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The Belinskii-Khalatnikov-Lifshitz Discrete Evolution as an Approximation to Mixmaster Dynamics

  • Chapter
Deterministic Chaos in General Relativity

Part of the book series: NATO ASI Series ((NSSB,volume 332))

Abstract

Mixmaster dynamics is defined to be the evolution of vacuum, diagonal Bianchi IX spatially homogeneous cosmologies. Belinskii, Lifshitz, and Khalatnikov (BKL) developed a discrete approximation to Mixmaster dynamics. Here we analyze their approximation to discuss how well it represents the true solution. We show that sufficiently close to the singularity, the BKL approximation describes all the features of an accurately computed Mixmaster trajectory. Within the BKL approximation, the origination of any sensitivity to initial conditions can be explicitly identified. Since a precise analogy can be drawn between the BKL parameters and variables which describe the true Mixmaster dynamics, it can be argued that the trajectories inherit any chaos that characterizes the approximation. Since the discrete evolution which is the closest analog of the true dynamics is only “marginally chaotic,” recent controversies over whether or not the dynamics is chaotic may be more semantic than substantive.

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Author information

Authors and Affiliations

  1. Physics Department, Oakland University, Rochester, MI, 48309, USA

    Beverly K. Berger

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  1. Beverly K. Berger
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Editor information

Editors and Affiliations

  1. University of Calgary, Calgary, Alberta, Canada

    David Hobill

  2. Dalhousie University, Halifax, Nova Scotia, Canada

    Adrian Burd  & Alan Coley  & 

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© 1994 Springer Science+Business Media New York

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Berger, B.K. (1994). The Belinskii-Khalatnikov-Lifshitz Discrete Evolution as an Approximation to Mixmaster Dynamics. In: Hobill, D., Burd, A., Coley, A. (eds) Deterministic Chaos in General Relativity. NATO ASI Series, vol 332. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-9993-4_19

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  • DOI: https://doi.org/10.1007/978-1-4757-9993-4_19

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4757-9995-8

  • Online ISBN: 978-1-4757-9993-4

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