Gravitational Energy of Kerr and Kerr Anti-de Sitter Space-Times in the Teleparallel Geometry

Transcription

1 Gravitational Energy of Kerr and Kerr Anti-de Sitter Space-Times in the Teleparallel Geometry arxiv:gr-qc/ v2 4 Jun 2002 J. F. da Rocha-Neto and K. H. Castello-Branco Instituto de Física Teórica, Universidade Estadual Paulista Rua Pamplona São Paulo Brazil Instituto de Física, Universidade de São Paulo São Paulo, Brazil , Abstract In the context of the Hamiltonian formulation of the teleparallel equivalent of general relativity we compute the gravitational energy of Kerr and Kerr Anti-de Sitter (Kerr-AdS) space-times. The present calculation is carried out by means of an expression for the energy of the gravitational field that naturally arises from the integral form of the constraint equations of the formalism. In each case, the energy is exactly computed for finite and arbitrary space volumes, without any restriction on the metric parameters. In particular, we evaluate the energy at the outer event horizon of the black holes. PACS numbers: Cv, Fy (*) rocha@ift.unesp.br

2 I. Introduction The problem of defining a consistent and unequivocal expression for the energy of the gravitational field is still an open and important question in general relativity. It is well known that the principle of equivalence has led to the belief that the gravitational energy cannot be localized [1]. However, the argument based on this principle regarding the nonlocalizibility of the gravitational energy is controversial and not generally acepted [2]. Therefore it is legitimate to conjecture that the difficulties associated to the problem of defining the gravitational energy is related to the geometrical description of the gravitational field, rather being an intrinsic nuisance of the theory [2]. The first attempts to define the energy of the gravitational field were based on pseudotensors [3], which make use of coordinate dependent expressions. More recently the idea of quasi-local energy, i.e., energy associated to a closed spacelike two-surface, in the context of the Hilbert-Einstein action integral, has emerged as a tentative description of the gravitational energy [4]. Teleparallel theories of gravity, whose basic entities are tetrad fields e aµ ( a and µ are SO(3, 1) and space-time indices, respectively) have been considered long time ago by Møller [5] in connection also with attempts to define the energy of the gravitational field. Teleparallel theories of gravity are defined on the Weitzenböck space-time [6], which is endowed with the afinne connection Γ λ µν = eaλ µ e aν. The curvature tensor constructed out of this connection vanishes identically. This connection defines a space-time with an absolute parallelism or teleparallelism of vector fields [7]. In this geometrical framework the gravitational effects are due to the torsion tensor corresponding to the above mentioned connection. It was in this context that Einstein tried to unify gravity and electromagnetism [8]. Although there exists an infinity of gravity theories in this geometrical framework [9], here we will consider only the teleparallel equivalent of general relativity 1

3 (TEGR) [10, 11, 12, 13, 14, 15]. The TEGR is a particular class of teleparallel gravity theories whose corresponding tetrad fields satisfy the Einstein s equations in tetrad form. The TEGR is an alternative formulation of Einstein s general relativity. As remarked by Hehl [16], by considering Einstein s general relativity as the best available alternative theory of gravity, its teleparallel equivalent is the next best one. Therefore it is interesting to perform studies of the space-time structure as described by the TEGR. A simple definition for the gravitational energy field has been established in the Hamiltonian formulation of the TEGR [17] in the framework of Schwinger s time gauge condition [18]. The gravitational energy is given by an integral of a scalar density in the form of a total divergence that appears in the Hamiltonian constraint of the theory. Such definition has led to consistent and relevant results when applied to important configurations of the gravitational field, such as the evaluation of the irreducible mass of the Kerr black hole [19]. The Hamiltonian formulation of the TEGR, with no a priori restriction on the tetrad fields, has recently been established [20]. Again, in this formulation, an expression for the gravitational energy naturally arises [21], in strict similarity with the procedure adopted in [17]. The literature concerning on black holes is mainly devoted to treat these objects in asymptotically flat space-times. Among these, the Kerr metric is one of the most important known configutations of the gravitational field, consisting in the only vacuum rotating black hole solution of Einstein s equations [22]. Nevertheless, the ideal case in considering a black hole space-time as being asymptotically flat is not always appropriate, and therefore it is important to analyse non-asymptotically flat black hole space-times [23], such as a Kerr-anti-de Sitter (Kerr-AdS) space-time. Recently, AdS spaces have received special attention due to the AdS/CFT correspondence [24] and studies in braneworld scenarios [25]. AdS spaces have been also studied earlier in the literature, as 2

