Thermodynamical Metrics and Black Hole Phase Transitions

TL;DR

This paper introduces a new thermodynamical metric that accurately signals black hole phase transitions through curvature singularities, addressing limitations of previous metrics like Ruppeiner's and Weinhold's.

Contribution

A novel thermodynamical metric based on the Hessian of free energy is proposed, which correctly indicates phase transitions in black holes, and its relation to other potential metrics is analyzed.

Findings

The new metric's curvature singularity coincides with specific heat divergence.

All thermodynamical potential metrics can be embedded into a flat (n,n)-dimensional space.

Generalized Ruppeiner metrics are conformal to metrics from thermodynamical potentials.

Abstract

An important phase transition in black hole thermodynamics is associated with the divergence of the specific heat with fixed charge and angular momenta, yet one can demonstrate that neither Ruppeiner's entropy metric nor Weinhold's energy metric reveals this phase transition. In this paper, we introduce a new thermodynamical metric based on the Hessian matrix of several free energy. We demonstrate, by studying various charged and rotating black holes, that the divergence of the specific heat corresponds to the curvature singularity of this new metric. We further investigate metrics on all thermodynamical potentials generated by Legendre transformations and study correspondences between curvature singularities and phase transition signals. We show in general that for a system with n-pairs of intensive/extensive variables, all thermodynamical potential metrics can be embedded into a flat (n,n)-dimensional space. We also generalize the Ruppeiner metrics and they are all conformal to the metrics constructed from the relevant thermodynamical potentials.