# Teleparallel Version of the Stationary Axisymmetric Solutions and their Energy Contents

## TL;DR

This paper develops the teleparallel equivalent of stationary axisymmetric solutions, analyzing torsion components, energy-momentum distribution, and comparing results with general relativity, revealing additional contributions in teleparallel theory.

## Contribution

It provides the teleparallel formulation of stationary axisymmetric solutions, including torsion analysis and energy distribution, and compares these with general relativity results.

## Key findings

- Axial-vector torsion has components only along 2 and z directions.
- Energy and momentum densities in teleparallel theory match GR plus an extra term.
- Special cases of solutions are discussed with specific properties.

## Abstract

This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along $\rho$ and $z$ directions. The three possibilities of the axial vector depending on the metric function $B$ are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of M$\ddot{o}$ller prescription to find the energy-momentum distribution of the solutions. It is interesting to note that (for $\lambda=1$) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions.