Where is the PdV term in the first law of black hole thermodynamics?

TL;DR

This work argues that the cosmological constant should be treated as a thermodynamic pressure $P$ with a conjugate volume $V$, promoting the black hole mass to enthalpy $H(S,P,J,Q)$ and extending the first law to $dU=T dS+\Omega dJ+\Phi dQ-P dV$. It derives a consistent thermodynamic framework where the thermodynamic volume coincides with the geometric volume for non-rotating AdS black holes and yields a positive, rotation-dependent volume for Kerr–AdS spacetimes. The analysis reveals a Van der Waals–type, mean-field critical behavior for rotating AdS black holes, including explicit critical exponents $\alpha=0$, $\beta=\tfrac{1}{2}$, $\gamma=1$, $\delta=3$, and a Maxwell construction, with a line of critical points when charge is nonzero. The results deepen the connection between black hole thermodynamics and standard fluid thermodynamics, offer new insights into Penrose-process efficiencies, and highlight open questions for de Sitter space and higher-dimensional generalizations.

Abstract

Traditional treatments of the first law of black hole thermodynamics do not include a discussion of pressure and volume. We give an overview of recent developments proposing a definition of volume that can be used to extend the first law to include these appropriately. New results are also presenting relating to the critical point and the associated second order phase transition for a rotating black-hole in four-dimensional space-time which is asymptotically anti-de Sitter. In line with known results for a non-rotating charged black-hole, this phase transition is shown to be of Van der Waals type with mean field exponents.

Figures (5)

• Figure 1: $T=0$ and $C_P\rightarrow\infty$ curves in $J-S$ plane.
• Figure 2: Construction associated with Maxwell's equal area law.
• Figure 3: Gibbs free energy as a function of pressure and temperature, at fixed angular momentum.
• Figure 4: The line of second order critical points in the $J-Q$ plane, CCK.
• Figure 5: The speed of sound for an electrically neutral, extremal, black hole (with $c=1$).