The Generalized Jacobi Equation

TL;DR

This paper extends the classical Jacobi equation in pseudo-Riemannian geometry to include relative velocities, analyzes its Hamiltonian structure, and explores astrophysical implications such as jet terminal speeds.

Contribution

It introduces the generalized Jacobi equation accounting for arbitrary relative velocities and investigates its Hamiltonian structure and astrophysical applications.

Findings

Existence of an attractor of uniform relative radial motion at about 0.7c.

Analysis of tidal accelerations in gravitational wave and rotating mass fields.

Implications for the terminal speed of relativistic astrophysical jets.

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.