Geometric curvatures of plane symmetry black hole

TL;DR

This paper investigates the geometric properties and thermodynamic phase transitions of plane symmetry black holes using Weinhold, Ruppeiner, and Quevedo metrics, revealing insights into phase transition points and stability.

Contribution

It introduces a unified geometric approach to analyze phase transitions in plane symmetry black holes, incorporating Legendre invariance and logarithmic corrections.

Findings

Weinhold curvature indicates phase transitions at N=1.

Ruppeiner curvature shows phase transitions for N≠1.

Unified Quevedo metric accurately describes thermodynamic interactions.

Abstract

In this paper, we study the properties and thermodynamic stability of the plane symmetry black hole from the viewpoint of geometry. Weinhold metric and Ruppeiner metric are obtained, respectively. The Weinhold curvature gives phase transition points, which correspond to the first-order phase transition only at N=1, where $N$ is a parameter in the plane symmetry black hole. While the Ruppeiner one shows first-order phase transition points for arbitrary $N\neq 1$. Both of which give no any information about the second-order phase transition. Considering the Legendre invariant proposed by Quevedo et. al., we obtain a unified geometry metric, which gives a correctly the behavior of the thermodynamic interactions and phase transitions. The geometry is also found to be curved and the scalar curvature goes to negative infinity at the Davies' phase transition points when the logarithmic correction is included.