General Relativity and Gravitation

, Volume 24, Issue 1, pp 59–85 | Cite as

Structure of the generalized Friedmann problem

  • R. T. Jantzen
  • C. Uggla
Research Articles


An investigation of those cases of the generalized Friedmann equation which are solvable in terms of elementary or elliptic functions is undertaken together with a study of the time gauges which allow this to occur. This is accomplished by examining the natural choices of independent and dependent variables in this problem using manipulations like those of the Kepler problem, which is shown to be equivalent to a generalized Friedmann problem, thus clarifying the similarities between the simplest solutions of each.


Differential Geometry Simple Solution Elliptic Function Natural Choice Friedmann Equation 
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  1. 1.
    Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973).Gravitation (W. H. Freeman, San Francisco).Google Scholar
  2. 2.
    York, J. W., Jr. (1972).Phys. Rev. Lett. 28, 1082.Google Scholar
  3. 3.
    Smarr, L., and York, J. W., Jr. (1978).Phys. Rev. D 17, 2529.Google Scholar
  4. 4.
    Landau, L. D., and Lifshitz, E. M. (1975).The Classical Theory of Fields (Pergamon Press, Oxford).Google Scholar
  5. 5.
    Moncrief, V. (1983).Geom. Phys.,1, 107; (1982).Ann. Phys. (N.Y.),141, 83.Google Scholar
  6. 6.
    Wainwright, J. (1980).Phys. Rev. D 22, 1906.Google Scholar
  7. 7.
    Szekeres, P. (1975).Commun. Math. Phys.,41, 55.Google Scholar
  8. 8.
    Kramer, D., Stephani, H., MacCallum, M. A. H., and Herlt, E. (1980).Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge).Google Scholar
  9. 9.
    Landau, L. D., and Lifshitz, E. M. (1965).Mechanics (Pergamon Press, Oxford).Google Scholar
  10. 10.
    Jantzen, R. T. (1984). InCosmology of the Early Universe, R. Ruffini and L. Z. Fang, eds. (World Scientific, Singapore);id. (1987). InGamov Cosmology, R. Ruffini and F. Melchiorri, eds. (North Holland, Amsterdam).Google Scholar
  11. 11.
    Rosquist, K., Uggla, C., and Jantzen, R. T. (1990).Class. Quant. Grav. 7, 611.Google Scholar
  12. 12.
    York, J. W., Jr. (1979). InSources of Gravitational Radiation, L. Smarr, ed. (Cambridge University Press, Cambridge).Google Scholar
  13. 13.
    Jantzen, R. T. (1987).Phys. Lett. 186B, 290.Google Scholar
  14. 14.
    Taub, A. H. (1951).Ann. Math.,53, 472.Google Scholar
  15. 15.
    Arnowit, R., Deser, S., and Misner, C. W. (1962). InGravitation: An Introduction to Current Research, L. Witten, ed. (John Wiley, New York).Google Scholar
  16. 16.
    Rosquist, K., Fišer, K., and Uggla, C. (1991). Preprint.Google Scholar
  17. 17.
    Jantzen, R. T. (1988).Phys. Rev. 37, 3472.Google Scholar
  18. 18.
    Belinsky, V. A., and Khalatnikov, I. M. (1973).Sov. Phys. JETP,36, 591.Google Scholar
  19. 19.
    Bogoyavlensky, O. I., and Novikov, S. P. (1974).Sov. Phys. JETP,37, 747;id. (1976).Trudy Sem. Petrovxk,1, 7 [English transl., 1982, inSel. Math. Sov.,2, 159];id. (1976).Trudy Sem. Petrovxk,2, 159 [English transl., 1985, inAm. Math. Soc. Trans.,125, 83].Google Scholar
  20. 20.
    Jantzen, R. T. (1980).Ann. Phys. (N.Y.),127, 302.Google Scholar
  21. 21.
    Barrow, J. D., and Stein-Schabes, J. (1984).Phys. Lett. 103A, 315.Google Scholar
  22. 22.
    Jacobs, K. C. (1968).Astrophys. J.,153, (1969).Astrophys. J.,155, 379.Google Scholar
  23. 23.
    Assad, M. J. D., and Sales de Lima, J. A. (1988).Gen. Rel. Grav. 20, 527.Google Scholar
  24. 24.
    Gotay, M. C., and Demaret, J. (1983).Phys. Rev. 28, 2402.Google Scholar
  25. 25.
    Robertson, H. P. (1933).Rev. Mod. Phys.,5, 62.Google Scholar
  26. 26.
    Stabell, R., and Refsdal, S. (1966).Mon. Not. R. Astr. Soc.,132, 379.Google Scholar
  27. 27.
    Harrison, E. R. (1967).Mon. Not. R. Astr. Soc.,137, 69.Google Scholar
  28. 28.
    Coquereaux, R., and Grossmann, A. (1982).Ann. Phys. (N.Y.),143, 296.Google Scholar
  29. 29.
    Dabrowski, M., and Stelmach, J. (1986).Ann. Phys. (N.Y.),166, 422.Google Scholar
  30. 30.
    Whittaker, E. T. (1927).Analytical Dynamics (Cambridge University Press, Cambridge).Google Scholar
  31. 31.
    Goldstein, H. (1950).Classical Mechanics (Addison-Wesley, Reading, Mass.).Google Scholar
  32. 32.
    Arnold, V. I. (1989).Mathematical Methods of Classical Mechanics (Springer-Verlag, New York).Google Scholar
  33. 33.
    Rauch, H. E., and Lebowitz, A. (1973).Elliptic Functions, Theta Functions, and Riemann Functions (Williams & Wilkins, Baltimore).Google Scholar
  34. 34.
    Gradshteyn, I. S., and Ryzhik, I. M. (1965).Table of Integrals, Series and Products (Academic Press, New York).Google Scholar
  35. 35.
    Whittaker, E. T., and Watson, G. N. (1952).A Course of Modern Analysis (Cambridge University Press, Cambridge).Google Scholar

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • R. T. Jantzen
    • 1
    • 2
  • C. Uggla
    • 3
    • 4
  1. 1.Department of Mathematical SciencesVillanova UniversityVillanovaUSA
  2. 2.International Center for Relativistic Astrophysics, Department of PhysicsUniversity of RomeRomeItaly
  3. 3.Department of PhysicsSyracuse UniversitySyracuseUSA
  4. 4.Department of PhysicsUniversity of StockholmStockholmSweden

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