The geometrical treatment of space, time, and gravity only requires as its basis the equivalence of inertial and gravitational mass. Given this assumption, we can describe the trajectory of any free-falling test particle as a geodesic. Equivalence of inertial and gravitational mass holds for Newtonian gravity, so it is indeed possible to redo Newtonian gravity as a theory of curved spacetime.
We think of a frame of reference as a body of measurements or possible measurements to be made by some observer. Ordered geometry lacks measure. The following argument shows that merely by adding a notion of parallelism to our geometry, we automatically gain a system of measurement.
We now want to pin down the properties of the Lorentz geometry that are left unspecified by the affine treatment. We need some further input from experiments in order to show us how to proceed.
In Newtonian physics, causal relationships fell into two classes: (1) could potentially cause any event that lay in its future or (2) could have been caused by any event in its past. In a Lorentz spacetime, there is a third class of events that are too far away from in space, and too close in time, to allow any cause and effect relationship, since causality’s maximum velocity is c . The boundary of this set is formed by the lines with slope ±1 on a (t,x) plot. This is referred to as the light
We’re now in a position to discuss tests of relativity more quantitatively. One such test is that relativity requires the speed of light to be the same in all frames of reference, for the following reasons. There are two types of tests we could do: (1) test whether photons of all energies travel at the same speed, i.e., whether the vacuum is dispersive; (2) test whether observers in all frames of reference measure the same speed of light.