# Non-linear charged AdS black hole in massive gravity

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## Abstract

In the context of the non-linear electrodynamics and the Einstein-massive gravity, we have obtained a \(4\hbox {D}\) non-linear charged AdS black hole solution. Then, we investigated its horizon structure. In addition, the thermodynamics and phase structure of this black hole solution have been studied in details. We have computed various thermodynamic quantities of the black hole, such as the temperature, entropy, the heat capacity at constant pressure, or the Gibbs free energy. The black hole can undergo the first-order, second-order phase transitions which depend crucially on the effective horizon curvature, the sign of the coupling parameter \(c_1\), the characteristic parameter of the non-linear electrodynamics, as well as the pressure. Finally, we derived the equation of state and studied \(P-V\) criticality in the case of the positive effective horizon curvature.

## 1 Introduction

Black holes are of the most important objects in the physics, which exhibit many interestingly physical consequences. Bekenstein and Hawking showed that the black holes are the thermodynamic systems whose entropy and temperature are determined by the area of the event horizon and the surface gravity at the event horizon, respectively [1, 2, 3, 4, 5]. Since their pioneering works, the thermodynamics of the black holes have been studied extensively. In 1983, Hawking and Page discovered a phase transition between Schwarzschild anti-de Sitter (AdS) black hole and thermal gas or thermal AdS space, well-known as the Hawking–Page transition in the literature [6]. According to AdS/CFT correspondence, the Hawking–Page transition can be explained as the gravitational dual of the confinement/deconfinement phase transition [7, 8, 9, 10]. The AdS/CFT correspondence thus has motivated the study of the black holes and their thermodynamics in the AdS spacetime. Chamblin et al. found that the Reissner–Nordström AdS (RN-AdS) black hole can undergo a first-order phase transition which is similar to the liquid–gas phase transition [11, 12]. Recently, the cosmological constant was considered as the thermodynamic pressure associated with its conjugate variable which is the thermodynamic volume [13, 14, 15, 16, 17]. In the extended phase space, the black hole mass is most naturally the enthalpy, rather than the internal energy. Accordingly, many interesting phenomena of the black hole thermodynamics have been found, such as \(P-V\) criticality [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40], multiply reentrant phase transitions [41, 42, 43, 44], the black holes as heat engines [45, 46, 47, 48, 49, 50, 51, 52, 53, 54], or the Joule-Thomson expansion of the black holes [55, 56, 57, 58, 59, 60].

The black hole solutions in general relativity (GR) have a central singularity surrounded by the event horizon [61]. It is widely believed that the black hole singularity does not exist in nature but shows the limitations of GR. On the other hand, it would be removed by a more fundamental theory of the gravitation, e.g. quantum gravity. However, a complete theory of quantum gravity has so far not been achieved. Thus, how to avoid the black hole singularity at the semi-classical level still gets many attentions. The regular black hole was first proposed by Bardeen [62], however, at that time what is the source leading to this regular black hole was not realized. In 2000, Ayon-Beato and Garcia reobtained the Bardeen black hole as a gravitational collapse of some magnetic monopole, in the scenario of Einstein gravity coupled to the non-linear electrodynamics [63]. Because the non-linear electrodynamics can lead to the regular black hole solutions, it has received many recent studies [19, 33, 39, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85].

With the progress in the understanding of the massive gravity, various black hole solutions and their thermodynamics have been extensively studied in the presence of the graviton mass [32, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109]. In this paper, inspired by the non-linear electrodynamics, we would like to study the charged AdS black hole with the non-linear source in Einstein-dRGT gravity.

This paper is organized as follows. In Sect. 2, we introduce a system of Einstein-dRGT gravity coupled to the non-linear electromagnetic field in the four-dimensional AdS spacetime. Then, we derive a spherically-symmetric and static black hole solution of carrying magnetic charge and investigate the horizon properties of this black hole solution in details. In Sect. 3, we calculate the thermodynamic quantities and study the thermodynamic stability, phase transitions and \(P-V\) criticality. Finally, we make conclusions in the last section, Sect. 4. Note that, in this paper, we use units in \(G_N=\hbar =c=k_B=1\) and the signature of the metric \((-,+,+,+)\).

