On the orbital motion description with a proper reference frame in
a complete Schwarzchild field
J.M. Gambi, P. Romero, P. Zamorano, M.L. Garcia del Pino
(Departamento de Ingenieria. Universidad Carlos III. Madrid. Spain
Departamento de Astronomia y Geodesia. Universidad Complutense. Madrid.
Spain )
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.