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Symmetrical twin paradox

  1. Jan 11, 2013 #1
    i can't get my head around the twin paradox at all.

    so twins A and B move relative to each other and as a result, according to special relativity, from twin A's perspective twin B will of aged less but from twin B's perspective twin A will of aged less. when they meet up which one will of actually aged less?

    as far as i can tell the most common answer is that the twins are not symmetric. i.e. if Twin A stays on earth and twin B travels in a spaceship then the paradox is resolved since twin B undergoes acceleration and twin A does not.

    but what if both twins each get in a space ship and fly in opposite directions, at the same velocity, and travel the same distance, before turning around and coming home?
     
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  3. Jan 11, 2013 #2

    PeterDonis

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    There have been umpteen to the umpteenth power threads on PF lately about the twin paradox. I would strongly recommend searching the forums and reading the latest threads before starting another one, if only to save wear and tear on the keyboards (and blood pressure) of those of us on PF who have been responding to all those threads. Also, I would recommend reading the Usenet Physics FAQ on the twin paradox:

    http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

    That said, the short answer to the question at the end of your post...

    ...is that this is a different scenario from the standard twin paradox, so the answer will be different; in the case you describe, both twins will have aged the same when the meet up again--but both of them will have aged *less* than a third "twin" (say they're triplets instead) who has stayed at home the whole time. (I believe there's been a recent thread on a "triplet paradox" as well which discussed a similar scenario.)
     
  4. Jan 11, 2013 #3

    Nugatory

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    They will both be the same age - and that age will be younger than that of their third sibling (if they were really triplets :smile:) who stayed at home.
     
  5. Jan 11, 2013 #4
    sorry i still don't understand at all. i did look through all the other threads on this and the math.ucr.edu page and still don't understand.

    are you saying that the time dilation in the twin paradox is entirely due to the acceleration and there is no time dilation at all as a result of their relative velocity?
     
  6. Jan 11, 2013 #5

    PeterDonis

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    Ok, then we'll have to back up until we reach a starting point that you're comfortable with. Unfortunately, I don't think that this...

    ...is such a starting point. The short answer to this question is "no"; but that probably isn't very helpful. :wink:

    Let's try this: the Doppler shift analysis. This analysis has the advantage of focusing entirely on what the two twins actually *see*, and it makes clear that in the standard scenario, there *is* an asymmetry between them; Stella (the traveling twin) sees Terence's (the stay at home twin's) light signals switch from redshift to blueshift halfway through her trip (when she turns around), but Terence sees Stella's light signals switch from redshift to blueshift a short time before Stella returns. So Stella sees half redshift + half blueshift, while Terence sees almost all redshift + just a short period of blueshift.

    In your modified scenario, Stella goes off to the right, Terence stays at home, and Ursula (the triplet who has just been discovered :smile:) goes off to the left. Ursula and Stella both see the same thing as Stella does in the standard scenario with Terence's light signals: half redshift + half blueshift. The aging that Terence gains on Stella and Ursula during the blueshift more than makes up for the amount he falls behind during the redshift. Terence sees the same thing as in the standard scenario with both Stella's and Ursula's light signals: almost all redshift + just a short period of blueshift. The short blueshift at the end isn't enough for Stella and Ursula to make up in aging for the amount they fell behind Terence during the redshift. So all three of them will predict that Stella and Ursula will age the same, and will both age less than Terence.

    What do Ursula and Stella see with each other's light signals? They see almost all redshift + a short period of blueshift: but the redshifts and blueshifts are both *larger* than those that Terence sees (because Ursula and Stella have a higher relative velocity with respect to each other). So Ursula and Stella each see the other aging slower during the redshift period than Terence does; but they also each see the other aging *faster* during the blueshift period than Terence does. This turns out to make just enough of a difference that Ursula and Stella each expect the other to be the same age as themselves when they meet up again (each one sees the other "catching up" just enough during the blueshift to make up for the amount the other "fell behind" during the redshift). So all three of them will also predict that Stella and Ursula will have aged the same amount (less than Terence) when they meet up again.

    If the above makes sense to you, that should provide a good starting point for looking at the other ways of interpreting what's going on.
     
  7. Jan 11, 2013 #6

    Nugatory

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    No. The time dilation is completely caused by the relative velocity.

    This problem is one of those that is most easily understood using the relativistic Doppler argument: Both traveling twins carry a light that flashes once a second, so they can keep track of time (both their own and the other traveler's time) by counting flashes. Both travelers see the same thing: During the outbound leg and well after turnaround they see flashes arriving from the other spaceship less often than their own light flashes; then towards the end of the inbound leg flashes arrive much more quickly than their own light flashes. Add the number of flashes received during the long period of slow flash arrival and the short period of fast flash arrival, and total number of flashes each twin receives from the other twin ends up equal to the total number of times their own light flashes - same elapsed time for both. The stay-at-home triplet receives the same number of flashes from both travelers, but fewer than the flashes of his own light - more time has passed for him.

