# Charged AdS black holes in Gauss–Bonnet gravity and nonlinear electrodynamics

## Abstract

New five-dimensional charged AdS black hole solutions are found in Einstein–Gauss–Bonnet gravity and the nonlinear electrodynamics. These solutions include regular black holes as well as extremal black holes. The first law of the black hole thermodynamics is confirmed in the extended phase space where the cosmological constant is treated as the pressure. The first and second order phase transitions are investigated by observing the behavior of the heat capacity at constant pressure and the Gibbs free energy. In addition, the equation of state for the black holes and their *P*–*V* criticality are studied. Finally, the critical exponents are found to be the same as those of the Van der Waals fluid.

## 1 Introduction

Black holes are the solutions of Einstein’s equation and one of the most important and fascinating objects in physics. They themselves possess a lot of interesting physical properties. As found by Bekenstein and Hawking, the black holes are thermodynamic objects with the temperature and entropy determined by the surface gravity at the event horizon and by the area of the event horizon, respectively [1, 2, 3, 4, 5]. Since the pioneering work of Bekenstein and Hawking, the thermodynamics and the phase structure of black holes have been studied extensively [6, 7, 8, 9]. In particular, the black hole thermodynamics in the asymptotically anti-de Sitter (AdS) spacetime has attracted a lot of attention. Hawking and Page [10] discovered a phase transition between Schwarzschild AdS black hole and thermal AdS space, so-called, the Hawking–Page transition. Interestingly, according to the AdS/CFT correspondence, the Hawking–Page transition is interpreted as the confinement/deconfinement phase transition in the dual conformal field theory at the boundary [11, 12, 13, 14]. Chamblin et al. found that charged AdS black holes can undergo a first-order phase transition which is very similar to that of the liquid/gas phase system [15, 16].

Recently, there have been attempts to treat the cosmological constant as the thermodynamic pressure *P* whose conjugate variable is the thermodynamic volume *V*. In this extension, the first law of black hole thermodynamics is modified with including a new term *VdP* [17, 18, 19, 20, 21, 22] and the black hole mass is naturally interpreted as the enthalpy rather than the internal energy. Accordingly, the study of the thermodynamics and phase transition for the AdS black holes has been generalized to the extended phase space. *P*–*V* criticality of charged AdS black holes was found by Kubizňák and Mann, at which the critical exponents were obtained and shown to coincide with those of the Van der Waals system [23]. Following this remarkable work, *P*–*V* criticality has been studied in various black holes [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]. In addition, many interesting phenomena of the AdS black hole thermodynamics have been found, such as multiply reentrant phase transitions [24, 51, 52, 53], the heat engine efficiency of black holes [54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64], or the Joule–Thomson expansion of black holes [65, 66, 67, 68, 69, 70].

Einstein–Gauss–Bonnet gravity is a natural generalization of Einstein gravity in (4+1) dimensions or higher in the gravity model which includes the higher order curvature corrections. This theory corresponds to first three terms of Lovelock gravity, which consists of the cosmological constant, Einstein–Hilbert action and the Gauss–Bonnet term [71]. The Gauss–Bonnet term is also naturally obtained from the low-energy limit of heterotic string theory [72, 73, 74, 75, 76, 77, 78]. Noble features of the theory including the Gauss–Bonnet term are that the theory is ghost-free and the corresponding field equations do not contain more than second order derivatives of the metric [79, 80, 81, 82]. In Einstein–Gauss–Bonnet gravity, the black hole solutions and their interesting properties have been studied in the literature [83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94].

One of the mysterious properties of black holes is that they have a curvature singularity in their interior [95]. This curvature singularity is inevitable in General Relativity under some physically reasonable conditions because of the famous singularity theorem proved by Hawking and Penrose [95]. The presence of the curvature singularity may be regarded as signaling the breakdown of General Relativity and would be removed by quantum gravity, which is not known yet. Since lots of efforts have been dedicated to understand how to avoid the curvature singularity at the semi-classical level. The first example of such singularity-free, or so-called regular, black hole was found by Bardeen [96], which was realized later as a gravitational collapse of some magnetic monopole in the framework of the nonlinear electrodynamics by Ayón-Beato and García [97].

