Advances in High Energy Physics

Volume 2016 (2016), Article ID 3265968, 13 pages

http://dx.doi.org/10.1155/2016/3265968

## Thermodynamics of Charged Rotating Dilaton Black Branes Coupled to Logarithmic Nonlinear Electrodynamics

^{1}Physics Department and Biruni Observatory, College of Sciences, Shiraz University, P.O. Box 55134-441, Shiraz 71454, Iran^{2}Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran

Received 1 March 2016; Accepted 20 April 2016

Academic Editor: Elias C. Vagenas

Copyright © 2016 A. Sheykhi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We construct a new class of charged rotating black brane solutions in the presence of logarithmic nonlinear electrodynamics with complete set of the rotation parameters in arbitrary dimensions. The topology of the horizon of these rotating black branes is flat, while due to the presence of the dilaton field the asymptotic behavior of them is neither flat nor (anti-)de Sitter [(A)dS]. We investigate the physical properties of the solutions. The mass and angular momentum of the spacetime are obtained by using the counterterm method inspired by AdS/CFT correspondence. We derive temperature, electric potential, and entropy associated with the horizon and check the validity of the first law of thermodynamics on the black brane horizon. We study thermal stability of the solutions in both canonical and grand-canonical ensemble and disclose the effects of the rotation parameter, nonlinearity of electrodynamics, and dilaton field on the thermal stability conditions. We find the solutions are thermally stable for , while for the solutions may encounter an unstable phase, where is dilaton-electromagnetic coupling constant.

#### 1. Introduction

Thermodynamics of black holes plays a central role in the attractive modern method relating classical gravity and high energy physics, namely, gauge/gravity duality. The issue was first taken into consideration by Bekenstein [1] and Hawking [2] and encountered increasing interests rapidly. Among different frameworks, the thermodynamics of black solutions has been studied; dilaton gravity possesses a significant predominancy. This preference has at least two reasons. From one side, the dilaton gravity which is one of the modified gravities is able to justify the accelerating expansion of the universe confirmed from observations [3–9] while Einstein gravity (General Relativity) requires exotic matter violating energy conditions to justify this phase of universe. From another side, the dilaton gravity appears in the low energy limit of string theory [10] and therefore can provide a good laboratory for testing this theory in the low energy limit through gauge/gravity duality. Since string theory proposes higher than four dimensions [10], it is natural to consider higher dimensional solutions within the gravity theories come from string theory.

Exact asymptotically flat solutions of Einstein-Maxwell-dilaton gravity have been constructed in the absence of dilaton potential in [11–20]. However, breaking of spacetime supersymmetry in ten dimensions may cause one or more Liouville-type potentials in the action of dilaton gravity. This type of dilaton potential changes the asymptotic behavior of solutions [17–19, 21–26]. In general, these solutions are neither asymptotically flat nor (anti-)de Sitter [(A)dS]. Thermodynamics of topological dilaton black holes in Einstein-Maxwell gravity has been explored in [27]. Asymptotically nonflat and non-(A)dS linearly charged rotating black branes were under investigation in [28, 29]. Slowly rotating charged black holes have been studied from thermodynamics point of view as well [30].

A natural interesting extension of such solutions is to change the electrodynamics Lagrangian from linear Maxwell to nonlinear ones. Some efforts have been done to construct exact solution in dilaton gravity with nonlinear electrodynamics. For example, thermodynamics of static black hole solutions in the presence of nonlinear power-law Maxwell (PLM) [31], Born-Infeld (BI) [32], and exponential [33, 34] electrodynamics has been investigated. As well, thermodynamic properties of rotating black brane solutions have been studied in the presence of PLM [35], BI [36], and exponential [37] nonlinear electrodynamics. Any one of the abovementioned nonlinear electrodynamics has its own importance and motivations. For instance, the PLM electrodynamics extends the conformal invariance property of Linear Maxwell Lagrangian in four dimensions to higher dimensional spacetimes, while BI [38] electrodynamics, which comes from open string theory [39–43], solves the problem of infinite self-energy of charged point-particle that appears in linear Maxwell case. The latter problem is also overcome by logarithmic nonlinear electrodynamics proposed for the first time in [44]. Although this type of nonlinear electrodynamics has no direct relation with superstring theory, it can be motivated from different sides. First, it can be regarded as a toy model showing that certain nonlinear field theories can produce particle-like solutions that can realize the limiting curvature hypothesis in cosmological theories [44]. Second, the behavior of logarithmic electrodynamics and BI electrodynamics Lagrangians are the same for large values of nonlinear parameter . Third, from gauge/gravity duality point of view, the ratio of holographic viscosity to entropy density is not affected by nonlinear terms rising from nonlinear electrodynamics in contrast with gravitational corrections [45]. Fourth, the values of important parameters of a holographic superconductor system such as critical temperature and order parameter are firmly sensitive to choice of electrodynamics [46, 47]. Other studies on the nonlinear electrodynamics have been carried out in [36, 39–41, 48–59].

