Thermodynamic analysis of topological black holes in Gauss-Bonnet gravity with nonlinear source
aa r X i v : . [ g r- q c ] O c t Thermodynamic analysis of topological black holesin Gauss-Bonnet gravity with nonlinear source
S. H. Hendi , , S. Panahiyan and E. Mahmoudi Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran
Employing two classes of nonlinear electrodynamics, we obtain topological black hole solutions ofGauss-Bonnet gravity. We investigate geometric properties of the solutions and find that there is anintrinsic singularity at the origin. We investigate the thermodynamic properties of the asymptoti-cally flat black holes and also asymptotically adS solutions. Using suitable local transformation, wegeneralize static horizon-flat solutions to rotating ones. We discuss their conserved and thermody-namic quantities as well as the first law of thermodynamics. Finally, we calculate the heat capacityof the solutions to obtain a constraint on the horizon radius of stable solutions.
I. INTRODUCTION
Among generalizations of the Einstein action, the Gauss-Bonnet (GB) gravity has some particular interests becauseit is ghost-free and emerges in the effective low-energy action of string theory [1]. The effects of GB gravity on thespeed of graviton propagation and appearance of potentially superluminal modes have been investigated in [2] andrecently the problem of its unusual causal structure has been solved [3]. In addition, the influences of GB gravityhave been investigated as regards various physical phenomena such as superconductors [4], hydrodynamics [5], LHCblack holes [6], dark matter [7], dark energy [8] and shear viscosity [9].Although one may consider the (Wald) entropy effect of GB gravity in four dimensions [10], it is notable that thevariation of GB Lagrangian is a total derivative in four dimension and therefore it does not affect the four dimensionalfield equations as well as black hole solutions. Thus, in order to study the GB contributions, we may look for the fiveand higher dimensional solutions.In addition to the higher derivative curvature terms, one would also expect to analyze the higher derivative gaugefield contributions. The Born-Infeld (BI) nonlinear electrodynamics (NLED) is the first nonlinear higher derivativegeneralization of the Maxwell theory [11] and its nonlinearity power is characterized by an arbitrary real positiveparameter β . Replacing the BI NLED with linear Maxwell theory in related topics, one can investigate the effects ofnonlinearity on the physical consequences in electrodynamics [12–16].Motivated by the recent results mentioned above, it is natural to investigate GB gravity in the presence of NLED [17,18]. For the first time, spherically symmetric black hole solutions of GB and GB-Maxwell gravity were, respectively,obtained in 1985 [19] and 1986 [20]. Then various properties of (charged) GB black holes were studied (for the veryincomplete list of references, see [21]). Besides, coupling of the BI theory with general relativity have been derivedby Hoffmann [22]. He removed the Reissner-Nordstr¨om divergency, but a conical singularity remained. Due to thecomplexity of the nonlinear field equations, BI theory has long been discarded. In 1984, Garcia and his colleaguescould obtain a BI black hole solution without conical singularity [23]. Moreover, black hole solutions of GB gravityin the presence of NLED have been investigated in Refs. [17, 18].The objective of this paper is to find topological black hole solutions of the GB gravity coupled to new classes ofNLED theory. In present work, we discuss the thermodynamics of the asymptotically flat black holes with sphericalhorizon as well as asymptotically adS rotating black branes with flat-horizon and specially the role played by GBand NLED terms to stability of the solutions. Recently, one of us [24] considered other kinds of BI type Lagrangiansto examine the possibility of black hole solutions. Although there are some analogies between the BI theory andLogarithmic or Exponential forms of NLED, there exist some differences between them as well.For the sake of simplicity, we consider five dimensional topological black holes and investigate the geometric as wellas thermodynamic properties of the solutions. Appendix is devoted to the higher dimensional generalization. II. EQUATIONS OF MOTION AND TOPOLOGICAL BLACK HOLE SOLUTIONS
We are interested in the GB gravity coupled to a nonlinear U (1) gauge field. The action is I G = − π Z M d n +1 x √− g [ R −
