Canonical Quantization and the Statistical Entropy of the Schwarzschild Black Hole

The canonical quantization of a Schwarzschild black hole yields a picture of the black hole that is shown to be equivalent to a collection of oscillators whose density of levels is commensurate with that of the statistical bootstrap model. Energy eigenstates of definite parity exhibit the Bekenstein mass spectrum, $M \sim \sqrt{\cal N} M_p$, where ${\cal N} \in {\bf N}$. From the microcanonical ensemble, we derive the statistical entropy of the black hole by explicitly counting the microstates corresponding to a macrostate of fixed total energy.Comment: 21 pages, PHYZZX macros. The counting of physical states has been corrected. Revised version to appear in Phys. Rev.

The Quantum States and the Statistical Entropy of the Charged Black Hole

We quantize the Reissner-Nordstr\"om black hole using an adaptation of Kucha\v{r}'s canonical decomposition of the Kruskal extension of the Schwarzschild black hole. The Wheeler-DeWitt equation turns into a functional Schroedinger equation in Gaussian time by coupling the gravitational field to a reference fluid or dust. The physical phase space of the theory is spanned by the mass, $M$, the charge, $Q$, the physical radius, $R$, the dust proper time, $\tau$, and their canonical momenta. The exact solutions of the functional Schroedinger equation imply that the difference in the areas of the outer and inner horizons is quantized in integer units. This agrees in spirit, but not precisely, with Bekenstein's proposal on the discrete horizon area spectrum of black holes. We also compute the entropy in the microcanonical ensemble and show that the entropy of the Reissner-Nordstr\"om black hole is proportional to this quantized difference in horizon areas.Comment: 31 pages, 3 figures, PHYZZX macros. Comments on the wave-functional in the interior and one reference added. To appear in Phys. Rev.

Spectrum and Statistical Entropy of AdS Black Holes

Popular approaches to quantum gravity describe black hole microstates differently and apply different statistics to count them. Since the relationship between the approaches is not clear, this obscures the role of statistics in calculating the black hole entropy. We address this issue by discussing the entropy of eternal AdS black holes in dimension four and above within the context of a midisuperspace model. We determine the black hole eigenstates and find that they describe the quantization in half integer units of a certain function of the Arnowitt-Deser-Misner (ADM) mass and the cosmological constant. In the limit of a vanishing cosmological constant (the Schwarzschild limit) the quantized function becomes the horizon area and in the limit of a large cosmological constant it approaches the ADM mass of the black holes. We show that in the Schwarzschild limit the area quatization leads to the Bekenstein-Hawking entropy if Boltzmann statistics are employed. In the limit of a large cosmological constant the Bekenstein-Hawking entropy can be recovered only via Bose statistics. The two limits are separated by a first order phase transition, which seems to suggest a shift from "particle-like" degrees of freedom at large cosmological constant to geometric degrees of freedom as the cosmological constant approaches zero.Comment: 14 pages. No figures. Some references added. Version to appear in Phys. Rev.

Spectrum of Charged Black Holes - The Big Fix Mechanism Revisited

Following an earlier suggestion of the authors(gr-qc/9607030), we use some basic properties of Euclidean black hole thermodynamics and the quantum mechanics of systems with periodic phase space coordinate to derive the discrete two-parameter area spectrum of generic charged spherically symmetric black holes in any dimension. For the Reissner-Nordstrom black hole we get $A/4G\hbar=\pi(2n+p+1)$, where the integer p=0,1,2,.. gives the charge spectrum, with $Q=\pm\sqrt{\hbar p}$. The quantity $\pi(2n+1)$, n=0,1,... gives a measure of the excess of the mass/energy over the critical minimum (i.e. extremal) value allowed for a given fixed charge Q. The classical critical bound cannot be saturated due to vacuum fluctuations of the horizon, so that generically extremal black holes do not appear in the physical spectrum. Consistency also requires the black hole charge to be an integer multiple of any fundamental elementary particle charge: $Q= \pm me$, m=0,1,2,.... As a by-product this yields a relation between the fine structure constant and integer parameters of the black hole -- a kind of the Coleman big fix mechanism induced by black holes. In four dimensions, this relationship is $e^2/\hbar=p/m^2$ and requires the fine structure constant to be a rational number. Finally, we prove that the horizon area is an adiabatic invariant, as has been conjectured previously.Comment: 21 pages, Latex. 1 Section, 1 Figure added. To appear in Class. and Quant. Gravit

