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Physics Letters B 801 (2020) 135191

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

AdS

black holes with nonlinear source in rainbow gravity

M. Dehghani

Department of Physics, Razi University, Kermanshah, Iran

a r t i c l e i n f o a b s t r a c t

Article history:

Received

28 July 2019

Received

in revised form 27 October 2019

Accepted

23 December 2019

Available

online 3 January 2020

Editor: N.

Lambert

Keywords:

Four-dimensional

black hole

Nonlinear

electrodynamics

Anti-de

Sitter black hole

We explore new four-dimensional black holes, with the Einstein- theory coupled to nonlinear

electrodynamics, in an energy-dependent spacetime. By considering the power-law model of nonlinear

electrodynamics and solving the coupled ﬁeld equations, with properly ﬁxing the parameters of the

theory, we obtain two new classes of nonlinearly charged black holes with anti-de Sitter (AdS) asymptote.

With the aim of studying the thermodynamic properties of the novel AdS black holes, we calculate the

conserved and thermodynamic quantities such as black hole mass, electric charge, electric potential,

temperature, and entropy in the presence of rainbow functions. Although some of these quantities are

affected by the rainbow functions, interestingly, they satisfy the ﬁrst law of black hole thermodynamics

in its standard form. Based on the canonical ensemble method and by calculating the black hole heat

capacity, we analyze the black hole local stability or thermodynamic phase transitions. The horizon

radius of those black holes which experience type one or type two phase transition as well as the

conditions under which the black holes remain stable are determined. Finally, via consideration of

black hole thermal ﬂuctuations, we examine the impacts of quantum gravitational corrections on the

thermodynamic properties of our new AdS rainbow black holes.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Combining the gravity with quantum mechanics to establish the quantum theory of gravity is still one of the most important problems

in the context of theoretical physics [1–3]. Gravity’s rainbow, in which the quantum gravitational effects are taken into account, is an

attempt in this direction. This theory is constructed out by extending the doubly special relativity to the curved spacetimes, and can be

considered as the Planck-scale generalization of the usual gravity theory [4,5]. Besides, doubly special relativity is the deformed version of

usual special relativity, in which the usual dispersion relation is promoted to the well-known modiﬁed dispersion relation. Although, the

modiﬁed dispersion relation is expected to play an important role in constructing the quantum theory of gravity, it is no longer Lorentz

invariant. Doubly or deformed special relativity, which is accomplished based on a class of nonlinear Lorentz transformations, preserves

Lorentz invariance of the modiﬁed dispersion relation. In this theory, light speed and Planck energy are two invariant quantities which

determine the upper bounds of the speed and energy that a particle can attain, respectively [6–9].

However,

doubly special relativity, as the Planck-scale formalism of the special relativity, has now been extended to the curved space

times, and it is named conventionally as doubly general relativity or gravity’s rainbow. In rainbow gravity, the geometry of space time

depends on the energy of the test particle, which probes the geometry. Thus, the geometry of space time seems different for the particles

having different amounts of energy. Therefore, instead of a single metric, there is a family of energy-dependent metrics. This is why the

doubly general relativity is named as gravity’s rainbow [10,11].

the other hand, Maxwell’s classical electromagnetic theory, as one of the most successful theories, seems to have some shortcom-

ings.

The related failures are attempted to be addressed by modifying the original action to that of the so-called nonlinear electrodynamics.

Among the various theories of nonlinear electrodynamics, which are nonlinear functions of Maxwell invariant F = F

αβ

, the Born-

Infeld,

power-law, exponential and logarithmic models are of most interest [12–16]. All of the models of nonlinear electrodynamics reduce

to Maxwell’s electrodynamics as a special case [16]. Indeed, nonlinear theories of electrodynamics are more suitable when the electromag-

netic

ﬁelds are of high strength. They contain higher powers of F , which are believed that indicate the photon-photon interactions. For

E-mail address: m.dehghani@razi.ac.ir.

https://doi.org/10.1016/j.physletb.2019.135191

0370-2693/

SCOAP

2 M. Dehghani / Physics Letters B 801 (2020) 135191

the case of enough weak electromagnetic ﬁelds the photon self-interactions are negligible, and Maxwell’s/or linear electromagnetic theory

can be used successfully [17,18]. One of the challenges of the linear or Maxwell’s theory of electrodynamics is that it violates the con-

formal

invariant symmetry in the spacetimes with the dimensions other than four. The power-law theory of nonlinear electrodynamics is

the only model of proposed nonlinear electrodynamics that preserves its conformal invariance in all spacetimes with arbitrary dimensions

[19–21].

