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Entropy of Kerr--Newman--AdS black holes with torsion

The canonical approach to black hole entropy in Poincar\'e gauge theory without matter is extended to include Maxwell field as a matter source. The new formalism is used to calculate asymptotic charges and entropy of Kerr--Newmann--AdS black holes with torsion. The result implies that the first law, with a nontrivial contribution of the Maxwell field, takes the same form as in general relativity.


Introduction
The analysis of black hole spacetimes in general relativity (GR) shows that astrophysically, the most significant among them are those produced by rotating massive bodies [1]. The simplest spacetime of this type is the rotating, asymptotically flat solution found by Kerr [2]. The Kerr metric has been further generalized by including first the electric charge, and then a nonvanishing cosmological constant [3,4]. The final result of these generalizations is the Kerr-Newman-Anti de Sitter (KN-AdS) black hole, which is the most general stationary, asymptotically anti-de Sitter solution of Einstein-Maxwell field equations [5].
Starting from the early 1980s, many well-known exact solutions of GR have been generalized to solutions of the Poincaré gauge theory (PG), a modern gauge theory of gravity in which both the curvature and the torsion have their own dynamical roles [6]. Successful constructions of exact solutions with torsion [7,8,9] have been followed, inter alia, by an intensive investigation of the concept of conserved charge [10,11]. In contrast to that, a systematic investigation of black hole entropy in PG has long been neglected, although some incomplete attempts could have been noticed in the literature, as noted in [12].
A few years ago, a general canonical approach to black hole entropy in PG was proposed in [12]. The approach is based on a canonical formulation of the idea developed in GR, according to which entropy is just the Noether charge on horizon [13]. Applying this approach to a number of black holes with or without torsion [14,15,16], we found somewhat unexpected result: in spite of many geometric and dynamic differences with respect to GR, entropy of black holes in PG without matter, as well as the associated first law, follow essentially the same pattern as in GR, up to a multiplicative constant. In the present paper, we extend our investigation of entropy by introducing Maxwell field as a matter source for gravity (PG-Maxwell system). The analysis is focussed on exploring thermodynamic properties of the generalized KN-AdS black hole, constructed by Baekler et al. [8] in the late 1980s.
The paper is organized as follows. In section 2, we present a brief account of the general thermodynamic aspects of the PG-Maxwell system. In particular, a new definition of the black hole entropy in the presence of Maxwell field is introduced as a natural generalization of the earlier definition, valid in vacuum. In section 3, we describe geometric aspects of the KN-AdS black hole as a solution of the PG-Maxwell system. Next, in sections 4 and 5, we use these results to calculate energy, angular momentum, and entropy. The thermodynamic role of the Maxwell field and the resulting first law are clarified in section 6, section 7 is devoted to concluding remarks, and appendixes contain some important technical details.
Our conventions are the same as in Ref. [16]. The Latin indices (i, j, . . . ) are the local Lorentz indices, the Greek indices (µ, ν, . . . ) are the coordinate indices, and both run over 0, 1, 2, 3. The orthonormal coframe (tetrad) ϑ i and the metric compatible (Lorentz) connection ω ij = −ω ji are 1-forms, the dual basis (frame) is e i = e i µ ∂ µ , the interior product of e i with ϑ j is e i ϑ j = δ j i , and A is the electromagnetic potential 1-form. The metric components in the local Lorentz and coordinate basis are η ij = (1, −1, −1, −1) and g µν = η ij ϑ i µ ϑ j ν , respectively, and ε ijmn is the totally antisymmetric symbol with ε 0123 = 1. The Hodge dual is marked by a star ⋆ , and the wedge product of forms is implicit.

PG-Maxwell system
We begin with an overview of the general Lagrangian and thermodynamic aspects of the PG dynamics in the presence of Maxwell field; for more details, see Refs. [12,17].

