# Scale-dependent (\(2+1\))-dimensional electrically charged black holes in Einstein-power-Maxwell theory

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## Abstract

In this work we extend and generalize our previous work on the scale dependence at the level of the effective action of black holes in the presence of non-linear electrodynamics. In particular, we consider the Einstein-power-Maxwell theory without a cosmological constant in (\(2+1\)) dimensions, assuming a scale dependence of both the gravitational and the electromagnetic coupling and we investigate in detail how the scale-dependent scenario affects the horizon and thermodynamic properties of the classical black holes for any value of the power parameter. In addition, we solve the corresponding effective field equations imposing the “null energy condition” in order to obtain analytical solutions. The implications of quantum corrections are also briefly discussed.

## 1 Introduction

Three-dimensional gravity is attracting a lot of attention for several reasons. On one hand due to the deep connection to Yang–Mills and Chern–Simons theory [1, 2, 3]. On the other hand in this lower dimensional gravitational theory, there are no propagating degrees of freedom, which makes analytic manipulations much more accessible. Furthermore, three-dimensional black holes are characterized by properties also found in their four-dimensional counterparts, such as horizon radius, temperature, entropy etc. Therefore, three-dimensional gravity allows to get deep insight into the corresponding systems that live in four-dimensions.

The main motivation to study non-linear electrodynamics (NLED) was to overcome certain problems present in the standard Maxwell’s theory. Initially, the so called Born–Infeld non-linear electrodynamics was introduced in the 30’s in order to obtain a finite self-energy of point-like charges [4]. During the last decades, these type of models reappear in the open sector of superstring theories [5, 6, 7, 8] as their describe the dynamics of D-branes [9, 10]. Similarly, in heterotic string theory a Gauss–Bonnet term coupled to quartic contractions of the Maxwell field strength appears [11, 12, 13, 14, 15].

Also, this kind of electrodynamics has been coupled to gravity in order to obtain, for example, regular black hole solutions [16, 17, 18], semiclassical corrections to the black hole entropy [19], and novel exact solutions with a cosmological constant acting as an effective Born–Infeld cut-off [20].

A particularly interesting class of NLED theories is the so called power-Maxwell theory (EpM hereafter). There are several reasons to study the Einstein-power-Maxwell electrodynamics, as it was recently pointed out in [21]: “In recent years, the use of power Maxwell fields has attracted considerable interest. It has been used for obtaining solutions in d-spacetime dimensions [22], Ricci flat rotating black branes with a conformally Maxwell source [23], Lovelock black holes [24], Gauss–Bonnet gravity [25], and the effect of power Maxwell field on the magnetic solutions in Gauss–Bonnet gravity [26]”.

The EpM theory is described by a Lagrangian density of the form \({\mathcal {L}}(F)=F^{\beta }\), where \(F=F_{\mu \nu }F^{\mu \nu }/4\) is the Maxwell invariant, and \(\beta \) is an arbitrary rational number. When \(\beta =1\) one recovers the standard linear electrodynamics, while for \(\beta =D/4\), with *D* being the dimensionality of space time, the electromagnetic energy momentum tensor is traceless [27, 28]. In three dimensions the generic black hole solution without imposing the traceless condition has been found in [21], while black hole solutions in linear Einstein–Maxwell theory are given in [29, 30]. Other interesting solutions and properties of black holes in the presence of power-Maxwell theory have been found in [22, 25, 31, 32, 33, 34], whereas some topological black hole solutions with power-law Maxwell fields have been investigated in [35, 36, 37], as well as Born–Infeld theory in [38, 39]. Interesting features arise from a study of the thermodynamic properties of EpM black holes, as discussed in [32].

It is well-known that one of the open issues in modern theoretical physics is a consistent formulation of quantum gravity. Although there are several approaches to the problem (for an incomplete list see e.g. [40, 41, 42, 43, 44, 45, 46, 47, 48] and references therein), most of them have something in common, namely that the basic parameters that enter into the action, such as Newton’s constant, the cosmological constant or the electromagnetic coupling, become scale-dependent quantities. As scale dependence at the level of the effective action is a generic result of quantum field theory, the resulting effective action of scale-dependent gravity is expected to modify the properties of classical black hole backgrounds.

It is the aim of this work to study the scale dependence at the level of the effective action of three-dimensional charged black holes in the presence of the Einstein-power-Maxwell non-linear electrodynamics for any value of the power parameter, extending and generalizing previous work [49], where we imposed the traceless condition \(\beta =3/4\). We will use the formalism and notation of [49].

