Kerr-Schild ansatz in Einstein-Gauss-Bonnet gravity: An exact vacuum solution in five dimensions

TL;DR

This paper explores Kerr-Schild metrics in five-dimensional Einstein-Gauss-Bonnet gravity, deriving simplified expressions for the tensors involved and analyzing solutions to the field equations, especially in maximally symmetric spacetimes.

Contribution

It provides new analytical expressions for the Einstein-Gauss-Bonnet tensor in five dimensions and examines conditions for solutions to the field equations with a cosmological term.

Findings

Simplified the Gauss-Bonnet tensor for Kerr-Schild metrics in five dimensions.

Derived explicit solutions for the trace of the Einstein-Gauss-Bonnet equations.

Found that solutions generally do not satisfy all field equations unless specific conditions are met.

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 different than 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.