Abstract

We consider spacetimes satisfying the vacuum Einstein equations with impulsive gravitational waves without symmetry assumptions. These are spacetimes such that the Riemann curvature tensor has a delta singularity across a null hypersurface. We prove local existence and uniqueness for the characteristic initial value problem with initial data that has a delta singularity in the curvature tensor. A precise description of the propagation of singularity is also given. The proof introduces a new type of energy estimates for the vacuum Einstein equations, allowing the $L^2$ norm of some components of the curvature tensor to be infinite. The new estimate allows us to prove local existence for a general class of initial data which is non-regular along a null direction. We will discuss extensions of this theorem, which can be applied to understand colliding impulsive gravitational waves and the formation of trapped surface. This is joint work with I. Rodnianski.