4 for example in [26], where it was showed that a (large) Schwarzschild-AdS black hole is thermodynamically stable, whereas in [27] the conserved charges of AdS spaces was studied. In this paper we apply the definition of gravitational energy that arises in the context of the Hamiltonian formulation of the TEGR for the cases of the Kerr and Kerr-AdS space-times. It is an important issue to investigate the generality of the energy definition of Ref. [21] by applying it to nontrivial manifolds of the AdS type. In the Sec. II we summarize the Hamiltonian formulation of TEGR developed in Ref. [20]. In the Secs. III and IV we compute the gravitational energy within an arbitrary volume V of space for the Kerr and Kerr-AdS black holes. In each case we obtain the gravitational energy for an arbitrary volume in exact form, without any condition on the metric parameters. In particular we compute the energy at the outer event horizon. The last section is devoted to concluding remarks. Notation: space-time indices µ, ν,... and SO(3,1) indices a, b,... run from 0 to 3. Time and space indices are indicated according to µ = 0, i, a = (0), (i). The tetrad field e a µ yields the definition of the torsion tensor: T a µν = µ e a ν ν e a µ. The flat, Minkowski space-time metric is fixed by η ab = e aµ e bν g µν = ( + ++). II. The Hamiltonian Formulation of the TEGR In this section we summarize the Hamiltonian formulation obtained in Ref. [20]. The latter was carried out without posing any a priori restriction on the tetrad fields. The Hamiltonian formulation is obtained from the Lagrangian density in empty space-time, given by 3

5 ( 1 L(e) = k e 4 T abc T abc + 1 ) 2 T abc T bac T a T a where e = det(e a µ), T abc = e b µ e c ν T aµν, T b = T a ab, k = 1 16πG, (2.1) and G is the gravitational constant. By explicit calculations [14] it is possible to show that the variation of the action integral with respect to e aµ yields the Einstein s equations in tetrad form δl δe k { aµ 2 e R aµ (e) 1 }. 2 R(e) The Hamiltonian is obtained by the prescription L = p q H and without making any kind of projection of metric or tetrad variables to the three-dimensional spacelike hypersurface. Since there is no time derivative of e a0 in (2.1), the corresponding momenta canonically conjugated Π a0 vanish identically. They constitute primary constraints that induce to the secundary constraints C a. Dispensing with surface terms, the total Hamiltonian density reads [20] H(e ai, Π ai ) = e a0 C a + α ik Γ ik + β k Γ k, (2.2) where C a = k Π ak + e a0 [ 1 ( 4g 00ke g ik g jl P ij P kl 1 ) 2 P 2 ( 1 +ke 4 gim g nj T b mnt bij gnj T i mnt m ij g ik T m mit n nk 1 ( 2g 00ke g ik g jl γ aij P kl 1 ) ( 2 g ijγ aij P ke e ai g 0m g nj T b ijt bmn ) +g nj T 0 mnt m ij + g 0j T n mjt m ni 2g 0k T m mkt n ni 2g jk T 0 ijt n nk, (2.3) )] Γ ik and Γ k are primary constraints defined by { } Γ ik = Γ ki = Π [ik] k e g im g kj T 0 mj + (g im g 0k g km g 0i )T j mj, 4

