# Dynamical and thermal stabilities of nonlinearly charged AdS black holes

## Abstract

In this paper, we study an extended phase space thermodynamics of a nonlinearly charged AdS black hole. We examine both the local and global stabilities, and possible phase transition of the black hole solutions. Finally, we compute quasi-normal modes via scalar perturbations and compare the obtained results with those of Reissner-Nordström black hole.

## 1 Introduction

Nonlinear electrodynamics was first proposed by Born and Infeld in order to remove the central singularity of the point-like charges and obtain finite energy solutions for particles by extending Maxwell theory [1]. Later, Plebanski and Perzanowski extended the model and presented other examples of nonlinear electrodynamics Lagrangians in the framework of special relativity [2]. Recently, studying various models of nonlinear electrodynamics has been under active investigation, mainly because these theories appear as effective theories at low energy limits of string theory [3]. In the framework of AdS/CFT, nonlinear electrodynamics has been used to obtain solutions describing baryon configurations which are consistent with confinement [4]. In other frameworks, in order to remove curvature singularities various important results have been obtained. For example, in cosmological models, one can use the nonlinear electrodynamics for explaining the inflationary epoch and the late-time accelerated expansion of the universe [5, 6]. In the field of black holes, different classes of regular black hole solutions in general relativity coupled to nonlinear electrodynamics have been found, in which the nonlinear electrodynamics is a source of field equations satisfying the weak energy condition, and recovering the Maxwell theory in the weak field limit [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. There are some nonlinear electrodynamics Lagrangians that can not remove singularity from the black holes, but they generalize the RN solution. For example, Born–Infeld Lagrangian gives rise to a singular spherically symmetric black hole and other solutions [19, 20, 21, 22, 23, 24, 25]. The progress in this direction and demand to investigate various aspects of nonlinear models are the main motivations for the present study. In this paper, we study an interesting class of nonlinear electrodynamics, which regularizes the metric function. Regarding such class of nonlinear electrodynamics, we find that although this model can regularize the metric function, the curvature scalars diverge at the origin.

Thermodynamic properties of black holes were reinforced by the discovery of Hawking radiation [26]. Bekenstein considered the concept of entropy for black holes and made it quantitative, namely the area law, \(S=A/4\) [27]. However, there are some differences between black hole thermodynamics and conventional thermodynamics, such as the black hole entropy, which is proportional to the horizon area and not volume, and the heat capacity of black holes which might be negative.

Thermodynamic properties of AdS black holes are more interesting due to some reasons: one of them comes from the main work of Hawking and Page, who discovered a first order phase transition between the Schwarzschild-AdS black hole and thermal AdS space [28]. This phenomenon, known as the Hawking–Page phase transition, is explained as the gravitational dual of the QCD confinement/deconfinement transition [29, 30]. Another important reason for the study of thermodynamics of AdS black holes was the discovery of phase transitions similar to Van der Waals liquid/gas phase transitions in the Reissner–Nordström/anti-de Sitter (RN-AdS) black holes by Chambline et al. [31, 32]. They studied the RN-AdS black holes in canonical ensemble and discovered a first order phase transition between small and large black holes. Another motivation of considering AdS black holes is due to the AdS/CFT correspondence [33, 34]. This duality has been recently used to study the behavior of quark-gluon plasmas and the qualitative description of various condensed matter phenomena. According to the AdS/CFT correspondence [29], a large static black hole in asymptotically AdS spacetime corresponds to a thermal state in the CFT living on the boundary.

Actually, the interests in studies of thermodynamics of the AdS black holes is partly due to the rich structures found by treating the cosmological constant as a thermodynamic variable. In the presence of a varying cosmological constant, the first law of black hole thermodynamics becomes consistent with the Smarr relation. The first law of black holes is modified by including a *VdP* term where the pressure *P* is given by \(-\dfrac{\Lambda }{8\pi }\). In this framework, the mass of the black hole *M* is considered as the enthalpy of the system instead of the internal energy [35, 36]. One of the first works to explore the extended phase space thermodynamics in AdS black holes was written by Kubiznak and Mann who showed the existence of a certain phase transition in the phase space of the Schwarzschild/AdS black hole [37]. Similar critical behavior is found in the spacetimes of a rotating AdS black hole and a higher dimensional RN-AdS black hole [38]. The same qualitative properties are also found in the AdS black hole spacetime with the Born–Infeld electrodynamics [39, 40], with the power-Maxwell field and with Gauss–Bonnet correction [41, 42]. There are many other works related to this concept including [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54].

