Hypertime Formalism for Spherically Symmetric Black Holes and Wormholes

Abstract

Recent work on an approach to the geometrodynamics of cylindrical gravity waves in the presence of interacting scalar matter fields, based on the Kucha\v{r} hypertime formalism, is extended to the analogous spherically symmetric system. This produces a geometrodynamical formalism for spherical black holes and wormholes in which the metric variables are divided into two classes, dynamical and redundant. The redundant variables measure the embedding of a spacelike hypersurface into the spacetime, and proper time in the asymptotically flat regions. All the constraints can be explicitly solved for the momenta conjugate to the embedding variables. The dynamical variables, including an extra ADM mass for wormhole topologies, can then be considered as functionals of the redundant ones, including the proper time variable. The solution of the resulting constraint system determines the momentum conjugate to the proper time as a function of the other variables, producing Unruh's Hamiltonian formalism for the spherical black hole, whilst extending it to an arbitrary foliation choice. The resulting formalism is appropriate as a starting point for the construction of a hypertime functional Schr\"odinger equation for quantized spherically symmetric black holes and wormholes.