HAMILTONIAN REDUCTION OF EINSTEINS EQUATIONS OF GENERAL-RELATIVITY

A program is outlined which resolves the problem of the Hamiltonian re duction of Einstein's vacuum field equations in (3 + 1)-dimensions. Th e problem involves writing Einstein's vacuum field equations as an unc onstrained Hamiltonian dynamical system where the variables of the unc onstrained system are the true degrees of freedom of the gravitational field. Our analysis is applicable to vacuum spacetimes that admit con stant mean curvature compact spacelike hypersurfaces M that satisfy ce rtain 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 cotangent bun dle TT-M of the Teichmuller space T-M = M/P/D-0 Of conformal structur es on M and one gets reduction of the full classical Hamiltonian syste m with constraints to a non-local time-dependent reduced Hamiltonian s ystem without constraints on the contact manifold, R x TT-M. For this reduced system, the time parameter is the parameter of a family of mo notonically increasing constant mean curvature compact spacelike hyper surfaces in a neighborhood of the given initial one and the Hamiltonia n is the volume functional of these hypersurfaces expressed in terms o f the canonical variables of the hypersurface.