Cambridge Journal of Mathematics

Volume 5 (2017)

Number 4

Nonlinear interaction of impulsive gravitational waves for the vacuum Einstein equations

Pages: 435 – 570



Jonathan Luk (Department of Mathematics, Stanford University, Stanford, California, U.S.A.)

Igor Rodnianski (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)


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.

J. Luk was supported by the NSF Postdoctoral Fellowship DMS-1204493.

I. Rodnianski was supported by the NSF grant DMS-1001500 and the FRG grant DMS-1065710.

Received 9 February 2017

Published 8 January 2018