Thermodynamics of Riemannian Kerr-AdS black holes in Poincar\'e gauge theory

A Hamiltonian approach to black hole entropy is used to study Riemannian Kerr-AdS solutions in the general, parity-violating Poincar\'e gauge theory. Entropy and the asymptotic charges are entirely determined by the parity-even sector of the theory, whereas the parity-odd contributions vanish. Entropy is found to be proportional to the horizon area, and the first law of black hole thermodynamics is confirmed.


Introduction
From the experience with general relativity (GR), we know that exact solutions play an essential role in the physical interpretation of gravitational theories [1]. In Poincaré gauge theory (PG), where both the torsion T i and the curvature R ij define the gravitational dynamics [2], exact solutions have been often constructed by suitably "incorporating" torsion into the known solutions of GR. In particular, an advanced technique was used by Baekler et al. [3] to construct a Kerr-AdS black hole with propagating torsion, in the sector of parity-invariant PG Lagrangians. Recently, Obukhov [4] made one more step in the same direction by extending the construction to the most general, parity-violating PG [5]. Thus, there are at least three versions of Kerr-AdS spacetimes, one in GR and the other two in PG; they possess the same metric but live in different dynamical setups.
Investigations of black hole thermodynamics have given rise to a deeper insight into both the classical and quantum nature of gravity [6]. In the 1990s, classical black hole entropy was most compactly described as the diffeomorphism Noether charge on horizon [7,8,9]. In a recently proposed Hamiltonian approach to black hole entropy [10], the same idea was extended to PG, where diffeomorphisms are replaced by the Poincaré gauge symmetry. It offers a unified description of the asymptotic charges (energy and angular momentum) and entropy of black holes with or without torsion, in terms of certain boundary integrals at infinity and on horizon, respectively. The approach was successfully applied to Kerr-AdS thermodynamics in GR [11], as well as to parity-invariant and general PG models [12,13]. In the present work, we extend the Hamiltonian analysis to the class of Riemannian Kerr-AdS solutions in the general PG. A comparison to the results found in Riemannian theories with quadratic curvature Lagrangians [8,14], as well as in the general PG models [13], reveals how black holes react to different dynamical frameworks.
The paper is organized as follows. In section 2, we give a short account of the Hamiltonian approach to black hole thermodynamics in the context of the general, parity-violating PG. In section 3, we introduce the tetrad formulation of the Kerr-AdS geometry, needed in the Hamiltonian analysis, and discuss limitations of Boyer-Lindquist coordinates. Sections 4 and 5 contain the main results of the paper-derivation of the asymptotic charges and entropy for Riemannian Kerr-AdS blck holes in PG. Section 6 is devoted to discussion.
Our basic notation is the same as in Refs. [11,13]. The latin indices (i, j, . . . ) refer to the local Lorentz frame, the greek indices (µ, ν, . . . ) refer to the coordinate frame, b i is the orthonormal coframe (tetrad) and ω ij is a metric compatible connection (1-forms), h i is the dual basis (frame) such that h i b k = δ k i , and η ij = (1, −1, −1, −1) is the local Lorentz metric. The wedge symbol in the exterior products is omitted, the volume 4-form isǫ = b 0 b 1 b 2 b 3 , the Hodge dual of a form α is denoted by ⋆ α, with ⋆ 1 =ǫ, and the totally antisymmetric symbol ε ijmn is normalized to ε 0123 = +1.

