2002/03/31 by C. Chicone, C Chicone, B. Mashhoon +1 · 3 citations
Physics and Astronomy · #Advanced Differential Geometry Research #Attractor #Geodesic #Gravitational field #Hamiltonian (control theory) #Hamilton–Jacobi equation #Linearization #Plane (geometry) #Quantum chaos and dynamical systems #Relativity and Gravitational Theory #Riccati equation #Vector field #astro-ph #gr-qc #nlin.CD
paper · pdf · doi:10.1088/0264-9381/19/16/301
published as Class.Quant.Grav. 19 (2002) 4231-4248 · LaTeX file, 4 PS figures, 28 pages, revised version, accepted for publication in Classical and Quantum Gravity
arxiv created 2002/06/28 · openalex publication_date 2002/07/30 · arxiv updated 2009/11/30 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 neighbouring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analysed 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 ≈ 0.7 c is pointed out. The astrophysical implication of this result for the terminal speed of a relativistic jet is briefly explored.