Dyonic (A)dS Black Holes in Einstein-Born-Infeld Theory in Diverse Dimensions

We study Einstein-Born-Infeld gravity and construct the dyonic (A)dS planar black holes in general even dimensions, that carry both the electric charge and magnetic fluxes along the planar space. In four dimensions, the solution can be constructed with also spherical and hyperbolic topologies. We study the black hole thermodynamics and obtain the first law. We also classify the singularity structure.


Introduction
In 1934, Born and Infeld [1] proposed an elegant nonlinear version of electrodynamics that successfully removes the divergence of self-energy of a point-like charge in Maxwell's theory of electrodynamics. The Lagrangian density of the Born-Infeld (BI) theory in D-dimensional Minkowski spacetime is given by where η µν = diag(−1, 1, 1, 1) is the Minkowski metric, F µν = 2∂ [µ A ν] is the Faraday tensor and A = A µ dx µ is the Maxwell gauge potential. BI theory contains a dimensionful parameter b, and in the limit b → ∞, BI theory reduces to the Maxwell theory, (1.2) In the limit b → 0, the Lagrangian in four dimensions becomes F ∧ F which is a total derivative. The limit is generally singular in higher dimensions.
BI theory has enjoyed further attentions since the invention of string theory. It turns out that the BI action can arise from string theory [2], describing the low energy dynamics of Dbranes [3]. We refer to e.g. [4,5] for some comprehensive reviews on the BI theory in string theory. The special Born-Infeld-like nonlinear form is also very useful to construct analogous new theories, such as Dirac-Born-Infeld (DBI) inflation theory [6,7] and Eddington-inspired Born-Infeld (EiBI) cosmologies [8]. BI theory can also be adopted to explore issues of dark energy [9,10].
In this paper, we focus on the study of black holes in Einstein-Born-Infeld (EBI) theory.
(A)dS black hole solutions in BI theory considered in literature typically involves only either the electric or magnetic charges. Although the dyonic black hole in EBI theory was constructed in [11], it is written in the general (static) type-D form. The global structure in the spherically symmetric form was analysed in [25] for the asymptotically-flat case. In this paper, we shall first study the dyonic (A)dS black holes in the EBI theory in four dimensions with general topologies, focus on analysing the black hole thermodynamics and singularity structure. We then construct dyonic AdS planar black holes in arbitrary even dimensions, where the solutions carry both the electric flux as well as the magnetic 2-form flux along the planar space.
Interestingly in almost all the previous works on constructing black holes, the equivalent action in four dimensions was used, rather than the original one. In D = 4, the Lagrangian can be equivalently expressed as [1] where 4) in which E and B are electric and magnetic fields, and where ε µνρσ is a tensor density with ε 0123 = 1. we generalize the results to all even dimensions. We conclude the paper in Sec. 5.

EBI and its equations of motion
In this section, we consider the EBI theory. The Lagrangian of BI theory can be naturally generalized to curved spacetimes and the Lagrangian is given by where g µν is the metric. The Lagrangian of the EBI theory with a bare cosmological constant Λ 0 can be written by Here, Λ is the effective cosmological constant. The variation of Lagrangian (2.2) gives rise to where g = det(g µν ), J µ is the surface term and in which G µν = R µν − g µν R/2, h µν = g µν + F µν /b, h ≡ det(h µν ), and (h −1 ) µν denotes the inverse of h µν , satisfying We further defined The equations of motion are then given by E µν = 0 and E µ A = 0. These equations are derived from the original Lagrangian (2.2) of the EBI theory and hence are applicable in all dimensions and for all charge configurations.

Dyonic black hole in four dimensions
In the previous section, we obtained the equations of motion of the EBI theory. We now construct the static dyonic (A)dS black hole solution with a general topological horizon in four dimensions. We shall then study the global structure and the black hole thermodynamics.

