The global stability of the Kaluza-Klein spacetime

TL;DR

This paper proves the global stability of the flat Kaluza-Klein spacetime under small perturbations, demonstrating that such perturbations do not lead to instability, using purely physical space techniques.

Contribution

It establishes the classical global stability of the Kaluza-Klein spacetime with a novel PDE analysis involving coupled wave and Klein-Gordon equations.

Findings

Small perturbations decay over time, maintaining stability.

Coupled wave and Klein-Gordon equations are effectively controlled.

Purely physical space methods are successfully applied.

Abstract

In this paper we show the classical global stability of the flat Kaluza-Klein spacetime, which corresponds to Minkowski spacetime in $\m R^{1+4}$ 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 vectorfield method.