The role of observers in the measurement of the Teleparallel Gravitoelectromagnetic fields in different geometries
Ednardo P. Spaniol, Leandro R. A. Belo, Juliano A. de Deus, Vanessa C. de Andrade
aa r X i v : . [ g r- q c ] N ov The role of observers in themeasurement of the TeleparallelGravitoelectromagnetic fields indifferent geometries
E.P. Spaniol a ∗ , L.R.A. Belo a † , J.A. de Deus a ‡ , and V.C. de Andrade a § a Physics Institute, Brazilia University70.917-910, Bras´ılia, Federal District, Brazil
Abstract
In the context of the Teleparallel Equivalent of General Relativity (TEGR) we haveinvestigated the role of local observers, associated with tetrad fields, in descriptionof the gravitational interaction through the concepts of the gravitoelectric (GE) andgravitomagnetic (GM) fields. We start by analyzing the gravitoelectromagnetic (GEM)fields obtained from an observer freely falling in the Schwarzschild space-time. Then,we investigated whether it is possible to distinguish between this situation and to be atrest in the Minkowski space-time. We conclude that, although there are non-zero com-ponents for the fields obtained for the case of free fall, its dynamical effect, measuredby the gravitational Lorentz force, is null. Moreover, the gravitational field energyobtained from the GEM fields for an observer freely falling in the Schwarzschild space-time is zero. These results are in complete agreement with the equivalence principle. keywords:
Teleparallel Equivalent of General Relativity, Gravitoelectromagnetism,Free falling frame.
PACS: ∗ E-mail: spaniol@fis.unb.br † E-mail: leandrobelo@fis.unb.br ‡ E-mail: julianoalves@fis.unb.br § E-mail: andrade@fis.unb.br Introduction
The study of gravitation through the GEM fields can bring some new insight [1]. To givea contemporary example, the interpretation of objects recently defined by the literature, asthe vortex and tendex lines [2] can be facilitated if we use the fields to describe them .Recently, at the linear regime of gravitation, we had a new confirmation of the existence ofthe GEM fields, through the Gravit Probe B experiment [3].In the literature, we found several studies that address the stationary GEM fields dueto the fact that the similarity between the field equations of electromagnetic theory and ofgravitation is reached in this context [4]. There are still some works that deal with the time-dependent GEM and the issue about the Faraday’s law in the context of general relativity(GR) [5]. However, there is no studies about the behavior of GEM fields for cases in whichthe observer is moving with respect to the source.On the other hand, with the advent of relativistic mechanics, the relativity principleswere extended to the electrodynamics in analyzing the behavior of electric and magneticfields in relation to different inertial frames. Just as in electrodynamics, we expect that,in gravitation, the GEM fields proposed assume different expressions depending on how thesource is observed, i.e., different observers note different GEM fields. Thus, it is natural tostudy the physical consequences of these different fields upon the observer itself.In a previous work [6], motivated by the fact that the TEGR can be described as a gaugetheory, we have proposed a new way to define the gravitoelectric and the gravitomagneticfields. These definitions, that are conceptually different from those that arise in the RG,were made in a very similar way to what is done on the Yang-Mills theory and the Electro-magnetism, being based on the field strength components. On that work, in the weak fieldlimit we have obtained the analogous Maxwell equations and for a set of tetrads which isadapted to a stationary observer relative to Schwarzschild, the gravitoelectric componentscalculated were in total agreement with the newtonian field.According to [7] we can interpret the extra degrees of freedom of the tetrad field as achoice of reference system. Two sets of tetrad fields may represent the same spacetime,though they are physically different. That is, besides being the fundamental object of thetheory, we can interpret them as ideal observers in spacetime. This subtleness is not presentin the metric description of gravity.In this paper, on the context of the TEGR, we discuss the issue of how different observersfeel the GEM fields. For two different observers, we