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Volume in the Extensive Thermodynamics of Black Holes

Réka Somogyfoki, Péter Ván

TL;DR

This work investigates how to define a meaningful thermodynamic volume for AdS black holes by contrasting Hawking–Page thermodynamics with Dolan’s enthalpy-based volume framework, introducing a pressure $p = -3\Lambda/(8\pi)$ and treating the mass as enthalpy with $V = \partial H/\partial p|_S$. It derives stability criteria from $C_V \ge 0$ and $(\partial p/\partial V)_T \le 0$, reproduces the Hawking–Page boundary with $T_{HP} = \sqrt{-\Lambda}/\pi$, and extends the setup to a generalized pressure that depends on $E$ and $S$, applying it to Kiselev black holes to obtain a $w = -5/3$ solution with $V \propto R^5$. A scaling analysis with $R(E,V) = R_S E^{\alpha} V^{\beta}$ shows that a radiation-like choice $\alpha = 1/2$, $\beta = 1/6$ yields extensivity, while a Kiselev-type non-extensive case with $\alpha = -3/4$, $\beta = 5/4$ arises, illustrating the sensitivity to environmental assumptions. The results suggest that simple average-pressure models may be insufficient for mixed fluid–radiation environments, motivating a more general, environment-aware thermodynamics for black holes.

Abstract

Since black holes lack a straightforward notion of geometrical volume due to their event horizon structure and coordinate dependence, various approaches have been proposed to introduce a meaningful geometric and thermodynamic volume. In this work we investigate the stability conditions of AdS black holes with and without volume.

Volume in the Extensive Thermodynamics of Black Holes

TL;DR

This work investigates how to define a meaningful thermodynamic volume for AdS black holes by contrasting Hawking–Page thermodynamics with Dolan’s enthalpy-based volume framework, introducing a pressure and treating the mass as enthalpy with . It derives stability criteria from and , reproduces the Hawking–Page boundary with , and extends the setup to a generalized pressure that depends on and , applying it to Kiselev black holes to obtain a solution with . A scaling analysis with shows that a radiation-like choice , yields extensivity, while a Kiselev-type non-extensive case with , arises, illustrating the sensitivity to environmental assumptions. The results suggest that simple average-pressure models may be insufficient for mixed fluid–radiation environments, motivating a more general, environment-aware thermodynamics for black holes.

Abstract

Since black holes lack a straightforward notion of geometrical volume due to their event horizon structure and coordinate dependence, various approaches have been proposed to introduce a meaningful geometric and thermodynamic volume. In this work we investigate the stability conditions of AdS black holes with and without volume.

Paper Structure

This paper contains 5 sections, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hawking--Page phase boundaries
  3. Volume and extensivity
  4. Discussion
  5. Acknowledgement

Figures (1)

  • Figure 1: The dimensionless temperature $(T)$ expressed in terms of radius $(R)$ for AdS black holes with the Hawking--Page(--Dolan) stability and evaporation limits on the left and the same functions for Kiselev(--CR) black holes on the right.