Teleparallel equivalent of general relativity and local Lorentz transformation: Revisited
aa r X i v : . [ phy s i c s . g e n - ph ] A ug Teleparallel equivalent of general relativity and local Lorentz transformation:Revisited ∗ Gamal G.L. Nashed , , and B. Elkhatib Centre for theoretical physics, the British University in Egypt, 11837 - P.O. Box 43,Egypt. Mathematics Department, Faculty of Science, Ain Shams University, Cairo, Egypt. Egyptian Relativity Group (ERG).e-mail:[email protected] is well known that the field equations of teleparallel theory which is equivalent togeneral relativity (TEGR) completely agree with the field equation of general relativity(GR). However, TEGR has six extra degrees of freedom which spoil the true physics. Theseextra degrees are related to the local Lorentz transformation. In this study, we give threedifferent tetrads of flat horizon space-time that depend only on the radial coordinate. One ofthese tetrads contains an arbitrary function which comes from local Lorentz transformation.We show by explicate calculations that this arbitrary function spoils the calculations ofthe conserved charges. We formulate a skew-symmetric tensor whose vanishing value put aconstraint on the arbitrary function. This constraint makes the conserved charges are freefrom the arbitrary function.
In the theory of GR, the gravitational field is gained by the curvature of space-times. Par-ticles are obliged by the curvature of the space-times to move on geodesics. Therefore, thetheory of GR is indeed geometrized by the gravitational field. The theory of GR has someconstraints on the classical level. The forecasts of GR are in agreement with the experimen-tal data accessible till now. Unification of the main four forces has continued as a favorite ∗ PACES numbers: 04.50. Kd, 04.70.Bw, 04.20. JbKeywords: Teleparallel equivalent of general relativity, local Lorentz transformation, total conservedcharge T . The dynamics of this Lagrangian are equal to GR and this follows from the result R = − T − B, with R being Ricci scalar and B is a boundary term linked to the divergence of the torsion.Due to the fact that B is a total derivative, it does not contribute to the equations of motionand therefore, the Lagrangian of TEGR is equal to that of GR.The aim of this study is to discuss the effect of the extra degrees of TEGR on the truephysics and how one can fix these degrees so that they do not contribute to the true physics.In §
2, an introduction to TEGR theory is presented. In §
3, several tetrad fields having flathorizons are given and application to the field equations of TEGR is explained. New analytic,solutions are derived in §
3. In §
4, calculation of the conserved quantities of each solutionhave been carried out and we have shown how the unphysical extra degrees contribute tothe true physics. In §
5, we gave a skew-symmetric tensor which constrains the extra degreeand shows the effect of this tensor on the acceleration components of an observer at infinity.Main results are discussed in final section. 2
Introduction to TEGR theory
Teleparallel theory equivalent to general relativity considers as another construction of GRof Einstein. The main entity of TEGR theory is the tetrad fields † ( e iα ). In TEGR theory,the metric can be built using the tetrad: g αβ = λ ij e iα e j β , with λ ij = diag(1 , − , − , − αβγ canbe constructed. Nevertheless, it is likely to build Weitzenb¨ock non-symmetric connectionΓ αβγ = e iα ∂ γ e iβ [1]. The Weitzeinb¨ock spacetime is labled as a pair ( M, e µ ), whereas M is a D -dimensional manifold and e β ( β = 0 , · · · ,
3) are D -linear independent vector definedglobally on M . The covariant derivative of the tetrad using Weitzenb¨ock connection isvanishing, i.e. ∇ α e iβ ≡
