EINSTEINS EQUATIONS AND THE CHIRAL MODEL

The vacuum Einstein equations for spacetimes with two commuting spacel ike Killing field symmetries are studied using the Ashtekar variables. The case of compact spacelike hypersurfaces which are three-tori is c onsidered, and the determinant of the Killing two-torus metric is chos en as the time gauge. The Hamiltonian evolution equations in this gaug e may be rewritten as those of a modified SL(2) principal chiral model with a time-dependent ''coupling constant,'' or equivalently, with ti me-dependent SL(2) structure constants. The evolution equations have a generalized zero-curvature formulation. Using this form, the explicit time dependence of an infinite number of spatial-diffeomorphism-invar iant phase-space functionals is extracted, and it is shown that these are observables in the sense that they Poisson commute with the reduce d Hamiltonian. An infinite set of observables that have SL(2) indices is also found. This determination of the explicit time dependence of a n infinite set of spatial-diffeomorphism-invariant observables amounts to the solutions of the Hamiltonian Einstein equations for these obse rvables.