On the generalized Jacobi equation

TL;DR

This paper explores the generalized Jacobi equation, which linearizes geodesic deviation with respect to coordinates only, extending its analysis from affine manifolds to lightlike geodesics in Lorentzian manifolds, including specific spacetime examples.

Contribution

It extends the study of the generalized Jacobi equation to affine manifolds and lightlike geodesics in Lorentzian manifolds, with explicit examples in Schwarzschild and plane-wave spacetimes.

Findings

Analysis of the generalized Jacobi equation on affine manifolds.

Application to lightlike geodesics in Lorentzian manifolds.

Illustrations in Schwarzschild and plane-wave spacetimes.

Abstract

The standard text-book Jacobi equation (equation of geodesic deviation) arises by linearizing the geodesic equation around some chosen geodesic, where the linearization is done with respect to the coordinates and the velocities. The generalized Jacobi equation, introduced by Hodgkinson in 1972 and further developed by Mashhoon and others, arises if the linearization is done only with respect to the coordinates, but not with respect to the velocities. The resulting equation has been studied by several authors in some detail for timelike geodesics in a Lorentzian manifold. Here we begin by briefly considering the generalized Jacobi equation on affine manifolds, without a metric; then we specify to lightlike geodesics in a Lorentzian manifold. We illustrate the latter case by considering particular lightlike geodesics (a) in Schwarzschild spacetime and (b) in a plane-wave spacetime.