On the completeness of impulsive gravitational wave space-times

TL;DR

This paper investigates a class of impulsive gravitational wave space-times, extending impulsive pp-waves, and proves their geodesic completeness when the Riemannian component is complete.

Contribution

It establishes a completeness result for impulsive gravitational wave space-times with a complete Riemannian base, generalizing previous models.

Findings

Proves geodesic completeness for a broad class of impulsive wave space-times.

Extends the understanding of impulsive pp-waves to more general geometries.

Shows that completeness depends on the completeness of the Riemannian part.

Abstract

We consider a class of impulsive gravitational wave space-times, which generalize impulsive pp-waves. They are of the form $M=N\times\mathbb{R}^2_1$, where $(N,h)$ is a Riemannian manifold of arbitrary dimension and $M$ carries the line element $ds^2=dh^2+ 2dudv+f(x)\delta(u)du^2$ with $dh^2$ the line element of $N$ and $\delta$ the Dirac measure. We prove a completeness result for such space-times $M$ with complete Riemannian part $N$.