International Journal of Theoretical Physics

, Volume 30, Issue 4, pp 555–565 | Cite as

Cauchy boundary andb-incompleteness of space-time

  • Jacek Gruszczak
  • Michael Heller
  • Zdzisław Pogoda


It is shown that if a space-time (M, g) is time-orientable and its Levi-Civita connection [in the bundle of orthonormal frames over (M, g)] is reducible to anO(3) structure, one can naturally select a nonvanishing timelike vector fieldξ and a Riemann metricg+ onM. The Cauchy boundary of the Riemann space (M, g+) consists of “endpoints” ofb-incomplete curves in (M, g); we call it theCauchy singular boundary. We use the space-time of a cosmic string with a conic singularity to test our method. The Cauchy singular boundary of this space-time is explicitly constructed. It turns out to consist of what should be expected.


Endpoint Field Theory Elementary Particle Quantum Field Theory Cosmic String 
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. Beem, J., and Ehrlich, P. (1981).Global Lorentzian Geometry, Dekker.Google Scholar
  2. Bosshard, B. (1976).Communications in Mathematical Physics,46, 263.Google Scholar
  3. Clarke, C. J. S. (1979).General Relativity and Gravitation,10, 977.Google Scholar
  4. Crittenden, R. (1962).Quarterly Journal of Mathematics (Oxford),13, 285.Google Scholar
  5. Dodson, C. T. J. (1978).International Journal of Theoretical Physics,17, 389.Google Scholar
  6. Dodson, C. T. J. (1979).General Relativity and Gravitation,10, 969.Google Scholar
  7. Ellis, G. F. R., and Schmidt, B. (1977).General Relativity and Gravitation,8, 915.Google Scholar
  8. Gancarzewicz, J. (1987).Differential Geometry, Polish Scientific Publishers [in Polish].Google Scholar
  9. Geroch, R., and Horowitz, G. T. (1979). InGeneral Relativity—An Einstein Centenary Survey, S. W. Hawking and W. Israel, eds., Cambridge University Press, Cambridge, p. 212.Google Scholar
  10. Geroch, R., Kronheimer, E. H., and Penrose, R. (1972).Proceedings of the Royal Society of London A,327, 545.Google Scholar
  11. Gruszczak, J. (1990).International Journal of Theoretical Physics,29(1), (1990).Google Scholar
  12. Hawking, S. W., and Ellis, G. F. R. (1973).Large Scale Structure of Space-Time, Cambridge University Press, Cambridge.Google Scholar
  13. Johnson, R. A. (1977).Journal of Mathematical Physics,18, 898.Google Scholar
  14. Kobayashi, S., and Nomizu, K. (1963).Foundations of Differential Geometry, Vol. 1, Interscience, New York.Google Scholar
  15. Penrose, R. (1978).Theoretical Principles in Astrophysics and Relativity, N. R. Lebovitz, W. H. Rein, and P. O. Vandervoort, eds., University of Chicago Press, Chicago, Illinois.Google Scholar
  16. Sachs, R. K. (1973).Communications in Mathematical Physics,33, 215.Google Scholar
  17. Schmidt, B. G. (1971).General Relativity and Gravitation,1, 269.Google Scholar
  18. Schmidt, B. G. (1979).General Relativity and Gravitation,10, 981.Google Scholar
  19. Staruszkiewicz, A. (1963).Acta Physica Polonica,24, 734.Google Scholar
  20. Steenrod, N. (1951).Topology of Fiber Bundles, Princeton University Press, Princeton, New Jersey.Google Scholar
  21. Szabados, L. B. (1988).Classical and Quantum Gravity,5, 121.Google Scholar
  22. Szabados, L. B. (1989).Classical and Quantum Gravity,6, 77.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • Jacek Gruszczak
    • 1
    • 2
  • Michael Heller
    • 3
    • 2
  • Zdzisław Pogoda
    • 4
  1. 1.Institute of PhysicsPedagogical UniversityCracowPoland
  2. 2.Cracow Group of CosmologyPoland
  3. 3.Vatican Astronomical ObservatoryVatican City State
  4. 4.Institute of MathematicsJagiellonian UniversityCracowPoland

Personalised recommendations