Penrose quasi-local energy and Kerr-Schild metrics

TL;DR

This paper proposes a modified Penrose quasi-local energy for Kerr-Schild spacetimes, connecting surface integrals to Einstein tensor volume integrals and establishing a twistor correspondence in special cases.

Contribution

It introduces a new modification of the Penrose quasi-local energy tailored for Kerr-Schild spacetimes, utilizing their Minkowski background structure.

Findings

Modified surface integral reduces to Einstein tensor volume integral.

Established a twistor correspondence for special Kerr-Schild surfaces.

Applicable to Kerr black holes and gravitational wave solutions.

Abstract

Specialising to the case of Kerr-Schild spacetimes, which include the Kerr black hole and gravitational wave solutions, we propose a modification of the Penrose quasi-local energy. The modification relies on the existence of a natural Minkowski background for Kerr-Schild spacetimes. We find that the modified surface integral reduces to a volume integral of the Einstein tensor, which has been proposed previously as an appropriate definition for quasi-local energy for Kerr-Schild backgrounds. Furthermore, in the special case that the Kerr-Schild null vector is normal to the surface of interest, we construct a 1-1 map between the 2-surface twistors in the Kerr-Schild background and Minkowski twistors projected onto the surface.