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arXiv · 0710.5692

Describing general cosmological singularities in Iwasawa variables

Thibault Damour·Sophie de Buyl
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Abstract

Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that the description of the asymptotic behavior of a generic solution of Einstein equations near a spacelike singularity could be drastically simplified by considering that the time derivatives of the metric asymptotically dominate (except at a sequence of instants, in the `chaotic case') over the spatial derivatives. We present a precise formulation of the BKL conjecture (in the chaotic case) that consists of basically three elements: (i) we parametrize the spatial metric $g_{ij}$ by means of \it{Iwasawa variables} $β^a, {\cal N}^a{}_i$); (ii) we define, at each spatial point, a (chaotic) \it{asymptotic evolution system} made of ordinary differential equations for the Iwasawa variables; and (iii) we characterize the exact Einstein solutions $β, {\cal{N}}$ whose asymptotic behavior is described by a solution $β_{[0]}, {\cal N}_{[0]}$ of the previous evolution system by means of a `\it{generalized Fuchsian system}' for the differenced variables $\bar β= β- β_{[0]}$, $\bar {\cal N} = {\cal N} - {\cal N}_{[0]}$, and by requiring that $\bar β$ and $\bar {\cal N}$ tend to zero on the singularity. We also show that, in spite of the apparently chaotic infinite succession of `Kasner epochs' near the singularity, there exists a well-defined \it{asymptotic geometrical structure} on the singularity : it is described by a \it{partially framed flag}. Our treatment encompasses Einstein-matter systems (comprising scalar and p-forms), and also shows how the use of Iwasawa variables can simplify the usual (`asymptotically velocity term dominated') description of non-chaotic systems.

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Thibault Damour, Sophie de Buyl. 2007-10-30. Describing general cosmological singularities in Iwasawa variables. https://doi.org/10.1103/physrevd.77.043520

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