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

Abstract

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.

Keywords

Manifold Differential Geometry Conjugate Point Index Theorem Morse Theory 

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Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

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

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