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 CJ,Φ and CΩ,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|.