Advertisement

General Relativity and Gravitation

, Volume 23, Issue 12, pp 1385–1402 | Cite as

Comments on the computation of Liapunov exponents for the Mixmaster universe

  • Beverly K. Berger
Research Articles

Abstract

To explain the discrepancy between recently computed vanishing Liapunov exponents for the evolution of Mixmaster universes and the positive Liapunov exponent for the associated 1-dimensional map first discussed by Belinskii, Khalatnikov, and Lifshitz, the Liapunov exponents computed from a numerical universe evolution are compared using several time variables. Previous numerical results of vanishing Liapunov exponents were obtained with time variables which increased roughly exponentially in each epoch. Here it is found that minisuperspace proper time, which increases by a fixed amount during each epoch, yields nonvanishing Liapunov exponents within the limited number of epochs numerically accessible. The map parameteru as measured along the trajectory attains the values predicted by the map to very high accuracy (except near the maximum of expansion) even though the metric coefficients deviate in some cases from idealized Mixmaster behavior. The number of consecutive single epoch eras is shown to be related to the presence of u in an interval bounded by ratios of Fibonacci numbers.

Keywords

Time Variable Differential Geometry Proper Time Fixed Amount Fibonacci Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Francisco, G., and Matsas, G. E. A. (1988).Gen. Rel. Grav. 20, 1047.Google Scholar
  2. 2.
    Burd, A. B., Buric, N., and Ellis, G. F. R. (1990).Gen. Rel. Grav. 22, 349.Google Scholar
  3. 3.
    Hobill, D., Bernstein, D., Welge, M., and Simkins, D. (1990). “The Mixmaster Cosmology as a Dynamical System,” National Center for Supercomputing Applications preprint.Google Scholar
  4. 4.
    Rugh, S. E., and Jones, B. J. T. (1990).Phys. Lett. A147, 353.Google Scholar
  5. 5.
    Berger, B. K. (1990).Class. Quant. Grav. 7, 203.Google Scholar
  6. 6.
    Pullin, J. (1990). “Time and Chaos in General Relativity,” Syracuse University preprint.Google Scholar
  7. 7.
    Lifshitz, E. M., and Khalatnikov, I. M. (1963).Adv. Phys. 12, 185; Lifshitz, E. M., and Khalatnikov, I. M. (1963).Usp. Fiz. Nauk. 80, 391; Khalatnikov, I. M., and Lifshitz, E. M. (1969).Phys. Rev. Lett. 24,76; Belinskii, V. A. and Khalatnikov, I. M. (1970).Zh. Eksp. Teor. Fiz. 59, 314; Belinskii, V. A., Khalatnikov, I. M., and Lifshitz, E. M. (1970).Adv. Phys. 19, 525; Belinskii, V. A., Khalatnikov, I. M., and Lifshitz, E. M. (1982).Adv. Phys. 31, 639; Belinskii, V. A., and Khalatnikov, I. M. (1969).Zh. Eksp. Teor. Fiz. 56, 1700; Belinskii, V. A., Khalatnikov, I. M., and Lifshitz, E. M. (1982).Adv. Phys. 31, 639.Google Scholar
  8. 8.
    Misner, C. W. (1969).Phys. Rev. 186, 1319; Misner, C. W. (1969).Phys. Rev. Lett. 22, 1071; Misner, C. W. (1970). InRelativity, ed. M. Carmeli et al. (Plenum, New York); Misner, C. W. (1972). InMagic without Magic, ed. J. Klauder (Freeman, San Francisco); Ryan, M. P. (1971).Ann. Phys. (N.Y.) 68, 541; Ryan, M. P. (1972).Hamiltonian Cosmology (Springer-Verlag, Berlin); Ryan, M. P. (1972).Ann. Phys. (N.Y.) 70, 301; Ryan, M. P., and Shepley, L. C. (1975).Homogeneous Relativistic Cosmologies (Princeton University, Princeton).Google Scholar
  9. 9.
    Barrow, J. D., and Chernoff, D. (1983).Phys. Rev. Lett. 50, 134; Barrow, J. D. (1984). InClassical General Relativity, ed. W. B. Bonnor, J. N. Islam, and M. A. H. MacCallum (Cambridge University Press, Cambridge).Google Scholar
  10. 10.
    Barrow, J. D. (1982).Phys. Rep.,85, 1.Google Scholar
  11. 11.
    Ma, P. K.-H., and Wainwright, J. (1991). InProceedings of the Third Hungarian Relativity Workshop, in press; Wainwright, J., and Hsu, L. (1989).Class. Quant. Grav. 6, 1409.Google Scholar
  12. 12.
    Jantzen, R. T. (1984). InCosmology of the Early Universe, ed. R. Ruffini and L. Z. Fang (World Scientific, Singapore).Google Scholar
  13. 13.
    Moser, A. R., Matzner, R. A., and Ryan, M. P. (1973)Ann. Phys. (N.Y.) 79, 558.Google Scholar
  14. 14.
    Bugalho, M. H., Rica da Suva, A., and Sousa Ramos, J. (1986).Gen. Rel. Grav. 18, 1263.Google Scholar
  15. 15.
    Lichtenberg, A. J., and Lieberman, M. A. (1983).Regular and Stochastic Motion (Springer-Verlag, New York).Google Scholar
  16. 16.
    Kasner, E. (1921).Am. J. Math. 43, 217.Google Scholar
  17. 17.
    Burd, A. B., Buric, N., and Tavakol, R. K. (1991).Class. Quant. Grav. 8, 123.Google Scholar
  18. 18.
    Shockley, J. E. (1967).Introduction to Number Theory (Holt, Rinehart, and Winston, New York), p. 191.Google Scholar
  19. 19.
    Wolf, A., Swift, J., Swinney, H., and Vastano, J. (1985).Physica 16D, 285.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Beverly K. Berger
    • 1
  1. 1.Physics DepartmentOakland UniversityRochesterUSA

Personalised recommendations