2007/10/14 by Volker Perlick · 1 citation
Physics and Astronomy · #Advanced Differential Geometry Research #Astrophysical Phenomena and Observations #Relativity and Gravitational Theory #gr-qc
paper · pdf · doi:10.1007/s10714-007-0589-x
published as Gen.Rel.Grav.40:1029-1045,2008 · Contribution to Mashhoon Festschrift (Special Issue of Gen. Rel. Grav.). 17 pages, 2 figures
arxiv created 2007/10/14 · openalex publication_date 2008/01/08 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
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.