Vacuum Kerr-Schild metrics generated by nontwisting congruences

Abstract

The Kerr-Schild pencil of metrics $\tilde g_{ab}=g_{ab}+V l_al_b$, with $g_{ab}$ and $\tilde g_{ab}$ satisfying the vacuum Einstein equations, is investigated in the case when the null vector $l$ has vanishing twist. This class of Kerr-Schild metrics contains two solutions: the Kasner metric and a metric wich can be obtained from the Kasner metric by a complex coordinate transformation. Both are limiting cases of the K\'ota-Perj\'es metrics. The base space-time is a pp-wave.