Observables for spacetimes with two Killing field symmetries

The Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied from a Hamiltonian point of view. The complexified Ashtekar canonical variables are used, and the symmetry reduction is performed directly in the Hamiltonian theory. The reduced system corresponds to the field equations of the SL(2,R) chiral model 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 equations as a single ‘‘zero curvature’’ equation, and then employing techniques used in the study of two-dimensional integrable models. Two infinite 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 another. The restriction of the (complex) observables to the Euclidean and Lorentzian sectors is discussed. It is also shown that the sl(2,R) observables can be associated with a solution-generating technique which is linked to that given by Geroch.