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Stable Cosmological Kaluza–Klein Spacetimes

  • Volker BrandingEmail author
  • David Fajman
  • Klaus Kröncke
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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.

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). We thank the anonymous referee for his remarks and suggestions that helped to improve the paper. D.F. has been supported by the Austrian Science Fund (FWF) project P29900-N27 Geometric Transport equations and the non-vacuum Einstein flow. V.B. gratefully acknowledges the support of the Austrian Science Fund (FWF) through the project P30749-N35 Geometric variational problems from string theory.

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© The Author(s) 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.Faculty of PhysicsUniversity of ViennaViennaAustria
  3. 3.Faculty of MathematicsUniversity of HamburgHamburgGermany

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