ABSTRACT
In this chapter we shall study geodesics on nonstationary Lorentzian manifolds; in this case the coefficients of the metric depend on the time coordinate. Unfortunately the variational principle of Chapter 3 does not hold, so we cannot reduce the search of geodesies to the study of a “Riemannian” functional. We have to apply the critical point theory for indefinite functionals, recently developed by many authors. The Saddle Point Theorem of Rabinowitz and the Relative Category, which is an extension of the classical Ljusternik-Schnirelmann category for unbounded functionals, are the abstract tools which we shall use. However, they work when the Morse index of the critical points of a functional is finite. As we know, the Morse of a geodesic of a Lorentzian manifold (as critical point of the action integral) is + ∞. To avoid this problem we shall use a Galerkin finite dimensional approximation argument (we point out that the gradient of the action integral is not a compact perturbation of the identity, so we can not apply the linking results of Benci-Rabinowitz, see [BR]).
