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

Abstract

Printed in SingaporeThe Global Nonlinear Stability of Minkowski Space for Self-Gravitating Massive Fields Downloaded from www.worldscientific.com by 54.213.91.117 on 05/10/18. For personal use only.The theory presented in this monograph establishes the first mathematically rigorous result on the global nonlinear stability of self-gravitating matter under small perturbations of an asymptotically flat, spacelike hypersurface of Minkowski spacetime. It allows one to exclude the existence of dynamically unstable, self-gravitating massive fields and, therefore, solves a long-standing open problem in General Relativity. By a significant extension of the Hyperboloidal Foliation Method they introduced in 2014, the authors establish global-in-time existence for the Einstein equations expressed as a coupled wave-Klein-Gordon system of partial differential equations. The metric and matter fields are sought for in Sobolev-type functional spaces, suitably defined from the translations and the boosts of Minkowski spacetime. PrefaceThe theory presented in this Monograph establishes the first mathematically rigorous result about the global nonlinear stability of self-gravitating matter under small perturbations. It allows us to exclude the existence of dynamically unstable, self-gravitating massive fields and, therefore, solves a long-standing open problem in General Relativity. 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 asymptotically 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.Our method of proof which we refer to as the Hyperboloidal Foliation Method, extends the standard 'vector field method' developed for massless fields and, importantly, does not use the scaling vector field of Minkowski space. We construct a foliation (of the interior of a light cone) by spacelike and asymptotically hyperboloidal hypersurfaces and we rely on a decomposition of the Einstein equations expressed in wave gauge and in a semi-hyperboloidal frame, in a sense defined in this Monograph. We focus here on the problem of the evolution of a spatially compact matter field, and 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 one of the unknowns).The main challenge is to establish a global-in-time existence theory for coupled wave-Klein-Gordon systems in Sobolev-type spaces defined from the translations and the boosts of Minkowski spacetime, only. To this end, we rely on the following novel an...

Global solutions of nonlinear wave–Klein–Gordon system in one space dimension

“…Similar formulation leads to a 2 + 1 dimensional wave-Klein-Gordon system (to be written as W-KG system in the follows), which contains the essential quasi-null structure of Einstein equation. However, since the decay of both wave and Klein-Gordon equations in 2 + 1 dimension is weaker than in 3 + 1 case, the analysis on this system, compared with our previous work [3], [4] (see also [5], [6]) in 3 + 1 case, will be much more delicate. This article and its successor can be considered as technical preparations, in which we will regard (1.1) as a model and concentrate firstly on the nonlinear terms which do not concern the quasi-null structure and/or (generalized-)wave gauge conditions enjoyed by Einstein-scalar system.…”

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

“…We refer to [39,40,41] for earlier work by the authors and to the companion work [42] for an extension to more general data and to the theory of modified gravity. We focus on p3`1qdimensional problems since this is the dimension of main interest.…”

“…The main challenge overcome by the hyperboloidal foliation method applied to (1.6) concerns the part of the solution supported in the region K r2,`8q or, more precisely, the global evolution of initial data posed on an asymptotically hyperbolic hypersurface. (See [42] for further details.) To guarantee this, the initial data posed on the hypersurface tt " 2u should have its support contained in the unit ball tr ă 1u.…”

“…We present here our method for a restricted class of initial data, while more general data as well as the theory of f pRq-modified gravity are treated in[42].…”

Global stability of the Dirac-Klein-Gordon system in two and three space dimensions

“…The hyperboloidal method used in [13] was introduced by Klainerman in [18] to prove global existence results for nonlinear Klein-Gordon equations by using commuting vector fields. Later this method was developed by Klainerman, Wang and Yang [19,24] and by LeFloch and Ma [20,21]; see also [10,11,14].…”