6 Γ k = Π 0k + 2k e (g kj g 0i T 0 ij g 0k g 0i T j ij + g 00 g ik T j ij). {C a, Γ k, Γ ik } is a set of first class constraints, α ik and β k are Lagrange multipliers and [..] denotes anti-symetrization. More details are given in Ref. [20]. The first term of the constraint C a is given by a total divergence in the form k Π ak. As in Ref. [17] we identify this total divergence on the three-dimensional spacelike hypersurface as the energy-momentum density of the gravitational field. The gravitational energy-momentum is then defined by P a = d 3 x i Π ai, (2.4) V where V is an arbitrary space volume. It is invariant under coordinate transformations on the spacelike manifold, and transforms as a vector under the global SO(3,1) group. The definition above generalizes the analogous energy expression (11) of the Ref. [17] to tetrad fields that are not restricted by the time gauge condition. However, as shown in Ref. [21] both expressions coincide if the time gauge condition is imposed. In order to compute the gravitational energy we need the explicit expression of the momenta Π ak. In terms of torsion tensor it is given by { Π ak = k e g 00 ( g kj T a 0j e aj T k 0j + 2e ak T j 0j) +g 0k (g 0j T a 0j + e aj T 0 0j) + e a0 (g 0j T k 0j + g kj T 0 0j) 2(e a0 g 0k T j 0j + e ak g 0j T 0 0j) } g 0i g kj T a ij + e ai (g 0j T k ij g kj T 0 ij) 2(g 0i e ak g ik e a0 )T j ji. (2.5) For asymptotically flat space-times, in the limit r the tetrad fields have the asymptotic behaviour e aµ η aµ + 1 ( ) 1 2 h aµ, (2.6) r 5

7 where η aµ is the Minkowski s metric tensor and h aµ is the first term in the asymptotic expansion of g µν. Asymptotically flat space-times are defined by (2.6) together with µ g λν = O( 1 ), or r 2 µ e aν = O( 1 ). Considering the a = (0) component in Eq. (2.4) and integrating over the whole three-dimensional spacelike hypersurface, after a straightforward r 2 calculation, we arrive at [21] P (0) = E g = V = 1 16πG d 3 x k Π (0)k = 2k d 3 x k (eg ik e (0)0 T j ji) V S ds k ( i h ik k h ii ) = E ADM, (2.7) which is the ADM energy [28]. The above result motivates the definition of the gravitational energy enclosed by an arbitrary volume V of the three-dimensional space as III. Kerr space-time E g = d 3 x i (Π (0)i ) = ds i Π (0)i. (2.8) V S In this section we investigate the application of the expression (2.8) in the analysis of the energy of the Kerr black hole. In Ref. [21] the energy of the Kerr black hole was computed only for a volume defined by the outer event horizon. Differently of [21] here we compute the gravitational energy within an arbitrary space volume V without any restriction on the metric parameters. In terms of Boyer-Lindquist [29] coordinates the Kerr metric tensor is given by ds 2 = ψ2 ρ 2 dt2 2χ sin2 θ dφ dt + ρ2 ρ 2 dr2 + ρ 2 dθ 2 + Σ2 sin 2 θ dφ 2, (3.1) ρ 2 where ρ 2 = r 2 + a 2 cos 2 θ, = r 2 + a 2 2mr, χ = 2amr and Σ 2 = (r 2 + a 2 ) 2 a 2 sin 2 θ, 6

8 ψ 2 = a 2 sin 2 θ. The construction of tetrads that correspond to a given metric has been carefully analysed in [21]. A set of tetrads that satisfy the boundary condition at infinity given by Eq. (2.6) and that is associated to the metric (3.1) is ψ 2 + χ2 sin 2 θ Σ ρ χ ρ sin θ sin φ Σρ e aµ = χ ρ sin θ cosφ Σρ 0 sin θ cosφ ρ cosθ cosφ Σ ρ Σ sin θ sin φ ρ cosθ sin φ sin θ cosφ ρ ρ cosθ ρ sin θ 0 sin θ sin φ. (3.2) Lets us now apply Eq. (2.8) to an arbitrary, fixed region of radius r of the Kerr spacetime. For this purpose we need to compute the component Π (0)1 of the momentum. After a long, but straigthforward calculation, we can show that Π (0)1 is given by where and ( Π (0)1 = 2k sin θ ρ + 1 ) α + β cos ρ 2 θ ρ r α + β cos 2 θ, (3.3) α = (r 2 + a 2 ) 2 a 2 β = a 2. Therefore, the gravitational energy contained inside an arbitrary spherical volume V of a three-dimesional spacelike hypersurface of the Kerr space-time is given by E g = 1 2π π ( dφ dθ sin θ ρ + 1 r ) α + β cos 8π 0 0 ρ 2 θ ρ r α + β cos 2 θ. (3.4) In Refs. [19, 21] the integration is performed only to the surface defined by r = r + (outer event horizon). However, the knowledge of the distribution of the gravitational energy for 7

