The Global Nonlinear Stability of Minkowski Space for Self-gravitating Massive Fields

Abstract

Abstract. The Hyperboloidal Foliation Method (introduced by the authors in 2014) is extended here and applied to the Einstein equations of general relativity. Specifically, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields, while existing methods only apply to massless scalar fields. First of all, by analyzing the structure of the Einstein equations in wave coordinates, we exhibit a nonlinear wave-Klein-Gordon model defined on a curved background, which is the focus of the present paper. For this model, we prove here the existence of global-in-time solutions to the Cauchy problem, when the initial data have sufficiently small Sobolev norms. A major difficulty comes from the fact that the class of conformal Killing fields of Minkowski space is significantly reduced in presence of a massive scalar field, since the scaling vector field is not conformal Killing for the Klein-Gordon operator. Our method relies on the foliation (of the interior of the light cone) of Minkowski spacetime by hyperboloidal hypersurfaces and uses Lorentz-invariant energy norms. We introduce a frame of vector fields adapted to the hyperboloidal foliation and we establish several key properties: Sobolev and Hardy-type inequalities on hyperboloids, as well as sup-norm estimates which correspond to the sharp time decay for the wave and the Klein-Gordon equations. These estimates allow us to control interaction terms associated with the curved geometry and the massive field, by distinguishing between two levels of regularity and energy growth and by a successive use of our key estimates in order to close a bootstrap argument.

Small Data Solutions of the Vlasov-Poisson System and the Vector Field Method

“…Proof The first part of the proposition is a consequence of Lemma 2.6, Lemma 2.4 and the global Klainerman Sobolev inequality (14). The second part of the proposition follows similarly, using that…”

“…Two distinct steps lead to the proof of (14), the L 1 Klainerman-Sobolev inequality of Lemma 3.1 and the special commutation properties of the velocity averaging operator as described in Lemma 2.4. To improve upon (14), the strategy is to try to use at the same time arguments similar to those of Lemma 3.1 and Lemma 2.4, instead of applying them one after the other.…”

“…To improve upon (14), the strategy is to try to use at the same time arguments similar to those of Lemma 3.1 and Lemma 2.4, instead of applying them one after the other. This will lead to us to the following improvement.…”

Back Matter

“…Doing so has no effect on the derivation of the sup-norm bounds (in Section 6.2, on which Section 7 is based), since in the application of the Klainerman-Sobolev inequality one uses one boost at least, and the additional growth allowed by (A.1) is negligible. Note in passing also that, in Section 6.5 of [41], the Hardy-based estimate (6.20a) is valid for k = |J| ≥ 1 only, while we already pointed out in [41] the next inequality (6.20b) is never used. In Lemma 8.1, the estimate (8.4) can be improved to 2) which is checked for |I| + |J| ≤ N − 1 by writing…”

“…This model 1 provided to the authors a simple, yet highly not trivial, example of coupling between a wave equation and a Klein-Gordon equation, before developing the method for the full Einstein system, as we do in the present monograph. We revisit here the proof of existence in [41] since our presentation missed one bootstrap condition in the list (5.1) which however turns out to be necessary for dealing with the (comparatively easier) wave component when k = 0 in (5.1).…”