Lagrangian formalism
In PG, the structure of spacetime is characterised by a Riemann-Cartan geometry, in which the torsion T i = dϑ i + ω i k ϑ k and the curvature R ij = dω ij + ω i k ω kj (2-forms) are the gravitational field strengths, associated to the Poincaré (translational and Lorentz) gauge potentials, the tetrad ϑ i and the Lorentz connection ω ij , respectively. Moreover, our physical system contains also the Maxwell field characterised by the field strength F = dA (2-form), where A is the electromagnetic gauge potential (1-form).
Dynamical properties of the PG-Maxwell system are defined by the total Lagrangian is a parity even PG Lagrangian, assumed to be at most quadratic in the field strengths, and L M = L M (ϑ i , F ) describes the Maxwell field interacting with gravity. The gravitational field equations are obtained by varying L with respect to the gravitational potentials ϑ i and ω ij . Introducing the gravitational covariant momenta, H i := ∂L G /∂T i and H ij := ∂L G /∂R ij , and the associated energy-momentum and spin currents, E i = ∂L G /∂b i and E ij := ∂L G /∂ω ij , these equations can be written in a compact form as The source term on the right-hand side of (2.2a) is the Maxwell energy-momentum current τ i := ∂L M /∂ϑ i , while the related spin current vanishes, σ ij := ∂L M /∂ω ij = 0. Similarly, the variation of L with respect to the electromagnetic potential A yields the Maxwell equation, where H := ∂L M /∂A is the electromagnetic covariant momentum. The PG part of the total Lagrangian (2.1) has the form where (a 0 , Λ, a n , b n ) are the gravitational coupling constants, and (n) T i , (n) R ij are irreducible parts of the field strengths. The Maxwell part reads where 4a 1 is a suitably normalized coupling constant.
In the analysis of black hole thermodynamics, we need the following explicit formulas:

Thermodynamics
The Hamiltonian approach to black hole entropy in PG [12] is based on the ideas developed originally in GR [18,13], according to which the asymptotic charges (energy and angular momentum) as well as entropy, can be defined by certain boundary terms. Here, we introduce an extended version of that approach, suitable for analysing non-vacuum solutions of the PG-Maxwell system. Consider a stationary black hole spacetime whose spatial section Σ has a two-component boundary, one component at infinity and the other at horizon, ∂Σ = S ∞ ∪ S H . Then, asymptotic charges and entropy of a PG-Maxwell black hole are determined by the boundary integral Γ := Γ ∞ − Γ H , determined by the following variational equations: By construction, δB is obtained from the canonical generator of local translations. It contains not only the gravitational term (upper line), but also the Maxwell term (bottom line), extending thereby the construction adopted in [12] to non-vacuum solutions. 1 Specific forms of the Killing vector ξ (ξ = ∂ t , ∂ ϕ or a linear combination thereof) are chosen so that the boundary integrals (Γ ∞ , Γ H ) could be physically interpreted in terms of the asymptotic charges, black hole entropy, and an external, Maxwell term. To have a consistent interpretation, we require the operation δ to satisfy the following rules: (r1) On S ∞ , the variation δ acts on the parameters of a black hole solution, but not on the parameters of the background configuration. (r2) On S H , the variation δ must keep surface gravity constant.
Moreover, mathematical consistency strongly depends on the boundary conditions: (r3) When the boundary terms (δΓ ∞ , δΓ H ) are δ-integrable and finite, they can be given the usual thermodynamic interpretation.
Finally, note that Γ ∞ and Γ H are introduced as a priory independent objects. However, the analysis of their construction from the canonical gauge generator reveals that the regularity of the generator can be expressed by the relation which is equivalent to the first law of black hole thermodynamics. The Maxwell contribution to δB is an essential part of the first law.

Geometry and dynamics
In this section, we analyse basic properties of KN-AdS black holes as solutions of the PG-Maxwell system [8].

Metric and tetrad
The metric of a KN-AdS black hole in Boyer-Lindquist coordinates has the form [1] Here, m, a and q are the parameters characterising energy (mass), angular momentum and electric charge of the solution, and λ = −Λ/3a 0 . The orthonormal tetrad associated to the metric is chosen in the form The larger root of ∆(r) = 0 defines the outer horizon, and the angular velocity and surface gravity have the same form as in GR [5,15], , (3.5) and the area of the horizon is The Riemannian connectionω ij is calculated in Appendix A.

