Communications in Mathematical Physics

, Volume 20, Issue 1, pp 57–84 | Cite as

A class of homogeneous cosmological models III: Asymptotic behaviour

  • M. A. H. MacCallum


The behaviour near to and far from an initial singularity in a broad subclass of the models studied in previous papers [1–3] is examined. The influence of the matter on the evolution at these times is discussed. The singularity types for the various models, which are mostly of cigar or oscillatory nature, are found. It is discovered that among these models, only those of the same Bianchi type as a Robertson-Walker model can become “approximately Robertson-Walker” in a sense defined in the paper. Qualitative conclusions concerning black-body isotropy, the Hubble relation, helium abundance and horizon structure are given.


Neural Network Statistical Physic Helium Complex System Asymptotic Behaviour 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ellis, G. F. R., MacCallum, M. A. H.: Commun. Math. Phys.12, 108 (1969).Google Scholar
  2. 2.
    Schmidt, B. G.: Commun. Math. Phys.15, 329 (1969).Google Scholar
  3. 3.
    MacCallum, M. A. H., Ellis, G. F. R.: Commun. Math. Phys.19, 31 (1970).Google Scholar
  4. 4.
    Kantowski, R., Sachs, R. K.: J. Math. Phys.7, 443 (1966); Kantowski, R.: Ph.D. thesis, University of Texas 1966.Google Scholar
  5. 5.
    Hawking, S. W., Ellis, G. F. R.: Phys. Letters17, 246 (1965).Google Scholar
  6. 6.
    Ellis, G. F. R.: Lectures at the International School of Physics, Corso XLVII, Varenna, to be published (1969).Google Scholar
  7. 7.
    Hawking, S. W., Ellis, G. F. R.: Astrophys. J.152, 25 (1968); Hawking, S. W., Penrose, R.: Proc. Roy. Soc. (London) A314, 529 (1970).Google Scholar
  8. 8.
    Bianchi, L.: Lezioni sulla teoria dei gruppi continui finiti transformazioni. Pisa: Editiozi Spoerri 1918.Google Scholar
  9. 9.
    Estabrook, F. B., Wahlquist, H. D., Behr, C. G.: J. Math. Phys.9, 497 (1968).Google Scholar
  10. 10.
    Misner, C. W.: Astrophys. J.151, 431 (1968).Google Scholar
  11. 11.
    —— Phys. Rev. Letters22, 1071 (1969): Phys. Rev.186, 1319 (1969).Google Scholar
  12. 12.
    Hawking, S. W.: Monthly Notices Roy. Astron. Soc.142, 129 (1969).Google Scholar
  13. 13.
    Lifschitz, E. M., Khalatnikov, I. M.: Z.E.T.F. Pisma11, 200 (1970), in Russian.Google Scholar
  14. 14.
    Gödel, K.: Proc. Int. Congr. Math. (Cambridge), Vol.I, 175 (1950).Google Scholar
  15. 15.
    Behr, C. G.: Astron. Abh. Hamburg. Sternwarte7, 5 (1965); Z. Astrophys.54, 268 (1962), and60, 286 (1965).Google Scholar
  16. 16.
    Ozsvath, I.: University of Texas preprint “Spatially Homogeneous World Models” (1970).Google Scholar
  17. 17.
    Misner, C. W.: in Relativity, ed. M. Carmeli, S. Fickler and L. Witten. New York: Plenum Press 1970.Google Scholar
  18. 18.
    Ellis, G. F. R.: J. Math. Phys.8, 1171 (1967); Stewart, J. M., Ellis, G. F. R.: J. Math. Phys.9, 1072 (1968).Google Scholar
  19. 19.
    MacCallum, M. A. H., Stewart, J. M., Schmidt, B. G.: Commun. Math. Phys.17, 343 (1970).Google Scholar
  20. 20.
