Kerr-Schild Symmetries

TL;DR

This paper investigates the structure of continuous symmetry groups generated by Kerr-Schild transformations in Lorentzian manifolds, characterizing their algebraic properties and explicit forms in various geometries.

Contribution

It provides an intrinsic characterization of Kerr-Schild symmetry vector fields and analyzes their Lie algebra structures across different manifold cases.

Findings

Kerr-Schild symmetry vector fields form a Lie algebra when the null direction is fixed.

These Lie algebras are generally finite-dimensional but can be infinite-dimensional in certain cases.

Explicit constructions are provided for 2D, spherically symmetric, and flat metrics with specific deformation directions.

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.