Thermodynamic geometric analysis of D-dimensional RN black hole

TL;DR

This paper explores the thermodynamic geometry of D-dimensional Reissner-Nordström black holes, revealing phase transitions in fixed-charge ensembles and linking geometric metrics to Euclidean path integral periodicity.

Contribution

It provides a detailed analysis of the Ruppeiner geometry for D-dimensional RN black holes and establishes a connection between thermodynamic geometry and Euclidean quantum gravity methods.

Findings

Curvature diverges at phase transition points for D > 4.

Geometry appears flat when all extensive variables fluctuate.

Thermodynamic metric relates to Euclidean path integral periodicity.

Abstract

This paper studies the thermodynamics and Ruppeiner geometry of D-dimensional RN black holes. We analyze the thermodynamic curvature scalar $R$ in various thermodynamic ensembles. It is found that in an ensemble of fixed charge (canonical ensemble), the Ruppeiner curvature is curved and diverges at a critical point, indicating the existence of a phase transition for $D > 4$. In contrast, when all extensive variables are allowed to fluctuate (for example, in a grand-canonical ensemble or with pressure fixed), the Ruppeiner geometry can appear flat. We also demonstrate that the thermodynamic geometric metric has a one-to-one correspondence with the periodicity of the Euclidean path integral method. In particular, the inverse temperature (the Euclidean time period) serves as a bridge connecting the thermodynamic geometry and the Euclidean action approach.