Phase transitions and geothermodynamics of black holes in dRGT massive gravity
Abstract
In this paper, we study the thermodynamics and geothermodynamics of spherical black hole solutions in dRGT massive gravity in a new extended phase space. Inspire by the work of Kastor et al. (Class Quantum Gravity 26:195011, 2009), by interpreting the graviton mass as a thermodynamical variable, we propose a first law of thermodynamics which include a mass term and establish a new Smarr Formula. Then, we perform a thermodynamical analysis to reveal the existence of a critical behavior for black holes in dRGT massive gravity with two different critical points through canonical and grand canonical ensembles. To consolidate these results, we make use of the thermodynamical geometry formalism, with the HPEM and the Gibbs free energy metrics, to derive the singularities of Ricci scalar curvatures and show that they coincide with those of the capacities. The effect of different values of the spacetime parameters on the stability conditions is also discussed.
1 Introduction
In Einstein’ theory of general relativity, the graviton is a massless particle. One might raise the natural and legitimate question of whether a selfconsistent gravity theory with massive graviton is achievable. A massive gravity came into existence as a straightforward modification of the general relativity by considering consistent interaction terms that can be interpreted as a graviton mass.
Massive gravity theories have become increasingly popular in the current literature. The first construction of a linear theory of massive gravity with a ghostfree theory of noninteracting massive gravitons goes back to Fierz and Pauli’ work (FP) [2]. By adding interaction terms in the linearized level of general relativity, FP showed that the mass term must be of the form \( m_{g}^{2}(h^{2}h_{\mu \nu }h^{\mu \nu })\). Unfortunately, the theory of Fierz and Pauli suffered from a discontinuity problem in the zero mass limit and which prevent recovery of the general relativity results at the low energy scales. This is called van Dam–Veltman–Zakharov (vDVZ) discontinuity [3, 4].
In addition, a ghost propagating degree of freedom, dubbed Boulware–Deser ghost (BD) [5, 6], shows up at the nonlinear level in a generic FP theory. This issue has been resolved recently by de Rham, Gabadadze and Tolley in the framework of dRGT massive gravity [7], who found a twoparameter family of nonlinear theories free from the BD ghost orderbyorder and to all orders, at least in the decoupling limit. Then, de Rham et al. proposed a candidate theory of massive gravity free of BD ghost [8]. The latter is automatically ghost free to all orders in the decoupling limit with the Hamiltonian constraint maintained at least up to quartic order away from the decoupling limit. Later, a generalization to a ghost free nonlinear massive gravity action for all orders was proposed in [9]. Until now, a fourdimensional black hole solution with a Ricci flat horizon has been constructed in [10]. The exact spherical black hole solution in de dRGT massive gravity used in this paper has been obtained in [11].
The graviton mass in massive gravity naturally generates the cosmological constant and the global monopole term. Hence such theory can provide the solution for describing our universe which is currently expanding with acceleration without introducing any dark energy or cosmological constant [12, 13]. Current experimental observations put constraints on the mass of the graviton \(m_{g}\), particularly from the recent detection of gravitational waves by advanced LIGO, which provided an upper limit : \(m_{g}<1.2\times 10^{22}\, \mathrm{eV/c}^{2}\) [13].
Recently, the study of phase transitions of black holes in asymptotically Anti deSitter spacetime has aroused growing interest [14, 15, 16, 17] since these transitions have been related with holographic superconductivity in the context of the AdS/CFT correspondence [18, 19, 20, 21, 22]. In this respect, many attempts have been made to establish relationship between the thermodynamic phase transitions and quasinormal modes of black holes [23, 24, 25]. The phase transitions are usually signaled by the existence of discontinuities of a state space variable, via either the analysis of the heat capacity [16] or by using the geothermodynamics approaches [26, 27, 28, 29, 30] proposed by Ruppeiner [31], Quevedo [32], HPEM [33] or by Liu–Lu–Luo–Shao [28].
Hence, our aim in this paper is to analyze the extended phase structure and investigate the critical behavior of both neutral and charged black holes in dRGT massive gravity by using the geometrical studies.
This paper is organized as follows: In the next section, we present a brief review of the thermodynamics of dRGT massive gravity black holes. In Sect. 3, by considering the graviton mass as a thermodynamic variable, we explore the critical behavior and the phase structure for neutral black holes, either in the canonical ensemble by fixing the graviton mass and in the grand canonical ensemble where the graviton potential is a constant. In Sect. 4, we extend the analyses of Sect. 3 to the charged black holes. In Sect. 5, we investigate the criticality from the geothermodynamical point of view with the HPEM and the Gibbs free energy metrics. The last section is devoted to our conclusion.
2 Massive gravity background and thermodynamics
2.1 General background
2.2 First law and Smarr formula
Kastor et al. have interpreted \(\Lambda \) as a thermodynamic variable and defined thermodynamically conjugate variable as the thermodynamic volume [1]. Here, we extend this idea by treating the graviton’s mass as a thermodynamic variable while its conjugate is identified with the graviton potential. Then, we perform a study of the thermodynamics and geothermodynamics of the neutral and charged black hole in dRGT massive gravity background.
3 Thermodynamic behaviour of neutral black holes
In this section, we study the thermodynamic phase transitions by considering the black hole either as a closed system (canonical ensemble), or as an open system (grand canonical ensemble).
3.1 The canonical ensemble
3.2 The grand canonical ensemble
4 Thermodynamic behaviour of charged black holes
In this section we aim to discuss the thermodynamic behaviour of charged black hole in the canonical ensemble as well as in the grand canonical ensemble.
4.1 The canonical ensemble
In Fig. 5, we plot T versus S and the function G as a function of the temperature for different values of \(m_g\), it can be seen that the characteristic “swallow tail” behaviour of \(GT\) curve appears for the first order phase transition where \(m_g\) satisfying \(m_g<m_{g_{c}}\) and \(m_g>m_{g_{c+}}\). However, within the range \(m_{g_{c}}<m_g<m_{g_{c+}}\) the swallow tail characteristic disappears. At \(m_g=m_{g_{c+}}\) and \(m_g=m_{g_{c}}\), we have the second order phase transition.
4.2 The grand canonical ensemble
Besides, we also analyse the behaviour of the heat capacity \(C_{\Phi ,\mu }\) as a function of the entropy S at constant \(\mu \) and \(\Phi \) in Fig. 8. From the left panel, we see that \(C_{\Phi ,\mu }\) is negative for small black holes (\(S<S_1\)) and as well as for large black holes (\(S>S_2\)), while the heat capacity turns positive when \(S_{1}<S<S_{2}\). Hence, one can conclude that the heat capacity diverges at the critical points \(S_1\) and \(S_2\), revealing a first order phase transition, whereas the black hole undergoes a second order phase transition at \(S_c\), as illustrated by the right panel.
5 Geometrical thermodynamics in dRGT massive gravity
In this section, we focus on the thermodynamical geometry of the black hole in dRGT massive gravity. Our objective is to check whether the thermodynamical curvature encodes the singularities of the heat capacities and consequently the critical behaviour of the black hole. To this end, two different geothermodynamics approaches are used: the first one, dubbed HPEM, has been proposed in [33] and the second one is the free energy metric introduced in [28].
5.1 HPEM metric
5.2 The freeenergy metric
6 Conclusion

