Energy and angular momentum of charged rotating black holes

TL;DR

This paper demonstrates that various pseudotensors yield consistent energy, momentum, and angular momentum distributions for Kerr-Schild metrics, providing new insights into energy localization for charged rotating black holes without approximation.

Contribution

It generalizes previous results by showing pseudotensor agreement for a broader class of metrics and computes these distributions explicitly for Kerr-Newman and Bonnor-Vaidya black holes.

Findings

Pseudotensors agree for Kerr-Schild metrics in energy and momentum calculations.

Energy distribution for Kerr-Newman black holes matches expected physical sharing.

Results align with Penrose quasi-local mass for Reissner-Nordström and Bonnor-Vaidya metrics.

Abstract

We show that the pseudotensors of Einstein, Tolman, Landau and Lifshitz, Papapetrou, and Weinberg (ETLLPW) give the same distributions of energy, linear momentum and angular momentum, for any Kerr-Schild metric. This result generalizes a previous work by G\"urses and G\"ursey that dealt only with the pseudotensors of Einstein and Landau and Lifshitz. We compute these distributions for the Kerr-Newman and Bonnor-Vaidya metrics and find reasonable results. All calculations are performed without any approximation in Kerr-Schild Cartesian coordinates. For the Reissner-Nordstr\"{o}m metric these definitions give the same result as the Penrose quasi-local mass. For the Kerr black hole the entire energy is confined to its interior whereas for the Kerr-Newman black hole, as expected, the energy is shared by its interior as well as exterior. The total energy and angular momentum of the Kerr-Newman black hole are $M$ and $ M a$, respectively ($M$ is the mass parameter and $a$ is the rotation parameter). The energy distribution for the Bonnor-Vaidya metric is the same as the Penrose quasi-local mass obtained by Tod.