Penrose quasi-local energy and Kerr–Schild metrics

Specialising in 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. (paper)