A scalar geodesic deviation equation and a phase theorem

Abstract

A scalar equation is derived for η, the distance between two structureless test particles falling freely in a gravitational field: η¨+(K−Ω2)η=0. An amplitude, frequency and a phase are defined for the relative motion. The phases are classed as elliptic, hyperbolic and parabolic according as K−Ω2>0,<0,=0. In elliptic phases we deduce a positive definite relative energy E and a phase-shift theorem. The relevance of the phase-shift theorem to gravitational plane waves is discussed.