There have been a lot of the regular black hole solutions obtained with the nonlinear electrodynamics. In Refs. [98, 99, 100], they proposed the regular black holes which behave asymptotically like the RN black hole but do not satisfy the weak energy condition (WEC). The Bardeen regular black hole [96, 97] satisfies WEC, but its asymptotic behavior does not look like the one of the RN black hole. The regular black holes, which satisfy WEC and behave asymptotically like the RN black hole, were obtained by authors in Refs. [101, 102, 103, 104]. In addition, the regular black hole solutions were found from the modification of the Ayón-Beato and García solution [105], in the lower dimensional spacetime [106], in the presence of some dark energy candidates [107, 108]. There exist other regular black hole solutions such as rotating regular black hole [109] and radiating Kerr-like regular black hole [110]. Extremal limit of the regular charged black holes was investigated in Ref. [111]. Beyond Einstein gravity, the regular black hole solutions were found in Einstein–Gauss–Bonnet gravity [112, 113, 114], in the quadratic gravity [115], in the *f*(*R*) gravity [116], and in the *f*(*T*) gravity [117].

Inspired by these works, we obtain new charged AdS black hole solutions with the nonlinear electrodynamics in the framework of Einstein–Gauss–Bonnet gravity. The solutions contain not only extremal charged black holes but also regular black holes without curvature singularity. This means there are a variety of scenarios for the fate of these black holes after Hawking radiation. We investigate the thermodynamics and the phase transitions of these black holes in the extended phase space including the cosmological constant as the pressure. We find that the solutions exhibit the Hawking–Page transition through the study of Gibbs free energy. We also find that the solutions have the liquid/gas-like phase transition below the critical temperature.

This paper is organized as follows. In Sect. 2, we introduce the action describing Einstein–Gauss–Bonnet gravity coupled to a nonlinear electromagnetic field in five-dimensional spacetime with negative cosmological constant. Then, we find static and spherically symmetric black hole solutions carrying the electric charge. In Sect. 3, we study the curvature singularity and find the regular black hole solutions. We investigate the thermodynamics and phase structure of the black holes in Sect. 4. We find the first and second order phase transitions. We study *P*–*V* criticality of the black holes in Sect. 5. We also obtain the critical exponents describing the behavior of the thermodynamic quantities of the black holes near the critical point. Finally, our conclusion is drawn in Sect. 6. 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

*l*as

^{1}and \(\mathcal {L}(F)\) is the function of the invariant \(F\equiv F_{\mu \nu }F^{\mu \nu }/4\) where \(F_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu \) is the field strength of the electromagnetic field \(A_{\mu }\).

*tr*) component is related to the conjugate momentum of the gauge field \(A_{\mu }\). The square invariant of the field strength \(P_{\mu \nu }\) satisfies

*U*(1) gauge field, where the first- and higher-order terms in the nonlinear coupling constant \(k_{0}\) are the higher order derivative corrections of the

*U*(1) gauge field originating from the integration of UV energy modes. The usual Maxwell electrodynamics is restored in the \(k_{0}\rightarrow 0\) limit.

*M*and the total electric charge

*Q*, given by the following ansatz:

*Q*is defined as

*q*related to the total charge

*Q*by \(q=\frac{4\pi }{S_3}Q\) in which \(S_3=2\pi ^2\) is the volume of the unit 3-sphere. The electrostatic potential \(A_t(r)\) of the black hole is obtained as

*tt*) component of Eq. (4) reads

*tt*) component Eq. (21) holds, as presented in the appendix. By integrating the equation, we obtain the black hole solution as

*m*is the integration constant. The ADM mass

*M*of the geometry can be obtained by following the expression given in Ref. [119] for the black holes in Einstein–Gauss–Bonnet gravity coupled to nonlinear electrodynamics and is given by

