Mass Spectrum and Statistical Entropy of the BTZ black hole from Canonical Quantum Gravity

TL;DR

This paper quantizes the BTZ black hole using canonical quantum gravity, revealing a discrete mass spectrum, deriving its entropy, and linking it to conformal field theory via boundary energy and asymptotic symmetries.

Contribution

It introduces a novel quantization of the BTZ black hole with a discrete mass spectrum and connects quantum gravity with conformal field theory through boundary energy analysis.

Findings

Discrete mass spectrum of the BTZ black hole derived

Canonical entropy matches Bekenstein-Hawking entropy

Boundary energy related to Virasoro algebra central charge

Abstract

In a recent publication we developed a canonical quantization program describing the gravitational collapse of a spherical dust cloud in 2+1 dimensions with a negative cosmological constant $-\Lambda\equiv -l^{-2}<0$. In this paper we address the quantization of the Banados--Teitelboim--Zanelli (BTZ) black hole. We show that the mass function describing the black hole is made of two pieces, a constant non-vanishing boundary contribution and a discrete spectrum of the form $\mu_n = \frac{\hbar}{l}(n+ \frac 12)$. The discrete spectrum is obtained by applying the Wheeler--DeWitt equation with a particular choice of factor ordering and interpreted as giving the energy levels of the collapsed matter shells that form the black hole. Treating a black hole microstate as a particular distribution of shells among the levels, we determine the canonical entropy of the BTZ black hole. Comparison with the Bekenstein--Hawking entropy shows that the boundary energy is related to the central charge of the Virasoro algebra that generates the asymptotic symmetry group of the three-dimensional anti-de Sitter space AdS$_3$. This gives a connection between the Wheeler--DeWitt approach and the conformal field theory approach.