TY - JOUR

T1 - Describing general cosmological singularities in Iwasawa variables

AU - Damour, Thibault

AU - de Buyl, Sophie

PY - 2008/2/20

Y1 - 2008/2/20

N2 - 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 gij by means of Iwasawa variables (βa,Nai); (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 β̄=β-β[0], N̄=N-N[0], and by requiring that β̄ and N̄ 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.

AB - 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 gij by means of Iwasawa variables (βa,Nai); (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 β̄=β-β[0], N̄=N-N[0], and by requiring that β̄ and N̄ 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.

UR - http://www.scopus.com/inward/record.url?scp=41049109358&partnerID=8YFLogxK

U2 - 10.1103/PhysRevD.77.043520

DO - 10.1103/PhysRevD.77.043520

M3 - Article

AN - SCOPUS:41049109358

VL - 77

JO - Physical Review D. Particles, Fields, Gravitation, and Cosmology

JF - Physical Review D. Particles, Fields, Gravitation, and Cosmology

SN - 1550-7998

IS - 4

M1 - 043520

ER -