Thermodynamics of black holes in massive gravity

Abstract

We present a class of charged black hole solutions in an (n + 2)-dimensional massive gravity with a negative cosmological constant, and study thermodynamics and phase structure of the black hole solutions both in grand canonical ensemble and canonical ensemble. The black hole horizon can have a positive, zero or negative constant curvature characterized by constant k. By using Hamiltonian approach, we obtain conserved charges of the solutions and find black hole entropy still obeys the area formula and the gravitational field equation at the black hole horizon can be cast into the first law form of black hole thermodynamics. In grand canonical ensemble, we find that thermodynamics and phase structure depends on the combination k − µ 2 /4 + c 2 m 2 in the four dimensional case, where µ is the chemical potential and c 2 m 2 is the coefficient of the second term in the potential associated with graviton mass. When it is positive, the Hawking-Page phase transition can happen, while as it is negative, the black hole is always thermodynamically stable with a positive capacity. In canonical ensemble, the combination turns out to be k + c 2 m 2 in the four dimensional case. When it is positive, a first order phase transition can happen between small and large black holes if the charge is less than its critical one. * E-mail: cairg@itp.ac.cn † E-mail: huyp@nuaa.edu.cn ‡ E-mail: panqiyuan@126.com § E-mail: ylzhang@itp.ac.cn 1 arXiv:1409.2369v1 [hep-th] Sep 2014In higher dimensional (n + 2 ≥ 5) case, even when the charge is absent, the small/large black hole phase transition can also appear, the coefficients for the third (c 3 m 2 ) and/or the fourth (c 4 m 2 ) terms in the potential associated with graviton mass in the massive gravity can play the same role as the charge does in the four dimensional case.

Massive charged BTZ black holes in asymptotically (a)dS spacetimes

“…Charged black hole solutions in the presence of this massive gravity have been studied in ref. [61]. Also, the existence of Van der Waals like behavior in extended phase space has been addressed in ref.…”

“…In order to obtain exact solutions, we should make a choice for the reference metric. We consider the following ansatz metric [61] …”

P−Vcriticality in the extended phase space of black holes in massive gravity

“…More recently, a nontrivial black hole solution was found in ghost-free massive gravity with a negative cosmological constant [29]. These black holes' corresponding thermodynamical properties and phase structure have been studied by [30]. The aim of this paper is to analyze the extended phase structure and investigate the P -V critical behavior of charged AdS black holes in canonical ensemble in that ghost-free massive gravity.…”

Influence of inhomogeneities on holographic mutual information and butterfly effect

“…Therefore, from this aspect the graviton mass plays the role of inhomogeneity in the boundary field theory, i.e., greater graviton mass is dual to greater inhomogeneity in the boundary. 1 There have been a number of papers investigating the inhomogeneous effects caused by massive gravity so far, see for example [35][36][37][38][39][40][41][42][43].…”

“…It is of great interest to investigate the effect of inhomogeneity on the holographic mutual information, in particular, we are going to explore the effects of the graviton mass on the mutual information with and without energy perturbations in the following context. The action of the massive gravity is as follows [32,35] …”