Black hole horizons can hide positive heat capacity

TL;DR

This paper explores conditions under which black hole horizons can exhibit positive heat capacity and thermal stability by considering various definitions of volume and thermodynamic variables, challenging traditional views.

Contribution

It introduces a framework where black hole horizons can be thermally stable with positive heat capacity by considering different volume definitions and thermodynamic assumptions.

Findings

Horizon thermodynamics can be stable with positive heat capacity under certain volume scalings.

The Christodoulou--Rovelli volume leads to a stable horizon with entropy 8/3 times the Bekenstein-Hawking entropy.

Thermal stability depends on the choice of volume and the equation of state at the horizon.

Abstract

Regarding the volume as independent thermodynamic variable we point out that black hole horizons can hide positive heat capacity and specific heat. Such horizons are mechanically marginal, but thermally stable. In the absence of a canonical volume definition, we consider various suggestions scaling differently with the horizon radius. Assuming Euler-homogeneity of the entropy, besides the Hawking temperature, a pressure and a corresponding work term render the equation of state at the horizon thermally stable for any meaningful volume concept that scales larger than the horizon area. When considering also a Stefan--Boltzmann radiation like equation of state at the horizon, only one possible solution emerges: the Christodoulou--Rovelli volume, scaling as $V\sim R^5$, with an entropy $S = \frac{8}{3}S_{BH}$.