# A class of black holes in dRGT massive gravity and their thermodynamical properties

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## Abstract

We present an exact spherical black hole solution in de Rham, Gabadadze, and Tolley (dRGT) massive gravity for a generic choice of the parameters in the theory, and also discuss the thermodynamical and phase structure of the black hole in both the grand canonical and the canonical ensembles (for the charged case). It turns out that the dRGT black hole solution includes other known solutions to the Einstein field equations, such as the monopole-de Sitter–Schwarzschild solution with the coefficients of the third and fourth terms in the potential and the graviton mass in massive gravity naturally generates the cosmological constant and the global monopole term. Furthermore, we compute the mass, temperature and entropy of the dRGT black hole, and also perform thermodynamical stability analysis. It turns out that the presence of the graviton mass completely changes the black hole thermodynamics, and it can provide the Hawking–Page phase transition which also occurs for the charged black holes. Interestingly, the entropy of a black hole is barely affected and still obeys the standard area law. In particular, our results, in the limit \(m_g \rightarrow 0\), reduced exactly to the results of general relativity.

## Keywords

Black Hole Black Hole Solution Canonical Ensemble Massive Gravity Grand Canonical Ensemble## 1 Introduction

The question of whether a mass term for the graviton field can be introduced existed in Einstein’s theory of general relativity since its inception. Massive gravity came into existence as a straightforward modification of general relativity by providing consistent interaction terms which are interpreted as a graviton mass. Such a theory can describe our Universe, which is currently undergoing accelerating expansion without introducing a bare cosmological constant. Massive gravity modifies gravity by weakening it at a large scale compared with general relativity, which allows the Universe to accelerate, while its predictions at small scales are the same as those in general relativity. Furthermore, if a solution exists in this theory, it may also elucidate the dark energy problem. Hence, in recent years there were numerous developments in the massive gravity theories [1, 2, 3, 4, 5, 6]. The first attempt was done, in 1939, by Fierz and Pauli [1]. They added the interaction terms at the linearized level of general relativity but later it was found that their theory suffered from a discontinuity in its predictions, which was pointed out by van Dam, Veltman, and Zakharov, the so-called van Dam–Veltman–Zakharov (vDVZ) discontinuity [2, 3, 4]. This discontinuity problem invoked further studies on the nonlinear generalization of Fierz–Pauli massive gravity. During the search for such generalizations, Vainshtein found that the origin of the vDVZ discontinuity is that the prediction made by the linearized theory cannot be trusted inside some characteristic “Vainshtein” radius and he also proposed the mechanism for the nonlinear massive gravity which can be used to recover the predictions made by general relativity [5]. At the same time, Boulware and Deser found that such nonlinear generalizations usually generate an equation of motion which has a higher derivative term yielding a ghost instability in the theory, later called a Boulware–Deser (BD) ghost [6]. However, these problems, arising in the construction of the massive gravity, have been resolved in the last decade by first introducing Stückelberg fields [7]. This permits a class of potential energies depending on the gravitational metric and an internal Minkowski metric. Furthermore, to avoid the reappearance of the ghost in massive gravity, the set of allowed mass terms was confined and furnished perturbatively by de Rham, Gabadadze, and Tolley (dRGT) [8, 9]. They summed these terms and found three possibilities, viz. quadratic, cubic, and quartic combinations of the mass terms. The dRGT massive gravity is constructed suitably so that the equations of motion have no higher derivative term, so that the ghost field is absent. However, these nonlinear terms lead to complexity in the calculations and hence, in general, finding exact solutions in this theory is strenuous. Nevertheless, recently, several interesting measures have been taken to obtain the spherically symmetric black holes in various massive gravities [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In particular, a spherically symmetric black hole solution with a Ricci flat horizon in four-dimensional massive gravity with a negative cosmological constant was obtained by Vegh [10] and was generalized to study the corresponding thermodynamical properties and phase transition structure [11, 12, 13]. The spherically symmetric solutions for dRGT were also addressed in [14, 15], the corresponding charged black hole solution was found in [16], and its bi-gravity extension was found in [17], which includes as particular cases the previously known spherically symmetric black hole solutions. (See [18, 19, 20], for reviews on black holes in massive gravity, see also [21] for the black hole solution in other classes of massive gravity.)

