Quasi-homogeneous black hole geometrothermodynamics in Einstein-Maxwell theory

TL;DR

This paper develops a geometrothermodynamics framework for black holes, linking curvature singularities in the thermodynamic space to phase transitions, and emphasizing the quasi-homogeneous nature of black hole thermodynamics.

Contribution

It introduces a Legendre-invariant formalism for black hole thermodynamics that accounts for quasi-homogeneity, connecting geometric singularities with phase transitions.

Findings

Curvature singularities correspond to phase transitions in black holes.

The framework applies to Reissner-Nordström, Kerr, and Kerr-Newman black holes.

Standard thermodynamic identities are invalidated by quasi-homogeneity.

Abstract

In this review, we establish the mathematical framework of geometrothermodynamics (GTD) as a formalism capable of describing non-extensive, quasi-homogeneous, self-gravitating systems in a Legendre-invariant manner. We argue that the fundamental equations of black holes are quasi-homogeneous functions, a property that invalidates the standard Euler identity of laboratory thermodynamics. We derive the metrics for the equilibrium manifold and analyze their curvature singularities for the Reissner-Nordstr\"om, Kerr, and Kerr-Newman black holes. Furthermore, we establish a direct correspondence between the curvature singularities of the equilibrium space and phase transitions, as determined by the divergences of the corresponding heat capacities.