# Non-linear charged dS black hole and its thermodynamics and phase transitions

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## Abstract

From solving the equations of the motion for a system of Einstein gravity coupled to a non-linear electromagnetic field in the dS spacetime with two integral constants, we derived a static and spherical symmetric non-linear magnetic-charged black hole. It is indicated that this black hole solution behaves like a dS geometry in the short-distance regime. And, thus this black hole is regular. The structure of the black hole horizons is studied in detail. Also, we investigated the thermodynamics and the thermal phase transition of the black hole in both the local and global views. By observing the discontinuous change of the specific heat sign and the swallowtail structure of the free energy, we showed that the black hole can undergo a thermal phase transition between a thermodynamically unstable phase and a thermodynamically stable phase.

## 1 Introduction

In General Relativity, the black hole solutions display a curvature singularity at the origin surrounded by an event horizon [1]. The presence of this curvature singularity is usually regarded as a sign of the breakdown of the classical gravity. It is widely believed that at the (very) short distances quantum gravity should become important to suppress the infinite growth of the spacetime curvature scalars and other physical quantities. And, thus the curvature singularity would be replaced by a regular spacetime region. However, so far there has no a consistent quantum theory of gravity. In the absence of such a theory, the resolution of the black hole singularity at the (semi-)classical level remains open.

In attempting to eliminate the problem of infinite energy of the electron, Born and Infeld proposed the non-linear electrodynamics as modifying the standard Maxwell theory at the short distances [2]. But, the non-linear electrodynamics did not solve this problem and thus was less popular. In recent years, the non-linear electrodynamics has received considerable attention because it leads to the regular black hole solutions. In 1998, Ayón-Beato and García studied a system of Einstein gravity coupled to a non-linear electromagnetic field in asymptotically flat spacetime and derived a regular electric-charged black hole solution [3]. Also, they reobtained the Bardeen black hole [4], which is regular, as a gravitational collapse of some magnetic monopole in the non-linear electrodynamics [5]. Later, many different regular black hole solutions in the non-linear electrodynamics have been derived [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

Astronomical observations show that our universe is undergoing accelerated expansion [18, 19, 20]. This accelerating expansion may be explained by unknown dark energy. There are various proposed explanations for dark energy, but a positive cosmological constant is usually considered as the simplest explanation for dark energy. Because of this fact, it is necessary to consider the black hole by including the positive cosmological constant, corresponding to the black hole in the de Sitter (dS) spacetime [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 89].

*V*[55, 56, 57, 58, 59]. This leads to an extended phase space at which the black hole mass is most naturally interpreted as the enthalpy [56]. Then, the thermodynamics and phase transitions of black holes in the extended phase space have been studied extensively in the literature, with new phenomena derived. The black hole can undergo the van der Waals-like phase transition which is similar to the liquid-gas phase transition [60, 61, 62, 63, 64, 65, 66, 67, 68, 69]. It was found that the black holes shows multiply reentrant phase transition and triple points [70, 71, 72]. It is also possible to consider a Carnot-cycle heat engine for the black hole, which is defined by a closed path in the

*P*–

*V*plane [73, 74, 75, 76, 77, 78, 79, 80, 81].

In this paper, motivated by the pioneering work of Ayón-Beato and García and the astrophysical observations of dark energy, we would like to study the charged dS black hole in the non-linear electrodynamics. We will introduce a system of Einstein gravity coupled to a non-linear electromagnetic field in the dS spacetime. Then, we successfully construct a static and spherical symmetric non-linear magnetic-charged black hole and study the large and short distance behaviors as well as the horizon properties of this black hole. These are given in Sect. 2. The thermodynamics and the phase transition of the black hole are investigated, in both the local and global views, in Sect. 3. Finally, we make the conclusion in Sect. 4.

## 2 Non-linear magnetic-charged black hole in the \(\text {d}\)S spacetime

*R*is the scalar curvature of the dS spacetime, \(\Lambda \) is the positive cosmological constant, and \(\mathcal {L}(F)\) is a function of the invariant \(F_{\mu \nu }F^{\mu \nu }/4\equiv F\) with \(F_{\mu \nu }=\partial _\mu A_\nu -\partial _\nu A_\mu \) to be the field strength of the non-linear electromagnetic field. In this paper, the non-linear electrodynamic term \(\mathcal {L}(F)\) is explicitly given as

*M*and

*Q*are mass and charge of the system, respectively. From the action, one can derive the equations of motion

*M*and the magnetic charge

*Q*(\(>0\)), given by ansatz

*Q*is defined as

*M*and

*Q*are two integral constants which should be used to integrate Eqs. (4) and (5). From Eqs. (5), (6) and (9), one can easily derive

*m*(

*r*) into

*f*(

*r*), we finally get

*R*, \(R_{\mu \nu }R^{\mu \nu }\), and \(R_{\mu \nu \rho \lambda }R^{\mu \nu \rho \lambda }\) which are indeed finite everywhere.

