Ruppeiner Geometry of Black Hole Thermodynamics

TL;DR

This paper explores the Ruppeiner geometric approach to black hole thermodynamics, revealing flat and curved geometries for different black hole types, which provides insights into their underlying statistical mechanics.

Contribution

It demonstrates the application of Ruppeiner geometry to various black holes, showing flatness for Reissner–Nordström and curvature singularities for Kerr black holes.

Findings

Reissner–Nordström black holes have flat Ruppeiner geometry.

Kerr black holes exhibit curvature singularities in Ruppeiner geometry.

Kerr black holes have flat Weinhold geometry.

Abstract

The Hessian of the entropy function can be thought of as a metric tensor on 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 these geometries for some families of black holes and find that the Ruppeiner geometry is flat for Reissner--Nordstr\"om black holes in any dimension, while curvature singularities occur for the Kerr black holes. Kerr black holes have instead flat Weinhold curvature.