Abstract
An idea which has been around in general relativity for more than 40 years is that in the approach to a big bang singularity solutions of the Einstein equations can be approximated by the Kasner map, which describes a succession of Kasner epochs. This is already a highly non-trivial statement in the spatially homogeneous case. There the Einstein equations reduce to ordinary differential equations and it becomes a statement that the solutions of the Einstein equations can be approximated by heteroclinic chains of the corresponding dynamical system. For a long time, progress on proving a statement of this kind rigorously was very slow but recently there has been new progress in this area, particularly in the case of the vacuum Einstein equations. In this paper we generalize some of these results to cases where the Einstein equations are coupled to matter fields, focussing on the example of a dynamical system arising from the Einstein–Maxwell equations with symmetry of Bianchi type VI0. It turns out that this requires new techniques since certain eigenvalues are in a less favourable configuration than in the vacuum case. The difficulties which arise in that case are overcome by using the fact that the dynamical system of interest is of geometrical origin and thus has useful invariant manifolds.
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References
Béguin F.: Aperiodic oscillatory asymptotic behavior for some Bianchi spacetimes. Class. Quantum Gravity 27, 185005 (2010)
Belinskii V., Khalatnikov I., Lifshitz E.: Oscillatory approach to a singular point in the relativistic cosmology. Adv. Phys. 19, 525–573 (1970)
Belinskii V., Khalatnikov I., Lifshitz E.: A general solution of the Einstein equations with a time singularity. Adv. Phys. 31, 639–667 (1982)
Heinzle J., Uggla C.: Mixmaster: fact and belief. Class. Quantum Gravity 26(7), 075016 (2009)
LeBlanc V.: Asymptotic states of magnetic Bianchi I cosmologies. Class. Quantum Gravity 14, 2281–2301 (1997)
LeBlanc V., Kerr D., Wainwright J.: Asymptotic states of magnetic Bianchi VI 0 cosmologies. Class. Quantum Gravity 12, 513–541 (1995)
Liebscher S., Härterich J., Webster K., Georgi M.: Ancient dynamics in Bianchi models: approach to periodic cycles. Commun. Math. Phys. 305, 59–83 (2011)
Reiterer, M., Trubowitz, E.: The BKL conjectures for spatially homogeneous spacetimes. arXiv:1005.4908 (2010)
Rendall A.: Partial Differential Equations in General Relativity. Oxford University Press, Oxford (2008)
Ringström H.: Curvature blow up in Bianchi VIII and IX vacuum solutions. Class. Quantum Gravity 17, 713–731 (2000)
Ringström H.: The Bianchi IX attractor. Ann. Henri Poincaré 2(3), 405–500 (2001)
Shilnikov L., Shilnikov A., Turaev D., Chua L.: Methods of Qualitative Theory in Nonlinear Dynamics I. Series on Nonlinear Science, Series A, vol. 4. World Scientific, Singapore (1998)
Wainwright, J., Ellis, G. (eds): Dynamical Systems in Cosmology. Cambridge University Press, Cambridge (1997)
Wainwright J., Hsu L.: A dynamical systems approach to Bianchi cosmologies: orthogonal models of a class A. Class. Quantum Gravity 6(10), 1409–1431 (1989)
Weaver M.: Dynamics of magnetic Bianchi type VI 0 cosmologies. Class. Quantum Gravity 17, 421–434 (2000)
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Communicated by Piotr T. Chrusciel.
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Liebscher, S., Rendall, A.D. & Tchapnda, S.B. Oscillatory Singularities in Bianchi Models with Magnetic Fields. Ann. Henri Poincaré 14, 1043–1075 (2013). https://doi.org/10.1007/s00023-012-0207-7
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DOI: https://doi.org/10.1007/s00023-012-0207-7