The main purpose of this paper is to present a new class of exact spherically symmetric black hole solutions including a generalization to the charged case in dRGT massive gravity, and also to discuss their thermodynamical properties. It turns out that the solution discussed in this paper represents a generalization of the Schwarzschild solution that includes most of the known dRGT black hole solutions. The paper is structured as follows. In the next section, we review dRGT massive gravity. We present the modified equations of motion for dRGT massive gravity and a class of exact black hole solutions in Sect. 3. The calculation of the thermodynamical quantities associated with the dRGT black hole solution and the study of the phase structure of a black hole in the canonical ensemble approach are the main subject of Sect. 4. The analyses in Sect. 4 are extended for the charged case in Sect. 5, and finally we summarize our results and evoke some perspectives to end the paper in Sect. 6. We have used units which fix the speed of light and the gravitational constant via \(8\pi G = c^4 = 1\).

## 2 dRGT massive gravity

*R*is the Ricci scalar and \(\mathcal{U}\) is a potential for the graviton which modifies the gravitational sector with the parameter \(m_g\) interpreted as graviton mass. Moreover, the action is written in units such that the Newtonian gravitational constant is unity (thus, \(\kappa ^2\equiv 8\pi \)). The effective potential \(\mathcal{U}\) in four-dimensional spacetime is given by

*g*and scalar fields \(\phi ^a\) are defined as

## 3 Black hole solution in dRGT massive gravity

The exact solutions for this ansatz are complicated and thus it is difficult to use these solutions to analyze the properties of black hole. Note that one may simplify the solution by choosing some specific relations of the parameters \(\alpha \) and \(\beta \) for example \(\alpha = - 3 \beta \) [16]. Furthermore, a class of parameters satisfying \(\beta =\frac{\alpha ^2}{3}\) simply yields the Schwarzschild–de Sitter solution [32] (see also the analyses of such solution in Refs. [33, 34, 35, 36]).

It is important to note that most solutions are asymptotically de Sitter or anti-de Sitter. This is not surprising since at large scale the theory should recover the cosmological solution in which the graviton mass will play the role of cosmological constant to drive the late-time acceleration of the Universe. However, a class of black hole solutions in dRGT massive gravity (or in other classes of massive gravity, e.g. the model in Ref. [37]) may encounter the issues of superluminality, the Cauchy problem, and strong coupling. (See [38] for the issues in dRGT and also [21, 39, 40, 41] for those in another model of massive gravity.)

*c*is a constant. With the choice of the fiducial metric, the action remains finite since it only contains non-negative powers of \(f_{\mu \nu }\) (see [10], for more details). It is important to note that the effective energy-momentum tensor in Eq. (9) is derived by requiring that the fiducial metric must be non-degenerate. From Eq. (12), it is obvious that the fiducial metric is degenerate and then one may not use the effective energy-momentum tensor expressed in Eq. (9). However, as pointed out in [42], one can use the Moore–Penrose pseudoinverse of the metric \(\mathcal{K}^\mu _\nu \) in order to find the effective energy-momentum tensor and it provides the same expression as in Eq. (9). Therefore, one can use the effective energy-momentum tensor defined in Eq. (9) for the form of the fiducial metric. Moreover, the results can be checked by using the mini-superspace action as usually done in cosmological analysis. One can obtain the equation of motion by using the Euler–Lagrangian equation for the variables

*n*and

*f*. As a result, we found that the equations of motion still valid.

*c*by which this effective energy-momentum tensor behaves like a cosmological constant. In particular, \(c=0\) simplifies each of the diagonal component of the effective energy-momentum tensor so that they depend only on the parameters of the theory, which are all equal constants. Actually, by setting \(c=0\) in Eq. (6), the tensor \(\mathcal {K}^\mu _\nu \) will be equal to the identity matrix leading to the fact that the mass terms are all constants at the Lagrangian level, which is corresponding to the cosmological-constant term. This is a crucially different point between our model and the one in Ref. [11]. In our model, the solution can be reduced to a Schwarzschild–AdS/dS solution where the cosmological constant-like term can be expressed in terms of the graviton mass, while the cosmological constant-like term in Ref. [11] is introduced by hand and is not related to the graviton mass.

