Black holes and Rindler superspace: Classical singularity and quantum unitarity

Abstract

Canonical quantization of spherically symmetric initial data appropriate to classical interior black hole solutions in four dimensions is solved exactly without gauge fixing the remaining kinematic Gauss law constraint. The resultant mini-superspace manifold is two dimensional, of signature $(+,\ensuremath{-})$, nonsingular, and can be identified with the first Rindler wedge. The associated Wheeler-DeWitt equation with evolution in intrinsic superspace time is a free massive Klein-Gordon equation, and the Hamilton-Jacobi semiclassical limit of plane wave solutions can be matched to the interiors of Schwarzschild black holes. Classical black hole horizons and singularities correspond to the boundaries of the Rindler wedge. Exact wave functions of the Dirac equation in superspace are also considered. Precise correspondence between Schwarzschild black holes and free-particle mechanics in superspace is noted. Despite the presence of classical singularities, Hermiticity of the Dirac Hamiltonian operator, and thus unitarity of the quantum theory, is equivalent to an appropriate boundary condition which must be satisfied by the quantum states. This boundary condition holds for quite generic quantum wave packets of energy eigenstates, but fails for the usual Rindler fermion modes which are eigenstates with zero uncertainty in energy.