Aspects of Black Hole Quantum Mechanics and Thermodynamics in 2+1 Dimensions

We discuss the quantum mechanics and thermodynamics of the (2+1)-dimensional black hole, using both minisuperspace methods and exact results from Chern-Simons theory. In particular, we evaluate the first quantum correction to the black hole entropy. We show that the dynamical variables of the black hole arise from the possibility of a deficit angle at the (Euclidean) horizon, and briefly speculate as to how they may provide a basis for a statistical picture of black hole thermodynamics.Comment: 20 pages and 2 figures, LaTeX, IASSNS-HEP-94/34 and UCD-94-1

A Simple, Approximate Method for Analysis of Kerr-Newman Black Hole Dynamics and Thermodynamics

In this work we present a simple, approximate method for analysis of the basic dynamical and thermodynamical characteristics of Kerr-Newman black hole. Instead of the complete dynamics of the black hole self-interaction we consider only such stable (stationary) dynamical situations determined by condition that black hole (outer) horizon circumference holds the integer number of the reduced Compton wave lengths corresponding to mass spectrum of a small quantum system (representing quant of the black hole self-interaction). Then, we show that Kerr-Newman black hole entropy represents simply the quotient of the sum of static part and rotation part of mass of black hole on the one hand and ground mass of small quantum system on the other hand. Also we show that Kerr-Newman black hole temperature represents the negative value of the classical potential energy of gravitational interaction between a part of black hole with reduced mass and small quantum system in the ground mass quantum state. Finally, we suggest a bosonic great canonical distribution of the statistical ensemble of given small quantum systems in the thermodynamical equilibrium with (macroscopic) black hole as thermal reservoir. We suggest that, practically, only ground mass quantum state is significantly degenerate while all other, excited mass quantum states are non-degenerate. Kerr-Newman black hole entropy is practically equivalent to the ground mass quantum state degeneration. Given statistical distribution admits a rough (qualitative) but simple modeling of Hawking radiation of the black hole too.Comment: 8 pages, no figure

A Simple, Approximate Method for Analysis of Kerr-Newman Black Hole Dynamics and Thermodynamics

In this work we present a simple, approximate method for analysis of the basic dynamical and thermodynamical characteristics of Kerr-Newman black hole. Instead of the complete dynamics of the black hole self-interaction we consider only such stable (stationary) dynamical situations determined by condition that black hole (outer) horizon circumference holds the integer number of the reduced Compton wave lengths corresponding to mass spectrum of a small quantum system (representing quant of the black hole self-interaction). Then, we show that Kerr-Newman black hole entropy represents simply the quotient of the sum of static part and rotation part of mass of black hole on the one hand and ground mass of small quantum system on the other hand. Also we show that Kerr-Newman black hole temperature represents the negative value of the classical potential energy of gravitational interaction between a part of black hole with reduced mass and small quantum system in the ground mass quantum state. Finally, we suggest a bosonic great canonical distribution of the statistical ensemble of given small quantum systems in the thermodynamical equilibrium with (macroscopic) black hole as thermal reservoir. We suggest that, practically, only ground mass quantum state is significantly degenerate while all other, excited mass quantum states are non-degenerate. Kerr-Newman black hole entropy is practically equivalent to the ground mass quantum state degeneration. Given statistical distribution admits a rough (qualitative) but simple modeling of Hawking radiation of the black hole too.Comment: 8 pages, no figure