc
+ Sk
m
m
k)
=0
Spin connection variation
Now vary the spin connection,
0
= δSˆ
= α (δΩab ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
= α
dδω ab − δω gb ω ag − ω gb δω ag εac...d εbe...f ec . . . ed fe . . . ff
ˆ
ˆ
= α εac...d εbe...f d δω ab ec . . . ed fe . . . ff + α εac...d εbe...f δω ab d ec . . . ed fe . . . ff
ˆ
−α
δω gb ω ag + ω gb δω ag εac...d εbe...f ec . . . ed fe . . . ff
ˆ
n−1
= α δω ab (n − 1) εac...d εbe...f dec . . . ed fe . . . ff + (−1)
(n − 1) εac...d εbe...f ec . . . ed dfe . . . ff
ˆ
−α εac...d εbe...f δω gb ω ag + ω gb δω ag ec . . . ed fe . . . ff
Now substitute from the structure equations and expand the spin connection and its variation as
Then
0
ω ag
=
a i
ωgi
e + ωgai fi
δω hk
=
Ahki ei + Bkhi fi
ˆ
n−1
δω ab (n − 1) εac...d εbe...f dec . . . ed fe . . . ff + (−1)
(n − 1) εac...d εbe...f ec . . . ed dfe . . . ff
ˆ
−α εac...d εbe...f δω gb ω ag + ω gb δω ag ec . . . ed fe . . . ff
ˆ
n−1
= α
Aabi ei + Bbai fi (n − 1) εach...d εbe...f eg ω cg + ωec + T̃c eh . . . ed fe . . . ff + (−1)
(n − 1) εac...d εbeh...f ec . . . ed
= α
12
ˆ
g j
a j
−α εac...d εbe...f Agbi ei + Bbgi fi ωgj
e + ωgaj fj + ωbj
e + ωbgj fj Aagi ei + Bgai fi ec . . . ed fe . . . ff
ˆ
ˆ
n−1
= α (n − 1) εach...d εbe...f Aabi ei eg ω cg + ωec + T̃c eh . . . ed fe . . . ff + α (−1)
(n − 1) εac...d εbeh...f Aabi ei ec . . . ed ω
ˆ
bj
−α Aabi εmc...d εbe...f ωamj ei fj ec . . . ed fe . . . ff + εac...d εme...f ωm
fj ei ec . . . ed fe . . . ff
ˆ
ˆ
n−1
+α (n − 1) εach...d εbe...f Bbai fi eg ω cg + ωec + T̃c eh . . . ed fe . . . ff + α (−1)
(n − 1) εac...d εbeh...f Bbai fi ec . . . ed
ˆ
m j
b
bj
−α Bbai εmc...d εbe...f fi ωaj
e + ωamj fj + εac...d εme...f ωmj
ej + ωm
fj fi ec . . . ed fe . . . ff
so we have two equations,
n−1
εach...d εbe...f ei T̃c eh . . . ed fe . . . ff + (−1)
εac...d εbeh...f ei ec . . . ed S̃e fh . . . ff
0 = α (n − 1) ∆ma
bn
n
be...f i
+α (n − 1) ∆ma
e ωec eh . . . ed fe . . . ff + (−1) α (n − 1) εac...d εbeh...f ei ec . . . ed ωfe fh . . . ff
bn εach...d ε
be...f i g cj
+α (n − 1) ∆ma
e e ωg fj eh . . . ed fe . . . ff + (−1)
bn εach...d ε
n−1
−α (−1)
0
n−1
α (n − 1) εac...d εbeh...f ei ec . . . ed ωegj fj fg fh . . . ff
n
be...f mj i c
bj i c
∆ma
ωa e e . . . ed fj fe . . . ff − α (−1) εac...d εme...f ωm
e e . . . ed fj fe . . . ff
bn εmc...d ε
be...f
= α (n − 1) ∆ma
fi T̃c eh . . . ed fe . . . ff + α (−1)
bn εach...d ε
n−1
(n − 1) εac...d εbeh...f fi ec . . . ed S̃e fh . . . ff
n
be...f
+α (n − 1) ∆ma
fi ωec eh . . . ed fe . . . ff + α (−1) (n − 1) εac...d εbeh...f fi ec . . . ed ωfe fh . . . ff
bn εach...d ε
n−1
be...f
c j h
+α (n − 1) ∆ma
fi eg ωgj
e e . . . ed fe . . . ff + α (−1)
bn εach...d ε
g j
(n − 1) εac...d εbeh...f fi ec . . . ed ωej
e fg fh . . . ff
