Accelerated Twins: The Answer

ResearchBlogging.orgYesterday’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:

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The article contains a full explanation, and also the following figure illustrating the result:

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The correct answer is indicated by the picture: Alice is the older of the two when they arrive in their new frame.

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Physics Quiz: Accelerated Twins

Just about everybody has heard of the Twin Paradox in relativity: one twin becomes as astronaut and sets off for Alpha Centauri, the other remains on Earth at mission control. Thanks to time dilation, the two age at different rates, and the one who made the trip out and back ends up younger than the one who stayed behind.

Of course, the paradox is not that the two twins have different ages– rather, it’s that from a simple approach to special relativity, you would think that each twin should see the other’s clock running slow, since it seems like getting into a rocket and flying off into space should be equivalent to sitting still in the rocket, and having the entire Earth go zipping off in the opposite direction. This is resolved by noting that the twin in the rocket experiences significant acceleration during the trip, while the other twin does not, and so the two frames of reference are not equivalent.

So, with that in mind, here’s a more subtle question:

Two twins, named Alice and Bob in keeping with convention, get into identical rocket ships separated by a distance L, with Alice in front and Bob behind her. At a pre-arranged time, they each start their rocket, and accelerate for a pre-determined time. At the end of the acceleration, they are each moving at a relativistic speed– 4/5ths the speed of light, say. Which of these twins is older at the end of the acceleration?

You can find the answer using Google, but that would be cheating. We’ll do this as a poll first, and I’ll give the answer probably tomorrow:

A second question, which may or may not help you think about the answer: At the end of their acceleration, what is the spacing between their ships as measured by Alice and Bob?

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