*f*which can be written as

*M*is an integration constant related to the mass of the black hole. This solution incorporates the cosmological-constant term, namely \(\varLambda \), naturally in terms of the graviton mass \(m_g\), which should not be surprising since the graviton mass serves as the cosmological constant in the self-expanding cosmological solution in massive gravity. Moreover, this solution can be identified with the known solutions in general relativity. Note that the solutions in alternative forms with other sets of parameters are shown explicitly in Appendix A. In the case \(m_g=0\) we have the Schwarzschild solution, as expected. For \(c=0\), which sets \(\gamma =\zeta =0\), the solution can be classified according to the values of \(\alpha \) and \(\beta \). If \(\left( 1+\alpha +\beta \right) <0\), the solution is in the form of Schwarzschild–de Sitter, while the case \(\left( 1+\alpha +\beta \right) >0\) yields the Schwarzschild–Anti-de Sitter solution. Using Eqs. (20) and (21), one finds that

*f*and

*n*differ only by a constant. One can choose the constant to obtain a black hole solution such that

However, this solution highly depends on the choice of the fiducial metric; changing to other forms of the fiducial metric will significantly affect the solution. This kind of dependency is one of the important properties of massive gravity. For example, from a cosmological point of view, one cannot have a nontrivial flat cosmological solution with a Minkowski fiducial metric [49]; only the open FLRW solution is allowed [43], where the FLRW solution with arbitrary geometry exists when the FLRW fiducial metric is considered [44]. By generalizing the form of the fiducial metric, the nontrivial cosmological solutions can be obtained [45].

## 4 Thermodynamics of the black hole

*f*(

*r*) in the right panel of this figure. We note that the gravitational mass of a black hole is determined by \(f(r_+) = 0\), which in terms of the outer horizon radius \(r_+\) reads

*T*of the black hole becomes

*S*). The black hole is supposed to obey the first law of thermodynamics, or \(\mathrm{d}M = T\mathrm{d}S\). To calculate the entropy, we use

*transition*between two states; from the

*non-black hole*or

*hot flat space*state to a black hole. The transition was realized by Hawking and Page [52] (see also [53]), who found the characteristics of the so-called Hawking–Page phase transition. The transition exists if it is thermodynamically spontaneous, or alternatively, globally thermodynamically stable, as can be found by evaluating the free energies between two states. In other words, this corresponds to evaluating the Euclidean actions of those two states where the temperature is treated as a period of the imaginary time. Since in this case, there is no particle transfer, we compute the Helmholtz free energy

*M*from Eq. (29) and then substitute this expression into the stability conditions to find stability regions in \((\alpha , \beta )\) space where

*M*is held fixed to be a positive constant. From this figure, one can see that there exists an allowed region for the transition phase with the parameters \(\alpha , \beta \sim O(1)\). This suggests that massive gravity can naturally provide the Hawking–Page phase transition without requiring fine-tuning of the parameters. For the right panel of Fig. 4, we combine three important regions according to the previous plots including the regions satisfying thermodynamical stability conditions (intersection region in the left panel of Fig. 4), \(T_{+(\text {min})} > 0\) and \(r_+ > 0\) (intersection region in left panel of Fig. 2), and the existence of three horizons (region in the left panel of Fig. 1). From this figure, one can see that the thermodynamically stable region of the black hole together with positive temperature and horizon size is not compatible with the region indicating the existence of three horizons. This is due to the fact that the temperature is proportional to \(f'(r_+)\) and the condition of existence of three horizons is that there exist two extremum points such that \(f'(r) = 0\). This implies that there always exists a range of horizons (\(r_+\)) at which the black hole temperature is negative if there exist three real black hole horizons. Therefore, we use a set of parameters which give rise to one or two horizons to illustrate the thermodynamical quantities such as the temperature and the heat capacity.

*c*. This means that the greater the value of the horizon size, the smaller the value of the parameter

*c*. In the left panel of Fig. 5, we explicitly show the stability region in \((\alpha ,\beta )\)-space with different values of the parameter

*c*such that \(c=0.8, c=1.0\), and \( c=1.2\). Furthermore, we also show the allowed region in \((\alpha , c)\)-space by fixing \(\beta = 0.1\) in the right panel. From this figure, it is found that the greater value of the parameter

*c*corresponds to the smaller value of the horizon size and the smaller allowed region in the parameter space. Therefore, the phase transition tends to occur more easily at large horizon size. Note that, in this case, the black hole is treated as a canonical ensemble system where particle transfer is prohibited. We will discuss the charged black hole case in the next section.

## 5 Charged black hole

*A*(

*r*). By requiring the spherically symmetric solution as in Eq. (13), one finds the equation of motion of

*A*(

*r*) as

*k*is an integration constant. The corresponding electric field is

*k*, one may consider an electric field from a charge

*Q*at large

*r*where the spacetime is asymptotically flat which, in Gaussian units, must take the form

*k*must be identified to the charge

*Q*. By solving Eqs. (43) and (44), one finds a charged dRGT black hole solution as

### 5.1 Grand canonical ensemble

### 5.2 Canonical ensemble

*Q*. The mass and temperature are given readily by Eqs. (51) and (52), respectively. Furthermore, the corresponding entropy still obeys the area law,

*Q*rather than the potential \(\mu \) as in the grand canonical ensemble case, as shown in the left panel of Fig. 9. Again, from the allowed region in this figure, the theory can provide the Hawking–Page phase transition naturally. From the right panel of this figure, it is also found that the existence of the charge makes the allowed region smaller, similar to the grand canonical case. This is implied by Eq. (52) where the region for which \(T > 0\) is reduced by the presence of charge. Similarly, the correspondence between the minimal temperature and the divergence of the heat capacity can be seen in Fig. 10.

