Future stability of the Einstein-non-linear scalar field system

TL;DR: In this article, the authors consider the case of non-linear scalar fields coupled with a nonlinear Gaussian scalar field and show that all causal geodesics in the maximal globally hyperbolic development that start in $B_{r_0}(p)$ are future complete.

Abstract: We consider the question of future global non-linear stability in the case of Einstein’s equations coupled to a non-linear scalar field. The class of potentials V to which our results apply is defined by the conditions V(0)>0, V’(0)=0 and V”(0)>0. Thus Einstein’s equations with a positive cosmological constant represents a special case, obtained by demanding that the scalar field be zero. In that context, there are stability results due to Helmut Friedrich, the methods of which are, however, not so easy to adapt to the presence of matter. The goal of the present paper is to develop methods that are more easily adaptable. Due to the extreme nature of the causal structure in models of this type, it is possible to prove a stability result which only makes local assumptions concerning the initial data and yields global conclusions in time. To be more specific, we make assumptions in a set of the form $B_{4r_0}(p)$ for some r 0>0 on the initial hypersurface, and obtain the conclusion that all causal geodesics in the maximal globally hyperbolic development that start in $B_{r_0}(p)$ are future complete. Furthermore, we derive expansions for the unknowns in a set that contains the future of $B_{r_0}(p)$ . The advantage of such a result is that it can be applied regardless of the global topology of the initial hypersurface. As an application, we prove future global non-linear stability of a large class of spatially locally homogeneous spacetimes with compact spatial topology.