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

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 ${\mu }_n=\frac{\hslash }{l}(n+\frac{1}{2})$. 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.

15 pages, no figures. Two explanatory paragraphs have been added. This version will appear in Physical Review D

References in corpus (4)

• Quantum geometry and microscopic black hole entropy
• Quantum Gravitational Collapse and Hawking Radiation in 2+1 Dimensions
• Classical and Quantum Gravitational Collapse in d-dim AdS Spacetime I. Classical Solutions
• Particle creation in (2+1) circular dust collapse

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• Classical and Quantum Gravitational Collapse in d-dim AdS Spacetime II. Quantum States and Hawking Radiation
• Reflection and Transmission at the Apparent Horizon during Gravitational Collapse
• A Rotating, Inhomogeneous Dust Interior for the BTZ Black Hole
• Signatures of an Emergent Gravity from Black Hole Entropy
• BTZ entropy from topological M-theory
• Canonical Chern-Simons Gravity