Gravitational energy of a magnetized Schwarzschild black hole - a teleparallel approach

TL;DR

This paper calculates the gravitational energy distribution of a Schwarzschild black hole in a magnetic field using teleparallel gravity, revealing magnetic field effects on energy and confirming physical consistency across different limits.

Contribution

It introduces a teleparallel gravity approach to evaluate gravitational energy of a magnetized black hole, extending previous methods to include magnetic field effects.

Findings

Magnetic field increases gravitational energy compared to Schwarzschild case.

Energy calculations are consistent in weak field and no black hole limits.

Results align with physical expectations across all scenarios.

Abstract

We investigate the distribution of gravitational energy on the spacetime of a Schwarzschild black hole immersed in a cosmic magnetic field. This is done in the context of the {\it Teleparallel Equivalent of General Relativity}, which is an alternative geometrical formulation of General Relativity, where gravity is describe by a spacetime endowed with torsion, rather than curvature, with the fundamental field variables being tetrads. We calculate the energy enclosed by a two-surface of constant radius - in particular, the energy enclosed by the event horizon of the black hole. In this case we find that the magnetic field has the effect of increasing the gravitational energy as compared to the vacuum Schwarzschild case. We also compute the energy (i) in the weak magnetic field limit, (ii) in the limit of vanishing magnetic field, and (iii) in the absence of the black hole. In all cases our results are consistent with what should be expected on physical grounds.