## 2 The black hole solution

*R*is the scalar curvature,

*l*is the curvature radius of the background AdS spacetime,

*m*is the graviton mass, \(c_i\) are the coupling parameters,

*f*is a fixed symmetric tensor usually called the reference metric, \(\mathcal {U}_i\) are symmetric polynomials in terms of the eigenvalues of the \(4\times 4\) matrix \({\mathcal {K}^\mu }_\nu =\sqrt{g^{\mu \lambda }f_{\lambda \nu }}\) given as

*k*is a fixed characteristic parameter of the non-linear electrodynamics by which the charge

*Q*and the mass

*M*of the system are related as, \(Q^2=Mk\). Indeed, one can see that the Lagrangian (4) provides a suitable modification of the Maxwell or linear electrodynamics. As the characteristic parameter of the non-linear electrodynamics goes to zero, \(k\rightarrow 0\), the Lagrangian (4) should clearly approach the Maxwell one. Also, in the weak field region \(|Q^2F|\ll 1\), we can expand the Lagrangian (4) as

Before proceeding, we stop here to more clarify the system described by the action (2) and indicate its suggestions. In this action, apart from considering the UV modification of the matter, we consider the modification of the gravity at the large distances or IR regime. Interestingly, at the short distances or UV regime, the gravity may obtain the corrections arising from a quantum theory of the gravity such as string theory. One of such UV modified gravity theories is the *f*(*R*) gravity where *f*(*R*) is a function of the scalar curvature *R* (see Ref. [110] for a review). Considering the modification of the gravity in both UV and IR regimes, corresponding to the *f*(*R*) nonlinear massive gravity, and its cosmological implications are studied in Refs. [111, 112]. In addition, in this work, the reference metric *f* is treated as a non-dynamical object. However, in general the reference metric *f* may be a dynamical object, and consequently we have a non-linear bimetric theory which consists of a massless spin-2 field coupling to a massive spin-2 field [113]. Therefore, it is significant to find the non-linear charged black hole solutions in the massive gravity with including the *f*(*R*) correction as well as considering the reference metric *f* to be a dynamical object. Also, an extension of the non-linear electrodynamics into the cosmology of the massive gravity [114, 115, 116, 117, 118, 119] is expected to lead to interesting implications. All of these points will be studied in our future works.^{1}

*c*is a positive constant. With given spacetime and reference metrics, one can easily write explicitly the massive gravity term as

*M*and magnetic charge

*Q*, with the ansatz of the magnetic field as

*Q*is defined by

*t*,

*t*) component of Eq. (9) reads

*M*

*m*(

*r*) into

*f*(

*r*), we finally get

*D*Schwarzschild-AdS black hole in the massive gravity. This is because, for \(k\rightarrow 0\), the squared charge \(Q^2\) of the black hole is extremely small compared to its mass

*M*and thus the black hole could be considered to be electrically neutral. In the limit of the large distances \(k/r\ll 1\), we have the following approximation

*D*RN-AdS black hole in the massive gravity [97, 98, 99, 100].

*R*, \(R_{\mu \nu }R^{\mu \nu }\), and \(R_{\mu \nu \rho \lambda }R^{\mu \nu \rho \lambda }\) given as

*r*approaches zero we have

*M*and its horizon radius \(r_H\), as

*M*approaches infinite as \(r_H\rightarrow 0\) or \(r_H\rightarrow \infty \), and

*M*is always larger than zero with the proper coupling parameter \(c_1\). This suggests that there actually exists the extremal black hole with the critical mass \(M_e\) below which there has no black hole. The extremal horizon radius \(r_e\) is a positive real root of the following equation

This figure shows that the extremal radius and mass both should increase when the characteristic parameter *k* of the non-linear electrodynamics increases. And, for the curvature radius *l* of the AdS spacetime decreasing, the extremal radius should decrease, whereas the extremal mass should increase. The black hole with the mass \(M>M_e\) has two different horizons corresponding to the inner horizon and the event horizon, as shown in the left panel of Fig. 3.

*M*approaches \(-\infty \) and \(+\infty \) as \(r_H\rightarrow 0\) and \(r_H\rightarrow \infty \), respectively. This means that there exits no the extremal black hole. But, the horizon radius of the black hole is bounded from below by

*k*of the non-linear electrodynamics and the coupling parameter \(c_2\). Also, with the proper parameters the black hole has possibly many inner horizons or does not have any inner horizon, as shown in Fig. 5. For the negative effective horizon curvature, \(1+m^2c^2c_2<0\), it is qualitatively the same as that of \(1+m^2c^2c_2=0\) and \(c_1<0\). However, in this case, the horizon radius of the black hole is bounded from below by

which is independent on the characteristic parameter *k* of the non-linear electrodynamics.

