Talk abstract details

The Hessian of the entropy function can be thought of as a metric tensor on the state space. In the context of thermodynamical fluctuation theory Ruppeiner has argued that the Riemannian geometry of this metric gives insight into the underlying statistical mechanical system; the claim is supported by numerous examples. Rather than studying the spacetimes of black holes as such we instead examine functions as entropy $S(M,Q,J)$ or mass $M(S,Q,J)$ where $Q$ is charge and $J$ angular momentum. The Ruppeiner metric is the Hessian of negative entropy $S$: $g^{R}_{ij} = - \partial_i \partial_jS $ while the Weinhold metric is the Hessian of $M$. The two metrics are conformally related as $ ds^2 = g^{R}_{ij}dM^idM^j = \frac{1}{T} g^W_{ij}dS^idS^j $ where $T$ denotes the Hawking temperature $ T = \frac{\partial M} {\partial S} \ . $ We study these geometries for some families of black holes and find that the Ruppeiner geometry is flat for Reissner--Nordstr\"om (RN) black holes, while curvature singularities occur for the Kerr black holes. Kerr black holes have instead flat Weinhold curvature. We also investigate thermodynamic curvatures of the corresponding spacetimes in dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four dimensional spacetime.