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

V.D. Ivashchuk and V.N. Melnikov1

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 M1,..., Mn. 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 a0 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 SN - 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 SN - 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.

References

  1. V. N. Melnikov, Multidimensional Classical and Quantum Cosmology and Gravitation. Exact Solutions and Variations of Constants, CBPF-NF-051/93, Rio de Janeiro, 1993; V. N. Melnikov, in: Cosmology and Gravitation, ed. M. Novello (Editions Frontieres, Singapore, 1994), p. 147.
  2. V. N. Melnikov, Multidimensional Cosmology and Gravitation, CBPF-MO-002/95, Rio de Janeiro, 1995, 210 p.; V. N. Melnikov, in: Cosmology and Gravitation. II, ed. M. Novello (Editions Frontieres, Singapore, 1996), p. 465.
  3. V. N. Melnikov. Exact Solutions in Multidimensional Gravity and Cosmology, III, CBPF-MO-03/02, Rio de Janeiro, 2002, 297 pp.
  4. V. N. Melnikov, Gravity as a Key Problem of the Millennium, Proc. 2000 NASA/JPL Conference on Fundamental Physics in Microgravity, CD-version, NASA Document D-21522, 2001, p. 4.1-4.17, (Solvang, CA, USA); gr-qc/0007067.
  5. V. D. Ivashchuk, On exact solutions in multidimensional gravity with antisymmetric forms, in: Proceedings of the 18th Course of the School on Cosmology and Gravitation: The Gravitational Constant. Generalized Gravitational Theories and Experiments (30 April-10 May 2003, Erice). Ed. G. T. Gillies, V. N. Melnikov, and V. de Sabbata (Kluwer Academic Publishers, Dordrecht, 2004), pp. 39-64; gr-qc/0310114.
  6. V. D. Ivashchuk and V. N. Melnikov, Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity, J. Math. Phys. 41, 6341 (2000); hep-th/9904077.
  7. V. D. Ivashchuk and V. N. Melnikov, Sigma model for the generalized composite p-branes, hep-th/9705036; Class. Quantum Grav. 14, 3001 (1997); Corrigenda: ibid., 15, 3941 (1998).
  8. V. D. Ivashchuk and V. N. Melnikov, Exact solutions in multidimensional gravity with antisymmetric forms, topical review, Class. Quantum Grav. 18, R82-R157 (2001); hep-th/0110274.
  9. V. D. Ivashchuk and V. N. Melnikov, Multidimensional classical and quantum cosmology with intersecting p-branes, hep-th/9708157; J. Math. Phys. 39, 2866 (1998).
  10. T. Damour and M. Henneaux, Chaos in superstring cosmology, Phys. Rev. Lett. 85, 920 (2000); hep-th/000313.
  11. H. Dehnen, V. D. Ivashchuk, and V. N. Melnikov, Billiard representation for multidimensional multi-scalar cosmological model with exponential potentials, Gen. Rel. Grav. 36, 1563 (2004); hep-th/0312317.
  12. V. D. Ivashchuk and S.-W. Kim, Solutions with intersecting p-branes related to Toda chains, J. Math. Phys., 41, 444 (2000); hep-th/9907019.
  13. C. M. Chen, D. M. Gal'tsov and M. Gutperle, S-brane solutions in supergravity theories, Phys. Rev. D 66, 024043 (2002); hep-th/0204071.
  14. V. D. Ivashchuk, Composite S-brane solutions related to Toda-type systems, Class. Quantum Grav. 20, 261 (2003); hep-th/0208101.
  15. V. D. Ivashchuk, On composite S-brane solutions with orthogonal intersection rules, hep-th/0309027.
  16. V. D. Ivashchuk, S-brane solutions with orthogonal intersection rules (invited paper to a fest of A. Garsia), Gen. Rel. Grav. 37, 751 (2005).
  17. V. D. Ivashchuk, V. N. Melnikov and A. I. Zhuk, Nuovo Cimento B 104, 575 (1989).
  18. E. Cremmer, B. Julia and J. Scherk, Phys. Lett. B 76, 409 (1978).
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