Communications in Mathematical Physics

, Volume 26, Issue 1, pp 24–38 | Cite as

Gravitational fields with groups of motions on two-dimensional transitivity hypersurfaces in a model with matter and a magnetic field

  • I. S. Shikin


For gravitational fields with metrics which admit of groups of motions multiply — transitive on 2-dimensional space-like invariant varieties, the exact solutions of the Einstein gravitational equations are given for the case when the sources of the gravitational field are dust-like matter and a magnetic field. A magnetic field is orientated along a direction orthogonal to transitivity hypersurface. The solutions contain arbitrary functions. In the case of transitivity hypersurface of positive curvature and in the absence of a magnetic field, the solution is reduced to the Tolman spherically symmetric solution for dust-like matter. The conditions are studied under which the solutions with a magnetic field become asymptotically isotropic and approach the flat and the open Friedmann models. The case of transitivity hypersurfaces with signature (+ −) is also considered.


Magnetic Field Neural Network Statistical Physic Exact Solution Complex System 
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  1. 1.
    Taub, A. H.: Ann. Math.53, 472 (1951).Google Scholar
  2. 2.
    Petrov, A. Z.: Einstein spaces. New York: Pergamon 1969.Google Scholar
  3. 3.
    Landau, L. D., Lifshitz, E. M.: Field theory. Moscow: Nauka 1967.Google Scholar
  4. 4.
    Bondi, H.: Monthly notices roy. Astron. Soc.107, 410 (1947).Google Scholar
  5. 5.
    Zeldovič, Ya. B.: Zh. Eksp. Teor. Fiz.48, 986 (1965);—Sov. Phys. JETP21, 656 (1965).Google Scholar
  6. 6.
    Doroshkevich, A. G.: Astrofizika1, 255 (1965), in Russian.Google Scholar
  7. 7.
    Shikin, I. S.: Dokl. Akad. Nauk SSSR171, 73 (1966);—Sov. Phys. Dokl.11, 944 (1967).Google Scholar
  8. 8.
    —— Dokl. Akad. Nauk SSSR176, 1048 (1967);—Sov. Phys. Dokl.12, 950 (1968).Google Scholar
  9. 9.
    Thorne, K. S.: Astrophys. J.148, 51 (1967).Google Scholar
  10. 10.
    Jacobs, K. C.: Astrophys. J.155, 379 (1969).Google Scholar
  11. 11.
    Goenner, H.: Commun. math. Phys.16, 34 (1970).Google Scholar
  12. 12.
    Kompaneets, A. S., Chernov, A. S.: Zh. Eksp. Teor. Fiz.47, 1939 (1964); — Sov. Phys. JETP20, 1303 (1965).Google Scholar
  13. 13.
    Kantowski, R., Sachs, R. K.: J. Math. Phys.7, 443 (1966).Google Scholar
  14. 14.
    Estabrook, F. B., Wahlquist, H. D., Behr, C. G.: J. Math. Phys.9, 497 (1968).Google Scholar
  15. 15.
    Goenner, H., Stachel, J.: J. Math. Phys.11, 3358 (1970).Google Scholar
  16. 16.
    Ellis, G. F. R.: J. Math. Phys.8, 1171 (1967).Google Scholar
  17. 17.
    Stewart, J. M., Ellis, G. F. R.: J. Math. Phys.9, 1072 (1968).Google Scholar
  18. 18.
    Vajk, J. P., Eltgroth, P. G.: J. Math. Phys.11, 2212 (1970).Google Scholar
  19. 19.
    Chap. 11 of Gravitation: An introduction to current research, ed. by L. Witten. New York: Wiley 1962.Google Scholar
  20. 20.
    Ellis, G. F. R., MacCallum, M. A. H.: Commun. math. Phys.12, 108 (1969).Google Scholar
  21. 21.
    Farnsworth, D. L.: J. Math. Phys.8, 2315 (1967).Google Scholar
  22. 22.
    Lichnerowicz, A.: Relativistic hydrodynamics and magneto-hydrodynamics. New York: Benjamin 1967.Google Scholar
  23. 23.
    Shikin, I. S.: Ann. Inst. Henri PoincaréA 11, 343 (1969).Google Scholar
  24. 24.
    Ruban, V. A.: Zh. Eksp. Teor. Fiz.56, 1914 (1969);—Sov. Phys. JETP29, 1027 (1969).Google Scholar
  25. 25.
    Hughston, L. P., Jacobs, K. C.: Astrophys. J.160, 147 (1970).Google Scholar
  26. 26.
    Melvin, M. A.: Phys. Rev.139, B 225 (1965).Google Scholar
  27. 27.
    Bertotti, B.: Phys. Rev.116, B 1331 (1959).Google Scholar
  28. 27a.
    Robinson, I.: Bull. Acad. Polon. Sci. Ser. Math.7, 351 (1959).Google Scholar

Copyright information

© Springer-Verlag 1972

Authors and Affiliations

  • I. S. Shikin
    • 1
  1. 1.Moscow State UniversityMoscowUSSR

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