Hamiltonian Formulation of the Teleparallel Equivalent of General Relativity Without Gauge Fixing

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HAMILTONIAN FORMULATION OF THE TELEPARALLEL EQUIVALENT OF GENERAL RELATIVITY WITHOUT GAUGE FIXING

J. W. Maluf ∗ and J. F. da Rocha-Neto Instituto de Física Universidade de Brasília C.P. 04385 70.919-970 Brasília, DF Brazil

Abstract

We establish the Hamiltonian formulation of the teleparallel equivalent of general relativity, without ﬁxing the time gauge condition, by rigorously performing the Legendre transform. The time gauge condition restricts the teleparallel geometry to the three-dimenional spacelike hypersurface. Geometrically, the teleparallel geometry is now ex- tended to the four-dimensional space-time. The resulting Hamiltonian formulation is diﬀerent from the standard ADM formulation in many aspects, the main one being that the dynamics is now governed by the Hamiltonian constraint H0 and a set of primary constraints. The vector constraint Hi is derived from the Hamiltonian constraint. The vanishing of the latter implies the vanishing of the vector constraint.

PACS numbers: 04.20.Cv, 04.20.Fy, 04.90.+e (*) e-mail: wadih@ﬁs.unb.br I. Introduction

Hamiltonian formulations, when consistently established, not only guarantee that field quantities have a well defined time evolution, but also allow us to understand physical theories from a different perspective. We have learned from the work of Arnowitt, Deser and Misner (ADM)[1] that the Hamilto- nian analysis of Einstein’s general relativity reveals the intrinsic structure of the theory: the time evolution of field quantities is determined by the Hamiltonian and vector constraints. Thus four of the ten Einstein’s equations acquire a prominent status in the Hamiltonian framework. Ultimately this is an essential feature for the canonical approach to the quantum theory of gravity. It is the case in general relativity that two distinct Lagrangian formulations that yield Einstein’s equations lead to completely different Hamiltonian constructions. An important example in this respect is the reformulation of the ordinary variational principle, based on the Hilbert-Einstein action, in terms of self-dual connections that define Ashtekar variables[2]. Under a Palatini type variation of the action integral constructed out of these field quantities one obtains precisely Einstein’s equations. The interesting features of this approach reside in the Hamiltonian domain. Einstein’s general relativity can also be reformulated in the context of the teleparallel geometry. As we will discuss ahead, novel features also arise in the Hamiltonian formulation. In this geometrical setting the dynamical field a quantities correspond to orthornormal tetrad fields e µ (a, µ are SO(3,1) and space-time indices, respectively). These fields allow the construction of the Lagrangian density of the teleparallel equivalent of general relativity (TEGR) [3, 4, 5, 6, 7, 8, 9, 10, 11], which offers an alternative geometrical framework for Einstein’s equations. The Lagrangian density for the tetrad field in the a TEGR is given by a sum of quadratic terms in the torsion tensor T µν = a a ∂µe ν − ∂ν e µ, which is related to the anti-symmetric part of the connection λ aλ Γµν = e ∂µeaν . The curvature tensor constructed out of the latter vanishes identically. This connection defines a space with teleparallelism, or absolute parallelism[12]. In a space-time with an underlying tetrad field two vectors at distant points may be called parallel[3] if they have identical components with respect to the local tetrads at the points considered. Thus consider a vector field V µ(x). At the point xλ its tetrad components are given by V a(x)=

1 a µ a e µ(x)V (x). For the tetrad components V (x + dx)itiseasytoshowthat a a a a a µ λ V (x + dx)=V (x)+DV (x), where DV (x)=e µ(∇λV )dx .Theco- λ aλ variant derivative ∇ is constructed out of the connection Γµν = e ∂µeaν . Therefore such connection defines a condition for absolute parallelism in space-time. Hence tetrad fields are required to transform under the global SO(3,1) group. In order to understand the equivalence of the TEGR with Einstein’s general relativity let us introduce an arbitrary connection ωµab.Thelat- 0 0 ter can be identically written as ωµab = ωµab(e)+Kµab,where ωµab(e) is the Levi-Civita connection and Kµab is the contorsion tensor defined by 1 λ ν − Kµab = 2 ea eb (Tλµν + Tνλµ Tµνλ), where now the torsion tensor is given a a a a b a b by T µν = ∂µe ν − ∂ν e µ + ωµ b e ν − ων b e µ. This connection leads to a the curvature tensor R bµν (ω). Substituting ωµab into the scalar curvature aµ bν R(e, ω)=e e Rabµν (ω) we obtain the identity

