The Generalized Jacobi Equation
نویسنده
چکیده
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 Hamilto-nian structure of this nonlinear equation is analyzed in this paper. The tidal accelerations for test particles in the field of a plane gravita-tional 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 ≈ 0.7c is pointed out. The astrophysical implications of this result for the terminal speed of a relativistic jet is briefly explored.
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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....
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