In this paper, we introduce a new charged AdS black hole solution, in the presence of a nonlinear electrodynamics. We then perform a detailed analysis of the thermodynamics of the obtained BH solution. Also, we study the behavior of a scalar field outside the black hole horizon by computing the complex frequencies associated with quasinormal modes. These modes are the resonant, non-radial perturbations of black holes which are the result of external perturbations of spacetime. Quasinormal modes are important for some reasons. First, because information about the black hole is encoded in there. By detection of the QNMs, one can obtain precise information related to the mass, charge, and other global parameters of the black hole. Another reason is that complex frequencies can provide information about dynamical stability of the black hole. Indeed, the real part of frequencies determines the oscillation frequency and the imaginary part determines the rate at which each mode is damped. The final reason comes from the fact that the QNMs frequencies of AdS black holes have an interpretation in the dual conformal field theory [55]. According to this correspondence, a large black hole in AdS corresponds to a thermal state in the CFT. Perturbing the black hole, corresponds to perturbing the thermal state and the decay of the perturbation describes the return to thermal equilibrium. Therefore, the time scale for the decay of the black hole perturbation, which is given by the imaginary part of its QNM, corresponds to the timescale to reach thermal equilibrium in the strongly coupled CFT [55, 56, 57, 58].

The paper is organized as follows: in Sect. 2, the basic equations are introduced. Section 3 is devoted to the calculation of conserved and thermal quantities and investigation of the first law of thermodynamics. Subsequently, thermal stability and phase transition are discussed. In Sect. 4, we compute QNMs and finally, our concluding remarks are given in Sect. 5.

## 2 Basic equations

*R*is the Ricci scalar and \( \Lambda =-\dfrac{3}{b^{2}} \) is the cosmological constant, in which

*b*denotes the radius of AdS space.

*L*(

*P*) is a function of \( P \equiv \dfrac{1}{4}P_{\mu \nu }P^{\mu \nu }\) and \( P_{\mu \nu } \) is an antisymmetric tensor related to the Faraday tensor \( F_{\mu \nu } \) according to \( P_{\mu \nu }= L_{F}F_{\mu \nu } \) and \( L_{F}\equiv \dfrac{\partial L}{\partial F} \) (\( F\cong F_{\mu \nu }F^{\mu \nu } \) is the Maxwell invariant). Applying the variational principle to the action (1), one can show that the field equations are given by [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63]

*Q*is an integration constant related to the electric charge. Here, we assume that the Lagrangian is as follows [8, 9, 10]:

*L*(

*P*) , to obtain the Lagrangian (5). So, this Lagrangian is appropriate only for the present black hole and by using this Lagrangian one can not obtain regular black holes such as Bardeen and other regular black holes. The particular form of the Lagrangian (5) thus leads to the interesting solution (7) which contrary to the RN solution of Einstein–Maxwell is more usual. System, behaves regularly as \( r\rightarrow 0 \). The associated electric field

*E*is given by

*M*and

*Q*are related to mass and charge according to

*M*is ADM mass and

*Q*is electric charge of the solution. Note that in contrast to the Einstein–Maxwell system,

*M*and

*Q*are not constants of integration, but rather are parameters present in EM Lagrangian (5). We plot diagrams for finding the possible roots of the metric function which correspond to horizons (Fig. 1). We find from this figure that our solution can represent a black hole, by suitable choices of parameters. For \( Q=0.827M \), the horizons degenerate to a single one, corresponding to an extreme black hole. Therefore, there is a minimal mass

*r*goes to zero. The electric field vanishes as \(r \rightarrow 0\) and \( r\rightarrow \infty \) and achieves its maximum \( E_{c} \) at \( r_{c} \) (see Fig. 2). In order to get information about the nature of \( r_{c} \), see [64] and the reference therein.

*r*goes to zero the Kretschmann scalar diverges as \( \frac{1}{r^{2}} \), for all values of

*Q*and

*M*. For comparison, \(K_{1}\propto \dfrac{1}{r^{6}}\) for the RN metric.

To derive the global structure of the metric, one can construct the Penrose diagrams. The Penrose diagram of the present metric is similar to the Reissner–Nordström-AdS metric. In the case of non-extreme black hole solution \( Q<0.827M \), we can split the space-time into three regions; \( I: r>r_{+} \), \( II:r_{-}<r<r_{+} \) and \( III: 0\le r<r_{-}\). In the extreme black hole case, \( Q=0.827M \), there arise two regions: \( r>r_{c} \) and \( 0<r<r_{c} \). For \( Q>0.827M \), there are no horizons.

