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Kerr-Schild Ansatz in Lovelock Gravity

We analyze the field equations of Lovelock gravity for the Kerr-Schild metric ansatz, $g_{ab}=\bar g_{ab} +λk_ak_b$, with background metric $\bar g_{ab}$, background null vector $k^a$ and free parameter $λ$. 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 $λ^2$. More general Gauss-Bonnet theories having two distinct vacua yield a pair of equations, at orders $λ$ and $λ^2$ 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 $λ^p$ 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 $λ^n$ with $1\le n \le p$. An examination of known static black hole solutions also supports this conclusion.