Acceleration in Special Relativity

I don't want to pretend to be an authority on general relativity, though I have studied it a little. The following thoughts represent my current understanding of one issue that seems to come up when the so-called paradoxes of special relativity are discussed.

There is a persistent myth that General Relativity is necessary to deal with accelerated reference frames. I even remember hearing that from one of my professors as an undergraduate. It's come up in discussion of the Ehrenfest Paradox below.

Misner, Thorne and Wheeler (in Gravitation), demolish the myth as follows:

...special relativity was developed precisely to predict the physics of accelerated objects-e.g. the radiation from an accelerated charge. Even the fantastic accelerations... of 28 g of a neutron in a nucleus...

The basis of the myth is somewhat obvious: special relativity treats inertial reference frames as "special" while general relativity purports to treat all reference frames on an equal basis. I say "purports" because it seems to me that locally inertial frames still have a special status.

General relativity is a theory of gravity. Gravity can't be reduced to special relativity because it's presence means that locally inertial reference frames can be and are accelerated with respect to one another.

The way one handles an accelerated reference frame in special relativity is to replace it at each instant with the inertial frame moving at the same velocity. Thus an accelerated object has its behavior calculated along a path by breaking that path into a sequence of inertial frames, each of which matches in frame of the object in velocity but not acceleration.

Does that work? Well yes, and there are tons of experimental data to prove. Moreover, it's an approach that respects the geometric character of Minkowski space.

In the presence of matter and its gravity, though, spacetime is no longer Minkowskian. A curved spacetime is necessary, and spacetime needs to become a manifold.