Thermodynamic curvature and phase transitions in Kerr-Newman black holes

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 $|R|$.Comment: 29 pages, 8 figure

Ehrenfest’s scheme and thermodynamic geometry in Born-Infeld AdS black holes

“…First of all from figures (8) and (9) we observe that the scalar curvature (R) diverges exactly at the points where the specific heat (C Φ ) diverges (also see figures (2) & (3)). This is an expected result, since according to Ruppeiner any divergence in R implies a corresponding divergence in the heat capacity C Φ (which is thermodynamic analog of C P ) that essentially leads to a change in stability [49]. On the other hand, no divergence in R could be observed at the Davis critical point [56], which is marked by the divergence in C Q (which is the thermodynamic analog of C V ).…”

“…In the state space geometry approach one aims to calculate the scalar curvature (R) [49] which suffers a discontinuity at the critical point for the second order phase transition.…”

Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature

“…Therefore, it is remarkable that the energy representation we have proposed yields a metric whose scalar curvature diverges only at absolute zero on both Riemannian submanifolds of E. Moreover, unlike the common result reported in literature [17,18,20,28], it remains finite at points where heat capacities diverge (cf. Ruppeiner [30]), whose physical situation is rather unclear [36].…”

“…Ruppeiner's (and Weinhold's) geometric approach to thermodynamics has been kindled recently in black hole thermodynamics [15,17,[20][21][22]26,[28][29][30][31][32]. The main goal of this standpoint is to analyze the scalar curvature of Ruppeiner's metrics for different "macroscopic" thermodynamic models of black holes, in order to obtain a hint via Ruppeiner's conjecture of the interactions that a microscopic model of gravity might involve.…”

The generalization of charged AdS black hole specific volume and number density

“…and the authors had given a natural interpretation for it from the holographic view [22,27,28]. The microscopic degrees of freedom of the black hole was proposed to be carried by the Planck area pixels.…”