Algebraic and optical properties of generalized Kerr-Schild spacetimes in arbitrary dimensions

We study the class of generalized Kerr-Schild (GKS) spacetimes in dimensions $n\geq 3$ and analyze their geometric and algebraic properties in a completely theory-independent setting. First, considering the case of a general null vector $\mathbf{k}$ defined by the GKS metric, we obtain the conditions under which it is geodesic. Assuming $\mathbf{k}$ to be geodesic for the remainder of the paper, we examine the alignment properties of the curvature tensors, namely the Ricci and Weyl tensors. We show that the algebraic types of the curvatures of the full (GKS) geometry are constrained by those of the respective background curvatures, thereby listing all kinematically allowed combinations of the algebraic types for the background and the full geometry. A notable aspect of these results is that, unlike the case of Kerr-Schild (KS) spacetimes, the Weyl types of the GKS spacetimes need not be type $II$ or more special. Then, focusing on the case of an expanding $\mathbf k$, we derive the conditions for it to satisfy the optical constraint, extending the previous results of KS spacetimes. We illustrate the general results using the example of (A)dS-Taub-NUT spacetimes in $n=4$, where we also comment on their KS double copy from a GKS perspective. Finally, as an application of our general results, we obtain the full family of GKS spacetimes with a geodesic, expanding, twistfree, and shearfree $\mathbf k$, satisfying the vacuum Einstein equations, and identify it with a subset of the higher-dimensional vacuum Robinson-Trautman solutions. In passing, we also determine the subcase of these solutions that manifests the KS double copy.