Perturbations of Minkowski spacetime with regular conformal compactification

Abstract

We construct perturbations of Minkowski spacetime in general relativity, when given initial data that decays inverse polynomially to initial data of a Kerr spacetime towards spacelike infinity. We show that the perturbations admit a regular conformal compactification at null and timelike infinity, where the degree of regularity increases linearly with the rate of decay of the initial data to Kerr initial data. In particular, the compactification is smooth if the initial data decays rapidly to Kerr initial data. This generalizes results of Friedrich, who constructed spacetimes with a smooth conformal compactification in the case when the initial data is identical to Kerr initial data on the complement of a compact set. Our results rely on a novel formulation of the Einstein equations about Minkowski spacetime introduced by the author, that allows one to formulate the dynamic problem as a quasilinear, symmetric hyperbolic PDE that is regular at null infinity and with null infinity being at a fixed locus. It is not regular at spacelike infinity, due to the asymptotics of Kerr. Thus the main technical task is the construction of solutions near spacelike infinity, using tailored energy estimates. To accomplish this, we organize the equations according to homogeneity with respect to scaling about spacelike infinity, which identifies terms that are leading, respectively lower order, near spacelike infinity, with contributions from Kerr being lower order.