Local solution
The static solution in the type-D form in the EBI theory was first constructed in [11]. The spherically-symmetric and asymptotically-flat solution was given in [25]. In this section, we study the properties of the dyonic (A)dS black holes. The most general static ansatz can be written as where k = 1, 0, −1 denotes the metric for the unit 2-spheres, 2-torus or the unit hyperbolic 2-space, and p is magnetic charge parameter. It turns out that the equations of motion of the 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 where and thereafter, we use a prime to denote a derivative with respect to r, and q is a integral constant that is related to the electric charge. The function f (r) satisfies Thus f (r) can be solved and expressed in terms of hypergeometric function 2 F 1 where µ is the integral constant corresponding to the mass of the solution. The electric potential φ(r) is expressed by In the limit b → ∞, the solution recovers the dyonic Reissner-Nordström-(A)dS black hole, On the other hand, in the limit b → 0, the Born-Infeld field vanishes and the solution is reduced to the Schwarzschild-(A)dS black hole in pure cosmological Einstein theory, In the large-r expansion, we have Thus we see that the first few leading-order expansions match those of the Reissner-Nordström-(A)dS black hole.

Thermodynamics
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 terms of r + , namely Since the metric is asymptotically (A)dS, according to the definition of mass in asymptotically (A)dS space by Abbott-Deser-Tekin (ADT) formalism [31], we find where ω 2 = dudϕ. For k = 1, corresponding the unit S 2 , we have ω 2 = 4π.
The temperature T and entropy S on the horizon are easily calculated as The electric and magnetic charges are given by Note that the above electric charge as a conserved quantity follows from the equation of motion (2.5). The electric and magnetic potentials are given by (3.14) The differential first law of black hole thermodynamics can be written as One can further treat the cosmological constant as a generalized "pressure" P Λ 0 = −Λ 0 /(8π) [32,33]. The conjugate quantity V can be viewed as a thermodynamical volume. The first law reads Since b is a dimensionful quantity, it will inevitably appearing in the Smarr relation. It is useful also to introduce it as a thermodynamical quantity. Since b 2 has the same dimension of the cosmological constant, we may define The corresponding thermodynamical potential is The extended differential first law of black hole thermodynamics is given by The above first law can also be expressed as 20) as was proposed in [22]. The integral first law of black hole thermodynamics , also called Smarr formula, is given by (See, also [34][35][36].) When the topological parameter k = 0, corresponding to AdS planar black holes, there exists an additional generalized Smarr relation [37]
Since the conserved charge of dyonic black hole has the same as that with pure electric case, so for simplicity we calculate the conserved charge for the back hole with the pure electric charge. The effective Lagrangian is
Considering the infinitesimal diffeomorphism x µ → x µ + ξ µ , one can get where i ξ denotes a contraction of ξ µ on the first index of the D-form ⋆L. One can thus define 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 the integration constants of a specific solution is given by where c denotes a Cauchy surface and Σ (D−2) is its boundary, which has two components, one at infinity and one on the horizon. Thus according to the Wald formalism, the first law of black hole thermodynamics is a consequence of For four dimensional EBI theory, we have 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 Choosing the gauge such that the electrostatic potential φ vanishes on the horizon, it is straightforward to verify that which yields the first law of black hole thermodynamics dM = T dS + Φ e dQ e .

Singularity structures
Although the vector field is singularity free, the general solution has a curvature singularity at the origin r = 0. To study the nature of the singularity, we consider small-r expansion near the origin: where The Riemann-tensor squared is given by -M = M ext : Extremal black hole with time-like 1/r 6 singularity.
-M = M * : A null singularity where the horizon and curvature singularity coincide.
It is worth noting that extremal black hole arises only for Q > 2/b.
As mentioned earlier, the matter field A is singularity free at r = 0, one would then expect that there exists a parameter like M = M * such that the spacetime solution is free from singularity. However, there is a singularity for the general solutions. To understand this phenomenon, we note from (2.4) that the matter energy-momentum tensor is It follows that even if h µν is non-singular and non-vanishing at r = 0, the matter energymomentum tensor diverges at r = 0 since √ −g vanishes there. The singularity however becomes much milder, and the solution with Q < 2/b and M = M * may be viewed as a quasi-soliton.

Generalization to higher dimensions
In previous sections, we studied the dyonic black hole solutions in the four-dimensional EBI theory, and obtained the first law of thermodynamics for these black holes. Now in this section, we will generalize these results to arbitrary even dimensions D = 2 + 2n.

