Kerr-Schild metrics revisited II. The complete vacuum solution

TL;DR

This paper derives the complete vacuum solutions of Einstein's equations within the Kerr-Schild class, extending previous flat-space solutions to include curved background metrics, thereby broadening the understanding of exact solutions in general relativity.

Contribution

It generalizes the Kerr-Schild solution framework from flat to curved background metrics, providing a comprehensive solution to Einstein's equations in this broader context.

Findings

Derived the full vacuum Kerr-Schild solutions with curved backgrounds

Extended the Kerr theorem to non-flat metrics

Provided explicit forms of the solutions in the curved case

Abstract

The complete solution of Einstein's gravitational equations with a vacuum-vacuum Kerr-Schild pencil of metrics $g_{ab}+V l_al_b$ is obtained. Our result generalizes the solution of the Kerr-Schild problem with a flat metric $g_{ab}$ (represented by the Kerr theorem) to the case when $g_{ab}$ is the metric of a curved space-time.