will analyze the behavior of GEM fieldsfor the Schwarzschild and the Minkowski spacetime. First, we will consider a free fallingobserver in Schwarzschild black hole, ie an observer who falls radially into a black hole dueto its gravitational force, then we will consider a second observer, but now, the observeris standing in the Minkowski spacetime. As expected, from the equivalence principle, wehave concluded in this approach that those observers are indistinguishable from the dynamicpoint of view, that is, being null the gravitational Lorentz force [8] felt by each one of them,they follow the same trajectory. Moreover, their energies were calculated, being zero in bothcases. It is interesting to understand how these results occur even though we have foundnon-zero GEM components, as can be viewed along the subsections 2.1 and 2.2. This possibility is still under investigation. η ab = ( − , , , µ, ν, σ, ... = 0 , , ,
3) and the tangent space indiceswill be denoted by the first half of the Latin alphabet ( a, b, c.. = 0 , , , i, j, k... )assume the values 1,2 and 3. Indices in parentheses will also be related to tangent space.We adopt the light velocity as c = 1. Let us present some of the more important expressions in TEGR that will be used in thewhole paper .The field strength of the theory is defined in the usual form F aµν = ∂ µ h aν − ∂ ν h aµ = h aρ T ρµν , (1)with h aµ being the components of the tetrad field. The object T ρµν is the torsion thatrepresents alone the gravitational field, defined by T ρµν ≡ Γ ρνµ − Γ ρµν , where Γ ρνµ is theWeitzenb¨ock connection given by Γ ρνµ ≡ h aρ ∂ µ h aν . Therefore, torsion can also be identifiedas the field strength written on the tetrad base.The dynamics of the gauge fields will be determined by the lagrangian [10] L G = h πG S ρµν T ρµν , (2)with h = det( h aµ ) and S ρµν = − S ρνµ ≡ (cid:2) K µνρ − g ρν T θµθ + g ρµ T θνθ (cid:3) (3)which is called superpotential, that will play an important role in theory, as we will see. Theobject K µνρ is the contorsion tensor defined by K µνρ = 12 T νµρ + 12 T ρµν − T µνρ . (4)The field equations resulting from this lagrangian are ∂ σ ( hS aσρ ) − πG ( hj aρ ) = 0 (5)with j aρ ≡ ∂ L ∂h aρ = h aλ ( F cµλ S cµρ − δ λρ F cµν S cµν ) , (6)being the gauge energy-momentum current of the gravitational field [9]. For detailed of the teleparallel fundamentals see Ref.[8, 9]. F aµν can be associated to the torsion tensor,in such way that we could use it to define our fields. On the other hand, the superpotential,defined above, assumes the role of the field strength in the field equations, similarly to whatoccurs in the electromagnetic equations. Therefore, inspired on the electromagnetism, we de-fine the gravitoelectric and gravitomagnetic fields in terms of the superpotential components.The gravitoelectric field (GE) is defined by S a i ≡ E ai (7)and the gravitomagnetic field (GM) is as follows S aij ≡ ǫ ijk B ak . (8)As already stated, these definitions were tested and passed on some important tests [6].Calculating these fields in specific configurations will allow us to better understand the roleof observers in gravitation. In this section we analyze the GEM fields obtained from an observer in free fall in Schwarzschildspacetime. Initially we consider the Schwarzschild metric which can be written as ds = − α − dt + α dr + r ( dθ + sinθ dφ ) (9)with α − = 1 − mr . (10)An observer that moves radially in free fall due to attraction of the Schwarzschild blackhole must have a four-velocity like [11] u ν = h(cid:16) − mr (cid:17) − , − (cid:16) mr (cid:17) / , , i . (11)A set of tetrad fields that satisfies the above condition is given by [7] h aµ = − − α β βsinθcosφ α sinθcosφ rcosθcosφ − rsinθsinφβsinθsenφ α sinθsenφ rcosθsinφ rsinθcosφβcosθ α cosθ − rsinθ , (12)where β is defined by β = r mr . (13)Through the expression of torsion written in terms of the tetrad fields T σµν = h aσ ∂ µ h aν − h aσ ∂ ν h aµ , (14)4e calculate the components of T σµν , of which the non null are T = − β∂ r β,T = − α ∂ r β,T = − rβ,T = − rβ sin θ,T = r (1 − α ) ,T = r (1 − α ) sin θ. (15)With these results, we can calculate the superpotential that will allow us to find theGEM fields. In this way, to get the GE fields, we need the componentes of S bµν following: S b i = 14 h h bk g g ij T j k + h bk g g ij T k j i + 12 h h b g ij g lk