Halc wrote:SR doesn't demand the short duration acceleration. It works fine with gradual acceleration, but that complicates the mathematics of what was a simple example. The simple example could be done with a tag team and no acceleration, with identical results, further evidence that the acceleration itself plays no role.
If by "tag team" you mean more than one travelling observer then, yes, that's right. The simplest scenario is to imagine another traveler, travelling at speed -v, who passes traveler B at C. Both B and C then remain in their respective inertial reference frames. But that's just the simplest scenario. We could have a series of inertial travelers all taking over the baton (as it were) in order to be able to consider the problem as a series of non-accelerating, but not co-moving, reference frames. Or, equivalently, we could imagine just the one traveler making a series of sudden changes of inertial reference frame. In other words, a series of changes, for each of which the time spent making the change tends to zero and the acceleration during that infinitesimal amount of time therefore tends to infinity.
Referring back to the Integral Calculus I mentioned earlier, this would effectively be a process of numerical integration. Remember, when you think of an arbitrary mathematical function y = F(x), the integral is the area under the graph. Numerical integration means finding that area by dividing it into columns. Integral Calculus means allowing the width of the columns to tend to zero. This is, essentially, what I was talking about earlier when I showed the outline of the mathematical process of going from the mathematics of Special Relativity to the mathematics of General Relativity.
As I've said before, I'm rusty on this. I haven't studied it for a long time. But I'm pretty sure the general idea is correct.
SR is special not because of use of limits, but because of being a model of flat space.
The idea of a model in which space is not flat, again, didn't come out of nowhere. It came out of a consideration of the implications of Special Relativity and the equivalence of inertial and gravitational mass, originating in Galileo's experiments. This led to the equivalence of accelerating reference frames and reference frames in the presence of a
uniform gravitational field (or, in other words, the equivalence of accelerating reference frames and reference frames in a small, "local", region of a real, radial gravitational field, within which the variation of the field can be neglected, and it can therefore be regarded as locally uniform).
This equivalence, together with considerations of the behaviour of light in an accelerating reference frame, is what led to the idea that light behaves in this same in a uniform gravitational field, and then to a consideration of the way that light behaves in a real gravitational field, and thereby to the idea that spacetime is not Euclidean. That's what was happening between 1905 and 1915.
So being "special" is not directly because spacetime is regarded as "flat". There is no direct intrinsic connection between the concepts of "specialness" and "flatness"!. In physics, being special means being a limiting case. That is, a case in which certain idealizing assumptions are made. i.e. a particular set of circumstances that the more general case doesn't require. In physics, the general
contains the special. Special Relativity is the limiting case of General Relativity for the particular simplifying idealization of reference frames that are far away from any gravitating bodies and (equivalently) are not accelerating. Or (Einstein later realized) are in free fall.
Likewise, for example, classical mechanics is the limiting case of quantum mechanics for the special case in which momenta are large enough that such things as the uncertainty principle can be neglected. The EM wave model of light is a limiting case in which the number of photons is large enough that the EM wave equations (Maxwell's Equations) are good enough. Classical mechanics is the limiting case in which relative velocities are low enough that the Lorentz transformations can be neglected. etc.