Sparling two-forms, the conformal factor and the gravitational energy density of the teleparallel equivalent of general relativity

It has been shown recently that within the framework of the teleparallel equivalent of general relativity (TEGR) it is possible to define the energy density of the gravitational field. The TEGR amounts to an alternative formulation of Einsteins general relativity, not to an alternative gravity theory. The localizability of the gravitational energy has been investigated in a number of space-times with distinct topologies, and the outcome of these analises agree with previously known results regarding the exact expression of the gravitational energy, and/or with the specific properties of the space-time manifold. In this article we establish a relationship between the expression for the gravitational energy density of the TEGR and the Sparling two-forms, which are known to be closely connected with the gravitational energy. We also show that our expression of energy yields the correct value of gravitational mass contained in the conformal factor of the metric field.

Gravitational energy of rotating black holes

In the teleparallel equivalent of general relativity the energy density of asymptotically flat gravitational fields can be naturaly defined as a scalar density restricted to a three-dimensional spacelike hypersurface $Sigma$. Integration over the whole $Sigma$ yields the standard ADM energy. After establishing the reference space with zero gravitational energy we obtain the expression of the localized energy for a Kerr black hole. The expression of the energy inside a surface of constant radius can be explicitly calculated in the limit of small $a$, the specific angular momentum. Such expression turns out to be exactly the same as the one obtained by means of the method preposed recently by Brown and York. We also calculate the energy contained within the outer horizon of the black hole for {it any} value of $a$. The result is practically indistinguishable from $E=2M_{ir}$, where $M_{ir}$ is the irreducible mass of the black hole.

Gravitational energy of conical defects

The energy density of asymptotically flat gravitational fields can be calculated from a simple expression involving the trace of the torsion tensor. Integration of this energy density over the whole space yields the ADM energy. Such expression can be justified within the framework of the teleparallel equivalent of general relativity, which is an alternative geometrical formulation of Einsteins general relativity. In this paper we apply this energy density to the evaluation of the energy per unit length of a class of conical defects of topological nature, which include disclinations and dislocations (in the terminology of crystallography). Disclinations correspond to cosmic strings, and for a spacetime endowed with only such a defect we obtain precisely the well known expression of energy per unit length. However for a pure spacetime dislocation the total gravitational energy is zero.