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 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 variablesβ = β − β [0] ,N = N − N [0] , and by requiring thatβ andN 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 non-chaotic systems.

Next-to-next-to-leading order post-Newtonian spin-orbit Hamiltonian for self-gravitating binaries

“…In this section we check that the global Poincaré algebra is fulfilled in a PN approximate way, see, e.g., [9,17]. Besides the Hamiltonian, the quantities entering the Poincaré algebra are the center-of-mass vector G, the total linear momentum P, and the total angular momentum tensor J ij = −J ji .…”

“…Besides the well-known Newtonian and 1PN Hamiltonians, previous results to this order for point-masses are the 2PN [1][2][3], 2.5PN [4,5], 3PN [6][7][8][9][10], and 3.5PN [11,12] Hamiltonians. For the spin part the leading order can be found in [13][14][15][16] and the next-to-leading order in [17][18][19]. At the 3.5PN level one also needs all Hamiltonians cubic in the spins derived in [20,21].…”

Cosmological singularity

“…This is so-called Iwasawa decomposition first used in [8] and thoroughly investigated in [26]. The difference is that in general inhomogeneous case instead of orthogonal matrix R it is more convenient to use an upper triangular matrix N (with components N A α where upper index A numerates the rows and lower index α corresponds to columns)…”

Gravity waves signatures from anisotropic preinflation

“…This growth can be ascribed to the instability of the Kasner geometry, either contracting or expanding, against gravitational waves which we are going to report in this Section. Therefore in this Section we consider Kasner geometry in very different formalism and language [30,31,32,33,35].…”