## 3 Thermodynamics and phase transitions

*k*, and the pressure.

*k*of the non-linear electrodynamics, with the parameters of the massive gravity fixed, as

*P*from (35), with parameters of the massive gravity and non-linear electrodynamics fixed, as

- 1.
If \(c_1>0\), the behavior of the isobar curves is qualitatively the same as that with \(1+m^2c^2c_2>0\) and \(P\ge P_c\) (or \(k\ge k_c\)).

- 2.If \(c_1<0\), the isobar curves should first decrease until a minimum and then increase when \(r_+\) increasing, where the minimum radius \(r_{\text {min}}\) and the minimum temperature \(T_{\text {min}}\) are given as$$\begin{aligned} r_{\text {min}}= & {} \frac{1}{4}\sqrt{-\frac{m^2cc_1k}{2\pi P}}, \end{aligned}$$(38)with the conditions \(r_{\text {min}}>r_{b0}\) and \(T_{\text {min}}>0\).$$\begin{aligned} T_{\text {min}}= & {} -\frac{kP}{3}+\frac{m^2cc_1}{4\pi } +\sqrt{-\frac{m^2cc_1kP}{2\pi }}, \end{aligned}$$(39)
- 3.
If \(c_1<0\) and \(r_{\text {min}}<r_{b0}\), the isobar curves always increase in the increasing of the event horizon radius \(r_+\).

- 4.
If \(c_1<0\), \(r_{\text {min}}>r_{b0}\), and \(T_{\text {min}}<0\), there are two temperature regions which are separated by a negative temperature region. (Note that, there has no the black hole with the event horizon radius corresponding to the negative temperature region). The first and second regions are the decreasingly and increasingly monotonous functions of the event horizon radius \(r_+\), respectively.

*V*and the conjugating quantities \(\mathcal {C}_{1,2}\) as

- 1.
If \(c_1>0\), the heat capacity is always positive, thus the black hole is thermodynamically stable.

- 2.
If \(c_1<0\), \(r_{\text {min}}>r_{b0}\), and \(T_{\text {min}}>0\), the small black holes of \(r_+<r_{\text {min}}\) are thermodynamically unstable because of the negative heat capacity. Whereas, because of the positive heat capacity, the large black holes of \(r_+>r_{\text {min}}\) are thermodynamically stable. In particular, there occurs a second-order phase transition at \(r_{\text {min}}\) between the unstable small and stable large black holes.

- 3.
If \(c_1<0\) and \(r_{\text {min}}<r_{b0}\), it is the same as the case of \(c_1>0\).

- 4.
If \(c_1<0\), \(r_{\text {min}}>r_{b0}\), and \(T_{\text {min}}<0\), there have two different phases corresponding to the unstable small and stable large black holes. However, unlike the case 2, there has no phase transition between these two phases because of the presence of the forbidden region separating these two phases. In other words, if the black hole exits in one of these two phases, then it always exits in that phase and is impossible to undergo a phase transition to another phase.

The numerical results for the radius \(r_{HP}\) and the temperature \(T_{HP}\) at the Hawking–Page phase transition, for the different values of the characteristic parameter *k* of the non-linear electrodynamics and the pressure, at \(m=c=c_1=c_2=1\)

\(k=0.1\) | \(k=0.15\) | \(k=0.2\) | \(k=0.25\) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

| \(r_{HP}\) | \(T_{HP}\) | | \(r_{HP}\) | \(T_{HP}\) | | \(r_{HP}\) | \(T_{HP}\) | | \(r_{HP}\) | \(T_{HP}\) |

0.3 | 0.8283 | 0.7447 | 0.1 | 1.4424 | 0.4656 | 0.01 | 4.7074 | 0.2053 | 0.01 | 4.6598 | 0.2041 |

0.5 | 0.6289 | 0.9216 | 0.2 | 0.9948 | 0.6124 | 0.05 | 2.0389 | 0.3524 | 0.04 | 2.2548 | 0.3211 |

0.7 | 0.5234 | 1.0602 | 0.3 | 0.7976 | 0.7202 | 0.1 | 1.4078 | 0.4567 | 0.08 | 1.5529 | 0.4124 |

0.9 | 0.4558 | 1.1765 | 0.4 | 0.6809 | 0.8079 | 0.2 | 0.9647 | 0.5959 | 0.12 | 1.2442 | 0.4792 |

