Dyonic (A)dS Black Holes in Einstein-Born-Infeld Theory in Diverse Dimensions
aa r X i v : . [ h e p - t h ] J u l Dyonic (A)dS Black Holes in Einstein-Born-Infeld Theory inDiverse Dimensions
Shoulong Li , H. L¨u , ∗ and Hao Wei School of Physics,Beijing Institute of Technology, 5 South Zhongguancun Street, Beijing 100081, China Center for Advanced Quantum Studies, Department of Physics,Beijing Normal University, 19 Xinjiekouwai Street, Beijing 100875, China
ABSTRACTWe study Einstein-Born-Infeld gravity and construct the dyonic (A)dS planar black holesin general even dimensions, that carry both the electric charge and magnetic fluxes alongthe planar space. In four dimensions, the solution can be constructed with also sphericaland hyperbolic topologies. We study the black hole thermodynamics and obtain the firstlaw. We also classify the singularity structure. sllee [email protected] ∗ [email protected] [email protected] ontents In 1934, Born and Infeld [1] proposed an elegant nonlinear version of electrodynamics thatsuccessfully removes the divergence of self-energy of a point-like charge in Maxwell’s theoryof electrodynamics. The Lagrangian density of the Born-Infeld (BI) theory in D -dimensionalMinkowski spacetime is given by L = − b s − det (cid:18) η µν + F µν b (cid:19) + b , (1.1)where η µν = diag( − , , ,
1) is the Minkowski metric, F µν = 2 ∂ [ µ A ν ] is the Faraday ten-sor and A = A µ dx µ is the Maxwell gauge potential. BI theory contains a dimensionfulparameter b , and in the limit b → ∞ , BI theory reduces to the Maxwell theory, L = − F + O (cid:18) b (cid:19) . (1.2)In the limit b →
0, the Lagrangian in four dimensions becomes F ∧ F which is a totalderivative. The limit is generally singular in higher dimensions.BI theory has enjoyed further attentions since the invention of string theory. It turns outthat the BI action can arise from string theory [2], describing the low energy dynamics of D-branes [3]. We refer to e.g. [4, 5] for some comprehensive reviews on the BI theory in string2heory. The special Born-Infeld-like nonlinear form is also very useful to construct analogousnew theories, such as Dirac-Born-Infeld (DBI) inflation theory [6,7] and Eddington-inspiredBorn-Infeld (EiBI) cosmologies [8]. BI theory can also be adopted to explore issues of darkenergy [9, 10].In this paper, we focus on the study of black holes in Einstein-Born-Infeld (EBI) theory.The most general static type- D metric of the BI theory in four dimensions was constructedin [11]. (See also [12] and [13].) The spherically-symmetric solution was generalized toarbitrary D dimensions in [14] where the black hole thermodynamics was studied. Theblack hole solutions was also generalized to include different topologies [15]. The Born-Infeld black hole solutions were also studied in Einstein theory with a dilaton field [16] andin the modified gravity theories such as Gauss-Bonnet theory [17], Lovelock theory [18],Brans-Dicke theory [19], f ( T ) theory [20], massive gravity [21], and so on. The extendedthermodynamics [14–20, 22–24], geodesics [25, 26], and AdS/CFT correspondence proper-ties [27–29] were studied too. Other Born-Infeld solutions are also studied, for example,thin-shell wormholes [30].