Kerr-Schild metrics revisited I. The ground state

TL;DR

This paper revisits Kerr-Schild metrics, proving that the generic shearing case does not smoothly limit to shear-free solutions and identifying a specific solution within the shearing class.

Contribution

It establishes a theorem that the shearing and shear-free Kerr-Schild metrics form distinct classes without a smooth transition, and identifies a particular solution in the shearing class.

Findings

Shearing Kerr-Schild metrics do not have a shear-free limit.

A specific Kóta-Perjés metric is a solution within the shearing class.

The shear-free subclass is not contained as a smooth limit in the generic shearing case.

Abstract

The Kerr-Schild pencil of metrics $g_{ab}+\La l_al_b$ is investigated in the generic case when it maps an arbitrary vacuum space-time with metric $g_{ab}$ to a vacuum space-time. The theorem is proved that this generic case, with the field $l$ shearing, does not contain the shear-free subclass as a smooth limit. It is shown that one of the K\'ota-Perj\'es metrics is a solution in the shearing class.