2Λ + αL GB + L ( F )] (1)where Λ is the cosmological constant and α is the GB coefficient. Furthermore, L GB and L ( F ) are, respectively, theLagrangians of GB and BI type theories, which can be defined as L GB = R µνγδ R µνγδ − R µν R µν + R , (2) L ( F ) = β (cid:16) exp( − F β ) − (cid:17) , EN EF − β ln (cid:16) F β (cid:17) , LN EF , (3)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 theFaraday tensor and A µ is the gauge potential. We should vary the action (1) with respect to g µν and A µ to obtainthe field equations G µν + Λ g µν − αH µν = 12 g µν L ( F ) − F µλ F λν L F , (4) ∂ µ (cid:0) √− gL F F µν (cid:1) = 0 , (5)where G µν is the Einstein tensor, L F = dL ( F ) d F and H µν is the divergence-free symmetric tensor H µν = 4 R ρσ R µρνσ − R ρσλµ R νρσλ − RR µν + 4 R µλ R λν + L GB g µν . (6)Now, we desire to obtain the 5-dimensional black hole solutions of Eqs. (4) and (5) with the following line element ds = − f ( r ) dt + dr f ( r ) + r d Ω k , (7)where d Ω k represents the metric of three dimensional hypersurface at r =constant and t =constant with constantcurvature 6 k and volume V . We can write d Ω k in the following explicit forms d Ω k = dθ + sin θ (cid:0) dφ + sin φdψ (cid:1) k = 1 dθ + sinh θ (cid:0) dφ + sin φdψ (cid:1) k = − dθ + dφ + dψ k = 0 . (8)Taking into account metric (7), we should consider a consistent gauge potential A µ with the following form A µ = h ( r ) δ µ . (9)where h ( r ) = R qr exp (cid:2) − L W (cid:3) dr, ENEF2 β R q Γ − dr, LNEF , (10)and q is an integration constant which is related to the electric charge. In addition, Γ = q q β r and L W = LambertW ( q β r ) [25]. We expand h ( r ) for large r to obtain the asymptotical behavior of the gauge potential h ( r ) | Large r = − q r + χq β r + O (cid:18) q β r (cid:19) , (11)where χ = 8 , r , the dominant (first) termof h ( r ) is the same as one in 5-dimensional linear Maxwell theory.Considering Eq. (9) with (10), 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 e = (cid:2) r − α ( f − k ) (cid:3) f ′ + 2 r ( f − k ) + 2Λ r β r χ Υ = 0 , (12) e (cid:2) r − α ( f − k ) (cid:3) f ′′ + 4 ( r − αf ′ ) f ′ + 2 ( f − k ) + 2Λ r + β r Υ = 0 (13)where the prime and double primes are, respectively, the first and second derivatives with respect to r andΥ = ( − (1 − L W ) e LW , ENEFΓ − − ln( Γ+12 ) , LNEF , (14)Υ = ( − e LW , ENEF − ln( Γ+12 ) , LNEF . (15)After some calculations, we find that de dr = e , so it is sufficient to solve e for each branch, yielding f ( r ) = k + r α (cid:16) − p Ψ( r ) (cid:17) , (16)where Ψ( r ) = 1 + 4 α (cid:18) Λ + 6 mr (cid:19) + 4 αβ , (17)Υ = ( + qβr R (cid:16) √ L W − √ L W (cid:17) dr, ENEF4 ln(2) − r R r [Γ − ln (1 + Γ)] dr, LNEF , (18)and m is an integration constant. In order to obtain the effect of nonlinearity parameter, one can expand the metricfunction for large values of β . Calculations show that the series expansion of Ψ( r ) for large values of β (or r ) isΨ( r ) = Ψ GBM ( r ) + χαq β r + O ( 1 β ) , (19)where the metric function of GB–Maxwell gravity is Ψ GBM ( r ) = 1 + α (cid:16) Λ + mr − q r (cid:17) .As one can confirm, these solutions are asymptotically adS; the same as those in GB-Maxwell theory. The secondterm on the right hand side of Eq. (19) is the leading NLED correction to the GB–Maxwell black hole solutions.Now, we want to compare the GB and NLED effects. As we know, the Maxwell theory, to a large extent in variousphysical scopes, has acceptable consequences. So, in the transition from the Maxwell theory to NLED, it is allowedto consider the effects of small nonlinearity variations, not strong effects.Considering the fact that we are working in gravitational framework and GB gravity is a natural generalization ofEinstein gravity (not a perturbation in general), the GB is dominant over nonlinear electrodynamics which should beconsidered as a perturbation to Maxwell field. Another reason for the majority of GB contributions is in the profoundinsight in the metric function. GB parameter is coupled with mass, cosmological constant and charge sector of themetric function and therefore its changing will be modified all sectors of the metric function, whereas modification ofnonlinearity parameter of electrodynamics affects charge part of the solutions. Another way to see the majority ofGB or NLED contributions is the series expansion of the metric function. One can consider GB and nonlinearity ofthe electrodynamics as corrections of Einstein-Maxwell black hole. Hence, we use series expansion of metric functionfor small values of α and also weak field limit of NLED ( β −→ ∞ ) to obtain f ( r ) = f EM + 2( k − f EM ) r α − χq r β + χq ( k − f EM )24 r αβ + O (cid:0) α , β − (cid:1) , (20)where the metric function of Einstein–Maxwell gravity is f EM = k − Λ r − mr + q r .In Eq. (20), the second and third terms are, respectively the GB and NLED corrections and fourth term is thecorrection of coupling between NLED and higher derivative gravity.Before proceeding, we should discuss about real solutions. Numerical evaluations show that depending on themetric parameters, the function Ψ( r ) may be positive, zero or negative. In order to have real solutions we can usetwo methods. First, we can restrict ourselves to the set of metric parameters, which lead to non-negative Ψ( r ) for0 ≤ r < ∞ . Second, we can focus on the r coordinate. One can define r as the largest root of Ψ( r = r ) = 0, inwhich Ψ( r ) is positive for r > r . One can use suitable coordinate transformation ( r −→ r ′ ) to obtain real solutionsfor 0 ≤ r ′ < ∞ (see the last reference in [17] 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 R αβγδ R αβγδ = f ′′ + 6 f ′ r + 12( f − k ) r . (21)After some algebraic manipulation with numerical analysis, we find that the Kretschmann scalar (21) with metricfunction (16) diverges at r = 0 and is finite for r = 0, and therefore there is a 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 metricfunctions, we find that the nonlinearity parameter, β , changes the value of the event horizon, r + . Furthermore, thereis a critical nonlinearity, β c , in which for β < β c , the horizon geometry of nonlinear charged solutions behaves likeSchwarzschild solutions (see Ref. [24] for more details). In addition, one can obtain the temperature of the blackholes with the use of surface gravity interpretation in the following form T = f ′ ( r + )4 π = 6 kr + − r + Υ ′ π (cid:0) r + 4 αk (cid:1) , Υ ′ = qβL − / W + (cid:0) − L W + (cid:1) − β r , ENEF8 β r h − Γ + + ln( + ) i , LNEF , (22)which shows that the nonlinearity parameter, β , and GB parameter can change the black hole temperature. Aswe discussed before, the singularity may be covered with two horizons for β > β c . Taking into account the metric(7) with a suitable local transformation, one can obtain the so-called Nariai spacetime [26]. Following the work byBousso-Hawking [27], the Nariai solution may be found by coincidence of two horizons. In this extremal regime, thetwo horizons have the same temperature and they are in the thermal equilibrium. Using a suitable choice of boundaryconditions one may discuss about anti-evaporation as it happens for Nariai anti-evaporating black holes [28]. III. THERMODYNAMICS OF ASYMPTOTICALLY FLAT BLACK HOLE SOLUTIONS (
Λ = 0 , k = 1 ) In this section, we set Λ = 0 and k = 1 to study the thermodynamic behavior of asymptotically flat solutions. It hasbeen shown that we could not use the so-called area law [29, 30] for higher derivative gravity [31]. For asymptoticallyflat black hole solutions, one can use the Wald formula for calculating the entropy S = 14 Z d x √ γ h α e R i = V (cid:18) αr (cid:19) r (23)where e R is the Ricci scalar for the induced metric γ ab on the 3-dimensional boundary. Eq. (23) shows that, in GBgravity, asymptotically flat black hole with spherical horizon violates the area law. We should note that although the(nonlinear) electromagnetic source changes the values of inner and outer horizons of charged black objects, it doesnot alter the entropy formula and area law (see Ref. [32] for more details). In order to obtain the electric charge perunit volume V of the black hole, we use the flux of the electric field at infinity, yielding Q = q π , (24)which shows that the total charge does not depend on the nonlinearity of the electrodynamics. The mentioned staticspacetime has a Killing vector ∂ t and therefore the electric potential Φ, measured at infinity (potential reference) withrespect to the event horizon, is defined byΦ = A µ χ µ | r →∞ − A µ χ µ | r = r + = βr + √ L W + h L W + F (cid:16) [1] , [ ] , L W + (cid:17)i , ENEF (1 − Γ + ) β r q − (cid:16) qβ Γ + (cid:17) / F (cid:16) [ , ] , [ ] , (cid:17) , LNEF , (25)where L W + = LambertW ( 4 q β r ) , and Γ + = s q β r . The ADM (Arnowitt-Deser-Misner) mass of black hole can be obtained by using the behavior of the metric at large r [33]. The mass per unit volume V of the black hole is M = 3 m π = 316 π r + 2 α − β r Υ | r = r + ! . (26)Eq. (26) shows that both NLED and GB terms may be changed the finite mass of the asymptotically flat black holesolutions. FIG. 1: Asymptotically flat solutions (ENEF): T + versus r + for q = 1. Left figure: α = 1, and β = 10 (bold line), β = 0 . β = 0 .
