Einstein's "special" or "restricted" theory of relativity was published in 1905, and can be considered a distillate of the equations are relationships of Lorentz Aether Theory ("LET", 1904). Lorentz aether theory was developed through the late 1800s, and was likely the most popular aether model in Germany when Einstein was a student ... his lecturer Minkowski would probably have impressed on Einstein the notion that these relationships had to be correct.
In 1909, Minkowski published his rendition of the Lorentz-Einstein equations as a pure spacetime geometry.
relativity + flat spacetime = special relativity
relativity + bodies with no gravitational fields = special relativity
relativity + time-symmetry = special relativity
However, if the existence of matter is associated with curvature, or if physics looks different in forward and reversed time, special relativity is ruled out.
All solutions to the relativity of inertial motion are related by Lorentzlike factors, and of these solutions, only special relativity looks the same in forward and reversed time. [] All other solutions are asymmetrical with respect to time: Newtonian theory gives E'/E = (c-v)/c for recession in forward time, and E'/E = c/(c+v) in reversed time. If, like Einstein, we find time-asymmetry abhorrent, we can average out these two predictions to give their geoemetric mean, sqrt[ (c-v) / (c+v) ], and this is then special relativity.
Special relativity therefore has the special status of being the time-average of Newtonian thery, and in fact, of all possible relativistic theories. It is the averaged, homogenised version of the real equations of motion.
Yes. Suppose that we thought that the Moon was a flat disc. Analysing photographs, we would find that if the Moon's radius is R, a crater located away from the centre of the Moon-disk by a distance r, appears contracted by the ratio L'/L = sqrt[1 - r2/R2] -- Lorentz contraction!
Does the appearance of the Lorentz factor mean that we have proved that the Moon really is a flat disc? No, the gamma factor only appeared because we did something wrong.
Under a working general theory, the Minkowski geometry is the wrong geometry. The Minkowski metric is fixed and absolute [], a GR metric needs to be dynamic and interactive. A proper general theory replaces the Schwarzschild and Minkowski metrics with a relativistic acoustic metric. [] Acoustic metrics didn't really become a subject until the late 1990s, and doesn't appear in most GR textbooks.
SR can be fundamental without being physics. SR uses an averaged concept of the speed of light. In experiments that deal solely with simple averaged parameters, it maye actually give correct answers, even if the one-way predictions are wrong.
In an acoustic metric physics, we shouldn't be using the "transformation" approach to try to derive basic properties. Transformations are the wrong tool for the job.
Under SR we can repeat the same experiment over and over, with differently moving objects and observers, and if the underlying shape of spacetime is assumed to be the same in all cases, we can arrive at the Lorentz transforms and special relativity.
But if the presence and motion of matter distorts spacetime, there is no single underlying geometry to derive, because each time we repeat the experiment with different parameters, we get a different geometry.
Once we know the laws of physics, transformations are good at describing how a known, non-interacting, distant geometry changes with the properties of a distant observer. But they're not good for deriving the laws and geometry when an interacting observer is up-close. Because then the geometry is variable.
The purpose of a velocity-addition formula is to allow us to calculate the results of a composite motion-shift in just one stage. Almost every theory has its own velocity-addition formula, as a function of its Doppler relationship: for instance, with Newtonian physics, where the recession shift formula is E'/E = (c-v)/c, a signal sent between two bodies receding at half lightspeed is halved, and if a third body then recedes at half lightspeed, the energy is halved again, giving E'/E = 0.25 . In a single stage, the "equivalent" velocity would be 0.75c. So under NM, we can say that according to the NM v.a.f., 0.5c+0.5c = 0.75c .
These different velocity-addition formula are superficially similar to that of special relativity: the difference is that under NM, the v.a.f. documents how the shape of spacetime is modified by objects with relative motion, whereas under SR, the v.a.f. is considered a structural property of spacetime itself.