A Commentary on Ruppeiner Metrics for Black Holes

TL;DR

This paper discusses the properties of Ruppeiner metrics in black hole thermodynamics, addressing controversies about their flatness and proposing a new interpretation, supported by calculations on rotating BTZ black holes.

Contribution

It offers a novel interpretation of the flatness of Ruppeiner metrics for certain black holes and explores potential modifications to the formalism.

Findings

Ruppeiner metrics for Reissner-Nordstrom black holes are flat.

A new interpretation clarifies the physical meaning of metric flatness.

Calculations on BTZ black holes support the proposed interpretation.

Abstract

There has been some recent controversy regarding the Ruppeiner metrics that are induced by Reissner-Nordstrom (and Reissner-Nordstrom-like) black holes. Most infamously, why does this family of metrics turn out to be flat, how is this outcome to be physically understood, and can/should the formalism be suitably modified to induce curvature? In the current paper, we provide a novel interpretation of this debate. For the sake of maximal analytic clarity and tractability, some supporting calculations are carried out for the relatively simple model of a rotating BTZ black hole.