## 3 Thermodynamic properties

*S*can be calculated via the area law

*P*is defined as [37]:

*M*is related to \( r_{+} \) and other BH parameters as

*M*will be interpreted as the enthalpy \(H=M\) [35, 36, 37, 50, 66, 68, 71]. So, the enthalpy in terms of thermodynamic quantities is given by

### 3.1 Specific heats and thermodynamic stability

*G*is a decreasing function for small/large event horizon, while it is an increasing function for intermediate \( r_{+} \). This behavior confirms that intermediate black holes are globally unstable. Large black holes have negative Gibbs free energy and therefore are more stable than small black holes. Also, one can see that for constant \( r_{+} \), for the larger pressure, the BH is more stable. Also, we compare the Gibbs free energy of the RN-AdS metric with the present solution. The Gibbs free energy could also have extremum by suitable choices of parameters. We shall discuss this issue in the next section when we study the phase transition.

Regarding Fig. 4, we find that the heat capacity is positive for large black holes and partly positive for the smaller ones. It means that the large black hole is thermodynamically more stable (locally). For medium black holes, since the heat capacity is negative, the black hole is locally unstable, which is similar to RN-AdS black hole. We shall discuss this issue in Sect. 3.2 when we consider the critical points in black hole phase diagram.

### 3.2 Phase transition

*T*and

*Q*fixed. The temperature of isotherm diagrams decreases from top to bottom. The two upper

*solid*lines correspond to the ideal gas phase, the critical isotherm is denoted by the

*dotted-dashed*line, lower solid lines correspond to temperatures smaller than the critical temperature and there is also a temperature below which the pressure becomes negative in a range of \( r_{+} \) (black dashed line). Again, the resemblance with the liquid/gas phase transition in Van der Waals gas and also for the RN-AdS black hole is clearly seen. The critical point is determined by \( \frac{\partial P}{\partial r_{+}}=0 \) and \( \frac{\partial ^{2}P}{\partial r_{+}^{2}}=0 \). By using Eq. (13), one can find \( \frac{P_{c}r_{c}}{T_{c}}=\frac{(0.002669)(2.54)}{0.0385}=0.176 \) which is independent of charge

*Q*[31, 32, 40, 67, 72].

## 4 Quasinormal modes

*x*and using spherical harmonics, one can rewrite the radial part of (28) in a Schrödinger form

*V*given by

*V*is positive and vanishes at the horizon (\( x=-\infty \)). It diverges at \( r=\infty \), which corresponds to a finite value of

*x*, and hence \( \varphi \) vanishes at infinity. This boundary condition is to be satisfied by the wave equation of the scalar field. In general, the frequency of waves must be complex, \( \omega =\omega _{r}-i\omega _{i} \). The imaginary and real parts are related to the damping time scale (\(\tau _{i}=1/\omega _{i}\)) and oscillation time scale (\(\tau _{r}=1/\omega _{r}\)), respectively. By using the numerical method suggested in [56, 57, 58], we have computed the QNMs frequencies via expanding the solution around the horizon and imposing the boundary condition that the solution vanishes at infinity. Here, we have considered large black holes, because large black holes in AdS/CFT correspond to a thermal state, and its perturbation corresponds to the perturbation of thermal state, and damping of perturbation in AdS is translated as return to equilibrium in thermal state. Small black hole is unstable and does not correspond to any thermal state in CFT [53]. For large black holes (\( r_{+}\gg b \)), the relation of the values of the lowest quasinormal frequencies for \( l=0 \) and selected values of \( r_{+} \) for different charge

*Q*are exhibited in Fig. 9. It can be seen that the real and the imaginary parts of the frequency are not linear functions of \( r_{+} \) similar to RN-AdS [58]. This can be better seen by calculation of diagram slope from Table 1.

From Fig. 9a, we learn that as *Q* increases, \( \omega _{i} \) and \( \omega _{r} \) decrease. According to the AdS/CFT correspondence, this means that for large *Q*, it is slower for the quasinormal ringing to settle down to thermal equilibrium and also, the frequency of the oscillation becomes smaller (Fig 9b).

*Q*increases, the linear relation between

*T*and \( r_{+} \) no longer holds. Again, the linear relation between real and imaginary parts of frequency with charge is changed (Fig. 11).

*l*for different values of

*Q*is exhibited. Unlike the case of RN-AdS [58] and Schwarzschild-AdS [56] black holes, by increasing

*l*, \( \omega _{i} \) and \( \omega _{r} \) increase and different values of

*Q*do not change the qualitative behavior of \( \omega _{r} \) and \( \omega _{i} \) with

*l*.

*Q*increases, the real and imaginary parts of the large black hole QNM frequencies decrease. Also, for fixed charge, as

*l*increases, QNM frequency increases, and this is consistent with Fig. 12.

