Nonlinear Stability of Self-Gravitating Massive Fields

Abstract

We consider the global evolution problem for Einstein’s field equations in the near-Minkowski regime and study the long-time dynamics of a massive scalar field evolving under its own gravitational field. We establish the existence of a globally hyperbolic Cauchy development associated with any initial data set that is sufficiently close to a data set in Minkowski spacetime. In addition to applying to massive fields, our theory allows us to cover metrics with slow decay in space. The strategy of proof, proposed here and referred to as the Euclidean-Hyperboloidal Foliation Method, applies, more generally, to nonlinear systems of coupled wave and Klein-Gordon equations. It is based on a spacetime foliation defined by merging together asymptotically Euclidean hypersurfaces (covering spacelike infinity) and asymptotically hyperboloidal hypersurfaces (covering timelike infinity). A transition domain (reaching null infinity) limited by two asymptotic light cones is introduced in order to realize this merging. On the one hand, we exhibit a boost-rotation hierarchy property (as we call it) which is associated with Minkowski’s Killing fields and is enjoyed by commutators of curved wave operators and, on the other hand, we exhibit a metric hierarchy property (as we call it) enjoyed by components of Einstein’s field equations in frames associated with our Euclidean-hyperboloidal foliation. The core of the argument is, on the one hand, the derivation of novel integral and pointwise estimates which lead us to almost sharp decay properties (at timelike, null, and spacelike infinity) and, on the other hand, the control of the (quasi-linear and semi-linear) coupling between the geometric and matter parts of the Einstein equations.