be...f
m j c
b
−α∆ma
fi ωaj
e e . . . ed fe . . . ff − αεac...d εme...f ωmj
ej fi ec . . . ed fe . . . ff
bn εmc...d ε
Look at the Weyl vector pieces,
n
be...f i
= α (n − 1) ∆ma
e ωec eh . . . ed fe . . . ff + (−1) α (n − 1) εac...d εbeh...f ei ec . . . ed ωfe fh . . . ff
bn εach...d ε
δ1
n
be...f i
= α (n − 1) ∆ma
e W j fj ec eh . . . ed fe . . . ff + (−1) α (n − 1) εac...d εbeh...f ei ec . . . ed W j fj fe fh . . . ff
bn εach...d ε
n−1
n
j
= α (n − 1) ∆ma
(−1)
εach...d εbe...f ei ec eh . . . ed fj fe . . . ff + (−1) εac...d ei ec . . . ed εbeh...f fj fe fh . . . ff
bn W
n
j
= (−1) α (n − 1) ∆ma
−εach...d εbe...f εich...d εje...f + εac...d εic...d εbeh...f εjeh...f Φnn
bn W
n
j
= (−1) α (n − 1) ∆ma
− (n − 1)! (n − 1)!δjb δai + (n − 1)! (n − 1)!δjb δai Φnn
bn W
=
0
and
δ10
n
be...f
= α (n − 1) ∆ma
fi ωec eh . . . ed fe . . . ff + α (−1) (n − 1) εac...d εbeh...f fi ec . . . ed ωfe fh . . . ff
bn εach...d ε
n
be...f
= α (n − 1) (−1) ∆ma
Wj ej ec eh . . . ed fi fe . . . ff − εac...d εbeh...f ej ec . . . ed Wj fi fe fh . . . ff
bn εach...d ε
n
be...f jch...d
= α (n − 1) (−1) Wj ∆ma
ε
εie...f − εac...d εbeh...f εjc...d εieh...f Φnn
bn εach...d ε
n
n
j b
j b
= (n − 1)! (n − 1)!α (n − 1) (−1) ∆ma
bn Wj δa δi − δa δi Φn
=
0
Now the spin connection terms, dropping the full volume form,
σ
n
n−1
cj
be...f igh...d
= α (n − 1) (−1) ∆ma
ε
εje...f + (−1)
bn ωg εach...d ε
−∆ma
bn α (−1)
n−1
n
α (n − 1) ωegj εac...d εbeh...f εic...d εjgh...f ∆ma
bn
bj
ωamj εmc...d εbe...f εic...d εje...f − α (−1) ωm
εac...d εme...f εic...d εje...f ∆ma
bn
n
cj ig b
gj be i
= α (n − 1) (−1) 2 (n − 1)! (n − 2)!∆ma
bn ωg δac εj − ωe δjg δa
n−1
mj b i
bj m i
−α (−1)
(n − 1)! (n − 1)!∆ma
bn ωa δj δm − ωm δj δa
13
n
n−1
bm i
= α (n − 1) (n − 1)! (n − 2)!∆ma
(−1) ωccb δai − ωaib − ωeeb δai + ωebe δai − (−1)
ωaib − ωm
δa
bn
n
cb i
ib
eb i
be i
ib
bm i
= α (n − 1) (n − 1)! (n − 2)! (−1) ∆ma
bn ωc δa − ωa − ωe δa + ωe δa + ωa − ωm δa
n
ib
be i
ib
bm i
= α (n − 1) (n − 1)! (n − 2)! (−1) ∆ma
bn −ωa + ωe δa + ωa − ωm δa
=
0
and
σ0
n
n
c
be...f gjh...d
a g
beh...f jc...d
= α (n − 1) (−1) ∆ma
ε
εie...f + α (n − 1) (−1) ∆ma
ε
εigh...f
bn ωgj εach...d ε
bn ωgj ωej εac...d ε
n
n−1
m
be...f jc...d
− (−1) α∆ma
ε
εie...f − α (−1)
bn ωaj εmc...d ε
b
me...f jc...d
∆ma
ε
εie...f
bn ωmj εac...d ε
n
n
c b gj
ma g j be
= α2 (n − 1)! (n − 1)! (−1) ∆ma
bn ωgj δi δac + α2 (n − 1)! (n − 1)! (−1) ∆bn ωej δa δig
n
n−1
m b j
− (−1) (n − 1)! (n − 1)!α∆ma
bn ωaj δi δm − α (n − 1)! (n − 1)! (−1)
n
= α (n − 1)! (n − 1)! (−1) ∆ma
bn
= 0
b
m j
∆ma
bn ωmj δi δa
c b
g b
e b
b
m b
b
ωac
δi − ωga
δi + ωea
δi − ωia
− ωam
δi + ωia
Therefore, we correctly find two relationships among the curvatures,
n−1
be...f i c h
d
beh...f i c
d
0 = α∆ma
ε
ε
e
T̃
e
.
