Hamiltonian reduction of Einstein's equations of general relativity

Abstract

A program is outlined which resolves the problem of the Hamiltonian reduction of Einstein's vacuum field equations in (3 + 1)-dimensions. The problem involves writing Einstein's vacuum field equations as an unconstrained Hamiltonian dynamical system where the variables of the unconstrained system are the true degrees of freedom of the gravitational field. Our analysis is applicable to vacuum spacetimes that admit constant mean curvature compact spacelike hypersurfaces M that satisfy certain topological restrictions. We find that for these spacetimes (3 + 1)-reduction can be completed much as in the (2 + 1)-dimensional case. In both cases, one gets as the reduced phase space the contangent bundle T ast;T M of the Teichm üller space T M= M P D 0 of conformal structures on M and one gets reduction of the full classical Hamiltonian system with constraints to a non-local time-dependent reduced Hamiltonian system without constraints on the contact manifold R × T ∗τ M. For this reduced system, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact spacelike hypersurfaces in a neighborhood of the given initial one and the Hamiltonian is the volume functional of these hypersurfaces expressed in terms of the canonical variables of the hypersurface.