01 (dashed line).
Right figure: β = 1, and α = 1 . α = 0 . α = 0 . Now, we obtain the total mass M as a function of the extensive quantities Q and S to check the first law ofthermodynamics. Using the expression for the entropy, the electric charge and the mass given in Eqs. (23), (24) and(26), one can obtain a Smarr-type formula M ( S, Q ) = 316 π (cid:20) r + 2 α − β r Θ6 (cid:21) , (27)Θ = + −
1) + 4 ln (cid:16) β r q (Γ + − (cid:17) − q F (cid:18) [ , ] , [ ] , − q β r (cid:19) β r , LNEF − q q L W + F (cid:18) [1] , [ ] , LW +6 (cid:19) βr + − qβr + √ L W + (cid:16) L W + (cid:17) , ENEF . Now, we regard the parameters Q and S as a complete set of extensive parameters and define the intensiveparameters conjugate to them. These quantities are the temperature and the electric potential, T = (cid:18) ∂M∂S (cid:19) Q = (cid:16) ∂M∂r + (cid:17) Q (cid:16) ∂S∂r + (cid:17) Q , (28)Φ = (cid:18) ∂M∂Q (cid:19) S = (cid:16) ∂M∂q (cid:17) r + (cid:16) ∂Q∂q (cid:17) r + . (29)Using Eqs. (23) and (24), one can show that the Eqs. (28) and (29) are equal to Eqs. (22) (with k = 1 and Λ = 0)and (25), respectively, and hence we conclude that these quantities satisfy the first law of thermodynamics dM = T dS + Φ dQ. (30)It has been shown that small Lovelock black holes (including GB gravity) are, dynamically, unstable [34]. In ourcase the mass M is a function of the entropy S and electric charge Q . In order to obtain thermodynamical stablesolutions, the heat capacity ( C Q = T + / (cid:0) ∂ M/∂S (cid:1) Q ) must be positive, a requirement usually referred to as canonicalensemble stability criterion. The canonical ensemble instability criterion for which the charge is a fixed parameter, issufficiently strong to veto some gravity toy models. We should leave out the analytical result for reasons of economyand one cannot prove the positivity of the heat capacity, analytically. In order to investigate the effects of NLED and FIG. 2: Asymptotically flat solutions (ENEF): C Q versus r + for q = 1, α = 1, and β = 1 (solid line), β = 0 . β = 0 . ”different ranges and scales” FIG. 3: Asymptotically flat solutions (ENEF): C Q versus r + for q = 1, β = 1, and α = 1 . α = 1 (dotted line) and α = 0 . ”different ranges and scales” GB gravity, we plot Figs. 1, 2 and 3. It is notable that we plot the figures of C Q for different ranges and scales formore clarifications and finding the roots of heat capacity. Fig. 1 shows that there is a minimum value of the horizonradius, r , for the physical black holes (positive temperature black holes). Also Figs. 2 and 3 show that small and largeblack holes are unstable. It means that there are two critical values r + min and r + max in which the asymptotically flatblack hole solutions are stable for r + min < r + < r + max . Numerical calculations