Geometry of black hole thermodynamics

TL;DR

This paper explores the geometric structure of black hole thermodynamics by analyzing the entropy Hessian as a metric, revealing flat and curved geometries for different black hole families, which provides insights into their underlying statistical mechanics.

Contribution

It applies Ruppeiner geometry to various black holes, identifying cases with flat and curved thermodynamic state space geometries, thus advancing understanding of black hole thermodynamics.

Findings

BTZ and Reissner-Nordstrom black holes have flat thermodynamic geometry.

Reissner-Nordstrom-anti-de Sitter and Kerr black holes exhibit curvature singularities.

The geometric approach offers insights into the statistical mechanics of black holes.

Abstract

The Hessian of the entropy function can be thought of as a metric tensor on the state space. In the context of thermodynamical fluctuation theory Ruppeiner has argued that the Riemannian geometry of this metric gives insight into the underlying statistical mechanical system; the claim is supported by numerous examples. We study this geometry for some families of black holes. It is flat for the BTZ and Reissner-Nordstrom black holes, while curvature singularities occur for the Reissner-Nordstrom-anti-de Sitter and Kerr black holes.