Canonical Quantum Statistics of an Isolated Schwarzschild Black Hole with a Spectrum E_n = sigma sqrt{n} E_P

TL;DR

This paper analyzes the quantum statistical properties of an isolated Schwarzschild black hole with a specific energy spectrum, revealing that the imaginary part of the partition function encodes its thermodynamics and reproduces Hawking entropy.

Contribution

It introduces a novel approach by extending the partition function into the complex plane and shows that the imaginary part encodes the black hole's thermodynamics, including Hawking entropy.

Findings

Partition function obeys a 1D heat equation.

Imaginary part of the partition function yields thermodynamic properties.

Derived entropy matches Hawking's area law with logarithmic corrections.

Abstract

Many authors - beginning with Bekenstein - have suggested that the energy levels E_n of a quantized isolated Schwarzschild black hole have the form E_n = sigma sqrt{n} E_P, n=1,2,..., sigma =O(1), with degeneracies g^n. In the present paper properties of a system with such a spectrum, considered as a quantum canonical ensemble, are discussed: Its canonical partition function Z(g,beta=1/kT), defined as a series for g<1, obeys the 1-dimensional heat equation. It may be extended to values g>1 by means of an integral representation which reveals a cut of Z(g,beta) in the complex g-plane from g=1 to infinity. Approaching the cut from above yields a real and an imaginary part of Z. Very surprisingly, it is the (explicitly known) imaginary part which gives the expected thermodynamical properties of Schwarzschild black holes: Identifying the internal energy U with the rest energy Mc^2 requires beta to have the value (in natural units) beta = 2M(lng/sigma^2)[1+O(1/M^2)], (4pi sigma^2=lng gives Hawking's beta_H), and yields the entropy S=[lng/(4pi sigma^2)] A/4 + O(lnA), where A is the area of the horizon.