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

TL;DR

This paper links Sparling two-forms with the gravitational energy density in the teleparallel equivalent of general relativity (TEGR), demonstrating that their energy expression correctly captures the gravitational mass in the conformal factor.

Contribution

It establishes a relationship between TEGR gravitational energy density and Sparling two-forms, providing a consistent expression for gravitational mass in the conformal factor.

Findings

The TEGR energy density relates to Sparling two-forms.

The energy expression correctly yields gravitational mass.

The approach aligns with known results for various space-times.

Abstract

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 Einstein's 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.