.
.
e
f
.
.
.
f
+
(−1)
ε
ε
e
e
.
.
.
e
S̃
f
.
.
.
f
ach...d
e
f
ac...d
e
h
f
bn
n−1
0 = α∆ma
εach...d εbe...f fi T̃c eh . . . ed fe . . . ff + α (−1)
εac...d εbeh...f fi ec . . . ed S̃e fh . . . ff
bn
Expanding the torsion and co-torsion and simplifying the first,
1
n−1
beh...f i c
d
mk
be...f ck
m i h
d
˜
0 = α∆ma
ε
ε
T
f
e
e
e
.
.
.
e
f
.
.
.
f
+
(−1)
ε
ε
e
e
.
.
.
e
S̃
f
f
f
.
.
.
f
ach...d
e
f
ac...d
m k h
f
bn
m k
e
2
1
n−1
n
S̃e mk εac...d εbeh...f εic...d εmkh...f Φnn
= α∆ma
(−1) T̃ ck m εach...d εbe...f εmih...d εke...f + (−1)
bn
2
1
n ck
n−1
n
b mi
mk i be
= α (n − 1)!2 (n − 2)!∆ma
T̃
δ
δ
+
S̃
δ
δ
(−1)
(−1)
bn
m k ac
e
a mk Φn
2
n
= α (n − 1)! (n − 2)! (−1) ∆ma
T̃ ib a − T̃ cb c δai − δai S̃e be Φnn
bn
and therefore
α∆ma
T̃ ib a − T̃ cb c δai
bn
T̃ imn − η ma ηbn T̃ ib a − δni T̃ cmc + η mi ηbn T̃ cb c
T̃ imn
−η
ma
ηbn T̃
ib
a
= α∆mi
bn S̃e
be
= δni S̃e
− η mi ηbn S̃e
me
δni S̃e me
=
There is only one independent trace; trace in,
α∆ma
T̃ nba − T̃ cb c δan
bn
α am
α
α
T̃ a − η ma ηbc T̃ bc a − (n − 1) T̃ cmc
2
2
2
ηbc T̃ bc a
−η
mi
ηbn S̃e
be
be
+ δni T̃ cmc − η mi ηbn T̃ cb c
be
= α∆mn
bn S̃e
α
=
(n − 1) S̃a ma
2
= − (n − 1) ηab S̃c bc − (n − 2) ηab T̃ cb c
So we have two results,
cm
2∆an
mb T̃ n
S̃e
be
= δnc S̃e me − η mc ηbn S̃e
1
= − (n − 1) T̃ ab a
n
14
be
+ δnc T̃ cmc − η mc ηbn T̃ cb c
For the second,
1
n
be...f c
k m h
d
beh...f c
d
k
m
0 = α
(−1) εach...d ε
T̃ km e e e . . . e fi fe . . . ff + εac...d ε
e . . . e S̃e m fk e fi fh . . . ff
2
1
n
n
be...f c
kmh...d
k
beh...f mc...d
(−1) εach...d ε
T̃ km ε
εie...f + (−1) S̃e m εac...d ε
ε
εkih...f Φnn
= α
2
1 c
n
km b
k
m be
T̃ km δac δi + S̃e m δa δki Φnn
= (−1) 2 (n − 1)! (n − 2)!α
2
b
1 c
n
k m
k m
k
b e
b e
T̃ km δa δc − δc δa δi + S̃e a δk δi − δi δk Φnn
= (−1) (n − 1)! (n − 2)!α
2
n
= (−1) (n − 1)! (n − 2)!α δib T̃ c ac + S̃i b a − S̃e e a δib Φnn
Therefore,
