# On the “Scattering Law” for kasner parameters appearing in asymptotics of an exact S-brane solution

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## Abstract

A multidimensional cosmological model with scalar and form fields [1–4] 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* ≥ 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 Kasnerlike asymptotics in the limits *τ* → +0 and *τ* → +∞, where *τ* is the synchronous time variable. A relation between two sets of Kasner parameters *α* _{∞} and *α* _{0} is found. This relation, called a “scattering law” (SL), coincides with the “collision law” (CL) obtained previously in [5] 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 “shadow” part of the Kasner sphere *S* ^{ N−2} (*N* = *n* + *l*) onto the “illuminated” part. This map is just a (generalized) inversion with respect to a point *ν* 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 *ν*. Explicit formulae for SL transformations corresponding to *SM*2- and *SM*5-brane solutions in 11-dimensional supergravity are presented.

## PACS numbers

04.50.-h## Preview

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