The above pointed out motivations are convincingly enough to satisfy one to seek for the effects of logarithmic electrodynamics on the solutions. Till now, exact rotating solutions of logarithmic electrodynamics in the context of dilaton gravity have not been constructed. In this paper, we would like to construct the rotating dilaton black branes in the presence of logarithmic nonlinear electrodynamics and investigate their thermodynamics as well as their thermal stability.

The outline of this paper is as follows. In Section 2, we present the basic field equations. In Section 3, we construct the rotating dilaton black branes with a complete set of rotation parameters in all higher dimensions and investigate their properties. In Section 4, we study thermodynamics of the spacetime, by calculating the conserved and thermodynamic quantities. In Section 5, we perform a stability analysis and show that the dilaton creates an unstable phase for the solutions. Section 6 is devoted to conclusions and discussions.

#### 2. Basic Field Equations

We consider an -dimensional action in which gravity is coupled to a dilaton and a nonlinear electrodynamic fieldwhere the Lagrangian of the logarithmic nonlinear electrodynamics coupled to the dilaton field (LNd) is chosen in the following form:In action (1), is the Ricci scalar curvature, is the dilaton field, and is a potential for . The dilaton parameter determines the strength of coupling of the scalar and LNd fields, , where is the electromagnetic tensor field, is the vector potential, and is the nonlinear parameter with dimension of mass. The last term in (1) is the Gibbons-Hawking boundary term which is chosen such that the variational principle is well-defined. The manifold has metric and covariant derivative . is the trace of the extrinsic curvature of any boundary(ies) of the manifold , with induced metric(s) . In this paper, we consider action (1) with a Liouville-type potential,where is a constant which may be referred to as the cosmological constant, since in the absence of the dilaton field () action (1) reduces to the action of Einstein gravity in the presence of nonlinear electrodynamics with cosmological constant. For later convenience, we redefine it as , where is a constant with dimension of length. The series expansion of (2) for large leads to For later convenience we rewrite where we have defined By varying action (1) with respect to the gravitational field , the dilaton field , and the gauge field , we findIn the limiting case where , we have . In this case the system of field equations (7)–(9) restore the well-known equations of EMd gravity [21–27], as expected.

##### 2.1. Finite Action in Canonical and Grand-Canonical Ensembles

In general, the total action given in (1) is divergent when evaluated on a solution. One way of dealing with the divergences of the action is adding some counterterms to action (1). The counterterms should contain a part which removes the divergence of the gravity part of the action and a part for dealing with the divergence of the matter action. Since the horizon of our solution is flat, the counterterm which removes the divergence of the gravity part should be proportional to . The counterterm for the matter part of the action in the presence of the dilaton is given bywhere is given by (16) and is a collection of terms involving derivatives of the boundary fields that could involve the curvature tensor constructed from the boundary metric. Since in our case the boundary is flat so is zero on the boundary. The variation of the total action about the solutions of the equations of motion iswhereEquation (11) shows that the variation of the total action with respect to will only give the equation of motion of the nonlinear massless field provided the variation is at fixed nonlinear massless gauge potential on the boundary. Thus, the total action, , given in (11) is appropriate for the grand-canonical ensemble, where on the boundary. But in the canonical ensemble, where the electric charge is fixed on the boundary, the appropriate action isThe last term in (13) is the generalization of the boundary term introduced by Hawking and Ross for linear electromagnetic field [60] and the results of [61, 62] for the nonlinear Lifshitz black holes to the exponential nonlinear gauge field coupled to the dilaton field. Thus, both in canonical and grand-canonical ensemble, the variation of total action about the solutions of the field equations isThat is, the nonlinear gauge field is absent in the variation of the total action both in canonical and grand-canonical ensembles.

