Geodesics for impulsive gravitational waves and the multiplication of distributions

TL;DR

This paper investigates particle trajectories in impulsive gravitational waves, resolving ambiguities in distributional profiles using covariant conditions, and aligning with Colombeau's multiplication of distributions.

Contribution

It introduces a covariant method to resolve distributional ambiguities in impulsive gravitational wave models, consistent with Colombeau's multiplication.

Findings

Ambiguity at $ heta(0)$ can be resolved by covariant constancy.

Method aligns with Colombeau's multiplication of distributions.

Provides a consistent framework for particle trajectories in impulsive waves.

Abstract

We consider particle trajectories in the gravitational field of an impulsive pp-wave. Due to the distributional character of the wave profile one inevitably encounters an ambiguous point value $\theta(0)$. We show that this ambiguity may be resolved by imposing covariant constancy of the square of the tangent. Our result is consistent with Colombeau's multiplication of distributions.