Local Propagation of Impulsive Gravitational Waves
Jonathan Luk, Igor Rodnianski
TL;DR
This work develops a rigorous framework for impulsive gravitational waves as solutions to the vacuum Einstein equations with delta-type curvature on a null hypersurface, constructed via a characteristic initial value problem on a double-null foliation. A key innovation is renormalizing curvature components to derive $L^2$ energy estimates without relying on the singular $\alpha$ component, enabling local existence and uniqueness for low-regularity data and precise description of how the delta singularity propagates along a null surface while the spacetime remains smooth elsewhere. The authors carefully design initial data with compact impulse and provide a detailed approximation scheme, proving convergence of smooth spacetimes to a limiting $C^0$ solution that exhibits impulsive behavior. The results extend the theory of impulsive gravitational waves beyond symmetric or plane-wave settings and offer a robust approach to non-regular characteristic data in general relativity.
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.