Black hole entropy as a boundary term
In this section, we give a short account of the Hamiltonian approach to the entropy of black holes, restricted to the class of Riemannian solutions in the general PG, see Refs. [11,13].
We begin by recalling some geometric aspects of PG. The gravitational field is described by two independent dynamical variables, the tetrad b i and the metric compatible spin connection ω ij (1-forms), which are associated to the translation and the Lorentz subgroups of the Poincaré group, respectively. The corresponding field strengths are the torsion T i = db i + ω i k b k and the curvature R ij := dω ij + ω i k ω kj (2-forms), and the underlying spacetime structure is characterized by a Riemann-Cartan geometry.
The general PG dynamics is determined by a Lagrangian L G (b i , T i , R ij ) which is at most quadratic in the field strengths and contains both even and odd parity modes. In this work, we are interested in the class of vacuum solutions with vanishing torsion. Their dynamics is effectively described by a simplified Lagrangian without torsion invariants, Here, (a 0 , Λ 0 , b n ) and (ā 0 ,b n ) are the Lagrangian parameters in the parity even and odd sectors, respectively, ⋆ R = ⋆ (b i b j )R ij is he Einstein-Hilbert and ⋆ X = (b i b j )R ij the Holst term [15], and (n) R ij are irreducible parts of the curvature [10]; for T i = 0, only those for n = (1, 4, 6) are nonvanishing. The gravitational field equations are derived by varying L G with respect to b i and ω ij . The status of Riemannian solutions in the framework of PG was clarified by Obukhov [5]: △ Any solution of GR with a cosmological constant is a torsion-free solution of the general PG field equations.
In particular, this is true for Kerr-AdS solutions, the subject of the present work.
In the Hamiltonian approach, the asymptotic charges are defined as certain boundary terms at spatial infinity, which make the associated canonical gauge generator G differentiable [16]. By extending this construction, one can naturally introduce entropy as a boundary term on horizon [10]. Consider a stationary, axisymmetric black hole, and let Σ be a spatial section of spacetime whose boundary consists of two components, one at infinity and the other at horizon, ∂Σ = S ∞ ∪ S H . The asymptotic charges and black hole entropy are defined by the variational equations for the respective boundary terms Γ ∞ and Γ H : where ξ is a Killing vector (∂ t or ∂ ϕ on S ∞ , and a linear combination thereof on S H , such that ξ 2 = 0), and H ij is the covariant momentum determined by the Lagrangian (2.1), The operation δ is assumed to be in accordance with the following two rules: (r1) The variation δΓ ∞ is defined by varying parameters of the black hole state over the boundary S ∞ , leaving the background configuration fixed. (r2) The variation δΓ H is defined by varying the parameters on horizon, but keeping surface gravity constant over the horizon, in accordance with the zeroth law.
When there exist finite solutions for Γ ∞ and Γ H (δ-integrability), they are interpreted as thermodynamic charges. Their values are strongly correlated to the adopted boundary conditions. The boundary terms Γ ∞ and Γ H in (2.2) are introduced as apriori independent objects. However, if the canonical generator G defining local symmeties of a black hole is differentiable, the corresponding boundary term Γ is not needed, it vanishes. Thus, assuming the boundary S H to have the opposite orientation with respect to S ∞ , we have which is nothing but the first law of black hole thermodynamics. As follows from Eq. (2.3), the covariant momentum H ij consists of two independent parts, defined by the even and odd parity sectors of L G . Consequently, each of the boundary terms Γ ∞ and Γ H can be represented as the sum of two parts with opposite parities.

.1 Tetrad formalism
The Kerr-AdS metric is a solution of GR with a cosmological constant. In Boyer-Lindquist coordinates (t, r, θ, ϕ), it can be formulated in terms of the orthonormal tetrad [11,12] Here, 0 ≤ θ < π and 0 ≤ ϕ < 2π, m and a are parameters of the solution, and λ is determiend by the PG field equations, 3a 0 λ = −Λ 0 . The metric ds 2 = η ij b i ⊗b j , which is stationary and axially symmetric, admits the Killing vectors ∂ t and ∂ ϕ . Many metric-related characteristics of geometry play an essential role in black hole thermodynamics, such as the location of the outher horizon r = r + , the horizon area A H , the angular velocity ω + and the surface gravity κ,

Limitations of Boyer-Lindquist coordinates
The background configuration for m = 0 is described by the AdS geometry but in somewhat non-standard coordinates, in which metric components depend on the parameter a. Hence, one cannot clearly distinguish whether the variation δa is related to the background or to the genuine black hole configuration. To avoid the variation of the AdS background, we introduce an improved version of the rule (r1).
(r1 ′ ) When δΓ ∞ is calculated for Kerr-AdS black holes in Boyer-Lindquist coordinates, first apply δ to all a's, then remove those δa terms that survive the limit m = 0, as they are associated to the background configuration.
However, this is not sufficient to make the Boyer-Lindquist coordinates well defined. Namely, as shown by Henneaux and Teitelboim [17], see also Carter [18], the metric in these coordinates does not have a proper, asymptotically AdS behavior, needed for the canonical identification of asymptotic charges. To avoid the problem, they introduced a suitable change of coordinates which brings the metric to the standard, asymptotically AdS form. In our variational approach (2.2), the problem shows up as δ-nonintegrability of the asymptotic charges. It can be resolved by using the reduced form of the Carter-Henneaux-Teitelboim coordinate transformations, see for instance [11], This change of coordinates and the improved rule (r1 ′ ) ensure the new asymptotic charges, , to be well defined. In addition to that, Black hole entropy is determined by δΓ H [ξ], where ξ = ∂ T − Ω + ∂ φ and Ω := g T φ g φφ = ω + λa , For large r, the new angular velocity Ω vanishes, as expected [9]. It turns out that the expression δΓ H [ξ] is invariant under the coordinate transformations (3.6).
In what follows, we will use the notation PG + and PG − for the even and odd parity sectors of PG, respectively.

