Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity

Abstract

Multidimensional model describing the cosmological evolution of n Einstein spaces in the theory with l scalar fields and forms is considered. When electromagnetic composite p-brane ansatz is adopted, and certain restrictions on the parameters of the model are imposed, the dynamics of the model near the singularity is reduced to a billiard on the (N − 1)-dimensional Lobachevsky space H N −1 , N = n + l. The geometrical criterion for the finiteness of the billiard volume and its compactness is used. This criterion reduces the problem to the problem of illumination of (N − 2)-dimensional sphere S N −2 by point-like sources. Some examples with billiards of finite volume and hence oscillating behaviour near the singularity are considered. Among them examples with square and triangle 2-dimensional billiards (e.g. that of the Bianchi-IX model) and a 4-dimensional billiard in "truncated" D = 11 supergravity model (without the Chern-Simons term) are considered. It is shown that the inclusion of the Chern-Simons term destroys the confining of a billiard.

Avoiding cosmological oscillating behavior forS-brane solutions with diagonal metrics

“…Moreover, the authors of [18] give a sketch of a proof of this conjecture. Although the ansatz for the cosmological metric in [18] contained d-beins leading to additional gravitational walls in the billiards and the Kasner parameters also depended on spatial coordinates, the analysis of [18], in essence, deals with the Bianchi-I type model with a maximal number of composite electric branes that form the billiard walls according to the prescription of [5]. The gravitational walls are irrelevant for the proof in [18].…”

“…It should be stressed that the billiard approach of [5] was derived in a class of block-diagonal metrics defined on the product of Einstein spaces (1) with certain restrictions imposed on the signatures of the Einstein metrics, g i . These restrictions are similar to the conditions > 0 for u 2 > 0 in the multicomponent fluid case.…”

“…In [5] the above analysis was repeated but with the matter sources given by scalar fields and form fields. The action for this system was given by…”

“…In [1][2][3][4][5][6] the behavior of certain multidimensional solutions with block-diagonal metrics was examined in terms of the billiard representation in generalized Lobachevsky spaces. The billiard representation is a graphical method of studying the asymptotical motion of the point described by the anisotropic minisuperspace coordinates, z a t, in a region enclosed by ''hard walls'' which come from the matter sources.…”

“…In [5] the form fields in (11) were taken to have an electric, composite brane ansatz, e.g. describing S branes (or spacelike branes).…”