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Conserved charges in 3D gravity

The covariant canonical expression for the conserved charges, proposed by Nester, is tested on several solutions in 3D gravity with or without torsion and topologically massive gravity. In each of these cases, the calculated values of energy-momentum and angular momentum are found to satisfy the first law of black hole thermodynamics.


Introduction
Topologically massive gravity with a cosmological constant (TMG Λ ) is an extension of the usual, Einstein's three-dimensional (3D) gravity, obtained by adding the gravitational Chern-Simons term to the action [1]. This step leads to an essential modification of Einstein's (topological) theory, giving rise to a propagating degree of freedom, the massive graviton. However, for generic values of the coupling constants, the physical interpretation of TMG Λ in the anti-de Sitter (AdS) sector suffers from serious difficulties, related to the instability of the AdS 3 vacuum [1,2,3]. To avoid the problem, Li et al. [3] introduced the so-called chiral version of TMG Λ and argued that it might be free from these difficulties, but there are also opposite opinions [4]. Another option would be to choose a new vacuum, such as the spacelike stretched AdS 3 , which could be a stable ground state of the theory [5].
It is clear that all these issues are closely related to the concept of conserved charges, which is essential for a clear understanding of the stability problem. The conserved charges were calculated for a number of different solutions of TMG Λ [6,7,8,9], but, as noted in [10], the methods used in these papers are not best suited for studying general structure of the boundary dynamics. An attempt to improve the situation was made by Mišković and Olea [10], who proposed a background-independent Noether construction of the conserved charges in TMG Λ . They found correct conserved charges for the Bañados-Teitelboim-Zanelli (BTZ) black hole and the logarithmic solution; however, when applied to the spacelike stretched black hole, their method yields the value of energy with an extra factor 1/2, as compared to [7,9], the result that is not supported by the first law of black hole thermodynamics [7,5]. In the present paper, we focus our attention on another idea-the covariant canonical formalism developed by Nester [11] and collaborators [12,13,14,15], which has had excellent results for a large class of the gravitational solutions in 4D. We intend to show that it leads to a rather general expression for the conserved charges in 3D gravity, valid not only in TMG Λ but also in 3D gravity with or without torsion.
The paper is organized as follows. In section 2, we give a brief account of the general first order Lagrangian formalism in 3D gravity, with a focus on (a) 3D gravity with or without torsion and (b) TMG Λ . In section 3, we introduce Nester's covariant canonical formalism and define general covariant expressions for the conserved charges. In section 4, we use these results to evaluate the conserved charges for several solutions in 3D gravity, including not only the BTZ black hole and the logarithmic solution, but also the spacelike stretched black hole. The conserved charges found here satisfy the first law of thermodynamics, and moreover, they are in agreement with the results obtained earlier in [7,8,9,16,17]. Section 5 is devoted to concluding remarks, and appendices contain some technical details.
Our conventions are as follows: the Latin indices (i, j, k, ...) refer to the local Lorentz frame, the Greek indices (µ, ν, λ, ...) refer to the coordinate frame, and both run over 0,1,2; the metric components in the local Lorentz frame are η ij = (+, −, −); totally antisymmetric tensor ε ijk is normalized to ε 012 = 1. Our notation follows the Poincaré gauge theory (PGT) framework in 3D [8,9]: fundamental dynamical variables are the triad field b i and the Lorentz connection ω i (1-forms), T i = db i + ε ijk ω j b k and R i = dω i + 1 2 ε i jk ω j ω k are the corresponding field strengths, the torsion and the curvature (2-forms), the wedge product signs (∧) are omitted for simplicity and the relation to the standard 4D notation is given by

