Some consequences of the Kerr-Schild gauge

TL;DR

This paper explores generalized Kerr-Schild deformations of arbitrary spacetimes, deriving exact relations between their Ricci tensors and introducing new examples beyond flat backgrounds, with potential implications for gravitational theories.

Contribution

It provides an exact formula relating Ricci tensors of deformed and background metrics and introduces new solutions for non-flat backgrounds.

Findings

Derived exact relation between Ricci tensors of deformed and background metrics.

Identified a new term in the Ricci tensor relation beyond the Fierz-Pauli operator.

Presented novel examples of Kerr-Schild deformations on non-flat backgrounds.

Abstract

Generalized Kerr-Schild nilpotent deformations of an arbitrary background spacetime $\overline{g}_{\m\n}(x)$ are considered. Those are of the form $\overline{g}_{\m\n}+l_\m l_\n$ with $l^2=0$. The relationship between the Ricci tensor of the background metric and the Ricci tensor of the deformed metric is found exactly. It consists of two terms, one is essentially the Fierz-Pauli operator, and the other is new. When the background is flat, the Kerr-Schild family is recovered. Novel examples for more general backgrounds (even including some simple sources) are discussed.