Symmetries of the Einstein Hamiltonian

Abstract

The vacuum Einstein equations for metrics that have two commuting spacelike Killing vector fields are studied from a Hamiltonian point of view using the Ashtekar variables. It is shown that the evolution equations are equivalent to those of a modified SL(2) principal chiral model with a time dependent ‘coupling constant’. This fact is used to extract an infinite set of symmetries of the Einstein Hamiltonian via a generalized zero-curvature formulation. These symmetries give evolving observables explicitly on the phase space, and may be viewed as providing an infinite set of solutions of the Hamiltonian Einstein equations. The possibility of quantization using these observables is briefly discussed.