  The theory obtained as a singular limit of General Relativity, if the
reciprocal velocity of light is assumed to tend to zero, is known to be not
exactly the Newton-Cartan theory, but a slight extension of this theory. It
involves not only a Coriolis force field, which is natural in this theory
(although not original Newtonian), but also a scalar field which governs the
relation between Newtons time and relativistic proper time. Both fields are or
can be reduced to harmonic functions, and must therefore be constants, if
suitable global conditions are imposed. We assume this reduction of
Newton-Cartan to Newton`s original theory as starting point and ask for a
consistent post-Newtonian extension and for possible differences to usual
post-Minkowskian approximation methods, as developed, for example, by
Chandrasekhar. It is shown, that both post-Newtonian frameworks are formally
equivalent, as far as the field equations and the equations of motion for a
hydrodynamical fluid are concerned.