α δib T̃ c
ac
+ S̃i b
a
− S̃e e a δib = 0
Trace bi,
nT̃ c
ac
T̃ c
ac
(n − 1) S̃b b a
1
=
(n − 1) S̃b b
n
=
a
Then
1 c b
b
α − S̃c a δi + S̃i a
=
n
S̃c b
a
=
0
1 b e
δ S̃
n c e
a
so we get two relations:
S̃i b
T̃ c
3.2.1
a
=
ac
=
1 b c
δ S̃
n i c a
1
(n − 1) S̃b b
n
a
Check
Vary,
0
= δSˆ
= α (δΩab ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
= α
dδω ab − δω gb ω ag − ω gb δω ag εac...d εbe...f ec . . . ed fe . . . ff
ˆ
= α εac...d εbe...f δω ab D ec . . . ed fe . . . ff
ˆ
n−1
na
be...f g c
d
bge...f c
d
= (n − 1) α δω m
∆
ε
ε
T̃
e
.
.
.
e
f
.
.
.
f
+
(−1)
ε
ε
e
.
.
.
e
S̃
f
.
.
.
f
agc...d
e
f
ac...d
g e
f
n
bm
ˆ
na
n−1
k
mk
be...f g c
= (n − 1) α
Am
T̃ e . . . ed fe . . . ff + (−1)
εac...d εbge...f ec . . . ed S̃g fe . . . ff
nk e + Bn fk ∆bm εagc...d ε
15
Then
0
ˆ
n−1
be...f g k c
d
bge...f k c
d
Aqpk ∆pa
ε
ε
T̃
e
e
.
.
.
e
f
.
.
.
f
+
(−1)
ε
ε
e
e
.
.
.
e
S̃
f
.
.
.
f
ac...d
g e
f
agc...d
e
f
bq
ˆ
n
be...f g c
+ (n − 1) α Bpqk ∆pa
T̃ e . . . ed fk fe . . . ff + εac...d εbge...f ec . . . ed S̃g fk fe . . . ff
bq (−1) εagc...d ε
ˆ
1
n
n−1
= (n − 1) α Aqpk ∆pa
(−1) εagc...d εbe...f T̃ gmn en ek ec . . . ed fm fe . . . ff + (−1)
εac...d εbge...f ek ec . . . ed S̃g mn fm fn fe .
bq
2
ˆ
1
n
n
+ (n − 1) α Bpqk ∆pa
(−1) εagc...d εbe...f T̃ g mn em en ec . . . ed fk fe . . . ff + (−1) εac...d εbge...f en ec . . . ed S̃g m n fm fk fe
bq
2
=
(n − 1) α
so the field equations are
0
0
1
(n − 1) α∆pa
εagc...d εbe...f T̃ gmn εnkc...d εme...f − εac...d εbge...f εkc...d S̃g
bq
2
pa
kb
k nb
k
bg
= (n − 1)! (n − 1)!α∆bq T̃ a − δa T̃ n − δa S̃g
pa
n bg
m
b1 c
= 2 (n − 1)! (n − 1)!α∆bq δk T̃ ac + δa δmk S̃g n
2
b c
b
b
m
= (n − 1)! (n − 1)!α∆pa
bq δk T̃ ac + S̃k a − δk S̃m a
=
mn
εmne...f
and therefore,
b c
b
b
m
α∆pa
δ
T̃
+
S̃
−
δ
S̃
k
ac
k a
k m a
bq
pa
kb
k nb
α∆bq T̃ a − δa T̃ n − δak S̃g bg
=
0
=
0
These agree with the previous results.
