Conformal Isometries and Curvature Collineations of an Impulsive Plane Wave: a distributional approach

TL;DR

This paper extends the concept of Lie derivatives to distribution-valued tensors to analyze symmetries of impulsive plane wave spacetimes with low regularity, revealing their conformal algebra and curvature collineations.

Contribution

It develops a distributional approach to Lie derivatives, enabling the study of geometric symmetries in low-regularity spacetimes, and characterizes the conformal algebra and curvature collineations of impulsive plane waves.

Findings

Conformal algebra of the impulsive plane wave has maximal dimension 7.

All conformal algebra vector fields are special curvature collineations.

Explicit form of conformal Killing vectors in distributional Brinkmann form.

Abstract

By extending the notion of Lie derivative to distribution-valued tensor fields of order $m$, Lie derivatives with respect to $C^k$ vector fields, $k\geqslant m+1$, can be shown to be well defined. Geometric symmetries, definable in terms of these Lie derivatives, can then be considered. In particular, for spacetimes of low regularity, geometric symmetries generated by vector fields of regularity as low as $C^1$ are definable. We find that the conformal algebra of the impulsive plane wave spacetime (with $+$ polarization) described by the continuous Baldwin-Jeffery-Rosen form of the metric has the maximal dimension seven. The curvature of this metric is well defined as a distribution and since this spacetime is Ricci-flat, we show that all the vector fields in the conformal algebra are special curvature collineations. We find also the general form of these last ones. Finally, we find the conformal Killing vector fields of this spacetime when using the distributional Brinkmann form of the metric.