*D*-dimensional Bardeen–AdS black holes in Einstein–Gauss–Bonnet theory

## Abstract

We present a *D*-dimensional Bardeen like Anti-de Sitter (AdS) black hole solution in Einstein–Gauss–Bonnet (EGB) gravity, viz., Bardeen–EGB–AdS black holes. The Bardeen–EGB–AdS black hole has an additional parameter due to magnetic charge (*e*), apart from mass (*M*) and Gauss–Bonnet parameter (\(\alpha \)). Interestingly, for each value of \(\alpha \), there exist a critical \(e = e_E\) which corresponds to an extremal regular black hole with degenerate horizons, while for \(e< e_E\), it describes non-extremal black hole with two horizons. Despite the complicated solution, the thermodynamical quantities, like temperature (*T*), specific heat(*C*) and entropy (*S*) associated with the black hole are obtained exactly. It turns out that the heat capacity diverges at critical horizon radius \(r_+ = r_C\), where the temperature attains maximum value and the Hawking-Page transition is achievable. Thus, we have an exact *D*-dimensional regular black holes, when evaporates lead to a thermodynamical stable remnant.

## 1 Introduction

The celebrated singularity theorems of Hawking and Penrose [1, 2, 3] have shown that under fairly general conditions, a sufficiently massive collapsing object will undergo continual gravitational collapse, resulting in the formation of a curvature singularity. However, the singularity is not visible to a far-away observer which essentially means that a black hole has formed. It is widely believed that these singularities do not exist in Nature, but that they are the artefact of classical general relativity. The existence of a singularity means spacetime ceases to exist, signal a breakdown of physics laws and that they must be resolved in a theory of quantum gravity [4, 5]. While we are far from a definite quantum gravity, attention has been shifted to regular models that are motivated by quantum arguments. The earliest idea of Sakharov [6] and Gliner [7], suggests that singularities could be avoided by matter, i.e., with a de Sitter core, with the equation of state \(p=-\rho \) obeyed by the cosmological constant.

*f*(

*r*) has a double root if \(\psi =\psi ^{*}\), two roots if \(\psi <\psi ^{*}\) and no root if \(\psi >\psi ^{*}\), with \(\psi =m/e\) [9]. These cases illustrate, respectively, an extreme black hole with degenerate horizons, a black hole with Cauchy and event horizons, and no black hole. Later, Ayon-Beato and Garcia [10, 11, 12, 13] invoked nonlinear electrodynamics a to generate the Bardeen model [8], i.e., an exact solution of general relativity coupled to nonlinear electrodynamics. Bronnikov [14, 15, 16, 17] purposed that it must be a magnetic field instead of an electric field as purposed in [10, 11], which serves as matter field to get regular black hole solutions of general relativity coupled to nonlinear electrodynamics. The Bardeen solution is regular everywhere, that can be realized from behaviour of the scalar invariants, Ricci scalar (

*R*), Ricci square (\(\mathcal {R} = R_{\mu \nu } R^{\mu \nu }\)) and Kretschmann scalar (\(\mathcal {K} = R_{\mu \nu \rho \sigma } R^{\mu \nu \rho \sigma }\)), which are given by

*m*and \(e \ne 0\).

Subsequently, also there has been intense activities in the investigation of regular black holes and more recently [18, 19, 20, 21]. But most of these solutions are more or less based on Bardeen’s model [8]. Also, some solutions in which generalized Bardeen model have been obtained in later years, which includes the Bardeen de Sitter solution [22], rotating or Kerr-like Bardeen’s solution [23] and noncommutative Bardeen solution [24].

The Bardeen’s regular metric is commonly used to compare with the classical black hole, in various applications which include thermodynamical properties [25, 26, 27], geodesics equations [28], quasinormal modes [29, 30, 31, 32, 33], Hawking evaporation [34, 35] and black hole’s remnant [36]. The rotating Bardeen regular metric has been tested with a black hole candidate in Cygnus X-1 [37], and also shown that the it can also act as a natural particle accelerator [38]. Lately, the Bardeen’s solution is extended to higher dimensional spacetime [39].

Last few decades gravity witnessed considerable activities in higher dimensions motivated by the superstring and field theories. In addition to higher-curvature corrections to Einstein theory, string theory makes several predictions about nature, the most important ones are the existence of extra dimensions [40]. The Einstein–Gauss–Bonnet gravity is a natural and most effective generalization of Einsteins general relativity, to higher dimensions, motivated by the heterotic string theory. It was discovered first by Lanczos [41], and rediscovered by David Lovelock [42]. The Einstein–Gauss–Bonnet (EGB) theory allow us to explore several conceptual issues of gravity in a broader setup and the theory is known to be free of ghosts while expanding about the flat space [43]. The effective field equations, in the EGB theory, are of second-order like in general relativity, but admit, in *D* \(> 5\), new black hole solutions [44] that are unavailable to the Einstein theory. The first black hole solutions of EGB theory was obtained by Boulware and Deser [43] which is similar to its general relativity counterpart with a curvature singularity at \(r=0\). Later several authors studied exact black hole solutions in EGB theory and their thermodynamical properties [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57].