(A)dS black hole solutions in BI theory considered in literature typically involves onlyeither the electric or magnetic charges. Although the dyonic black hole in EBI theory wasconstructed in [11], it is written in the general (static) type- D form. The global structurein the spherically symmetric form was analysed in [25] for the asymptotically-flat case. Inthis paper, we shall first study the dyonic (A)dS black holes in the EBI theory in fourdimensions with general topologies, focus on analysing the black hole thermodynamics andsingularity structure. We then construct dyonic AdS planar black holes in arbitrary evendimensions, where the solutions carry both the electric flux as well as the magnetic 2-formflux along the planar space.Interestingly in almost all the previous works on constructing black holes, the equivalentaction in four dimensions was used, rather than the original one. In D = 4, the Lagrangiancan be equivalently expressed as [1] L = b − b p I + I , (1.3)where I = 12 b F µν F µν = B − E b , I = − b (cid:16) F µν e F µν (cid:17) = − ( E · B ) b , (1.4)in which E and B are electric and magnetic fields, and e F µν = 12 ǫ µνρσ F ρσ = 12 p − det( η ab ) ε µνρσ F ρσ , (1.5)3here ε µνρσ is a tensor density with ε = 1.The equivalence of (1.3) and (1.1) is only true in four dimensions; it is no longer valid inhigher dimensions. However, if one considers only static solutions carrying electric charges,one can nevertheless use the reduced Lagrangian (1.3). In fact in this case, one can evenignore the I term. This was indeed done in many previous works, for example [12–15]. Sinceone of our purposes is to construct dyonic black holes in higher dimensions, the Lagrangian(1.3) is not suitable for this purpose and we shall use the original Lagrangian (1.1) insteadfor all our constructions.The paper is organized as follows. In Sec. 2, we review the EBI theory and then derivethe equations of motion for all dimensions. In Sec. 3, we obtain the exact dyonic (A)dSblack hole solutions in four dimensions with a generic topological horizon. Then we studythe global structure, black hole thermodynamics and the singularity structures. In Sec. 4,we generalize the results to all even dimensions. We conclude the paper in Sec. 5. In this section, we consider the EBI theory. The Lagrangian of BI theory can be naturallygeneralized to curved spacetimes and the Lagrangian is given by L = − b s − det (cid:18) g µν + F µν b (cid:19) + b q − det ( g µν ) , (2.1)where g µν is the metric. The Lagrangian of the EBI theory with a bare cosmological constantΛ can be written by L = √− g ( R − ) − b s − det (cid:18) g µν + F µν b (cid:19) , (2.2)where Λ = Λ − b /
2. Here, Λ is the effective cosmological constant. The variation ofLagrangian (2.2) gives rise to δ L = √− g ( − E µν δg µν + E νA δA ν + ∇ µ J µ ) , (2.3)where g = det( g µν ), J µ is the surface term and E µν = G µν + g µν Λ + b √− h √− g (cid:0) h − (cid:1) ( µν ) , (2.4) E νA = ∇ µ (cid:20) √− h √− g b (cid:0) h − (cid:1) [ µν ] (cid:21) , (2.5)4n which G µν = R µν − g µν R/ h µν = g µν + F µν /b , h ≡ det( h µν ), and ( h − ) µν denotes theinverse of h µν , satisfying ( h − ) µρ h ρν = δ µν , h νρ ( h − ) ρµ = δ µν . (2.6)We further defined (cid:0) h − (cid:1) ( µν ) = 12 (cid:2)(cid:0) h − (cid:1) µν + (cid:0) h − (cid:1) νµ (cid:3) , (cid:0) h − (cid:1) [ µν ] = 12 (cid:2)(cid:0) h − (cid:1) µν − (cid:0) h − (cid:1) νµ (cid:3) . (2.7)The equations of motion are then given by E µν = 0 and E µA = 0. These equations arederived from the original Lagrangian (2.2) of the EBI theory and hence are applicable inall dimensions and for all charge configurations. In the previous section, we obtained the equations of motion of the EBI theory. We now con-struct the static dyonic (A)dS black hole solution with a general topological horizon in fourdimensions. We shall then study the global structure and the black hole thermodynamics.