1 abc 1 abc a µ eR(e, ω)=eR(e)+e( T Tabc + T Tbac − T Ta) − 2∂µ(eT ) . (1) 4 2 a Thus if we require the vanishing of the curvature tensor R bµν (ω), the scalar curvature R(e) becomes equivalent to the combination of torsion squared terms that defines the Lagrangian density of the TEGR, as we will see. A a consistent implementation of the vanishing of R bµν (ω) has been carried out in the Hamiltonian analysis of the TEGR in ref. [11]. It is shown in the latter that a well established Hamiltonian formulation can only be achieved if the local SO(3,1) symmetry in identity (1) is turned into the global SO(3,1) a symmetry. Condition R bµν (ω) = 0 has the ultimate effect of discarding the connection. Therefore we can dispense with the connection ωµab and consider a teleparallel theory with tetrad fields that transform under the global SO(3,1) group, exactly along the lines of refs. [3, 10]. In the framework of the TEGR it is possible to make definite statements about the energy and momentum of the gravitational field. This fact constitutes the major motivation for considering such theory. In the 3+1 formulation of the TEGR, and by imposing Schwinger’s time gauge condition[14], we find that the Hamiltonian and vector constraints contain each one a di- vergence in the form of scalar and vector densities, respectively, that can be identified with the energy and momentum densities of the gravitational

2 field[13]. This identification has proven to be consistent, and has shown that the TEGR provides a natural setting for investigations of the gravitational energy. Several relevant applications have been presented in the literature. Among the latter we point out investigations on the gravitational energy of rotating black holes[15] (the evaluation of the irreducible mass of a Kerr black hole) and of Bondi’s radiating metric[16]. In this paper we carry out the Hamiltonian formulation of the TEGR without imposing the time gauge condition. We have rigorously performed the Legendre transform, and not found it necessary to establish a 3+1 decomposition for the tetrad field. We only assume g00 =6 0, a condition that ensures that t = constant hypersurfaces are spacelike. The Lagrange multipliers are given by the zero components of the tetrads, ea0. The constraints corresponding to the Hamiltonian (H0) and vector (Hi) constraints are obtained in the form Ca = 0. The dynamical evolution of the field quantities ik is completely determined by H0 and by a set of primary constraints Γ and k Γ , as we will show. The surprising feature is that if H0 = 0 in the subspace of the phase space determined by Γik =Γk = 0, then it follows automaticaly that Hi = 0. As we will see, Hi can be obtained from the very definition of H0. As a consequence of this analysis, we arrived at a scalar density that trans- forms as a four-vector in the SO(3,1) space, again arising in the expression of the constraints of the theory, and whose zero component is closely related to the energy of the gravitational field. The analysis developed here is similar to that developed in ref. [17], in which the Hamiltonian formulation of the TEGR in null surfaces was established. The 3+1 formulation of the TEGR has already been considered in ref. [9]. There are several differences between the latter and the present analysis. The investigation in [9] has not pointed out neither the emergence of the scalar densities mentioned above nor the relationship between H0 and Hi. Our approach is different and allowed us to proceed further into the constraint structure of the theory.

Notation: spacetime 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 = a (0), (i). The tetrad field e µ yields the definition of the torsion tensor: a a a T µν = ∂µe ν − ∂νe µ. The flat, Minkowski spacetime metric is fixed by µν ηab = eaµebν g =(− + ++).

3 II. Lagrangian formulation

The Lagrangian formulation of the TEGR in empty space-time is for- a mulated in terms of the tetrad ﬁeld e µ only, and displays invariance under coordinate and global SO(3,1) transformations. The Lagrangian density is given by

abc L(e)=−keΣ Tabc , (2) 1 a µ ν where k = 16πG, G is Newton’s constant, e = det(e µ), Tabc = eb ec Taµν and 1 1 Σabc = (T abc + T bac − T cab)+ (ηacT b − ηabT c) . (3) 4 2 Tetrads transform space-time into SO(3,1) indices and vice-versa. The trace of the torsion tensor is given by

a Tb = T ab . The tensor Σabc is deﬁned such that

abc 1 abc 1 abc a Σ Tabc = T Tabc + T Tbac − T Ta . 4 2 The ﬁeld equations obtained from (2) read