show that r = r + min . Figs. 1 and 3confirm that although the GB parameter affects the value of the heat capacity and temperature, it does not change r + min . This is due to the fact that the GB parameter appear in the denominator of T + in Eq. (22). Also, Figs. 1and 2 show that the nonlinearity parameter, β , changes both r + min and r + max . In other words, r + max decreases withincreasing (decreasing) β ( α ) and as the nonlinearity parameter, β , increases, r + min decreases, too.Furthermore, we use series expansion of the heat capacity for large values of β and also small values of α to see theeffects of corrections. After some calculations, we find C Q = C EinMax + αC GB + C NLED β + C coupled αβ + O (cid:18) α β (cid:19) , (31)where C EinMax is the heat capacity of the Einstein-Maxwell gravity and C GB , C NLED and C coupled are, respectively,the leading corrections of GB gravity, NLED theory and the nonlinear gauge-gravity coupling with the followingexplicit forms C EinMax = − (cid:0) q − r (cid:1) r q − r ) ,C GB = − (cid:0) q − r (cid:1) (cid:0) q − r (cid:1) r (5 q − r ) ,C NLED = − χ (cid:0) q − r (cid:1) q r (5 q − r ) ,C coupled = − χ (cid:0) q − r (cid:1) (cid:0) q − r (cid:1) q r (5 q − r ) . IV. THERMODYNAMICS OF ASYMPTOTICALLY ADS ROTATING BLACK BRANES
In this section, we take into account zero curvature horizon with a rotation parameter. In order to add angularmomentum to the metric (7), we perform the following boost t t q a l − aφ, φ φ q a l − al t. (32)Applying the mentioned boost in Eq. (7) with k = 0, one obtains 5-dimensional rotating spacetime with zero curvaturehorizon ds = − f ( r ) dt q a l − adφ + r l adt − l dφ q a l + dr f ( r ) + r ( dθ + dψ ) , (33)where the function f ( r ) is presented in Eq. (16). The consistent gauge potential for the rotating metric (33) is A µ = h ( r ) r a l δ µ − aδ φµ ! , (34)where the function h ( r ) is the same as Eq. (10). Now, we want to calculate the electric charge and potential of thesolutions. Calculations show that the electric charge per unit volume V is Q = q π r a l . (35)Considering the rotating spacetime (33), we find that the null generator of the horizon is χ = ∂ t + Ω ∂ φ . The electricpotential Φ for the rotating solutions may be written asΦ = Φ | asymptotically flat case q a l . (36)In addition, one may use the analytic continuation of the metric with the regularity at the horizon to obtain thetemperature and angular velocities in the following form T = f ′ ( r + )4 π Ξ = − r + Υ ′ π Ξ r , (37)Ω = a Ξ l . (38) FIG. 4: Asymptotically adS rotating solutions (ENEF): T + versus r + for q = 1, Λ = − . Left figure:
Ξ = 1 .
1, and β = 5 (bold line), β = 0 .
75 (solid line) and β = 0 . Right figure: β = 1, and Ξ = 1 .
01 (bold line), Ξ = 2 (solid line) and Ξ = 5 (dashed line).