*f*(

*r*) is given by

*M*of the black hole solution is related to the ‘reduced’ mass

*m*as

- 5
*D**Schwarzschild-AdS GB black holes*In the limit \(q\rightarrow 0\), the metric function is given bywhich corresponds to the metric of 5$$\begin{aligned} f(r)=1+\frac{r^2}{4\alpha }\left( 1-\sqrt{1-\frac{8\alpha }{l^2}+\frac{8\alpha m}{r^4}}\right) . \end{aligned}$$(27)*D*Schwarzschild-AdS GB black hole [85]. - 5
*D**charged AdS black holes with the GB term and Maxwell electrodynamics*In the limit \(k\rightarrow 0\), the Lagrangian for the gauge field becomes the usual Maxwell Lagrangian. In this limit, the solution becomes of the form,which describes the charged AdS black hole solution in Einstein–Gauss–Bonnet gravity. If we take additional limit, \(l\rightarrow \infty \), it becomes the Wiltshire black hole solution [84], in which the GB term plays the role of the effective cosmological constant. On the other hand, under the limit, \(l\rightarrow \infty \) and \(q\rightarrow 0\), it reduces to the Boulware–Deser black hole solution [83].$$\begin{aligned} f(r)=1+\frac{r^2}{4\alpha }\left( 1-\sqrt{1-\frac{8\alpha }{l^2}+\frac{8\alpha m}{r^4}-\frac{8\alpha q^{2}}{3r^6}}\right) ,\nonumber \\ \end{aligned}$$(28) - 5
*D**charged AdS black holes with higher order derivative electrodynamics*If we take the \(\alpha \rightarrow 0\) limit, we obtain another new charged black hole solution of Einstein gravity with higher order derivative*U*(1) gauge field asIn this solution, the lower bound of the reduced mass is also given by \(m = q^2/3k\), where it becomes the regular black hole solution. Since all the thermodynamic properties of this solution can be trivially obtained by taking \(\alpha \rightarrow 0\) limit, we will present the thermodynamic properties for the general \(\alpha \) case only.$$\begin{aligned} f(r)=\frac{r^2}{l^{2}}+1-\frac{m}{r^2}-\frac{q^2}{3kr^2}\Big (e^{-\frac{k}{r^2}}-1\Big ). \end{aligned}$$(29)

## 3 Properties of black hole solutions

*k*and the cosmological constant, there are two classes of the solutions. The first one is when the mass is in the region , \(m > \frac{q^{2}}{3k}\). In this class, the behavior of the curvature scalar

*R*near \(r=0\) is given by,

*f*(

*r*) behaves as

An essential property of the regular black hole solutions is that their interior near \(r=0\) looks like the vacuum solution. As just indicated, the core region of our regular black hole solution is the same as the one of AdS geometry, just like the Ayón-Beato–García regular black hole [101, 103]. Whereas, the standard Bardeen solution [96, 97] and the regular black hole solutions obtained by Dymnikova [102, 120, 121, 122] have the dS center.

