Thermodynamic Geometry and Critical Behavior of Black Holes

TL;DR

This paper explores the thermodynamic geometry of various black holes, revealing critical behavior and phase transitions through the analysis of scalar curvature in Ruppeiner geometry, aligning with known phase transition points.

Contribution

It introduces the use of Ruppeiner geometry based on internal energy and electric potential to study phase transitions in black holes, extending previous analyses.

Findings

Scalar curvature diverges at phase transition points

Geometry is curved for RN, Kerr, and RN-AdS black holes

Results agree with existing literature on phase transitions

Abstract

Based on the observations that there exists an analogy between the Reissner-Nordstr\"om-anti-de Sitter (RN-AdS) black holes and the van der Waals-Maxwell liquid-gas system, in which a correspondence of variables is $(\phi, q) \leftrightarrow (V,P)$, we study the Ruppeiner geometry, defined as Hessian matrix of black hole entropy with respect to the internal energy (not the mass) of black hole and electric potential (angular velocity), for the RN, Kerr and RN-AdS black holes. It is found that the geometry is curved and the scalar curvature goes to negative infinity at the Davies' phase transition point for the RN and Kerr black holes. Our result for the RN-AdS black holes is also in good agreement with the one about phase transition and its critical behavior in the literature.