Our work is organized as follows. After this introduction we present the model and the field equations. Section 3 is devoted to introduce the classical black hole background. In Sects. 4 and 5 we allow for scale dependent couplings, we impose the “null energy condition”, and after that we present our solution for the metric lapse function as well as for the couplings in the scale dependent scenario. In Sect. 7 we briefly discuss our main findings, concluding in the same section.

## 2 Classical Einstein-power-Maxwell theory

*R*is the Ricci scalar, \({\mathcal {L}}(F)\) is the electromagnetic Lagrangian density, \(\gamma \) is a proportionality constant,

*F*is the Maxwell invariant previously defined, and \(F_{\mu \nu } = \partial _{\mu }A_{\nu } - \partial _{\nu }A_{\mu }\) is the electromagnetic field strength tensor. We use the metric signature \((-, +, +)\), and natural units (\(c = \hbar = k_B = 1\)) such that the action is dimensionless. Note that \(\beta \) is an arbitrary rational number, which also appears in the exponent of the electromagnetic coupling in order to maintain the action dimensionless. It is easy to check that the special case \(\beta = 1\) reproduces the classical Einstein–Maxwell action, and thus the standard electrodynamics is recovered. For \(\beta \ne 1\) one can obtain Maxwell-like solutions. In the following we shall consider the general case, so that \(\beta \) is taken to be a free parameter. As our solution should reproduce the classical one, we restrict the values of this parameter by demanding the energy conditions to be satisfied. According to [21], we will only take into account the (naive) range \(\beta \in \mathfrak {R}^+\) (our solution, however, could have additional forbidden values of the parameter \(\beta \)). The classical equations of motion for the metric field are given by Einstein’s field equations

*E*(

*r*) is given by

## 3 Black hole solution for Einstein–Maxwell model of arbitrary power

*C*is related with the mass of the black hole \(M_0\) while

*B*takes into account the classical charge \(Q_0\) (the same for the parameter

*A*). In addition, note that the auxiliary parameter \(\alpha \) is defined as follow:

*S*, and the specific heat, \(C_Q\). Their corresponding expressions are given by

## 4 Scale dependent coupling and scale setting

*k*(

*x*). The effective action for this theory reads

*k*has to be set to a quantity characterizing the physical system under consideration. Thus, for background solutions of the gap equations, it is not constant anymore. However, having an arbitrarily chosen non-constant \(k=k(x)\) implies that the set of equations of motion does not close consistently. This implies that the stress energy tensor is most likely not conserved for almost any choice of the functional dependence \(k=k(x)\). This type of scenario has been largely explored in the context of renormalization group improvement of black holes in asymptotic safety scenarios [59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74]. The loss of a conservation laws comes from the fact that there is one consistency equation missing. This missing equation can be obtained from varying the effective action (16) with respect to the scale field

*k*(

*r*), i.e.

## 5 The null energy condition

## 6 Scale dependent Einstein-power-Maxwell theory

### 6.1 Solution

*G*(

*r*), i.e.

*E*(

*r*) and the electromagnetic coupling \(e(r)^{\alpha +1}\). Then, we have

*f*(

*r*) and the electromagnetic coupling \(e(r)^{\alpha + 1}\) gives the solution:

*D*is an auxiliary parameter given by

*a*,

*b*,

*c*, and

*z*as long as \(|z| <1\). Outside the circle \(|z| < 1\), the function is defined as the analytic continuation with respect to

*z*of this sum, the parameters

*a*,

*b*,

*c*held fixed [91]. Besides, the special case \(_2{\tilde{F}}_1(a,b;c;z) = 0\) is forbidden because we assume a non-null \(_2{\tilde{F}}_1\) in the computation of thermodynamic quantities. In general, the constants are chosen such that the solution matches the classical case when the running parameter is switched off \(\epsilon \rightarrow 0\). However, as the final result depends on the value of the free index \(\beta \) (or \(\alpha \)), we first need to take some particular values of these parameters. We must emphasize the number of integration constants involved into the problem. Firstly, the scale-dependent gravitational coupling introduce two of them, i.e. \(G_0\) and \(\epsilon \). This is because we are in the presence of a second order differential equation. The electromagnetic field gives and additional integration constant

*A*whereas the solution for the lapse function implies two additional integration constants

*B*and

*C*(for the same reason as is the gravitational coupling case). Thus, the integration constant

*C*can be associated with the classical mass of the black hole \(M_0\), an the constant

*B*encodes the classical charge \(Q_0\). Following Ref. [21] we can set the relation between our integration constants and the classical counterpart as:

