Reissner-Nordstr\"om Spacetime in the Tetrad Theory of Gravitation

TL;DR

This paper presents two exact spherically symmetric solutions in tetrad gravity that reproduce the Reissner-Nordström metric, analyzes their energy content, and verifies their consistency with Møller's energy-momentum complex conditions.

Contribution

It introduces two new classes of solutions in tetrad gravity that correspond to the Reissner-Nordström spacetime and examines their energy properties and Lorentz transformation behavior.

Findings

Both solutions reproduce the Reissner-Nordström metric.

The energy content depends on the arbitrary functions and parameters.

The second solution satisfies Møller's energy-momentum condition.

Abstract

We give two classes of spherically symmetric exact solutions of the couple gravitational and electromagnetic fields with charged source in the tetrad theory of gravitation. The first solution depends on an arbitrary function $H({R},t)$. The second solution depends on a constant parameter $\eta$. These solutions reproduce the same metric, i.e., the Reissner--Nordstr$\ddot{o}$m metric. If the arbitrary function which characterizes the first solution and the arbitrary constant of the second solution are set to be zero, then the two exact solutions will coincide with each other. We then calculate the energy content associated with these analytic solutions using the superpotential method. In particular, we examine whether these solutions meet the condition which M{\o}ller required for a consistent energy-momentum complex: Namely, we check whether the total four-momentum of an isolated system behaves as a four-vector under Lorentz transformations. It is then found that the arbitrary function should decrease faster than $1/\sqrt{R}$ for $R\to \infty$. It is also shown that the second exact solution meets the M{\o}ller's condition.