Symmetry reduced Einstein gravity and generalized sigma and chiral models

Details Symmetry reduced Einstein gravity and generalized sigma and chiral models Symmetry reduced Einstein gravity and generalized sigma and chiral models

1997-12-11

arxiv.org/abs/gr-qc/9712053v2

Int.J.Mod.Phys. D7 (1998) 549-566

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

17 pages, latex, reference added, a subtlety commented on; to appear in Int. J. Mod. Phys. D

Scientific paper

10.1142/S0218271898000371

Certain features associated with the symmetry reduction of the vacuum Einstein equations by two commuting, space-like Killing vector fields are studied. In particular, the discussion encompasses the equations for the Gowdy $T^3$ cosmology and cylindrical gravitational waves. We first point out a relation between the $SL(2,R)$ (or SO(3)) $\sigma$ and principal chiral models, and then show that the reduced Einstein equations can be obtained from a dimensional reduction of the standard SL(2,R) sigma-model in three dimensions. The reduced equations can also be derived from the action of a `generalized' two dimensional SL(2,R) sigma-model with a time dependent constraint. We give a Hamiltonian formulation of this action, and show that the Hamiltonian evolution equations for certain phase space variables are those of a certain generalization of the principal chiral model. Using these Hamiltonian equations, we give a prescription for obtaining an infinite set of constants of motion explicitly as functionals of the space-time metric variables.

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