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

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

2007-12-12

arxiv.org/abs/0712.1998v2

Phys.Rev.D77:064021,2008

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

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

Scientific paper

10.1103/PhysRevD.77.064021

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.

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