T kjl − h bi g g kj T j k i . (16)The GE radial components are obtained by making i = 1, that is S b = E b = 12 h h b g g T + h b g g T − h b g g T − h b g g T i . (17)To the angular components θ we perform i = 2 in the above expression (16) and we find S b = E b = − (cid:2) h b g g T + h b g g T (cid:3) . (18)The φ components are obtained by making i = 3 S b = E b = − (cid:2) h a g g T + h a g g T (cid:3) . (19)Let us now consider the internal index equal to zero in the above expressions, that is, b = 0: E (0) r = 0 ,E (0) θ = 0 ,E (0) φ = 0 . (20)Then we calculate the spacial components for b . Considering (17) and attributing b = 1 , , E (1) r = − βr sin θ cos φ, (21) E (2) r = − βr sin θ sin φ, (22) E (3) r = − β cos θr . (23)In the same way, assigning the values b = 1 , , E (1) θ = − α β r cos θ cos φ, (24)5 (2) θ = − α β r cos θ sin φ, (25) E (3) θ = − α β r sin θ, (26)and finally, making b = 1 , , E (1) φ = α β sin φ r sin θ , (27) E (2) φ = − α β cos φ r sin θ , (28) E (3) φ = 0 . (29)Let us now calculate the GM fields for this configuration. Writing the superpotencial interms of torsions, S bij = 14 (cid:2) h a g ik g jm ( T mk + T km − T km ) + h an g ik g jm ( T mkn + T nkm − T kmn ) (cid:3) + 12 (cid:2) − h aj g ik ( g nm T mkn − g T k ) + h ai g jl ( g nm T mln − g T l ) (cid:3) , (30)and using the definition (8) with the internal index b = 0 in the above expression we obtain: B (0) φ = 0 ,B (0) θ = 0 ,B (0) r = 0 . (31)In the sequence we consider b = 1 , , φ component: B (1) φ = m r cos θ cos φ, (32) B (2) φ = m r cos θ sin φ, (33) B (3) φ = − m r sin θ. (34)For θ component: B (1) θ = m r sin φ sin θ , (35) B (2) θ = − m r cos φ sin θ , (36) B (3) θ = 0 . (37)Finally, the remaining radial components B (1) r = B (2) r = B (3) r = 0 . (38)On the other hand, if we consider a static observer in Minkowski spacetime and perform asimilar calculation we obtain all the GEM field components equal to zero. However, assuming6alid the equivalence principle, we should not be able to discern between two observers, oneof then freely falling in Schwarzschild black hole, and other static in Minkowski spacetime.This apparent inconsistency should be clarified when investigating the role of the non-zerocomponents b = 1 , , definition for the GEM fields must be in accordance with theequivalence principle, that is, for a non-rotating and free fall observer there is no gravita-tional forces and therefore the GEM fields are zero. Our definition is in full agreement withthis since in the weak field limit mr << b = 0 components also vanish. This show that the operational definitions must be relatedwith b = 0 components, which is in agreement with a similar analysis in [6].Let us now verify the effects of the non-zero components of the GE and GM fields on thedynamics of the observers. As mentioned earlier, as a consequence of the equivalence principle, an observer representedby a not spinning tetrad field and freely falling in Schwarzschild spacetime, should not be ableto distinguish - at least from the dynamic point of view - if is freely falling in this spacetimeor at rest with respect to the Minkowski spacetime. A way to tackle this issue is to use theequation that describes the behavior of scalar particles in the presence of gravitation: theTEGR gravitational Lorentz force [8] h aµ du a ds = F aµν u a u ν , (40)in which the right side of the equation plays the role of force, analogous to the Lorentz forceof Electromagnetism. Alternatively, this equation can be rewritten as the geodesic equationin the context of RG [8]. From this expression, we can evaluate the consequences of thenon-zero GEM fields components previously obtained.Since the GEM fields are defined from the superpotencial S bρµ it is convenient to rewritethe above equation in terms of these quantities. For this, we should first rewrite the gravi-tational field strength tensor in terms of the superpotential, ie F aγδ = h bγ g ρδ h aµ S bµρ − h bδ g νγ h aµ S bµν − h aδ g νγ h bθ S bθν + 12 h aγ g ρδ h bθ S bθρ . (41)Thus, we obtain h aµ du a ds = − h bν S bρµ u ρ u ν − h bθ ( S bµθ u ρ u ρ − S bνθ u µ u ν ) . (42)By making use of the (11) and of the GEM fields we can calculate the right side of theequation (42) for an observer freely falling in Schwarzschild spacetime. Thus, we get a null That allow a direct analogy with electromagnetism.
