Describing general cosmological singularities in Iwasawa variables

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

50 pages, 4 figures

References in corpus (22)

• Quiescent cosmological singularities
• Chaos in Superstring Cosmology
• Harmonic coordinate method for simulating generic singularities
• E_(10), BE_(10) and Arithmetical Chaos in Superstring Cosmology
• The past attractor in inhomogeneous cosmology
• Analytic description of singularities in Gowdy spacetimes
• Hyperbolic Kac-Moody Algebras and Chaos in Kaluza-Klein Models
• Numerical simulations of generic singuarities
• Mixmaster Behavior in Inhomogeneous Cosmological Spacetimes
• Evidence for an oscillatory singularity in generic U(1) symmetric cosmologies on $T^3\times R$
• Einstein billiards and overextensions of finite-dimensional simple Lie algebras
• The Singularity in Generic Gravitational Collapse Is Spacelike, Local, and Oscillatory
• Cosmological solutions of the Vlasov-Einstein system with spherical, plane, and hyperbolic symmetry
• Pre-big bang bubbles from the gravitational instability of generic string vacua
• Billiard Representation for Multidimensional Cosmology with Multicomponent Perfect Fluid near the Singularity
• Oscillatory approach to the singularity in vacuum spacetimes with $T^2$ isometry
• Numerical simulations of general gravitational singularities
• Asymptotic silence of generic cosmological singularities
• Asymptotic Behavior of Polarized and Half-Polarized U(1) symmetric Vacuum Spacetimes
• Half polarized U(1) symmetric vacuum spacetimes with AVTD behavior
• Dynamics of inhomogeneities of metric in the vicinity of a singularity in multidimensional cosmology authors
• On the Nature of Singularities in Plane Symmetric Scalar Field Cosmologies

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• Arithmetic and Hyperbolic Structures in String Theory
• Nonadiabatic bounce and an inflationary phase in the quantum mixmaster universe
• A New Class of Asymptotically Non-Chaotic Vacuum Singularities
• Vacuum space-times with controlled singularities and without symmetries
• Onset of synchronization in coupled Mixmaster oscillators
• Timelike singularities and Hamiltonian cosmological billiards