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Twins Paradox - but with a different spin

  1. Jan 12, 2012 #1
    While attempting to wrap my head around the classical Twin Paradox, I asked myself the following. So far I have not been able to reach a well reasoned answer... which is no big surprise. :)

    Twins (A and B) are oriented in space such that A could be said to be standing on B but facing in the opposite direction. Both twins have a jetpack and clock. They are both tethered to a pole which is positioned 1ly/2∏ away. On this pole is mounted a third clock.

    All three clock are syncronized and the twins ignite their jetpacks and begin to accelerate in the opposite direction around the 1ly circumference path. They both finally reach and maintain a constant velocity of .5c in the first revolution and continue as this velocity for 100 years.

    Can one say the twins have equivalent accelerated reference frames? Maybe equivalent but opposite? I am unsure as to what constitutes a reference frame.

    From the perspective of the pole, the twin's clocks should be slower. But what of the twins with respect to each other? Shouldn't they both see the other's clock as slower and the other twin age at a slower rate?
     
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  3. Jan 12, 2012 #2
    If I understand what you are saying correctly, then they should both age equally because the situation is symmetrical from the pole's rest frame.
     
  4. Jan 12, 2012 #3

    PAllen

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    The two revolving observers are not inertial, and neither can meaningfully uses the Lorentz transform (except over a tiny region for a moment of time).

    If they all got back together, at the pole (for example; I assume the pole is inertial), the two revolving clocks would be slower by the same amount compared to the pole clock.

    As to perceptions, the pole clock would see the revolving clocks going slow at all times. The revolving clocks would see the pole clock going fast at all times. (Because of non-inertial motion, there is no symmetry in perception - each of these does not see the other slow).

    As for how the revolving clocks see each other, this would vary moment to moment, and also depend on what you mean by 'see'. You could mean 'directly see', in which case Doppler effect would would be a major contributor. You could also mean, factoring out Doppler, in which case you would still interpret periodic variation each other's clocks, but the function would be different (from the 'raw' observation, not factoring out Doppler).
     
  5. Jan 12, 2012 #4

    ghwellsjr

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    The pole and initial rest state of the twins is the only valid inertial frame (of those you mentioned) from which to analyze your scenario in Special Relativity. You are correct that in this frame, both twins will experience the same time dilation and will age the same and their clocks will always have the same time on them. They will be about 86.6 years old after 100 years according to the pole's clock. They will always see the pole clock running faster than their own. They will actually see each others clock as running slower than their own as they separate and then running faster as they approach but at the same time each time they get together.

    You cannot consider each twin as being in a valid frame according to SR because they are constantly accelerating which means they are not inertial. In the more normal Twin Paradox scenario, one twin travels away at a constant speed and during this time they each can consider the other ones clock to be time dilated but when the traveling twin stops and turns around, he becomes non-inertial and so is no longer at rest in the same frame.
     
  6. Jan 12, 2012 #5
    I would agree that if I were observing from the perspective of the pole, both twins should age the same to me. But what of A's perspective of B? It is my understanding that since there is no absolute positioning, it is equally fair for A to say he is not moving and both the pole and B are moving relative to him. This is the original Twin Pardox. The explanation for the paradox states that since one twin is actually undergoing acceleration (an accelerated reference frame) that the reference frames for the two twins is are not equivalent. Thus it is the accelerated twin which ages more slowly.

    However in my example, both twins are moving identically, except for direction which is opposite. I think each twin would observe the other as aging more slowly since they are moving relative to one another... but this can not be.
     
  7. Jan 12, 2012 #6
    No, this is not valid because A's reference frame is not inertial. He has no right to claim he is at rest because he is constantly accelerating (circular motion, even at constant speed, involves acceleration which can be measured).
     
  8. Jan 12, 2012 #7
    Yes, I agree and it is why I did not put it in my original post. But am I wrong to say that A could claim an inertial frame so long as everything else (pole and B) are accelerated equally in the opposite direction of A's acceleration? Something a kin to free fall?

    For example, if one plots the path of a projectile that undergoes acceleration due to gravity, one can also hold the projectile as motionless and move the paper in the opposite direction and successfully plot the exact trajectory? Think moving a pen along the path against stationary paper and then move the paper in the opposite direction against a stationary pen?
     
  9. Jan 12, 2012 #8

    Dale

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    No. If A carries an accelerometer it will read something other than 0. Therefore A is a non-inertial observer. A is definitely not in free fall.
     
  10. Jan 12, 2012 #9
    I am sorry, I failed to express my thoughts sufficently. I was not attempting to say that A was in a free fall.

    Yes, I agree A is being accelerated and it can be measured. And due to this, A is a non-inertial observer. That being said... can A's frame of reference to transformed to an inertial frame by adjusting the acceleration of all other relevent objects in the frame in the opposite direction.

    ie: When you plot the path of a projectile under the influence of gravity, you can move the pen over stationary paper or move the paper under a stationary pen... you obtain the same plot.
     
