Andrea Nützi: Maurer-Cartan perturbation theory in general relativity

Abstract: I will describe a differential graded Lie algebra tailored to study perturbations of Minkowski spacetime, including questions about regularity at infinity. The Maurer-Cartan equation is equivalent to the Einstein equations: the differential describes linearized gravity about the Minkowski background, the bracket describes the interaction of gravitational waves. One can then use Maurer-Cartan perturbation theory to construct formal power series solutions. The homology is computed using a contraction obtained via gauge fixing. The homology in degree two is the obstruction space, it is ten-dimensional and can be interpreted as the physical mass and angular momentum charges. The obstructions force us to supplement the basic Maurer-Cartan iteration scheme by a renormalization scheme, which is based on a parametrization of the obstruction space in terms of a family of Kerr black hole spacetime near spacelike infinity. I will indicate how the methods can also be used to obtain actual PDE solutions (as opposed to formal power series solutions) of the Einstein equations.