Pressure and volume in the first law of black hole thermodynamics

TL;DR

This work reframes black hole thermodynamics by treating the cosmological constant as a thermodynamic pressure $P$ with a conjugate volume $V$, so the mass corresponds to enthalpy $H$ and the internal energy is $U=H-PV$. It derives $U(S,V,J,Q)$ via a Legendre transform and obtains the equation of state, including a virial expansion that reveals a van der Waals–like critical point. The paper then analyzes energy extraction through Penrose-like processes, showing that a negative cosmological constant can, in principle, raise maximal efficiencies: up to about $0.75$ for extremal charged AdS black holes and about $0.5184$ for extremal neutral AdS black holes, surpassing the corresponding flat-space limits. It discusses why the study focuses on $\Lambda<0$ (AdS) due to ambiguities and instabilities for $\Lambda>0$, and highlights the broader implications for black hole thermodynamics and phase structure.

Abstract

The mass of a black hole is interpreted, in terms of thermodynamic potentials, as being the enthalpy, with the pressure given by the cosmological constant. The volume is then defined as being the Legendre transform of the pressure and the resulting relation between volume and pressure is explored in the case of positive pressure. A virial expansion is developed and a van der Waals like critical point determined. The first law of black hole thermodynamics includes a PdV term which modifies the maximal efficiency of a Penrose process. It is shown that, in four dimensional space-time with a negative cosmological constant an extremal charged rotating black hole can have an efficiency of up to 75%, while for an electrically neutral rotating back hole this figure is reduced to 52%, compared to the corresponding values of 50% and 29% respectively when the cosmological constant is zero.