0. Therefore, the vanishing of the covariant derivative of the tetradidentifies the auto parallelism or absolute parallelism restriction. Actually, the operator ∇ α has a big issue, that is not invariant under local Lorentz transformations (LLT). The issuepermits all LLT invariant quantities to rotate freely at each point of the space [4]. Thus,the symmetric metric is not able to guess one form of tetrad field; therefore, the additionaldegrees of freedom have to be restricted such that one physical frame can be identified. TheWeitzenb¨ock connection is has a vanishing curvature however it has a torsion defined byT αβγ := Γ αβγ − Γ αγβ . (1)The contortion is defined asK αβγ := − (cid:0) T αβ γ − T βαγ − T γαβ (cid:1) . (2)In the TEGR one can construct three Weitzenb¨ock invariants: I = T αβγ T αβγ , I = T αβγ T βαγ and I = T α T α , where T α = T βαβ . We next define the invariant T = A I + A I + A I ,where A k , k = 1 , , = 1 /
4, A = 1 / = − (˚Γ) , up to divergence term. In this sense, theteleparallel gravity will be identical with GR. The torsion scalar of TEGR is defined asT := T αβγ S βγα , (3)with S αβγ being the superpotential tensor defined asS αβγ := 12 (cid:0) K βγα + δ βα T ργ ρ − δ γα T ρβ ρ (cid:1) . (4)The tensor S αβγ is skew symmetric tensor in the last two indices.The identity between torsion scalar and Ricci one is given byR (˚Γ) = − T (Γ) − ∇ α T βαβ , (5) † The Greek symbols indicate the elements of tangent space and Latin components indicate the symbolsof the spacetime. T instead of the Ricci scalar. In this sense, the torsionand Ricci scalars are identical. Despite the quantitative equivalence, they, T an R , arequalitatively not equivalent. For instant, Ricci scalar tensor is invariant under LLT whilstthe divergence term ∇ α T βαβ is not invariant and therefore the torsion scalar. Therefore, thetheory of TEGR action is not form invariant with respect to LLT [5, 6, 7].The action of the gauge gravitational field Lagrangian is given by [8, 9]S = M Pl Z | e | [(T − M (Φ A )] d x, (6)with L M being the Lagrangian of matter fields Φ A and M Pl being the mass of Planck, thatis connected to gravitational constant G through M Pl = p ~ c/ πG . In this study we use theunits G = c = ~ = 1 and | e | = √− g = det ( e iα ). Making variation of Eq. (6) regard to thetetrad fields e iα give the following field equations [8] ∂ β ( e S iαβ ) = 4 π ee iβ (t βα + Θ βα ) , (7)with S iαβ = e iρ S ραβ , being the (pseudo) tensor t αβ andt αβ = 116 π [4 | rmT ργα S ρβγ − δ βα (T − , (8)is the energy-momentum tensor Θ αβ = e iα (cid:18) − e δ L M δ e iβ (cid:19) . (9)As the tensor S iαβ is anti-symmetric, i.e S iαβ = − S iβα , this leads to ∂ α ∂ β (eS iαβ ) ≡ ∂ β (cid:2) ee iρ (t ρβ + Θ αβ ) (cid:3) = 0 . The pseudo-tensor t βα is disappeared in the theory of GR. To prob its behavior we see thatthe previous equation leads us to the conservationddt Z V ee iα (t α + Θ α ) d x = − I Σ [ee iα (t jα + Θ jα )] d Σ j . The integration of the previous equation is carried out on 3-dimensional volume limited bythe surface. Therefore, t αβ represent the energy-momentum tensor of the gravitational field[8]. 4 Flat transverse solutions
We apply the TEGR field equations (7) to the first flat transverse section which gives thefollowing tetrad written in cylindrical coordinate ( t , r , φ , z ) as: (cid:0) e iµ (cid:1) = a(r) 0 0 00 b(r) 0 00 0 r
00 0 0 r , (10)Substituting from (10) into (3) we calculate torsion scalar as T = − a ′ r + a ) r ae , (11)where a ′ := da ( r ) dr . Using (10) in (7) we getΘ = Λ r e + 2 rb ′ − b r e , Θ = Λ r e a − ra ′ − a r e a ,T heta = Θ = Λ re a − rba ′′ + ra ′ b ′ + ab ′ − ba ′ re a . (12)The solution of the above differential equation has the form a ( r ) = c √ Λ r + 3 c √ r , b ( r ) = ∓ √ r √ Λ r + 3 c , (13)The second flat transverse section tetrad is given by (cid:0) e iµ (cid:1) = a ( r ) 0 0 00 b ( r ) cos φ − r sin φ b ( r ) sin φ r cos φ
00 0 0 r . (14)Tetrad (14) is related to (10) by a rotation matrix given by (cid:0) Λ ij (cid:1) = φ − sin φ
