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 $({\beta }^a,\mathcal{N}^a{}_i)$; (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 $\beta$, $\mathcal{N}$ whose asymptotic behavior is described by a solution ${\beta }_{[0]}$, ${\mathcal{N}}_{[0]}$ of the previous evolution system by means of a “generalized Fuchsian system” for the differenced variables $\overline{\beta }=\beta -{\beta }_{[0]}$, $\overline{\mathcal{N}}=\mathcal{N}-{\mathcal{N}}_{[0]}$, and by requiring that $\overline{\beta }$ 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.

Figure 1

(a) Nonchaotic behavior.—Sufficiently close to the singularity, the dynamics of the gravitational field can be approximated by a Kasner-like metric at each spatial point. Let us focus on one particular spatial point where asymptotically the metric is given by $d{s}_{\mathrm{spatial}}^2=(t^{2p_1}{l}_i{l}_j+t^{2p_2}{m}_i{m}_j+\dots +t^{2p_d}{r}_i{r}_j)d{x}^{i}d{x}^j$. When $t\to 0$, the directions for which the Kasner exponent $p_i$ is negative are stretched while the ones with positive exponent are squeezed. At the singularity, these directions are still defined. (b) Chaotic behavior.—Now instead of a Kasner-like metric at each spatial point, there is a never ending chaotic succession of Kasner epochs before reaching the singularity. Are there still some preferred directions at the singularity? Or is some other structure of the metric preserved asymptotically?

Figure 2

Nonchaotic behavior.—This picture is a schematic drawing of the asymptotic dynamics of the “diagonal variables” at a given spatial point $x$, this dynamics is represented in the $\beta$-space. The dashed arrow represents the asymptotic solution ${\beta }_{[0]}$ (which is valid after the last collision on a wall and corresponds to a free motion of the particle $\beta$). The exact solution is sketched as a continuous curve. The idea is that the approximate solution ${\beta }_{[0]}$ becomes better and better as $\tau \to +\infty$, this is formalized by the Fuchs theorem that tells us precisely how $\beta -{\beta }_{[0]}\to 0$ when $\tau \to +\infty$, see the text. Note that here we consider a nonchaotic system and that the “fundamental chamber” determined by the walls is not contained within the light cone.

Figure 3

Schematic drawing of the source term of the system of Eqs. (46).

Figure 4

Chaotic behavior.—This picture is a schematic drawing of the asymptotic dynamics of the “diagonal variables” at a given spatial point $x$, this dynamics is represented in the $\beta$-space. The dashed curve represents the zeroth order solution ${\beta }_{[0]}$. The exact solution is sketched as a continuous curve. The idea is that the approximate solution ${\beta }_{[0]}$ becomes better and better as $\tau \to +\infty$; this is formalized via a “generalized Fuchs theorem,” see the text. Note that here we consider a chaotic system and that the “fundamental chamber” determined by the walls is contained within the light cone.