\(m_g=m_{g_{c}}\) and \(mg=m_{g_{c+}}\): the black hole undergoes a second order phase transition;

\(m_g<m_{g_{c}}\) and \(mg>m_{g_{c+}}\): the first order phase transition shows up;

\(m_{g_{c}}<m_g<m_{g_{c+}}\), no phase transition.
At last, we would like to mention that a variety of methods exist in literature to study black hole thermodynamics geometry. As example, the main features of Ruppeiner and Quevedo metrics are briefly presented in the “Appendix”.
Footnotes
 1.
\(T=\frac{\kappa }{2\pi }=\left. \frac{f'\left( r\right) }{4\pi }\right _{r=r_{h}}=\frac{S m_g^2 (\pi c^2 \alpha '2 \sqrt{\pi } c \sqrt{S} \beta '+3 S \gamma ')+\pi \left( S\pi Q^2\right) }{4 \pi ^{3/2} S^{3/2}}\) [see S.G. Ghosh, S.D. Maharaj, PRD 89, 084027 (2014)].
 2.
Note here that if c is introduced as thermodynamical variable, then the Smarr formula becomes, \(M=2 T S+Q \Phi  \mu m_g +2\left( 1\lambda \right) \mu _{\alpha ^\prime } \alpha ^\prime + \left( 1\lambda \right) \mu _{\beta ^\prime } \beta ^\prime + \lambda \mu _{c} c\), where \(\lambda \) is an arbitrary constant. We can thus see that the contribution of \(\alpha '\) and \(\beta '\) are lost when \(\lambda =1\), a scenario that we would like to avoid.
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