00 sin φ cos φ
00 0 0 r . (15)5ubstituting from (14) into (3) we evaluate the torsion scalar as T = − rba ′ + ab − ra ′ − a ) r ae . (16)Using (14) in (7) we get get the same differential equation given by Eq. (12) which have thesame solution given by Eq. (13).The third flat transverse section tetrad is given by (cid:0) e iµ (cid:1) = − a ( r ) L − H b ( r ) cos φ − r H sin φ a ( r ) H cos φ b ( r )(sin φ + L cos φ ) r sin φ cos φ ( L −
1) 0 a ( r ) H sin φ b ( r ) sin φ cos φ ( L − r (cos φ + L sin φ ) 00 0 0 r , (17)where L = √ H and H is an arbitrary function of r . This arbitrary function perseveresthe flat horizon of tetrad (17). Tetrad (17) is related to (10) by a LLT matrix given by (cid:0) Λ iµ (cid:1) = − L − H cos φ − H sin φ H cos φ sin φ + L cos φ sin φ cos φ ( L −
1) 0 H sin φ sin φ cos φ ( L −
1) cos φ + L sin φ
00 0 0 1 . (18)Substituting from (17) into (3) we evaluate the torsion scalar as T = 2( ba H + ba + rb H a ′ + rba ′ − ab L − b L ra ′ + rab HH ′ − a L − ra ′ L )) r ae L . (19)Using (17) in (7) we get the same differential equation given by Eq. (12) which have thesame solution given by Eq. (13). Therefore, the field equations (7) are not able to give aspecific form of the arbitrary function H as expected due to the fact that the field equations(7) are equivalent to GR. In the next section we are going to discuss the physical relevanceof each tetrad. There are many modifications of GR. In the frame of theoretical physics, the theory ofEinstein-Cartan (EC), which is identified as “Einstein-Cartan-Sciama-Kibble theory”, is6lassic gravitation theory like GR in which its connection has no skew symmetric part.Thus, in EC, the torsion tensor can be accompanied to the spin of the matter, by the samepattern the curvature is joined to the momentum and energy of the matter. Actually, spinof matter using non-flat spacetime demands that the torsion tensor does not vanishing butbe a variable, i.e., stationary action. The theory of EC deals with the torsion tensor andmetric as independent which give the right extension of conservation law in the existence ofthe gravitational field. The EC theory constructed by ´Elie Cartan in [10] and recently it hasmany application [11]. The Lagrangian of EC has the form [12]: L ( ϑ i , Γ jk ) = − κ (cid:0) R ij ∧ η ij − η (cid:1) , (20)with ϑ i being the co-frame one form, Γ jk being the connection, κ and Λ are the gravitationaland the cosmological constants. The Lagrangian given by Eq. (20) is a form invariant underdiffeomorphism and Lorentz local transformation [12]. Carrying out the principle of leastaction to equation (20) we get [12, 13]E i := − κ (cid:0) R jk ∧ η ijk − η i (cid:1) , H ij := 12 κ η ij , (21)where η ij is a two form given in the Appendix A, E i is the energy-momentum and H ij isthe rotational gauge field momentum. The momentum of translation and the spin take thefollowing form H i := − ∂ L ∂ T i = 0 , E ij := − ϑ [ i ∧ H j ] = 0 . (22)The minimally coupling of matter is supposed such that ∂ L Matter ∂ T i = 0 and ∂ L matter ∂ R ij = 0. ‡ Theconserved current is given by [12] [ ξ ] = 12 κ d (cid:8) ∗ (cid:2) d k + ξ ⌋ (cid:0) ϑ i ∧ T j (cid:1)(cid:3)(cid:9) , wherek = ξ i ϑ i , and ξ i = ξ ⌋ ϑ i , (23)where ∗ is defined as the Hodge duality and ξ is an arbitrary vector field ξ = ξ i ∂ i . ξ i in thisstudy are four parameters ξ , ξ , ξ and ξ . Because this study is in the frame of theoryof TEGR which is equivalent to GR, thus, torsion is nil and conserved charge, Eq. (23), isgiven by Q[ ξ ] = 12 κ Z ∂ S ∗ dk . (24)Expression (24) was given by Komar [14]–[18] and is invariant under diffeomorphism.The coframe ϑ i of solution (13) using tetrad (10) has the form: ϑ ˆ0 = adt, ϑ ˆ1 = bdr, ϑ = rdθ, ϑ = rdφ. (25) ‡ The derivative of the coframe vanishes if ξ is a Killing vector field, i.e. L ξ ϑ i = 0 [12].