*M*and its horizon radius is established from the equation of the horizons \(f(r)=0\) as

*Q*and the cosmological constant \(\Lambda \), the black hole mass curve displays one local minimum following one local maximum which both are located above the horizontal axis. These suggest that the black hole possesses possibly the inner horizon \(r_-\), the event horizon \(r_+\) (\(\geqslant r_-\)) and the cosmological horizon \(r_c\) (\(\geqslant r_+\)). (The cases of black holes and no black hole are explicitly given in Fig. 1.) When the local minimum and local maximum points of the black hole mass curve merge into an inflexion point, these three horizons coincide together. In this case, the hole black is called the ultracold black hole whose horizon radius and mass are, with given magnetic charge

*Q*, given as

*Q*and the cosmological constant \(\Lambda \). As \(M=M_{\text {max}}\) the event horizon \(r_+\) and the cosmological horizon \(r_c\) coincide together, and such a black hole is called the Nariai black hole whose event horizon radius \(r_N\) is the largest positive real solution of the following equation

## 3 Thermodynamics and thermal phase transition

*S*, the magnetic charge

*Q*, and the thermodynamic pressure

*P*are regarded as a complete set of the extensive thermodynamic variables. Their corresponding conjugating quantities, which are intensive thermodynamic variables, are the temperature

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

*V*. In this way, the first law of the thermodynamics is established on the event and cosmological horizons, respectively

*M*should decrease whereas the entropy always does not decrease.) Note that, from Eqs. (1) and (16) the black hole mass can be expressed on the event and cosmological horizons, respectively

### 3.1 Local view

In this subsection, we consider that the event and cosmological horizons are located far away. Thus, one can analyze the thermodynamics and the thermal phase transition on these horizons in an independent way.

*Q*.

*Q*fixed, we can obtain the equation of state on the event and cosmological horizons, respectively

### 3.2 Global view

If the event and cosmological horizons are not located far away, one cannot analyze the thermodynamics and the thermal phase transition on them in an independent way. Because the temperatures on the event and cosmological horizons are different, except for the degenerate case \(r_+=r_c\), the black hole cannot in general be in thermodynamic equilibrium. Since the notion of the effective thermodynamics of the black hole has been emerged [36, 44, 84, 85, 86, 87, 88, 90].

*M*of the black hole and the cosmological constant \(\Lambda \) in terms of \(r_+\), \(r_c\), and

*Q*as

*Q*. For “large”

*Q*(the blue and red curves), there exist two regions (\(R_I\) and \(R_{II}\)) which are separated by a negative temperature region or forbidden region \(r_+\in (r_{div},r_0)\), where \(r_{div}\) and \(r_0\) are respectively a divergent point and a zero-temperature point of \(T_{eff}\). The first region \(R_I\) of \(T_{eff}\) is an increasingly monotonous function of \(r_+\). Whereas, with respect to the second region \(R_{II}\) of \(T_{eff}\) the effective temperature should first increase until a maximum and then decrease when \(r_+\) increasing. Note that, because of the forbidden region, if the black hole stays in one of the two regions then it always stays in that region and is impossible to transit another region. For “small”

*Q*(the green curve), the first region \(R_I\) of \(T_{eff}\) should disappear. For small

*Q*but below a certain value (the purple curve), the effective temperature is only a decreasingly monotonous function of \(r_+\).

The behavior of the heat capacity \(C_P\) is explicitly depicted in Fig. 11. From this figure, we can analyze the thermodynamic stability and the thermal phase transition of the black hole. For large magnetic charge (top left and top right), if the black hole stays in the first region \(R_I\), because of the negative and regular heat capacity \(C_P\) the black hole is thermodynamically unstable and there has no thermal phase transition. Otherwise, if the black hole stays in the second region \(R_{II}\), the heat capacity \(C_P\) should suffer a discontinuity. This suggests a thermal phase transition, at this discontinuous point, between a thermodynamically unstable large black hole (negative \(C_P\)) and a thermodynamically stable small black hole (positive \(C_P\)). For small magnetic charge (bottom left), the first region \(R_I\) should disappear and thus the black hole always stays in the second region \(R_{II}\). This means that the black hole can undergo the thermal phase transition between a thermodynamically unstable phase and a thermodynamically stable phase. For small magnetic charge but below a certain value (bottom right), like the black hole staying in the first region \(R_I\), the black hole is thermodynamically unstable and there has no thermal phase transition.

*F*given by

*F*is a multivalued function shown by the presence of the swallowtail structure. And, thus there should actually occur the thermal phase transition between a thermodynamically unstable phase (large black hole) and a thermodynamically stable phase (small black hole). Here, the black hole of the larger magnetic charge has the smaller critical effective temperature.

## 4 Conclusion

*M*,

*Q*, and \(\Lambda \) are the black hole mass, the black hole magnetic-charge, and the positive cosmological constant, respectively. By this, the singularity at the origin should be replaced by a core of the dS geometry, and thus the black hole solution is regular. Also, we indicated a critical value for the cosmological constant, above which there exists no black hole for any mass. At this critical value, all horizons of the black hole coincide together. Below this critical value, the black hole has possibly three horizons: the inner, event and cosmological horizons. In particular, for suitable black hole mass, two of three horizons possibly coincide together.

In the extended phase space, we have studied the thermodynamics of the black hole and analyzed its thermal phase transition based on the discontinuous change of the specific heat sign and the swallowtail structure of the free energy. If the event and cosmological horizons are located far away, the thermodynamics and the thermal phase transition on these horizons are investigated in an independent way. On the contrary, we use the notion of the effective thermodynamics of the black hole, which has been emerged in the recent years, to investigate.

## 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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