Kerr-Schild Ansatz in Lovelock Gravity

We analyze the field equations of Lovelock gravity for the Kerr-Schild metric ansatz, $g_{ab}=\bar g_{ab} +\lambda k_ak_b$, 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.