Flat Information Geometries in Black Hole Thermodynamics

TL;DR

This paper explores the geometric structure of black hole thermodynamics, revealing that certain state spaces are flat due to scale invariance, which clarifies thermodynamic properties like divergent specific heats.

Contribution

It demonstrates that the Hessian metrics of entropy or energy for specific black hole families are flat, linking geometric properties to scale invariance in Einstein-Maxwell equations.

Findings

State space is a flat wedge for certain black holes.

Hessian metrics are flat due to scale invariance.

Divergent specific heats are explained geometrically.

Abstract

The Hessian of either the entropy or the energy function can be regarded as a metric on a Gibbs surface. For two parameter families of asymptotically flat black holes in arbitrary dimension one or the other of these metrics are flat, and the state space is a flat wedge. The mathematical reason for this is traced back to the scale invariance of the Einstein-Maxwell equations. The picture of state space that we obtain makes some properties such as the occurence of divergent specific heats transparent.