OBSERVABLES FOR SPACETIMES WITH 2 KILLING FIELD SYMMETRIES

The Einstein equations for spacetimes with two commuting spacelike Kil ling field symmetries are studied from a Hamiltonian point of view. Th e complexified Ashtekar canonical variables are used, and the symmetry reduction is performed directly in the Hamiltonian theory, The reduce d system corresponds to the field equations of the SL(2,R) chiral mode l with additional constraints. On the classical phase space, a method of obtaining an infinite number of constants of motion, or observables , is given. The procedure involves writing the Hamiltonian evolution e quations as a single ''zero curvature'' equation and then employing te chniques used in the study of two-dimensional integrable models. Two i nfinite sets of observables are obtained explicitly as functionals of the phase space variables. One set carries sl(2,R) Lie algebra indices and forms an infinite-dimensional Poisson algebra, while the other is formed from traces of SL(2,R) holonomies that commute with one anothe r. The restriction of the (complex) observables to the Euclidean and L orentzian sectors is discussed. It is also shown that the sl(2,R) obse rvables can be associated with a solution-generating technique which i s linked to that given by Geroch.