The static solution in the type- D form in the EBI theory was first constructed in [11]. Thespherically-symmetric and asymptotically-flat solution was given in [25]. In this section, westudy the properties of the dyonic (A)dS black holes. The most general static ansatz canbe written as ds = − h ( r ) dt + dr f ( r ) + r (cid:18) du − ku + (1 − ku ) dϕ (cid:19) , A = φ ( r ) dt + pudϕ , (3.1)where k = 1 , , − p is magnetic charge parameter. It turns out that the equations of motion ofthe metric g µν imply that h ( r ) = f ( r ) and the equations of motion for A µ imply that φ ( r )can be expressed as φ ′ ( r ) = q q r + Q b , with Q = p p + q , (3.2)where and thereafter, we use a prime to denote a derivative with respect to r , and q is aintegral constant that is related to the electric charge. The function f ( r ) satisfies rf ′ ( r ) + f ( r ) = k − Λ r − b r r + Q b . (3.3)5hus f ( r ) can be solved and expressed in terms of hypergeometric function F f ( r ) = −
13 Λ r + k − µr − b r r + Q b + Q r F (cid:20) ,
12 ; 54 ; − Q b r (cid:21) , (3.4)where µ is the integral constant corresponding to the mass of the solution. The electricpotential φ ( r ) is expressed by φ ( r ) = Z ∞ r qd ˜ r q ˜ r + Q b = qr F (cid:20) ,
12 ; 54 ; − Q b r (cid:21) . (3.5)In the limit b → ∞ , the solution recovers the dyonic Reissner-Nordstr¨om-(A)dS black hole, f ( r ) = − Λ3 r + k − µr + Q r , φ ( r ) = qr . (3.6)On the other hand, in the limit b →
0, the Born-Infeld field vanishes and the solution isreduced to the Schwarzschild-(A)dS black hole in pure cosmological Einstein theory, f ( r ) = k − µr − Λ3 r . (3.7)In the large- r expansion, we have f ( r ) = − Λ3 r + k − µr + Q r + O (cid:18) r (cid:19) , φ ( r ) = qr + O (cid:18) r (cid:19) . (3.8)Thus we see that the first few leading-order expansions match those of the Reissner-Nordstr¨om-(A)dS black hole. Now we discuss black hole thermodynamics. The event horizon is defined through f ( r + ) = 0,where r + denotes the largest root of f . It is convenient to express the constant µ in termsof r + , namely µ = −
13 Λ r + kr + − b r + r r + Q b + Q r + 2 F (cid:20) ,
12 ; 54 ; − Q b r (cid:21) . (3.9)Since the metric is asymptotically (A)dS, according to the definition of mass in asymptoti-cally (A)dS space by Abbott-Deser-Tekin (ADT) formalism [31], we find M = ω π µ , (3.10)where ω = R dudϕ . For k = 1, corresponding the unit S , we have ω = 4 π .6he temperature T and entropy S on the horizon are easily calculated as T = f ′ ( r + )4 π = k − Λ r πr + − b πr + r r + Q b , (3.11) S = A r ω . (3.12)The electric and magnetic charges are given by Q e = ω π √− h ( h − ) [ tr ] | r →∞ = q π ω , Q m = ω π F uφ | r →∞ = p π ω . (3.13)Note that the above electric charge as a conserved quantity follows from the equation ofmotion (2.5). The electric and magnetic potentials are given byΦ e = qr + 2 F (cid:20) ,
12 ; 54 ; − Q b r (cid:21) , Φ m = pr + 2 F (cid:20) ,
12 ; 54 ; − Q b r (cid:21) . (3.14)The differential first law of black hole thermodynamics can be written as dM = T dS + Φ e dQ e + Φ m dQ m . (3.15)One can further treat the cosmological constant as a generalized “pressure” P Λ = − Λ / (8 π )[32, 33]. The conjugate quantity V can be viewed as a thermodynamical volume. The firstlaw reads dM = T dS + Φ e dQ e + Φ m dQ m + V d P Λ , (3.16)where V = ω r . (3.17)Since b is a dimensionful quantity, it will inevitably appearing in the Smarr relation. Itis useful also to introduce it as a thermodynamical quantity. Since b has the same di-mension of the cosmological constant, we may define P b = − b / (16 π ), The correspondingthermodynamical potential is V b = ω r s Q r b − Q b r