The black holes with higher derivative curvature in Anti-de Sitter (AdS) spaces have been considered in the recent years, e.g., static AdS black hole solutions in EGB gravity with several interesting features [58, 59, 60]. Bardeen [8] was the first to purpose the regular black hole solution by taking a magnetic field as matter field, which have a de-Sitter core instead of singularity at the center. The enormous advancement on the applications of regular black holes makes it important to study these types of solutions in a broader setup. Demand of extra dimensions by string theory has made it very important and interesting to study the solutions in higher dimensional manifolds. Although, the generalized Bardeen solution in general relativity has been already presented in [39], but the broader setup of EGB gravities motivated us to obtain generalized Bardeen solution in EGB. However, there are many solutions of EGB coupled with nonlinear electrodynamics [61, 62] already present in the literature, but those are not regular black hole solutions. Thus, the purpose of this paper is to obtain a *D*-dimensional spherically symmetric Bardeen-like black holes solution for the EGB gravity in AdS spacetimes, viz., Bardeen–EGB–AdS metric. It is shown that the Bardeen–EGB–AdS metric is an exact black hole solution of EGB coupled to nonlinear electrodynamics in AdS spacetime thereby generalizing the Boulware–Desser solution [43] which is encompassed as a special case. We analyze their thermodynamical properties to find a stable black hole remnant and also perform a thermodynamic stability analysis of the Bardeen–EGB–AdS black holes.

The paper is ordered as: we obtain *D*-dimensional Bardeen–EGB–AdS black hole metric for a nonlinear electrodynamics as a source in Sect. 2 and also give the basic equations governing EGB theory. We investigated the structure and location of the horizons of the *D*-dimensional Bardeen–EGB–AdS Black holes metric along with their energy conditions in Sect. 2. Section 3 is devoted to the study of the thermodynamical properties of *D*-dimensional Bardeen–EGB–AdS Black holes with a focus on the stability and also discuss black hole’s remnant. We end the paper with our concluding remarks in Sect. 5. We use the units such that \(G=c=1\).

## 2 Einstein–Gauss–Bonnet with nonlinear electrodynamics

*R*are respectively the Ricci tensors, Riemann tensors, and Ricci scalar. The variation of the action with respect to the metric \(g_{\mu \nu }\) gives the following EGB equations of motion [57]

*anstaz*for the Maxwell field [39]

## 3 Bardeen Anti de-Sitter black holes in EGB theory

*D*-dimensional static spherically symmetric solutions of Eq. (5). We assume the metric to be of the following form [57, 66]

*r*,

*r*) equation of field equation reduces to

*r*and \(\tilde{\alpha }=\) \(\left( D-3\right) \left( D-4\right) \alpha \). The Eq. (15) can be easily integrated to give general solution as

*M*with relation [57]

*f*(

*r*) must be real valued. So, we must consider the value of radial co-ordinate (

*r*) such that the term under square root in (16) must be non-negative. For the case, \(1-4\tilde{\alpha }/l^2\ge 0\), that will always be non-negative. But for \(1-4\tilde{\alpha }/l^2<0\), there will be some values of

*r*for which that term can be negative, so, for those values of

*r*our solution is no more real valued. Hence, we can say that the Gauss–Bonnet coupling constant \(\tilde{\alpha }\) must lie in the interval \([0,l^2/4]\). Besides, the causality of dual theory demands another constraint on Gauss–Bonnet coupling constant [67, 68]

*D*-dimensional solution of EGB theory coupled to nonlinear electrodynamics in an AdS spacetime thereby generalizing the Bardeen solution. The special case in which charge \(e=0\) and \(\Lambda =0\), one get the Boulware–Deser solution [43]. It see that solution (14) with (16) gets for other field equation. For definiteness, henceforth, we shall call solution (16) Bardeen–EGB–AdS black holes. In the case of no charge \(e = 0\), Eq. (16) reduces to

*D*-dimensional EGB–AdS black holes [59, 60, 63, 69, 70, 71, 72, 73, 74] and in the limit, \(\alpha \rightarrow 0\), the negative branch of (16) to

*D*- dimensional Bardeen–AdS black holes [39]

*D*-dimensional Bardeen–EGB–AdS black hole solution is well defined everywhere by its curvature invariants.

