Gravitational Energy-Momentum Density in Teleparallel Gravity

Abstract

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...

Teleparallel versions of Friedmann and Lewis–Papapetrou spacetimes

“…Theories of gravity based on the geometry of distance parallelism [3,[8][9][10][11][12][13][14] are commonly considered as the closest alternative to the GR. TP gravity models possess a number of attractive features both from the geometrical and physical viewpoints.…”

Stability of Reissner-Nordström Black Hole

“…(12), (13), and (14). It is proved that these tensors are vanishing identically for any spherically symmetric solutions [28,29].…”

“…This is a logic result since both solutions reproduce the same metric tensor and these scalars mainly depend on the metric tensor. The scalars of torsion tensor, basic vector, traceless part, and the axial vector part of the space-time given by solution (13) are given by…”

“…The tetrad theory of gravitation based on the geometry of absolute parallelism [5][6][7][8][9][10][11][12][13][14] can be considered as the closest alternative to general relativity, and it has a number of attractive features both from the geometrical and physical viewpoints. Absolute parallelism is naturally formulated by gauging space-time translations and underlain by the Weitzenböck geometry, which is characterized by the metricity condition and by the vanishing of the curvature tensor (constructed from the connection of the Weitzenböck geometry).…”

Brane world black holes in teleparallel theory equivalent to general relativity and their Killing vectors, energy, momentum and angular momentum

“…Therefore, it is only possible to define the energy of the gravitational filed in whole space-time region and not only in a small region. The Einstein's GR can also be reformulated in the context of teleparallel geometry [18]∼ [27]. In this geometrical setting the dynamical field quantities correspond to orthonormal tetrad field e a µ * (a, µ are SO(3,1) and space-time indices, respectively).…”