Non-linear interaction of three impulsive gravitational waves for the Einstein equations under polarised $\mathbb{U}(1)$ symmetry

Non-linear interaction of three impulsive gravitational waves for the Einstein equations under polarised $\mathbb{U}(1)$ symmetry

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Maxime Van De Moortel, University of Cambridge
Fine Hall 314

An impulsive gravitational wave is a weak solution of the Einstein vacuum equation whose metric admits a curvature delta singularity supported on a null hyper-surface and which can be thought as an idealization of gravitational waves emanating from a strongly gravitating source. In the presence of multiple sources, the impulsive waves eventually interact and it is interesting to study the space-time up to and after the interaction. For such singular solutions, the bounded $L^2$ curvature theorem is not applicable and it is not even clear a priori whether local existence/uniqueness holds.

Tremendous progress have been made in this direction by Luk and Rodnianski in 2013, who proved well- posedness for very general space-times featuring the interaction of two gravitational waves. One crucial idea is to exploit that the metric is very regular in two directions, those parallel to the intersection of the two singular planes corresponding to each impulsive wave. However, their method does not apply to the case where three or more impulsive waves interact transversally, since the space-time no longer admits two privileged directions.

I will present a new local existence result for $\mathbb{U}(1)$ polarised Cauchy data featuring three impulsive gravitational waves of small amplitude propagating towards each others. The proof is achieved with the help of an elliptic gauge constructed in the $\mathbb{U}(1)$ polarised class by Huneau and Luk (2017), together with localisation techniques inspired from Christodoulou’s short pulse method and new results in harmonic analysis that are tailored to the problem.

This is joint work with Jonathan Luk.