On the "Scattering Law" for Kasner Parameters Appearing in Asymptotics of an Exact S-Brane Solution
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 spacesM1, ... , 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 tau -> +0 and tau -> +infinity, where tau is the synchronous time variable. A relation between two sets of Kasner parameters alpha(infinity) and alpha(0) is found. This relation, called a "scattering law" (SL), coincides with the "collision law" (CL) obtained previously in  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 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.