Billiard representation for multidimensional cosmology with multicomponent perfect fluid near the singularity

Abstract

The multidimensional cosmological model describing the evolution of n Einstein spaces is considered in the presence of a multicomponent perfect fluid. When certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity are reduced to a billiard on the (n-1)-dimensional Lobachevsky space . The geometrical criterion for the finiteness of the billiard volume and its compactness is suggested. This criterion reduces the problem to the problem of illumination of an (n-2)-dimensional sphere by point-like sources. Some generalizations of the considered scheme (including scalar field and quantum generalizations) are considered.

Multidimensional Cosmological and Spherically Symmetric Solutions with Intersecting p-Branes

“…Multidimensional generalizations of this model were considered by many authors (see, for example, [29,76,77,78]). In [79,80,81] the billiard representation for multidimensional cosmological models near the singularity was considered and the criterion for the volume of the billiard to be finite was established in terms of illumination of the unit sphere by point-like sources. For perfect-fluid this was considered in detail in [81].…”

“…In [79,80,81] the billiard representation for multidimensional cosmological models near the singularity was considered and the criterion for the volume of the billiard to be finite was established in terms of illumination of the unit sphere by point-like sources. For perfect-fluid this was considered in detail in [81]. Some interesting topics related to general (non-homogeneous) situation were considered in [82].…”

“…In [81] the "quantum billiard" was obtained for multidimensional WDW solutions near the singularity. It should be also noted that the "third-quantized" multidimensional cosmological models were considered in several papers [59,103,97].…”

On Brane Solutions with Intersection Rules Related to Lie Algebras

“…This metric coincides with the power-law, inflationary solution in the model with a one-component perfect fluid when the following equation of state is adopted: p = 2 5 ρ , where p is pressure and ρ is the density of fluid [119,120]. …”

“…For D = 11 supergravity and ten-dimensional I I A , I IB supergravities, all (U s , U s ) = 2 [44,55] and corresponding KM algebras are simply laced. It was shown in our papers [29][30][31] that the inequality (U s , U s ) > 0 is a necessary condition for the formation of the billiard wall (ifone approaches the singularity) by the s-th matter source (e.g., a fluid component or a brane).…”