    Misner, C. W.: Phys. Rev.186, 1328 (1969).Google Scholar
  21. 21.
    ter Haar, D.: Elements of Hamiltonian mechanics. Amsterdam: North-Holland 1961.Google Scholar
  22. 22.
    Thorne, K. S.: Astrophys. J.148, 51 (1967).Google Scholar
  23. 23.
    Rindler, W.: Monthly Notices Roy. Astron. Soc.116, 662 (1956).Google Scholar
  24. 24.
    Penrose, R.: Battelle rencontres 1967, ed. C. M. de Witt and J. A. Wheeler. New York: Benjamin 1968.Google Scholar
  25. 25.
    Carswell, R.: Monthly Notices Roy. Astron. Soc.144, 279 (1969); Ph. D. thesis, University of Sussex, 1969.Google Scholar
  26. 26.
    Novikov, I. D.: Sov. Astron.-A.J.12, 427 (1969); Russian original, Astron. Zh.45, 538 (1968).Google Scholar
  27. 27.
    Saunders, P. T.: Ph.D. thesis, King's College, London (1967); Monthly Notices Roy. Astron. Soc.142, 213 (1969).Google Scholar
  28. 28.
    Jacobs, K. C.: Astrophys. J.153, 661 (1968) and155, 379 (1969).Google Scholar
  29. 29.
    Kasner, E.: Am. J. Math.43, 217 (1921).Google Scholar
  30. 30.
    Lifschitz, E. M., Khalatnikov, I. M.: Advanc. Phys.12, 185 (1963).Google Scholar
  31. 31.
    Belinsky, V. A., Khalatnikov, I. M.: J.E.T.P.29, 911 (1969) and to appear; Russian originals. Z.E.T.F.56, 1700 (1969) and57, 2163 (1969).Google Scholar
  32. 32.
    Khalatnikov, I. M., Lifschitz, E. M.: Phys. Rev. Letters24, 76 (1970).Google Scholar
  33. 33.
    Misner, C. W., Taub, A. H.: J.E.T.P.28, 22 (1969); Russian original Z.E.T.F.55, 233 (1968).Google Scholar
  34. 34.
    Taub, A. H.: Ann. Math.53, 472 (1951).Google Scholar
  35. 35.
    Newman, E. T., Tamburino, L., Unti, T.: J. Math. Phys.4, 915 (1963).Google Scholar
  36. 36.
    Harrison, E. R.: Astrophys. J.142, 1643 (1965); Chanmugam, G.: J. Phys. A3, L.19 (1970).Google Scholar
  37. 37.
    Hawking, S. W.: Adams Prize Essay, Cambridge University 1966.Google Scholar
  38. 38.
    Tomita, K.: Prog. Theor. Phys.40, 264 (1968).Google Scholar
  39. 39.
    Hawking, S. W., Tayler, R. J.: Nature209, 1278 (1966).Google Scholar
  40. 40.
    Ehlers, J.: Lectures at the International School of Physics, Varenna, Corso XLVII; to be published (1969).Google Scholar
  41. 41.
    Wagoner, R. V., Fowler, W. A., Hoyle, F.: Astrophys. J.148, 3 (1967). Tayler, R. J.: Nature217, 433 (1968).Google Scholar
  42. 42.
    Stewart, J. M.: Ph.D. thesis, Cambridge University (1969); Monthly Notices Roy. Astron. Soc.145, 347 (1969).Google Scholar
  43. 43.
    Matzner, R. A., Shepley, L. C., Warren, D. R.: Ann. Phys. (N. Y.)57, 401 (1970).Google Scholar
  44. 44.
    Doroshkevich, A. G., Novikov, I. D.: Preprint No. 15, Institute of Applied Mathematics, Acad. Sci. U.S.S.R., Moscow (1970).Google Scholar
  45. 45.
    Matzner, R. A.: Astrophys. J.157, 1085 (1969).Google Scholar
  46. 46.
    Taub, A. H.: In: Proceedings of the 1967 Colloque sur fluides et champ gravitationnel. Paris: Centre National de la Recherches Scientifiques 1969.Google Scholar

Copyright information

© Springer-Verlag 1971

Authors and Affiliations

  • M. A. H. MacCallum
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeUK

Personalised recommendations