Bianchi IX model: Reducing phase space

Abstract

The mathematical structure of higher-dimensional physical phase spaces of the nondiagonal Bianchi IX model is analyzed in the neighborhood of the cosmological singularity by using dynamical system methods. Critical points of the Hamiltonian equations appear at infinities and are of a nonhyperbolic type, which is a generic feature of the considered singular dynamics. The reduction of the kinematical symplectic 2-form to the constraint surface enables the determination of the physical Hamiltonian. This procedure lowers the dimensionality of the dynamics arena. The presented analysis of the phase space is based on canonical transformations. We test our method for the specific subspace of the physical phase space. The obtained results encourage further examination of the dynamics within our approach.Comment: 21 pages, no figures, published in Phys. Rev.

Robustness of the quantum BKL scenario

“…Thus, considered gravitational system evolves away from the singularity at t = 0. The range of the variables q 1 and q 2 results from the physical interpretation ascribed to them [8] so that (q 1 , q 2 ) ∈ R 2 . Thus, the physical phase space Π consists of the two half planes:…”

Quantum Belinski–Khalatnikov–Lifshitz scenario

“…The range of the variables q 1 and q 2 results from the physical interpretation ascribed to them [15]. Since 0 < a < +∞ and 0 < b < +∞, we have (q 1 , q 2 ) ∈ R 2 .…”

“…In what follows, the choice of H differs from the one presented in[15] by a factor minus one to fit properly the third term of the r.h.s. of(13).…”

Comparing the dynamics of diagonal and general Bianchi IX spacetime

“…The quantization of the Bianchi IX model requires full understanding of its classical dynamics in terms of variables convenient for quantization procedure. Our recent paper [6] has initiated such analysis. As far as we [7,8] and references therein).…”