Radius of Cauchy horizon (\(r_-\)), the event horizon (\(r_+\)) and \(\delta = r_+-r_-\) for different values of charge *e* and dimension *D*

Dimensions | \(\alpha =0.1\) | \(\alpha =0.2\) | ||||||
---|---|---|---|---|---|---|---|---|

| \(r_-\) | \(r_+\) | \(\delta \) | | \(r_-\) | \(r_+\) | \(\delta \) | |

\(D=5\) | \(e_E=0.493\) | 0.6242 | 0.6242 | 0 | \(e_E=0.373\) | 0.566 | 0.566 | 0 |

0.3 | 0.2519 | 0.8695 | 0.6176 | 0.2 | 0.2152 | 0.7632 | 0.548 | |

0.4 | 0.3746 | 0.8164 | 0.4418 | 0.3 | 0.3501 | 0.7182 | 0.3681 | |

\(D=6\) | \(e_E=0.52\) | 0.607 | 0.607 | 0 | \(e_E=0.372\) | 0.4773 | 0.4773 | 0 |

0.3 | 0.2619 | 0.8038 | 0.5419 | 0.2 | 0.1583 | 0.6276 | 0.4693 | |

0.4 | 0.3278 | 0.768 | 0.4402 | 0.3 | 0.2871 | 0.5969 | 0.3098 | |

\(D=7\) | \(e_E=0.547\) | 0.5939 | 0.5939 | 0 | \(e_E=0.423\) | 0.4656 | 0.4656 | 0 |

0.3 | 0.1767 | 0.7564 | 0.5797 | 0.2 | 0.1142 | 0.6062 | 0.4920 | |

0.4 | 0.2840 | 0.7442 | 0.4602 | 0.3 | 0.2182 | 0.5950 | 0.3768 | |

\(D=8\) | \(e_E=0.578\) | 0.5816 | 0.5816 | 0 | \(e_E=0.475\) | 0.4881 | 0.4881 | 0 |

0.3 | 0.1460 | 0.7380 | 0.5920 | 0.2 | 0.0860 | 0.6175 | 0.5315 | |

0.4 | 0.2442 | 0.7319 | 0.4877 | 0.3 | 0.1732 | 0.6090 | 0.4358 |

Next, we proceed to discuss the horizon structure of our Bardeen–EGB–AdS black holes. The horizons radius, if exists, are zeros of \(g^{rr}=f(r)=0\). The numerical analysis of \(f(r)=0\) reveals that it is possible to find non-vanishing value of \(\alpha \) and *e* for which metric function *f*(*r*) is minimum, i.e, \(f(r)=0\) admits two roots \(r_{\pm }\). The smaller and larger roots, respectively, corresponds to the Cauchy and event horizon of the black holes. We have shown that for a given value of \(\alpha \) and fixed \(\mu '\), there exists a critical charge parameter \(e_E\), and critical horizon radius \(r_E\), such that \(f(r_E)=0\) has a double root, i.e, \(r_E=r_{\pm }\). This case corresponds to the extremal Bardeen–EGB–AdS black holes with degenerate horizons. When \(e < e_E\) the two horizons \(r_{\pm }\) correspond to the non-extremal black hole and \(e>e_E\) has no horizon, i.e., no black holes (cf. Fig. 1 and Table 1). It is clear that the critical value of \(e_E\) and \(r_E\) depend upon the coupling constant \(\alpha \). For \(\alpha =0.1\), 0.2 the critical value of the charge corresponds to the degenerate horizon for \(D=5,6,7 \) and 8 are shown in Table 1. Also, the radius of the event horizon decreases with increase in Gauss–Bonnet coefficient \(\alpha \) and increases with charge *e* and dimensions *D* as shown in Fig. 1.

## 4 Black hole thermodynamics

*M*, charge

*e*and \(\Lambda \). The black hole mass can be determined by using \(f(r_+)=0\) in terms of horizon radius \(r_+\) as

*D*-dimensional Bardeen–AdS black hole [39], in the limit \(\alpha \rightarrow 0\), yields

*D*-dimensional Bardeen black hole temperature when \(\alpha = \Lambda =0\)

The maximum Hawking temperature \(T_+^{Max}\) at critical radius \(r_C^T\) for different values of charge *e* and different dimension \(D=5,6,7\) and 8 with fixed value of \(l=10\)

