Thermodynamic analysis of topological black holes in Gauss-Bonnet gravity with nonlinear source

Employing two classes of nonlinear electrodynamics, we obtain topological black hole solutions of Gauss-Bonnet gravity. We investigate geometric properties of the solutions and find that there is an intrinsic singularity at the origin. We investigate the thermodynamic properties of the asymptotically flat black holes and also asymptotically adS solutions. Using suitable local transformation, we generalize static horizon-flat solutions to rotating ones. We discuss their conserved and thermodynamic quantities as well as the first law of thermodynamics. Finally, we calculate the heat capacity of the solutions to obtain a constraint on the horizon radius of stable solutions.


I. INTRODUCTION
Among generalizations of Einstein action, the Gauss-Bonnet (GB) gravity has some particular interests because it is ghost-free and emerges in the effective low-energy action of string theory [1]. The effects of GB gravity have been investigated in various physical phenomena such as superconductors [2], hydrodynamics [3], LHC black holes [4], dark matter [5], dark energy [6] and Shear viscosity [7]. It is notable that the variation of GB Lagrangian is a total derivative in four dimensions and therefore it does not effect the four dimensional field equations. Thus, in order to find GB contributions, we should look for the five and higher dimensional solutions.
In addition to the higher derivative curvature terms, one would also expect to analyze the higher derivative gauge field contributions. The Born-Infeld (BI) NLEDs is the first nonlinear higher derivative generalization of the Maxwell theory [8] and its nonlinearity power is characterized by an arbitrary real positive parameter β. Replacing the BI NLEDs with linear Maxwell theory in related topics, one can investigate the effects of nonlinearity on the consequences [9][10][11][12][13].
Motivated by the recent results mentioned above, it is natural to investigate GB gravity in the presence of NLEDs [14,15]. For the first time, spherically symmetric black hole solutions of GB and GB-Maxwell gravity were, respectively, obtained in 1985 [16] and 1986 [17]. After that many authors have been investigated various features of (charged) GB black hole solutions with different topologies (for the very incomplete list of references, see [18]). Besides, coupling of the BI theory with general relativity have been derived by Hoffmann [19]. He removed the Reissner-Nordström divergency, but a conical singularity remained. Because of complicated nonlinear field equations, BI theory did not use for many years till Garcia et. al. obtained BI black hole without conical singularity [20]. Moreover, black hole solutions of GB-BI theory and GB-PMI gravity have been investigated in Refs. [14,15].
The objective of this paper is to find black hole solutions of the Gauss-Bonnet gravity coupled to new classes of NLEDs theory. Recently, one of us [21] considered other kinds of BI type Lagrangian to examine the possibility of black hole solutions. Although there are some analogues between the BI theory and Logarithmic or Exponential form of NLEDs, there exist some differences between them.
For the sake of simplicity, we consider five dimensional topological black holes and investigate geometric as well as thermodynamic properties of the solutions. Appendix devote to the higher dimensional generalization.

II. EQUATIONS OF MOTION
We are interested in the Gauss-Bonnet gravity coupled to a nonlinear U (1) gauge field. The action is where Λ = −n(n − 1)/l 2 is the negative cosmological constant for asymptotically AdS solutions and α is the Gauss-Bonnet coefficient with dimension (length) 2 . Furthermore, L GB and L(F ) are, respectively, the Lagrangians of Gauss-Bonnet and BI type NLEDs theories which can be defined as where β is called the nonlinearity parameter, the Maxwell invariant F = F µν F µν in which F µν = ∂ µ A ν − ∂ ν A µ is the Faraday tensor and A µ is the gauge potential. In order to obtain the field equations, we should vary the action (1). Due to the fact that variation of Eq. (1) is not well defined, one should add a boundary action I b to the bulk action [22] (for more details about I b one can see Ref. [18]). Varying the well defined action with respect to g µν and A µ , one can find that the boundary action does not effect on the following equations of motions where G µν is the Einstein tensor, L F = dL(F ) dF and H µν is the divergence-free symmetric tensor