3.3
Solder form variation
The solder form variation is:
0
= δS
ˆ
=
(αδΩab + βδba δΩ + γδea fb ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
+ (n − 1) (αΩab + βδba Ω + γea fb ) εacd...e εbf ...g δec ed . . . ee ff . . . fg
ˆ
d
f
k
f
=
α −∆ac
e − hde fe − ∆ac
δek − δhke fe εac...d εbe...f ec . . . ed fe . . . ff
db δhcf e + hck δe
kb fc + hcf e
ˆ
+
−βδba δek fk + γδek δka fb εac...d εbe...f ec . . . ed fe . . . ff
ˆ
+ (n − 1) (αΩab + βδba Ω + γea fb ) εakd...e εbf ...g δek ed . . . ee ff . . . fg
Expand the variations as
δek
δhke
= Ak m em + B km fm
= δek , ee + ek , δee
= hme Ak
hck δhke
=
−δhck hke hef
=
δhcf
=
km e
A m
m+h
me k
hck h A m + hck hkm Ae m
hck hme hef Ak m + hck hkm hef Ae m
−hck Ak f − hf k Ak c
16
this becomes
ˆ
m de
am
f de
ac
de m
ac
m
ac me
em
0 =
αAk m −∆ac
hcf ef fe + ∆ac
hcf ef fk εac.
db hck e h fe − ∆db hf k e h fe + ∆db hck h e fe − ∆kb fc e + ∆kb h
eb h
ˆ
+ Ak m −βδba δki + γδka δbi εac...d εbe...f em fi ec . . . ed fe . . . ff
ˆ
a i
a i
bf ...g
+ (n − 1) Ak m αΩai
fi ej em ed . . . ee ff . . . fg
b j + βδb Ω j − γδj δb εakd...e ε
ˆ
d
ac
f
be...f c
+ αB km −∆ac
e . . . ed fe . . . ff
db hck fm e − ∆kb hcf e fm εac...d ε
ˆ
1
1
n
+ (n − 1) B km α Ωabij + βδba Ωij (−1) εakd...e εbf ...g ei ej ed . . . ee fm ff . . . fg
2
2
resulting in two field equations,
n−1
aj m
j
dj m
am
dj
ac
dj m
ac mj
ac em
i b
0 = (−1)
α −∆ac
h
h
δ
−
∆
h
h
+
∆
h
h
δ
+
∆
δ
+
∆
h
h
+
∆
h
h
δ
ci
ci k δa δj
db ck
i
db ik
db ck
i
kb
eb
kb i
jm b
n−1
a i
a i
+ (−1)
−βδba δki + γδka δbi δib δam − 2 αΩai
b j + βδb Ω j − γδj δb δak δi
b i
1 a
ij b
an
j
an
j
a1
δm
0 = α ∆ib hnk δm − ∆kb hni δm δj δa − 2 α Ωbij + βδb Ωij δak
2
2
Expand the first,
0
=
1
1
1
1
1
1
− αδkm + αη mc hck ηdb hdb − αδkm + αη am hak ηdb hdb + αδkm − αη mc hck ηdb hdb
2
2
2
2
2
2
1
1 m 1
1 m 1 m 1
m
mb
ac
em
+ αnδk − αδk + αδk − αηkb h (η hca ) + αδk − αηek h (η ac hca )
2
2
2
2
2
2
m
mb
a
m
m
−βδkm + nγδkm − αΩab
δ
+
αΩ
−
βΩ
δ
+
βΩ
+
γ n2 − n δkm
b a k
b k
a k
k
ab
m
m
a
m
αΩmb
b k − αΩb a δk + βΩ k − βΩ a δk
1
1
+ αn − 2β + 2n2 γ δkm + αη mc hck ηdb hdb − αηkb hmb (η ac hca )
2
2
The field equation is therefore
=
0
=
ab
m
m
a
m
αΩmb
b k − αΩb a δk + βΩ k − βΩ a δk
1
1
+ αn − 2β + 2n2 γ δkm + αη mc hck ηdb hdb − αηkb hmb (η ac hca )
2
2
Check the trace,
0
a
= − (n − 1) αΩab
b a + βΩ a
1
1
+ n αn − 2β + 2n2 γ − α ηmb hmb (η ac hca )
2
2
Therefore, we may write
0
=
=
=
m
m
ab
a
αΩmb
b k + βΩ k − αΩb a + βΩ a δk
1
1
+ αn − 2β + 2n2 γ δkm + αη mc hck ηdb hdb − αηkb hmb (η ac hca )
2
2
1
ac
1
1
m
2
mb
αΩmb
+
βΩ
−
n
αn
−
2β
+
2n
γ
−
α
η
h
(η
h
)
δkm
mb
ca
b k
k
n−1 2
2
1
1
+ αn − 2β + 2n2 γ δkm + αη mc hck ηdb hdb − αηkb hmb (η ac hca )
2
2
m
αΩmb
b k + βΩ k
ac
m 1 mc
1
1
2
mb
+
β − n γ + α ηmb h
(η hca ) − n δk + αη hck ηdb hdb − αηkb hmb (η ac hca )
n−1
2
2
17
The second equation is
0
3.4
=
1
α ((n − 2) hmk + ηkm (η an hna )) − αΩamak − βΩmk
2
Variation of the co-solder form
Finally, we vary fa .
