Black holes and Rindler superspace: classical singularity and quantum unitarity

TL;DR

This paper develops a quantum model of black hole interiors using canonical quantization, revealing a non-singular superspace structure analogous to Rindler space and establishing conditions for unitarity despite classical horizons.

Contribution

It provides an exact quantization of black hole interiors, identifying the superspace with Rindler wedge and linking classical horizons to quantum boundary conditions for unitarity.

Findings

Superspace is two-dimensional, non-singular, and matches Rindler wedge.

Wheeler-DeWitt equation reduces to Klein-Gordon form.

Unitarity depends on boundary conditions, not on classical horizons.

Abstract

Canonical quantization of spherically symmetric initial data which is appropriate to classical interior black hole solutions in four dimensions is carried out and solved exactly without gauge fixing the remaining kinematic Gauss Law constraint. The resultant mini-superspace manifold and arena for quantum geometrodynamics is two-dimensional, of signature (+, -), non-singular, and can in fact be identified precisely with the first Rindler wedge. The associated Wheeler-DeWitt equation with evolution in intrinsic superspace time can be formulated as a free massive Klein-Gordon equation; and the Hamilton-Jacobi semiclassical limit of plane wave solutions can be matched precisely to the interiors of Schwarzschild black holes. Furthermore, classical black hole horizons and singularities correspond to the boundaries of the Rindler wedge. Exact wavefunctions of the first-order-in-superspace intrinsic time Dirac equation are also considered. Precise correspondence between Schwarzschild black holes and free particle mechanics in superspace is noted. Another intriguing observation is that, despite the presence of classical horizons and singularities, hermiticity of the Dirac hamiltonian operator for the mini-superspace, 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 wavepackets of energy eigenstates, but fails for the usual Rindler fermion modes which are precisely eigenstates with zero uncertainty in energy.