The global nonlinear stability of Minkowski space for self-gravitating massive fields

Philippe G Lefloch (1) , Yue Ma (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 Xjtu - Xi'an Jiaotong University

We establish that Minkowski spacetime is nonlinearly stable in presence of a massive scalar field under suitable smallness conditions (for, otherwise, black holes might form). We formulate the initial value problem for the Einstein-massive scalar field equations, when the initial slice is a perturbation of an asymptot-ically flat, spacelike hypersurface in Minkowski space, and we prove that this perturbation disperses in future timelike directions so that the associated Cauchy development is future geodesically complete. Hence, our theory excludes the existence of dynamically unstable, self-gravitating massive fields and, therefore, solves a long-standing open problem in general relativity. Our method of proof which we refer to as the Hyperboloidal Foliation Method, goes significantly beyond the standard 'vector field method', which only applies to mass-less scalar fields. Our approach does not use the scaling vector field of Minkowski spacetime. We rely on a foliation of the interior of a light cone by spacelike hyperboloidal hypersurfaces and on a decomposition of the Einstein equations expressed in wave gauge and in a semi-hyperboloidal frame, in a sense defined in this paper. We treat here the problem of the evolution of a spatially compact matter field, i.e. we consider initial data coinciding, in a neighborhood of spacelike infinity, with a spacelike slice of Schwarzschild spacetime. We express the Einstein equations as a system of coupled nonlinear wave-Klein-Gordon equations (with differential constraints) posed on a curved space (whose metric is the main unknown). Our main challenge is to establish a global existence theory for this system in suitably weighted Sobolev spaces. To this end, we rely on the following novel and robust techniques: a sharp decay estimate for wave equations, a sharp decay estimate for Klein-Gordon equations, Sobolev and Hardy inequalities on the hyperboloidal foliation, the quasi-null hyper-boloidal structure of the Einstein equations, as well as integration arguments along characteristics and radial rays.

Domaines: Equations aux dérivées partielles [math.AP]

Philippe G Lefloch, Yue Ma. The global nonlinear stability of Minkowski space for self-gravitating massive fields. 2016. ⟨hal-01423475v1⟩