A generalised Kerr–Schild perturbation has the form h a b = V l a l b , where V is a scalar and l a is a null vector with respect to the background spacetime. It can be shown that any perturbation of this form will only appear quadratically in the Einstein equations, and only linearly if the condition l a ∇ ∇ a l b = ϕ ϕ l b , where ϕ ϕ is a scalar, is imposed. This condition is equivalent to requiring that the orbits of l a are geodesics.