The Kerr-Schild ansatz revised

TL;DR

This paper revises the Kerr-Schild ansatz by treating Kerr-Schild metrics as exact linear perturbations of Minkowski spacetime, deriving the Kerr solution through perturbative methods and relaxing previous assumptions.

Contribution

It introduces a new perspective by considering Kerr-Schild metrics as linear perturbations, providing an alternative derivation of the Kerr solution without assuming shearfree congruences initially.

Findings

Kerr-Schild metrics can be viewed as exact linear perturbations.

The congruence must be geodesic and shearfree as a consequence of the equations.

An alternative derivation of the Kerr solution is achieved.

Abstract

Kerr-Schild metrics have been introduced as a linear superposition of the flat spacetime metric and a squared null vector field, say $\boldsymbol{k}$, multiplied by some scalar function, say $H$. The basic assumption which led to Kerr solution was that $\boldsymbol{k}$ be both geodesic and shearfree. This condition is relaxed here and Kerr-Schild ansatz is revised by treating Kerr-Schild metrics as {\it exact linear perturbations} of Minkowski spacetime. The scalar function $H$ is taken as the perturbing function, so that Einstein's field equations are solved order by order in powers of $H$. It turns out that the congruence must be geodesic and shearfree as a consequence of third and second order equations, leading to an alternative derivation of Kerr solution.