Thermodynamic curvature and phase transitions in Kerr-Newman black holes

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. Away from the extremal case, R diverges to +∞ along curves of diverging heat capacities C_J,Φ and C_Ω,Q, but not along the Davies curve of diverging CJ,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 $|R|$.