Dimensions | \(\alpha =0.1\) | \(\alpha =0.2\) | ||||
---|---|---|---|---|---|---|

| \(r_c^T\) | \(T_+^{Max}\) | | \(r_c^T\) | \(T_+^{Max}\) | |

\(D=5\) | 0.4 | 1.040 | 0.09785 | 0.3 | 1.088 | 0.08127 |

0.493 | 1.115 | 0.08723 | 0.373 | 1.367 | 0.0786 | |

0.6 | 1.330 | 0.08252 | 0.5 | 1.400 | 0.07480 | |

\(D=6\) | 0.4 | 1.083 | 0.12140 | 0.3 | 0.914 | 0.09704 |

0.52 | 1.179 | 0.11610 | 0.374 | 1.099 | 0.09477 | |

0.6 | 1.287 | 0.11180 | 0.5 | 1.356 | 0.09114 | |

\(D=7\) | 0.45 | 0.871 | 0.14500 | 0.36 | 0.668 | 0.13180 |

0.547 | 1.069 | 0.13760 | 0.423 | 0.814 | 0.12130 | |

0.65 | 1.274 | 0.13090 | 0.55 | 1.075 | 0.10830 | |

\(D=8\) | 0.52 | 0.869 | 0.17070 | 0.42 | 0.666 | 0.17440 |

0.578 | 0.964 | 0.16250 | 0.475 | 0.772 | 0.15880 | |

0.65 | 1.105 | 0.15500 | 0.6 | 0.983 | 0.13630 |

By numerical analysis, we conclude that the Hawking temperature vanishes at the radius of the black hole double horizon. The Hawking temperature diverges in the absence of charge (\(e=0\)), when \(r_+ \rightarrow 0\) except in 5*D* (see Fig. 2). However, it becomes finite for non zero value of charge *e* Fig. 2. The temperature depends on both charge *e* and Gauss–Bonnet parameter \(\alpha \). From Fig. 3, one can see that for small and large horizon radius (\(r_+\)), slope of the plot is positive, whereas it is negative for intermediate horizon radius, which is showing van der Waals like small-large black hole phase transition [80, 81].

*S*is the entropy of the black hole and \(\Phi \) is potential and

*e*is the constant charge. The entropy can be obtained by integrating Eq. (34) as

*D*-dimensional Bardeen black hole [39] in the limit \(\alpha \rightarrow 0\)

## 5 Local stability and black hole remnants

*e*, cosmological constant \(\Lambda \) and the dimension

*D*. In the absence of charge

*e*, we reduce to the expression for heat capacity of EGB–AdS black hole [44, 59, 60, 63, 69, 69, 70, 71, 72, 73, 74], which reads

The state and stability of black hole with horizon radius \(r+\)

Region | State | Stability |
---|---|---|

\(r_0<r_+<r_1\) | Small | Unstable |

\(r_1<r_+<r_C\) | Intermediate | Stable |

\(r_+>r_C\) | Large | Unstable |

*D*-dimensional Bardeen black hole [39] specific heat in the limit \(\alpha =\Lambda = 0\)

The remnant size \(r_0\) and the remnant mass term \(\mu '_0\) for different values of parameter *e* with Gauss–Bonnet coupling constant \(\alpha =0.1\) with \(l=10\)

Charge | \(D=5\) | \(D=6\) | \(D=7\) | \(D=8\) | ||||
---|---|---|---|---|---|---|---|---|

| \(r_0\) | \(\mu '_0\) | \(r_0\) | \(\mu '_0\) | \(r_0\) | \(\mu '_0\) | \(r_0\) | \(\mu '_0\) |

1 | 1.12 | 3 | 1.04 | 3.80 | 1.02 | 5.70 | 0.98 | 6.70 |

2 | 2.00 | 10.90 | 1.92 | 22.37 | 1.86 | 47.93 | 1.82 | 106.9 |

3 | 2.89 | 25.10 | 2.71 | 72.20 | 2.69 | 215.5 | 2.67 | 660.4 |

4 | 3.76 | 46.80 | 3.48 | 173.83 | 3.46 | 668.2 | 3.45 | 2636 |

5 | 4.55 | 77.4 | 4.32 | 353.56 | 4.26 | 1669.56 | 4.24 | 8090 |

The remnant size \(r_0\) and the remnant mass term \(\mu '_0\) for different values of parameter *e* with fixed value of Gauss–Bonnet coupling constant \(\alpha =0.2\) with \(l=10\)

Charge | \(D=5\) | \(D=6\) | \(D=7\) | \(D=8\) | ||||
---|---|---|---|---|---|---|---|---|

| \(r_0\) | \(\mu '_0\) | \(r_0\) | \(\mu '_0\) | \(r_0\) | \(\mu '_0\) | \(r_0\) | \(\mu '_0\) |

