This talk will concern the interaction problem in General Relativity in a specific setting. More precisely, we will describe a research program aimed at a rigorous understanding of the interaction of gravitational waves originating from several localized and distant sources.
First, we will restrict our attention to a model problem (quasilinear wave equations with the null condition) and describe a global stability result obtained in collaboration with John Anderson (Stanford). We show global stability in a class of small initial data localized around a collection of points whose mutual distances are large. Our approach relies on a bilinear spacetime estimate which quantifies precisely the nonlinear interaction.
We will then turn our attention to the problem for the Einstein equations and focus on the existence of suitable initial data sets. In joint work with John Anderson and Justin Corvino (Lafayette College), we construct solutions to the time-symmetric Einstein constraint equations with a finite (or countably infinite) number of localized (and smooth) sources.
In the last part of the talk, we will comment on the corresponding evolution problem for the Einstein vacuum equations arising from the above initial data sets.