General Relativity and Gravitation

, Volume 20, Issue 4, pp 371–382 | Cite as

Gravity and the frame field

  • J. S. R. Chisholm
  • R. S. Farwell
Research Articles


The covariant derivative of a single massive fermion field on a Riemannian manifold is defined. The standard method of defining free bosonic Lagrangians from the fermion covariant derivative does not give the usual Lagrangian density for the free gravitational field. We express the fermion Lagrangian mass term as a “frame field” term added to the covariant derivative; this “extended covariant derivative” defines a gravitational Lagrangian density proportional to the usual scalar curvatureR, plus a term quadratic in the curvature components. The quadratic term is expected to be negligible at distances much greater than the fermion Compton wavelength, and is of a general form widely studied in recent years. The frame field term used to derive this gravitational Lagrangian is essentially the same as that used previously to derive the electroweak interaction boson mass matrix without using the Higgs-Kibble mechanism.


Manifold Riemannian Manifold Scalar curvatureR Covariant Derivative Boson Mass 
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.
    Green, H. S. (1958).Nucl. Phys.,7, 373.Google Scholar
  2. 2.
    Halpern, L. (1977).Proceedings of the First Marcel Grossmann Meeting, Ruffini, R., ed. (North-Holland, Amsterdam), 113; (1978) AIP Conference Proceedings,48, Lannutti, J. E., and Williams, P. K., eds. (American Institute of Physics), 140.Google Scholar
  3. 3.
    Schrödinger, E. (1932). Sitz. Preuss. Akad. Wiss. Phys. Math. K1.,105.Google Scholar
  4. 4.
    Loos, H. G. (1963).Ann. Phys.,25, 91.Google Scholar
  5. 5.
    Rasevskii, P. K. (1957).Amer. Math. Soc. Trans.,2, 6, 1.Google Scholar
  6. 6.
    Chisholm, J. S. R., and Farwell, R. S. (1984).Il Nuovo Cim. A.,82, 2, 145.Google Scholar
  7. 7.
    Chisholm, J. S. R., and Farwell, R. S. (1987).J. Phys. A,20, 6561.Google Scholar
  8. 8.
    Drechsler, W., and Mayer, M. E. (1977). Lecture Notes in Physics 67 (Springer-Verlag, Berlin); (1987). Drechsler, W. U(4) spin gauge theory over a Riemann-Cartan space-time, Max-Planck-Institut preprint, MPI-PAE/PTh41/87.Google Scholar
  9. 9.
    Chisholm, J. S. R., and Farwell, R. S. (1981).Proc. Roy. Soc. Land. A,377, 1.Google Scholar
  10. 10.
    Chisholm, J. S. R., and Farwell, R. S. (1984).Il Nuovo Cim.,82A, 2, 185.Google Scholar
  11. 11.
    Chisholm, J. S. R., and Farwell, R. S. (1984).Il Nuovo Cim.,82A, 2, 210.Google Scholar
  12. 12.
    Eddington, A. S. (1930).Mathematical Theory of Relativity C. U. P.Google Scholar
  13. 13.
    Lanczos, C. (1969).J. Math. Phys.,10, 1059.Google Scholar
  14. 14.
    Bicknell, G. V. (1974).J. Phys.,A7, 1061.Google Scholar
  15. 15.
    Boulware, D. G. (1984).Quantum Theory of Gravity, Christensen, S. M. (d.), 267.Google Scholar
  16. 16.
    Isham, C. J. (1987). InProceedings of GR 11, M. A. H. MacCallum, ed. (C.U.P.).Google Scholar
  17. 17.
    Steele, K. S. (1978).Gen. Rel. Grav.,9, 353.Google Scholar
  18. 18.
    Storobinskii, A. A. (1980).J.E.T.P. Lett.,30, 682.Google Scholar
  19. 19.
    Barrow, J., and Ottewill, A. C. (1983).J. Phys. A,16, 2757.Google Scholar
  20. 20.
    Fradkin, E. S., and Tseytlin, A. A. (1982).Nucl. Phys. B,201, 469.Google Scholar
  21. 21.
    Barth, N. H., and Christensen, S. M. (1983).Phys. Rev. D,28, 1876.Google Scholar
  22. 22.
    Whitt, B. (1984).Phys. Lett. B,145, 176.Google Scholar
  23. 23.
    Boulware, D. G., Deser, S., and Stelle, K. S. (1986).Phys. Lett. B,168, 336.Google Scholar

Copyright information

© Plenum Publishing Corporation 1988

Authors and Affiliations

  • J. S. R. Chisholm
    • 1
  • R. S. Farwell
    • 2
  1. 1.University of KentCanterburyEngland
  2. 2.St. Mary's CollegeTwickenhamEngland

Personalised recommendations