Mass Quantization of the Schwarzschild Black Hole

TL;DR

This paper investigates the quantum states of Schwarzschild black holes using the Wheeler-DeWitt equation, revealing a discrete mass spectrum for states of definite parity that aligns with Bekenstein's conjecture.

Contribution

It introduces a quantization scheme for black hole mass based on parity considerations, providing a novel quantum description of Schwarzschild black holes.

Findings

Discrete energy spectrum for definite parity states

Mass quantization consistent with Bekenstein's conjecture

Indefinite parity states lack a quantized mass spectrum

Abstract

We examine the Wheeler-DeWitt equaton for a static, eternal Schwarzschild black hole in Kucha\v r-Brown variables and obtain its energy eigenstates. Consistent solutions vanish in the exterior of the Kruskal manifold and are non-vanishing only in the interior. The system is reminiscent of a particle in a box. States of definite parity avoid the singular geometry by vanishing at the origin. These definite parity states admit a discrete energy spectrum, depending on one quantum number which determines the Arnowitt-Deser-Misner (ADM) mass of the black hole according to a relation conjectured long ago by Bekenstein, $M \sim \sqrt{n}M_p$. If attention is restricted only to these quantized energy states, a black hole is described not only by its mass but also by its parity. States of indefinite parity do not admit a quantized mass spectrum.