Geometric thermodynamics: black holes and the meaning of the scalar curvature

TL;DR

This paper challenges the interpretation of scalar curvature in thermodynamic geometry, showing it does not reliably indicate ideal behavior or lack of interactions, and proposes an alternative approach for black hole thermodynamics.

Contribution

It demonstrates limitations of Ruppeiner's metric in characterizing thermodynamic interactions and introduces a new energy representation for black holes to improve geometric analysis.

Findings

Scalar curvature vanishes for non-ideal systems.

Flatness does not imply absence of microscopic interactions.

New black hole energy representation yields non-degenerate metrics.

Abstract

In this paper we show that the vanishing of the scalar curvature of Ruppeiner-like metrics does not characterize the ideal gas. Furthermore, we claim through an example that flatness is not a sufficient condition to establish the absence of interactions in the underlying microscopic model of a thermodynamic system, which poses a limitation on the usefulness of Ruppeiner's metric and conjecture. Finally, we address the problem of the choice of coordinates in black hole thermodynamics. We propose an alternative energy representation for Kerr-Newman black holes that mimics fully Weinhold's approach. The corresponding Ruppeiner's metrics become degenerate only at absolute zero and have non-vanishing scalar curvatures.