Generalized Kerr-Schild gauge

TL;DR

This work extends the Kerr-Schild gauge to non-null deformations by introducing an inverse metric ansatz $g^{\mu\nu}=\bar{g}^{\mu\nu}+\xi\kappa^{2}A^{\mu}A^{\nu}$ with a rank-one deformation $h_{\mu\nu}=\kappa^{2}A_{\mu}A_{\nu}$. It demonstrates that the usual Fierz-Pauli condition $FP_{\mu\nu}=0$ is no longer sufficient to guarantee Ricci-flatness, and derives a condition for curvature invariance: $R=\bar{R}+\frac{\kappa^{2}}{2}(1+\xi)\bar{\nabla}_{\alpha}[A^{\beta}\bar{\nabla}_{\beta}A^{\alpha}]$. The authors identify three possibilities to maintain $R_{\mu\nu}=\bar{R}_{\mu\nu}$, with the physically meaningful geodesic case $A^{\alpha}\bar{\nabla}_{\alpha}A_{\beta}=0$ ensuring Ricci-flatness if the FP equation holds; a nonzero $\phi$ in the second case generally spoils this, unless $\phi=0$. The main result is a theorem: for geodesic deformations, the Ricci tensor is governed by the linear part, so Ricci-flat backgrounds remain Ricci-flat under the non-null Kerr-Schild transformation, significantly broadening the class of exact Ricci-flat metrics obtainable from a background.

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 geodesic of the background spacetime.