Describing general cosmological singularities in Iwasawa variables

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 (β^a,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 β, N whose asymptotic behavior is described by a solution β_[0], N_[0] of the previous evolution system by means of a “generalized Fuchsian system” for the differenced variables β̅ =β-β_[0], N̅ =N-N_[0], and by requiring that β̅ and N̅ 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.