Symmetry reduced Einstein gravity and generalized sigma and chiral models

TL;DR

This paper explores the symmetry reduction of vacuum Einstein equations, relating them to sigma and chiral models, and introduces a Hamiltonian framework to identify conserved quantities in these reduced models.

Contribution

It establishes a connection between Einstein gravity reductions and generalized sigma models, providing a Hamiltonian formulation and explicit conserved quantities.

Findings

Reduced Einstein equations derived from SL(2,R) sigma-model

Hamiltonian formulation of the generalized sigma-model

Explicit constants of motion for the reduced equations

Abstract

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.