Ruppeiner geometry and thermodynamic phase transition of the black hole in massive gravity

Bin Wu,Chao Wang,Zhen-Ming Xu,Wen-Li Yang

doi:10.1140/epjc/s10052-021-09407-y

Abstract

The phase transition and thermodynamic geometry of a 4-dimensional AdS topological charged black hole in de Rham, Gabadadze and Tolley (dRGT) massive gravity have been studied. After introducing a normalized thermodynamic scalar curvature, it is speculated that its value is related to the interaction between the underlying black hole molecules if the black hole molecules exist. We show that there does exist a crucial parameter given in terms of the topology, charge, and massive parameters of the black hole, which characterizes the thermodynamic properties of the black hole. It is found that when the parameter is positive, the singlet large black hole phase does not exist for sufficient low temperature and there is a weak repulsive interaction dominating for the small black hole which is similar to the Reissner–Nordström AdS black hole; when the parameter is negative, an additional phase region describing large black holes also implies a dominant repulsive interaction. These constitute the distinguishable features of dRGT massive topological black hole from those of the Reissner–Nordström AdS black hole as well as the Van der Waals fluid system.

Highlights

As the combined parameter x decreases, a smaller change in thermodynamical volume at a fixed phase transition temperature would cause the transition from singlet small black hole phase to singlet large black hole phase
We studied the phase transition and thermodynamic geometry of a four-dimensional charged topological black hole in massive gravity
We find that there does exist a combined parameter in terms of the topology, charge, and massive parameter of the massive black hole, which characterizes the thermodynamic properties of the black hole

Summary

Introduction

One of the most prominent features that the thermodynamical behaviors of dRGT massive gravity have provided is that the so-called small/large black hole phase transition for the charged AdS black holes always exist, no matter the horizon topology is spherical (k = 1), Ricci flat (k = 0) or hyperbolic (k = −1) [79,80]. In the literature [88], the authors find that the singularities of scalar curvatures which are constructed from HPEM metric and the Gibbs free energy metrics coincide with the critical point of phase transition for the spherical black hole in dRGT massive gravity Motivated by these interesting results, in this paper, our aim is to investigate the Ruppeiner geometry with normalized scalar curvature proposed in [60] and the phase transition of topological black holes in dRGT massive gravity.

Ruppeiner thermodynamic geometry

Thermodynamics of dRGT massive gravity

Phase transition of charged dRGT black hole

Ruppenier geometry of the black hole

Summary and discussion