13/11/2015 - Philippe G. LeFloch - Lecture 6 - The geometry of weakly regular spacetimes

This course offers an introduction to some aspects of the dynamics of self-gravitating matter described by the Einstein equations of general relativity. The notion of "weakly regular spacetime" is introduced and the initial value problem for the Einstein equations is solved when the initial data set has solely weak regularity and enjoys T2 or spherical symmetry. The proposed theory allows for impulsive gravitational waves, as well as shock waves, and despite the presence of such singular waves propagating in the spacetime, its global causal geometry can be analyzed: geodesic completeness, crushing singularity property, formation of trapped surfaces, etc. This course also studies the dynamics of self-gravitating massive fields and provides a proof that Minkowski spacetime is nonlinearly stable in presence of massive fields with sufficiently small mass. A small perturbation of an asymptotically flat, spacelike hypersurface in Minkowski space is proven to disperse toward future timelike directions, so that the spacetime is future geodesically complete. This course is part of the Trimester Program “Mathematical general relativity” which took place at the Institut Henri Poincaré in order to celebrate the 100th anniversary of general relativity. See http://philippelefloch.org for further information.