Do Einstein's two postulates allow for handling acceleration in special relativity or is something else needed? When I was taught special relativity, we started with Einstein's two postulates and worked from there.  However we were also taught that a proper resolution of the twin paradox required general relativity - because one twin accelerates.  Apparently this was Einstein's opinion as well.
However modern texts, such as M,T&W's Gravitation, state that special relativity can handle the paradox.  Specifically they state that when a uniformly accelerating observer momentarily passes a non-accelerating observer travelling at the same velocity, they will agree that their clocks are running at the same speed. With that statement, if accepted as part of special relativity, the twin paradox can be resolved.
However, I do not see how this last statement follows from Einstein's two postulates.  Does it?  Or is special relativity, as understood now-a-days, reliant on more than the two postulates?
 A: I would guess you are referring to the clock hypothesis. Annoyingly I cannot find a web page that neatly describes this, but it is discussed in detail in the question What is the history of adding the Clock Hypothesis to Special Relativity? on the History of Science and Mathematics Stack Exchange.
As far as I know Einstein never stated this and did not include it in his postulates. However Minkowski's formulation of special relativity as a metric theory implicitly assumes it is true.
The way we approach special relativity these days owes more to Minkowski than Einstein, and if we use the geometric approach handling acceleration is straightforward. For example the chapter in MTW (chapter 6 in the first edition) derives the coordinate transformations straightforwardly from the fact the four-velocity and four-acceleration have to be normal to each other.
If you are interested, the resolution to the twin paradox using the geometric approach is described in What is the proper way to explain the twin paradox? though this will be challenging for the novice.
A: It's a philosophical question what the "proper resolution" to the twin paradox may be. Einstein did think that the treatment of it in SR was philosophically unsatisfactory. He thought that GR solved the problem through what he called the "general principle of relativity", but that view isn't popular today.
There are many people today who seem to think that SR can't handle acceleration at all, even mathematically. That isn't true.
There also seem to be many who think that you need to add a "clock postulate" saying that acceleration doesn't affect clocks, which is also untrue.
Many clocks are affected by acceleration. The reason it's possible for them to be affected by acceleration is that acceleration is absolute. It can be measured locally. Because it can be measured locally, you can measure it and use the measurement to compensate for the effect of acceleration on a clock and get the correct elapsed time.
This does require the assumption that it's possible to determine the correct elapsed time for calibration. That assumption is implicit in every pre-quantum theory: it's a special case of the assumption that the fundamental quantities of the theory can be (passively) measured. You could argue that it should be explicit in modern presentations. But a special "clock postulate" isn't needed.
A: About the twin scenario: the name 'twin paradox' is awkward, because there isn't actually a paradox.
The twin scenario goes against our everyday life intuition, of course, but that does not a paradox make.

Discussions of the twin scenario are for the purpose of showing that no self-contradiction can be construed; the mathematics checks out.
So yeah, showing that no self-contradiction can be construed is possible within the realm of special relativity.

There is no such thing as "resolving" the twin scenario because there isn't actually a paradox.

(There is the psychological phenomenon of 'hidden assumption'. A person may bring a hidden assumption to the table, one that is incompatible with the mathematics of special relativity. If a person feels that in fact a self-contradiction can be construed then the actual clash is with the hidden assumption.)


In physics the word 'postulate' is not used in the same way as in mathematics. Generally there are, besides the explicitly stated 'postulates' also implicit assumptions. It's not rigorous the way mathematics is.
In physics words such as 'postulate' and 'axiom' should be understood as expressing: this is important, this touches on the core of what we are doing here.
As mentioned by John Rennie, special relativity in terms of Minkowski spacetime is straightforward, expecially compared to developing SR in terms of the two 1905 postulates.
In my opinion: it would have been better if Special Relativity would have been introduced in terms of Minkowski spacetime right from the start.
While the two 1905 postulates are historically important, in my opinion: once the concepts of Minkowski spacetime and Minkowski metric were introduced the two 1905 postulates were in effect rendered obsolete.
A: In the twin paradox, which is a veridical paradox, meaning that the conclusion is correct even if on the face of it it seems paradoxical, is a consequence of special relativity even if in the natural interpretation accelerations are required and so one might think GR.
As you suggest, both Einstein and Born did think of it this way but we can see that we can do without the acceletation by formulating the problem with the following device:
There is another auxilary spacecraft which travels towards earth at exactly the same speed and in the opposite direction as the outward going spacecraft. At the moment they pass each other, the clock reading from the outward going twin is transferred to that of auxilary observor. In this way we replicate an instant turn around with no acceleration.
Analysing this situation in special relativity shows that the paradox relies on the inertial frame switch of the twin in flight.
By the way, velocity along a curve on any curved manifold can always be defined in a natural generalisation of the derivative. But to define acceleration requires a connection (equivalently, a covariant derivative) and then acceleration is the covariant derivative of the velocity in the direction of the velocity.
A: I think you have stated the situation carefully and well.
The first thing to say is that no postulate which mentions only constant-velocity motion can have anything to say about accelerated motion. This is why, for example, the rest of electromagnetism cannot be deduced from Coulomb's law and the principles of relativity.
So it is true that the agreement of an accelerated clock with an inertially moving clock (over some small time interval) does not itself logically follow from the two postulates. However one can give many convincing physical arguments to show that it would be very odd for two such clocks to disagree in the limit of low acceleration, and low acceleration (for a long enough time) is all that is required in the twin paradox. Therefore I think most physicists would judge that it is not needed to add a further principle. One just takes the two principles and adds to them the general notion of elegance and simplicity.
In a similar way, although the rest of electromagnetism cannot be deduced from Coulomb's law plus relativity, it can be
argued to be the simplest field theory which respects relativity and gives rise to a pure force (one which does not change the rest mass of particles acted upon).
I mention this other example to illustrate the wider way in which physics often operates. We choose to give the status of 'principle' to certain very wide-ranging and useful observations. People with a more philosophical approach tend to want to add lots of further 'principles', but physicists don't always find that to be useful for gaining insight, so those extra 'principles' don't find their way into physics textbooks.
A: The twin paradox can be explain straightforwardly as a consequence of the geometry of flat spacetime. There is no reason whatsoever to invoke general relativity- special relativity alone is sufficient. General relativity is required only when there is significant curvature of spacetime (ie when gravitation needs to be taken into account).
