ABSTRACT
This chapter explains the theory of timelike loop spaces for globally hyperbolic space-times, summarizing some results of Uhlenbeck. Since completeness is necessary to develop the Riemannian loop space theory, it is not surprising that global hyperbolicity is needed for the Lorentzian theory. The chapter gives a proof of the Morse Index Theorem for timelike geodesics in an arbitrary space-time. Several different approaches to Morse index theory for timelike geodesics under various causality conditions have been published by Uhlenbeck, Woodhouse, Everson and Talbot, and Beem and Ehrlich. The chapter discusses the Morse theory of the path space of future directed timelike curves joining chronologically related points in a globally hyperbolic space-time following Uhlenbeck. Both of these treatments are modeled on Milnor’s exposition of the Morse theory for the path space of a complete Riemannian manifold in which the full path space is approximated by piecewise smooth geodesics.
