Advertisement

International Journal of Theoretical Physics

, Volume 30, Issue 9, pp 1171–1215 | Cite as

Holonomy and path structures in general relativity and Yang-Mills theory

  • J. W. Barrett
Article

Abstract

This article is about a different representation of the geometry of the gravitational field, one in which the paths of test bodies play a crucial role. The primary concept is the geometry of the motion of a test body, and the relation between different such possible motions. Space-time as a Lorentzian manifold is regarded as a secondary construct, and it is shown how to construct it from the primary data. Some technical problems remain. Yang-Mills fields are defined by their holonomy in an analogous construction. I detail the development of this idea in the literature, and give a new version of the construction of a bundle and connection from holonomy data. The field equations of general relativity are discussed briefly in this context.

Keywords

Manifold Field Theory Crucial Role General Relativity Elementary Particle 
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. Aharonov, Y., and Bohm, D. (1959).Physical Review,115, 485–491.Google Scholar
  2. Anandan, J. (1983). Holonomy groups in gravity and gauge fields, inProceedings Conference Differential Geometric Methods in Physics, Trieste 1981, G. Denardo and H. D. Doebner, eds., World Scientific, Singapore.Google Scholar
  3. Atiyah, M. F. (1980). Geometrical aspects of gauge theories, inProceedings International Congress Mathematics, O. Lento, ed., Helsinki, pp. 881–885.Google Scholar
  4. Babelon, O., and Viallet, C. M. (1981).Communications in Mathematical Physics,81, 515–525.Google Scholar
  5. Barrett, J. W. (1985). The holonomy description of classical Yang-Mills theory and general relativity, Ph.D. thesis, University of London.Google Scholar
  6. Barrett, J. W. (1986).Classical and Quantum Gravity,3, 203–206.Google Scholar
  7. Barrett, J. W. (1987).Classical and Quantum Gravity,4, 1565–1576.Google Scholar
  8. Barrett, J. W. (1988).Classical and Quantum Gravity,5, 1187–1192.Google Scholar
  9. Barrett, J. W. (1989).General Relativity and Gravitation,21, 457–466.Google Scholar
  10. Bialynicki-Birula, I. (1963).Bulletin de l'Académie Polonaise des Sciences,11, 135.Google Scholar
  11. Cartan, E. (1922).Comptes Rendus,174, 437–439.Google Scholar
  12. Chan, H.-M., and Tsou, S. T. (1986).Acta Physica Polonica B,17, 259–276.Google Scholar
  13. Chan, H.-M., Scharbach, P., and Tsou, S. T. (1986).Annals of Physics,166, 396–421.Google Scholar
  14. Dirac, P. A. M. (1931).Proceedings of the Royal Society of London A,133, x-xxi.Google Scholar
  15. Dugundji, J. (1966).Topology, Allyn and Bacon, Boston.Google Scholar
  16. Durhuus, B. (1980).Letters in Mathematical Physics,4, 515–522.Google Scholar
  17. Einstein, A. (1922).The Meaning of Relativity, 6th ed., Chapman and Hall, London.Google Scholar
  18. Fischer, A. E. (1986).General Relativity and Gravitation,18, 597–608.Google Scholar
  19. Giles, R. (1981).Physical Review D,24, 2160–2168.Google Scholar
  20. Hawking, S. W., and Ellis, G. F. R. (1973).The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge.Google Scholar
  21. Isham, C. J. (1981). Quantum gravity—An overview, inQuantum Gravity 2, A Second Oxford Symposium, C. J. Isham, R. Penrose, and D. W. Sciama, eds., Clarendon Press, Oxford.Google Scholar
  22. Isham, C. J. (1984). Topological and global aspects of quantum theory, in1983 Les Houches Summer School Lectures “Relativity Groups and Topology”, North-Holland, Amsterdam.Google Scholar
  23. Kelly, R. M., Tod, K. P., and Woodhouse, N. M. J. (1986).Classical and Quantum Gravitation,3, 1151–1167.Google Scholar
  24. Kibble, T. W. B. (1961).Journal of Mathematical Physics,2, 212–221.Google Scholar
  25. Kobayashi, S. (1954).Comptes Rendus,238, 443–444.Google Scholar
  26. Kobayashi, S., and Nomizu, K. (1963).Foundations of Differential Geometry, Volume 1, Interscience, New York.Google Scholar
  27. Lashof, R. (1956).Annals of Mathematics,64, 436–446.Google Scholar
  28. Mandelstam, S. (1962a).Annals of Physics,19, 1–24.Google Scholar
  29. Mandelstam, S. (1962b).Annals of Physics,19, 25–66.Google Scholar
  30. Mandelstam, S. (1968a).Physical Review,175, 1580–1603.Google Scholar
  31. Mandelstam, S. (1968b).Physical Review,175, 1604–1623.Google Scholar
  32. Miller, W. A. (1986).Foundations of Physics,16, 143–169.Google Scholar
  33. Milnor, J. (1956).Annals of Mathematics,63, 272–284.Google Scholar
  34. Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1972).Gravitation, Freeman, San Francisco.Google Scholar
  35. Narasimhan, M. S., and Ramadas, T. R. (1979).Communications in Mathematical Physics,67, 121–136.Google Scholar
  36. Penrose, R. (1982).Proceedings of the Royal Society of London A,381, 53–63.Google Scholar
  37. Penrose, R., and MacCullum, M. A. H. (1973).Physics Reports,6C, 241–316.Google Scholar
  38. Polyakov, A. M. (1979).Nuclear Physics B,164, 171–188.Google Scholar
  39. Regge, T. (1961).Nuovo Cimento,19, 558–571.Google Scholar
  40. Sciama, D. W. (1962). On the analogy between charge and spin in general relativity, inRecent Developments in General Relativity, Pergamon, Oxford.Google Scholar
  41. Singer, I. M. (1981).Physica Scripta,24, 817–820.Google Scholar
  42. Spanier, E. H. (1966).Algebraic Topology, McGraw-Hill, New York.Google Scholar
  43. Teleman, C. (1960).Annales Scientifiques de l'École Normale Supérieure 3,77, 195–234.Google Scholar
  44. Teleman, C. (1963).Annali di Matematica, Pura ed Applicata,LXII, 379–412.Google Scholar
  45. Teleman, C. (1969a).Indagationes Mathematicae,31, 89–103.Google Scholar
  46. Teleman, C. (1969b).Indagationes Mathematicae,31, 104–112.Google Scholar
  47. Wilson, K. G. (1974).Physical Review D,10, 2445–2459.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • J. W. Barrett
    • 1
  1. 1.Department of PhysicsThe UniversityNewcastle upon TyneEngland

Personalised recommendations