Stable cosmological Kaluza-Klein Spacetimes

TL;DR

This paper proves the future stability of the Milne universe in a class of cosmological Kaluza-Klein spacetimes derived from Einstein's equations on product manifolds, introducing a new gauge for Maxwell equations.

Contribution

It establishes the stability of a broad class of Kaluza-Klein vacua by analyzing Einstein flow with a novel gauge for Maxwell equations in a cosmological setting.

Findings

Future stability of the Milne universe in Kaluza-Klein spacetimes.

Effective Einstein equations coupled with wave map and Maxwell equations.

Introduction of slice-adapted gauge for Maxwell equations.

Abstract

We consider the Einstein flow on a product manifold with one factor being a compact quotient of 3-dimensional hyperbolic space without boundary and the other factor being a flat torus of fixed arbitrary dimension. We consider initial data symmetric with respect to the toroidal directions. We obtain effective Einsteinian field equations coupled to a wave map type and a Maxwell type equation by the Kaluza-Klein reduction. The Milne universe solves those field equations when the additional parts arising from the toroidal dimensions are chosen constant. We prove future stability of the Milne universe within this class of spacetimes, which establishes stability of a large class of cosmological Kaluza-Klein vacua. A crucial part of the proof is the implementation of a new gauge for Maxwell-type equations in the cosmological context, which we refer to as slice-adapted gauge.