On the geometry of impulsive gravitational waves

TL;DR

This paper models impulsive gravitational waves using Colombeau's generalized functions, enabling rigorous handling of distributional products and deriving distributional limits, with applications to impulsive plane waves and particle motion.

Contribution

It introduces a distributional framework for impulsive gravitational waves using Colombeau's theory, providing regularization-independent results and comparing with continuous metric approaches.

Findings

Derived distributional limits for impulsive gravitational waves

Handled products of distributions in geodesic equations rigorously

Compared impulsive plane wave results with continuous metric solutions

Abstract

We describe impulsive gravitational pp-waves entirely in the distributional picture. Applying Colombeau's nonlinear framework of generalized functions we handle the formally ill-defined products of distributions which enter the geodesic as well as the geodesic deviation equation. Using a universal regularization procedure we explicitly derive regularization independent distributional limits. In the special case of impulsive plane waves we compare our results with the particle motion derived from the continuous form of the metric.