1 | 1.14 | 3.41 | 1.12 | 5.12 | 1.05 | 7.70 | 1.01 | 11.17 |

2 | 2.10 | 11.45 | 1.95 | 25.47 | 1.93 | 59.60 | 1.90 | 143.7 |

3 | 2.93 | 25.60 | 2.78 | 77.30 | 2.75 | 243.3 | 2.72 | 796.7 |

4 | 3.79 | 47.30 | 3.56 | 181.20 | 3.51 | 721.6 | 3.51 | 2972 |

5 | 4.57 | 77.96 | 4.34 | 363.58 | 4.32 | 1757 | 4.31 | 8775 |

*e*and \(\alpha \), which is identified as the critical radius \(r^{C}_+\). Further, we noticed that the heat capacity changes its sign around \(r^{C}_+\). Thus, we can say the black hole is thermodynamically stable for \(r_1<r_+ < r^{C}_+\), whereas it is thermodynamically unstable for \(r_+>r_+^C\), and there is a second order phase transition at \(r_+=r^{C}_+\) from the stable to unstable phases. So, the heat capacity of Bardeen–EGB–AdS black hole, in any dimension for different values of

*e*, \(\alpha \) and \(\Lambda \), is positive for \(r_1<r_+ < r^{C}_+\), and it is negative for \(r_1>r_+>r_+^C\). Here, we noticed from the Fig. 4 that the value of critical radius \(r^{C}_+\) increases with the increase in the charge

*e*, for given value of Gauss–Bonnet coupling constant \(\alpha \) and cosmological constant. Thus, the change in the value of charge affected the thermodynamical stability of the black hole. One can find that the Bardeen–EGB–AdS black holes have two unstable regions and a stable region as shown in Table 3.

*D*-dimensional Bardeen black hole [39] in the limit \(\alpha \rightarrow 0\)

*e*grows for given value of Gauss–Bonnet coupling parameter \(\alpha \) and \(\Lambda \). The peak also increase as the value of \(\alpha \) grows. The behaviour of free energy suggests it is mostly positive for larger \(r_+\). From Fig. 5, one can see that Bardeen–EGB–AdS black hole more stable for smaller \(r_+\).

The remnant of a black hole is a localized late stage of the black hole after the Hawking evaporation, which is either absolutely stable or long-lived [85]. It is very important to study the black hole remnant as it is a candidate to be the source of dark matter [86] as well to resolve the information loss paradox of black hole [87]. We can get the radius \(r_E\) of black hole remnant from \(f^{\prime }(r)|_{r=r_E} = 0\). Here \(r = r_E\) corresponds to the extremal black hole with the degenerate horizon. As it is very tedious to solve \(f^{\prime }(r_E)=0\) analytically, so we tabulated the numerical results of remnant size and remnant term \(\mu '\) in Table 4 and Table 5 in various dimensions for different values of charge *e* with Gauss–Bonnet coupling constant \(\alpha =0.1\) and \(\alpha =0.2\) respectively. The remnant mass \(M_0\) can be calculated very easily by inserting the value of \(\mu '_0\) in Eq. (17). In order to analyze the emitted features of Bardeen–EGB–AdS black hole, we plotted the metric function given in Eq. (16) as a function of radius for extremal Bardeen–EGB–AdS black hole in Fig. 6 for different values of *e* and \(\alpha \). From Fig. 6, we can say that at the minimal non zero mass \(M_0\), there is a possibility of the extremal configuration with one degenerate event horizon. So, \(M = M_0\) is the condition for having one degenerate event horizon and there will be no event horizon for \(M<M_0\).

## 6 Conclusion

The EGB gravity is a higher curvature generalization of general relativity which is also considered as quantum corrected gravity model and AdS black holes help us to understand the idea from quantum gravity as well as general relativity. Further, the holography beyond the AdS/CFT continues to exist in true quantum gravity that requires the inclusion of higher order curvature derivative term. Motivated by this, we studied exact static spherical *D*-dimensional Bardeen–EGB–AdS black holes and discuss their properties. Thus, we obtained an exact black hole in EGB gravity for a static and spherically symmetric *D*-dimensional AdS spacetime with energy-momentum given by a nonlinear electrodynamics. The solution is characterized by analyzing horizons which could be at the most two.

Later, we compute exact expressions for Hawking temperature, entropy, heat capacity and free energy associated with the black holes, also demonstrate that Hawking-Page transition is achievable. We perform a detailed analysis of the thermodynamical specific heat with focus on the local and global stability. It turns out that heat capacity can be negative or positive depending on the choice of parameters *e* and \(\alpha \), which further, respectively, tells us that the black hole is unstable or stable.

