Schwarzschild black hole as a grand canonical ensemble

TL;DR

This paper reinterprets Schwarzschild black holes as grand canonical ensembles, resolving previous issues with temperature and fluctuations, and deriving corrected entropy and energy distributions.

Contribution

It introduces a grand canonical ensemble framework for Schwarzschild black holes, addressing inconsistencies and deriving new entropy and energy fluctuation results.

Findings

Logarithmic correction to Bekenstein-Hawking entropy

Gaussian distribution for energy levels

Resolution of temperature and fluctuation issues

Abstract

For long black holes have been considered as endowed with a definite temperature. Yet when the Schwarzschild black hole is treated as a canonical ensemble three problems arise: incompatibility with the Hawking radiation, divergence of the partition function, and a formally negative mean-square fluctuation of the energy. We solve all three problems by considering the Schwarzschild black hole as a grand canonical ensemble, with the Hamiltonian (the ADM mass) and the horizon surface area, separately, as observable parameters. The horizon area simulates the number of particles in statistical mechanics since its spectrum is assumed to be discrete and equally spaced. We obtain a logarithmic correction to the Bekenstein-Hawking entropy and a Gaussian type distribution for the energy levels.