Completeness of general pp-wave spacetimes and their impulsive limit

TL;DR

This paper studies the geodesic completeness of pp-wave spacetimes, including impulsive and gyratonic types, deriving criteria for completeness and showing all impulsive geometries are complete regardless of profile behavior.

Contribution

It provides new completeness criteria for extended pp-waves and proves all impulsive pp-waves are geodesically complete, regardless of profile functions.

Findings

Completeness depends on spatial asymptotics of the profile function for extended pp-waves.

All impulsive pp-wave geometries are complete, independent of profile function behavior.

Derived explicit criteria linking geometry asymptotics to geodesic completeness.

Abstract

We investigate geodesic completeness in the full family of pp-wave or Brinkmann spacetimes in their extended as well as in their impulsive form. 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 NPWs, which generalise 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.