Torsion, connection and curvature
Riemann-Cartan geometry of PG is characterized by a nonvanishing torsion. For KN-AdS black holes, the ansatz for torsion is formally the same as for the Kerr-AdS case [8,14], where ϑ − = ϑ 0 − ϑ 1 , but the metric function N and the torsion functions V n are modified by the nonvanishing electric charge parameter q 2 , Having introduced torsion, the Riemann-Cartan connection can be expresses as where K ij is the contortion 1-form, implicitly defined by the relation The curvature 2-form R ij = dω ij + ω i k ω kj has only two nonvanishing irreducible parts: The quadratic invariants (Euler, Pontryagin and Nieh-Yan) are given by

PG-Maxwell field equations
Since the only nonvanishing parts of the gravitational field strengths are (1) T i , (2) T i and (4) R ij , (6) R ij , the "effective" form of the gravitational Lagrangian reads The covariant momenta H i and H ij , appearing in the field equations (2.2), are given by 13) and the corresponding spin currents are 14) The contribution of the electromagnetic sector to Eqs. (2.2) is described by the Maxwell energy-momentum current [17] The form of τ i depends on the Maxwell potential in a KN-AdS spacetime [19], where q e is the electromagnetic charge parameter. This expression is a natural generalization of the spherically symmetric form A = −(q e /r)dt. The related field strength and the covariant momentum are F = − q e ρ 4 (r 2 − a 2 cos 2 θ)ϑ 0 ϑ 1 + 2ar cos θ ϑ 2 ϑ 3 , (3.17a) When all the previous results taken into account, the explicit calculation shows that basic dynamical variables (ϑ i , ω ij , A) of a KN-AdS black hole, which are defined in Eqs. (3.2a), (3.9a) and (3.16), solve the PG-Maxwell field equations (2.2) if the Lagrangian parameters (a n , b n , Λ) and the solution parameters (λ, q, q e ) satisfy the relations Thus, according to our conventions, the electromagnetic charge parameter q e differs from the metric charge parameter q. However, none of them coincides with the asymptotic Maxwell charge, as will be shown in section 6.

Asymptotic boundary terms
The asymptotic values of energy and angular momentum are defined by the boundary term δB(ξ) in (2.5). Two aspects of explicit calculations deserve a special attention. First, Carter [20] and Henneaux and Teitelboim [21] demonstrated that the asymptotic metric of Kerr-AdS spacetimes cannot be properly described in Boyer-Lindquist coordinates. They found a new set of coordinates in which this deficiency is brought under control. However, our variational approach (2.5) allows a simpler procedure [14,15], in which only the subset (t, ϕ) of the Boyer-Lindquist coordinates is transformed to the "untwisted" form, Under these transformations, the components (v t , v ϕ ) of a 4-vector v µ transform as In particular, g T ϕ = g tϕ + g ϕϕ , And second, the background configuration, defined by m = q = 0, depends on the parameter a. To avoid the variation of those a's that are associated to the background, we introduce a more precise formulation of the rule (r1) for the variation δ, given below Eq. (2.5): (r1 ′ ) In calculating δΓ ∞ (ξ), first apply δ to all the parameters (m, a, q), then subtract those δa terms that survive the limit m = q = 0, as they come from the background.
Before continuing, it is interesting to note that the lower line in the expression δB(ξ), Eq. (2.5), which refers to the contribution of the Maxwell field, yields vanishing boundary terms at infinity, but not at horizon. This follows from the asymptotic behavior of the variables A and H, defined by Eqs. (3.16) and (3.17). Hence, nontrivial energy and angular momentum are generated only by the contributions stemming from the gravitational sector.