Local solutions
The general ansatz for AdS planar black holes in D = 2 + 2n dimensions is given by (4.1) The equations of motion of A ν imply that It reduces to the previous D = 4 case when n = 1. The Einstein equations imply that Note that Λ 0 = Λ − 1 2 b 2 and hence there is a smooth b → ∞ limit. However, the limit b → 0 is singular for n ≥ 3.
Thus we see that the solution to the metric function can be expressed in terms of a quadrature. In order to read off the thermodynamical quantities, we would like to write the solution in terms of a well-defined quadrature as a definite integration . To do so, we may define a function U n (r), which is convergent at r = 0, such that U n (r) − (r 4 + p 2 b 2 ) n + q 2 b 2 has a falloff that is faster than 1/r. This choice is not unique, and we may choose U n = (r 4 + p 2 /b 2 ) n 2 . Making use of the identity we find that the function f can now be expressed as

Thermodynamics
Now we study the thermodynamics of the dyonic AdS planar black holes in D = 2 + 2n dimensions, constructed in the previous subsection. In the large-r expansion, the term associated with graviton condensation has the falloff of 1/r 2n−1 . It follows from (4.5) that its coefficient is −µ, with no other terms giving any further contribution. Although there are slower falloffs due to the presence of the magnetic charges, one can nevertheless, following from the Wald formalism, define a "gravitional mass" associated with only the condensation of the graviton modes [37]. It is given by (4.7) Here, for simplicity, we assume that 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 these parameters, the temperature and entropy are given by The electric and magnetic charges are given by It follows from (4.2) that the electric potential is given by It is easy to very that Φ e = ∂M/∂Q e . By assuming the differential first law of black hole thermodynamics (3.15) is still hold, we can obtain the magnetic potential (4.12) The generalized "pressure" (4.13) The conjugate term of P b = −b 2 /(16π) is given by 14) The extended differential first law of black hole thermodynamics is given by The above first law can also be expressed as The Smarr formula is given by The generalized Smarr formula is given by In order to show the above identities, we need to use Note that the definite integrations in all the above equations are well-defined with no divergence. It is remarkable that although the general solution is given up to a well-defined quadrature, the first law of thermodynamics, and Smarr relations can nevertheless be fully established.

Some explicit examples
We obtained the general dyonic AdS planar black holes, up to a quadrature. Here we present some explicit examples where the quadrature can be integrated in terms of some special functions.

Pure electric solutions
In this case, we set p = 0, and we find Although our ansatz is for even D = 2n + 2 dimensions, the above solution is applicable for odd dimensions as well so we rewrite it in terms of D: Furthermore, we can add a topological parameter k to f so that f → f + k. The solution becomes that for general topologies and was obtained in [15].
The function f (r) is given by where F 1 is the Appell hypergeometric function. This form of the solution is not convenient for extracting the asymptotic infinite behavior. Another equivalent form of the solution is given by (4.29) The large-r expansion of f (r) is given by (4.30) It is then clear that the parameter µ is related to the gravitional mass. Since this solution is a special case of the general solutions, we shall not discuss its thermodynamics further.

A more general topology
We may consider more general ansatz with the following general topologies in D = 2n + 2 dimensions ds 2 = −f (r)dt 2 + dr 2 f (r) + r 2 Σ n i=1 dΩ 2 i,k , (4.31) where The φ(r) is given again the same as (4.2) and f (r) is given by (4.34) It reduces to the previous result (4.5) for k = 0. The horizon topology now becomes M 2 × M 2 × · · · × M 2 , where M 2 can be sphere, torus or hyperbolic 2-space.

Conclusion
In this paper, we studied the EBI theory and derived the equations of motion that is valid 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 then constructed the dyonic AdS black holes in four dimensions with a general topology. We analyzed the global structure and obtained the first law of thermodynamics. We classified the singularity structure of these solutions. We then constructed the dyonic AdS black holes in general even dimensions, where the solutions carry both the electric charge and also the magnetic fluxes along the planar space. The general solutions were given up to a quadrature; nevertheless, we show that the first law of black hole thermodynamics can be established. We also give many special examples where the quadrature can be integrated in terms of special functions. These solutions provide new gravity duals to study the AdS/CFT correspondence.