Geodesics in nonexpanding impulsive gravitational waves with Λ, part I

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 five-dimensional embedding formalism and a general regularisation technique, we prove the existence and uniqueness of the 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. (paper)