# Geodesics in nonexpanding impulsive gravitational waves with $\Lambda$, II

## TL;DR

This paper rigorously analyzes all geodesics in nonexpanding impulsive gravitational waves within (anti-)de Sitter universes, establishing global existence, uniqueness, and geodesic completeness using advanced distributional methods.

## Contribution

It extends the regularization approach to a full nonlinear distributional analysis, proving geodesic completeness for low regularity impulsive wave spacetimes.

## Key findings

- Proves global existence and uniqueness of geodesics crossing impulsive waves.
- Establishes geodesic completeness in low regularity spacetimes.
- Provides a rigorous foundation for the physical equivalence of continuous and distributional metrics.

## Abstract

We investigate all geodesics in the entire class of nonexpanding impulsive gravitational waves propagating in an (anti-)de Sitter universe using the distributional metric. We extend the regularization approach of part I, [SSLP16] to a full nonlinear distributional analysis within the geometric theory of generalized functions. We prove global existence and uniqueness of geodesics that cross the impulsive wave and hence geodesic completeness in full generality for this class of low regularity spacetimes. This, in particular, prepares the ground for a mathematically rigorous account on the 'physical equivalence' of the continuous with the distributional `from' of the metric.