4+ 2 F (cid:20) ,
12 ; 54 ; − Q b r (cid:21)! . (3.18)The extended differential first law of black hole thermodynamics is given by dM = T dS + Φ e dQ e + Φ m dQ m + V d P Λ + V b d P b . (3.19)The above first law can also be expressed as dM = T dS + Φ e dQ e + Φ m dQ m + V d P Λ + b π ( V − V b ) db , (3.20)7s was proposed in [22]. The integral first law of black hole thermodynamics , also calledSmarr formula, is given by M = 2 ( T S − VP Λ − V b P b ) + Φ e Q e + Φ m Q m . (3.21)(See, also [34–36].) When the topological parameter k = 0, corresponding to AdS planarblack holes, there exists an additional generalized Smarr relation [37] M = 23 ( T S + Φ e Q e + Φ m Q m ) . (3.22) Now we will calculate the conserved charge M by using Wald formalism [31]. The conservedcharges of AdS black hole has been calculated by many different methods such as thecovariant phase space approach [38,39] developed by Wald, ADT formalism [31], and quasi-local ADT formalism [40–45]. The Wald formalism has been used to study the first law ofthermodynamics for asymptotically-AdS formalism in lots of theories, including Einstein-scalar theoy [46, 47], Einstein-Proca [48], Einstein-Yang-Mills [49], Einstein-Horndeski [50–52], in gravities extended with quadratic-curvature invariants [53], and also for Lifshitzblack hole [54].Since the conserved charge of dyonic black hole has the same as that with pure electriccase, so for simplicity we calculate the conserved charge for the back hole with the pureelectric charge. The effective Lagrangian is L = √− gL , L = R − − b r F b , (3.23)A general variation of the Lagrangian (3.23) was given in (2.3). The equations of motionare given by E µν = G µν + g µν Λ + 12 g µν b r F b − F µρ F µρ q F b , (3.24) E νA = ∇ µ F µν q F b . (3.25)The surface term J µ = J µg + J µA is given by J µg = g µρ g νσ ( ∇ σ δg νρ − ∇ ρ δg νσ ) ,J µA = − F µν δA ν q F b . (3.26)8rom this one can define a 1-form J (1) = J µ dx µ and its Hodge dual Θ ( D − = ( − ( D − ⋆J (1) .Considering the infinitesimal diffeomorphism x µ → x µ + ξ µ , one can get J ( D − ≡ Θ ( D − − i ξ ⋆ L = E Φ δ Φ − d ⋆ J (2) , (3.27)where i ξ denotes a contraction of ξ µ on the first index of the D -form ⋆ L . One can thus definean ( D − Q ( D − = ⋆J and J ( D − = d Q D − . Here we use the subscript notation“ ( p ) ” to denote a p -form. To make contact with the first law of black hole thermodynamics,we take ξ µ = ( ∂ t ) µ . Wald shows that the variation of the Hamiltonian with respect to theintegration constants of a specific solution is given by δ H = 116 π δ Z c J ( D − − π Z c d (cid:0) i ξ Θ ( D − (cid:1) = 116 π Z Σ ( D − (cid:0) δ Q ( D − − i ξ Θ ( D − (cid:1) , (3.28)where c denotes a Cauchy surface and Σ ( D − is its boundary, which has two components,one at infinity and one on the horizon. Thus according to the Wald formalism, the first lawof black hole thermodynamics is a consequence of δ H ∞ = δ H + . (3.29)For four dimensional EBI theory, we have J α α