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
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I answered the question correctly by imagining the rocket ships being very far apart, which meant that Bob would be flying for a longer amount of time.
So if you extend this example, and have them decelerate at the same speed - when they come to a stop, will they be the same age?
I thought the elevation effect was because of the force (and hence acceleration) of gravity is lower further from the center of mass. In this case, the two are undergoing the same acceleration, so how is that analogous to different elevations?
Nope, all that matters in gravitational time dilation is a difference in potential. You might be thinking about tidal forces.
I've to admit, I'm still lost, especially because the comparison to a gravitational problem doesn't hold. The time dilation from the "high tower" and earth orbit problems is related to local differences in the gravitational force, so the local acceleration is not the same.
This violates the premise of the problem.
I don't get it.
So far as I can see they remain at rest WRT each other the entire time. If x0 is sufficiently small (e.g. they are at opposite ends of a single rocket) does it still happen?
Think of it in terms of redshifts and blueshifts, then. When any transmission is made, it moves outward from its origin point in a spherical shell at the speed of light. However, Bob is always approaching the origin points of Alice's signals, so he always receives blueshifted transmissions from Alice. Similarly, Alice is always receding from the origin points of Bob's signals, so she always receives redshifted transmissions. Bob thinks Alice's clock is running fast, and Alice thinks Bob's clock is running slow. They'll both agree that Alice has experienced more elapsed time in the acceleration phase of the experiment.
This is also what happens in the experiment with clocks at different heights in a gravitational well. Assume a point mass with stationary observers and adopt spherical coordinates with r = 0 at the location of the mass. If Alice is at a larger radius, she beams her signals "downwell" towards Bob, and Bob receives blueshifted transmissions. Bob sends his signal "upwell," and Alice receives redshifted transmissions.
Essentially, the direction of the acceleration tells you which direction the "mass" is located, if you're trying to draw equivalence between the two situations. This is true even if both rockets have the same acceleration.
Ah! Yes, that makes it clear.
So if Alice and Bob don't observe each other, they age at the same rate, but if they do observe each other, they age at different rates? If they were to somehow return to their original mutual reference frame, they would still be the same age but returning to that reference frame is impossible, right?
What would happen if they launched from the same point in space, traveled the same distance with identical acceleration and deceleration, but started at different times?
Physics, schmysics. This is actually a basic record-keeping question which can be posed as: "If you subtract the difference in Alice and Bob's age due to the effects of their space travel, from the difference in Alice and Bob's age due to the circumstances of their birth, which will be older?" References to their birth records should clear this right up.
The better question to ask is, who will be older when Jane walks over to Joe to wish him happy birthday.
Nope, I wasn't confusing it with tidal forces, I was just wrong. Thanks for the explanations, but the world made a lot more sense before today, I'll have to spend some more time thinking this over.