The generalization of the ADM gravitational energy-momentum

TL;DR

This paper demonstrates that the teleparallel approach to gravitational energy-momentum generalizes the ADM formalism, providing consistent results in cases where ADM fails, and discusses advantages and challenges of the teleparallel stress-energy tensor.

Contribution

It proves that teleparallel energy-momentum extends the ADM formalism and coincides with it under certain conditions, offering a more robust framework in specific scenarios.

Findings

Teleparallel 4-momentum can match ADM 4-momentum when applicable.

Teleparallel approach succeeds where ADM formalism fails.

Advantages of teleparallel stress-energy tensor over Landau-Lifshitz pseudo-tensor.

Abstract

In this paper, it is proved that the teleparallel energy-momentum generalizes that of the ADM formalism. In doing so, it is shown that the teleparallel $4$-momentum can be made to coincide with that of the ADM approach whenever the ADM $4$-momentum is applicable. The only assumptions are the time gauge for the teleparallel frame and the well-known restrictions for the coordinate system used in the calculation of the ADM $4$-momentum. Then, examples where the ADM formalism fails to give consist results, but the teleparallel approach does not, are given. The advantages of the teleparallel stress-energy tensor (density) over the pseudo-tensor of Landau-Lifshitz are exhibited. Finally, the difficulties in identifying the gravitational angular momentum density is discussed; it is shown that the spatial part of the proposed angular momentum density $M^{ab}$ vanishes when the teleparallel frame satisfies the time gauge condition.