Another physical evidence that enables us to face the issue of non-zero components for thecase of the freely falling reference frame in the Schwarzschild spacetime is the gravitationalfield energy. Again, being valid the equivalence principle, we should not to be able to discernbetween two observers, one freely falling in Schwarzschild black hole, and another static inMinkowski spacetime. Thus, being zero the gravitational field energy associated with thesecond situation, an equal result should occur with the energy measured by the observerin the first situation. We can calculate the gravitational field energy as given by the zerocomponent of (6), from the GEM fields obtained by an observer represented by the tetradfield (12), since they are defined from the superpotential which appears in the definition ofenergy momentum tensor. Let us consider then j (0)0 = h (0) λ ( F ciλ S ci − δ λ F cµν S cµν ) . (43)Substituting (41) in (43) we get j (0)0 = h (0) λ (cid:16) h bi g ρλ h cγ S bγρ − h bλ g ρi h cγ S bγρ − h cλ g ρi h bγ S bγρ + 12 h ci g ρλ h bγ S bγρ (cid:17) S c i + 14 h (0)0 (cid:16) h bµ g ρλ h cγ S bγρ − h bλ g ρµ h cγ S bγρ − h cλ g ρµ h cγ S bγρ + 12 h cµ g ρλ h bγ S bγρ (cid:17) S cµλ . (44)Using the definitions (7) and (8) we can rewrite the expression above in terms of E ai and B ai . Note that as it is quadratic in the superpotential, it is also quadratic in the GEM fields.After a lengthy calculation, we found out the following result for the above component j (0)0 = 0 , (45)ie the gravitational field energy written in terms of the GEM fields are zero for a freely fallingobserver in the Schwarzschild black hole. While there are non-zero components of the GEMfield, they combine in such a way that do not change the expected result of the gravitationalfield energy, in a similar way with what happened in gravitational Lorentz force calculation.As consequence, it is not possible - at least from the dynamical point of view - for a localobserver to distinguish between to be in free fall in the Schwarzschild black hole or to be atrest in the Minkowski spacetime. Perhaps the components E (1 , , i and B (1 , , k have a measurable physical sense in a semi- classicalscenario and allow a differentiation between the frames. The analysis of this issue will be presented elsewhere. P = Z h(cid:0) E (0) i (cid:1) + (cid:0) B (0) i (cid:1) i d x. (46)It is important to stress out that this definition was inferred based only on the case of areference in free fall in a Schwarzschild black hole, being the extension of its validity stillunder investigation. As obtained in a previous work [6], for a set of tetrads which is adapted to a stationaryobserver relative to Schwarzschild spacetime, it has been showed that in the weak fieldlimit the gravitoelectric components are in total agreement with the Newtonian field and,in addition, all GM components are zero. The conceptual definitions of what we expectto be analogous to electromagnetic fields were identified as the zero internal components ofGEM fields, that is b = 0, what we have called ”operational definitions”. Here in this work,when we consider a freely falling observer in the Schwarzschild black hole we obtained a nullresult for all the GE and GM components with zero internal index, which corroborate theidea of ”operationality” for b = 0 component fields. We would like to emphasize that thechoice of coordinate systems is the same in both cases above mentioned, through the use ofappropriate tetrad fields.We have obtained as main conclusion in this work that through the use of GEM fieldsit is not possible for a local observer to distinguish between free falling in the Schwarzschildblack hole or at resting in the Minkowski spacetime. Although this idea seems to be natural,due to the equivalence principle, it emerged in this approach after a deeper analysis, sincewe found out that non-null fields arise in the free fall case. One possibility would be toconsider only the operational components b = 0, since they are all equal to zero and simplyto discard the other non null components that came from spacial internal indices. Then itwould be straightforward to postulate the equivalence between the references. But these nonull components could store some important information that would violate the central idea.To investigate the role of non null components of GEM fields in dynamics we haveused the gravitational Lorentz force written in terms of them and we concluded that theircontributions cancel each other resulting in a null total force measured by the free fallingobserver in the Schwarzschild geometry, in the same way it were placed at rest in Minkowskispacetime . Thus, any experiment which make use of dynamical effects from the gravitationalfield will not be able to distinguish between those two references. Moreover, in order tosupport the results, we showed that the gravitational field energy measured by the referencein free fall is zero, as expected if compared with the field energy associated with the flatspacetime. We also should like to stress out that all the calculations were done using theGEM fields and outside the weak field limit, that is, we have obtained exact results that canalso be applicable to treat intense fields like, for example, jet formations in supermassiveblack holes. 9 cknowledgements The authors thank CAPES and CNPq (Brazilian agencies) for the financial support.
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