Advertisement

Foundations of Physics

, Volume 13, Issue 5, pp 501–527 | Cite as

Gravitational field equations based on Finsler geometry

  • G. S. Asanov
Article

Abstract

The analysis of a previous paper (see Ref. 1), in which the possibility of a Finslerian generalization of the equations of motion of gravitational field sources was demonstrated, is extended by developing the Finslerian generalization of the gravitational field equations on the basis of the complete contractionK = K lj lj of the Finslerian curvature tensorK l j hk (x, y). The relevant Lagrangian is constructed by the replacement of the directional variabley i inK by a vector fieldy i (x), so that the notion of osculation may be regarded as the key concept on which the approach is based. The introduction of the auxiliary vector fieldy i (x) is shown to be of physical significance, for the field equations refer not only to the proper field variables but also to a special coordinate system associated withy i (x) through the Clebsch representation of the latter. The status of the conservation laws proves to be similar to that in the theory of the Yang-Mills field. By choosing a special Finslerian metric function we elucidate in detail the structure of the field equations in the static case.

Keywords

Coordinate System Field Equation Gravitational Field Static Case Physical Significance 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. S. Asanov,Found. Phys. 11, 137 (1981).Google Scholar
  2. 2.
    H. Rund,The Differential Geometry of Finsler Spaces (Springer-Verlag, Berlin, 1959). (Russian translation: Nauka, Moscow, 1981.)Google Scholar
  3. 3.
    H. Rund and J. H. Beare,Variational Properties of Direction-Dependent Metric Fields (University of South Africa, Pretoria, 1972).Google Scholar
  4. 4.
    G. S. Asanov,Inst. Math. Polish Academy of Sci. 195, 1 (1979).Google Scholar
  5. 5.
    G. S. Asanov and E. G. Kirnasov,Rep. Math. Phys. 19, 35 (1981).Google Scholar
  6. 6.
    A. Einstein,Preuss. Akad. Wiss., Phys.-Math. Kl. Sitzungsber. (1928), pp. 217–221.Google Scholar
  7. 7.
    H. Rund, inTopics in Differential Geometry (Academic Press, New York, 1976), pp. 111–133.Google Scholar
  8. 8.
    G. S. Asanov,Found. Phys. 10, 855 (1980).Google Scholar
  9. 9.
    D. Lovelock and H. Rund,Tensors, Differential Forms and Variational Principles (Wiley, New York, 1975).Google Scholar
  10. 10.
    A. Krasiński,Acta Phys. Polon. B5, 411 (1974).Google Scholar
  11. 11.
    R. Baumeister,Utilitas Mathematica 16, 43 (1979).Google Scholar
  12. 12.
    G. Cavalleri and G. Spinelli,Rivista del Nuovo Cim. 3, 8 (1980).Google Scholar
  13. 13.
    A. Trautman, inGravitation (Wiley, London, 1962), pp. 169–198.Google Scholar
  14. 14.
    J. L. Synge,Relativity: The General Theory (North-Holland, Amsterdam, 1960), formula (8.179).Google Scholar
  15. 15.
    M. Sachs,Found. Phys. 11, 329 (1981).Google Scholar

Copyright information

© Plenum Publishing Corporation 1983

Authors and Affiliations

  • G. S. Asanov
    • 1
  1. 1.Department of Theoretical PhysicsMoscow State UniversityMoscowUSSR

Personalised recommendations