Energy Distribution associated with Static Axisymmetric Solutions

TL;DR

This paper evaluates energy and momentum densities for static axisymmetric solutions in General Relativity, specifically Erez-Rosen and gamma metrics, using four different energy-momentum prescriptions, revealing discrepancies in energy density calculations.

Contribution

It compares four energy-momentum prescriptions for static axisymmetric solutions, highlighting their differences and conditions for matching results.

Findings

Energy densities differ across prescriptions

Momentum remains constant in all cases

Results match under specific boundary conditions

Abstract

This paper has been addressed to a very old but burning problem of energy in General Relativity. We evaluate energy and momentum densities for the static and axisymmetric solutions. This specializes to two metrics, i.e., Erez-Rosen and the gamma metrics, belonging to the Weyl class. We apply four well-known prescriptions of Einstein, Landau-Lifshitz, Papaterou and M$\ddot{o}$ller to compute energy-momentum density components. We obtain that these prescriptions do not provide similar energy density, however momentum becomes constant in each case. The results can be matched under particular boundary conditions.