0
= δS
ˆ
=
(αδΩab + βδba δΩ + γea δfb ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
+ (αΩab + βδba Ω + γea fb ) (n − 1) εac...d εbge...f ec . . . ed δfg fe . . . ff
ˆ j
i
i
=
α −∆ag
e − hjk fk − ∆ag
−δhjk fk − hjk δfk − βδba eg δfg + γea δfb εac...d εbe...f ec .
jb δfg + δhgi e
jb fg + hgi e
ˆ
+ (αΩab + βδba Ω + γea fb ) (n − 1) εac...d εbge...f ec . . . ed δfg fe . . . ff
To find how hab varies we have
hδfa , fb i + hfa , δfb i = −δhab
so with δfa = Cab eb + Da b fb ,
Da
c
hcb + Db
c
hac
= δhab
and therefore,
hna Da
c
hcb + Db
na
h Da
−hna Da
c
m
c na
h hac
= hna δhab
hcb + Db
n
= −δhna hab
− hmb Db
n
= δhnm
Now substitute for the variations and collect terms.
ˆ
ˆ
1 ars 1 a rs
n−1
n−1
js
jg as
be...f r c
d
αΩ
βδ
+
Ω
(−1)
(n − 1
0 =
Cgr α∆ag
h
−
αh
∆
(−1)
ε
ε
e
e
.
.
.
e
f
f
.
.
.
f
+
C
ac...d
s e
f
gm
jb
jb
2 b
2 b
ˆ s
m
m
sm
jm
jk
+
α∆ag
+ α∆ag
hmr hjs + α∆ag
hgm hjs − α∆ag
Dm j − α∆ag
Dm s + α∆ag
rb Dg
jb Dg
jb Dr
jb hgr h
jb hgr h
jb hgr h D
ˆ
n
a s
a s
bge...f r c
+ Dg m (αΩas
e e . . . ed fs fm fe . . . ff
b r + βδb Ω r − γδr δb ) (n − 1) (−1) εac...d ε
The two field equations are:
1
+ βΩmg
= − α (n − 2) hmg + η mg ηab hab + αΩmbg
b
2
g
1
1
ab
g
g
a g
0 = αΩag
2γn2 + nα − 2β δm
− αη ag hma ηjb hjb + αηmb hbg η ak hka
m a − αΩb a δm + βΩ m − βΩ a δm +
2
2
0
or (from the check, and with better indices),
0
0
1
1
(n − 2) αhki − αη ki ηjb hjb
2
2
nm
r
r
a
r
= αΩar
s a − αΩm n δs + βΩ s − βΩ a δs
1
1
+ nα − 2β + 2n2 γ δsr − αη rc hsc ηab hab + αηsc hrc η ab hab
2
2
ki
= αΩkbi
b + βΩ −
18
3.4.1
Check
From the action,
0
= δSˆ
= δ (αΩab + βδba Ω + γea fb ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
=
(αδΩab + βδba δΩ + γea δfb ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
+ (n − 1) (αΩab + βδba Ω + γea fb ) εac...d εbef ...g ec . . . ed δfe ff . . . fg
ˆ
j
k
k
=
−α∆ai
e − hjm fm − α∆ai
−δhjm fm − hjm δfm εac...d εbe...f ec . . . ed fe . . . ff
jb δfi + δhik e
jb fi + hik e
ˆ
+ (−βδba eg δfg + γea δfb ) εac...d εbe...f ec . . . ed fe . . . ff
ˆ
+ (n − 1) (αΩab + βδba Ω + γea fb ) εac...d εbef ...g ec . . . ed δfe ff . . . fg
where
dω ab
f
= ω cb ω ac + ∆ac
db fc + hcf e
dω
= e a fa + Ω
δfa
= Cab eb + Da b fb
= fa , Cbd ed + Db
ed − hde fe + Ωab
Now substitute variations,
δhab
c
= Db
δhde
to get
0
ˆ
+
ˆ
+
ˆ
c
fc , fb i
hcb
− had Da
e
k
−α∆ai
jb Cik e + Di
k
−α∆ai
