I've done a lot of writing for simplicity in the  ( v, t ) coordinate sheet of SR and GR. It seems to suppose that the cases represented are operating on a fully contracted tensor domain. 

d(x1) = d(r)

d(x2) = 0

d(x3) = 0

d(x4) = d(r) / v

The question arises about whether or not general principles can be tested in this coordinate system, or are the results only relevant in the reference system in which they were developed.

I can imagine many other possible coordinate systems with axes that mathematically do not look like geodesics, but am not sure whether or not such reference frames can be physically constructed as a central force coordinate system.

I seem to have made assumptions of symmetry in the contracting of tensors, a type of rotational transformation, leaving the possibility of other cases where the symmetries do not exist.

Is Every Axis Of Every Cenrtal Force Coordinate System A Geodesic?