Future Stability of the Einstein-Maxwell-Scalar Field System

Ringström managed (in Invent Math 173(1):123–208, 2008) to prove future stability of solutions to Einstein’s field equations when matter consists of a scalar field with a potential creating an accelerated expansion. This was done for a quite wide class of spatially homogeneous space–times. The methods he used should be applicable also when other kinds of matter fields are added to the stress-energy tensor. This article addresses the question whether we can obtain stability results similar to those Ringström obtained if we add an electromagnetic field to the matter content. Before this question can be addressed, more general properties concerning Einstein’s field equation coupled to a scalar field and an electromagnetic field have to be settled. The most important of these questions are the existence of a maximal globally hyperbolic development and the Cauchy stability of solutions to the initial value problem.