The global nonlinear stability of Minkowski space for the Einstein equations in presence of a massive field

TL;DR

This paper extends the Hyperboloidal Foliation Method to prove the global nonlinear stability of Minkowski space in the Einstein equations coupled with massive scalar fields, advancing understanding of gravitational stability with matter.

Contribution

It introduces a new formulation of the Hyperboloidal Foliation Method capable of handling strong interactions in wave-Klein-Gordon systems within Einstein equations.

Findings

Proves the nonlinear stability of Minkowski space with massive scalar fields.

Develops a generalized method for Einstein equations with matter.

Establishes global existence results for coupled wave-Klein-Gordon systems.

Abstract

We provide a significant extension of the Hyperboloidal Foliation Method introduced by the authors in 2014 in order to establish global existence results for systems of quasilinear wave equations posed on a curved space, when wave equations and Klein-Gordon equations are coupled. This method is based on a (3+1) foliation (of the interior of a future light cone in Minkowski spacetime) by spacelike and asymptotically hyperboloidal hypersurfaces. In the new formulation of the method, we succeed to cover wave-Klein-Gordon systems containing "strong interaction" terms at the level of the metric, and then generalize our method in order to establish a new existence theory for the Einstein equations of general relativity. Following pioneering work by Lindblad and Rodnianski on the Einstein equations in wave coordinates, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields.