Local Propagation of Impulsive Gravitational Waves

TL;DR

This paper rigorously constructs impulsive gravitational wave solutions to Einstein's equations, demonstrating delta singularity propagation along characteristic hypersurfaces without symmetry assumptions, advancing mathematical understanding of such spacetime phenomena.

Contribution

It provides the first construction of compact impulsive gravitational waves without symmetry, using characteristic initial data with delta singularities, and extends to broader non-regular data.

Findings

Delta singularity propagates along characteristic hypersurface

Spacetime remains smooth away from the singularity

First construction of compact impulsive gravitational wave without symmetry

Abstract

In this paper, we initiate the rigorous mathematical study of the problem of impulsive gravitational spacetime waves. We construct such spacetimes as solutions to the characteristic initial value problem of the Einstein vacuum equations with a data curvature delta singularity. We show that in the resulting spacetime, the delta singularity propagates along a characteristic hypersurface, while away from that hypersurface the spacetime remains smooth. Unlike the known explicit examples of impulsive gravitational spacetimes, this work in particular provides the first construction of an impulsive gravitational wave of compact extent and does not require any symmetry assumptions. The arguments in the present paper also extend to the problem of existence and uniqueness of solutions to a larger class of non-regular characteristic data.