Gravitational Energy in Van Stockum Space-Time

TL;DR

This paper compares energy and momentum distributions of Van Stockum space-time in general relativity and teleparallel gravity, showing their equivalence and the variability of energy-momentum complexes.

Contribution

It demonstrates the equivalence of general relativity and teleparallel gravity for Van Stockum space-time and analyzes differences in energy-momentum complexes within these theories.

Findings

Total gravitational energy is zero for homogeneous space-times.

Different energy-momentum complexes yield different results.

Both theories are equivalent for the space-time considered.

Abstract

The purpose of this paper is to illustrate the problem of energy and momentum distributions of Van Stockum space-time within the framework of two different theories of gravity, general relativity and teleparallel gravity. We have shown that for all homogeneous space-times with metric components $g_{\mu\nu}$ being functions of time variable, $t$, alone and independent of space variables the total gravitational energy for any finite volume is identically zero. By working with general relativity, we have calculated the energy-momentum density for Van Stockum space-time using double index complexes and in the framework teleparallel gravity, we used the energy-momentum complexes of Einstein, Bergmann-Thomson and Landau-Lifshitz. In our analysis, we sustained that general relativity and teleparallel gravity are equivalent theories of space-time under consideration. For space-time under consideration, we have shown that different complexes of energy-momentum density do not provide the same results neither in general relativity nor in teleparallel gravity.