III. TOPOLOGICAL BLACK HOLE SOLUTIONS
In this section, we desire to obtain the 5-dimensional black hole solutions of Eqs. (4) and (5). We consider static spherically symmetric spacetimes given by the following line element where dΩ 2 k represents the metric of three dimensional hypersurface at r =constant and t =constant with constant curvature 6k and volume V 3 in which k is a horizon curvature constant. We can write dΩ 2 k with the following explicit form Taking into account metric (7), we should consider a consistent gauge potential A µ with the following form Inserting the mentioned gauge potential in the electromagnetic field equation (5), one obtains where prime and double prime are, respectively, the first and second derivative with respect to r. The solution of the Eq. (10) can be written as where q is an integration constant which is related to the electric charge and Γ = 1 + q 2 β 2 r 6 . In addition, L W = LambertW ( 4q 2 β 2 r 6 ) which satisfies LambertW (x) exp [LambertW (x)] = x (for more details, see [23]). One may expand Eq. (11) for large r (or large β) to obtain the asymptotic behavior of the gauge potential χ = 8, 1 for ENEF and LNEF, respectively. Eq. (12) shows that for large values of r (or β), the dominant (first) term of h(r) is the same as one in 5-dimensional linear Maxwell theory. Regarding Eq. (9) with (11), we can solve the gravitational field equation (4). After some cumbersome calculations, we find that the nonzero (independent) components of the field equation (4) may be written as where After some calculations, we find that de1 dr = e 2 and so it is sufficient to solve e 1 for each branch. After some simplifications, one can obtain the solutions of Eq. (4) with the following relation where and M is an integration constant. In order to obtain the effect of nonlinearity parameter, one can expand the metric function for large value of β. Calculations show that the series expansion of Ψ(r) for large values of β (or r) is where the metric function of GB-Maxwell gravity is As one can confirm, these solutions are asymptotically AdS which is the same as that in GB-Maxwell theory. The second term on the right hand side of Eq. (20) is the leading NLEDs correction to the GB-Maxwell black hole solutions. In addition, one can consider GB and nonlinearity of the electrodynamics as corrections of Einstein-Maxwell black hole. Hence, we use series expansion of metric function for small values of α and also weak field limit of NLEDs (β −→ ∞) to obtain: where the metric function of Einstein-Maxwell gravity is In Eq. (22), the second and third terms are, respectively the GB and NLEDs corrections and fourth term is the correction of coupling between NLEDs and higher derivative gravity. Before proceeding, we should discuss about real solutions. Numerical evaluations show that depending on the metric parameters, the function Ψ(r) may be positive, zero or negative. In order to have real solutions we can use two methods. First, we can restrict ourselves to the set of metric parameters, which leads to non-negative Ψ(r) for 0 ≤ r < ∞. Second, we can focus on r coordinate. One can define r 0 as the largest root of Ψ(r = r 0 ) = 0, in which Ψ(r) is positive for r > r 0 . One can use suitable coordinate transformation (r −→ r ′ ) to obtain real solutions for 0 ≤ r ′ < ∞ (see last reference in [14] for more details). In this paper we use the first method. Now, we should look for the black hole interpretation. We should make an analysis of the essential singularity(ies) and horizon(s). Calculations show that the Kretschmann scalar is After some algebraic manipulation with numerical analysis, we find that the Kretschmann scalar (24) with metric function (17) diverges at r = 0 and is finite for r = 0, and therefore there is an curvature singularity located at r = 0. Seeking possible black hole solutions, one may determine the real root(s) of g rr = f (r) = 0 to find the of horizon(s).
Here, we should explain the effects of the nonlinearity on the event horizon. Taking into account the metric functions, we find that the nonlinearity parameter, β, changes the value of the event horizon, r + . Furthermore, there is critical nonlinearity, β c , in which for β < β c , the horizon geometry of nonlinear charged solutions behaves like as Schwarzschild solutions (see Ref. [21] for more details). In addition, one can obtain the temperature of the black holes with the use of surface gravity interpretation with the following form where in which shows that the nonlinearity parameter, β, and GB parameter can change the black hole temperature.