### 6.2 Asymptotic behaviour

#### 6.2.1 Asymptotics for \(r \rightarrow 0\)

#### 6.2.2 Asymptotics for \(r \rightarrow \infty \)

### 6.3 Horizons

*f*(

*r*), it is required to select a certain value of the index \(\beta \) (or \(\alpha \)). Note that the effect of scale dependence (\(\epsilon \ne 0\)) can be understood as a non-trivial deviation from the classical solution (\(\epsilon =0\)). As we commented before, we will focus on models where \(\alpha \ge 2\). The corresponding lapse functions in that regime has a polynomial structure and the roots usually have a complex form. As a benchmark point, we will revisit the solution for \(\alpha =2\) which was previously discussed in Ref. [49]. Note that, although we are able to produce physical solutions for \(\alpha \ge 2\), only a single case will be shown here explicitly. Figure 2 show the behavior of that solution plus two additional cases assuming \(\alpha = \{3,4\}\). The scale dependent lapse function \(f(r;\alpha )\) is, for \(\alpha =2\),

### 6.4 Thermodynamic properties

*M*, which unfortunately in the present work is unknown. In spite of that, to get some insight into the underlying physics, we take the case where \(\alpha =2\) to exemplify how the new Smarr-like relation looks like. It is straightforward to check that in the classical theory one obtains:

Before we conclude our work a final comment is in order here. The no-go theorem of [93], which links the existence of smooth black hole horizons to the presence of a negative cosmological constant, does not apply in the given case. First, the theorem is based on unmodified classical Einstein Field equations, which is not the case in scale-dependent scenarios. Second, the no-go theorem assumes the dominant energy condition which is not part of our assumptions. Instead, we take advantage of the so-called null energy condition. Furthermore, and most importantly, given the solutions previously presented one can check that they do have smooth horizons and well behaved asymptotic spacetimes, and therefore they are black holes. Note that even the classical solution in [21] was shown to be a black hole in this sense, even for a vanishing cosmological constant.

## 7 Conclusions

In the present article we have studied the effect of scale dependent couplings on charged black holes in the presence of three-dimensional Einstein-power-Maxwell non-linear electrodynamics for any value of the power parameter, extending and generalizing a previous work. First we presented the model and the classical black hole solution assuming static circular symmetry, and then we allowed for a scale dependence of the couplings, both the electromagnetic and the gravitational one. We solved the corresponding effective field equations applying the same formalism already used in our previous work, namely by imposing the “null energy condition”. Black hole properties, such as horizon structure, Hawking temperature, Bekenstein–Hawking entropy as well as asymptotic properties, are discussed in detail. In order to show how the scale-dependent scenario modifies the classical solution, we have considered three different benchmark cases taking \(\alpha = \{2, 3, 4 \}\) which are shown in Figs. 1, 2 and 3. The aforementioned solutions have a manageable mathematical structure which allows to obtain analytical expressions for the physical quantities. The solutions obtained in this work and our main numerical results show that the scale-dependent scenario allows us to induce deviations from classical black hole solutions, confirming a result already reported in [21]. In particular, it is worth mentioning that the behavior of the electromagnetic coupling depends drastically on the choice of the parameter \(\alpha \). Regarding the basic black hole properties, we have found that for a fixed classical black hole mass, the Hawking temperature increases with \(\epsilon \), while both the event horizon radius and the Bekenstein–Hawking entropy decrease when the strength of the scale dependence increases. Our findings imply that quantum corrections may have an remarkable effect, i.e. the black hole becomes hotter and at the same time loses less information compared to its classical counterpart. This is in agreement with the findings in [59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74]. Finally, it is well-known that a black hole, viewed as a thermodynamical system, is locally stable if its heat capacity is positive [94]. We have found that the black holes studied here are unstable (\(C_Q < 0\)), both classically and in the scale dependent scenario. To conclude, our results allow us to gain a solid understanding of the most important modifications that a possible scale dependence would imply for the Einstein–Maxwell black holes of arbitrary power in \(2+1\) dimensions.

## Notes

### Acknowledgements

The author A.R. was supported by the CONICYT-PCHA/ Doctorado Nacional/2015-21151658. The author P.B. was supported by the Faculty of Science and Vicerrectoría de Investigaciones of Universidad de los Andes, Bogotá, Colombia. The author B.K. was supported by the Fondecyt 1161150. The author G.P. thanks the Fundação para a Ciência e Tecnologia (FCT), Portugal, for the financial support to the Center for Astrophysics and Gravitation-CENTRA, Instituto Superior Técnico, Universidade de Lisboa, through the Grant No. UID/FIS/00099/2013.

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