General Relativity and Gravitation

, Volume 18, Issue 8, pp 781–804 | Cite as

Teleparallelism as a universal connection on null hypersurfaces in general relativity

  • P. O. Mazur
  • L. M. Sokolowski
Research Articles


We show that there exists a close relationship between inner geometry of a null hypersurfaceN3 and the Newman-Penrose (NP) spin coefficient formalism. Projecting the null complexNP tetrad ontoN3 we get two triads of basis vectors inN3. Inner geometry ofN3 is based on the assumption that these vectors are parallelly transported along the surface; this gives rise to the teleparallel connection as a metric nonsymmetric affine connection. The gauge freedom for the choice of the basis triads is given by the isotropy subgroup of the local Lorentz group leaving invariant the direction of the null generators ofN3, and teleparallelism is determined by the equivalence class of the basis triads with respect to the global gauge group. Nine of the twelve NP coefficients are identified as the triad components of the torsion and the second fundamental form ofN3. The resulting generalized Gauss-Codazzi equations are identical to 9 of the NP equations, i.e., to the half of the Ricci identities. This result gives a geometrical meaning to the entire formalism. Finally we present a general proof of Penrose's theorem that the shear of the null generators ofN3 is the only initial null datum for a gravitational field onN3.


Isotropy Subgroup Affine Connection Null Hypersurface Global Gauge Coefficient Formalism 
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.
    Newman, E. T., and Penrose, R. (1962).J. Math. Phys.,3, 566.Google Scholar
  2. 2.
    Newman, E. T., and Penrose, R. (1963).J. Math. Phys.,4, 998.Google Scholar
  3. 3.
    Papapetrou, A. (1971a).C. R. Acad. Sci. Paris,272, 1537.Google Scholar
  4. 4.
    Papapetrou, A. (1971b).C. R. Acad. Sci. Paris,272, 1613.Google Scholar
  5. 5.
    Chandrasekhar, S. (1983).Mathematical Theory of Black Holes (Clarendon Press, Oxford/New York), pp. 39 and 51.Google Scholar
  6. 6.
    Eisenhart, L. P. (1966).Riemannian Geometry (Princeton, New Jersey), chapter IV.Google Scholar
  7. 7.
    Vogel, W. O. (1965).Archiv der Mathem.,16, 106.Google Scholar
  8. 8.
    Lemmer, G. (1965).Nuovo Cim.,37, 1659.Google Scholar
  9. 9.
    Galstyan, N. G. (1967).Doklady Akad. Nauk Armyan. SSR,45, 97.Google Scholar
  10. 10.
    Galstyan, N. G. (1968). InGravitation and Relativity, Kazan, in Russian.Google Scholar
  11. 11.
    Galstyan, N. G. (1969).Doklady Akad. Nauk Armyan. SSR,49, 166.Google Scholar
  12. 12.
    Galstyan, N. G. (1970).Doklady Akad. Nauk Armyan. SSR,50, 257 (all in Russian).Google Scholar
  13. 13.
    Kammerer, J. B. (1967a).Compt. Rend. Acad. Sci., Paris,264, 86.Google Scholar
  14. 14.
    Kammerer, J. B. (1967b).Rend. Circ. Matem. Palermo II,16, 129.Google Scholar
  15. 15.
    Bonnor, W. B. (1972).Tensor N.S.,24, 329.Google Scholar
  16. 16.
    Hajiček, P. (1973).Commun. Math. Phys.,34, 37.Google Scholar
  17. 17.
    Schouten, J. (1954).Ricci Calculus (Berlin) 2nd ed.Google Scholar
  18. 18.
    Dautcourt, G. (1967).J. Math. Phys.,8, 1492.Google Scholar
  19. 19.
    Dautcourt, G. (1968).Mathem. Nach.,36, 311.Google Scholar
  20. 20.
    Smallwood, J. (1979).J. Math. Phys.,20, 459.Google Scholar
  21. 21.
    Jankiewicz, C. (1954).Bull. Acad. Polon. Sci.,III2, 301.Google Scholar
  22. 22.
    Staruskiewicz, A. (1973a).Intern. J. Theor. Phys.,8, 247.Google Scholar
  23. 23.
    Staruszkiewicz, A. (1973b).Acta Phys. Polon. Ser. B,4, 57.Google Scholar
  24. 24.
    Sokolowski, L. M. (1975).Acta Phys. Polon. Ser. B,6, 657.Google Scholar
  25. 25.
    Sokolowski, L. M. (1976).The Geometry of Null Hypersurfaces, preprint TPJU, unpublished.Google Scholar
  26. 26.
    Mazur, P. O. (1978). Unpublished.Google Scholar
  27. 27.
    Geroch, R. (1977). InAsymptotic Structure of Space-Time, F. Esposito and L. Witten eds. (Plenum Press, New York).Google Scholar
  28. 28.
    Vanstone, J. (1964).Canad. J. Math.,16, 549.Google Scholar
  29. 29.
    Wong, Y. (1964).Nagoya Math. J.,24, 67.Google Scholar
  30. 30.
    Weitzenböck, R. (1923).Invariantentheorie (Moordhoff, Groningen).Google Scholar
  31. 31.
    Einstein, A. (1928).Sitzungsber. Akad. Wiss. (Berlin), Phys.-Math. Kl.,217, 224.Google Scholar
  32. 32.
    Hehl, F. W., Von Der Heyde, P., Kerlick, G., and Nester, J. (1976).Rev. Mod. Phys.,48, 393.Google Scholar
  33. 33.
    Müller-Hoissen, F., and Nitsch, J. (1983).Phys. Rev. Ser. D,28, 718.Google Scholar
  34. 34.
    Wigner, E. P. (1939).Am. Math.,40, 149.Google Scholar
  35. 35.
    Han, D., Kim, Y., and Son, D. (1982).Phys. Rev. Ser. D,26, 3717.Google Scholar
  36. 36.
    Rashevsky, P. K. (1967).Riemannian Geometry and Tensor Analysis, Moskva, pp. 570–584, in Russian.Google Scholar
  37. 37.
    Geroch, R., Held, A., and Penrose, R. (1973).J. Math. Phys.,14, 874.Google Scholar
  38. 38.
    Sachs, R. K. (1962).J. Math. Phys.,3, 908.Google Scholar
  39. 39.
    Penrose, R. (1980).Gen. Rel. Grav.,12, 225.Google Scholar
  40. 40.
    Müller Zum Hagen, H., and Seifert, H.-J. (1977).Gen. Rel. Grav.,8, 259.Google Scholar
  41. 41.
    Müller Zum Hagen, H., and Seifert, H.-J. (1979). InProceedings of the International School of Physics E. Fermi, Isolated Gravitating Systems in General Relativity, J. Ehlers, ed. (North Holland, Amsterdam).Google Scholar
  42. 42.
    Hehl, F. W., Nitsch, J., and Von Der Heyde, P. (1980). InGeneral Relativity and Gravitation. One Hundred Years After the Birth of Albert Einstein, A. Held, ed. (Plenum Press, New York), Vol. 1.Google Scholar

Copyright information

© Plenum Publishing Corporation 1986

Authors and Affiliations

  • P. O. Mazur
    • 1
  • L. M. Sokolowski
    • 2
  1. 1.Institute for Theoretical PhysicsUniversity of CaliforniaSanta Barbara
  2. 2.Astronomical ObservatoryJagellonian UniversityKrakowPoland

Personalised recommendations