jb fi + hik e
k
fk + Dk
hnj Dn
−βδba eg Cgk ek − βδba eg Dg
m
m
k
hmi ek + Di
m
hmk ek
ej − hjm fm
εac...d εbe...f ec . . . ed fe . . . ff
fm + hnm Dn j fm − hjm Cmk ek − hjm Dm k fk
fk + γea Cbk ek + γea Db
k
εac...d εbe...f ec . . . ed fe . . . ff
fk εac...d εbe...f ec . . . ed fe . . . ff
(n − 1) (αΩab + βδba Ω + γea fb ) εac...d εbef ...g ec . . . ed Cek ek + De
+
0
d
c
fd + hCac ec + Da
ˆ
=
Then
hac + Da
= −hae Da
d
k
fk ff . . . fg
ˆ
=
ˆ
+
ˆ
jm k
α∆ai
e fm + α∆ai
jb Cik h
jb Di
nj
−α∆ai
jb h Dn
m
k j
e fk + α∆ai
jb Dk
m
hmi hjn ek fn + α∆ai
jb Di
m
hmk hjn ek fn εac...d εbe...f ec . . . ed fe . . . ff
nm
jm
jm
hik ek fm − α∆ai
Dn j hik ek fm − α∆ai
Cmk ek fi + α∆ai
Dm k hin en fk εac...d εbe...f ec . .
jb h
jb h
jb h
+ γDb k δga εac...d εbe...f eg fk ec . . . ed fe . . . ff
ˆ
1 amn 1 a mn
n−1
αΩb
+ βδb Ω
(−1)
εac...d εbef ...g ek ec . . . ed fm fn ff . . . fg
+ Cek (n − 1)
2
2
ˆ
+ De k (n − 1) (αΩab + βδba Ω + γea fb ) εac...d εbef ...g ec . . . ed fk ff . . . fg
+
−βδba Dg
k
19
so that
0
ˆ
=
ˆ
jn
α∆ai
+ α∆ai
jb Cik h
kb Di
+
ˆ
nj
−α∆ai
jb h Dn
m
n
+ α∆ai
jb Dk
m
hmi hjn + α∆ai
jb Di
m
n−1
εac...d εbe...f ek ec . . . ed fn fe . . . ff
hmk hjn (−1)
nm
ji
ai jn
hik − α∆ai
Dn j hik − α∆am
jb h
jb h Cik + α∆jb h Dn
m
n−1
hik (−1)
εac...d εbe...f ek ec . . . ed fm fe
n−1
εac...d εbe...f eg ec . . . ed fk fe . . . ff
+ γDb k δga (−1)
ˆ
1 amn 1 a mn
n−1
αΩb
+ βδb Ω
(−1)
εac...d εbef ...g ek ec . . . ed fm fn ff . . . fg
+ Cek (n − 1)
2
2
ˆ
n−1
a m
a m
+ De k (n − 1) (−αΩam
εac...d εbef ...g en ec . . . ed fm fk ff . . . fg
b n − βδb Ω n + γδn δb ) (−1)
−βδba Dg
+
k
Next,
1 amn 1 a mn k bi
0 =
−
+2
αΩ
+ βδb Ω
δa δmn
2 b
2
n
ai r m
jn
jn
0 = α∆ar
+ α∆ar
δnb δak
kb δs + α∆jb δk δs hmi h
jb hsk h
b k
rj m
ai rm
m jr
+ −α∆ai
hik + α∆ai
jb h δs hik − α∆sb h
jb δs h hik δm δa
+ −βδba δgr + γδbr δga δsb δag
jn b k
α∆ai
jb h δn δa
ji b k
α∆am
jb h δm δa
a m
a m
br n
+2 (−αΩam
b n − βδb Ω n + γδn δb ) δms δa
so that
0
0
4
1
1
(n − 2) αhki − αη ki ηjb hjb
2
2
nm
r
r
a
r
= αΩar
s a − αΩm n δs + βΩ s − βΩ a δs
1
+ nα − 2β + 2n2 γ δsr
2
1
−αη rc hsc ηab hab + αηsc hrc η ab hab
2
ki
= αΩkbi
b + βΩ −
Summary of field equations
Collecting the field equations:
β S̃k jk = β T̃ jk k − T̃ kj k
β T̃ c jc = β S̃h h j − S̃j k k
b c
b
b
m
0 = α∆pa
δ
T̃
+
S̃
−
δ
S̃
k
ac
k a
k m a
bq
pa
0 = α∆bq T̃ kb a − δak T̃ nbn − δak S̃g bg
