Generalized Kerr-Schild gauge

TL;DR

This paper extends the Kerr-Schild gauge to non-null vectors, demonstrating that such vectors can produce finite curvature expansions and establishing conditions for Ricci-flatness.

Contribution

It generalizes the Kerr-Schild gauge to non-null vectors and proves conditions under which the deformed metric remains Ricci flat.

Findings

Non-null vectors generate finite curvature expansions.

Deformation vector must be irrotational for Ricci-flatness.

The deformation vector is geodesic when irrotational.

Abstract

The Kerr-Schild gauge is generalized to the case that the vector generating the deformation is not null. Contrary to naive expectations, this vector generates a finite expansion for the curvature tensor. We prove a theorem on the conditions for the deformed metric being Ricci flat, namely that the deformation vector must be irrotational (then geodesic) in the background spacetime.