# Mass Spectrum and Statistical Entropy of the BTZ black hole from Canonical Quantum Gravity - General Relativity and Quantum Cosmology

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Abstract: In a recent publication we developed a canonical quantization programdescribing the gravitational collapse of a spherical dust cloud in 2+1dimensions with a negative cosmological constant $-\Lambda\equiv -l^{-2}<0$. Inthis paper we address the quantization of the Banados-Teitelboim-ZanelliBTZ black hole. We show that the mass function describing the black hole ismade of two pieces, a constant non-vanishing boundary contribution and adiscrete spectrum of the form $\mu n = \frac{\hbar}{l}n+ \frac 12$. Thediscrete spectrum is obtained by applying the Wheeler-DeWitt equation with aparticular choice of factor ordering and interpreted as giving the energylevels of the collapsed matter shells that form the black hole. Treating ablack hole microstate as a particular distribution of shells among the levels,we determine the canonical entropy of the BTZ black hole. Comparison with theBekenstein-Hawking entropy shows that the boundary energy is related to thecentral charge of the Virasoro algebra that generates the asymptotic symmetrygroup of the three-dimensional anti-de Sitter space AdS$ 3$. This gives aconnection between the Wheeler-DeWitt approach and the conformal field theoryapproach.

Author: ** Cenalo Vaz, Sashideep Gutti, Claus Kiefer, T. P. Singh, L.C.R. Wijewardhana**

Source: https://arxiv.org/