Charged Axially Symmetric Solution and Energy in Teleparallel Theory Equivalent to General Relativity

TL;DR

This paper derives an exact charged, axially symmetric solution in teleparallel gravity equivalent to general relativity, analyzes its singularities, and computes its energy content, showing the regularized energy expression yields consistent results.

Contribution

It introduces a new coordinate transformation to simplify the structure function and computes the energy in teleparallel gravity, providing a novel exact solution with detailed energy analysis.

Findings

The structure function can be factorized for easier root identification.

The energy calculated depends on the radial coordinate without regularization.

Regularized energy expression yields a consistent total energy value.

Abstract

An exact charged solution with axial symmetry is obtained in the teleparallel equivalent of general relativity (TEGR). The associated metric has the structure function $G(\xi)=1-{\xi}^2-2mA{\xi}^3-q^2A^2{\xi}^4$. The fourth order nature of the structure function can make calculations cumbersome. Using a coordinate transformation we get a tetrad whose metric has the structure function in a factorisable form $(1-{\xi}^2)(1+r_{+}A\xi)(1+r_{-}A\xi)$ with $r_{\pm}$ as the horizons of Reissner-Nordstr$\ddot{o}$m space-time. This new form has the advantage that its roots are now trivial to write down. Then, we study the singularities of this space-time. Using another coordinate transformation, we obtain a tetrad field. Its associated metric yields the Reissner-Nordstr$\ddot{o}$m black hole. In Calculating the energy content of this tetrad field using the gravitational energy-momentum, we find that the resulting form depends on the radial coordinate! Using the regularized expression of the gravitational energy-momentum in the teleparallel equivalent of general relativity we get a consistent value for the energy.