I Why Is the Green Observer Always the Bent One in the Twin Paradox?

[SiennaTheGr8]

It's maybe worth emphasizing that constructing whole coordinate systems for an accelerating traveler is a different exercise from tracking the accelerating traveler's aging (elapsed proper time). For the latter, I believe it's always okay to divide the trip into infinitely many instantaneous inertial frames (for which the traveler is momentarily at rest) and integrate over the traveler's corresponding proper-time infinitesimals.

 

[stevendaryl]

It's maybe worth emphasizing that constructing whole coordinate systems for an accelerating traveler is a different exercise from tracking the accelerating traveler's aging (elapsed proper time). For the latter, I believe it's always okay to divide the trip into infinitely many instantaneous inertial frames (for which the traveler is momentarily at rest) and integrate over the traveler's corresponding proper-time infinitesimals.

Yes, that is true. What that construction does not do, however, is to give an unambiguous answer to the question: How old is this traveler compared with that traveler? (Unless they are in the same place and time.)

 

[Nugatory]

Yes, that is true. What that construction does not do, however, is to give an unambiguous answer to the question: How old is this traveler compared with that traveler? (Unless they are in the same place and time.)

Is there any construction that can give an unambiguous answer when the two travelers are not colocated? To remove the ambiguity we need an additional arbitrary choice of a simultaneity convention, do we not?

 

[PAllen]

I think debates about when, in the history of a general noninertial worldline, does the exra time of the inertial path correspond, are exactly as meaningful as asking:

Given a straight line and curved line between two points in a plane, which part of the curved line is the extra distance?

That is, it is nonsense. In both cases you can match up points on the two lines to get any answer you want.

 

[stevendaryl]

Is there any construction that can give an unambiguous answer when the two travelers are not colocated?

No. I don't think so.

 

[jbriggs444]

Is there any construction that can give an unambiguous answer when the two travelers are not colocated? To remove the ambiguity we need an additional arbitrary choice of a simultaneity convention, do we not?

Depends on what you are willing to consider as "arbitrary". For inertial travelers, the frame where their velocities are equal and opposite would be one obvious non-arbitrary choice.

 

[Ibix]

Depends on what you are willing to consider as "arbitrary". For inertial travelers, the frame where their velocities are equal and opposite would be one obvious non-arbitrary choice.

I think this is one source of confusion in this type of scenario. From the point of view of any given scenario there is always (or often, at least) a fairly narrow selection of sensible frames to use. One where a velocity is zero, or two velocities are equal, and suddenly some bit of maths is trivial. But from the point of view of the physical laws any choice is completely arbitrary.