How are the twins distinguished?

[Passionflower]

WHich was the explanation regarding the loop?

The explanation you were not interested in.

 

[Al68]

I think you could come up with other non-inertial systems where this was true. At any event E on the accelerating observer's worldline consider the surface of simultaneity in the observer's instantaneous inertial rest frame at E, then as long as the surface of simultaneity for the non-inertial frame is defined in such a way that every point on the surface of simultaneity for the same event E is guaranteed to be on or "below" (in the past of) that inertial surface of simultaneity, then coordinates assigned to events shouldn't depend on the observer's future motion.

edit: actually on second thought, lying "under" the inertial plane of simultaneity isn't really the issue, all that matters is that the plane of simultaneity for any point on the worldline is defined in such a way that the future behavior of the worldline doesn't matter--for example, at any point we could define a plane of simultaneity by imagining what would happen if we had particles which could move FTL, and imagining sending them out in all directions such that they moved at 200c in the accelerating observer's instantaneous inertial rest frame, so the spacelike paths of all these particles would define a surface of simultaneity.

Maybe I'm missing something here, but if a non-inertial frame is defined so that the coordinates assigned to events are independent of the observer's future motion, then don't the coordinates assigned necessarily have to be the same coordinates assigned for the case of all future motion being inertial motion, ie the same coordinates assigned by a co-moving inertial observer, to be consistent with the SR simultaneity convention?

It seems to me that "independent of future motion" necessarily means "the same as the ICMIF", since the latter is one of the possible future motions of the accelerated observer.

 

[JesseM]

Maybe I'm missing something here, but if a non-inertial frame is defined so that the coordinates assigned to events are independent of the observer's future motion, then don't the coordinates assigned necessarily have to be the same coordinates assigned for the case of all future motion being inertial motion, ie the same coordinates assigned by a co-moving inertial observer, to be consistent with the SR simultaneity convention?

I don't see why--after all, you can design a non-inertial rest frame for an inertial observer, keeping in mind that in SR "non-inertial frame" doesn't necessarily mean that objects at fixed coordinate positions in the frame are moving non-inertially, it just means a frame that doesn't satisfy the two postulates of SR. For a simple example, suppose an inertial observer is moving at 0.8c in the +x direction of an inertial (x,t) frame. Then we can obtain a non-inertial (x',t') frame where that observer is at rest using the Galilei transformation:

x' = x - 0.8c*t
t' = t