Geodesics in nonexpanding impulsive gravitational waves with $\Lambda$, Part I

TL;DR

This paper rigorously analyzes geodesics in nonexpanding impulsive gravitational waves within (anti-)de Sitter spaces, establishing existence, uniqueness, and explicit forms using advanced mathematical techniques.

Contribution

It provides a rigorous proof of geodesic behavior in these spacetimes, confirming previous heuristic results with a formal mathematical framework.

Findings

Existence and uniqueness of geodesics crossing the impulsive wave

Explicit formulas for geodesics in the studied spacetime

Completeness of the geodesic structure in these models

Abstract

We investigate the geodesics in the entire class of nonexpanding impulsive gravitational waves propagating in an (anti-)de Sitter universe using the distributional form of the metric. Employing a 5-dimensional embedding formalism and a general regularisation technique we prove existence and uniqueness of geodesics crossing the wave impulse leading to a completeness result. We also derive the explicit form of the geodesics thereby confirming previous results derived in a heuristic approach.