First order Lagrangians in 3D gravity
Using Nester's ideas [11], we show in this section that the following gravitational models in three dimensions: (a) 3D gravity with torsion [18,16], including also Einstein's 3D gravity, and (b) topologically massive gravity [1], can be described by a generic first order Lagrangian (density), with Here, not only b i and ω i , but also τ i and ρ i are independent dynamical variables (1-forms), and V is a conveniently chosen term for each of the above two cases, as described below.
(a) Three-dimensional gravity with torsion is a topological theory in Riemann-Cartan spacetime, defined by the Mielke-Baekler Lagrangian [18,16]: where a = 1/16πG, and L CS (ω) = ω i dω i + 1 3 ε ijk ω i ω j ω k is the Chern-Simons Lagrangian for the Lorentz connection. The Lagrangian can be rewritten in the form where τ i and ρ i are defined as the covariant field momenta, conjugate to the field strengths T i and R i : Now, instead of using the original Lagrangian (2.2), it is more convenient to reformulate it so that τ i and ρ i become independent dynamical variables. This goal is achieved by the first order Lagrangian (2.1), where the constraints (2.2c) are enforced via Lagrange multipliers in V [12,14].
(b) A simple description of TMG Λ is given by the Lagrangian [8] where λ i (1-form) is a Lagrange multiplier that ensures the vanishing of torsion. This theory can also be transformed into the form (2.1), with a choice of V that ensures the correct on-shell values of τ i , ρ i : The explicit form of V in the generic Lagrangian (2.1) is not important for our discussion of the conserved charges; all we need to remember is that V does not contain derivatives of dynamical variables,

Covariant canonical formalism for conserved charges
Starting with the first order Lagrangian (2.1) and using the field-theoretic analogue of the classical mechanics relation Ldt = (pq − H)dt, one can introduce a timelike vector field ξ and define the Hamiltonian 2-form H(ξ) on the spatial hypersurface Σ of spacetime [11]. In analogy with old Dirac's ideas [19], these considerations can be generalized by allowing ξ to be either timelike or spacelike, whereupon H(ξ) becomes the generalized Hamiltonian density, associated with the dynamical evolution along ξ.
The Hamiltonian density H(ξ) contains a boundary term dB, but the requirement that H(ξ) generates the correct equations of motion allows a freedom in the choice of B. As shown by Regge and Teitelboim [20], the proper form of B is determined by the requirement that the functional derivatives of H(ξ) = Σ H(ξ) are well defined or, equivalently, that the boundary term in δH(ξ) vanishes. The verification of this criterion is closely related to the specific form of the asymptotic conditions.
If ξ is asymptotically a Killing vector, the related conserved charge is naturally identified as the on-shell value of H(ξ). Since the generalized Hamiltonian has the form where H µ is found to be proportional to the field equations, the value of H(ξ) on a solution of the field equations reduces to the boundary integral ∂Σ B(ξ). Hence, the corresponding conserved charge is defined as the value of ∂Σ B(ξ).
The construction of the correct boundary term for specific boundary conditions can be a rather complicated task. However, to find a unique expression for B that is compatible with a number of different boundary conditions is much more difficult. Ideally, we would like to have a universal expression for B that holds for all (physically acceptable) boundary conditions. Nester [11] started a search for an ideal B in the context of PGT by proposing an expression for it, compatible with solutions having either flat or constant curvature asymptotic behavior. Later modifications of this boundary term were intended to make it valid in more general gravitational theories and for a larger set of boundary conditions, and also to improve its covariance properties [12,13,14,15].
In order to properly define the boundary term, it is necessary to choose a reference dynamical configuration. This configuration is most naturally linked to the minimal value of the conserved charge [20]. Let us denote the difference between any variable X and its reference valueX by ∆X = X −X. In 3D, the boundary term B is a 1-form. With a suitable set of boundary conditions for the fields, the proper boundary term reads [13]: and the relation to our 3D notation is defined by If ξ is asymptotically a Killing vector, this formula can be simplified by choosing b i so that Lξ b i vanishes on the boundary. In that case, the identity One should note that when triad and connection are independent dynamical variables, the concept of Killing vector differs from the usual one, as defined in GR; see, for instance [21]. By our convention, since ω ij and ρ ij are antisymmetric objects, the summation over i, j in the last two terms goes over i < j. After returning to our 3D notation, we obtain In our calculations of the boundary integrals, we shall use the Schwarzschild-like coordinates x µ = (t, r, ϕ). For solutions with Killing vectors ∂ t and ∂ ϕ , the conserved charges are energy and angular momentum, respectively: where ∂Σ is a circle (which may be located at infinity), described by coordinate ϕ.

Conserved charges in 3D gravity
In 4D, the expression (3.1a) for B was shown to be valid for general gauge theories of gravity, such as PGT or metric-affine gravity (MAG), and for a large set of known solutions with different boundary conditions [11,12,13,14,15]. Here, we wish to verify the correctness of B in 3D gravity by evaluating the conserved charges for several solutions in either 3D gravity with torsion or topologically massive gravity.

