Black hole horizons can hide positive heat capacity

Tamás S Biró,Viktor G Czinner,Hideo Iguchi,Péter Ván

doi:10.1016/j.physletb.2018.05.035

Tamás S Biró, Viktor G Czinner + Show 2 more

Open Access

https://doi.org/10.1016/j.physletb.2018.05.035

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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∼R5, with an entropy S=83SBH.

Highlights

The irreducible mass of black holes is connected to an entropy function in black hole thermodynamics [1,2,3,4,5]
Due to the fact that the irreducible mass of black holes is proportional to the radius of their event horizon, the entropy, proportional to its surface, S(M ) ∼ M 2, is seemingly convex and the heat capacity derived from it is negative
This is common in all bound systems where the total energy is negative and the kinetic energy is positive, due to an increase of the temperature via an increase in the kinetic energy, – in a stationary state satisfying a virial theorem, – the total energy will decrease, displaying formally a negative heat capacity

Summary

Introduction

The irreducible mass of black holes is connected to an entropy function in black hole thermodynamics [1,2,3,4,5]. Due to the fact that the irreducible mass of black holes is proportional to the radius of their event horizon, the entropy, proportional to its surface, S(M ) ∼ M 2, is seemingly convex and the heat capacity derived from it is negative. This is common in all bound systems where the total energy is negative and the kinetic energy is positive, due to an increase of the temperature via an increase in the kinetic energy, – in a stationary state satisfying a virial theorem, – the total energy will decrease, displaying formally a negative heat capacity. We derive the thermodynamic properties of Schwarzschild black holes by including the usual work term in the first law based only on the assumption that the entropy is a first order homogeneous (extensive) function of the volume.

Black hole EoS with volume term

T dS dE

Specific heat and stability

Causality and the third law of thermodynamics

Conclusions