Pseudo focal points along Lorentzian geodesics and Morse index

TL;DR

This paper introduces a new concept of pseudo conjugate points along Lorentzian geodesics, establishing a Morse index theorem that parallels Riemannian results, with applications to stationary and static spacetimes.

Contribution

It defines $\\mathcal{Y}$-pseudo conjugate points, proves their finiteness and relation to Morse index, extending Morse theory to Lorentzian geometry with specific cases for stationary and static manifolds.

Findings

$\\mathcal{Y}$-pseudo conjugate instants are finite in number.

Number of pseudo conjugate points equals the Morse index.

Extension of Morse index theorem to Lorentzian manifolds.

Abstract

Given a Lorentzian manifold $(M,g)$, a geodesic $\gamma$ in $M$ and a timelike Jacobi field $\mathcal Y$ along $\gamma$, we introduce a special class of instants along $\gamma$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field $\mathcal Y$ is obtained as the restriction of a globally defined timelike Killing vector field.