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 Iwasawa variables $({\ensuremath{\beta}}^{a},\mathcal{N}^{a}{}_{i})$; (ii) we define, at each spatial point, a (chaotic) asymptotic evolution system made of ordinary differential equations for the Iwasawa variables; and (iii) we characterize the exact Einstein solutions $\ensuremath{\beta}$, $\mathcal{N}$ whose asymptotic behavior is described by a solution ${\ensuremath{\beta}}_{[0]}$, ${\mathcal{N}}_{[0]}$ of the previous evolution system by means of a ``generalized Fuchsian system'' for the differenced variables $\overline{\ensuremath{\beta}}=\ensuremath{\beta}\ensuremath{-}{\ensuremath{\beta}}_{[0]}$, $\overline{\mathcal{N}}=\mathcal{N}\ensuremath{-}{\mathcal{N}}_{[0]}$, and by requiring that $\overline{\ensuremath{\beta}}$ and $\overline{\mathcal{N}}$ 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 asymptotic geometrical structure on the singularity: it is described by a 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 nonchaotic systems.
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Dec 31, 2018
Yeong-Jeu Sun + 1
Open Access
