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.
Highlights
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 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
Summary
A prerequisite for the stability analysis of the class of Kaluza–Klein spacetimes with Minkowski space as their macroscopic part is the corresponding nonlinear stability result for the classical 4-dimensional vacuum Einstein equations [CK,LR]. The Milne model is future complete and past incomplete and its future nonlinear stability problem has been resolved in the vacuum setting by Andersson and Moncrief [AMb] This result covers the higher-dimensional case, not in the sense of compactified dimensions. The background solution being investigated is −4dt2 + 2t2σ + d x2, where σ is a metric of constant negative scalar curvature on They prove future stability of this solution considered within the set of solutions to the 4-dimensional vacuum Einstein equations obeying a U (1)-symmetry in the S1 direction.
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