Thermodynamic curvature and phase transitions in Kerr-Newman black holes

TL;DR

This paper investigates the thermodynamic curvature of Kerr-Newman black holes to gain microscopic insights, finding similarities with Fermi gases and analyzing divergence behaviors of the curvature scalar.

Contribution

It introduces the use of thermodynamic Riemannian curvature to connect black hole thermodynamics with microscopic models, highlighting cases with positive curvature and exact correspondences with Fermi gases.

Findings

The curvature scalar R is mostly positive for Kerr-Newman black holes.

Exact correspondences are found between the extremal Kerr-Newman black hole and the two-dimensional Fermi gas.

R diverges along certain heat capacity curves, indicating phase transition behaviors.

Abstract

Singularities in the thermodynamics of Kerr-Newman black holes are commonly associated with phase transitions. However, such interpretations are complicated by a lack of stability and, more significantly, by a lack of conclusive insight from microscopic models. Here, I focus on the later problem. I use the thermodynamic Riemannian curvature scalar $R$ as a try to get microscopic information from the known thermodynamics. The hope is that this could facilitate matching black hole thermodynamics to known models of statistical mechanics. For the Kerr-Newman black hole, the sign of $R$ is mostly positive, in contrast to that for ordinary thermodynamic models, where $R$ is mostly negative. Cases with negative $R$ include most of the simple critical point models. An exception is the Fermi gas, which has positive $R$. I demonstrate several exact correspondences between the two-dimensional Fermi gas and the extremal Kerr-Newman black hole. $R$ diverges to $+\infty$ along curves of diverging heat capacities $C_{J,\Phi}$ and $C_{\Omega,Q}$, but not along the Davies curve of diverging $C_{J,Q}$. Finding statistical mechanical models with like behavior might yield additional insight into the microscopic properties of black holes. I also discuss a possible physical interpretation of $|R|$.