Thermodynamic charges in PG +
To simplify technical exposition of our analysis of Kerr-AdS thermodynamics, we first analyse Eqs. (2.2) in the PG + sector, leaving PG − for the next section.
The PG + sector is effectively described by the Lagrangian and the corresponding covariant momentum is The expressions for angular momentum, energy and entropy, produced by the first term in (4.1b), are of the GR form, but with a 0 → (a 0 − λb 6 ), see [11]: where T = κ/2π. Hence, to obtain the complete result, one needs to calculate only an additional contributions from the Weyl curvature term H W ij := 2b 1 ⋆ W ij . In the calculations that follow, the integration over the boundaries is implicit.

Asymptotic charges
Angular momentum. We start with the additional contribution to angular momentum, determined by the W -reduced relation (2.2) with ξ = ∂ φ = ∂ ϕ , Here, there are only two nonvanishing terms, After completing the integration, one obtains Summing up this expression with the GR-like term (4.2a), the complete PG + contribution to angular momentum takes the form Energy. Consider now the W -contribution to energy, defined by the variational equation There are three nonvanishing contributions to δE W t , defined by (i, j) = (0, 1), (1, 2) and (1, 3). A direct calculation yields the result which is not δ-integrable. Such an inconsistency of Boyer-Lindquist coordinates was noted also in GR [11]. Transition to the well-behaved (T, φ) coordinates via (4.6a) yields Then, adding the GR-like term (4.2b) yields the complete PG + contribution to energy,

Entropy and the first law
For ξ = ∂ T − Ω + ∂ φ , the W -contribution to entropy is given by where ω ij ξ := ξ ω ij and similarly for H W ijξ . It contains only one nonvanishing contribution, In (4.11a), we used ω 01 ξ = −κ, and in (4.11b), the first term is obtained by integration, and the last equality follows from the identity 1 + λr 2 + = α + λ(r 2 + + a 2 ). Summing the above result with (4.2c), one obtains the complete PG + expression for entropy, Each of the Kerr-AdS thermodynamic charges (E φ , E T , S) in the PG + sector can be obtained from the corresponding GR expression by a 0 → A 0 . Hence, the first law is automatically satisfied, δE T − Ω + δE φ = T δS .

Thermodynamic charges in PG −
The analysis starts with the effective Lagrangian and the corresponding covariant momentum 1. The PG − expression for angular momentum is defined by Here, there are only two nonvanishing terms, ω 23 ϕ δH 23 + δω 23 H 23ϕ = δ ω 23 ϕ (H 23 ) θϕ dθdϕ , ω 02 ϕ δH 02 + δω 02 H 02ϕ = δ ω 02 ϕ (H 02 ) θϕ dθdϕ . For each of these terms, the integration over θ yields an expression of the general form where the change of variables x = cos θ implies I = 0. Hence, the complete angular momentum stemming from the PG − sector vanishes, The PG − contribution to energy is determined by the relations

Concluding remarks
In the present paper, we analyzed thermodynamic properties of Riemannian Kerr-AdS solutions in the context of general, parity-violating PG models, using the Hamiltonian approach proposed in [10]. Black hole entropy and asymptotic charges are completely determined by the contributions stemming from the PG + sector, whereas those from the PG − sector vanish. The general form of the thermodynamic charges guarantees that the first law is automatically satisfied. Using the identity W ij = R ij − λb i b j , one can rewrite the complete covariant momentum (4.1b) + (5.1b) in an equivalent form as whereĀ 0 = (ā 0 − λb 6 ) + λb 1 . The last terms in the upper and lower line are associated to the Euler and Pontryagin topological invariants, R ij ⋆ R ij and R ij R ij , respectively, in the Lagrangian. Our calculations show that these two terms produce vanishing contributions to the termodynamic charges. Note also that the Holst term is not a topological invariant, but it also has no impact on the Kerr-AdS thermodynamic charges. These conclusions are in agreement with those obtained by Jacobson and Mohd [14] in their analysis of the tetrad form of higher curvature gravity.
Comparing the results obtained here to those describing Kerr-AdS solutions with a nonvanishing torsion [13], one can conclude that they are characterized by different characteristic constants A 0 and a 1 , respectively, This difference can be understood as a consequence of different dynamical settings in the two cases, or, more specifically, as an effect of torsion on the Riemann-Cartan connection.