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Triplet Paradox

  1. Oct 17, 2012 #1
    A variant of the classic twin paradox.

    There are three triplets, who have the boring names of Adam, Bob and Charles.

    In Charles' "rest" frame, Adam and Bob get into identical rocketships, jet off in opposite directions and return a year later.

    To Charles, Adam and Bob's paths are symmetrical.

    At their reunion, who will be older and why?

    This problem may remove some of the asymmetries of the original paradox, and I would like to hear from you guys.
  2. jcsd
  3. Oct 17, 2012 #2


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    Adam and Bob are the same age, and less aged than Charles.
  4. Oct 18, 2012 #3


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    Nugatory answered your first question but the reason why is simply that time dilation (the slowing of a clock) is based only on the speed of that clock as defined in any particular reference frame. Therefore, since Charles remains at rest in your chosen reference frame, his clock will not be time dilated and since Adam and Bob both move identically, their clocks will be time dilated to the same extent and so will end up with less time on them at the grand reunion.

    But I'm curious, this is nothing more than two classic Twin Paradoxes, why did you think it would remove any asymmetries?
  5. Oct 18, 2012 #4
    now that you say it, it is like two twin paradoxes, but more complicated.

    The two twin paradoxes are Adam-Charles and Bob-Charles, but now we also have to deal with Adam-Bob.
  6. Oct 18, 2012 #5
    I'm still uncertain about the Doppler explanation as a resolution to the paradox.

    The Time-Gap objection seems to be the most rational explanation, despite its bizarre predictions.
  7. Oct 18, 2012 #6


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    I did deal with them. I said both their clocks are time dilatated to the same extent and so they end up with less time on them. Do you have any doubt about this conclusion?
  8. Oct 18, 2012 #7


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    Have you made any progress in going through the Doppler explanation?
    Have you made any progress in going through the Time-Gap explanation?
  9. Oct 18, 2012 #8
    But isn't that only from Charles point of view? From Adam or Bob's point of view their other two brothers would age more slowly.

    I'm currently reading

    which is pretty informative. Still can't quite get the hang of it though.
    Last edited: Oct 18, 2012
  10. Oct 18, 2012 #9


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    No, a point of view is what someone sees. Each person sees something different because they are at different points of the scenario at different times except at the beginning and at the end. Doppler analysis is how you determine what each person sees and has nothing to do with a frame of reference which is how we specify and calculate things like time dilation. No one can see time dilation. If they could, then because each person has a different speed in each different frame of reference, they would see a different time dilation and that doesn't make sense, does it? Remember, all frames of reference are equally valid and you can use any frame of reference to calculate what each person.

    So if you are asking about each person's point of view, you're asking about what they actually see, correct? And this can be done most easily using Doppler analysis. Have you attempted to do this? Do you know the formula for the Relativistic Doppler factor? Do you know the formula for Velocity Addition?

    Do you want to put some numbers on your example, like how fast do Adam and Bob travel? You already said they return after a year so I assume they travel away for a half year (according to the rest frame of Charles) and then instantly turn around and travel back at the same speed for the other half of the year?
  11. Oct 18, 2012 #10
    Yup I know the formulas.

    I always thought that observing time dilation is like watching high-speed footage. Something like this.

    Putting in some values would be good.

    But put yourself in Adam's shoes. Before you left, both you and Charles were handsome strapping young men.

    After you've returned, you can still easily pick up babes, but Charles has become a lecherous old fool with Einstein-hair.

    The question is, what happened in-between? It seems like the reverse effect of time dilation.
    Last edited by a moderator: Sep 25, 2014
  12. Oct 18, 2012 #11


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    You said that Adam and Bob reunite with Charles after one year. Even if they traveled at an extremely high speed, they're only going to end up one year younger than Charles. If you want Charles to age by say 50 years, you better make the trip last 50 years. And then if you want Adam and Bob to age by just a couple years, they're going to have to travel faster than 99.9%c.