7y using equation (25) in equation (24) we obtain k = a ξ dt − b ξ dr − r [ ξ dθ + ξ dφ ] . (26)Total derivative of Eq. (26) gives d k = 2[ aa ′ ξ ( dr ∧ dt ) + rξ ( dθ ∧ dr ) − rξ ( dr ∧ dφ )] . (27)From the inverse of (25) using (27) in (24) and taking the Hodge-duality to d k, we getQ[ ξ t ] = Q[ ξ r ] = Q[ ξ θ ] = Q[ ξ φ ] = 0 . (28)Equation (28) indicates in clear way that the total conserved charge of (13) are nil. Carryingout the same procedure to tetrads (14) and (17) we get the same result of Eq. (28). Thus,equation (24) must redefined to get a well defined value, i.e. Eq. (24) needs a regularization. Expression (24) is form invariant under diffeomorphism and Lorentz local transformation.However, it is demonstrated that plus to diffeomorphism and Lorentz local transformationthere exists other defect in the form of conserved quantities. This defect lies in equations ofmotion which permit a relocalization [12]. Therefore, conserved quantities can be altered us-ing relocalization. Relocalization appeared from amercement of the gravitational Lagrangianthrough a total derivative term has the formS ′ = S + d Φ , where Φ = Φ( ϑ i , Γ ij , T i , R ij ) , (29)with Γ ij is a 1-form connection. The second term in Eq. (29), i.e. d Φ, amendments theboundary part of the Lagrangian, premitting the equations of motion to be in a covariantform ( [12] and references therein). It has been explained that the total conserved chargescould be regularized by applying a relocalization procedure. It has been explained that tosolve the odd result derived in Eq. (28), one has to employ relocalization given by theboundary expression that appears in the Lagrangian. We use the relocalization in the form H ij → H ′ ij = H ij − βη ijkl R kl , that is originated from the modification of the Lagrangian [12] L → S ′ = S + βd Φ , where H ′ ij = (cid:18) κ − β Λ3 (cid:19) η ij − βη ijkl (cid:18) R kl − Λ3 ϑ k ϑ l (cid:19) . (30)8ssuming β , which exists in Eq. (30) has the value κ in 4-dimension to confirm thecancelation of the vanishing value (that comes from inertia) which exists in Eq. (28). Thus,the conserved charges after regularization has the form [12, 19, 20] J [ ξ ] = − κ Λ Z ∂S η ijkl Ξ ij W kl , (31)with W ij is the Weyl 2-form described byW ij = 12 C klij ϑ k ∧ ϑ l , (32)where C ijkl = e iµ e j ν e kα e lβ C µναβ is the Weyl tensor and Ξ ij is denoted byΞ ij = 12 e j ⌋ e i ⌋ dk. (33)The conserved currents J [ ξ ] given by Eq (31) are form invariant under diffeomorphism andLorentz local transformation. These currents J [ ξ ] are linked to the vector field ξ on thespacetime manifold.calculating the necessary components of Eq. (31) we get Ξ = − a ′ ξ b , Ξ = − ξ b . (34)Using Eq. (34), we get the value of η ijkl Ξ ij W kl in 4-dimension in the form η ijkl Ξ ij W kl = 2 √ c c ξ (2Λ r − c )3 r . (35)Substituting Eq. (35) in (31) we get J [ ξ t ] = √ ξ πc c + (cid:18) r (cid:19) , J [ ξ r ] = J [ ξ z ] = J [ ξ φ ] = 0 . (36)Equation (36) shows that the two constants c and c may take the values c = √ π and c = M in which total mass and angular momentum takes the form [21, 22] E = M + (cid:18) r (cid:19) , J [ ξ r ] = J [ ξ z ] = J [ ξ φ ] = 0 . (37)Repeat the same calculation we get the non-vanishing components Ξ ij of tetrad (14) tohave the form Ξ = a ′ cos φξ b , Ξ = − a ′ sin φξ b , Ξ = − cos φξ b , Ξ = − sin φξ b . (38)Using Eq. (31), we get the value of η ijkl Ξ ij W kl in 4-dimension in the same form of Eq.(35) which gives the same conserved quantities as given by Eq. (37).9he survive components Ξ ij of tetrad (17) are Ξ = − ξ a ′ [(2 H + 1) cos φ + L sin φ ] b , Ξ = − ξ a ′ sin φ cos φ (2 H + 1 − L ) b , Ξ = − ξ cos φ H b , Ξ = ξ a ′ H sin φ (2[ L −