Indeed, the phase transition of black hole is characterized by the divergence of its specific heat at a critical horizon radius \(r^{C}_+\) which is varying with the spacetime dimension *D* and parameter \(\alpha \). The black holes are thermodynamically stable with a positive heat capacity for the range \(r_1<r_+<r^{C}_+\) and unstable for \(r_1>r_+> r^{C}_+\) (cf. Fig. 4). We discussed the black hole remnant and tabulated the numerical values of black hole remnant size and mass. The results presented here are the generalization of the previous discussions and in the appropriate limits, go over to AdS-EGB black holes and EGB black holes. The possibility of a further generalization of these results to Lovelock gravity is an interesting problem for future research.

## Notes

### Acknowledgements

D.V.S. acknowledges the University Grant Commission, India, for financial support through the D. S. Kothari Post Doctoral Fellowship (Grant no. BSR/2015-16/PH/0014). S.G.G. would like to thanks SERB-DST Research Project Grant no. SB/S2/HEP-008/2014 and DST INDO-SA bilateral project DST/INT/South Africa/P06/2016 and also to IUCAA, Pune for the hospitality while this work was being done.

## References

- 1.R. Penrose, Phys. Rev. Lett.
**14**, 57 (1965)ADSMathSciNetCrossRefGoogle Scholar - 2.S.W. Hawking, Proc. R. Soc. Lond. A
**300**, 187 (1967)ADSCrossRefGoogle Scholar - 3.S.W. Hawking, R. Penrose, Proc. R. Soc. Lond. A
**3**(14), 529 (1970)ADSCrossRefGoogle Scholar - 4.R. Penrose, Riv. Nuovo Cimento
**1**, 252 (1969)Google Scholar - 5.R. Penrose, in
*General Relativity, an Einstein Centenary Volume*, ed. by S.W. Hawkingand, W. Israel (Cambridge University Press, Cambridge, 1979)Google Scholar - 6.A.D. Sakharov, Sov. Phys. JETP
**22**, 241 (1966)ADSGoogle Scholar - 7.E.B. Gliner, Sov. Phys. JETP
**22**, 378 (1966)ADSGoogle Scholar - 8.J. Bardeen, in
*Proceedings of GR5*(U.S.S.R, Tiflis, 1968)Google Scholar - 9.S. Ansoldi, arXiv:0802.0330 [gr-qc]
- 10.E. Ayon-Beato, A. Garcia, Phys. Rev. Lett.
**80**, 5056 (1998)ADSCrossRefGoogle Scholar - 11.E. Ayon-Beato, A. Garcia, Gen. Relativ. Gravit.
**31**, 629 (1999)ADSCrossRefGoogle Scholar - 12.E. Ayon-Beato, A. Garcia, Phys. Lett. B
**493**, 149 (2000)ADSMathSciNetCrossRefGoogle Scholar - 13.E. Ayon-Beato, A. Garcia, Gen. Relativ. Gravit.
**37**, 635 (2005)ADSCrossRefGoogle Scholar - 14.K.A. Bronnikov, Phys. Rev. Lett.
**85**, 4641 (2000)ADSCrossRefGoogle Scholar - 15.K.A. Bronnikov, Phys. Rev. D
**63**, 044005 (2001)ADSMathSciNetCrossRefGoogle Scholar - 16.K.A. Bronnikov, Int. J. Mod. Phys. D
**27**, 1841005 (2018)ADSCrossRefGoogle Scholar - 17.K.A. Bronnikov, Gravit. Cosmol.
**23**, 343–348 (2017)ADSMathSciNetCrossRefGoogle Scholar - 18.L. Xiang, Y. Ling, Y.G. Shen, Int. J. Mod. Phys. D
**22**, 1342016 (2013)ADSCrossRefGoogle Scholar - 19.H. Culetu, Int. J. Theor. Phys.
**54**, 2855 (2015)MathSciNetCrossRefGoogle Scholar - 20.L. Balart, E.C. Vagenas, Phys. Lett. B
**730**, 14 (2014)ADSMathSciNetCrossRefGoogle Scholar - 21.L. Balart, E.C. Vagenas, Phys. Rev. D
**90**, 124045 (2014)ADSCrossRefGoogle Scholar - 22.S. Fernando, Int. J. Mod. Phys. D
**26**, 1750071 (2017)ADSCrossRefGoogle Scholar - 23.C. Bambi, L. Modesto, Phys. Lett. B
**721**, 329 (2013)ADSMathSciNetCrossRefGoogle Scholar - 24.M. Sharif, W. Javed, Can. J. Phys.