Quasinormal frequencies of the scalar perturbations for \(l=0 \) and \( b=1 \)

\(r_{+}\) | Q = 5 | Q = 25 | Q = 40 |
---|---|---|---|

50 | 70.71 + 168.3 i | 27.93 + 156.8 i | |

60 | 84.00 + 198.1 i | 47.64 + 18.2 i | |

70 | 102.40 + 240.7 i | 72.10 + 232.4 i | 17.91 + 220.0 i |

80 | 115.00 + 269.7 i | 87.60 + 262.3 i | 44.70 + 250.8 i |

90 | 131.00 + 306.5 i | 106.60 + 299.0 i | 69.50 +289.7 i |

100 | 144.30 + 337.3 i | 122.0 + 331.3 i | 88.00 + 321.8 i |

150 | 217.10 + 505.8 i | 201.90 + 501.8 i | 178.20 + 495.4 i |

200 | 290.00 + 674.9 i | 278.50 + 679.1 i | 260.00 + 667.0 i |

*P*to obtain the phase transitions between small-large black holes. In Fig. 14, we have plotted the behavior of an isobar. The crossing curves in the left plot indicates the coexistence of two phases in equilibrium that correspond to the separated points in the right plot for \(T_{c}=0.0306\). The QNMs frequencies corresponding to small-large black holes have been plotted in Fig. 15. For the case of small black holes, one can see that by increasing the radius of event horizon the imaginary part increases and real part of frequencies decreases (Table 2). While, for the large black hole, we find that when the black hole radius increases, the real part together with the imaginary part of QNMs frequencies increase [73, 74].

The QNM frequencies of massless scalar perturbations for \(l=0 \) and \(P=0.0015\)

T | \(r_{+}\) | \(\omega \) |
---|---|---|

0.0265 | 1.30 | 2.3562 + 0.9616 i |

0.0280 | 1.35 | 2.2392 + 1.1242 i |

0.0290 | 1.40 | 2.1612 + 1.1845 i |

0.0299 | 1.45 | 2.1420 + 1.2171 i |

0.0305 | 1.49 | 2.0415 + 1.3105 i |

0.0307 | 6.00 | 5.7229 + 2.8845 i |

0.0310 | 6.50 | 5.9928 + 2.9997 i |

0.0330 | 7.50 | 6.0015 + 3.0003 i |

0.0340 | 8.00 | 6.1204 + 3.0572 i |

0.0350 | 8.50 | 6.1297 + 3.0149 i |

0.0360 | 9.00 | 6.3184 + 3.1765 i |

## 5 Conclusion

Although black hole singularities seem to be inevitable within general theory of relativity and through the collapse of ordinary matter which respects energy conditions, there are alternative models which avoid the spacetime singularity by ignoring energy conditions or other assumptions of the singularity theorems [7, 8, 9]. In the present work, we demonstrated that a non-linear electromagnetic Lagrangian can lead to black hole solutions which behave like a Reissner–Nordestr\(\ddot{o}\)m-AdS BH at large distance, while having a quasi non-singular core in the sense that the metric function behaves like \( f(r)\approx 1-\dfrac{2\alpha }{\beta ^{2}}r+O(r^{2}) \) as \( r \rightarrow 0\). The electromagnetic Lagrangian, although having a relatively complicated form, reduces to the Maxwellian form as \( F\rightarrow 0 \). The BH solutions presented here have the interesting property that their ADM mass and charge are not free parameter, but depend on the Lagrangian parameters \( \alpha \) and \(\beta \).

Since, thermodynamical behavior are of great importance in search for a quantum theory of gravitation, we have also managed to perform a thermodynamic investigation of the BH solutions. The conserved and thermodynamic quantities were calculated and the validity of the first law was checked. Global stability of the BH was examined by plotting the Gibbs free energy, and the heat capacities were studied to check the local stability. We showed that the present solutions admitted small/large phase transitions similar to the Van der Waals liquid/gas phase transition. Then, by writing and solving the wave equation for a scalar field in the BH background spacetime, the QNMs were calculated. We pointed out that the frequencies of QNMs behave somehow differently form those of RN-AdS and Schwarzschild-AdS black holes. The effect of the BH charge on eigen-frequencies were demonstrated through several diagrams which were calculated numerically.

Finally, we have obtained the QNMs frequencies of massless scalar perturbations around small and large black hole. We found that when the Van der Waals-like phase transition happens, as the horizon radius increase, the slopes of the QNMs frequency change differently in the small and large black holes.

As a further remark, in order to study the thermodynamics of black hole, one can also use the geometrothermodynamical formalism.

