On the “Scattering Law” for kasner parameters appearing in asymptotics of an exact S-brane solution
V.D. Ivashchuk and V.N. Melnikov^{1}
(1) Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya Str., Moscow 119361, Russia
Abstract
A multidimensional cosmological model with scalar and form fields is studied. An exact S-brane solution (either electric or magnetic) in a model with l scalar fields and one antisymmetric form of rank m ³ 2 is considered. This solution is defined on a product manifold containing n Ricci-flat factor spaces M_{1},..., M_{n}. In the case where the kinetic term for scalar fields is positive-definite, we have singled out a special solution governed by the function cosh. It is shown that this special solution has Kasner-like asymptotics in the limits t® +0 and t® +¥, where t is the synchronous time variable. A relation between two sets of Kasner parameters a_{¥} and a_{0} is found. This relation, called a ßcattering law" (SL), coincides with the "collision law" (CL) obtained previously in the context of a billiard description of S-brane solutions near the singularity. A geometric sense of the SL is clarified: it is shown that the SL transformation is a map of a ßhadow" part of the Kasner sphere S^{N - 2} (N = n + l) onto the ïlluminated" part. This map is just a (generalized) inversion with respect to a point v located outside the Kasner sphere S^{N - 2}. The shadow and illuminated parts of the Kasner sphere are defined with respect to a pointlike source of light located at v. Explicit formulae for SL transformations corresponding to SM2- and SM5-brane solutions in 11-dimensional supergravity are presented.
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