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The Generalized Jacobi Equation
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abstract
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 3 Pith papers
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Geodesic deviation to all orders via a tangent bundle formalism
A tangent-bundle flow formalism yields an explicit all-orders formula for the Jacobi propagators in geodesic deviation, with the Lagrangian and equation of motion given explicitly up to tenth order.
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Cosmic Jets in General Relativity
Free particles and scalar waves in Kasner and plane-wave spacetimes asymptotically align into jets that move at nearly the speed of light relative to fiducial observers.
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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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