In order to obtain the conserved charges of the spacetime, we use the counterterm method [63–67] inspired by (A)dS/CFT correspondence. For asymptotically AdS solutions this method works very well [64–67]. However, in our paper we have the scalar dilaton field with a Liouville potential. It was argued that the presence of Liouville-type dilaton potential, which is regarded as the generalization of the cosmological constant, changes the asymptotic behavior of the solutions to be neither asymptotically flat nor (A)dS. It has been shown that no dilaton dS or AdS black hole solution exists with the presence of only one Liouville-type dilaton potential [17–19]. But, as in the case of asymptotically AdS spacetimes, according to the domain-wall/QFT (quantum field theory) correspondence [68–70], there may be a suitable counterterm for the stress-energy tensor which removes the divergences. In this paper, we deal with the spacetimes with zero curvature boundary [], and therefore the counterterm for the stress-energy tensor should be proportional to . We find the finite stress-energy tensor in -dimensional Einstein-dilaton gravity with Liouville-type in the form [36, 59]where is given by

In the particular case , the effective of (16) reduces to of the AdS spacetimes. The first two terms in (15) are the variation of action (1) with respect to , and the last term is the counterterm which removes the divergences. One may note that the counterterm has the same form as in the case of asymptotically AdS solutions with zero curvature boundary, where is replaced by . If we choose the Killing vector field on spacelike surface in with metric , then the quasilocal conserved quantities may be obtained from the following relation [36, 59]:where is the determinant of the boundary metric and is the unit normal vector on the boundary . In our case, the boundary has two Killing vector fields timelike () and rotational (). The corresponding conserved charges are the quasilocal mass and angular momentum may be obtained as

#### 3. Rotating Dilaton Black Branes in Higher Dimensions

In this section, we would like to construct the rotating black brane solutions of the field equations (7)–(9) with rotation parameters. The number of independent rotation parameters for an -dimensional localized object is equal to the number of Casimir operators, which is , where is the integer part of [71]. The metric of -dimensional rotating solution with cylindrical or toroidal horizons and rotation parameters can be written as [72, 73]where ’s are rotation parameters. There are two unknown functions and in the above metric which should be determined by solving the field equations. The range of the angular coordinates is and is the Euclidean metric on the -dimensional submanifold with volume .

First of all, we integrate the electromagnetic field equation (9). The result iswhere is an integration constant related to the electric charge of the brane. When , reduces to the electric field of -dimensional black brane of Einstein-Maxwell-dilaton gravity [71]In order to solve the system of (7) and (8) for three unknown functions , , and , we make the ansatz [71]In order to justify this choice for the metric function , let us note that is indeed added to the metric (20) in order to increase the degrees of freedom for obtaining solutions in the presence of the dilaton field. Choosing in the form of (23) is an ansatz. However, it is chosen such that, in the absence of the dilaton field , we have , as expected. With this ansatz, we are able to solve the field equation, analytically.

Substituting (23), the electromagnetic fields (21), and the metric (20) into the field equations (7) and (8), one can obtain the following solutions:where and are constant of integration. We find that these solutions will fully satisfy the system of (7) and (8) provided we choose . Note that has the dimension of [Length] to make the argument of logarithmic function dimensionless. In the above solutions , andIn the above expression, appears as an integration constant and is related to the mass of the black hole. The integration of (24) can be performed using the MATHEMATICA software. The resulting solution can be writtenwhere is the hypergeometric function [74, 75]. It is worth mentioning that the solutions are ill-defined for . We expect that, for large , the function reduces to the -dimensional charged rotating dilaton black brane solutions given in [71]. Indeed, if we expand (27) for large , we arrive atSetting in (28), we reach The last term in the right hand side of the above expression is the leading nonlinear correction to the AdS black brane with dilaton field. In the absence of a nontrivial dilaton (), the above solutions reduce to the asymptotically AdS charged rotating black brane solutions of Einstein gravity in the presence of EN electrodynamics [52, 53]. Finally, in the limit and , the solution given by (28) has the form of the asymptotically AdS black brane solutions [72, 76, 77]. Figures 1 and 2 depict the behavior of given by (27) for different ’s and ’s, respectively.