A Morse Complex for Lorentzian Geodesics

Abstract

We prove the Morse relations for all geodesics connecting two non-conjugate points on a class of globally hyperbolic Lorentzian manifolds. We overcome the difficulties coming from the fact that the Morse index of every geodesic is infinite, and from the lack of the Palais-Smale condition, by using the Morse complex approach.

Comparison results for conjugate and focal points in semi-Riemannian geometry via Maslov index

“…Estimates on the number of geodesics in terms of conjugate points can be done either via Morse theory or via bifurcation theory. Recent results in this direction are available also in the case of non-Riemannian metrics; see for instance [Abbondandolo et al 2003;Abbondandolo and Majer 2008] for the Morse theory of geodesics in globally hyperbolic Lorentzian manifolds, [Giannoni et al 2001] for that theory in stationary Lorentzian manifolds, and [Piccione et al 2004] for the bifurcation theory of geodesics in arbitrary semi-Riemannian manifolds.…”

Spectral flow and iteration of closed semi-Riemannian geodesics

“…Hilbert space H = H z in (4.1) and to the closed finite codimensional subspace V = H o defined in (4.2), obtaining: Proposition 4.6 For all z ∈ S 1 , the following equality holds:3 …”

“…Recall that a semi-Riemannian manifold is a manifold M endowed with a nondegenerate, but possibly non positive definite, metric tensor g. In this context, the geodesic variational theory is extremely more involved, even in the fixed endpoint case [1,3], due to the strongly indefinite character of the action functional. When the metric tensor is Lorentzian, i.e., it has index equal to 1, and the metric is stationary, i.e., time invariant, then it is possible to perform a certain reduction of the geodesic variational problem that yields existence results similar to the positive definite case [8,10,12,13,18].…”

“…For the reader's convenience, in Sect. 3 we will review briefly this definition, that is given in terms of the choice of a periodic frame along the geodesic. An explicit computation shows that the periodic spectral flow equals the fixed endpoint spectral flow plus a concavity index, as in the original paper by Morse [19], plus a certain degeneracy term (see Theorem 3.2).…”

On a formula for the spectral flow and its applications

“…This fact is probably more interesting in se than the formula itself. Further developments of the theory are to be expected in the realm of Morse theory for semi-Riemannian periodic geodesics, which at the present stage is a largely unexplored field (see [6] for the stationary Lorentzian case, or [5] for the fixed endpoints Lorentzian case). A natural conjecture would be that, under suitable nondegeneracy assumptions, the difference of spectral flows at two distinct geodesics is equal to the dimension of the intersection between the stable and the unstable manifolds of the gradient flow at the two critical points in the free loop space.…”