**89**, 1027 (2011)ADSCrossRefGoogle Scholar - 25.K. Ghaderi, B. Malakolkalami, Gravit. Cosmol.
**24**, 61 (2018)ADSMathSciNetCrossRefGoogle Scholar - 26.N. Bretn, S.E. Perez Bergliaffa, A.I.P. Conf, Proc.
**1577**, 112 (2014)Google Scholar - 27.J. Man, H. Cheng, Gen. Relativ. Gravit.
**46**, 1660 (2014)ADSCrossRefGoogle Scholar - 28.Z. Stuchlk, J. Schee, Int. J. Mod. Phys. D
**24**, 1550020 (2014)ADSCrossRefGoogle Scholar - 29.S. Fernando, J. Correa, Phys. Rev. D
**86**, 064039 (2012)ADSCrossRefGoogle Scholar - 30.C.F.B. Macedo, L.C.B. Crispino, E.S. de Oliveira, Int. J. Mod. Phys. D
**25**, 1641008 (2016)ADSCrossRefGoogle Scholar - 31.S.C. Ulhoa, Braz. J. Phys.
**44**, 380 (2014)ADSCrossRefGoogle Scholar - 32.W. Wahlang, P.A. Jeena, S. Chakrabarti, Int. J. Mod. Phys. D
**26**, 1750160 (2017)ADSCrossRefGoogle Scholar - 33.M. Saleh, B.B. Thomas, T.C. Kofane, Eur. Phys. J. C
**78**, 325 (2018)ADSCrossRefGoogle Scholar - 34.D.V. Singh, N.K. Singh, Ann. Phys.
**383**, 600 (2017)ADSCrossRefGoogle Scholar - 35.H. Huang, M. Jiang, J. Chen, Y. Wang, Gen. Relativ. Gravit.
**47**, 8 (2015)ADSCrossRefGoogle Scholar - 36.S.H. Mehdipour, M.H. Ahmadi, Astrophys. Space Sci.
**361**, 314 (2016)ADSCrossRefGoogle Scholar - 37.C. Bambi, Phys. Lett. B
**730**, 59 (2014)ADSCrossRefGoogle Scholar - 38.S.G. Ghosh, M. Amir, Eur. Phys. J. C
**75**, 553 (2015)ADSCrossRefGoogle Scholar - 39.Md Sabir Ali, S.G. Ghosh, Phys. Rev. D
**98**, 084025 (2018)ADSCrossRefGoogle Scholar - 40.D. Gross, E. Witten, Nucl. Phys. B
**277**, 1 (1986)ADSCrossRefGoogle Scholar - 41.C. Lanczos, Ann. Math.
**39**, 842 (1938)MathSciNetCrossRefGoogle Scholar - 42.D. Lovelock, J. Math. Phys. (N.Y.)
**12**, 498 (1971)ADSCrossRefGoogle Scholar - 43.D.G. Boulware, S. Deser, Phys. Rev. Lett.
**55**, 2656 (1985)ADSCrossRefGoogle Scholar - 44.R.G. Cai, Phys. Rev. D
**65**, 084014 (2002)ADSMathSciNetCrossRefGoogle Scholar - 45.J.T. Wheeler, Nucl. Phys. B
**268**, 737 (1986)ADSCrossRefGoogle Scholar - 46.J.T. Wheeler, Nucl. Phys. B
**273**, 732 (1986)ADSCrossRefGoogle Scholar - 47.S.G. Ghosh, D.W. Deshkar, Phys. Rev. D
**77**, 047504 (2008)ADSMathSciNetCrossRefGoogle Scholar - 48.S.G. Ghosh, Phys. Lett. B
**704**, 5 (2011)ADSMathSciNetCrossRefGoogle Scholar - 49.R.C. Myers, J.Z. Simon, Phys. Rev. D
**38**, 2434 (1988)ADSMathSciNetCrossRefGoogle Scholar - 50.M.H. Dehghani, R.B. Mann, Phys. Rev. D
**72**, 124006 (2005)ADSMathSciNetCrossRefGoogle Scholar - 51.S.G. Ghosh, S.D. Maharaj, Phys. Rev. D
**89**, 084027 (2014)ADSCrossRefGoogle Scholar - 52.H. Maeda, N. Dadhich, Phys. Rev. D
**75**, 044007 (2007)ADSCrossRefGoogle Scholar - 53.S.G. Ghosh, D.V. Singh, S.D. Maharaj, Phys. Rev. D
**97**, 104050 (2018)ADSMathSciNetCrossRefGoogle Scholar - 54.G. Kofinas, R. Olea, Phys. Rev. D
**74**, 084035 (2006)ADSMathSciNetCrossRefGoogle Scholar - 55.S.G. Ghosh, Class. Quantum Gravity
