Age divergence of Identical accelerating twins

In summary, the conversation discusses the Twin Paradox in special relativity and poses a question about two twins in separate rocket ships accelerating at the same time. The claim is that Alice ages more than Bob, but this is incomplete due to the lack of specific details and assumptions made. The conversation also mentions the idea of synchronized clocks at different locations and the issue of invariance in the equation used to calculate the delay in aging. Ultimately, the lack of specification means there is no definite answer to the question.
  • #1
kmarinas86
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1
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:

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

This is the same as:

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

This does not look like an invariant to me. [itex]L[/itex] and [itex]\gamma_v[/itex] may be inversely related, but then why would [itex]2v\gamma_v / c^2[/itex] 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 [itex]\delta t'_{desync}=0[/itex]?" In that case, you would have [itex]v=0[/itex] and [itex]\gamma_v=1[/itex], but with respect to what inertial frame is [itex]v[/itex]?
 
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  • #2
kmarinas86 said:
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.
 
  • #3
kmarinas86 said:
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:

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

This is the same as:

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

This does not look like an invariant to me. [itex]L[/itex] and [itex]\gamma_v[/itex] may be inversely related, but then why would [itex]2v\gamma_v / c^2[/itex] 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 [itex]\delta t'_{desync}=0[/itex]?" In that case, you would have [itex]v=0[/itex] and [itex]\gamma_v=1[/itex], but with respect to what inertial frame is [itex]v[/itex]?

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?).
 
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Related to Age divergence of Identical accelerating twins

1. How does "age divergence of identical accelerating twins" occur?

The age divergence of identical accelerating twins occurs due to the effects of time dilation, which is a phenomenon predicted by Einstein's Theory of Relativity. This means that time moves slower for objects that are moving at high speeds, causing one twin to age slower than the other.

2. What is the significance of studying "age divergence of identical accelerating twins"?

Studying the age divergence of identical accelerating twins allows us to better understand the effects of time dilation and the concept of time itself. It also has practical applications in fields such as space travel, where time dilation must be taken into account for accurate calculations.

3. Is the age difference between the twins significant?

Yes, even a small difference in age between the twins can have significant implications. For instance, if one twin traveled for 10 years at a speed close to the speed of light, they would age only 1 year while their twin on Earth would age 10 years. This could lead to noticeable physical and cognitive differences between the twins.

4. Can time dilation also occur in non-identical twins?

Yes, time dilation can occur for any objects that are moving at high speeds relative to each other. However, the age difference may not be as significant for non-identical twins due to genetic and environmental factors that already contribute to age differences.

5. Are there any real-life examples of the age divergence of identical accelerating twins?

One famous example is the "Twin Paradox" experiment conducted by physicist Richard Feynman, where he used the concept of time dilation to explain the age difference between the two twins in a thought experiment. In addition, astronauts who have traveled in space for extended periods of time have also experienced minor age differences compared to their counterparts on Earth.

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