Stabilité de l’espace-temps de de-Sitter pour le système d’Einstein-mMxwell-Klein-Gordon avec constante cosmologique positive

Abstract

The Maxwell-Klein-Gordon (MKG) equations are a system of nonlinear equations that model the motion of a charged particle in an electromagnetic field. In this work, we consider the Einstein-Maxwell-Scalar Field system with a positive cosmological constant (E M S C ), which is thus a general form of the MKG equations, and we study the asymptotic stability of the solutions to this system in the future. This study appears as a generalization of the work by Dongho Chae [7] and Costa et al. [15]. Indeed, we modify the framework developed by Chae in [7] to study the Einstein-Maxwell-Scalar Field system without a cosmological constant by introducing a positive cosmological constant Λ > 0. On the other hand, the Einstein-Scalar Field system with a positive cosmological constant (E C S ) studied by Costa et al. in [15] becomes a special case of the E M C S system examined here, since we consider a complex scalar field, unlike Costa et al.’s choice of a real scalar field. To reach this goal, we choose the Bondi spherically symmetric spacetime as the geometric framework in which we write the E M C SΛ system in Bondi coordinates, and we remark that the full content of this system is encoded in a single nonlinear first-order partial differential (integro-differential) equation equivalent to the system. To study the associated Cauchy problem for this equivalent integro-differential equation, we prescribe the initial data on an isotropic cone. With such initial data chosen sufficiently small, we demonstrate that this equation admits a unique local and global solution in Bondi time. We also show that the spacetime is asymptotically de Sitter and is geodesically complete in the future : this is a result of nonlinear de Sitter stability for the E M C SΛ system as well as a realization of cosmic censorship.