BTZ black hole in 3D gravity with or without torsion
The BTZ black hole [22], a well-known solution of Einstein's 3D gravity in the AdS sector (with Λ = −1/ℓ 2 ), is also a solution of 3D gravity with torsion [22,16]. In the Schwarzschildlike coordinates, the black hole metric is given by Our approach to 3D gravity with torsion is based on the PGT formalism; see [16]. Given the metric, we choose the triad field to have the simple, "diagonal" form: while the solution for Cartan's connection reads whereω i is Riemannian connection, and p = (α 3 Λ + α 4 a)/(α 3 α 4 − a 2 ). Equations (4.1) define the analogue of the BTZ black hole in Riemann-Cartan spacetime. The black hole solution (4.1) possesses two Killing vectors, ξ (0) = ∂ t and ξ (2) = ∂ ϕ , which leave the triad field b i form invariant, Lξ b i = 0. As the reference configuration, we take the solution (4.1) with m = J = 0 (the black hole vacuum). The conserved charges are defined in (3.3), with the field momenta τ i and ρ i given in (2.2c). Using the asymptotic behavior of b i and ω i (Appendix A), we obtain the following expressions for the energy and angular momentum, respectively: The result was derived in [22] 1 using Nester's covariant method, and rederived in [16] using Dirac's canonical formalism. Recalling the expressions for the black hole entropy and angular velocity [16], one can verify the validity of the first law of black hole thermodynamics. For α 3 = α 4 = 0, 3D gravity with torsion reduces to Einstein's 3D gravity, and we have E = m and M = J, as expected.

BTZ black hole in TMG Λ
The BTZ black hole is a trivial solution of TMG Λ , since the related Cotton tensor identically vanishes. The solution for b i and ω i has the same form as in (4.1) but with p = 0, while the Lagrange multiplier is given by [8] To find the conserved charges, we combine (2.3b) with (4.3c) to obtain Note that the solution (4.3) and the covariant field momenta (4.4) can be obtained from the corresponding expressions (4.1) and (2.2c) in 3D gravity with torsion, by implementing the following limit on parameters: Hence, the same limit yields the conserved charges in TMG Λ : in complete agreement with the results found in [2,8,10]. The first law of black hole thermodynamics is a direct consequence of the arguments given in the previous subsection. At the chiral points µℓ = ∓1, the BTZ charges (4.5) satisfy the chirality relations ℓE = ±M, independently of the values of m and J. Specifically, for the extreme black holes J = ∓mℓ at the chiral points µℓ = ∓1, both energy and angular momentum vanish: (4.6)

Spacelike stretched black hole in TMG Λ
After introducing a convenient notation, Λ = −1/ℓ 2 and ν = µℓ/3, the metric of the spacelike stretched black hole can be written in the form [7,5] where The solution is not of constant curvature. Going over to the PGT formalism [9], we choose the triad field as and calculate the corresponding Riemannian connectionω i : Finally, the solution for λ m has the form The result follows from the general formula where (Ric) mn = −ε ij m R ijn is the Ricci tensor, and R = −ε ijm R ijm the scalar curvature, see Eqs. (2.5c) and (B.2b) of Ref. [9]. Now, using the expressions (2.3b) for τ i and ρ i and the asymptotic behavior of N, B, K and N ϕ (Appendix B), with reference configuration defined by r − = r + = 0, Eq. (3.3) yields the energy and angular momentum of the spacelike stretched black hole (4.8): These values coincide with those found in [7,9,5], and moreover, they are in agreement with the first law of black hole thermodynamics [7,5]. On the other hand, a comparison with the results of Ref. [10] shows a discrepancy by a factor of 1/2 in energy.

