On the generalized Jacobi equation

Details On the generalized Jacobi equation On the generalized Jacobi equation

2007-10-14

arxiv.org/abs/0710.2667v1

Gen.Rel.Grav.40:1029-1045,2008

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Contribution to Mashhoon Festschrift (Special Issue of Gen. Rel. Grav.). 17 pages, 2 figures

Scientific paper

10.1007/s10714-007-0589-x

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.

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