Age divergence of Identical accelerating twins

[kmarinas86]

http://scienceblogs.com/principles/2010/02/physics_quiz_accelerated_twins.php

Physics Quiz: Accelerated Twins

Category: Education • Physics • Polls • Relativity • Science
Posted on: February 2, 2010 11:28 AM, by Chad Orzel

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:

The answer, they claim, is that Alice ages more than Bob.

But say this were true, it would also mean that of two synchronized clocks placed on opposite sides of the earth, one at sunrise, and the other at sunset, the one at sunset would age slower than the one at sunrise. Alternatively, two clocks at opposite ends of a circular orbit around the sun would experience a difference in aging too. http://www.phil-inst.hu/~szekely/PIRT_Budapest/abstracts/Ghosal_abst.pdf says the delay is this:

\delta t'_{desync}=2v\left(\gamma_v\right)^2 \frac{L}{c^2}

This is the same as:

\delta t'_{desync}=2\frac{v}{1-\frac{v^2}{c^2}}\frac{L}{c^2}

This does not look like an invariant to me. L and \gamma_v may be inversely related, but then why would 2v\gamma_v / c^2 be a constant?
My second problem with this is, "How would you define the case where to two objects separated by distance satisfy the condition \delta t'_{desync}=0?" In that case, you would have v=0 and \gamma_v=1, but with respect to what inertial frame is v?

 

[Dale]

The answer, they claim, is that Alice ages more than Bob.

The correct answer is that the problem is incompletely specified. However, assuming that "At a pre-arranged time" refers to simultaneity in the original inertial rest frame, and assuming that "they each start their rocket, and accelerate" means that they each have identical coordinate acceleration profiles in the inertial rest frame, and assuming that "At the end of the acceleration" also refers to simultaneity in the original inertial rest frame, and "Which of these twins is older" also refers to simultaneity in the original rest frame, then their times are equal. I don't know which of these assumptions they are thinking of differently in order to claim that Alice ages more than Bob.

 

[PAllen]

http://scienceblogs.com/principles/2010/02/physics_quiz_accelerated_twins.php



The answer, they claim, is that Alice ages more than Bob.

But say this were true, it would also mean that of two synchronized clocks placed on opposite sides of the earth, one at sunrise, and the other at sunset, the one at sunset would age slower than the one at sunrise. Alternatively, two clocks at opposite ends of a circular orbit around the sun would experience a difference in aging too. http://www.phil-inst.hu/~szekely/PIRT_Budapest/abstracts/Ghosal_abst.pdf says the delay is this:

\delta t'_{desync}=2v\left(\gamma_v\right)^2 \frac{L}{c^2}

This is the same as:

\delta t'_{desync}=2\frac{v}{1-\frac{v^2}{c^2}}\frac{L}{c^2}

This does not look like an invariant to me. L and \gamma_v may be inversely related, but then why would 2v\gamma_v / c^2 be a constant?
My second problem with this is, "How would you define the case where to two objects separated by distance satisfy the condition \delta t'_{desync}=0?" In that case, you would have v=0 and \gamma_v=1, but with respect to what inertial frame is v?

I claim there is no answer as stated. You have to either specify how Alice and Bob get beck together to compare their clocks; alternatively you have to specify for what particular observer you are comparing their ages (according to Alice? according to Bob? according to the planet they left?).