    Or you can use the time dilation explanation: Both travelers know that the other one's clock is running slow relative to their own during the outbound leg. They can't actually SEE the other clock, but they can calculate its tick rate from the Lorentz transforms. The same is true during the inbound return leg. But during the turnaround, the same calculations show the other clock running very very fast, so fast that it completely makes up for the time dilation on the outbound leg and on the return leg. How can this be? Well, the basic time dilation formula only works for observers moving at a constant velocity relative to one another; during the turnaround they are changing their velocity relative to one another and you can't just use the time dilation formula. Instead, you have to use more general Lorentz transformations to calculate the relationship between the two clocks.

    [edit: looks like I'm going to have to learn to type faster... I'm 0-for-2 against PeterDonis here]
     
    Last edited: Jan 11, 2013
  8. Jan 15, 2013 #7
    ok i think i get it!

    if this isn't right then i think i'm just going to give up for now and come back to it another day!

    i did understand the other explanations before, except why you couldn't just swap twin A and B's names round and get the opposite answer. but i think i have come up with an explanation in which i understand why you can't do that.

    twin A travels
    twin B stays on earth

    i think it can be explained with length contraction:
    twin A travels and so experiences length contraction relative to proper length but twin B having not moved continues to observes proper length. as a result twin A feels like he has travelled for example 1 light year but from twin B's perspective he has travelled 2 light years.
    since both of them observe the same relative velocity but different distances, they observe different amounts of time for the journey i.e. the journey takes longer from twin B's perspective than from twin A's.

    am i right?
     
  9. Jan 15, 2013 #8

    PeterDonis

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    This is not the right way to put it. A better way to put it would be that the proper length between A's starting point and his turnaround point, as seen by A, is shorter than the proper length between A's starting point and his turnaround point, as seen by B.

    This is fine, but *both* lengths are proper lengths: they are just proper lengths as measured by different observers.

    Not really, because from twin A's perspective, twin B is moving and so twin B should see a shorter length than twin A does. Length contraction alone isn't sufficient to capture what's going on.

    One other thing you might try is to look at the Spacetime Diagram Analysis. This is the analysis I personally prefer. In this analysis, the difference between the twins is simple: they travel on two different paths through spacetime, and those paths have different lengths, just as different paths through ordinary space can have different lengths. In spacetime, the "length" of a path is just the proper time experienced by an observer traveling on that path, so the difference in path lengths means a difference in experienced proper time. Perhaps I should have tried this one first. :wink:
     
  10. Jan 15, 2013 #9
    Yup that's Right.

    Peter seems to say the same thing as you, but worded differently.
     
  11. Jan 15, 2013 #10

    PeterDonis

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    Not really; see the end of my previous post.

    No, I'm not. I said explicitly that length contraction alone is not sufficient to capture what's going on. In fact, it's not even necessary; the approaches I have suggested in this thread (the doppler effect analysis and the spacetime diagram analysis) make no use of length contraction at all.

    The basic problem with trying to analyze relativistic scenarios using length contraction, and/or time dilation, is that length contraction and time dilation are derived concepts, not fundamental concepts. To really understand what's going on you have to get down to the fundamental concepts, which are the invariants--the things that don't change when you change reference frames--and the direct observables--the things that you don't need to even define a reference frame to measure. (These are not disjoint, by the way; proper time is both an invariant and a direct observable, for example.) The doppler effect analysis uses the doppler redshift/blueshift, which is a direct observable; the spacetime diagram analysis uses the path length along a curve, which is an invariant.
     
  12. Jan 15, 2013 #11

    ghwellsjr

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    Here are some Spacetime Diagrams to illustrate the Inertial Reference Frames (IRFs) discussed in this thread. I use a velocity for the two traveling sisters of 0.6c so that the Relativistic Doppler shifts will be factors of 2 to make it easy to keep track of them in the diagrams. Each triplet sends out a signal each month to the other two triplets. The sisters travel for 12 months and then turn around and spend another 12 months coming home. When they meet, they have both aged 2 years but their stay-at-home brother has aged 30 months.

    I am taking the triplet scenario from post #5 in which I show Stella as a thick black line with dots every month of her Proper Time. She sends out signals shown as thin black lines. She is traveling to the right for a distance of 9 light-months and then turns around.

    Ursula is shown as a thick red line with dots every month where she sends out a signal shown as a thin red line. She travels to the left for 9 light-months and then comes back.

    Terence stays home and is shown as a thick blue line with dots every month. His signals are shown as thick yellow lines going in both directions. I do it this way because there are times when his signal overlaps with a signal from one of his traveling sisters and I wanted to be able to show both signals traveling together.

