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.

arXiv comment15 pages, no figures. Two explanatory paragraphs have been added. This version will appear in Physical Review D

References in corpus (31)

• The Black Hole in Three Dimensional Space Time
• Microscopic Origin of the Bekenstein-Hawking Entropy
• Geometry of the 2+1 Black Hole
• Black Hole Entropy from Near-Horizon Microstates
• Quantum Geometry and Black Hole Entropy
• Black hole entropy in Loop Quantum Gravity
• Black Hole Entropy from Loop Quantum Gravity
• A Correspondence Principle for Black Holes and Strings
• The asymptotic dynamics of three-dimensional Einstein gravity with a negative cosmological constant
• Black hole entropy from Quantum Geometry
• Geometrodynamics of Schwarzschild Black Holes
• Exact Counting of Black Hole Microstates
• The Statistical Mechanics of the (2+1)-Dimensional Black Hole
• AdS/CFT Duals of Topological Black Holes and the Entropy of Zero-Energy States
• DeSitter entropy, quantum entanglement and ADS/CFT
• Gravitationally Collapsing Dust in $(2+1)$ Dimensions
• Logarithmic Corrections to Black Hole Entropy and AdS/CFT Correspondence
• Statistical Entropy and AdS/CFT Correspondence in BTZ Black Holes
• Classical and quantum LTB model for the non-marginal case
• Quantum geometry and microscopic black hole entropy
• Semiclassical Black Hole States and Entropy
• Symmetries, Horizons, and Black Hole Entropy
• Canonical Quantization and the Statistical Entropy of the Schwarzschild Black Hole
• Quantum Black Holes from Quantum Collapse
• Quantum Gravitational Collapse and Hawking Radiation in 2+1 Dimensions
• Black Hole Thermodynamics from Euclidean Horizon Constraints
• Gravitational Collapse of Inhomogeneous Dust in (2+1) Dimensions
• Quantum general relativity and Hawking radiation
• Thermodynamics and Evaporation of the 2+1-D Black Hole
• 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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• Conceptual Problems in Quantum Gravity and Quantum Cosmology
• Quantum geometrodynamics: whence, whither?
• The Role of Angular Momentum and Cosmic Censorship in the (2+1)-Dimensional Rotating Shell Collapse
• Indefinite oscillators and black-hole evaporation
• Quantum gravitational correction to the Hawking temperature from the Lemaitre-Tolman-Bondi model
• Spectrum and Statistical Entropy of AdS Black Holes
• Canonical Quantization of Spherically Symmetric Dust Collapse
• Classical and Quantum Gravitational Collapse in d-dim AdS Spacetime II. Quantum States and Hawking Radiation
• Discrete quantum spectrum of black holes
• Throat quantization of the Schwarzschild-Tangherlini(-AdS) Black Hole
• Quantum Gravitational Collapse in the Lemaitre-Tolman-Bondi Model with a Positive Cosmological Constant
• Reflection and Transmission at the Apparent Horizon during Gravitational Collapse
• A Rotating, Inhomogeneous Dust Interior for the BTZ Black Hole
• Decoding infrared imprints of quantum origins of black holes
• Bose Condensation and the BTZ Black Hole
• Quantum dust collapse in 2+1 dimensions
• BTZ entropy from topological M-theory
• Signatures of an Emergent Gravity from Black Hole Entropy
• Canonical Chern-Simons Gravity