δL bλν bν 1 bcd = eaλebµ∂ν(eΣ ) − e Σ aTbνµ − eaµTbcdΣ =0, (4) δeaµ 4 It can be shown by explicit calculations[11] that these equations yield Ein- stein’s equations: δL 1 1 ≡ e Raµ(e) − eaµR(e) . δeaµ 2 2 In order to carry out the 3+1 decomposition we need a first order differential formulation of (2). For this purpose we introduce an auxiliary field quantity φabc = −φacb that ultimately will be related to the torsion tensor. a The first order differential formulation of (2) is now given in terms of e µ and φabc. It is described by the Lagrangian density

4 abc L(e, φ)=keΛ (φabc − 2Tabc) , (5) where Λabc is deﬁned in terms of φabc exactly as Σabc in terms of T abc: 1 1 Λabc = (φabc + φbac − φcab)+ (ηacφb − ηabφc) . (6) 4 2 Variation of the action constructed out of (5) with respect to φabc yields

Λabc =Σabc , (7) which, after some manipulations, can be reduced to

φabc = Tabc . (8) The equation above may be split into two equations:

φa0k = Ta0k = ∂0eak − ∂kea0 , (9a)

φaik = Taik = ∂ieak − ∂keai . (9b) In view of eq. (8), it can be shown that the second ﬁeld equation, the variation of the action integral with respect to eaµ, leads precisely to (4). Therefore (2) and (5) describe the same physical system.

III. Legendre transform and the 3+1 decomposition

The Hamiltonian density will be obtained by the standard prescription L = pq˙ − H0 and by properly identifying primary constraints. We have not found it necessary to establish any kind of 3+1 decomposition for the tetrad ﬁelds. Therefore in the following both eaµ and gµν are space-time ﬁeld quantities. We will follow here the procedure presented in [17]. Lagrangian density (5) can be expressed as

a0k a0k aij abc L(e, φ)=−4ke Λ e˙ak +4ke Λ ∂kea0 − 2ke Λ Taij + keΛ φabc , (10) where the dot indicates time derivative, and

5 a0k abc 0 k Λ =Λ eb ec ,

aij abc i j Λ =Λ eb ec .

Therefore the momentum canonically conjugated to eak is given by

Πak = −4keΛa0k , (11) In terms of (11) expression (10) reads

ak ak aij abc L =Π e˙ak − Π ∂kea0 − 2ke Λ Taij + ke Λ φabc . (12) By working out the last two terms of expression above we can further rewrite the Lagrangian density (12) according to

ak ak ak L =Π e˙ak + ea0∂kΠ − ∂k(ea0Π )

1 im nj a 1 nj i m ik j n −ke g g φ mnφaij + g φ mnφ ij − g φ jiφ nk 4 2

1 ak 0i kj a ai 0j k kj 0 ak 0i a0 ki j − φa0k Π +ke g g φ ij −e (g φ ij −g φ ij)+2(e g −e g )φ ji . 2 (13) The Hamiltonian formulation is established once we rewrite the Lagrangian ak density (13) in terms of eak,Π and further nondynamical field quantities. It is carried out in two steps. First, we take into account equation (9b) in (13) so that half of the auxiliary fields, φaij , are eliminated from the La- grangian by means of the identification φaij = Taij . Second, we must express the remaining field quantities, the “velocities” φa0k, in terms of the momenta Πak. This is the nontrivial step of the Legendre transform. We need to consider the full expression of Πak.Itisgivenby ak 00 kj a aj k ak j Π = ke g (−g φ 0j − e φ 0j +2e φ 0j)

0k 0j a aj 0 a0 0j k kj 0 a0 0k j ak 0j 0 +g (g φ 0j + e φ 0j)+e (g φ 0j + g φ 0j) − 2(e g φ 0j + e g φ 0j)

6 0i kj a ai 0j k kj 0 0i ak ik a0 j −g g φ ij + e (g φ ij − g φ ij) − 2(g e − g e )φ ji . (14)

Denoting (..)and[..] as the symmetric and anti-symmetric parts of ﬁeld quantities, respectively, we can decompose Πak into irreducible components:

ak a (ik) a [ik] a 0k Π = e i Π + e i Π + e 0 Π , (15) where

(ik) 00 kj i ij k ik j 0k 0j i ij 0 0i j Π = ke g (−g φ 0j−g φ 0j+2g φ 0j)+g (g φ 0j+g φ 0j−g φ 0j)

