Teleparallel Version of the Levi-Civita Vacuum Solutions and their Energy Contents

TL;DR

This paper derives the teleparallel equivalent of the Levi-Civita vacuum solutions, analyzes torsion components, and compares energy-momentum distributions in teleparallel gravity and General Relativity using various prescriptions.

Contribution

It presents the teleparallel version of Levi-Civita solutions and compares energy contents with GR, highlighting the Möller prescription's independence from coupling constants.

Findings

Energy in teleparallel equals GR in Möller prescription.

Momentum remains constant in both theories.

Energy discrepancies exist in other prescriptions.

Abstract

In this paper, we find the teleparallel version of the Levi-Civita metric and obtain tetrad and the torsion fields. The tensor, vector and the axial-vector parts of the torsion tensor are evaluated. It is found that the vector part lies along the radial direction only while the axial-vector vanishes everywhere because the metric is diagonal. Further, we use the teleparallel version of M$\ddot{o}$ller, Einstein, Landau-Lifshitz and Bergmann-Thomson prescriptions to find the energy-momentum distribution of this metric and compare the results with those already found in General Relativity. It is worth mentioning here that momentum is constant in both the theories for all the prescriptions. The energy in teleparallel theory is equal to the corresponding energy in GR only in M$\ddot{o}$ller prescription for the remaining prescriptions, the energy do not agree in both theories. We also conclude that M$\ddot{o}$ller's energy-momentum distribution is independent of the coupling constant $\lambda$ in the teleparallel theory.