9 an arbitrary surface is of importance in the description of other properties of the black hole space-time, such as a thermodynamic analysis not restricted to the event horizon. By the change of varible y = cosθ, we find that it is possible to integrate (3.4) in terms of elliptical functions in analytic, exact form for arbitrary values of r and no restriction on the metric parameters. After the integration we arrive at the final expression of the gravitational energy enclosed by a surface of arbitrary radius r. It reads E g = 1 4 [ ( r 2 + a 2 + r2 2a ln r2 + a 2 ) a r2 + a 2 + a + i2 ( α E i a β a 2 r, α ) r i ( β a a 2 α rαf i a β r, α ) r a where i 2 = 1, and + i [ ( α a 2 β rβ F i a β r, α ) ( r E i a β a r, α )]] r, (3.5) a E(x, k) = x 0 dy 1 k2 y 2 1 y 2, x F(x, k) = 1 dy 1 y 2 1 k 2 y 2 0 are the definitions of the elliptic functions E(x, y) and F(x, y), respectively. In the assymptotic limit r, the gravitational energy given by (3.5) approaches the ADM energy m, and for a = 0, we obtain the energy of the Schwarzschild black hole [4]. For the surface defined by r = r +, the above expression for the gravitational energy E g reduces to [ 2p E g (r + ) = m + 4 6p k2 4k ( )] 2p + k ln, (3.6) p where p = k 2, a = km, 0 k 1. The result in (3.6) is the same obtained in Ref. [21]. This result is in excellent agreement with the Martinez conjecture [30], as analysed in Ref. [21]. 8

10 Recently, the gravitational energy of the Kerr black hole has been analysed in the context of quasi-local energy by means of the counterterm method [31]. In [31], the gravitational energy is computed in exact, closed analytic form only at the surface defined by r = r +. The result obtained in [31] is different from that given here by Eq. (3.6) (see Eq. (30) of the latter Ref.). In the case of slow angular momentum the gravitational energy E g (r + ) can be expanded in powes of a as ( a 2 )) E g (r + ) = 2M i (1 + O, Mi 4 where M i = 1 2 r+ 2 + a 2 = m 2p 2 is the irreducible mass of the Kerr black hole [32], which is the maximal amount of energy that can be extracted from the black hole via Penrose s process [33]. Another interesting case that can be obtained from Eq. (3.4) is the case of slow angular momentum for an arbitrary radius r. Expanding Eq. (3.4) and integrating in φ and θ we arrive at E g = r [1 + A2 3 (1 + M) )] 1 + A 2 2M (1 A2 6 (1 + 2M) + O(A 4 ), (3.8) where A = a/r and M = m/r. This result is strictly the same obtained by Dehghani and Mann in [31]. IV. Kerr-AdS space-time Since the Kerr-AdS space-time is a non-compact space, let us now briefly consider the Hamiltonian formulation of the TEGR for non-compact geometries. For space-times with different topologies the appropriate gravitational action integrals require a surface term 9

