General Relativity and Gravitation

, Volume 7, Issue 7, pp 609–621 | Cite as

Morse theory on timelike and causal curves

  • J. Everson
  • C. J. Talbot
Research Articles


It is shown that the set of timelike curves in a globally hyperbolic space-time manifold can be given the structure of a Hubert manifold under a suitable definition of “timelike.” The causal curves are the topological closure of this manifold. The Lorentzian energy (corresponding to Milnor's energy, except that the Lorentzian inner product is used) is shown to be a Morse function for the space of causal curves. A fixed end point index theorem is obtained in which a lower bound for the index of the Hessian of the Lorentzian energy is given in terms of the sum of the orders of the conjugate points between the end points.


Manifold Differential Geometry Conjugate Point Index Theorem Morse Theory 
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.
    Palais, R. S. (1963).Topology,2, 299.Google Scholar
  2. 2.
    Woodhouse, N. M. J. (1973). Ph.D. thesis (King's College, London).Google Scholar
  3. 3.
    Milnor, J. (1969).Morse Theory (Princeton University Press, Princeton, New Jersey).Google Scholar
  4. 4.
    Hawking, S. W. (1971).Gen. Rel. Grav.,1, 393.Google Scholar
  5. 5.
    Fischer, A. E., and Marsden, J. E. (1974).Gen. Rel. Grav.,5, 71.Google Scholar
  6. 6.
    Clarke, C. J. S. (1970).Proc. Roy. Soc. A,314, 417.Google Scholar
  7. 7.
    Hawking, S. W., and Ellis, C. F. R. (1973).The Large Scale Structure of Spacetime. (Cambridge University Press, Cambridge).Google Scholar
  8. 8.
    Penrose, R. (1972).Techniques of Differential Topology in Relativity. (Siam).Google Scholar
  9. 9.
    Synge, J. L. (1965).Relativity: the Special Theory. (North Holland, Amsterdam).Google Scholar
  10. 10.
    Sternberg, S. (1964).Lectures on Differential Geometry. (Prentice-Hall, Englewood Cliffs, New Jersey).Google Scholar
  11. 11.
    Lang, S. (1972).Differential Manifolds. (Addison Wesley, Reading, Massachusetts).Google Scholar
  12. 12.
    Uhlenbeck, K. (1975).Topology,14, 69.Google Scholar
  13. 13.
    Everson, J. (1975). Proposed D.Phil Thesis. (The Polytechnic, Huddersfield, England).Google Scholar
  14. 14.
    Eisenhardt, L. P. (1974).An Introduction to Differential Geometry, (Princeton University Press, Princeton, New Jersey), p. 246.Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • J. Everson
    • 1
  • C. J. Talbot
    • 1
  1. 1.Department of MathematicsThe PolytechnicHuddersfieldEngland

Personalised recommendations