Extended Kerr-Schild spacetimes: general properties and some explicit examples

We study extended Kerr-Schild (xKS) spacetimes, i.e. extensions of the Kerr-Schild (KS) ansatz where, in addition to the null KS vector, a spacelike vector field appears in the metric. In contrast to the KS case, we obtain only a necessary condition under which the KS vector is geodetic. However, it is shown that this condition becomes sufficient if we appropriately restrict the geometry of the null KS and spacelike vector. It turns out that xKS spacetimes with a geodetic KS vector are of Weyl type I and the KS vector has the same optical properties in the full and background spacetimes. In the case of Kundt xKS spacetimes, the compatible Weyl types are further restricted and a few examples of such metrics are provided. The relation of pp-waves to the class of Kundt xKS spacetimes is briefly discussed. We also show that the CCLP black hole, as an example of an expanding xKS spacetime, is of Weyl type I-i, which specializes to type D in the uncharged or non-rotating limit, and that its optical matrix satisfies the optical constraint.