Nonlinear stability of self-gravitating massive fields. A wave-Klein-Gordon model

TL;DR

This paper introduces the Euclidean-hyperboloidal foliation method to analyze the nonlinear stability of self-gravitating massive fields, demonstrated through a wave-Klein-Gordon model that captures key challenges of Einstein-matter systems.

Contribution

The paper develops and proves the Euclidean-hyperboloidal foliation method for a wave-Klein-Gordon model, advancing the understanding of nonlinear stability in self-gravitating massive fields.

Findings

Established sharp decay estimates for wave and Klein-Gordon equations in curved spacetime.

Provided a full proof of stability for a wave-Klein-Gordon model.

Demonstrated the method's potential for Einstein-matter systems.

Abstract

Significant advances were made in recent years on the global evolution problem for self-gravitating massive matter in the small-perturbative regime close to Minkowski spacetime. To study the coupling between a Klein-Gordon equation and Einstein's field equations, we introduced the ``Euclidean-hyperboloidal foliation method'', which is based on the construction of a spacetime foliation adapted to the derivation of sharp decay estimates for wave and Klein-Gordon equations in a curved spacetime. We give here an outline of this method, together with a full proof for a wave-Klein-Gordon model which retains some main challenges arising with the Einstein-matter system.