An Extended Kerr-Schild Ansatz

TL;DR

This paper extends the Kerr-Schild ansatz by including a spacelike vector, simplifying vacuum Einstein equations and analyzing conditions for their truncation, with applications to electrovac solutions.

Contribution

It introduces a generalized Kerr-Schild ansatz with a spacelike vector and derives conditions for vacuum equations to truncate at quadratic order.

Findings

Many Kerr-Schild simplifications persist in the extended ansatz.

Derived sufficient conditions for vacuum equations truncation.

Extended analysis to electrovac solutions with gauge fields.

Abstract

We present an analysis of the vacuum Einstein equations for a recently proposed extension of the Kerr-Schild ansatz that includes a spacelike vector field as well as the usual Kerr-Schild null vector. We show that many, although not all, of the simplifications that occur in the Kerr-Schild case continue to hold for the extended ansatz. In particular, we find a simple set of sufficient conditions on the vectors such that the vacuum field equations truncate beyond quadratic order in an expansion around a general vacuum background solution. We extend our analysis to the electrovac case with a related ansatz for the gauge field.