## Notes

### Acknowledgements

We are grateful to the anonymous referee for the insightful comments and suggestions, which have allowed us to improve this paper significantly. SNS and NR acknowledge the support of Shahid Beheshti University. SHH wishes to thank Shiraz University Research Council. The work of SHH has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha, Iran.

## References

- 1.M. Born, Proc. R. Soc. Lond. A
**143**(849), 410 (1934). https://doi.org/10.1098/rspa.1934.0010 ADSCrossRefGoogle Scholar - 2.J.F. Plebanski, M. Przanowski, Int. J. Theor. Phys.
**33**, 1535 (1994). https://doi.org/10.1007/BF00670696 CrossRefGoogle Scholar - 3.N. Seiberg, E. Witten, JHEP
**9909**, 032 (1999). https://doi.org/10.1088/1126-6708/1999/09/032. arXiv:hep-th/9908142 ADSCrossRefGoogle Scholar - 4.O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri, Y. Oz, Phys. Rep.
**323**, 183 (2000). https://doi.org/10.1016/S0370-1573(99)00083-6. arXiv:hep-th/9905111 ADSMathSciNetCrossRefGoogle Scholar - 5.R. Garcia-Salcedo, N. Breton, Int. J. Mod. Phys. A
**15**, 4341 (2000). https://doi.org/10.1016/S0217-751X(00)00216-9. https://doi.org/10.1142/S0217751X00002169. https://doi.org/10.1142/S0217751X00002160. arXiv:gr-qc/0004017 - 6.M. Novello, S.E. Perez Bergliaffa, J. Salim, Phys. Rev. D
**69**, 127301 (2004). https://doi.org/10.1103/PhysRevD.69.127301. arXiv:astro-ph/0312093 ADSCrossRefGoogle Scholar - 7.E. Ayon-Beato, A. Garcia, Phys. Lett. B
**493**, 149 (2000). https://doi.org/10.1016/S0370-2693(00)01125-4. arXiv:gr-qc/0009077 ADSMathSciNetCrossRefGoogle Scholar - 8.E. Ayon-Beato, A. Garcia, Gen. Relativ. Gravit.
**37**, 635 (2005). https://doi.org/10.1007/s10714-005-0050-y. arXiv:hep-th/0403229 ADSCrossRefGoogle Scholar - 9.E. Ayon-Beato, A. Garcia, Phys. Rev. Lett.
**80**, 5056 (1998). https://doi.org/10.1103/PhysRevLett.80.5056. arXiv:gr-qc/9911046 ADSCrossRefGoogle Scholar - 10.K.A. Bronnikov, Phys. Rev. Lett.
**85**, 4641 (2000). https://doi.org/10.1103/PhysRevLett.85.4641 ADSCrossRefGoogle Scholar - 11.K.A. Bronnikov, Phys. Rev. D
**63**, 044005 (2001). https://doi.org/10.1103/PhysRevD.63.044005. arXiv:gr-qc/0006014 ADSMathSciNetCrossRefGoogle Scholar - 12.I.G. Dymnikova, Int. J. Mod. Phys. D
**5**, 529 (1996). https://doi.org/10.1142/S0218271896000333 ADSCrossRefGoogle Scholar - 13.I.G. Dymnikova, Phys. Lett. B
**472**, 33 (2000). https://doi.org/10.1016/S0370-2693(99)01374-X. arXiv:gr-qc/9912116 ADSMathSciNetCrossRefGoogle Scholar - 14.I.G. Dymnikova, A. Dobosz, M.L. Fil’chenkov, A. Gromov, Phys. Lett. B
**506**, 351 (2001). https://doi.org/10.1016/S0370-2693(01)00174-5. arXiv:gr-qc/0102032 ADSCrossRefGoogle Scholar - 15.I. Dymnikova, Int. J. Mod. Phys. D
**12**, 1015 (2003). https://doi.org/10.1142/S021827180300358X. arXiv:gr-qc/0304110 ADSMathSciNetCrossRefGoogle Scholar - 16.I. Dymnikova, Class. Quantum Gravity
**21**, 4417 (2004). https://doi.org/10.1088/0264-9381/21/18/009. arXiv:gr-qc/0407072 ADSCrossRefGoogle Scholar - 17.I. Dymnikova, E. Galaktionov, Class. Quantum Gravity
**22**, 2331 (2005). https://doi.org/10.1088/0264-9381/22/12/003. arXiv:gr-qc/0409049 ADSCrossRefGoogle Scholar - 18.M. Novello, S.E. Perez Bergliaffa, J.M. Salim, Class. Quantum Gravity
**17**, 3821 (2000). https://doi.org/10.1088/0264-9381/17/18/316. arXiv:gr-qc/0003052 ADSCrossRefGoogle Scholar - 19.D.L. Wiltshire, Phys. Rev. D
**38**, 2445 (1988). https://doi.org/10.1103/PhysRevD.38.2445 ADSMathSciNetCrossRefGoogle Scholar - 20.M.H. Dehghani, S.H. Hendi, A. Sheykhi, H. Rastegar Sedehi, JCAP
**0702**, 020 (2007). https://doi.org/10.1088/1475-7516/2007/02/020. arXiv:hep-th/0611288 ADSCrossRefGoogle Scholar - 21.S.H. Hendi, Ann. Phys.
**333**, 282 (2013). https://doi.org/10.1016/j.aop.2013.03.008. arXiv:1405.5359 [gr-qc]ADSCrossRefGoogle Scholar - 22.M. Hassaine, C. Martinez, Phys. Rev. D
**75**, 027502 (2007). https://doi.org/10.1103/PhysRevD.75.027502. arXiv:hep-th/0701058 ADSMathSciNetCrossRefGoogle Scholar - 23.S.H. Hendi, B.E. Panah, Phys. Lett. B
**684**, 77 (2010). https://doi.org/10.1016/j.physletb.2010.01.026. arXiv:1008.0102 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 24.H.P. de Oliveira, Class. Quantum Gravity
**11**, 1469 (1994). https://doi.org/10.1088/0264-9381/11/6/012 ADSCrossRefGoogle Scholar - 25.H.H. Soleng, Phys. Rev. D
**52**, 6178 (1995). https://doi.org/10.1103/PhysRevD.52.6178. arXiv:hep-th/9509033 ADSCrossRefGoogle Scholar - 26.J.M. Bardeen, B. Carter, S.W. Hawking, Commun. Math. Phys.
**31**, 161 (1973). https://doi.org/10.1007/BF01645742 ADSCrossRefGoogle Scholar - 27.J.D. Bekenstein, Phys. Rev. D
**9**, 3292 (1974). https://doi.org/10.1103/PhysRevD.9.3292 ADSCrossRefGoogle Scholar - 28.S.W. Hawking, D.N. Page, Commun. Math. Phys.
**87**, 577 (1983). https://doi.org/10.1007/BF01208266 ADSCrossRefGoogle Scholar - 29.E. Witten, Adv. Theor. Math. Phys.
**2**, 253 (1998). https://doi.org/10.4310/ATMP.1998.v2.n2.a2. arXiv:hep-th/9802150 ADSMathSciNetCrossRefGoogle Scholar - 30.E. Witten, Adv. Theor. Math. Phys.
**2**, 505 (1998). https://doi.org/10.4310/ATMP.1998.v2.n3.a3. arXiv:hep-th/9803131 MathSciNetCrossRefGoogle Scholar - 31.A. Chamblin, R. Emparan, C.V. Johnson, R.C. Myers, Phys. Rev. D
**60**, 064018 (1999). https://doi.org/10.1103/PhysRevD.60.064018. arXiv:hep-th/9902170 ADSMathSciNetCrossRefGoogle Scholar - 32.A. Chamblin, R. Emparan, C.V. Johnson, R.C. Myers, Phys. Rev. D
**60**, 104026 (1999). https://doi.org/10.1103/PhysRevD.60.104026. arXiv:hep-th/9904197 ADSMathSciNetCrossRefGoogle Scholar - 33.J.M. Maldacena, Int. J. Theor. Phys.
**38**, 1113 (1999)CrossRefGoogle Scholar - 34.J.M. Maldacena, Adv. Theor. Math. Phys.
**2**, 231 (1998). https://doi.org/10.1023/A:1026654312961. https://doi.org/10.4310/ATMP.1998.v2.n2.a1. arXiv:hep-th/9711200 - 35.D. Kastor, S. Ray, J. Traschen, Class. Quantum Gravity
**27**, 235014 (2010). https://doi.org/10.1088/0264-9381/27/23/235014. arXiv:1005.5053 [hep-th]ADSCrossRefGoogle Scholar - 36.B.P. Dolan, Class. Quantum Gravity
**28**, 125020 (2011). https://doi.org/10.1088/0264-9381/28/12/125020. arXiv:1008.5023 [gr-qc]ADSCrossRefGoogle Scholar - 37.D. Kubiznak, R.B. Mann, JHEP
**1207**, 033 (2012). https://doi.org/10.1007/JHEP07(2012)033. arXiv:1205.0559 [hep-th]ADSCrossRefGoogle Scholar - 38.S. Gunasekaran, R.B. Mann, D. Kubiznak, JHEP
**1211**, 110 (2012). https://doi.org/10.1007/JHEP11(2012)110. arXiv:1208.6251 [hep-th]ADSCrossRefGoogle Scholar - 39.A. Belhaj, M. Chabab, H. El Moumni, M.B. Sedra, Chin. Phys. Lett.
