Energy-Momentum Distribution: A Crucial Problem in General Relativity

TL;DR

This paper investigates the problem of defining energy-momentum in General Relativity by applying various prescriptions to specific solutions, revealing inconsistencies that support the view that localization should rely on the energy-momentum tensor rather than pseudo-tensors.

Contribution

It compares multiple energy-momentum prescriptions on exact solutions, highlighting their inconsistencies and supporting the tensor-based approach to energy localization in GR.

Findings

Prescriptions do not yield consistent energy-momentum densities for the solutions.

Inconsistencies support the idea that localization is better described by the energy-momentum tensor.

Results align with the Hamiltonian approach's perspective on energy localization.

Abstract

This paper is aimed to elaborate the problem of energy-momentum in General Relativity. In this connection, we use the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and M\"{o}ller to compute the energy-momentum densities for two exact solutions of Einstein field equations. The spacetimes under consideration are the non-null Einstein-Maxwell solutions and the singularity-free cosmological model. The electromagnetic generalization of the G\"{o}del solution and the G\"{o}del metric become special cases of the non-null Einstein-Maxwell solutions. It turns out that these prescriptions do not provide consistent results for any of these spacetimes. These inconsistence results verify the well-known proposal that the idea of localization does not follow the lines of pseudo-tensorial construction but instead follows from the energy-momentum tensor itself. These differences can also be understood with the help of the Hamiltonian approach.