1] cos φ + 1 b , Ξ = ξ (sin φ + cos φ L ) b , Ξ = ξ sin φ cos φ ( L − b . (39)Using Eq. (31), we get the value of η ijkl Ξ ij W kl in 4-dimension in the form η ijkl Ξ ij W kl = 2 √ c c ξ (2 H [ L −
1] sin φ cos φ − H − r − c )3 r . (40)Substituting Eq. (40) in (31) we get J [ ξ t ] = √ c c ξ (2 H [ L −
1] sin φ cos φ − H − r − c )4 r , J [ ξ r ] = J [ ξ θ ] = J [ ξ φ ] = 0 . (41)Equation (41) shows that the arbitrary function H which describes inertia contributes to thetrue physics. As we discussed in the previous sections that we have three tetrad fields reproduce the samemetric. The TEGR field equations give the same solution of the two unknown functionshowever, they can not able to give any specific form of the arbitrary function. Also we haveshown that the scalar torsion depend on the tetrad as is known in the literature which meansthat it is not local Lorentz transformation. In the previous section we try to calculate theconserved quantities and show that they depend on the inertia. In this section, we are goingto assume specific form of skew-tensor and see if this tensor will help in solving the aboveproblem or not? It is well known that the field equations of f ( T ) is non-symmetric [23]–[35] § S ( µν ) ρ T ,ρ f ( T ) T T − (cid:2) e − e aµ ∂ ρ ( be aα S αρν ) − T αλµ S ανλ (cid:3) f ( T ) T − g νµ ( f ( T ) − π T νµ ,S [ µν ] ρ T ,ρ f ( T ) T T = 0 . (42)Therefor, in TEGR the skew-symmetric is satisfied automatic due to the fact that f T T = 0.Now let us check if the skew symmetric tensor S [ µν ] ρ T ,ρ , (43)is vanishing for the above tetrad fields or not? For the first tetrad given by Eq. (10) Eq.(43) is satisfied automatic. Therefore, tetrad (10) we call it a physical tetrad. For the secondtetrad field, given by (14), Eq. (43) is satisfied identically. § We will denote the symmetric part by ( ), for example, A ( µν ) = (1 / A µν + A νµ ) and the antisymmetricpart by the square bracket [ ], A [ µν ] = (1 / A µν − A νµ ). S [10]1 T , = H cos φ L / abr (cid:18) b (cid:20) r ab L a ′′ + r H a b L H ′′ − r b L a ′ − raa ′ L ( r L b ′ − b [ r HH ′ − L ]) − a [ rb ′ L ( r HH ′ + L ) − b ( r H ′ − L )] (cid:21) − L / [ r ab ( b + 2) a ′′ − r b ( b + 2) a ′ − raa ′ ( r ( b + 4) b ′ + b + 2 b ) − a ( rb ′ ( b + 2) + 2 b + 2 b ] (cid:19) ,S [20]1 T , = H sin φ L / ab r (cid:18) b (cid:20) r ab L a ′′ + r H a b L H ′′ − r b L a ′ − raa ′ L ( r L b ′ − b [ r HH ′ − L ]) − a [ rb ′ L ( r HH ′ + L ) − b ( r H ′ − L )] (cid:21) − L / [ r ab ( b + 2) a ′′ − r b ( b + 2) a ′ − raa ′ ( r ( b + 4) b ′ + b + 2 b ) − a ( rb ′ ( b + 2) + 2 b + 2 b ] (cid:19) . (44)Solution of Eq. (44) has the form H = ± √ − r c Λ − rc c c p r ( r Λ + 3 c ) . (45)Equation (45) is a solution of Eq. (44) and when we use it Eq. (41) we get E = M + (cid:18) r (cid:19) . (46)which is identical with Eq. (37). This means that the solution of the skew-symmetric tensorremoves the inertia from the true physics. 11 Discussion and conclusion
We have discussed the TEGR theory and its 6 extra degrees of freedom. For this purpose wehave studied three tetrad fields, with the flat horizon, reproduce the same metric. The firsttetrad field has two unknown functions and the field equations of TEGR give an analytic formof these functions. The conserved quantities of this tetrad are calculated and we have gota finite conserved quantity in the temporal coordinate after using the Regularization usingrelocalization. We coined this tetrad as the physical tetrad. The reason for this name comesfrom the fact that we have defined a skew-symmetric tensor that is