Before we proceed, let us pause to clarify the differences between our charged AdS black hole solutions and those obtained in various gravity theories in the four- and higher-dimensional spacetime. Many charged AdS black hole solutions have been obtained in Einstein/modified gravity with/without the higher order derivative corrections of the *U*(1) gauge field. For example, the charged AdS black hole solutions were obtained in Einstein gravity with power-law Maxwell field [123, 124, 125, 126], in Gauss–Bonnet gravity’s rainbow [127, 128], in Gauss–Bonnet gravity with power-law Maxwell field [129, 130] or with quadratic nonlinear electrodynamics [131], in third order Lovelock gravity with Born–Infeld type nonlinear electrodynamics [132], in third order quasitopological gravity with power-law Maxwell source [133], in Lovelock gravity coupled to Born–Infeld nonlinear electrodynamics [134], in Einsteinian cubic gravity [135], and in the *f*(*R*) gravity [136, 137]. One may expect that the higher order curvature corrections of quantum gravity as well as the higher order derivative corrections of the *U*(1) gauge field can smooth the spacetime geometry. However, for all these black hole solutions, there is always a curvature singularity at \(r=0\). The corrections of quantum gravitation as well as the nonlinear electrodynamics only lead to the modification of the usual black holes or decreasing the strength of the curvature singularity. In our black hole solutions, there exists a lower bound for the black hole mass where the combination of the corrections for both gravity and the electrodynamics can smooth the spacetime geometry completely. As a result, the curvature singularity disappears and the black hole corresponding to this lower bound is regular. The lower bound of the mass of the regular black hole is determined in terms of its charge and the nonlinear coupling. This relation is analogous to the one in the regular black hole solutions obtained in Refs. [138, 139, 140, 141, 150]. One of the main virtues of our black hole solutions, in contrast to the above solutions, is that they contain not only regular black hole but also extremal black hole as a lower bound of mass, which gives a variety of scenarios for the end points of Hawking radiation.

Furthermore, for the regular black hole solutions mentioned in the introduction, the nonlinear coupling constant should be fine-tuned and related to the charge and mass of the regular black holes. In contrast to these regular black hole solutions, the nonlinear coupling constant in our theory is given as a free parameter and the usual black hole solutions with curvature singularity exist. The regular black hole emerges as a special limit of these solutions with the mass given in terms of the charge and the nonlinear coupling constant.

*m*and the horizon radius \(r_+\) as

*k*, \(\alpha \),

*q*and

*l*.

*q*is small enough for given coupling constants

*k*and \(\alpha \).

^{2}More specifically, it corresponds to the case that the charge

*q*is smaller than a critical value \(q_c\) for given

*l*and

*k*, where \(q_c\) is determined in terms of

*l*and

*k*by the following relation

^{3}

*q*, the mass function develops the local minimum, which corresponds to the extremal black hole. The horizon radius \(r_e\) of the extremal black hole is a large positive root of the equation \(F_{1}(r_{e})=0\), where

*k*makes both extremal radius and extremal mass decreasing. On the other hand, increasing the cosmological constant leads to the decrease of the extremal radius and the increase of the mass. In this situation, we have two different cases. For intermediate region of the charge

*q*, the reduced mass of the extremal black hole is larger than the one of regular black hole \(m=\frac{q^{2}}{3k}\). In this case the regular black hole solution may never be reached by the Hawking radiation and the extremal black hole would be the end point of the radiation. It is depicted in the middle panel of Fig. 2 and will be called the case II.

*q*is large enough for given coupling constants

*k*and the cosmological constant, the reduced mass of extremal black hole becomes smaller than the reduced mass of the regular black hole. It is depicted in the right panel of Fig. 2 and will be denoted as the case III. Since the regular black hole mass is larger than the global minimum of the curve \(m(r_+)\), there is no black hole solutions with the mass in the dashed part. One may note that the horizon radius of the regular black hole is the root of the equation,

*q*through the radiation, there is a chance that it ends up to be regular as well as extremal.

The numerical results for the local maximum temperature \(T_{\text {max}}\) and the local minimum temperature \(T_{\text {min}}\) corresponding to the radius \(r_{\text {max}}\) and the radius \(r_{\text {min}}\) are given in terms of the pressure under various values of Gauss–Bonnet coupling constant \(\alpha \), at \(k =0.35\) and \(q=\sqrt{2}\)

\(\alpha =0.5\) | \(\alpha =1\) | ||||||||
---|---|---|---|---|---|---|---|---|---|

| \(r_{\text {max}}\) | \(T_{\text {max}}\) | \(r_{\text {min}}\) | \(T_{\text {min}}\) | | \(r_{\text {max}}\) | \(T_{\text {max}}\) | \(r_{\text {min}}\) | \(T_{\text {min}}\) |