    As I said before, time dilation applies to the one who is traveling at a high speed in a given frame. Adam and Bob are the ones who are traveling in [STRIKE]Bob[/STRIKE]
    Charles's rest frame so they are the ones that age more slowly. Why does that seem like the reverse effect of time dilation?
    Last edited: Oct 18, 2012
  13. Oct 18, 2012 #12
    I think you meant to write "Charles"?

    okay so we make them travel for 50 years before the reunion.
    and both Adam and Bob are travelling at 0.99999999999999999c, as seen by Charles.

    Adam can't feel time passing more slowly for himself. Time seems to be flowing normally.

    However, to Adam, Charles was the same age before and perhaps a good 40 years older when they reunite. So it does seem like the reverse effect of time dilation to Adam.
  14. Oct 18, 2012 #13


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    Yes, thanks.
    I don't have the computational power to deal with a number that close to 1. And I doubt that you do, too. Let's go with 99.9%c, OK?
    True, but he will feel an extreme acceleration. It will be much worse than getting punched in the face.
    I told you, nobody can see time dilation--it's a calculation based on the speed in a given frame of reference. Adam and Bob are the only ones moving in your chosen frame of reference so they are the ones whose clocks are running slow in that frame.

    Now at 0.999c, the speed that Adam and Bob are moving away from Charles, you need to use the Velocity Addition formula to calculate the relative speed between Adam and Bob. Can you do that? Tell me what you get.
  15. Oct 18, 2012 #14
    so unfortunately we can't see cool things like

    the relative velocity between Adam and Bob is 0.9999994994997501c.

    but the fact of the matter is, to Adam, Charles did age faster.
    Last edited by a moderator: Sep 25, 2014
  16. Oct 18, 2012 #15
    Good! Thus the usual twin paradox discussions are relevant such as the one that is still going on (and with links to earlier ones):

    Did you go through it? What is still unclear, when applying it to a triplet?
  17. Oct 18, 2012 #16


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    Good, now can you calculate the Relativistic Doppler factor at that speed and also at the relative speed between Adam and Charles, 0.999c?
  18. Oct 18, 2012 #17


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    It will be helpful for you to think in terms of geometry. This scenario is equivalent to the following:

    Take a square and draw a diagonal from one corner to the opposite. There are now three paths connecting the two corners, which is shortest and why?
  19. Oct 20, 2012 #18
    at 0.999c, 44.71017781 and 0.02236627204

    and between Adam and Bob 1999 and 5.00250125×10^-4
  20. Oct 20, 2012 #19


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    I said in post #11 that Adam and Bob are going to age (about) a couple years so let's say they travel away for exactly one year according to their own clocks and then turn around and get back to Charles in exactly one more year. We'll first deal with what happens between Adam and Bob and when we get done with that we'll figure out what goes on between each of them and Charles.

    Now according to the Doppler Analysis, Adam and Bob will each see the other ones clock running slower than their own by the factor of 5.00250125×10^-4 (which is [STRIKE]1/1900[/STRIKE] 1/1999). So the first question we want to answer is what time will each of them see on the other ones clock when they reach the point of turnaround? The answer is simple--we multiply 1 year by 5.00250125×10^-4 (or divide it by [STRIKE]1900[/STRIKE] 1999), which is just a little over four and a half hours.

    The next question is what Doppler Factor will apply at the moment of turn around? How fast will they each see the other ones clock ticking immediately after they each turn around? What do you think?
    Last edited: Oct 21, 2012
  21. Oct 20, 2012 #20
    so, they should see each other's clocks ticking very quickly during the turnaround, and this will offset the previous effect?
  22. Oct 20, 2012 #21


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    No, that is not correct.

    This would be correct for how Adam and Bob see Charles (but not how Charles sees Adam and Bob) so let's work on that relationship. Use the correct Doppler factors for how Adam and Bob see Charles's clock at the end of one year on their clocks and calculate how much they see Charles's clock progress. Then use the reciprocal factor for the return trip and see how much they see Charles's clock progress during their one year return. Add the two numbers together and you will have determined how much Charles has aged during their trips. What do you get?

    Now think about what Charles sees when he looks at Adam and Bob. Using the same Doppler factors, figure out what time is on Charles's clock when he sees Adam and Bob reach one year and when he sees them turn around. Then for the remaining time that you calculated in the previous paragraph, you can figure out how much time on Charles's clock goes by while he's watching them return.

