Future asymptotics and geodesic completeness of polarized $T^2$-symmetric spacetimes

Abstract

We investigate the late-time asymptotics of future-expanding, polarized vacuum Einstein spacetimes with $T^2$-symmetry on $T^3$, which, by definition, admit two spacelike Killing fields. Our main result is the existence of a stable asymptotic regime within this class; that is, we provide here a full description of the late-time asymptotics of the solutions to the Einstein equations when the initial data set is close to the asymptotic regime. Our proof is based on several energy functionals with lower-order corrections (as is standard for such problems) and the derivation of a simplified model that we exhibit here. Roughly speaking, the Einstein equations in the symmetry class under consideration consist of a system of wave equations coupled to constraint equations plus a system of ordinary differential equations. The unknowns involved in the system of ordinary equations are blowing up in the future timelike directions. One of our main contributions is the derivation of novel effective equations for suitably renormalized unknowns. Interestingly, this renormalization is not performed with respect to a fixed background, but does involve the energy of the coupled system of wave equations. In addition, we construct an open set of initial data that are arbitrarily close to the expected asymptotic behavior. We emphasize that, in comparison, the class of Gowdy spacetimes exhibits a very different dynamical behavior to the one we uncover in the present work for general polarized $T^2$-symmetric spacetimes. Furthermore, all the conclusions of this paper are valid within the framework of weakly $T^2$-symmetric spacetimes previously introduced by the authors.