Although,

the various models of nonlinear electrodynamics are motivated for solving the problems of the appearance of the inﬁnite

ﬁeld and self-energy for the pointlike charged particles in Maxwell’s theory, this issue has now provided an interesting subject area

for studying the charged exact solutions of the alternative theories of gravity. Now, there are a variety of novel and interesting works

in which Maxwell’s theory is promoted to nonlinear electrodynamics. Motivated by the interesting results provided through the study

of physics in the energy-dependent spacetimes such as black hole remnant [22,23], nonsingular universe [24], this work is devoted to

the study of nonlinearly charged AdS black holes in the presence of rainbow functions. Three-dimensional exact black hole solutions

have been obtained, and the related thermodynamic properties have been studied in my last work [25]. Here, this idea is extended to

the case of four-dimensional AdS spacetimes. The main object of this paper is to obtain new exact solutions of the four-dimensional

Einstein-nonlinear electrodynamics with the cosmological constant in energy-dependent spherically symmetric spacetime. Also, the study

of the thermodynamic properties and especially thermodynamic stability or phase transition of the black holes are of interest. Besides,

quantum gravitational effects on the black hole thermodynamics are examined by consideration of the black hole thermal ﬂuctuations.

The

paper is outlined as follows: In Sec.2, by varying the suitable action of four-dimensional Einstein- gravity theory, in the pres-

ence

of the power-law nonlinear electrodynamics, the coupled ﬁeld equations have been solved in an energy-dependent and spherically

symmetric geometry. As a result, two novel classes of nonlinearly charged black holes have been found with the AdS asymptotic behavior.

Sec.3 is devoted to the study of thermodynamic properties of the black hole solutions we just obtained. The temperature, entropy, electric

potential, conserved mass, and electric charge of the new AdS black holes have been calculated in the presence of rainbow functions. It

has been shown that, although some of these quantities get modiﬁed by the rainbow functions, the ﬁrst law of black hole thermodynamics

is still valid for either of the AdS black hole solutions. Then, from the viewpoint of the canonical ensemble method and regarding the

signature of the black hole heat capacity, a thermodynamic stability analysis has been performed. The location of the type one and type

two phase transition points and the conditions under which the black holes are stable have been characterized. Finally, the corrections

arisen from the consideration of quantum gravity effects have been investigated by studying the black hole thermal ﬂuctuations. Sec.4 is

dedicated to summarizing and discussing the results.

2. AdS rainbow black hole solutions

The action of nonlinearly charged four-dimensional black holes in the Einstein- gravity theory is written as [26]

I =−

16π



√

−g

[

R −2 +L(F )

]

x. (2.1)

Note that R is the Ricci scalar,  is the cosmological constant which is written as  =−3

−2

in AdS spacetimes. The last term de-

notes

the electromagnetic Lagrangian density as a function of Maxwell’s invariant F = F

μν

with F

μν

= ∂

− ∂

and A

is the

electromagnetic potential. Here, we are interested on the power-law nonlinear electrodynamics in the following form [20,26]

L(F ) = (−F )

, (2.2)

which states the electromagnetic Lagrangian as the power of Maxwell invariant, with p, as the power, sometimes is named as the nonlin-

earity

parameter. By varying the action (2.1)with respect to gravitational ﬁeld we get the Einstein’s ﬁeld equations as

μν

−

μν

+g

μν

(−F )

+2p(−F )

p−1

μα

. (2.3)

Also, varying the action (2.1)with respect to the electromagnetic ﬁeld one obtains

∂



√

−gL



(F )F

μν



0, (2.4)

where prime means derivative with respect to the argument. Note that F

is the only non-vanishing element of the Faraday’s tensor. Now

we explore the solutions of the gravitational and electromagnetic ﬁeld equations as given in Eqs. (2.3) and (2.4). In this regard, we start

with a four-dimensional static and spherically symmetric energy-dependent line element as an ansatz. It takes the following general form

[27]

=−

V (r)

(ε)



V (r)

(dθ

+sin

θdϕ

)



. (2.5)

Here, f (ε) and g(ε) are known as the so-called temporal and spatial rainbow functions, respectively. The argument ε is a dimensionless

quantity deﬁned by ε = E/E

, in which E

is the Planck-scale energy and E denotes the test particle’s energy. They are restricted to fulﬁll

following requirements

lim

ε→0

f (ε) = 1, and lim

ε→0

g(ε) = 1. (2.6)

With the help of above conditions one can recover the infrared limit of the results letting ε → 0. There are some proposed forms for f (ε)

and g(ε) as functions of ε, among them are [28,29]

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