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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Andersson, L., Moncrief, V.: Elliptic–hyperbolic systems and the Einstein equations. Ann. Henri Poincaré 4 (2003)
Andersson L., Moncrief V.: Einstein spaces as attractors for the Einstein flow. J. Differ. Geom. 89, 1–47 (2011)
Andersson, L., Fajman, D.: Nonlinear Stability of the Milne Model with Matter. arXiv:1709.00267 (2017)
Branding V., Kröncke K.: Global existence of wave maps and some generalizations on expanding spacetimes. Calc. Var. 57, 119 (2018)
Bieri L., Zipser N.: Extensions of the stability theorem of the Minkowski space in general relativity American Mathematical Society. International Press, Vienna (2009)
Christodoulou, D., Klainerman, S.: The global nonlinear stability of the Minskowski space
Choquet-Bruhat, Y., Moncrief, V.: Future global in time einsteinian spacetimes with U(1) isometry group. Ann. Henri Poincaré 2 (2001)
Choquet-Bruhat, Y.: General Relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford Science Publications
Fajman, D., Kröncke, K.: Stable fixed points of the Einstein flow with a positive cosmological constant to appear in Commun. Geom. Anal.
Valiente Kroon J.A.: Conformal Methods in General Relativity. Cambridge University Press, Cambridge (2016)
Kröncke K.: On the stability of Einstein manifolds. Ann. Glob. Anal. Geom. 47, 81–98 (2015)
Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. 2, 171 (2010)
Overduin J.M., Wesson P.S.: Kaluza–Klein gravity. Phys. Rep. 283(5-6), 303–378 (1997)
Polchinski J.: String Theory, Volume I, An Introduction to the bosonic string. Cambridge University Press, Cambridge
Rendall, A.D.: Partial Differential Equations in General Relativity. Oxford Graduate Texts in Mathematics (2008)
Ringström H.: Future stability of the Einstein-non-linear scalar field system. Invent. Math. 173, 123–208 (2008)
Rodnianski I., Speck J.: A regime of linear stability for the Einsetin-scalar field system with applications to nonlinear Big Bang formation. Ann. Math. 187, 1 (2018)
Rodnianski, I., Speck J.: On the nature of Hawking’s incompleteness for the Einstein-vacuum equations: the regime of moderately spatially anisotropic initial data. arXiv:1804.06825 (2018)
Speck J.: The global stability of the Minkowski spacetime solution to the Einstein-nonlinear system in wave coordinates. Anal. PDE 7(4), 771–901 (2014)
Witten E.: Instability of the Kaluza–Klein vacuum. Nucl. Phys. B195, 481–492 (1982)
Wyatt, Z.: The Weak Null Condition and Kaluza–Klein Spacetimes. arXiv:1706.00026 (2017)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by W. Schlag
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
OpenAccess This 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.
About this article
Cite this article
Branding, V., Fajman, D. & Kröncke, K. Stable Cosmological Kaluza–Klein Spacetimes. Commun. Math. Phys. 368, 1087–1120 (2019). https://doi.org/10.1007/s00220-019-03319-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-019-03319-5