Now, we are in a position to calculate the mass and angular momentum of the solutions. We calculate the actionand conserved quantities of the black brane solutions. In general the action and conserved quantities of the spacetimeare divergent when evaluated on the solutions. One of the systematic methods for calculating the finite conservedcharges of asymptotically adS solutions is the counterterm method inspired by the anti-de Sitter/conformal fieldtheory (AdS/CFT) correspondence [35]. Considering two Killing vectors ∂/∂t and ∂/∂φ and by using the Brown-York method [36], one can find that the conserved quantities associated with the mentioned Killing vectors are themass and angular momentum with the following relations M = m π (cid:18) a l + 3 (cid:19) , (39) J = ma π r a l , (40)where m comes from the root of metric function with the following form m = − r (cid:16) Λ + β Υ | r = r + (cid:17) . (41)Eq. (41) shows that unlike the NLED, the GB term does not change the finite mass and angular momentum of theblack hole solutions with zero curvature horizon.The final step is devoted to the entropy calculation. Since the asymptotical behavior of the solutions is adS, wecan use the Gibbs-Duhem relation to calculate the entropy S = 1 T ( M − Γ i C i ) − I, (42)where I is the finite action evaluated by the use of counterterm method, C i and Γ i are the conserved charges andtheir associate chemical potentials, respectively. Surprisingly, calculations show that the entropy obeys the area lawfor our case where the horizon curvature is zero, i.e. S = r r a l (43)Conserved and thermodynamic quantities at hand, we can check the first law of thermodynamics. We regard theentropy S , the electric charge Q and the angular momentum J as a complete set of extensive quantities for the mass M ( S, Q, J ) and define the intensive quantities conjugate to them. These conjugate quantities are the temperature,
FIG. 5: Asymptotically adS rotating solutions (ENEF): C Q,J and 50 T + versus r + for q = 1, Λ = − . . Left figure: ” C Q,J : β = 5 (solid line), β = 1 (dotted line) and β = 0 . T + : β = 5 (solid-bold line), β = 1(dotted-bold line) and β = 0 . Right figure: ” C Q,J : β = 0 . β = 0 . β = 0 . T + is positive definite electric potential and angular velocities T = (cid:18) ∂ M ∂S (cid:19) Q,J , (44)Φ = (cid:18) ∂ M ∂Q (cid:19) S,J , (45)Ω = (cid:18) ∂ M ∂J (cid:19) S,Q . (46)Considering f ( r = r + ) = 0 with Eqs. (44), (45) and (46) and after some straightforward calculations, we find that theintensive quantities calculated by Eqs. (44), (45) and (46) are consistent with Eqs. (37), (36) and (38), respectively,and consequently one can confirm that the relevant thermodynamic quantities satisfy the first law of thermodynamics dM = T dS + Φ dQ + Ω dJ. (47)Now, our task is investigation of thermodynamic stability with respect to small variations of the thermodynamiccoordinates. For rotating case the mass M is a function of the entropy S , the angular momentum J and theelectric charge Q . As we mentioned before, in the canonical ensemble, the positivity of the heat capacity C Q,J = T + .(cid:0) ∂ M/∂S (cid:1) Q,J is sufficient to ensure the thermodynamic stability.As we mentioned before, we may leave out the analytical result for reasons of economy and investigate the numericalcalculations. We also plot Figs. 5 and 6 for more clarification. Fig. 5 and 6 show that there is a critical value for thenonlinearity parameter, β c , in which for β > β c there is a minimum value of the horizon radius ( r + min ) for physicalblack branes and the temperature is positive definite for β < β c (see the left figure in Fig. 4 for more details). Inaddition, Figs. 5 and 6 indicate that rotating physical adS black branes are stable for β > β c . Also, for β < β c ,although the solutions are physical for all values of r + , there is a minimum horizon radius for the stable adS blackbranes. Moreover, Fig. 5 confirms that for a fixed value of the rotation parameter, β affects the value of the r + min .It means that for β > β c ( β < β c ), decreasing the nonlinearity parameter, β , leads to decreasing (increasing) r + min .Furthermore, Fig. 6 shows that, although for β > β c , the rotation parameter does not affect r + min , it changes theminimum value of r + min for β < β c . In other words, for β < β c , increasing the rotation parameter leads to decreasingthe value of r + min .0 FIG. 6: Asymptotically adS rotating solutions (ENEF): C Q,J and 50 T + versus r + for q = 1, Λ = − . Left figure: ” C Q,J : β = 1, Ξ = 1 .
01 (solid line), Ξ = 1 . T + : β = 1, Ξ = 1 . . Right figure: ” C Q,J : β = 0 .
5, Ξ = 1 .
