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
1Department of Mathematics, Imperial College, London SW7 2AZ, UK
2Department of Mathematics, Auburn University, Alabama 36849, USA
3Department of Mathematics, Chelsea College, University of London, 552, King's Road, London SW10 OUA, UK
Received 23 September 1982
Copyright © 1983 P. Choudhury et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.