A new approach toward geometrical concept of black hole thermodynamics

TL;DR

This paper introduces a new geometric metric for black hole thermodynamics that accurately identifies phase transition points, improving upon previous metrics by aligning divergences with heat capacity phase transitions.

Contribution

The paper proposes a novel thermodynamic metric that correctly correlates Ricci scalar divergences with phase transitions, addressing limitations of existing metrics.

Findings

New metric's Ricci scalar divergence matches phase transitions

Previous metrics' divergences do not coincide with phase transition points

Generalization of the new metric for multiple parameters is successful

Abstract

Motivated by the energy representation of Riemannian metric, in this paper we study different approaches toward the geometrical concept of black hole thermodynamics. We investigate thermodynamical Ricci scalar of Weinhold, Ruppeiner and Quevedo metrics and show that their number and location of divergences do not coincide with phase transition points arisen from heat capacity. Next, we introduce a new metric to solve these problems. We show that the denominator of the Ricci scalar of the new metric contains terms which coincide with different types of phase transitions. We elaborate the effectiveness of the new metric and shortcomings of the previous metrics with some examples. Furthermore, we find a characteristic behavior of the new thermodynamical Ricci scalar which enables one to distinguish two types of phase transitions. In addition, we generalize the new metric for the cases of more than two extensive parameters and show that in these cases the divergencies of thermodynamical Ricci scalar coincide with phase transition points of the heat capacity.