11 that is specific to each topology. Therefore the correspoding Hamiltonian also acquires a surface term that is determined by the topological boundary conditions [34]. However the Hamiltonian constraint for a space-time foliated by spacelike hypersufaces always has the same basic structure, irrespective of bondaury conditions. Additional terms such as the cosmological constant may appear in the Hamiltonian constraint, as we will see ahead. Some relevant and encouraging results obtained in the context of the TEGR for noncompact geometries are (i) the calculation of the energy per unit length of defects of topological nature [35], which agrees with the standard known result in the literature, and (ii) the result obtained in the analysis of the gravitational energy of the Schwarszchildde Sitter space-time developed in Ref. [36]. These results encourage us to apply the energy expression obtained in the context of Hamiltonian formulation of the TEGR to the Kerr-AdS space-time. Specifically, in this section we consider the expression (2.8), obtained from the Lagrangian density (2.1) for the case that the space-time has a negative cosmological constant Λ < 0. In order to do this we must add to (2.1) the term 2eΛ, where e is the determinant of the tetrad field e aµ. Because of this additional term the action integral has an extra term, as well as the Hamiltonian constraint. Therefore in the constraint given by Eq. (2.3) there appears an additional term given by 2e a0 eλ. The Kerr-AdS metric in Boyer-Lindquist coordinates is given by ds 2 = ( r dt a ) 2+ ρ 2 ρ 2 χ sin2 θdφ dr 2 + ρ2 dθ 2 r θ + θ sin 2 [ θ adt (r2 + a 2 ) 2 dφ], (4.1) ρ 2 χ 10

12 where r = (r 2 + a 2 )(1 + r 2 /l 2 ) 2mr, θ = 1 a 2 cos 2 θ/l 2, χ = 1 a 2 /l 2, ρ 2 = r 2 + a 2 cos 2 θ, l being the AdS radius, related to the cosmological constant by Λ = 3/l 2. The parameters m and a are related to the mass and angular momentum of the black hole, respectively. The metric above describes a rotating black hole solution of the Einstein s field equations with a negative cosmological constant Λ. This metric has two horizons located at r+, provided the parameter m satisfies the condition given by [31] m l ( ) ( ) a2 6 l + a4 2 l a a2 4 l 2 l + a4 2 l 1 + a2. 4 l 2 In the limit l, Eq. (4.1) reduces to the Kerr asymptotically flat solution. In the case that m = 0, the metric (4.1) is that of an empty AdS space-time, and for a = 0, it is the Schwarzschild-AdS space-time metric. The application of (2.8) to the Kerr-AdS metric requires the construction of a set of tetrad fields associated to (4.1). A set of tetrads that corresponds to (4.1) is given by where A B sin θ sin φ C sin θ cosφ Dr cosθ cosφ Er sin θ sin φ e aµ = B sin θ cos φ C sin θ sin φ Dr cosθ sin φ Er sin θ cos φ 0 C cosθ Dr sin θ 0 A = A + B2 C,, (4.2) 11

13 B = B, Csin θ C = ρ r, and D = ρ r θ, A = r ρ a2 θ sin 2 θ, 2 ρ 2 B = a ) sin2 θ ( r (r 2 + a 2 ) θ, C = sin2 θ ρ 2 χ 2 ρ 2 χ ( θ (r 2 + a 2 ) 2 r sin 2 θ ). When l the above tetrad field reduces to that given by (3.2). In order to calculate the energy we need to determine the expression of Π (0)1 corresponding to Eq. (4.2). After a long but straightforward calculation we arrive at [ ρ sin θ Π (0)1 = 2k θ + sin θ χρ ( ) 1/2 θ (r 2 + a 2 ) 2 r a 2 sin 2 θ ( ) r sin θ 1/2 ] χρ r θ (r 2 + a 2 ) 2 r a 2 sin 2 θ. (4.3) θ The gravitational energy contained within a surface of constant radius r is then given by E g = 1 2π π [ ρ dφ dθ sin θ + 1 ( ) 1/2 α l + β l cos 2 θ 8π 0 0 θ χρ where r ( ) 1/2 ] χρ r α l + β l cos 2 θ, (4.4) θ α l = (r 2 + a 2 ) 2 r a 2, β l = r a 2 (r2 + a 2 ) 2 a 2 l 2. 12