0i 0j k kj 0 0k j ik 0j 0 ik +g (g φ 0j + g φ 0j − g φ 0j) − 2g g φ 0j +∆ , (16a)

ik 0m kj i ij k ik j km 0i im 0k j ∆ = −g (g φ mj +g φ mj −2g φ mj )−(g g +g g )φ mj , (16b)

[ik] im kj 0 im 0k km 0i j Π = ke −g g φ mj +(g g − g g )φ mj , (17)

0k kj 0i 0 0k 0i j 00 ik j Π = −2ke(g g φ ij − g g φ ij + g g φ ij) . (18) The crucial point in this analysis is that only the symmetrical components (ij) [ij] Π depend on the “velocities” φa0k. The other six components, Π and 0k Π depend on φaij . Therefore we can express only six of the “velocity” ﬁelds (ij) φa0k in terms of the momenta Π . With the purpose of ﬁnding out which components of φa0k can be inverted in terms of the momenta we decompose φa0k identically as

a ai ai a0 φ 0j = e ψij + e σij + e λj , (19) with the following deﬁnitions: 1 ψij = ψji = (φi0j + φj0i) , 2

7 1 σij = −σji = (φi0j − φj0i) , 2

λj = φ00j , a where φµ0j = e µφa0j. Next we substitute (19) in (16a). By deﬁning 1 P ik = Π(ik) − ∆ik , (20) ke ik we ﬁnd that P depends only on ψij:

ik 00 im kj ik P = −2g (g g ψmj − g ψ)

0i km 0j 0k im 0j ik 0m 0j 0i 0k +2(g g g + g g g )ψmj − 2(g g g ψmj + g g ψ) , (21)

mn where ψ = g ψmn. We recall that we are not assuming any type of 3+1 decomposition for gµν and eaµ. ik We can now invert ψmj in terms of P . After a number of manipulations we arrive at 1 ik 1 ψmj = − gimgkjP − gmj P , (22) 2g00 2 ik where P = gikP . We need now to work out the last line of Lagrangian density (13). By making use of deﬁnition (14) for the momenta Πak we can rewrite

1 ak 0i kj a ai 0j k kj 0 ak 0i a0 ki j − φa0k Π +ke g g φ ij −e (g φ ij −g φ ij)+2(e g −e g )φ ji 2 in the form

1 00 kj a aj k ak j 0k 0j a aj 0 − φa0k ke g (−g φ 0j − e φ 0j +2e φ 0j)+g (g φ 0j + e φ 0j) 2 a0 0j k kj 0 a0 0k j ak 0j 0 +e (g φ 0j + g φ 0j) − 2(e g φ 0j + e g φ 0j)

8 Substitution of (19) in the expression above yields 00 kj mi mk ij 0k 0j mi 0i 0j mk ke g (g g − g g ) − 2g g g +2g g g ψmkψij .

Making use of (22) we arrive ultimately at 1 ij kl 1 2 ke gikgjlP P − P . 4g00 2

Therefore we can finally obtain the primary Hamiltonian H0: ak 1 ij kl 1 2 H0 = −ea0∂kΠ − ke gikgjlP P − P 4g00 2 1 im nj a 1 nj i m ik j n +ke g g T mnTaij + g T mnT ij − g T jiT nk . (23) 4 2 In the expression above we have already taken into account (9b) and identified φaij = Taij . We may now write the total Hamiltonian density. For this purpose we have to identify the primary constraints. They are given by expressions (17) ak and (18), which represent relations between eak and the momenta Π .Thus we define

ik ki [ik] im kj 0 im 0k km 0i j Γ = −Γ =Π − ke −g g φ mj +(g g − g g )φ mj , (24)

k 0k kj 0i 0 0k 0i j 00 ik j Γ =Π +2ke(g g φ ij − g g φ ij + g g φ ij) . (25) Therefore the total Hamiltonian density is given by

ik k ak H = H0 + αikΓ + βkΓ + ∂k(ea0Π ) , (26) where αik and βk are Lagrange multipliers.

