Yesterday’s post on a variation of the “Twin Paradox” with both twins accelerating was very successful– 337 people voted in the first poll question, as of a little before 9am, and the comments to the original post are full of lively discussion. That’s awesome.
I wish I could take credit for it, but the problem posed is not original to me. It comes from a 1989 paper in the American Journal of Physics, which also includes the following illustration setting up the situation:
The article contains a full explanation, and also the following figure illustrating the result:
The correct answer is indicated by the picture: Alice is the older of the two when they arrive in their new frame.
The key issue here is the question of synchronization of clocks in relativity, and an aspect of the problem that doesn’t get quite as much attention as the usual time paradoxes. The timing of events depends not only on the speed of the observer but also on the positions of the events.
Several commenters to the original post got the right answer. Buddha Buck has the right description, and miller has the right numbers. According to an observer in their original rest frame, their rockets start and stop simultaneously, but an observer in the frame where they end up sees those events happen at different times. Specifically, Alice stops first, and Bob stops a little later. That difference in timing explains the difference in spacing, and also how Alice is “older”– they each arrive in the new frame at the same time on their local clocks, but Alice gets there before Bob, according to an observer waiting in that frame.
Several commenters on that post, and also in the literature note correctly that there is some ambiguity about the meaning of statements about the timing of events that are not at the same position. Strictly speaking, for Alice and Bob to compare ages, they need to be at the same position, or at least to communicate with each other about their ages. The follow-up comment from AJP goes into detail about the mechanics of this, but agrees with the basic conclusion– when they go through the whole business of properly comparing times in the new frame, Alice ends up older.
I also want to highlight the comment from dr. dave, which hits on the reason why this is pedagogically useful: it leads into general relativity. A uniformly accelerating frame is indistinguishable from a gravitational field, and thus the case of the accelerating twins can be understood as analogous to the case of two clocks at different positions in a gravitational field. The timing difference that shows up because of the spatial separation of the twins is analogous to the “gravitational redshift,” which causes clocks at different elevations to run at slightly different speeds. This is the most important practical consequence of general relativity– the satellite-based atomic clocks in the GPS system need to be corrected for the timing difference between the clocks in the satellites and clocks on the ground. In From Eternity to Here (which is sitting next to my computer), Sean Carroll gives the number as 38 extra microseconds per day for a clock in orbit compared to one on the ground, and because I’m lazy, I’ll go with his number.
I’ve been using this article as an assignment in our sophomore-level modern physics class for several years now– I ask students to read it, and explain the timing. It generally works pretty well, not only for illustrating the issues involved with timing of events, but also for showing that there are problems in relativity that give even faculty members pause.
Boughn, S. (1989). The case of the identically accelerated twins American Journal of Physics, 57 (9) DOI: 10.1119/1.15894
Desloge, E. (1991). Comment on ”The case of the identically accelerated twins,” by S. P. Boughn [Am. J. Phys. 57, 791-793 (1989)] American Journal of Physics, 59 (3) DOI: 10.1119/1.16580