The effect of charge on the stability regions of both the grand canonical ensemble and the canonical ensemble compared with the non-charged case are shown in Fig. 11. From this figure, one can see that the effect of the chemical potential \(\mu \) in the grand canonical ensemble and the charge *Q* in the canonical ensemble decreases the size of the allowed region of the parameters. This can be seen from Eqs. (52) and (53), since the contribution from the charge and the chemical potential makes the positive temperature region smaller. In the canonical ensemble, we also found that the parameter region that satisfies the conditions for the existence of four horizons is not compatible with the stability region, similar to the non-charged case. The effect of the parameter *c* on the region of stability is also similar to the non-charged case. The stability region will increase where the horizon size increases or the parameter *c* decreases.

## 6 Concluding remarks

The dRGT massive gravity is a natural extension of Einstein’s theory of general relativity, providing mass to the graviton, and it is a great arena for theoretical physics research. The dRGT massive gravity describes nonlinear interaction terms as a correction of the Einstein–Hilbert action and hence admits general relativity as a particular case. It is believed that dRGT massive gravity may provide a possible explanation for the accelerated expansion of the Universe that does not require any dark energy or cosmological constant. Hence, dRGT massive gravity has received significant attention [54] including searches for black holes [18]. In this paper, we have obtained a class of black hole solutions in dRGT massive gravity, and we studied the thermodynamics and phase structure of the black hole solutions. In the dRGT massive gravity outlined in this paper, there are three terms in the effective potential associated with the graviton mass. Furthermore, we note that due to the inclusion of the massive gravity term in the action, the Schwarzschild solution in general relativity is modified. Interestingly, it turns out that solutions to the Einstein field equations, such as the monopole-de Sitter–Schwarzschild model, become solutions in dRGT massive gravity for suitable choices of the parameters of the theory, where the coefficients for the third and fourth terms in the potential and the graviton mass in massive gravity naturally generate the cosmological constant and the global monopole term. The corresponding thermodynamical quantities are also changed. However, the black hole entropy is not affected significantly by the existence of the graviton mass, and it still obeys the standard area law as in general relativity. Moreover, we consider the charged black hole solution in both the grand canonical and the canonical ensembles to analyze the thermodynamics and phase transition. We have demonstrated through the calculation of the heat capacity and the free energy that there is a critical point where the heat capacity diverges and a phase transition is possible without requiring fine-tuning of the parameters as shown in Fig. 4. Even though it is possible to obtain three horizons in some region of parameter space, the region is still not compatible with the stability region. This implies that the phase transition will not occur when the black hole has three horizons. The presence of the charge will not change this argument, the phase transition will not occur when the black hole has four horizons. The presence of the charge affects the appearance of the Hawking–Page phase transition such that the allowed parameter region decreases as shown in Fig. 11. We also found that the phase transition tends to occur in the large horizon size in both the charged and the non-charged cases.

The black hole solutions obtained are immensely simplified due to the choice of the fiducial metric as \(f_{\mu \nu }= \text {diag}(0,0,c^2 ,c^2 \sin ^2\theta )\) and the choice of the Stückelberg scalars. It will be interesting to apply the technique discussed here in other massive gravities to get black holes. It will also be interesting to consider the motion of particles in the background of the dRGT massive black holes considered and to see how the graviton mass affects the equations of motion. These and related areas are for future investigation.

## Notes

### Acknowledgments

P. W. and L. T. are supported by the Thailand Toray Science Foundation (TTSF) science and technology research grant. P.W is also supported by the Naresuan University Research Fund through Grant No. R2557C083. S. G. G. would like to thank SERB-DST, government of India for Research Project Grant No. SB/S2/HEP-008/2014. Moreover, this project is partially supported by the ICTP through Grant No. OEA-NET-76. Furthermore, we would like to thank the Institute for Fundamental Study (IF), Naresuan University for hospitalities during the process of this work. Last but not least, we would like to thank Matthew James Lake for reading through the manuscript and correcting some grammatical errors.

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