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Scattering of three impulsive gravitational waves and other low regularity problems

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An impulsive gravitational wave is a weak solution of the Einstein equation, which models gravitational waves emanating from a strongly gravitating source. In the presence of multiple sources, the impulsive waves eventually interact transversally and it is interesting to study their scattering.

The interaction of two general impulsive waves has been understood locally in the seminal work of Luk and Rodnianski. They prove that non-linear singularities do not occur in the scattering, in the sense that space-time stays smooth away from the two waves immediately after the interaction.

Impulsive gravitational waves belong to the class of low regularity space-times, which appear in many interesting topics of classical General Relativity such as the formation of trapped surfaces, the structure of Black Hole interiors and high-frequency metrics. In fact, the techniques used to address two impulsive waves by Luk and Rodnianski have found spectacular applications to all the above-mentioned problems.

However, the case where three or more impulsive waves interact transversally is not covered by earlier work. The methods of Luk-Rodnianski do not apply in this case and one conjecturally expects a qualitatively different behaviour. I will present a new local existence and stability result for U(1) polarised Cauchy data featuring three impulsive gravitational waves of small amplitude propagating towards each other. This result establishes a framework to subsequently understand the non-linear singularity of three waves, and eventually the scattering of more singular solutions.

This is joint work with Jonathan Luk.

This talk is part of the DAMTP Friday GR Seminar series.

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