**35**, 085008 (2018)ADSCrossRefGoogle Scholar - 56.D.V. Singh, M.S. Ali, S.G. Ghosh, Int. J. Mod. Phys. D
**27**, 1850108 (2018)ADSCrossRefGoogle Scholar - 57.S.G. Ghosh, U. Papnoi, S.D. Maharaj, Phys. Rev. D
**90**, 044068 (2014)ADSCrossRefGoogle Scholar - 58.T. Torii, H. Maeda, Phys. Rev. D
**71**, 124002 (2005)ADSMathSciNetCrossRefGoogle Scholar - 59.Y.M. Cho, I.P. Neupane, Phys. Rev. D
**66**, 024044 (2002)ADSMathSciNetCrossRefGoogle Scholar - 60.I.P. Neupane, Phys. Rev. D
**69**, 084011 (2004)ADSCrossRefGoogle Scholar - 61.S.H. Hendi, S. Panahiyan, M. Momennia, Int. J. Mod. Phys. D
**25**, 1650063 (2016)ADSCrossRefGoogle Scholar - 62.S. H. Hendi, S. Panahiyan, B. Eslam Panah, PTEP
**103E01**(2015)Google Scholar - 63.M.H. Dehghani, S.H. Hendi, Int. J. Mod. Phys. D
**16**, 1829 (2007)ADSCrossRefGoogle Scholar - 64.D.J. Gross, J.H. Sloan, Nucl. Phys. B
**291**, 41 (1987)ADSCrossRefGoogle Scholar - 65.M.C. Bento, O. Bertolami, Phys. Lett. B
**368**, 198 (1996)ADSMathSciNetCrossRefGoogle Scholar - 66.S.H. Hendi, S. Panahiyan, B. Eslam Panah, J. High Energy Phys.
**01**, 129 (2016)ADSCrossRefGoogle Scholar - 67.X. Zeng, W. Liu, Phys. Lett. B
**726**, 481 (2013)ADSMathSciNetCrossRefGoogle Scholar - 68.Y. Sun, H. Xu, L. Zhao, J. High Energ Phys.
**09**, 060 (2016)ADSCrossRefGoogle Scholar - 69.N. Deruelle, L. Farina-Busto, Phys. Rev. D
**41**, 3696 (1990)ADSMathSciNetCrossRefGoogle Scholar - 70.M.H. Dehghani, Phys. Rev. D
**69**, 064024 (2004)ADSMathSciNetCrossRefGoogle Scholar - 71.N. Deruelle, J. Katz, S. Ogushi, Class. Quantum Gravity
**21**, 1971 (2004)ADSCrossRefGoogle Scholar - 72.M.H. Dehghani, S.H. Hendi, Phys. Rev. D
**73**, 084021 (2006)ADSMathSciNetCrossRefGoogle Scholar - 73.A. Padilla, Class. Quantum Gravity
**20**, 3129 (2003)ADSCrossRefGoogle Scholar - 74.M.H. Dehghani, G.H. Bordbar, M. Shamirzaie, Phys. Rev. D
**74**, 064023 (2006)ADSCrossRefGoogle Scholar - 75.F.R. Tangherlini, Nuovo Cim.
**27**, 636 (1963)ADSMathSciNetCrossRefGoogle Scholar - 76.D. Wiltshire, Phys. Rev. D
**38**, 2445 (1988)ADSMathSciNetCrossRefGoogle Scholar - 77.S.W. Hawking, D.N. Page, Commun. Math. Phys.
**87**, 577 (1983)ADSCrossRefGoogle Scholar - 78.C. Sahabandu, P. Suranyi, C. Vaz, L.C.R. Wijewardhana, Phys. Rev. D
**73**, 044009 (2006)ADSMathSciNetCrossRefGoogle Scholar - 79.P. Kanti, Lect. Notes Phys.
**769**, 387–423 (2009)ADSCrossRefGoogle Scholar - 80.D. Kubiznak, R.B. Mann, J. High Energ Phys.
**033**, 1207 (2012)Google Scholar - 81.S.-W. Wei, Y.-X. Liu, Phys. Rev. D
**90**, 044057 (2014)ADSCrossRefGoogle Scholar - 82.J.M. Bardeen, B. Carter, S.W. Hawking, Commun. Math. Phys.
**31**, 161–170 (1973)ADSCrossRefGoogle Scholar - 83.J.D. Bekenstein, Phys. Rev. D
**7**, 2333 (1973)ADSMathSciNetCrossRefGoogle Scholar - 84.E. Herscovich, M.G. Richarte, Phys. Lett. B
**689**, 192200 (2010)CrossRefGoogle Scholar - 85.P. Chen, Y.C. Ong, D.H. Yeom, Phys. Rep.
**603**, (2015)Google Scholar - 86.J.H. MacGibbon, Nature
**329**, 308 (1987)ADSCrossRefGoogle Scholar - 87.J. Preskill, arXiv:hep-th/9209058

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