The Energy-Momentum Problem in General Relativity

TL;DR

This paper reviews the challenges of defining and localizing energy-momentum in general relativity, compares various energy-momentum complexes, and demonstrates their effectiveness in calculating energy distributions across different space-times, supporting Einstein's complex as the most reliable.

Contribution

The study systematically compares multiple energy-momentum complexes in general relativity and affirms Einstein's complex as the most effective for energy distribution calculations.

Findings

Several energy-momentum complexes yield consistent results for a given space-time.

Einstein's energy-momentum complex is identified as the most reliable tool.

Results support the Cooperstock hypothesis on energy localization in GR.

Abstract

Energy-momentum is an important conserved quantity whose definition has been a focus of many investigations in general relativity. Unfortunately, there is still no generally accepted definition of energy and momentum in general relativity. Attempts aimed at finding a quantity for describing distribution of energy-momentum due to matter, non-gravitational and gravitational fields resulted in various energy-momentum complexes whose physical meaning have been questioned. The problems associated with energy-momentum complexes resulted in some researchers even abandoning the concept of energy-momentum localization in favour of the alternative concept of quasi-localization. However, quasi-local masses have their inadequacies, while the remarkable work of Virbhadra and some others, and recent results of Cooperstock and Chang {\it et al.} have revived an interest in various energy-momentum complexes. Hence in this work we use energy-momentum complexes to obtain the energy distributions in various space-times. We elaborate on the problem of energy localization in general relativity and use energy-momentum prescriptions of Einstein, Landau and Lifshitz, Papapetrou, Weinberg, and M{\o}ller to investigate energy distributions in various space-times. It is shown that several of these energy-momentum complexes give the same and acceptable results for a given space-time. This shows the importance of these energy-momentum complexes. Our results agree with Virbhadra's conclusion that the Einstein's energy-momentum complex is still the best tool for obtaining energy distribution in a given space-time. The Cooperstock hypothesis for energy localization in GR is also supported.