1.1353 | 0.4080 | 1.3054 | 0.4950 | 0.6052 | 0.8795 | 0.2732 | 0.8122 | 0.6720 | 0.1715 | 1.0219 | 0.5476 |

2 | 0.2931 | 1.6288 | 1 | 0.4106 | 1.1603 | 0.4 | 0.6578 | 0.7782 | 0.2 | 0.9386 | 0.5800 |

*G*is defined as

For the positive effective horizon curvature, if \(P<P_c\) and \(k<k_c\) the Gibbs free energy exhibits a swallowtail structure which suggests the occurrence of a first-order phase transition between small and large black holes. This first-order phase transition should disappear when \(P\ge P_c\) or \(k\ge k_c\). In addition, we can point out the occurrence of the Hawking–Page phase transition between the thermal radiation and the black hole, at which the Gibbs free energy vanishes [6]. The radius \(r_{HP}\) and the temperature \(T_{HP}\), corresponding to the Hawking–Page phase transition, are numerically given in Table 1.

*k*of the non-linear electrodynamics and the parameters of the massive gravity. Note that, \(v_c=2r_c-k/3\) is critical specific volume. Interestingly, in the case of the coupling parameter \(c_1=0\), the universal constant becomes

*k*of the non-linear electrodynamics approaches zero. In Fig. 52, we plot the universal constant with respect to the coupling parameter \(c_1\) or \(c_2\), under the different values of the characteristic parameter

*k*of the non-linear electrodynamics.

This figure shows that, for fixed but negative coupling parameter \(c_1\), the universal constant should increase as the characteristic parameter *k* of the non-linear electrodynamics increases or as the coupling parameter \(c_2\) decreases. This happens to the contrary if \(c_1>0\). Whereas, for \(c_2\) fixed, the universal constant is an decreasingly monotonic function of \(c_1\) independently of the sign of \(c_2\).

## 4 Conclusion

In this paper, we have derived a solution of 4D non-linear charged AdS black hole, which is spherically-symmetric and static, in the context of the non-linear electrodynamics and the Einstein-massive gravity. The non-linear electrodynamics is characterized by a fixed parameter *k* by which the charge *Q* and the mass *M* of the system are related as, \(Q^2=Mk\). As the characteristic parameter *k* approaches zero, the Lagrangian of the non-linear electrodynamics should become the usual Maxwell one. In the massive gravity theory proposed by de Rham, Gabadadze and Tolley, the potential associated with the graviton mass consists of the four terms. However, in the four-dimensional spacetime, only two terms corresponding to the coupling parameters \(c_1\) and \(c_2\) appear in the black hole solution.

We have studied the horizon properties of the black hole, which are crucially dependent on the sign of both \(c_1\) and \(c_2\). Interesting, the presence of \(c_2\) yields an effective horizon curvature which can be positive, zero, or negative corresponding to the sphere, flat, or hyperbolic effective horizon. The extremal black hole only exists for the case of the positive effective horizon curvature or that of the zero effective horizon curvature and \(c_1>0\). On the contrary, the black hole solution can exist with arbitrary mass but its horizon radius is bounded from below. Also, with the proper parameters the black hole possibly has many inner horizons or does not have any inner horizon.

We have studied the thermodynamics, the thermodynamic stability and the phase transitions of the black hole. Interestingly, unlike the usual charged black hole, the usual term \(\Phi dQ\) disappears naturally in the first law of the present black hole. Also, the black hole entropy is affected only by the non-linear electrodynamics at which the area law is broken. But, the area law of the black hole entropy should be approximately restored in the regime of the large horizon radius. Based on the heat capacity at constant pressure, we studied the thermodynamic stability of the black hole and pointed to its second-order phase transitions which occur at the maximum temperature and the minimum one. Whereas, based on the Gibbs free energy, we pointed to a firs-order phase transition of the black hole and the Hawking–Page phase transition. In the case of the positive effective horizon curvature, we have also shown a critical point in the \(P-r_+\) diagram. For the temperature below the critical value, a first-order phase transition between the small and large black holes occurs, which is analogous to the van der Waals phase transition between the liquid and gas.

## Footnotes

- 1.
We would like to thank Reviewer for indicating these points.

## Notes

### Acknowledgements

This work was supported by the National Research Foundation of Korea(NRF) grant with the grant number NRF-2016R1D1A1A09917598 and by the Yonsei University Future-leading Research Initiative of 2017(2017-22-0098).

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