# Gravitational Impulsive Waves

# Gravitational Impulsive Waves

We consider spacetimes satisfying the vacuum Einstein equations with gravitational impulsive waves without symmetry assumptions. These are spacetimes such that some components of the Riemann curvature tensor have delta singularities on a null hypersurface. We prove local existence for the characteristic initial value problem with initial data that has a delta singularity in some components of the curvature tensor. We also prove local existence in the case where two gravitational impulsive waves collide. The proof introduces a new type of energy estimates for the vacuum Einstein equations, allowing the $L2$ 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 rough in one direction. It also gives an improvement to the theorem of Christodoulou on the formation of trapped surface. This is joint work with I. Rodnianski.