The Hamiltonian reduction of Einstein's equations to
the cotangent bundle of Teichmüller space
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 cotangent bundle T*
TM of the Teichmüller
space 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 x T*TM. 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.
Fall 1997 meeting of the PNGS