SYMMETRY REDUCED EINSTEIN GRAVITY AND GENERALIZED-SIGMA AND CHIRAL MODELS

Certain features associated with the symmetry reduction of the vacuum Einstein equations by two commuting, spacelike Killing vector fields a re studied. In particular, the discussion encompasses the equations fo r the Gowdy T-3 cosmology and cylindrical gravitational waves. We firs t point out a relation between the SL(2,R) (or SO(3)) sigma and princi pal chiral models, and then show that the reduced Einstein equations c an be obtained from a dimensional reduction of the standard SL(2,R) si gma-model in three dimensions. The reduced equations can also be deriv ed from the action of a 'generalized' two dimensional SL(2,R) sigma-mo del with a time dependent constraint. We give a Hamiltonian formulatio n of this action, and show that the Hamiltonian evolution equations fo r certain phase space variables are those of a certain generalization of the principal chiral model. Using these Hamiltonian equations, we g ive a prescription for obtaining an infinite set of constants of motio n explicitly as functionals of the spacetime metric variables.