Logarithmic solution of TMG Λ
Let us now consider another genuine solution to TMG Λ , the logarithmic solution at the chiral point µℓ = −1 [23]. The metric of the solution is stationary and spherically symmetric: where: The metric depends on two constants m and k, whereas r 0 is a normalization factor that does not influence the values of the conserved charges, but does have an impact on the geometric properties of the solution. For k = 0, (4.11) reduces to the extreme BTZ black hole with J = −mℓ, at the chiral point µℓ = −1.
In the PGT formalism, we start with the corresponding Riemannian connection ω i =ω i takes the form and the solution for λ m reads: For a detailed derivation of (4.12), see Appendix C. Using our basic formula (3.3) for the conserved charges, the expressions (2.3b) for the field momenta τ i , ρ i , the asymptotic behavior as described in Appendix D and taking the reference configuration with m = k = 0, we find that the energy and angular momentum of the logarithmic solution (4.12) are: To verify (4.13), we first used the standard canonical approach and found that the energy and angular momentum of the logarithmic solution (4.12) are given by the formulas which are the same as those describing the AdS sector of TMG Λ , taken at µℓ = −1 [8]. The evaluation of these expressions produces the same result as in (4.13). Moreover, since the Hawking temperature of the logarithmic solution vanishes, 4πT = (A 2 ) ′ | r + = 0, and the angular velocity is Ω = C| r + = −1/ℓ (with r 2 + = 4Gmℓ 2 ), it follows that the charges (4.13) satisfy the first law of black hole thermodynamics: The results (4.13) essentially agree with those obtained by Clement [17], up to some differences in conventions. Indeed, his angular momentum and Ω have oposite signs as compared to ours, but they still satisfy the first law. Our conserved charges also agree with the results found in [10], but they differ by an overall factor 3/2 when compared to [23].
Let us observe that in parallel with the logarithmic solution at the chiral point µℓ = −1, there exists another logarithmic solution at µℓ = 1, obtained from (4.12) by changing the sign of C. For k = 0, the new solution reduces to the extreme BTZ black hole with J = mℓ. An analogous evaluation of its conserved charges leads tō Since nowΩ = 1/ℓ, the first law is again satisfied. Note that this "antichiral" solution is different from the solution displayed in Eq. (25) of Ref. [23].

Concluding remarks
In this paper, we examined Nester's covariant canonical expression (3.1) for the conserved charges in 3D gravity. It is shown that the evaluated energy and angular momentum of the BTZ black hole (in both 3D gravity with torsion and TMG Λ ), spacelike stretched black hole and the logarithmic solution, are in complete agreement with the results obtained by different methods in [2,7,8,9,16,17]. Moreover, for each of these solutions, the calculated conserved charges are seen to satisfy the first law of black hole thermodynamics. On the other hand, the authors of [10], working in the context of TMG Λ , obtained the same results as ours in the case of the BTZ black hole and the logarithmic solution, but their treatment of the spacelike stretched black hole features certain difficulties. Thus, our approach based on (3.1) exhibits a wider range of validity.
The Lagrangians considered in the present paper are linear in torsion and/or curvature. As a further test of universality of the formula (3.1) in 3D gravity, it would be interesting to apply it to the Lagrangians quadratic in torsion and/or curvature. After this paper has been completed, we started studying the conserved charges in new massive gravity [24]. As a first result in this direction, we found that the formula (3.3) gives the correct conserved charges for the BTZ solution [25].

A Asymptotics of the BTZ black hole
In this appendix, we display the asymptotic form of the BTZ black hole solution in 3D gravity with torsion and in TMG Λ . The subleading terms are given up to the order that contributes to the values of conserved charges.
BTZ black hole in 3D gravity with torsion. Starting with (4.1a), we find the asymptotic form of the triad field: Recalling that Cartan's connection has the form (4.1b), it follows that its asymptotics is determined by (A.1a) and BTZ black hole in TMG Λ . Looking at (4.3), we see that the asymptotic form of the basic dynamical variables in TMG Λ is determined by the formulas (A.1) taken at p = 0, where the torsion vanishes.

B Asymptotics of the spacelike stretched black hole
Dynamical variables of the spacelike stretched black hole solution (4.8) are expressed in terms of the functions N, K and N ϕ . Consequently, the asymptotic behavior of the solution is determined by the following asymptotic relations:

C Derivation of the logarithmic solution
To derive the logarithmic solution (4.12), we start with the triad field (4.12a), and calculate the Riemannian connection ω i =ω i : This result implies (4.12b). Note that the coefficients α, β and γ are not independent: In the next step, we calculate the Riemann curvature: The solution for λ m , based on Eq. (4.9), takes the form which implies (4.12c).
In order to show that (4.12) is indeed a solution of TMG Λ , we calculated the Cotton 2-form C i = (µ/2a)∇λ i , and verified the basic field equation of TMG Λ :