In the context of a gauge theory for the translation group, a conserved energy-momentum gauge current for the gravitational field is obtained. It is a true spacetime and gauge tensor, and transforms covariantly under global Lorentz transformations. By rewriting the gauge gravitational field equation in a purely spacetime form, it becomes the teleparallel equivalent of Einstein's equation, and the gauge current reduces to the Møller's canonical energy-momentum density of the gravitational field.PACS numbers: 04.50. + hThe definition of an energy-momentum density for the gravitational field is one of the oldest and most controversial problems of gravitation. As a true field, it would be natural to expect that gravity should have its own local energy-momentum density. However, it is usually asserted that such a density cannot be locally defined because of the equivalence principle [1]. As a consequence, any attempt to identify an energy-momentum density for the gravitational field leads to complexes that are not true tensors. The first of such attempts was made by Einstein who proposed an expression for the energy-momentum density of the gravitational field which was nothing but the canonical expression obtained from Noether's theorem [2]. Indeed, this quantity is a pseudotensor, an object that depends on the coordinate system. Several other attempts have been made, leading to different expressions for the energy-momentum pseudotensor for the gravitational field [3].Despite the existence of some controversial points related to the formulation of the equivalence principle [4], it seems true that, in the context of general relativity, no tensorial expression for the gravitational energy-momentum density can exist. However, as our results show, in the gauge context, the existence of an expression for the gravitational energy-momentum density which is a true spacetime and gauge tensor turns out to be possible. Accordingly, the absence of such an expression should be attributed to the general relativity description of gravitation, which seems to not be the appropriate framework to deal with this problem [5].In spite of some skepticism [1], there has been a continuous interest in this problem [6]. In particular, a quasilocal approach has been proposed recently which is highly clarifying [7]. According to this approach, for each gravitational energy-momentum pseudotensor, there is an associated superpotential which is a Hamiltonian boundary term. The energy momentum defined by such a pseudotensor does not really depend on the local value of the reference frame, but only on the value of the reference frame on the boundary of a region-then its quasilocal character. As the relevant boundary conditions are physically acceptable, this approach validates the pseudotensor approach to the gravitational energy-momentum problem. It should be mentioned that these results were obtained in the context of the general relativity description of gravitation.In the present work a different approach will be used to reexamine the gravitation...
The fundamentals of the teleparallel equivalent of general relativity are presented, and its main properties described. In particular, the field equations, the definition of an energy-momentum density for the gravitational field, the teleparallel version of the equivalence principle, and the dynamical role played by torsion as compared to the corresponding role played by curvature in general relativity, are discussed in some details.
A new expression for the spin connection of teleparallel gravity is proposed, given by minus the contorsion tensor plus a zero connection. The corresponding minimal coupling is covariant under local Lorentz transformation, and equivalent to the minimal coupling prescription of general relativity. With this coupling prescription, therefore, teleparallel gravity turns out to be fully equivalent to general relativity, even in the presence of spinor fields. DOI: 10.1103/PhysRevD.64.027502 PACS number͑s͒: 04.20.Cv, 04.50.ϩh In the absence of spinor fields, the equivalence of teleparallel gravity ͓1͔ with general relativity is a well established subject ͓2͔. In the presence of spinor fields, however, it is widely believed that this equivalence is spoiled ͓3͔. The reason for this is that, in teleparallel gravity, the spin connection is believed to vanish. The basic purpose of this Brief Report is to show that this conclusion is not appropriate, and that it comes from a particular choice of the teleparallel spin connection. A new expression for the spin connection is then proposed, according to which teleparallel gravity becomes much more consistent and fully equivalent to general relativity, even in the presence of spinor fields.We use the Greek alphabet (,,, . . . ϭ0,1,2,3) to denote indices related to spacetime ͑base space͒, and the Latin alphabet (a,b,c, . . . ϭ0,1,2,3) to denote indices related to the tangent space ͑fiber͒, assumed to be a Minkowski spacetime. The spin connection A° can thus be written aswhere S ab is an element of the Lorentz Lie algebra written in an appropriate representation. In general relativity, the spin connection is ͓4͔where h a is a tetrad field, and ⌫° is the Levi-Civita connection. In order to obtain the teleparallel spin connection, inspired by the definition ͑2͒, it is usual to start by making the following attempt:with ⌫ the Weitzenböck connection. However, as a consequence of the absolute parallelism condition ͓2͔, we have A a b ϭ0. This does not mean that in teleparallel gravity the spin connection vanishes. It only means that we have not made an appropriate guess.Let us then adopt a different procedure to look for the teleparallel spin connection. Our basic guideline will be to find a coupling prescription that results equivalent to the coupling prescription of general relativity. This can be achieved by taking the relation between the Levi-Civita and the Weitzenböck connections ͓5͔,with K the contorsion tensor, and rewriting it in the tetrad basis. The result iswhere we have already used the fact that A a b ϭ0. Notice that the zero connection appearing in Eq. ͑5͒ is crucial in the sense that it is the responsible for making the right-hand side a true connection. Therefore, based on these considerations, we can say that the teleparallel spin connection Accordingly, the teleparallel Fock-Ivanenko derivative operator is to be written in the formIt is important to remark that, differently from A°a b which depends on the Levi-Civita connection only, a b depends on the Weitzenböck ...
Relying upon the equivalence between a gauge theory for the translation group and general relativity, a teleparallel version of the original Kaluza-Klein theory is developed. In this model, only the internal space (fiber) turns out to be five dimensional, spacetime being kept always four dimensional. A five-dimensional translational gauge theory is obtained which unifies, in the sense of Kaluza-Klein theories, gravitational and electromagnetic interactions. 04.50.+h; 12.10.-g
Due to its underlying gauge structure, teleparallel gravity achieves a separation between inertial and gravitational effects. It can, in consequence, describe the isolated gravitational interaction without resorting to the equivalence principle, and is able to provide a tensorial definition for the energy-momentum density of the gravitational field. Considering the conceptual conflict between the local equivalence principle and the nonlocal uncertainty principle, the replacement of general relativity by its teleparallel equivalent can be considered an important step towards a prospective reconciliation between gravitation and quantum mechanics.Comment: 9 pages. Contribution to the proceedings of the Albert Einstein Century International Conference, Paris, 18-22 July, 200
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