A Path to the Geometrization of Gravity--'Suggested' by Experiments
...This is a 'new' organization of ideas...somewhat different from what is in the book...
We take the point of view that whatever theoretical foundation we put down for gravitation and other phenomena, we base it as closely as possible to experimental results.
We take the following experimental underpinnings as given [i.e. well tested and verified]:
Thus the local spacetime structure is given by the interval between events in a local region:
ds2 = - c2dt2 + dx2 + dy2 + dz2
Note that Minkowski spacetime possesses the causal structure imposed by the lightcone.
Using g = G M/ R2e we could show that the form is
ds2 = - c2 ( 1 -2GM/(c2 r))dt2 + dx2 + dy2 + dz2
where the combination rs = 2GM/c2 is called the Schwarzschild radius and will play an important role later. Note that as we get far from the object whose mass is M we approach the Minkowksi spacetime model.
The Radar-ranging experiment measured the time it takes for the radar to go from the earth to Mercury and back.
Among the various possibilities
of things to change, we take the Einstein point of view that it is the
geometric structure which is to encode the information related to gravitation.
This point of view suggests that we consider warping space in addition
This means that we consider the following form for the interval between events:
ds2 = - c2 ( 1 -2GM/(c2 r))dt2 + gxxdx2 + gyydy2 + gzzdz2
where the gxx, gyy, and gzz are functions to be determined so that the resulting spacetime structure is consistent both with the Pound-Rebka experiment and the Shapiro radar-ranging experiments.
The important point to make is: The 1915 geometric theory of gravitation of A. Einstein does this without any freely adjustable parameters!