Extended phase space thermodynamics for black holes in a cavity

Peng Wang,Houwen Wu,Feiyu Yao,Haitang Yang

doi:10.1007/jhep09(2020)154

Abstract

In this paper, we extend the phase space of black holes enclosed by a spherical cavity of radius rB to include V=4pi {r}_B^3/3 as a thermodynamic volume. The thermodynamic behavior of Schwarzschild and Reissner-Nordstrom (RN) black holes is then investigated in the extended phase space. In a canonical ensemble at constant pressure, we find that the aforementioned thermodynamic behavior is remarkably similar to that of the anti-de Sitter (AdS) counterparts with the cosmological constant being interpreted as a pressure. Specifically, a first-order Hawking-Page-like phase transition occurs for a Schwarzschild black hole in a cavity. The phase structure of a RN black hole in a cavity shows a strong resemblance to that of the van der Waals fluid. We also display that the Smarr relation has the same expression in both AdS and cavity cases. Our results may provide a new perspective for the extended thermodynamics of AdS black holes by analogy with black holes in a cavity.

Highlights

JHEP09(2020)154 the Hawking-Page phase transition in the P -T diagram is semi-infinite and reminiscent of the solid/liquid phase transition [36]
We extended the phase space of black holes enclosed in a cavity to include 4πrB3 /3 as a thermodynamic volume, where rB2 f (rB) is the cavity radius
Such extension is largely motivated by the extended phase space of anti-de Sitter (AdS) black holes, in which the cosmological constant is interpreted as a pressure

Summary

Phase structure of a Schwarzschild black hole in a cavity

We consider a thermodynamic system with a Schwarzschild black hole enclosed in a cavity. Solving eq (2.3) for x in terms of Tgives x = x(T), which is plotted in the left panel of figure 1 It shows that for T ≥ Tmin, x = x(T) is multivalued and consists of two branches, namely Small BH and Large BH. To study the thermodynamic stability of the two branches against thermal fluctuations in an isobaric process, we consider the heat capacity at constant pressure, CP = CP /P = T. which is presented in the left panel of figure 2. Which is presented in the left panel of figure 2 It shows that Small (Large) BH is thermally unstable (stable). Similar to a Schwarzschild-AdS black hole, a first-order Hawking-Page-like phase transition between the thermal spacetime and Large BH occurs at T = Tp, where these two phases are of equal Gibbs free energy. In the limit of large V T 3, the equation of state becomes

T3 V T3

Phase structure of a RN black hole in a cavity

Smarr relation

Discussion and conclusion