Abstrakt
We investigate geodesic completeness in the full family of pp-wave (or Brinkmann) spacetimes in their extended and impulsive forms. This class of geometries contains the recently studied gyratonic pp-waves, modelling the exterior field of a spinning beam of null particles, as well as N-fronted waves with parallel rays, which generalize classical pp-waves by allowing for a general wave surface.
The problem of geodesic completeness reduces to the question of completeness of trajectories on a Riemannian manifold under an external force field. Building upon respective recent results, we derive completeness criteria in terms of the spatial asymptotics of the profile function in the extended case.
In the impulsive case, we use a fixed-point argument to show that, irrespective of the behaviour of the profile function, all geometries in the class are complete.