14 As in the Kerr case, we perform the integration in terms of elliptic functions. By means of the change of variable y = cosθ, the integration is performed in exact, closed form for arbitrary values of r and of the metric parameters m and a. The final expression for E g is E g = 1 4 [ 2rl ( a a E l, i l ) + i 2 ( α l r χa E i a ) r, βl r α l a where r l r α l i aχ α l (r 2 + l 2 ) F r l 3 r β l + i χa α 3 l (r 2 + a 2 ) ( r2 + a i 2 a l 2 a 2 r, a2 + l 2 β l /α l r 2 + l 2 [ ( r2 + l F i 2 a l 2 + a 2 r, a2 + l 2 β l /α l r 2 + a 2 ( r2 + l + Π i 2 a l 2 + a 2 r, r 2 a2 + l 2 β l /α l r 2 + l 2, r 2 + l 2 Π(x, y, k) = x is the defination of the elliptic function Π(x, y, k). 0 1 dz (1 yz 2 ) 1 z 2 1 k 2 z 2 ) r a ) r a )]] r, (4.5) a In the absence of angular momentum (a = 0) it is easy to check that (4.5) reduces to ( E g = r 1 1 2m r + r2 l 2 ), (4.6) which is the energy within an arbitrary surface of fixed radius r in the Schwarschild-AdS space-time. The result (4.6) has been obtained by Brown et al in the context of the quasilocal energy, when the background subtraction method is used [23], where the reference term is taken as ǫ 0 (R) = 1/4πR ( in the notation of Ref. [23]). It has also been obtained in Ref. [36], in the context of TEGR when Schwinger s time gauge is imposed a priori. 13

15 This is an expected result, since the tetrad field (4.2) satisfies the time gauge condition. In the limit r, we find that (4.5) gives E g. This result is expected, since the anti-de Sitter space is a non-compact manifold with constant negative curvature. For the especial case in which E g is evaluated for the surface defined by r = r +, i.e, for the external horizon of the Kerr-AdS black hole, we find that (4.5) reduces to E g = 1 4 [ ( 2lr+ a a E l, i l ) r + + i 2(r2 + + ( a2 ) E i a, i r )] +. (4.7) χa r + l This result differs from that obtained in Ref. [31] in the context of the counterterms method. In the latter reference the gravitational energy inside a volume defined by the outer horizon is divergent in the limit l (see Eq. (49) in [31]). Differently of [31], in this limit, expressions (4.5) and (4.7) reduces to those of the Kerr case, given by (3.5) and (3.6), respectively. This is a prominent result of the energy expression (2.8), and is consistent with the fact that the metric given by Eq. (4.1) reduces to that given by Eq. (3.1) in the limit l. This suggests that the flat space limit of AdS space can be defined as the limit l. Finally, let us now consider the particular case of small angular momentum in (4.4). For a fixed radius r, the gravitational energy E g can be expanded in powers of a as E g = r [1 + (M A ) 3 L 2 (1 + A 2 ) (1 + 1 ) 2M (1 (1 A2 + 2M 4 )) L 2 6 L 2 ] + O(A 4 ), (4.8) where A = a/r, M = m/r and L = l/r. We note that this result is very similar to the corresponding one obtained in [31]. In the limit l the above expression reduces to that of the Kerr case, given by (3.8). 14

16 V. Concluding remarks By applying the definition of gravitational energy that arises in the Hamiltonian formulation of the TEGR developed in [20], we computed the gravitational energy in the space-times of Kerr and Kerr-AdS black holes. In each case the gravitational energy has been evaluated exactly, in analytic closed form for an arbitrary volume of space without any restriction on the metric parameters. This is the major result of this work and constitutes an advantage of the TEGR approach for the description of the gravitational field energy as compared to those based on the Hilbert-Einstein action integral, where the energy has been calculated exactly only in particular cases of the Kerr and Kerr-AdS space-times [31]. Our results for the gravitational energy of the Kerr-AdS black hole reduce naturally to those of Kerr in the limit l (vanishing cosmological constant). This is a self-consistent result of our energy expression and indicates that it is possible to define the flat space limit of the AdS space as l. Finally, in view of the above results we expect that the expression (2.8) for the gravitational field energy to be useful in the study of the thermodynamics of self-graviting systems. We hope to consider this issue in the future. Acknowledgements This work was supported by FAPESP-Brazil. The authors are grateful to J. W. Maluf for the critical reading of the paper and valuable comments. J. F. R-N would like to thank J. G. Pereira for the hospitality at the Instituto de Física Teórica-IFT/UNESP-Brazil. References [1] C. W. Misner, K. S. Thorne, and J. A. Wheller, Gravitation (Freeman, San Francisco, 1973), sec

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