Kerr-Schild Ansatz in Lovelock Gravity

Abstract

We analyze the field equations of Lovelock gravity for the Kerr-Schild metric ansatz, $g_{ab}={\bar{g}}_{ab}+\lambda k_a{k}_b$, with background metric ${\bar{g}}_{ab}$, background null vector $k^a$ and free parameter $\lambda$. Focusing initially on the Gauss-Bonnet case, we find a simple extension of the Einstein gravity results only in theories having a unique constant curvature vacuum. The field equations then reduce to a single equation at order ${\lambda }^2$. More general Gauss-Bonnet theories having two distinct vacua yield a pair of equations, at orders $\lambda$ and ${\lambda }^2$ that are not obviously compatible. Our results for higher order Lovelock theories are less complete, but lead us to expect a similar conclusion. Namely, the field equations for Kerr-Schild metrics will reduce to a single equation of order ${\lambda }^p$ for unique vacuum theories of order $p$ in the curvature, while non-unique vacuum theories give rise to a set of potentially incompatible equations at orders ${\lambda }^n$ with $1\le n\le p$. An examination of known static black hole solutions also supports this conclusion.

14 pages; v2 reference added

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• Slowly rotating black holes in Quasi-topological gravity
• Generalized Lense--Thirring metrics: higher-curvature corrections and solutions with matter