Proper Time Adjusted Special Relativity with Gravitation
The next step in the development of Proper Time Adjusted Special Relativity will be to try to introduce gravity in this formulation. The usual objections to this is that Special Relativity does not and cannot handle gravitational forces. This is most probably true for Standard Special Relativity and for the Minkowski spacetime, but not necessarily for Proper Time Adjusted Special Relativity, which has already introduced a curved expanding space dimension in Special Relativity.
The introduction of gravity in Proper Time Adjusted Special Relativity seems relatively straightforward. We start from the form of curvature described in Forms of Spacetime Curvature and Spacetime Curvature and Gravitation, and we use the same formulation, but instead of having a straight (Galilean) spaceline, we have a closed curved expanding one (Lorentzean). This means that we will have to derive the equations that describe the existence of such a curvature on an ellipse.
Remember that curvature of the spaceline means that its points have different time coordinates. The equations have to be of such a form so that the projection of the Moving Body on the x axis of the Stationary Body will obey the inverse square law.
The first step will be to derive the Equation of Motion of a Particle in a Gravitation Field as a function of time. First in its classical newtonian form, and then taking into account the definition of acceleration in Special Relativity. Hopefully, by comparing the two we will be able to determine the shape of spacetime curvature that produces gravitation.