Angular momentum
The angular momentum is defined by δE ϕ := δΓ ∞ (∂ ϕ ). The calculation is performed by ignoring (m, q)-independent δa terms (background), even when they are divergent, and by omitting asymptotically vanishing terms. The nonvanishing contributions are Summing up the two terms, one obtains

Energy
Going over to energy, we calculate the nonvanishing contributions to δE t = δΓ ∞ (∂ t ), Hence, The result is not δ-integrable but, as we mentioned above, it can be corrected by moving to the untwisted coordinates (T, φ): The expressions (4.2) and (4.3) are proportional to the corresponding GR values.

Entropy
In this section, we analyse the PG part of the boundary term at horizon, δΓ H , where the Killing vector ξ is given by As well be shown, this part defines the black hole entropy. The Maxwell contribution to δΓ H will be discussed in the next section.

Nonvanishing terms
The calculation entropy is organised in two technical steps.
The only nonvanishing contributions stemming from the first element of δΓ 1 are Here, the symbol [=] stands for an equality up to the factor dθdϕ.

Simplifications
The above contributions can be simplified using the following properties (see Appendix B): S1. The sum of the terms proportional to δN/N in (5.2)-(5.4) vanishes.
Hence, the original contributions (5.2)-(5.4) can be simplified as follows: Next, we use the relation A ′ 0 = λb 4 + a 1 to express these contributions in terms of only two independent constants, λb 4 and a 1 . The analysis of the λb 4 part leads to an additional simplification (Appendix B).
The conclusions S1, S2 and S3 are the KN-AdS extensions of the results found for the Kerr-AdS black holes in [14].

Maxwell boundary term and the first law
The standard canonical analysis of the Maxwell sector implies that the asymptotic electric charge Q can be defined by the boundary integral The minus sign is just a matter of convention. Next, following Ref. [19], we define the electric potential Φ by The result is confirmed by the identity (C.2). After removing the common factor 16πa 1 , the first law (6.4) becomes identical to its GR form.

Concluding remarks
The canonical approach to black hole entropy proposedd in [12] has been successfully applied to a number of vacuum solutions of PG [14,15,16]. In the present paper, we introduce its natural extension to non-vacuum solutions, by including Maxwell field as a matter source of gravity. Using this formalism, we study thermodynamic properties of KN-AdS black holes, encoded in the boundary terms at infinity and horizon, δΓ ∞ and δΓ H , respectively. Analysing energy and angular momentum as the boundary terms at infinity, we found that their KN-AdS values are exactly the same as for the uncharged Kerr-AdS solution [4,14]. This is in agreement with the fact that the asymptotic Maxwell contribution vanishes. Moreover, these asymptotic charges are proportional to the related GR expressions.
The boundary term at horizon produces entropy and an external, Maxwell term, such that both of them are also proportional to the corresponding GR expressions [4,15]. Then, the first law is described by the general relation δΓ ∞ = δΓ H , which follows from the way the boundary terms are constructed, see subsection 2.2. Apart from this general argument, we give an explicit proof of the first law based on the identity derived in Appendix B. After removing the overall multiplicative factor, the first law becomes identical to its GR form.
Thus, although PG has a rather different dynamical structure from GR, the present description of the KN-AdS thermodynamics is rather close to the GR results. A reason for this "accidental" similarity might be hidden in the identity found in Appendix B.

Acknowledgments
This work was partially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia.
Explicit form of the covariant momenta H i and H ij is given by and

B On the evaluation of entropy
In this appendix, we discuss certain technical details of the derivation of entropy.
B. r 2 + + a 2 sin θ , one can directly conclude that their sum vanishes, as a consequence of A ′ 0 ≡ a 1 + λb 4 . There are two more contributions of this type, sin 3 θ , sin 3 θ , whose sum also vanishes. Hence, all (δN)/N terms in entropy can be simply ignored. After removing δN/N terms, one finds that the sum of δP/P terms also vanishes:

C Proof of the first law
In this appendix, we derive an identity which is of essential importance for understanding the kinematic origin of the first law.
Then, a direct comparison shows that the relation L = R is identically satisfied: This identity coincides with the first law in GR.