0.001 | 1.84532 | 0.05307 | 10.5445 | 0.02864 | 0.0005 | 2.28472 | 0.0394 | 14.9138 | 0.02025 |

0.002 | 1.90893 | 0.05466 | 7.16091 | 0.03976 | 0.001 | 2.36655 | 0.0403 | 10.133 | 0.02812 |

0.003 | 1.98842 | 0.05636 | 5.57573 | 0.04776 | 0.0015 | 2.46662 | 0.04124 | 7.89979 | 0.03378 |

0.004 | 2.0948 | 0.0582 | 4.55215 | 0.05398 | 0.002 | 2.59544 | 0.04228 | 6.47005 | 0.0382 |

## 4 Thermodynamics and phase transitions

*S*, the total charge

*Q*and the pressure

*P*are a complete set of the thermodynamic variables for the mass or enthalpy function

*M*(

*S*,

*Q*,

*P*), while the temperature

*T*, the chemical potential \(\Phi \) and the thermodynamic volume

*V*are conjugate variables to

*S*,

*Q*and

*P*, respectively, and are defined as

*q*, respectively.

*V*can also be obtained from the first law as

*S*and the chemical potential \(\Phi \) are not affected by the cosmological constant and remain the same. The first law of the black hole thermodynamics is satisfied as

If the heat capacity \(C_P\) is positive, the black hole is thermodynamically stable and if it is negative, the black hole is unstable. The black hole undergoes the second-order phase transition between these stable and unstable phases. This is due to the fact that the heat capacity \(C_{P}\) suffers from discontinuity at the extremal points \(r_\mathrm{ex}\) satisfying \(F_{2}(r_\mathrm{ex})=0\). In Fig. 6, we plot the black hole heat capacity in terms of the radius of the event horizon under various values of the pressure and the charge *q*.

The numerical results for the radius \(r_{HP}\) and the temperature \(T_{HP}\) at the Hawking–Page phase transition, under the different values of the pressure and the nonlinear parameter, at \(\alpha =0.5\) and \(q=\sqrt{2}\)

| \(k=0.4\) | \(k=0.8\) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.002 | 0.005 | 0.010 | 0.015 | 0.020 | 0.003 | 0.008 | 0.012 | 0.018 | 0.025 | |

\(r_{HP}\) | 9.89057 | 4.97141 | 2.07537 | 1.76487 | 1.62517 | 7.59375 | 2.43735 | 1.88212 | 1.64399 | 1.50632 |

\(T_{HP}\) | 0.04161 | 0.06025 | 0.06955 | 0.0731 | 0.07582 | 0.04961 | 0.05534 | 0.071 | 0.07454 | 0.07026 |

*G*, we can obtain another kind of phase transition. The Gibbs free energy

*G*is defined by

*G*in terms of the temperature for various values of the pressure and the charge

*q*is plotted in Fig. 7. The Gibbs free energy

*G*changes its sign at which the Hawking–Page (HP) phase transition between the black hole and the thermal AdS space occurs [10]. We give the numerical results for the radius \(r_{HP}\) and the temperature \(T_{HP}\) for the Hawking-Page phase transition in Table 2.

From this table, one may see that the increase of the pressure or cosmological constant results in the decrease of the radius of the HP phase transition and the increase of the associated temperature.

In particular, there are swallowtail structures meaning that the black hole undergoes the first-order phase transition between the thermodynamically stable phases. More specifically, for the black holes belonging in the cases II and III, there is a first-order phase transition which disappears if the pressure is larger than the critical one \(P_c\). On the other hand, for the black hole belonging in the case I, there exists, at least, one first-order phase transition point, irrespective of the pressure.

## 5 *P*–*V* criticality and Critical Exponents

*P*–

*V*criticality and critical exponents of the black holes in the extended phase space. From Eq. (41), the equation of state \(P=P(T,V)\) is given by

*V*, \(r_{+}=\left( \frac{2}{\pi ^{2}}V\right) ^{1/4}\), from Eq. (47). By comparing with the Van der Waals equation [23], one may identify the specific volume

*v*of the fluid with the event horizon radius \(r_+\) as

*P*–

*V*criticality, we present the relation between the critical quantities and the nonlinear coupling, given numerically in Table 3.