α = E.O.M. + ǫ α α α µ ∇ ν ∇ [ ν ξ µ ] − F µν A λ ξ λ q F b . (3.30)To specialise to our static black hole ansatz (3.1) in D = 4 dimensions (note that h ( r ) = f ( r )), the result for Lagrangian is well established and is given by δ Q − i ξ Θ = − ω r δfr + " − φ ′ b − r φδφ ′ . (3.31)Choosing the gauge such that the electrostatic potential φ vanishes on the horizon, it isstraightforward to verify that δ H + = T δS , δ H ∞ = δM − Φ e δQ e , (3.32)which yields the first law of black hole thermodynamics dM = T dS + Φ e dQ e . Although the vector field is singularity free, the general solution has a curvature singularityat the origin r = 0. To study the nature of the singularity, we consider small- r expansionnear the origin: f = − M − M ∗ ) r + k − bQ + ( b − ) r − b Q r + O ( r ) , (3.33)9here M ∗ = Γ( ) √ b √ π Q . (3.34)The Riemann-tensor squared is given by R µνρσ R µνρσ = 48( M − M ∗ ) r + 8 bQ ( M − M ∗ ) r + b Q r + O ( 1 r ) . (3.35)Thus we see that when M > M ∗ , the spacetime has a space-like singularity analogous tothe Schwarzschild black hole, whilst it has a time-like singularity. Note that the time-likesingularity arising from M < M ∗ is different from that in the Reissner-Nordstr¨om blackhole which has a 1 /r divergence. When M = M ∗ , the solution has a conical singularitywhere g tt is non-vanishing. Thus, for spherically-symmetric solutions with k = 1, we havethe following classifications: • Q > b : – M > M ∗ : Schwarzschild-like black hole with space-like 1 /r singularity. – M = M ∗ : Black hole with space-like 1 /r conical singularity. – M ext < M < M ∗ : Black hole with time-like 1 /r singularity, with outer andinner horizons. – M = M ext : Extremal black hole with time-like 1 /r singularity. – M < M ext : Naked time-like 1 /r singularity. • Q < b : – M > M ∗ : Schwarzschild-like black hole with space-like 1 /r singularity. – M = M ∗ : Naked time-like 1 /r singularity. – M < M ∗ : Naked time-like 1 /r singularity. • Q = b : – M > M ∗ : Schwarzschild-like black hole with space-like 1 /r singularity. – M = M ∗ : A null singularity where the horizon and curvature singularity coin-cide. – M < M ∗ : Naked time-like 1 /r singularity.It is worth noting that extremal black hole arises only for Q > /b .As mentioned earlier, the matter field A is singularity free at r = 0, one would thenexpect that there exists a parameter like M = M ∗ such that the spacetime solution is free10rom singularity. However, there is a singularity for the general solutions. To understandthis phenomenon, we note from (2.4) that the matter energy-momentum tensor is T µν mat = − b √− h √− g (cid:0) h − (cid:1) ( µν ) . (3.36)It follows that even if h µν is non-singular and non-vanishing at r = 0, the matter energy-momentum tensor diverges at r = 0 since √− g vanishes there. The singularity howeverbecomes much milder, and the solution with Q < /b and M = M ∗ may be viewed as aquasi-soliton. In previous sections, we studied the dyonic black hole solutions in the four-dimensional EBItheory, and obtained the first law of thermodynamics for these black holes. Now in thissection, we will generalize these results to arbitrary even dimensions D = 2 + 2 n . The general ansatz for AdS planar black holes in D = 2 + 2 n dimensions is given by ds = − f ( r ) dt + dr f ( r ) + r (cid:0) dx + dx + · · · + dx n − + dx n (cid:1) ,F = φ ′ ( r ) dr ∧ dt + p ( dx ∧ dx + · · · + dx n − ∧ dx n ) . (4.1)The equations of motion of A ν imply that φ ( r ) = Z ∞ r qdr q ( r + p b ) n + q b . (4.2)It reduces to the previous D = 4 case when n = 1. The Einstein equations imply that( r n − f ) ′ = − Λ n r n − b n s(cid:18) r + p b (cid:19) n + q b . (4.3)Note that Λ = Λ − b and hence there is a smooth b → ∞ limit. However, the limit b → n ≥ U n ( r ), which is convergent at r = 0, such that U n ( r ) − q ( r + p b ) n + q b /r . This choice is not unique, and we may choose U n =( r + p /b ) n . Making use of the identity Z r d ˜ r (cid:18) ˜ r n − (cid:16) ˜ r + p b (cid:17) n (cid:19) = r n +1 n + 1 (cid:18) − F (cid:20) − − n , − n − n − p b r (cid:21)(cid:19) , (4.4)we find that the function f can now be expressed as f ( r ) = − µr n − − Λ n (2 n + 1) r − b r n (2 n + 1) F (cid:20) − − n , − n − n − p b r (cid:21) + b nr n − Z r ∞ d ˜ r ((cid:16) ˜ r + p b (cid:17) n − s(cid:18) ˜ r + p b (cid:19) n + q b ) . (4.5) Now we study the thermodynamics of the dyonic AdS planar black holes in D = 2 + 2 n dimensions, constructed in the previous subsection. In the large- r expansion, the termassociated with graviton condensation has the falloff of 1 /r n − . It follows from (4.5) thatits coefficient is − µ , with no other terms giving any further contribution. Although there areslower falloffs due to the presence of the magnetic charges, one can nevertheless, followingfrom the Wald formalism, define a “gravitional mass” associated with only the condensationof the graviton modes [37]. It is given by M = nω n π µ , (4.6)where µ = − Λ r +2 n +1 n (2 n + 1) − b r n +1+ n (2 n + 1) F (cid:20) − − n , − n − n − p b r (cid:21) + b n Z r + ∞ dr ((cid:16) r + p b (cid:17) n − s(cid:18) r + p b (cid:19) n + q b ) . (4.7)Here, for simplicity, we assume that R dx dx = R dx dx = · · · = R dx n − dx n ≡ ω .The relation between the mass and the horizon radius r + can be determined by f ( r + ) = 0.We can now treat ( q, p, r + ) as independent parameters of the solution. In terms of theseparameters, the temperature and entropy are given by T = f ′ ( r + )4 π = − Λ nπ r + − b r − n + nπ s(cid:18) r + p b (cid:19) n + q b , (4.8) S = A ω n r n + . (4.9)The electric and magnetic charges are given by Q e = ω n π √− h ( h − ) [ tr ] | r →∞ = q π ω n , Q m = nω π Z F = np π ω , (4.10)12t follows from (4.2) that the electric potential is given byΦ e = Z ∞ r + qdr r(cid:16) r + p b (cid:17) n + q b . (4.11)It is easy to very that Φ e = ∂M/∂Q e . By assuming the differential first law of black holethermodynamics (3.15) is still hold, we can obtain the magnetic potentialΦ m = ∂M∂Q m = 2 ω n − " b r n +1+ np (cid:20) p b r (cid:21) n − F (cid:20) − − n , − n − n − p b r (cid:21)! + p Z r + ∞ (cid:18) r + p b (cid:19) n − − vuuut (cid:16) r + p b (cid:17) n (cid:16) r + p b (cid:17) n + q b dr . (4.12)The generalized “pressure” P Λ = − Λ / (8 π ), its conjugate quantity VV = ∂M∂ P Λ = ω n n + 1 r n +1+ . (4.13)The conjugate term of P b = − b / (16 π ) is given by V b = − nω n " − r n +1+ n (cid:18) p b r (cid:19) n + 2 n − n (2 n + 1) r n +1+ 2 F (cid:20) − − n , − n − n − p b r (cid:21) + 12 n Z r + ∞ ((cid:16) r + p b (cid:17) n − s(cid:18) r + p b (cid:19) n + q b ) dr + q nb Z r + ∞ dr r(cid:16) r + p b (cid:17) n + q b − p b Z r + ∞ (cid:18) r + p b (cid:19) n − − vuuut (cid:16) r + p b (cid:17) n (cid:16) r + p b (cid:17) n + q b dr , (4.14)The extended differential first law of black hole thermodynamics is given by dM = T dS + Φ e dQ e + Φ m dQ m + V d P Λ + V b d P b . (4.15)The above first law can also be expressed as dM = T dS + Φ e dQ e + Φ m dQ m + V d P Λ + b π ( V − V b ) db . (4.16)The Smarr formula is given by M = 2 n n − T S − n − VP + Φ e Q e + 12 n − m Q m − n − V b P b . (4.17)13he generalized Smarr formula is given by M = 2 n n + 1 ( T S + Φ e Q e ) + 22 n + 1 Φ m Q m . (4.18)In order to show the above identities, we need to use Z r + ∞ ((cid:16) r + p b (cid:17) n − s(cid:18) r + p b (cid:19) n + q b ) dr = r + n + 1 "(cid:18) r + p b (cid:19) n − s(cid:18) r + p b (cid:19) n + q b − nq (2 n + 1) b Z r + ∞ dr r(cid:16) r + p b (cid:17) n + q b + 2 np (2 n + 1) b Z r + ∞ (cid:18) r + p b (cid:19) n − − vuuut (cid:16) r + p b (cid:17) n (cid:16) r + p b (cid:17) n + q b dr . (4.19)Note that the definite integrations in all the above equations are well-defined with no di-vergence. It is remarkable that although the general solution is given up to a well-definedquadrature, the first law of thermodynamics, and Smarr relations can nevertheless be fullyestablished. We obtained the general dyonic AdS planar black holes, up to a quadrature. Here wepresent some explicit examples where the quadrature can be integrated in terms of somespecial functions.
In this case, we set p = 0, and we find f ( r ) = − µr n − − Λ r n (2 n + 1) − b r n (2 n + 1) F (cid:20) − , − − n ; 12 − n ; − q b r n (cid:21) . (4.20)Although our ansatz is for even D = 2 n + 2 dimensions, the above solution is applicable forodd dimensions as well so we rewrite it in terms of D : f ( r ) = − µr D − − r ( D − D − b r ( D − D − F (cid:20) − , − D − D −
2) ; D − D −
2) ; − q b r D − (cid:21) . (4.21)Furthermore, we can add a topological parameter k to f so that f → f + k . The solutionbecomes that for general topologies and was obtained in [15].14 .3.2 Pure magnetic solutions In this case, we set q = 0, and we find f ( r ) = − µr n − − Λ n (2 n + 1) r − b r n (2 n + 1) F (cid:20) − − n , − n − n − p b r (cid:21) . (4.22)Note that when n = 1, corresponding to four dimensions, the two solutions (4.22) and (4.21)takes the same form with q ↔ p , indicating electric and magnetic duality.Note also that when n = 2 m is even, corresponding to D = 4 m + 2 = 6 , , . . . dimen-sions, the hypergeometric function in (4.22) solution reduces to some polynomial functions.Here are some low-lying examples in 6, 10, 14 respective dimensions: n = 2 : f ( r ) = − Λ10 r − µr − p r ; n = 4 : f ( r ) = − Λ36 r − µr − p r − p b r ; n = 6 : f ( r ) = − Λ78 r − µr − p r − p b r − p b r . (4.23)Note that when n = 2, corresponding to D = 6, the metric is independent of b , whichimplies that the energy-momentum tensor for the Born-Infeld model is the same as that ofthe Maxwell theory.Here we present the thermodynamical properties of pure magnetic AdS planar blackholes µ = − Λ r +2 n +1 n (2 n + 1) − b r n +1+ n (2 n + 1) F (cid:20) − − n , − n − n − p b r (cid:21) , (4.24) T = − Λ nπ r + − b r − n + nπ (cid:18) r + p b (cid:19) n , S = ω n r n + , Q m = np π ω , (4.25)Φ m = 2 ω n − " b r n +1+ np (cid:20) p b r (cid:21) n − F (cid:20) − − n , − n − n − p b r (cid:21)! , (4.26) V b = − nω n − r n +1+ n (cid:18) p b r (cid:19) n + 2 n − n (2 n + 1) r n +1+ 2 F (cid:20) − − n , − n − n − p b r (cid:21) ! . (4.27) The quadrature cannot be integrated for general n in terms of a special function, exceptfor n = 1 and n = 2. The n = 1 example was discussed earlier. Now let us consider n = 2.15he function f ( r ) is given by f ( r ) = − Λ r − ˜ µr − r p p + b q F "
