Iteration of closed geodesics in stationary Lorentzian manifolds

TL;DR

This paper extends Bott's classical results by analyzing the Morse index of iterated closed geodesics in stationary Lorentzian manifolds, revealing a structured relationship with eigenvalues of the Poincaré map.

Contribution

It introduces a locally constant integer-valued map that describes the Morse index of iterated geodesics, generalizing Bott's formula to Lorentzian geometry.

Findings

Morse index of iterated geodesics relates to roots of unity via a specific map.

Discontinuities in the map occur at eigenvalues of the Poincaré map.

The theory has applications in the study of Lorentzian geodesics.

Abstract

Following the lines of a celebrated result by R. Bott (Comm. Pure Appl. Math. 9, 1956) we study the Morse index of the iterated of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic $\gamma$, we prove the existence of a locally constant integer valued map $\Lambda_\gamma$ on the unit circle with the property that the Morse index of the iterated $\gamma^N$ is equal, up to a correction term $\epsilon_\gamma\in\{0,1\}$, to the sum of the values of $\Lambda_\gamma$ at the $N$-th roots of unity. The discontinuities of $\Lambda_\gamma$ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincar\'e map of $\gamma$. We discuss some applications of the theory.