The numerical results for the critical radius, temperature and pressure are given in terms of the nonlinear coupling under various values of the electric charge, at \(\alpha =0.4\)

\(q=0.9\) | \(q=1.5\) | ||||||
---|---|---|---|---|---|---|---|

| \(r_c\) | \(T_c\) | \(P_c\) | | \(r_c\) | \(T_c\) | \(P_c\) |

0.0 | 2.42 | 0.0706 | 0.00763 | 0.0 | 2.65 | 0.0681 | 0.00692 |

0.5 | 2.39 | 0.0708 | 0.00769 | 0.7 | 2.60 | 0.0685 | 0.00705 |

1.0 | 2.36 | 0.0710 | 0.00775 | 1.4 | 2.53 | 0.0670 | 0.00718 |

1.5 | 2.33 | 0.0711 | 0.00781 | 2.1 | 2.47 | 0.0694 | 0.00731 |

2.0 | 2.31 | 0.0713 | 0.00787 | 2.8 | 2.41 | 0.0698 | 0.00744 |

2.5 | 2.28 | 0.0715 | 0.00792 | 3.5 | 2.35 | 0.0702 | 0.00757 |

3.0 | 2.26 | 0.0716 | 0.00797 | 4.2 | 2.30 | 0.0706 | 0.00770 |

3.5 | 2.25 | 0.0717 | 0.00801 | 4.9 | 2.25 | 0.0710 | 0.00782 |

4.0 | 2.23 | 0.0719 | 0.00806 | 5.6 | 2.21 | 0.0713 | 0.00793 |

As the nonlinear coupling increases, both the critical temperature and the critical pressure increase and approach eventually at the values \(r_c=2.19\), \(T_c=0.0726\), and \(P_c=0.00829\) for \(\alpha =0.4\), which correspond to the \(q\rightarrow 0\) (or \(k\rightarrow \infty \)) limit.

*S*is a function of the volume

*V*only, independent of the temperature

*T*. This means that \(C_V=0\) and thus the critical exponent \(\alpha =0\).

## 6 Conclusion

One of the main motivation of this work has been to find the regular black hole solution which does not have the curvature singularity. One way to obtain such a solution is to smooth out the geometry by adding nontrivial source like *U*(1) gauge field with higher order derivatives. In this work, we have found a new class of 5*D* charged AdS black hole solutions in Einstein–Gauss–Bonnet gravity coupled to a nonlinear electromagnetic field. In particular, the solutions admit not only extremal black holes, but also regular black hole solution.

In general, the regular black hole has non-zero Hawking temperature and thus radiates. Our solution suggests various interesting scenarios on the fate of the black holes after Hawking radiation. Depending on the initial state of black holes, the end point of our black holes would be the extremal black hole with curvature singularity or would be the regular extremal black hole. The perspective to obtain extremal black holes without any curvature singularity, even with small possibility, is very exciting.

We have investigated the thermodynamics and the phase transitions of these black holes, in the extended phase space where the cosmological constant is taken into account as the pressure. Based on the behavior of heat capacity at constant pressure, we have found the second-order phase transitions. And by studying the behavior of the Gibbs free energy, we have found the Hawking-Page phase transition. Through the \(P-r_+\) diagram we have shown that the black hole can undergo liquid/gas-like phase transition if the black hole temperature is below a critical value. Finally, we have computed the critical exponents which govern the behavior of the specific heat at constant volume, the order parameter, the isothermal compressibility and the critical isotherm near the critical point and found that they have the same values as those for the Van der Waals fluid.

## Footnotes

## Notes

### Acknowledgements

We would like to thank Dr. Kyung Kiu Kim for useful discussion. 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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