14 ; − , −
12 ; 54 ; − b r p − b q + p ; b r p − b q − p , (4.28)where F is the Appell hypergeometric function. This form of the solution is not convenientfor extracting the asymptotic infinite behavior. Another equivalent form of the solution isgiven by f ( r ) = − Λ r − µr − p + b r r s q b + (cid:18) r + p b (cid:19) + 3 p + b q b r F "
34 ; 12 ,
12 ; 74 ; p − b q − p b r , − p − b q − p b r + 3 p ( p + b q )35 b r F "
74 ; 12 ,
12 ; 114 ; p − b q − p b r , − p − b q − p b r . (4.29)The large- r expansion of f ( r ) is given by f ( r ) = − Λ10 r − p r − µr + q r − p q b r − q ( b q − p )352 b r + p q (3 b q − p )480 b r + · · · . (4.30)It is then clear that the parameter µ is related to the gravitional mass. Since this solutionis a special case of the general solutions, we shall not discuss its thermodynamics further. We may consider more general ansatz with the following general topologies in D = 2 n + 2dimensions ds = − f ( r ) dt + dr f ( r ) + r Σ ni =1 d Ω i,k , (4.31) F = φ ′ ( r ) dr ∧ dt + p Σ ni =1 dx i ∧ dy i , (4.32)where d Ω i,k = dx i − kx i + (1 − kx i ) dy i . (4.33)The φ ( r ) is given again the same as (4.2) and f ( r ) is given by f ( r ) = k n − − µr n − − Λ n (2 n + 1) r − b r n (2 n + 1) F (cid:20) − − n , − n − n − p b r (cid:21) + b nr n − Z r ∞ d ˜ r ((cid:16) ˜ r + p b (cid:17) n − s(cid:18) ˜ r + p b (cid:19) n + q b ) . (4.34)16t reduces to the previous result (4.5) for k = 0. The horizon topology now becomes M × M × · · · × M , where M can be sphere, torus or hyperbolic 2-space. In this paper, we studied the EBI theory and derived the equations of motion that isvalid in all dimensions and for all charge configurations. By contrast, the Lagrangian (1.3)considered in many previous works has limited application in higher dimensions. We thenconstructed the dyonic AdS black holes in four dimensions with a general topology. Weanalyzed the global structure and obtained the first law of thermodynamics. We classifiedthe singularity structure of these solutions. We then constructed the dyonic AdS blackholes in general even dimensions, where the solutions carry both the electric charge andalso the magnetic fluxes along the planar space. The general solutions were given up to aquadrature; nevertheless, we show that the first law of black hole thermodynamics can beestablished. We also give many special examples where the quadrature can be integrated interms of special functions. These solutions provide new gravity duals to study the AdS/CFTcorrespondence.
ACKNOWLEDGEMENTS
S.L.L. is grateful to Shuang-Qing Wu for useful discussions. He also thanks Xing-Hui Fengfor kind help and discussions. S.L.L. and H.W. are supported in part by NSFC underGrants No. 11575022 and No. 11175016. H.L. is supported in part by NSFC grants NO.11175269, No. 11475024 and No. 11235003.
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