The Generalized Jacobi Equation

Details The Generalized Jacobi Equation The Generalized Jacobi Equation

2002-03-21

arxiv.org/abs/gr-qc/0203073v2

Class.Quant.Grav. 19 (2002) 4231-4248

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

LaTeX file, 4 PS figures, 28 pages, revised version, accepted for publication in Classical and Quantum Gravity

Scientific paper

10.1088/0264-9381/19/16/301

The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighboring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analyzed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed $2^{-1/2}c\approx 0.7 c$ is pointed out. The astrophysical implications of this result for the terminal speed of a relativistic jet is briefly explored.

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