Black holes as gases of punctures with a chemical potential: Bose-Einstein condensation and logarithmic corrections to the entropy

TL;DR

This paper investigates black hole thermodynamics within loop quantum gravity by modeling black holes as gases of punctures with a chemical potential, revealing conditions for Bose-Einstein condensation and logarithmic entropy corrections.

Contribution

It introduces a detailed analysis of black hole microstates as Bose gases with holographic degeneracy, highlighting the role of chemical potential and conditions for semi-classical regimes.

Findings

Logarithmic corrections to entropy in the holographic limit

Bose-Einstein condensation of punctures at ground state spin

Semi-classical regime characterized by minimal spin punctures

Abstract

We study the thermodynamical properties of black holes when described as gases of indistinguishable punctures with a chemical potential. In this picture, which arises from loop quantum gravity, the black hole microstates are defined by finite families of half-integers spins coloring the punctures, and the near-horizon energy measured by quasi-local stationary observers defines the various thermodynamical ensembles. The punctures carry excitations of quantum geometry in the form of quanta of area, and the total horizon area $a_\text{H}$ is given by the sum of these microscopic contributions. We assume here that the system satisfies the Bose-Einstein statistics, and that each microstate is degenerate with a holographic degeneracy given by $\exp\big(\lambda a_\text{H}/\ell_\text{Pl}^2\big)$ and $\lambda>0$. We analyze in detail the thermodynamical properties resulting from these inputs, and in particular compute the grand canonical entropy. We explain why the requirements that the temperature be fixed to the Unruh temperature and that the chemical potential vanishes do not specify completely the semi-classical regime of large horizon area, and classify in turn what the various regimes can be. When the degeneracy saturates the holographic bound ($\lambda=1/4$), there exists a semi-classical regime in which the subleading corrections to the entropy are logarithmic. Furthermore, this regime corresponds to a Bose-Einstein condensation, in the sense that it is dominated by punctures carrying the minimal (or ground state) spin value $1/2$.