must vanish identicallyin the framework of TEGR. The first tetrad satisfied this property.For the second tetrad which is obtained from the first one by multiplied it by a rotationmatrix. We have calculated the scalar torsion of this tetrad and have shown that the scalartorsion is not invariant under local Lorentz transformation. We have calculated the fieldequations and show that they are not different from the first tetrad and consequently thesolution of the two unknown functions are the same as of the first tetrad. The discussion ofthe conserved quantities and the skew-symmetric tensors are the same of the first tetrad.For the third tetrad, we have shown that it is related to the first tetrad through alocal Lorentz transformation that contains an arbitrary function, H . We have calculatedthe scalar torsion of this tetrad and have shown that it depends on the arbitrary func-tion. As usual, the field equations of TEGR can not fix any form of the arbitrary function.We have calculated the conserved quantities of this tetrad and have shown that the tem-poral component of the coordinate depends on the arbitrary function. This shows in aclear way that the inertia contributes to the true physics. We have calculated the skew-symmetric tensor of this tetrad and have shown that some of its components are not van-ishing. We have solved these non-vanishing components and have derived a form of thearbitrary function. When we have substituted this solution in the form of the conservedquantities we have shown that the energy will coincide with the value of the physical tetrad. Appendix ANotation
The indices i , j , · · · are the (co)-frame components whilst µ , ν , · · · are the holonomic spacetimecoordinates. The hats ˆ0,ˆ1, · · · c indicate special frame components. The exterior productis denoted by ∧ . The interior product of ξ and Ψ is described by ξ ⌋ Ψ. The vector dual tothe 1-forms ϑ i is labeled by e i and their inner product satisfy e i ⌋ ϑ j = δ ij . Employing localcoordinates x µ , we have ϑ i = e iµ dx µ and e i = e iµ ∂ µ where e iµ and e iµ are the componentscovariant and contravariant of the tetrad fields. The volume η := ϑ ˆ0 ∧ ϑ ˆ1 ∧ ϑ ˆ2 ∧ ϑ ˆ3 defines4-form. Moreover, we can altered η i := e i ⌋ η = 13! ǫ ijkl ϑ j ∧ ϑ k ∧ ϑ l , ǫ ijkl is totally antisymmetric with ǫ = 1. η ij := e j ⌋ η i = 12! ǫ ijkl ϑ k ∧ ϑ l , η ijk := e k ⌋ η ij = 11! ǫ ijkl ϑ l . Finally, we can define η ijkl := e l ⌋ η ijk = e l ⌋ e k ⌋ e j ⌋ e i ⌋ η, which is the tensor density of Levi-Civita. The following useful identities ϑ i ∧ η j := δ ij η, ϑ i ∧ η jk := δ ik η j − δ ij η k , ϑ i ∧ η jkl := δ ij η kl + δ ik η lj + δ il η jk ,ϑ i ∧ η jkln := δ in η jkl − δ il η jkn + δ ik η jln − δ ij η kln are holds using the η -forms.The line element ds := g ij ϑ i N ϑ j is defined by the spacetime metric g ij . Appendix B: Calculations of Weyl tensor and the object W µν The non-vanishing components of Weyl tensor using solution of tetrad (10) have theform: C = − C = C = − C = −√ c c r ,C = − C = C = − C = C = − C = C = − C = −√ Λ r + 3 c c c r / ,C = − C = C = − C = − C = C = − C = C = −√ c r / √ Λ r +3 c ,C = − C = − C = C = − c r . (47)The non-vanishing components of W µν are given by W = − √ c c r ( dr ∧ dt )] , W = − c c √ Λ r + 3 c r / ( dθ ∧ dt ) ,W = − c c √ Λ r + 3 c r / ( dφ ∧ dt ) , W = − √ c r / √ Λ r + 3 c ( dr ∧ dθ ) ,W = − √ c r / √ Λ r + 3 c ( dr ∧ dφ ) , W = − c r ( dθ ∧ dφ ) . (48)By the same method we can calculate the non-vanishing of Weyl tensor and of the object W µν of the second and third tetrad. References [1] R. Weitzenbock,
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