Kerr–Schild perturbations

A generalised Kerr–Schild perturbation has the form ${\displaystyle h_{ab}=Vl_{a}l_{b}}$, where ${\displaystyle V}$ is a scalar and ${\displaystyle l_{a}}$ is a null vector with respect to the background spacetime.[2] It can be shown that any perturbation of this form will only appear quadratically in the Einstein equations, and only linearly if the condition ${\displaystyle l^{a}\nabla _{a}l_{b}=\phi l_{b}}$, where ${\displaystyle \phi }$ is a scalar, is imposed. This condition is equivalent to requiring that the orbits of ${\displaystyle l^{a}}$ are geodesics.[2]

While the form of the perturbation may appear very restrictive, there are several black hole metrics which can be written in Kerr–Schild form, such as Schwarzschild (stationary black hole), Kerr (rotating), Reissner–Nordström (charged) and Kerr–Newman (both charged and rotating).[2][3]