IV. Secondary constraints

Since the momenta {Πa0} vanish identically they also constitute primary constraints that induce the secondary constraints

9 δH Ca ≡ =0. (27) δea0 a In order to obtain the expression of C we have only to vary H0 with re- ik k spect to ea0, because variations of Γ and Γ with respect to ea0 yield the constraints themselves:

δΓik 1 = − (eaiΓk − eakΓi) , (28a) δea0 2 δΓk = −ea0Γk . (28b) δea0 In (28a,b) we have made use of variations like

δebµ = −eaµeb0 . δea0 In the process of obtaining Ca we need the variation of P ij with respect to ea0.Itreads

δP ij = −ea0P ij + γaij , δea0 with γaij deﬁned by

aij 1 ai j aj i ak 00 jm i im j ij m γ = − (e Γ + e Γ ) − e g (g φ km + g φ km +2g φ mk) 2ke

0m 0j i 0i j 0i 0j m jm 0i im 0j ij 0m 0 +g (g φ mk + g φ mk) −2g g φ mk +(g g + g g −2g g )φ mk , (29) aij aij Note that γ satisﬁes ea0γ =0. After a long calculation we arrive at the expression of Ca: a ak a0 1 ij kl 1 2 C = −∂kΠ + e − ke gikgjlP P − P 4g00 2 1 im nj b 1 nj i m ik m n +ke g g T mnTbij + g T mnT ij − g T miT nk 4 2

10 1 aij kl 1 aij ai 0m nj b − ke gikgjlγ P − gijγ P − ke e g g T ijTbmn 2g00 2

nj 0 m 0j n m 0k m n jk 0 n +g T mnT ij + g T mj T ni − 2g T mkT ni − 2g T ijT nk . (30)

Inspite of the fact that expression above is somehow intricate, we imme- diately notice that

a ea0C = H0 . (31) Therefore the total Hamiltonian becomes

a ik k ak H = ea0C + αikΓ + βkΓ + ∂k(ea0Π ) . (32)

We observe that {ea0} arise as Lagrange multipliers. We note in addition that Ca satisfy δCa ea0 =0, δeb0 a relation that follows from (31).

V. Simpliﬁcation of the constraints and Poisson brackets

The first two terms of the expression of Ca yield the primary Hamiltonian a0 in the form e H0. This fact can be easily verified by expressing the first term of (30) as

ak a0 bk aj bk −∂kΠ = e (−eb0∂kΠ )+e (−ebj∂kΠ ) . If we substitute deﬁnitions (16b) and (29) for ∆ij and γaij , respectively, into (30) we obtain after a long calculation a simpliﬁed form for Ca:

a a0 ai C = e H0 + e Fi , (33) with the following deﬁnitions:

11 m lm 1 kl 1 j Fi = Hi +Γ T0mi +Γ Tlmi + (gikgjlP − gijP )Γ , (34) 2g00 2

bk bk Hi = −ebi∂kΠ − Π Tbki . (35)

We denote H0 as the Hamiltonian constraint. As we will discuss later, Hi is the vector constraint. It amounts to a SO(3,1) version of the vector constraint of ref. [11]. In view of (31) and (32), the time evolution of ﬁeld quantities is given by the simpler Hamiltonian density (26). Dispensing with the surface term, it reads

ik k H = H0 + αikΓ + βkΓ . (36) The Poisson bracket between two quantites F and G is deﬁned by Z { } 3 δF δG − δF δG F, G = d x ai ai , δeai(x) δΠ (x) δΠ (x) δeai(x) by means of which we can write down the evolution equations. The ﬁrst set of Hamilton’s equations is given by

Z 3 δ ik k e˙aj (x)={eaj (x),H} = d y H0(y)+αik(y)Γ (y)+βk(y)Γ (y) , δΠaj(x) (37) This equation can be worked out to yield

1 k im 1 i 0 Ta0j = − ea (gikgjmP − gkjP )+ea αij + ea βj , (38) 2g00 2 from which we obtain

1 1 km 1 (Ti0j + Tj0i)=ψij = − (gikgmjP − gijP ) , (39a) 2 2g00 2 1 (Ti0j − Tj0i)=σij = αij , (39b) 2

T00j = λj = βj , (39c)

12 according to the definitions in equation (19). Thus the Lagrange multipliers in (36) acquire a well defined meaning. Expression (39a) is in total agreement with (22). Consequently we can obtain an expression for Π(ij) in terms of velocities via equations (20) and (21). The dynamical evolution of the field quantities is completed with Hamilton’s equations for Π(ij),

Z (ij) (ij) ˙ (ij) { (ij) } 3 δΠ (x) δH − δΠ (x) δH Π (x)= Π (x),H = d y ak ak , δeak(y) δΠ (y) δΠ (y) δeak(y) (40) together with

Γik =0, (41a)

Γk =0. (41b) The Hamiltonian density (36) determines the time evolution of any ﬁeld quantity f(x): Z

