Linear Einstein equations and Kerr-Schild maps

Abstract

We prove that given a solution of the Einstein equations $g_{ab}$ for the matter field $T_{ab}$, an autoparallel null vector field $l^a$ and a solution $(l_a{l}_c,{\mathcal{T}}_{ac})$ of the linearized Einstein equation on the given background, the Kerr-Schild metric $g_{ac}+\lambda l_a{l}_c$ ($\lambda$ arbitrary constant) is an exact solution of the Einstein equation for the energy-momentum tensor $T_{ac}+\lambda {\mathcal{T}}_{ac}+{\lambda }^2{l}_{(a}{\mathcal{T}}_{c)b}{l}^b$. The mixed form of the Einstein equation for Kerr-Schild metrics with autoparallel null congruence is also linear. Some more technical conditions hold when the null congruence is not autoparallel. These results generalize previous theorems for vacuum due to Xanthopoulos and for flat seed space-time due to Gürses and Gürsey.

9 pages, accepted by Class. Quant. Grav

References in corpus (4)

• Kerr-Schild metrics revisited I. The ground state
• Kerr-Schild metrics revisited II. The complete vacuum solution
• Vacuum Kerr-Schild metrics generated by nontwisting congruences
• Solution of the vacuum Kerr-Schild problem

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• Approach of background metric expansion to a new metric ansatz for gauged and ungauged Kaluza-Klein supergravity black holes
• Exact Kerr--Schild spacetimes from linearized kinetic gravity braiding
• K-essence sources of Kerr-Schild spacetimes
• Emergent Gravity from an Augmented Variational Principle