I'm glad that you're serious about this. I'll go back over some of those points and try to explain more completely.
"I asked what the SR permanent aging difference would be after the ten years of constant velocity travel for twin B outbound, and 10 more years inbound."
That's precisely what we answered, doing the SR calculation using the reference frame at rest with respect to the initial and final traveler positions.
"My original question was whether constant velocity caused real aging, and 'everyone must agree' does not address that, but begs the question. "
What we have here is a failure to communicate philosophically. The things we consider 'real' are the ones which don't depend on point of view. When we conclude that the differences in age between the travelers don't depend on point of view, we're saying those are real differences.
"Second, I was not asking about calculating the aging difference due to GR." True, and we calculated it using only SR, picking a reference frame in which SR is ok. However, if you instead use the reference frame of either traveler, which are not close to being inertial, you have to use some GR ingredients. We described how, even if you were to do that, your results would be consistent with our simple SR calculation.
"Third, I truly want to understand your statement that the GR time changes involve not only the acceleration but also the displacement between the objects. You said that B is accelerating back toward A when the displacement is larger, so B sees a bigger GR speed-up of A's aging. We're talking about permanent aging, not just what B observes, right?"
Again, we know no truth outside the collection of observables from different points of view. When A and B return to the same spot, so that all observers must then agree on which is older, since any can see them simultaneously,
the net result from B's point of view agrees with A's and everybody else's. This is a real age difference. Which relativistic effects it's attributed to depends on which reference frame you use.
As for the technical question, a frame accelerating at a
with respect to a good inertial frame sees the aging rate of an object at position r
change (to lowest order in a
) by a factor (1+a•r
), where "•" means vector dot product. If you are on an accelerating rocket, far from gravity sources, a clock at the head will run faster than one at the tail. In gravity, the equivalent real effect is that clocks in attics run faster than ones in basements.
In your example, although each traveler has the same magnitude of a, one has it while r is big and the other while r is small, so they see different effects on the other's aging.
But I think the key point here is the philosophical one, not the technical one.
(published on 08/13/09)