01 (solid line), Ξ = 1 . T + is positive definite One finds that although C Q,J does not depend on the GB parameter, it depends on the nonlinearity and rotationparameters. In order to check the nonlinearity effect, we can use the series expansion for large value of β to comparethe mentioned C Q,J with that of GB-Maxwell gravity. One finds C Q,J = C EinMax + C NLED β + O (cid:18) β (cid:19) , (48)where the heat capacity of the Einstein-Maxwell gravity C EinMax , (which is equal to the GB-Maxwell gravity forhorizon-flat solutions) and its first nonlinear correction, C NLED , are C EinMax = − (cid:2) Λ r (6Ξ −
5) + Λ q r (cid:0) − (cid:1) + 2 q (6Ξ − (cid:3) r ( q + Λ r )(2 q − Λ r )(2Ξ + 1)Ξ ,C NLED = 3 χq (cid:2) r − Λ r q (6Ξ + 1) + 4 q Ξ (cid:0) Λ r + q (cid:1)(cid:3) r ( q + Λ r ) (2 q − Λ r ) (2Ξ + 1)Ξ V. CLOSING REMARKS
In this paper we obtained 5-dimensional black hole solutions of GB gravity in the presence of NLED with varioushorizon topology. We considered two classes of Maxwell modification, named logarithmic and exponential forms ofNLED as source of gravity and found that for weak field limit ( β → ∞ ) all relationes reduce to GB–Maxwell gravity.At first, we investigated the thermodynamic properties of the asymptotically flat black holes. We found thatalthough the NLED and GB gravity change some properties of the solutions such as entropy, mass, electric charge,and the temperature, all conserved and thermodynamic quantities satisfy the first law of thermodynamics. Weanalyzed the thermodynamic stability of the solutions and found that there is a minimum horizon radius, r + min , inwhich the black holes have positive temperature for r + > r + min . Besides, we showed that there is an upper limithorizon radius for the stable black holes. It means that the asymptotically flat black hole solutions are stable for r + min < r + < r + max . In addition, we found that although the GB parameter affects the values of the heat capacity,the temperature and r + max , it does not change r + min . We also showed that β changes both r + min and r + max .Then we focused on the horizon-flat black hole solutions and obtained rotating adS black branes by use of a suitablelocal transformation. We used Gauss’ law, the counterterm method, and the Gibbs-Duhem relation to calculate theconserved and thermodynamic quantities. We found that these quantities satisfy the first law of thermodynamics.We investigated the effects of various parameters and found that although T + and C Q,J do not depend on the GB1parameter, they depend on the β and the rotation parameter. Calculations showed that there is a critical value for thenonlinearity parameter, β c , in which for β > β c there is a minimum value of the horizon radius ( r + min ) for physicallystable adS black branes. We found that for β < β c , although the temperature is positive definite for all values of r + , there is a minimum horizon radius for the stable adS black branes. Also, we showed that for a fixed value of therotation parameter and β > β c ( β < β c ), decreasing the nonlinearity parameter, β , leads to decreasing (increasing) r + min . Besides, we indicated that although for β > β c , the rotation parameter does not affect r + min , it changes theminimum value of r + min for β < β c . More precisely, for β < β c , increasing the rotation parameter leads to decreasingthe value of r + min .In this paper, we only considered static and rotating solutions. Therefore, it is worthwhile to use a suitable localtransformation to obtain the so-called Nariai spacetime and investigate the anti-evaporation process. One moresubject of interest for further study is regarding a model of time dependent FRW spacetime and investigate the effectsof considering the NLED and fifth dimensions. Acknowledgments
We would like to thank the referees for their fruitful suggestions. We also thank A. Poostforush for reading themanuscript. The authors wish to thank Shiraz University Research Council. This work has been supported financiallyby Research Institute for Astronomy & Astrophysics of Maragha (RIAAM), Iran.
Appendix
Here, we generalize three dimensional d Ω k (Eq. (8)) to the ( n − d ˆ g k to obtain ( n + 1)-dimensionalsolutions. The explicit form of d ˆ g k is d ˆ g k = dθ + n − P i =2 i − Q j =1 sin θ j dθ i k = 1 dθ + sinh θ dθ + sinh θ n − P i =3 i − Q j =2 sin θ j dθ i k = − n − P i =1 dφ i k = 0 . (49)Taking into account the electromagnetic and gravitational field equations, we find that h ( r ) = R E ( r ) dr in which E ( r ) = Qr n − × ( exp (cid:16) − L Wn (cid:17) , EN EF n +1 , LN EF , (50)with L W n = LambertW ( 4 Q β r n − ) , Γ n = s Q β r n − and f ( r ) = k + r n − n − α (cid:16) − p Ψ n ( r ) (cid:17) , (51)whereΨ n ( r ) = 1 + 8( n − n − α Λ n ( n −
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