Kerr–Schild ansatz in Einstein–Gauss–Bonnet gravity: an exact vacuum solution in five dimensions

Abstract

As is well known, Kerr–Schild metrics linearize the Einstein tensor. We shall see here that they also simplify the Gauss–Bonnet tensor, which turns out to be only quadratic in the arbitrary Kerr–Schild function f when the seed metric is maximally symmetric. This property allows us to give a simple analytical expression for its trace, when the seed metric is a five-dimensional maximally symmetric spacetime in spheroidal coordinates with arbitrary parameters a and b. We also write in a (fairly) simple form the full Einstein–Gauss–Bonnet tensor (with a cosmological term) when the seed metric is flat and the oblateness parameters are equal, a = b. Armed with these results we give in a compact form the solution of the trace of the Einstein–Gauss–Bonnet field equations with a cosmological term and a ≠ b. We then examine whether this solution for the trace does solve the remaining field equations. We find that it does not in general, unless the Gauss–Bonnet coupling is such that the field equations have a unique maximally symmetric solution.