Extended Phase Space Thermodynamics of Black Holes in Massive Gravity

TL;DR

This paper explores the thermodynamic phase transitions of black holes in massive gravity within an extended phase space framework, revealing second order phase transitions analogous to liquid-gas systems.

Contribution

It introduces the analysis of black hole critical behavior in massive gravity using extended phase space, deriving equations of state and confirming second order phase transitions.

Findings

Black holes exhibit second order phase transitions at the critical point.

The critical ratio of black holes matches that of liquid-gas systems.

Ehrenfest equations are satisfied, indicating a second order phase transition.

Abstract

We study the extended phase space thermodynamics of black holes in massive gravity. Particularly, we examine the critical behaviour of this black hole using the extended phase space formalism. Extended phase space in a sense that in which the cosmological constant should be treated as a thermodynamic pressure and its conjugate variable as a thermodynamic volume. In this phase space, we derive the black hole equation of state, the critical pressure, the critical volume and the critical temperature at the critical point. We also derive the critical ratio of this black hole. Moreover, we derive the black hole reduced equation of state in terms of the reduced pressure, the reduced volume and the reduced temperature. Furthermore, we examine the Ehrenfest equations of black holes in massive gravity in the extended phase space at the critical point. We show that the Ehrenfest equations are satisfied of this black hole and the black hole encounters a second order phase transition at the critical point in the said phase space. This is re-examined by evaluating the Pregogine-Defay ratio~($\varPi$). We determine the value of this ratio is $\varPi=1$. The outcome of this study is completely analogous to the nature of liquid-gas phase transition at the critical point. This investigation also further gives us the profound understanding between the black hole of massive gravity with the liquid-gas system.