**29**, 100401 (2012). https://doi.org/10.1088/0256-307X/29/10/100401. arXiv:1210.4617 [hep-th]ADSCrossRefGoogle Scholar - 40.R. Banerjee, D. Roychowdhury, Phys. Rev. D
**85**, 104043 (2012). https://doi.org/10.1103/PhysRevD.85.104043. arXiv:1203.0118 [gr-qc]ADSCrossRefGoogle Scholar - 41.S.H. Hendi, M.H. Vahidinia, Phys. Rev. D
**88**(8), 084045 (2013). https://doi.org/10.1103/PhysRevD.88.084045. arXiv:1212.6128 [hep-th]ADSCrossRefGoogle Scholar - 42.S.W. Wei, Y.X. Liu, Phys. Rev. D
**87**(4), 044014 (2013). https://doi.org/10.1103/PhysRevD.87.044014. arXiv:1209.1707 [gr-qc]ADSCrossRefGoogle Scholar - 43.J. Xu, L.M. Cao, Y.P. Hu, Phys. Rev. D
**91**(12), 124033 (2015). https://doi.org/10.1103/PhysRevD.91.124033. arXiv:1506.03578 [gr-qc]ADSCrossRefGoogle Scholar - 44.J.X. Mo, W.B. Liu, Eur. Phys. J. C
**74**(4), 2836 (2014). https://doi.org/10.1140/epjc/s10052-014-2836-0. arXiv:1401.0785 [gr-qc]ADSCrossRefGoogle Scholar - 45.J.X. Mo, G.Q. Li, X.B. Xu, Phys. Rev. D
**93**(8), 084041 (2016). https://doi.org/10.1103/PhysRevD.93.084041. arXiv:1601.05500 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 46.M. Zhang, W.B. Liu, arXiv:1610.03648 [gr-qc]
- 47.G.Q. Li, Phys. Lett. B
**735**, 256 (2014). https://doi.org/10.1016/j.physletb.2014.06.047. arXiv:1407.0011 [gr-qc]ADSCrossRefGoogle Scholar - 48.J. Mo, G. Li, X. Xu, Eur. Phys. J. C
**76**, 545 (2016)ADSCrossRefGoogle Scholar - 49.R.G. Cai, L.M. Cao, L. Li, R.Q. Yang, JHEP
**1309**, 005 (2013). https://doi.org/10.1007/JHEP09(2013)005. arXiv:1306.6233 [gr-qc]ADSCrossRefGoogle Scholar - 50.R.G. Cai, Y.P. Hu, Q.Y. Pan, Y.L. Zhang, Phys. Rev. D
**91**(2), 024032 (2015). https://doi.org/10.1103/PhysRevD.91.024032. arXiv:1409.2369 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 51.D. Kubiznak, R.B. Mann, M. Teo, Class. Quantum Gravity
**34**(6), 063001 (2017). https://doi.org/10.1088/1361-6382/aa5c69. arXiv:1608.06147 [hep-th]ADSCrossRefGoogle Scholar - 52.R. Monteiro, https://doi.org/10.17863/CAM.16097. arXiv:1006.5358 [hep-th]
- 53.R.A. Konoplya, Phys. Rev. D
**66**, 084007 (2002). https://doi.org/10.1103/PhysRevD.66.084007. arXiv:gr-qc/0207028 ADSMathSciNetCrossRefGoogle Scholar - 54.M.S. Ma, R. Zhao, Phys. Lett. B
**751**, 278 (2015). https://doi.org/10.1016/j.physletb.2015.10.061. arXiv:1511.03508 [gr-qc]ADSCrossRefGoogle Scholar - 55.D. Birmingham, I. Sachs, S.N. Solodukhin, Phys. Rev. D
**67**, 104026 (2003). https://doi.org/10.1103/PhysRevD.67.104026. arXiv:hep-th/0212308 ADSMathSciNetCrossRefGoogle Scholar - 56.G.T. Horowitz, V.E. Hubeny, Phys. Rev. D
**62**, 024027 (2000). https://doi.org/10.1103/PhysRevD.62.024027. arXiv:hep-th/9909056 ADSMathSciNetCrossRefGoogle