Ruppeiner Geometry of RN Black Holes: Flat or Curved?

TL;DR

This paper investigates the Ruppeiner geometry of Reissner-Nordström black holes, demonstrating that considering the full thermodynamic variable set reveals a non-flat state space with curvature divergences at extremal limits.

Contribution

It shows that using the complete thermodynamic variables results in a non-flat Ruppeiner geometry, contrasting previous findings of flatness, and highlights curvature divergence at extremal limits.

Findings

Ruppeiner curvature is non-zero with complete variables

Curvature diverges at extremal black hole limits

Previous flatness results are due to incomplete variable sets

Abstract

In some recent studies \cite{aman1, aman2, aman3}, Aman {\it et al.} used the Ruppeiner scalar as a measure of underlying interactions of Reissner-Nordstr\"{o}m black holes, indicating that it is a non-interacting statistical system for which classical thermodynamics could be used at any scale. Here, we show that if we use the complete set of thermodynamic variables, a non-flat state space will be produced. Furthermore, the Ruppeiner curvature diverges at extremal limits, as it would for other types of black holes.