0
=
ab
αΩmb
b k − αΩb
m
a δk
+ βΩm k − βΩa
m
a δk
+
1
1
αn − 2β + 2n2 γ δkm − αηkb hmb (η ac hca ) + αη mc hck ηdb hdb
2
2
1
= αΩamak + βΩmk − α ((n − 2) hmk + ηkm (η an hna ))
2
1
0 = αΩmbk
+ βΩmk − α (n − 2) hmk + η mk ηab hab
b
2
1
g
1
ab
g
g
a g
0 = αΩag
nα − 2β + 2n2 γ δm
− αη ag hma ηjb hjb + αηmb hbg η ak hka
m a − αΩb a δm + βΩ m − βΩ a δm +
2
2
0
20
Notice that the first four equations involve only the torsion and co-torsion, while the remaining four depend
only on the Lorentz and dilatational curvatures.
References
[1] Jeffrey S Hazboun and James T Wheeler, Time and dark matter from the conformal symmetries of
Euclidean space, Class. Quantum Grav. 31 (2014) 215001 (34pp), doi:10.1088/0264-9381/31/21/215001
[2] Spencer J A and Wheeler J T 2011 The existence of time Int. J. Geom. Method Mod. Phys. 8 273–301
[3] Jeffrey S Hazboun and James T Wheeler, Curved biconformal space, in preparation.
[4] James T. Wheeler, Weyl gravity as general relativity, Physical Review D 90, 025027 (2014), DOI:
10.1103/PhysRevD.90.025027
[5] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry (John Wiley and Sons, New York,
1963).
[6] E. A. Ivanov and J. Niederle, Phys. Rev. D 25, 988 (1982).
[7] J. T. Wheeler, New conformal gauging and the electromagnetic theory of Weyl, J. Math. Phys. 39, 299
(1998).
[8] A. Wehner and J. T. Wheeler, Conformal actions in any dimension, Nucl. Phys. B557, 380 (1999).
[9] Westman H F adn Zlosnik T G 2014 Exploring Cartan gravity with dynamical symmetry breaking,
arXiv: 1302:1103v2[gr-qc]
[10] Fernando J, Barbero G, From Eulidean to Lorentzian general relativity: the real way, Phys. Rev D54
(1996) 1492-9.
[11] Fernando J, Barbero G and Villasenor E J S, Lorentz violations and Euclidean signature metrics, Phys.
Rev. D 68 (2003) 087501
[12] Jacobson T 2007 Einstein-aether gravity: a status report, PoSQG-PH 020 arXiv:0801.1547[gr-qc]
[13] Anderson L B and Wheeler J T 2004 Biconformal supergravity and the AdS/CFT conjecture Nucl. Phys.
B686 285–309
[14] Anderson L B and Wheeler J T 2007 Yang–Mills gravity in biconformal space Class. Quantum Grav.
24 475–96
[15] Wheeler J T 2007 Gauging Newton’s law Can. J. Phys. 85 307–44
[16] Anderson L B and Wheeler J T 2006 Quantum mechanics as a measurement theory on biconformal
space Int. J. Geom. Math. Mod. Phys. 3 315–40
21