International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 4, Pages 795-802
doi:10.1155/S0161171283000678
A scalar geodesic deviation equation and a phase theorem
P. Choudhury1
, P. Dolan1
and N.S. Swaminarayan3
1Department of Mathematics, Imperial College, London SW7 2AZ, UK
3Department of Mathematics, Auburn University, Alabama 36849, USA
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.