Scholar - 57.G.T. Horowitz, Class. Quantum Gravity
**17**, 1107 (2000). https://doi.org/10.1088/0264-9381/17/5/320. arXiv:hep-th/9910082 ADSCrossRefGoogle Scholar - 58.B. Wang, C.Y. Lin, E. Abdalla, Phys. Lett. B
**481**, 79 (2000). https://doi.org/10.1016/S0370-2693(00)00409-3. arXiv:hep-th/0003295 ADSMathSciNetCrossRefGoogle Scholar - 59.M. Novello, V.A. De Lorenci, J.M. Salim, R. Klippert, Phys. Rev. D
**61**, 045001 (2000). https://doi.org/10.1103/PhysRevD.61.045001. arXiv:gr-qc/9911085 ADSCrossRefGoogle Scholar - 60.K.A. Bronnikov, J.C. Fabris, Phys. Rev. Lett.
**96**, 251101 (2006). https://doi.org/10.1103/PhysRevLett.96.251101. arXiv:gr-qc/0511109 ADSMathSciNetCrossRefGoogle Scholar - 61.J. Matyjasek, D. Tryniecki, M. Klimek, Mod. Phys. Lett. A
**23**, 3377 (2009). https://doi.org/10.1142/S0217732308028715. arXiv:0809.2275 [gr-qc]ADSCrossRefGoogle Scholar - 62.S.I. Kruglov, Phys. Rev. D
**94**(4), 044026 (2016). https://doi.org/10.1103/PhysRevD.94.044026. arXiv:1608.04275 [gr-qc]ADSMathSciNetCrossRefGoogle Scholar - 63.W. Berej, J. Matyjasek, D. Tryniecki, M. Woronowicz, Gen. Relativ. Gravit.
**38**, 885 (2006). https://doi.org/10.1007/s10714-006-0270-9. arXiv:hep-th/0606185 ADSCrossRefGoogle Scholar - 64.S.N. Sajadi, N. Riazi, Gen. Relativ. Gravit.
**49**(3), 45 (2017). https://doi.org/10.1007/s10714-017-2209-8 ADSCrossRefGoogle Scholar - 65.L. Balart, S. Fernando, Mod. Phys. Lett. A
**32**(39), 1750219 (2017). https://doi.org/10.1142/S0217732317502194. arXiv:1710.07751 [gr-qc]ADSCrossRefGoogle Scholar - 66.B.P. Dolan, https://doi.org/10.5772/52455. arXiv:1209.1272 [gr-qc]
- 67.C. Niu, Y. Tian, X.N. Wu, Phys. Rev. D
**85**, 024017 (2012). https://doi.org/10.1103/PhysRevD.85.024017. arXiv:1104.3066 [hep-th]ADSCrossRefGoogle Scholar - 68.D.C. Zou, S.J. Zhang, B. Wang, Phys. Rev. D
**89**(4), 044002 (2014). https://doi.org/10.1103/PhysRevD.89.044002. arXiv:1311.7299 [hep-th]ADSCrossRefGoogle Scholar - 69.S.H. Hendi, B. Eslam Panah, M. Momennia, S. Panahiyan, Eur. Phys. J. C
**75**(9), 457 (2015). https://doi.org/10.1140/epjc/s10052-015-3677-1. arXiv:1509.03081 [hep-th]ADSCrossRefGoogle Scholar - 70.S.H. Hendi, S. Panahiyan, B. Eslam Panah, Adv. High Energy Phys.
**2015**, 743086 (2015). https://doi.org/10.1155/2015/743086. arXiv:1509.07014 [gr-qc]CrossRefGoogle Scholar - 71.D. Kubiznak, F. Simovic, Class. Quantum Gravity
**33**(24), 245001 (2016). https://doi.org/10.1088/0264-9381/33/24/245001. arXiv:1507.08630 [hep-th]ADSCrossRefGoogle Scholar - 72.N. Altamirano, D. Kubiznak, R.B. Mann, Z. Sherkatghanad, Galaxies
**2**, 89 (2014). https://doi.org/10.3390/galaxies2010089. arXiv:1401.2586 [hep-th]ADSCrossRefGoogle Scholar - 73.R.A. Konoplya, A. Zhidenko, JHEP
**1709**, 139 (2017). https://doi.org/10.1007/JHEP09(2017)139. arXiv:1705.07732 [hep-th]ADSCrossRefGoogle Scholar - 74.P. Prasia, V.C. Kuriakose, Gen. Relativ. Gravit.
**48**(7), 89 (2016). https://doi.org/10.1007/s10714-016-2083-9. arXiv:1606.01132 [gr-qc]ADSCrossRefGoogle Scholar

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