How to determine whether a metric meets the Kerr-Schild form?

Question

A Kerr-Schild metric can be expressed as $$g_{\mu\nu}=\eta_{\mu\nu}+\phi k_\mu k_\nu,$$ where $\eta_{\mu\nu}$ is Minkowski metric, $\phi$ is a scalar function and $k_\mu$ is a (co)vector field which is null with respect to both the Minkowski and full metric, i.e. $g_{\mu\nu}k^\mu k^\nu=\eta_{\mu\nu}k^\mu k^\nu=0$. There are many solutions meet Kerr-Schild form, such as Schwarzschild solution, Kerr solution, pp-waves and so on. But it is difficult to determine whether a metric meets the Kerr-Schild form. Even for Schwarzschild solution, the Kerr-Schild form must be obtained through coordinate transformation in 4.2.1 of this paper. So is there a general way to determine whether a metric meets the Kerr-Schild form?

Answer 1

A necessary condition for a metric of a vacuum solution to admit Kerr–Schild form is the existence of double principal null direction, so the spacetime must be algebraically special in the sense of Petrov's classification. This is proven in the paper

• Gürses, M., & Gürsey, F. (1975). Lorentz covariant treatment of the Kerr–Schild geometry. Journal of Mathematical Physics, 16(12), 2385-2390, doi:10.1063/1.522480.

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A set of necessary and sufficient conditions for the metric to be of Kerr–Schild type formulated in terms of Newman–Penrose spin coefficients could be found in a paper by Debever

• Debever, R. (1974). Sur une classe d'espaces Lorentziens. Bulletins de l'Académie Royale de Belgique, 60(1), 998-1011, link with OA pdf.

Abstract (Google translated from French)

We determine necessary and sufficient conditions for a Lorentzian manifold $(V_4,g)$ to have a metric $g$ equivalent to $$η + l ⊗ l$$ where $η$ is the Minkowskian metric and $l$ is a field of real isotropic vectors. The trajectories of $l$ are moreover geodesic, without distortion and with divergence for the metric $η$ and consequently for the metric $g$.

Metrics of this type, solutions of vacuum equations have been studied by R. Kerr and A. Schild [$1$].