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Gravitation and Cosmology

, Volume 25, Issue 1, pp 18–23 | Cite as

Hidden Symmetries in a Mixmaster-Type Universe

  • A. E. PavlovEmail author
Article
  • 16 Downloads

Abstract

A model of multidimensional mixmaster-type vacuum universe is considered. It belongs to a class of pseudo-Euclidean chains characterized by root vectors. An algebraic approach of our investigation is founded by a construction of the Cartan matrix of spacelike root vectors in Wheeler–DeWitt space. Kac-Moody algebras can be classified according to their Cartan matrix. In this way a hidden symmetry of the model considered is revealed. It is known that gravitational models which demonstrate a chaotic behavior are associated with hyperbolic Kac–Moody algebras. The algebra considered in our paper is not hyperbolic. The square of the Weyl vector is negative. The mixmaster-type universe is associated with a simply-laced Lorentzian Kac–Moody algebra. Since the volume of the billiard table is infinite, the model is not chaotic.

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References

  1. 1.
    C. W. Misner, in Deterministic Chaos in General Relativity. Ed. D. Hobill, (Plenum, 1994).Google Scholar
  2. 2.
    V. D. Ivashchuk and V. N. Melnikov, “Billiard representation for multidimensional cosmology with multicomponent perfect fluid near the singularity,” Class. Quantum Grav. 12, 809 (1995).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    V. D. Ivashchuk and V. N. Melnikov, “Billiard representation for multidimensional cosmology with intersected p-branes near the singularity,” J. Math. Phys. 41, 6341 (2000); hep-th/9904077.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    O. I. Bogoyavlensky, “On perturbations of the periodic Toda lattice,” Commun. Math. Phys. 51, 201 (1976).ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    A. E. Pavlov, “The mixmaster cosmological model as a pseudo-Euclidean generalized Toda chain,” Reg. Chaot. Dyn. 1, 111 (1996); gr-qc/9504034.MathSciNetzbMATHGoogle Scholar
  6. 6.
    S. de Buyl, G. Pinardi, and Ch. Schomblond, “Einstein billiards and spatially homogeneous cosmological models,” Class. Quantum Grav. 20, 3595 (2003); hep-th/0306280.MathSciNetzbMATHGoogle Scholar
  7. 7.
    M. Henneaux, D. Persson, and Ph. Spindel, “Spacelike singularities and hidden symmetries of gravity,” Living Rev. Relativity 11, 1–232 (2008).ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    V. Belinski and M. Henneaux, The cosmological singularity (Cambridge University Press, Cambridge, 2018).zbMATHGoogle Scholar
  9. 9.
    T. Damour et al., “Einstein billiards and overextensions of finite-dimensional simple Lie algebras,” J. High Energy Phys. 2002 (08), 030 (2002); hepth/0206125.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M. Demiański et al., “The group-theoretical classification of the 11-dimensional classical homogeneous Kaluza–Klein cosmologies,” J. Math. Phys. 28, 171 (1987).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M. Szydłowski and G. Pajdosz, “The dynamics of homogeneous multidimensional cosmological models,” Class. Quantum Grav. 6, 1391 (1989).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    W. M. Stuckey, L. Witten, and B. Stewart, “Dynamics of the mixmaster-type vacuum universe with geometry R × S3 × S3 × S3,” Gen. Rel. Grav. 22, 1321 (1990).ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    V. R. Gavrilov, V. D. Ivashchuk, and V. N. Melnikov, “Multidimensional integrable vacuum cosmology with two curvatures,” Class. Quantum Grav. 13, 3039 (1996).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    A. E. Pavlov, “Mixmaster model associated to a Borcherds algebra,” Grav. Cosmol. 23, 20 (2017).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    V. Kac, Infinite-dimensional Lie algebras (Cambridge University Press, Cambridge, 1990).CrossRefzbMATHGoogle Scholar
  16. 16.
    V. G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinite-dimensional Lie algebras (World Scientific, 1987).zbMATHGoogle Scholar
  17. 17.
    V. D. Ivashchuk and V. N. Melnikov, “Quantum billiards in multidimensional models with branes,” Europ. Phys. J. C 74, 2805 (2014).ADSCrossRefGoogle Scholar
  18. 18.
    L. Carbone et al., “Classification of hyperbolic Dynkin diagrams, root lengths andWeyl group orbits,” arXiv: 1003. 0564.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Bogoliubov Laboratory for Theoretical PhysicsJoint Institute of Nuclear ResearchDubnaRussia
  2. 2.Institute of Mechanics and EnergeticsRussian State Agrarian University—Moscow Timiryazev Agricultural AcademyMoscowRussia

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