Kerr–Schild Symmetries

Abstract

We study continuous groups of generalized Kerr-Schild transformations and the vector fields that generate them in any n-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixed. The properties of these Lie algebras are briefly analyzed and we show that they are generically finite-dimensional but that they may have infinite dimension in some relevant situations. The most general vector fields of the above type are explicitly constructed for the following cases: any two-dimensional metric, the general spherically symmetric metric and deformation direction, and the flat metric with parallel or cylindrical deformation directions.

On Kerr-Schild spacetimes in higher dimensions

“…The relation to our work will be discussed elsewhere. Let us also recall that properties of KS transformations in arbitrary dimensions have been studied in [19]. This does not overlap significantly with our contribution.…”

Note on trapped surfaces in the Vaidya solution

“…On the other hand we know from the companion paper [10], using an argument based on the interplay between the Kerr-Schild vector field [12] and the trapped surfaces, that it is impossible to construct a closed trapped surface entering the flat region if…”

Spacetime Ehlers group: transformation law for the Weyl tensor

“…With this expression at hand, the covariant Laplacian of F αβ follows from (15). The result takes a simpler form if the Riemann tensor is decomposed in terms of the Ricci tensor and the Weyl tensor C αβγδ .…”