Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations

TL;DR

This paper proves the existence and uniqueness of spacetimes resulting from the nonlinear interaction of impulsive gravitational waves in vacuum Einstein equations, extending previous models and showing the persistence of smoothness away from wave interactions.

Contribution

It establishes a rigorous mathematical framework for the nonlinear interaction of impulsive gravitational waves with delta singularities in vacuum Einstein equations, extending prior results to more general initial data.

Findings

Existence and uniqueness of the resulting spacetime.

Propagation of curvature delta singularities along null hypersurfaces.

Spacetime remains smooth away from wave interactions.

Abstract

In this paper, we study the problem of the nonlinear interaction of impulsive gravitational waves for the Einstein vacuum equations. The problem is studied in the context of a characteristic initial value problem with data given on two null hypersurfaces and containing curvature delta singularities. We establish an existence and uniqueness result for the spacetime arising from such data and show that the resulting spacetime represents the interaction of two impulsive gravitational waves germinating from the initial singularities. In the spacetime, the curvature delta singularities propagate along 3-dimensional null hypersurfaces intersecting to the future of the data. To the past of the intersection, the spacetime can be thought of as containing two independent, non-interacting impulsive gravitational waves and the intersection represents the first instance of their nonlinear interaction. Our analysis extends to the region past their first interaction and shows that the spacetime still remains smooth away from the continuing propagating individual waves. The construction of these spacetimes are motivated in part by the celebrated explicit solutions of Khan-Penrose and Szekeres. The approach of this paper can be applied to an even larger class of characteristic data and in particular implies an extension of the theorem on formation of trapped surfaces by Christodoulou and Klainerman-Rodnianski, allowing non-trivial data on the initial incoming hypersurface.