On the orbital motion description with a proper reference frame in a complete schwarzschild field

Abstract

The orbital motion of test particles about a central body at rest is described in terms of a coordinate system that, unlike the usual coordinates, can be materialized by a proper (Fermi) reference frame with origin located at the body's centre world line. By means of the world function in the post-Newtonian approximation of the exterior Schwarzchild solution for this system it is demonstrated that the radial coordinate, which initially is shown to be asymptotically equal to the invariant relativistic distance between any test particle's world line and that of the body's centre, when calculated for two particles placed in conjunction with respect to the central body, becomes exactly their euclidean distance. It is also shown that the differential equation in the reciprocal of the radial coordinate for any bounded orbit is governed by a function which is asymptotically, but is not, a cubic, and its solution is obtained in terms of elliptic functions. It is finally shown, when the post-Newtonian polar coordinates equations are derived, that a relevant relativistic correction appearing in the usual post-Newtonian approximations of the exterior Schwarzchild solution disappears or, to be more precise, becomes exactly zero.