    The purpose of these diagrams is to show how Time Dilation, Length Contraction, Relativity of Simultaneity and Relativistic Doppler all work in the different IRF's. The first three are different in each IRF but Relativistic Doppler is the same in each IRF. Once I have set up the scenario in the original IRF, I simply use the Lorentz Transformation to create the other IRF's.

    Time Dilation is the ratio of the Coordinate Time (marked by the grid lines) to each triplet's Proper Time (marked by the dots). Time Dilation is dependent on the speed of an observer relative to the IRF, not to another observer. At a speed 0.6c the Time Dilation factor is the gamma factor which is 1.25. In the last two diagrams, there are times when some of the triplets travel at 0.882c (the relativistic addition of 0.6c + 0.6c) and for them the Time Dilation factor is 2.125.

    The Relatistic Doppler factor is also dependent on relative speed but this time it is not between an observer and the IRF but rather it is between two observers, keeping in mind the time delay between when one observer sends the signal and the other observer receives the signal. The Relativistic Doppler factors at 0.6c are 2 for approaching and 0.5 for receding. At 0.882c they are 4 for approaching and 0.25 for receding.

    Note that relative direction matters for Relativistic Doppler but not for Time Dilation.

    Ok, we start with the IRF in which Terence remains at rest and in which the two sisters start and end up at rest:

    attachment.php?attachmentid=54763&stc=1&d=1358278576.png

    Note that during the outbound leg of both sisters' trips, they receive signals from Terence at one-half the rate that they send them and they receive signals from the other sister at one-quarter the rate that they send them. As soon as they each turn around, they start receiving the signals from Terence at double the rate that they send them but they receive the signals from each other at the same rate that they send them because they are both traveling at the same speed when both sending and receiving. Then near the end of their trips, they start seeing the signals from the other sister at four times the rate they are sending them.

    Terence, on the other hand, receives signals from his two sisters at one-half the rate he is sending them for most of the time they are away but near the end, he starts receiving them at double the rate he sends them.

    Between the sisters and their brother, the Doppler shifts are either 0.5 or 2 but this ratio switches near the end of the scenario for the brother but at the turn-around point half way through the scenario for the sisters. So the brother sees his sisters time running slow for more than half the scenario which accounts for why they end up with less time accumulated than he does.

    Please read the previous posts and links for additional information on how to interpret these diagrams.

    Now for the IRF in which Stella is at rest during the outbound portion of her trip and in which Ursula is at rest during the inbound portion of her trip:

    attachment.php?attachmentid=54764&stc=1&d=1358278576.png

    Note that this IRF makes no difference to the Doppler shifts of the received signals as recieved by each triplet but it does make a difference to the Time Dilation that each triplet is subject to. For example, it is Terence who is traveling at 0.6c and subject to a Time Dilation of 1.25 while his sisters spend some of their time at rest (no Time Dilation) and some of their time traveling at 0.882c and subject to a Time Dilation of 2.125.

    You can see the effect of Length Contraction if you look at the distance between one of the sisters at the moment of turn around compared to their brother. That distance contracts from 9 light-months to 9/gamma = 1/1.25 = 7.2. Look at the Coordinate Time of 12 months and observe that Stella (black) is at a Coordinate Location of 0 while Terence (blue) is at -7.2 months. Similarly, at Coordinate Time of about 25.5 months, Ursula (red) is at -22.5 light-months while Terrence (blue) is at -15.3 for a delta of 7.2 light-months.

    The Relativity of Simultaneity is demonstrated all over these diagrams when you compare pairs of events in the two IRFs. For example, in the original IRF, both sisters turned around at Coordinate Time of 15 months but in this IRF, Stella (black) turns around at Coordinate Time of 12 months while Ursula (red) turns around at a Coordinate Time of 25.5
    months.

    Finally, the IRF in which Stella is at rest during the inbound portion of her trip and in which Ursula is at rest during the outbound portion of her trip:

    attachment.php?attachmentid=54765&stc=1&d=1358278576.png

    Hope this helps. Any questions?
     

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    Last edited: Jan 15, 2013
  13. Jan 15, 2013 #12

    PeterDonis

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    George, great diagrams (once again). What tools are you using to make these?
     
  14. Jan 15, 2013 #13

    ghwellsjr

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    I made my own.
     
  15. Jan 15, 2013 #14

    PeterDonis

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    Yes, but using what software? Just graphs in an Excel spreadsheet, or something like that?
     
  16. Jan 15, 2013 #15
    Here is another fundamental concept in the scenario;

    twin A travels
    twin B stays on earth

    You however choose to ignore it, so yes I suppose only the invariants matter to you in analysis.
     
  17. Jan 15, 2013 #16

    ghwellsjr

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  18. Jan 15, 2013 #17

    PeterDonis

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