3 f˙(x)= d y{f(x),H(y)} . (42) Γik=Γk=0

Physical quantities take values in the subspace of the phase space PΓ deﬁned by (41). In this subspace constraints Ca become

a a0 ai C = e H0 + e Hi . (43) a Restricting considerations to PΓ we note that if H0 vanishes, then ea0C a also vanishes. Since {ea0} are arbitrary, it follows that C = 0 (from (43) we δCa can easily obtain = 0). Consequently we must have Hi = 0. Therefore δeb0 the vanishing of the Hamiltonian constraint H0 implies the vanishing of the vector constraint Hi. Moreover we observe from (43) that Hi can be obtained from H0 according to δ eai H0 = Hi . (44) δea0

Thus Hi is derived from H0. In PΓ we ﬁnd that H0 is a conserved quantity. After a long calculation we obtain

13 {H0(x),H0(y)} =0. (45) We have not veriﬁed the consistency of conditions (41) by calculating the Poisson brackets {Γik(x),H(y)}, {Γk(x),H(y)},whereH(x) is the total Hamiltonian given by (36). However (39a) and (40) are expected to determine the time evolution of eaj in agreement with Einstein’s equations, which have a well deﬁned initial value problem. Thus we foresee no problem re- garding the consistency of (41). The evaluation of the complete constraint ik k algebra between H0,Γ and Γ is by no means trivial and will be presented elsewhere. Here we just provide the simple calculation of the Poisson bracket {Hj(x),Hk(y)}.Itisgivenby

∂ ∂ {Hj(x),Hk(y)} = −Hk(x) δ(x − y)+Hj(y) δ(x − y) , ∂xj ∂yk which is the standard relation for the vector constraint in the Hamiltonian formulation of gravity theories.

VI. Comments

We have seen that the vector constraint Hi can be obtained from the Hamiltonian constraint H0. Although not completely clear to us, this feature may be related to the nature of the teleparallel geometry, according to which eaµ constitutes a set of autoparallel vector fields. In contrast, in the ADM formulation the Hamiltonian and vector constraints are not mutually related, and in practice one has to consider both constraints for the dynamical evolution via Hamilton equations. In the present analysis the Hamiltonian constraint yields the four constraints Ca by means of relation (31). Therefore H0 and the primary constraints (24) and (25) determine the time evolution of field quantities via equation (42), and in particular of the metric tensor gij of three-dimensional spacelike hypersurfaces. This property might simplify approaches to a canonical, nonperturbative quantization of gravity provided we manage to construct the reduced phase space determined by (41). After implementing the primary constraints via equations (36), the first a ai term of C is given by −∂iΠ . From our previous experience (cf. ref. [13]) we are led to conclude that this term is intimately related to energy and

14 momentum of the gravitational field. In similarity with the previous analysis of ref. [16], and following Møller[3], in the case of asymptotically flat spacetimes we adopt the boundary conditions 1 1 eaµ ≈ ηaµ + haµ( ) , (46) 2 r in the limit r →∞. In the expression above ηaµ is the Minkowski metric tensor and haµ = hµa is the first term of the asymptotic expansion of the metric tensor gµν .Sincehaµ is asymptotically a symmetric tensor, these boundary conditions impose 6 conditions on the tetrads. These conditions are not fixed in the body of the theory because the SO(3,1) is a global (rather than local) symmetry group. Boundary conditions (46) uniquely associate a tetrad field to a given metric tensor for asymptotically flat space-times. We recall that similar conditions for triads restricted to the three-dimensional spacelike hypersurface were essential in order to arrive at the ADM energy[13]. In the present approach we obtain likewise the ADM energy. Asymptoti- 1 cally flat spacetimes are defined by (46) together with ∂µgλν = O( r2 ), or 1 ∂µeaν = O( r2 ). Thus considering the a = (0) component in (14) and inte- (0)k grating −∂kΠ over the whole three-dimensional spacelike hypersurface we a find that all terms like T 0j cancel out, and eventually only the last term in (14) contributes to the integration. Hence we obtain Z Z 3 (0)k 3 ik (0)0 j E ≡− d x∂kΠ = −2k d x∂k(eg e T ji) V →∞ V →∞ Z 1 = dSk(∂ihik − ∂khii)=EADM . 16πG S→∞ The energy expression above can be applied to finite volumes of space in order to obtain, for instance, the irreducible mass of rotating black holes. Certainly such analysis has to be compared with the result of ref. [15], which was obtained under the imposition of the time gauge condition.

Acknowledgements J. F. R. N. is supported by CAPES, Brazil.

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