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The Generalized Jacobi Equation
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The Generalized Jacobi Equation
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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.
Forward citations
Cited by 2 Pith papers
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Generic Peculiar Motions in FLRW Spacetimes
Boosted cosmic test masses in FLRW spacetimes produce Fermi metrics containing a circular gravitomagnetic field around their direction of motion.
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Generic Peculiar Motions in FLRW Spacetimes
Fermi coordinates around a boosted cosmic test mass in FLRW spacetime produce a circular gravitomagnetic field absent in the comoving frame.
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