IV. THERMODYNAMICS OF ASYMPTOTICALLY ADS ROTATING BLACK BRANES WITH ZERO CURVATURE HORIZON
In this section, we take into account zero curvature horizon with a rotation parameter. In order to add angular momentum to the metric (7), we perform the following boost Applying the mentioned boost in Eq. (7) with k = 0, one obtains 5-dimensional rotating spacetime with zero curvature horizon where the function f (r) is presented in Eq. (17). The consistent gauge potential for the rotating metric (27) is where the function h(r) is the same as Eq. (11). Now, we want to calculate the electric charge and potential of the solutions. Calculating the flux of the electric field at infinity, we can write the electric charge per unit volume V 3 as Considering the rotating spacetime (27), we find that the null generator of the horizon is χ = ∂ t + Ω∂ φ . The electric potential Φ, measured at infinity (potential reference) with respect to the event horizon, is defined by where In addition, one may use the analytic continuation of the metric with the regularity at the horizon to obtain the temperature and angular velocities with the following form Now, we are in a position to calculate the mass and angular momentum of the solutions. we calculate the action and conserved quantities of the black brane solutions. In general the action and conserved quantities of the spacetime are divergent when evaluated on the solutions. One of the systematic methods for calculating the finite conserved charges of asymptotically AdS solutions is the counterterms method inspired by the anti-de Sitter/conformal field theory (AdS/CFT) correspondence [24]. Using the Brown-York method [25], it has been shown that for asymptotically AdS solutions of Lovelock gravity with flat boundary, R abcd (γ) = 0, the finite energy-momentum tensor is where γ µν and K are, respectively, induced metric and trace of extrinsic curvature of boundary and J is the trace of and the last term comes from the counterterm method. In our solutions, usually referred to as zero curvature horizon, l ef f is where we should note that l ef f reduces to l as α goes to zero, as it should be. Taking into account the Killing vectors ∂/∂t and ∂/∂φ, one can find that the conserved quantities associated with them are the mass and angular momentum with the following relations Final step is devoted to the entropy calculation. Generally we could not use the so-called area law [26,27] for higher derivative gravity [28]. Since the asymptotic behavior of the solutions is AdS, we can use the Gibbs-Duhem relation to calculate the entropy where I is the finite action evaluated by the use of counterterm method, and C i and Γ i are the conserved charges and their associate chemical potentials, respectively. Surprisingly, calculations show that the entropy obeys the area law for our case where the horizon curvature is zero, i.e.
Conserved and thermodynamic quantities at hand, we can check the first law of thermodynamics. We regard the entropy S, electric charge Q and angular momentum J as a complete set extensive quantities for the mass M (S, Q, J) and define the intensive quantities conjugate to them. These conjugate quantities are the temperature, electric potential and angular velocities Considering f (r = r + ) = 0 with Eqs. (41), (42) and (43) and after some straightforward calculations, we find that the intensive quantities calculated by Eqs.
Now, our task is investigation of thermodynamic stability with respect to small variations of the thermodynamic coordinates. In order to obtain stable solutions, the heat capacity (C Q,J = T + ∂ 2 M/∂S 2 Q,J ) must be positive, a requirement usually referred to as canonical ensemble stability criterion. The canonical ensemble instability criterion, which the charge and the angular momenta are fixed parameters, is sufficiently strong to veto some gravity toy models. We should leave out the analytical result for reasons of economy and one cannot prove the positivity of the heat capacity, analytically. But numerical calculations show that there is a lower limit for stable solutions (see Fig.  1 for more clarifications). In other words, there is an r min in which for r + > r min the heat capacity is positive and so the solutions are stable. Furthermore, we use series expansion of the heat capacity for large values of β to see the effects of nonlinearity corrections. After some calculations, we find where the heat capacity of the Einstein-Maxwell gravity C EinMax , and its first nonlinear correction, C corr , are Taking into account Eq. (45), we find that the heat capacity of the horizon-flat black branes does not depend on the GB parameter up to the second order of the nonlinearity parameter, β.

V. CLOSING REMARKS
In this paper we obtained 5-dimensional black hole solutions of Gauss-Bonnet gravity in the presence of NLEDs with various horizon topology. We considered two classes of Maxwell modification, named logarithmic and exponential forms of NLEDs as source of gravity and found that for weak field limit (β → ∞) all relationes reduce to Gauss-Bonnet Maxwell gravity.
We focused on horizon flat black hole solutions and obtained rotating black hole by use of a suitable local transformation. We used Gauss' law, counterterm method and Gibbs-Duhem relation to calculate conserved and thermodynamic quantities. We found that these quantities satisfy the first law of thermodynamics.
Finally, we studied Hawking phase transition in canonical ensemble. We calculated heat capacity and found that there is a minimum size for stable black holes. In other word, there is a minimum radius, r min , in which the solutions are stable for r + > r min .

Appendix
Here, we generalize our solutions to (n + 1)-dimensional ones. Considering the following metric where dĝ 2 k is the metric of (n − 1)-dimensional hypersurface of constant curvature (n − 1)(n − 2)k with the following explicit form Taking into account the electromagnetic and gravitational field equations, we find that h(r) = E(r)dr in which with L Wn = LambertW ( 4Q 2 β 2 r 2n−2 ), Γ n = 1 + Q 2 β 2 r 2n−2 and f (r) = k + r 2 2(n − 2)(n − 3)α 1 − Ψ n (r) , where Ψ n (r) = 1 + 8(n − 2)(n − 3)αΛ n(n − 1) + 4(n − 2)(n − 3)αM r n + Υ n , , LNEF , In addition, one can take into account zero curvature horizon with global rotation. We know that the rotation group in n + 1 dimensions is SO(n) and therefore the maximum number of independent rotation parameters is [n/2], where [x] is the integer part of x. In order to add angular momentum to the static spacetime (46), one can perform the following rotation boost for i = 1... [n/2]. Applying the mentioned boost in Eq. (46) with k = 0, we can obtain asymptotically AdS rotating spacetime with zero curvature horizon and p ≤ [n/2] rotation parameters where Ξ = 1 + k i a 2 i /l 2 and the function f (r) is presented in Eq. (49). The consistent gauge potential for the rotating metric (46) is where the function h(r) is the same as Eq. (48). One can use the same method of 5-dimensional case to obtain conserved and thermodynamic quantities of (n + 1)-dimensional solutions.