    If you do all that correctly and if you understand what you are doing and why, you should be able to figure out why it is different when Adam and Bob look at each other. Can you do that?
  23. Oct 24, 2012 #22

    Analysing Adam-Charles, wouldn't the effects of Doppler shifting be mutual?
  24. Oct 24, 2012 #23


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    The Doppler factors that you calculated in post #18, 44.71017781 and 0.02236627204, are mutual but they don't apply symmetrically.

    In our example, when they first depart, Adam and Charles both see each others clock ticking at 0.02236627204 times their own. After one year on Adam's clock, he sees that Charles's clock has advanced by 0.02236627204 years (8.1636892946 days), correct? Then he turns around and now he sees Charles's clock ticking 44.71017781 times his own so in one more year he sees Charles's clock advance by 44.71017781 more years for a total of 44.73254408204 years. So Adam sees Charles's clock advance by 44.73254408204 years while his own clock advances by just 2 years.

    Now what does Charles see? He is going to watch Adam's clock ticking slower than his own until it reaches one year because that is the time he sees on Adam's clock when he turns around, correct? So what time is on Charles's clock when that happens? Well, it would be the reciprocal of 0.02236627204, wouldn't it, which is 44.71017781 years. Now he sees Adam's clock ticking faster than his own for another year, correct? How much time progresses on his clock while that happens? It is the reciprocal of 44.71017781 which is 0.02236627204 years, correct? The sum of these two numbers, 44.73254408204 years, is how much time progresses on Charles's clock while he watches 2 years progress on Adam's clock.

    So can you see how even though the same two Doppler factors apply for both Adam and Charles in watching the other ones clock, they don't apply for the same length of time according to each observer and that's why they end up with different times on their own clocks when they reunite?

    Do you have any questions on what happens between Adam and Charles? If not, can you see why we can't do a similar analysis between Adam and Bob?
    Last edited: Oct 24, 2012
  25. Oct 25, 2012 #24
    Aha brilliant! You nearly had me there. :)

    So first, with regards to how Charles sees Adam. At first I thought there was a flaw because to Charles, Adam took less time to return. Then i realized, this was what Charles saw, and not what actually happened in Charles' frame.

    After drawing an ordinary displacement-time graph, i find that you're absolutely right.

    Now with regards to how Adam sees Charles. I have encountered your explanation three times already, from three different textbooks. Heh.

    Let's say that in Adam's frame Charles emits light pulses at a regular frequency. According to your explanation, Adam spends half his time receiving signals at a redshifted frequency, and the other half receiving signals at a blueshifted frequency.

    after drawing a graph, it shows that this is not the case. Like Charles, Adam should spend more than half the time receiving signals at a redshifted frequency.

    By overestimating the no. of signals received at a blueshifted frequency, the paradox is apparently solved.

    You were quite clever in connecting both frames together in order to demonstrate the asymmetry. However using that explanation i could also say that Adam only makes a turnaround after seeing 44 years elapse on Charles clock, by which 2000 years have elapsed for Adam.
    Do note that a displacement-time graph does show that 44 years elapses for Charles before he sees Adam make a turnaround. Just that the reason for that is different.

    Now some have tried to solve the problem by saying that distances for Adam are length contracted. I turned to a Minkowski diagram for the solution. However, it produces a time gap.

    Naturally, this raises three questions. is some part of Charles' life "event cloaked" to Adam?
    Why does the time gap only apply one way? Why is the time gap just the right amount, but not more or less?

    Other than the time-gap explanation, we can use the GR explanation. As Adam experiences acceleration, he perceives himself to be in a stronger gravitational field than Charles. Therefore time passes more slowly for him.

    Given that most members know nuts about GR, I think you should stick to it.
  26. Oct 25, 2012 #25
    Except for the calculations (nice!), this thread starts to look a little like the other spin-off from the last twin paradox thread:


    And already that one looked too much like all forgoing twin paradox threads.

    Time gap: explained many times, such as here:


    In a nutshell: you are wondering why a Lorentz transformation works by "just the right amount". :wink:

    The usual related issues also apply:


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