The Quantum States and the Statistical Entropy of the Charged Black Hole

TL;DR

This paper quantizes the charged Reissner-Nordström black hole using a canonical approach, deriving a quantum condition on horizon areas and calculating the black hole entropy, linking it to horizon area differences.

Contribution

It introduces a canonical quantization method for charged black holes and relates the entropy to quantized horizon area differences, extending Bekenstein's ideas.

Findings

Horizon area difference is quantized in integer units.

Black hole entropy is proportional to the quantized area difference.

Exact solutions of the quantum equation support the area quantization.

Abstract

We quantize the Reissner-Nordstr\"om black hole using an adaptation of Kucha\v{r}'s canonical decomposition of the Kruskal extension of the Schwarzschild black hole. The Wheeler-DeWitt equation turns into a functional Schroedinger equation in Gaussian time by coupling the gravitational field to a reference fluid or dust. The physical phase space of the theory is spanned by the mass, $M$, the charge, $Q$, the physical radius, $R$, the dust proper time, $\tau$, and their canonical momenta. The exact solutions of the functional Schroedinger equation imply that the difference in the areas of the outer and inner horizons is quantized in integer units. This agrees in spirit, but not precisely, with Bekenstein's proposal on the discrete horizon area spectrum of black holes. We also compute the entropy in the microcanonical ensemble and show that the entropy of the Reissner-Nordstr\"om black hole is proportional to this quantized difference in horizon areas.