Gravitational energy of rotating black holes

TL;DR

This paper derives a localized gravitational energy expression for Kerr black holes within teleparallel gravity, matching known results and providing new insights into energy distribution near black hole horizons.

Contribution

It presents a new method to compute localized gravitational energy for Kerr black holes in teleparallel gravity, aligning with previous approaches and extending to all angular momentum values.

Findings

Energy inside a surface of constant radius matches Brown and York's results.

Energy within the outer horizon approximates twice the irreducible mass.

The approach is consistent for any angular momentum value.

Abstract

In the teleparallel equivalent of general relativity the energy density of asymptotically flat gravitational fields can be naturaly defined as a scalar density restricted to a three-dimensional spacelike hypersurface $\Sigma$. Integration over the whole $\Sigma$ yields the standard ADM energy. After establishing the reference space with zero gravitational energy we obtain the expression of the localized energy for a Kerr black hole. The expression of the energy inside a surface of constant radius can be explicitly calculated in the limit of small $a$, the specific angular momentum. Such expression turns out to be exactly the same as the one obtained by means of the method preposed recently by Brown and York. We also calculate the energy contained within the outer horizon of the black hole for {\it any} value of $a$. The result is practically indistinguishable from $E=2M_{ir}$, where $M_{ir}$ is the irreducible mass of the black hole.