The global stability of the Kaluza–Klein spacetime

Abstract

In this paper, we show the classical global stability of the flat Kaluza–Klein spacetime, which corresponds to Minkowski spacetime in ${\mathbb{R}}^{1+4}$ with one direction compactified on a circle. We consider small perturbations which are allowed to vary in all directions including the compact direction. These perturbations lead to the creation of massless modes and Klein–Gordon modes. On the analytic side, this leads to a PDE system coupling wave equations to an infinite sequence of Klein–Gordon equations with different masses. The techniques we use are based purely in physical space using the vector field method. In addition to Kaluza–Klein stability, our techniques can be easily adapted to provide a new proof of the stability of the Minkowski solution to the Einstein–Klein–Gordon equations.