General Relativity and Gravitation

, Volume 9, Issue 7, pp 621–635 | Cite as

Classical theory of the interaction between a spinor field and the gravitational field: First-order field equations

  • N. C. T. Coote
  • A. J. Macfarlane
Research Articles


The classical theory of the interaction of a neutral spin-1/2 Dirac field and the gravitational field is studied. For the purely gravitational part of the Lagrangian, written in terms of a vierbein and the local connection coefficient ω aμb , (regarded as independent field variables), the usual first-order form is adopted. For the Dirac part, however, a different choice is made, in which the covariant derivative ofψ is built with the aid of the vierbein instead of with ω aμb . This still yields a first-order formalism, but one in which ω aμb is related to the vierbein in the same way as it would be in the absence ofψ. This ensures that the global connection Г μν α remains symmetric inμ andv in the presence ofψ. The way in which the vierbein field equation leads to a familiar Einstein equation with a symmetric and conserved stress tensor on its right side is also analyzed.


Field Equation Classical Theory Gravitational Field Einstein Equation Spinor Field 
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.
    Kibble, T. W. B. (1961).J. Math. Phys.,2, 212.Google Scholar
  2. 2.
    Kibble, T. W. B. (1963).J. Math. Phys.,4, 1433.Google Scholar
  3. 3.
    Schwinger, J. (1963).Phys. Rev.,130, 406.Google Scholar
  4. 4.
    Schwinger, J. (1963).Phys. Rev.,130, 1253.Google Scholar
  5. 5.
    Schwinger, J. (1970).Particles, Sources and Fields, Vol 1 (Addison Wesley, Reading, Massachusetts).Google Scholar
  6. 6.
    Macfarlane, A. J. (1961).Commun. Math. Phys.,2, 133.Google Scholar
  7. 7.
    Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973).Gravitation (Freeman, San Francisco).Google Scholar
  8. 8.
    Weinberg, S. (1972).Gravitation and Cosmology (Wiley, New York).Google Scholar
  9. 9.
    DeWitt, B. S. (1965).Dynamical Theory of Groups and Fields (Gordon and Breach, New York).Google Scholar
  10. 10.
    Deser, S., and van Nieuwenhuizen, P. (1974).Phys. Rev. D,10, 411.Google Scholar
  11. 11.
    Boulware, D. (1975).Phys. Rev. D,12, 350.Google Scholar

Copyright information

© Plenum Publishing Corporation 1978

Authors and Affiliations

  • N. C. T. Coote
    • 1
  • A. J. Macfarlane
    • 1
  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeEngland

Personalised recommendations