Dr. Lawrence B Crowell,
Quoting you:
"I would suspect that if you patch worked Newtonian mechanics in a way to make it fit with data which supports special relativity that you would end up in fact with special relativity --- of sorts. As for general relativity and gravitation, that might be harder or problematic."
Why would you use the word patchwork? I presume you are intent on demonstrating my lack of understanding about the reality of relativity theory. Actually, I don't use patchwork. I move step by step from fundamentals upward toward more comples theory. I choose not to use patchwork. In my opinion, that is something similar to what Einstein did. For example, why were transform equations necessary? They are what forces results. They are not safe mathematics, because the theorist can use them to patch things together.
"As for general relativity and gravitation, that might be harder or problematic."
Actually they fall into place very easily.
"Science never proves theories."
Nor does it confirm them.
"This in fact happened with Newtonian physics. It began to fail to account for physically observed phenomenon. On a small scale it was supplanted by quantum mechanics and for large velocities and large gravity fields it was replaced with general relativity. General relativity is likely to be incomplete when quantum physics enters the picture. This is the problem of quantum gravity."
Yes Newtonian physics failed at high speed and small dimansions. He did not have the empirical evidence of these two regions available to him or he would, I think, have known what to do very quickly. I think he might have helped us avoid adopting two very different theories and spending a great many years trying to force theoretical unity onto them. Of course, they can't be brought together, but they can be replaced by a new theory, perhaps string theory, that has free access to unlimited hidden dimensions and control of space and time. That God-like combination ought to be successful at forcing unity so long as the nature of that unity is kept out of our sight. With regard to quantum gravity, I think that is an example of trying to force two incompatible theories together. It seems, for now, a safe practice. It will be a long while before that one can be tested. However, theory is very flexible and can move in just about any direction required.
"I will just say that special relativity states that in order for the speed of light to be an invariant, or is always measured to be the same in all frames of reference, that space and time must transform into each other."
Yes it does and I think that is evidence of its failure to represent reality.
"... it turns out that time and spatial variables also transform into each other in this more general setting."
Yes that is one of the really weird results of trusting in transforms.
"... time dilation."
No one has ever conducted an experiment on time.
"This "rule" is extended for gravitation where spacetime curvature is on a small region sufficiently flat that special relativity holds. The curved spacetime then results by patching together these flat region, with connection coefficients and a calculus limit."
Like I said, I am not the one doing the patching.
"This part gets a bit abstract and requires Riemannian geometry to address."
So what am I to conclude from this: That reality is an abstraction?
James