Universality
of the Einstein theory of
gravitation
Jerzy Kijowski
Center for Theoretical Physics, Warsaw
Hilbert linear Lagrangian
Einstein theory of gravity:
Hilbert Lagrangian:
where
Field equations:
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Non-linear Lagrangians
for empty space:
Generalized theories:
Field equations of the 4-th differential order!
BUT!!!
Theorem: There exists a one-to-one change of variables:
and a new matter Lagrangian:
such that the old variables satisfy field equations derived from L if and only if
the new variables satisfy conventional ``Einstein + matter’’ field equations
derived from
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New metric and new matter fields
Decompose the Riemann tensor
and Weyl
.
into irreducible parts: Ricci
The new metric is defined as the ``momentum canonically conjugate'' to the
Ricci tensor:
The old metric tensor is ``downgraded’’ to the level of matter fields.
Moreover, there is an additional matter field defined as the ``momentum
canonically conjugate'' to the Weyl tensor:
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Higher order Lagrangians
Even more generalized theories:
Field equations for the metric are of the 2(k+2)-th differential order!
BUT!!!
Theorem: There exists a one-to-one change of variables:
and a new matter Lagrangian:
In collaboration with:
G. Moreno and K. Senger
such that the old variables satisfy field equations derived from L if and only if
the new variables satisfy conventional ``Einstein + matter’’ field equations
derived from
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Non-standard matter
Generalizations of GR based on non-standard Lagrangians are equivalent to
theories based on non-standard matter fields, with gravitational interaction
between them being standard (i.e. Einsteinian).
Advantage: stability (posive mass theorem) well established in Einstein
theory.
Advantage: derivation of matter fields from geometry.
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Example: Sacharov theory
First, physically well motivated example: the Sacharov theory
This example belongs to a more general class:
Consequently, the ``old metric’’
can be uniquely encoded by the
scalar field
(M. Ferraris 1986, J. Jakubiec and J. Kijowski 1987).
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Theories based on the Ricci tensor
Lagrangians depending (non-linearly) upon the Ricci tensor
were
thoroughly investigated by J. Jakubiec and J.Kijowski (1987 – 89):
Example:
The only ``new matter field’’ is the old metric. The new metric given again
by the universal formula :
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Idea of the proof
Idea of the proof: Legendre transformation.
Fundamental observation:
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Idea of the proof
Idea of the proof: Legendre transformation.
Fundamental observation:
where:
Hence:
where:
Hence:
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Idea of the proof
Similar identity for the matter Lagrangian:
Canonical momenta defined as usual:
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Symplectic relations (warm-up)
Variation of the total Lagrangian:
volume
boundary
„brachistochrone-like” philosophy
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Symplectic relations (warm-up)
Variation of the total Lagrangian:
volume
boundary
„on shell” philosophy
Jet space of these fields equipped with a ``God-given” symplectic structure:
Response parameters
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Control parameters
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Symplectic relations (warm-up)
Variation of the total Lagrangian:
Important computational simplification in conventional G.R.:
where
denotes a useful combination of connection coefficients (no specific
geometric interpretation!). For example:
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Symplectic relations (warm-up)
Variation of the total Lagrangian:
Important computational simplification in conventional G.R.:
Hence, symplectic reduction:
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Symplectic relations (warm-up)
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Symplectic relations (warm-up)
Second-order v.p. – equally correct but more complicated (constraints!):
First-order variational principle:
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Symplectic relations (seriously)
Proof of our Theorem:
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Symplectic relations (seriously)
Proof of our Theorem:
Formula:
no longer true!!!
Bianchi first type identity
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Symplectic relations (seriously)
where:
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Decomposition into irreducible parts
Proof of our Theorem:
Decomposition of the momentum
into irreducible parts
is dual to decomposition of the curvature tensor into trace and traceless part:
Ricci:
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Decomposition into irreducible parts
Proof of our Theorem:
where:
New metric:
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Legendre transformation
Proof of our Theorem:
new matter fields
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Legendre transformation
Remark: Metricity constraints
In a generic case: no constraints here!
Back to the Hilbert Lagrangian:
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Example – Weyl Lagrangian
Example:
constraints!
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Example – Weyl Lagrangian
Non-invariant!!! But invariant modulo boundary terms:
Can easily be made invariant if we allow for 2-nd order matter Lagrangians!
Field equations: Variation w.r. to
:
Matter eqs.:
Variation w.r. to
:
Einstein eqs.:
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Conclusions
1) Einsteinian version of the theory enables us to use standard tools of
G.R. to analyze the Cauchy problem.
2) Stability: ``energy conditions” for the right-hand-side of Einstein
equations.
3) Works also for higher order Lagrangians (i.e. depending not only upon
curvature tensor, but also upon its higher order derivatives).
4) Perspectives of the fundamental ``purely affine’’ theory (no metric in the
original Lagrangian).
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