  11. Jan 12, 2012 #10
    No. If you could find an inertial frame where A was at rest then A wouldn't be able to measure any acceleration.
     
  12. Jan 12, 2012 #11
    Ouch. Ghwellsjr, you don’t really mean that do you? You can’t use the Lorentz Transformation, but that’s not the only transformation in SR.
     
  13. Jan 12, 2012 #12

    Dale

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    No. By definition "A's frame" is a frame where A is at rest. And an intertial frame is one where any object at rest in the frame is inertial (has 0 reading on an accelerometer). Therefore, since A is at rest in A's frame and since A is non-inertial A's frame is non-inertial.
     
  14. Jan 12, 2012 #13
    Ok... let me see if I got this...
    If both A and B traveled in the same direction, they would both age the same with respect to the pole and both twin's clocks would have the same time.

    In this example, they are moving in opposite direction. They would perceive each other's clocks as moving slower for half of the trip, as they moved away from each other. As their paths start to converge, they would perceive the others clock as moving faster and as their paths converged, their clocks would once again be in sync. So the net difference would be 0 with respect to one another.

    The fact that they were accelerated negates any time differential that would have occurred due to traveling at different velocities relative to each other.

    My egg is scrambled. :)
     
  15. Jan 12, 2012 #14

    Dale

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    It is actually a little more complicated than that. The fact that they are accelerated means that you cannot use the standard time dilation formulas or Lorentz transform formulas to determine what happens in A and B's respective frames. Those formulas only apply to inertial frames. This is why it is important to always know what situations a given physics formula covers, no formula covers everything.

    In non-inertial frames all sorts of weird things can happen. In general, time dilation becomes a function of position as well as speed. There are ways to calculate what happens in non-inertial frames. The most systematic way is to use tensors and differential geometry, but it requires a little bit of notation and math concepts that you may never have encountered before.
     
  16. Jan 12, 2012 #15

    PAllen

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    Please see my post #3, and tell me what you don't understand.
     
  17. Jan 12, 2012 #16

    ghwellsjr

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    You have to make a distinction between perception (what each observer sees of the other clocks) and the times on the clocks as defined by a reference frame. Perception does not depend on any reference frame nor any theory such as Special Relativity or Lorentz Ether Theory or having any theory at all. So what you said about perception is correct, it's what I said in post #4, and to be precise, you should have said at the end, each time they came back together, their clocks would once again appear to be in sync because according to your chosen reference frame, that of the stationary pole, they are always in sync, they always have the same time on them. But this statement is dependent on a theory, in this case SR, and it is dependent on our chosen frame of reference because another frame of reference could have the twin's clocks out of sync during the major part of the cycle where they are not together.
    It's not just the fact that they were accelerated, it's the particular way that they were accelerated that results in them both being at the same speed (which is what time dilation depends on) according to our chosen reference frame and the fact that they are continually being brought back together that negates any time differential. If they were accelerated identically but in different directions such that they never came back to the same location, then in our chosen reference frame they would have clocks that remained in sync but in other reference frames this would not be the case.
     
    Last edited: Jan 12, 2012
  18. Jan 12, 2012 #17

    ghwellsjr

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    I keep hearing this but nobody ever comes through with such a transformation. Why don't you be the first to show the OP one of the infinite ways there are of doing this. And then we'll let the OP explain it to me.
     
  19. Jan 12, 2012 #18
    I do not fully understand the Lorentz transform but think I get the main jest... glad it is not applicable. :)

    I thought I understood this but I may not.. see below.

    Here is where I have trouble. Why do the non-inertial observers see the inertial clock as running faster? If they were not accelerating and thus inertial, the pole's clock would appear to be moving slower, correct? I do not understand how acceleration would change this.

    What I mean by "see" or "perceive" would be when the non-inertial twins path of travel converged, they would directly observe the other's clock and physical condition (aging). Here I am assuming they would detect a difference but it appears by your comments and the comments of other, my assumption is incorrect.
     
  20. Jan 12, 2012 #19
    Yep. I get frustrated with folks who do that too. I put a summary on my web site (because I'm terrible with latex).

    http://www.relativitysimulation.com/Documents%20/AccelerationTransformation%20in%20SR.html [Broken]

    The hard part is going to be when you ask me to solve a problem with it.
     
    Last edited by a moderator: May 5, 2017
  21. Jan 12, 2012 #20

    PAllen

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    Acceleration is fundamentally different because your concept of simultaneity (assuming you try to define it similar to an inertial frame) is changing moment to moment. This change in what events are considered simultaneous radically changes the way signals from a distance are interpreted.

    In this case, there is a simple way to see the validity of what I suggest. Imagine the pole observer sending out a signal every second. This signal is an expanding spherical wave front. The revolving observers get this signal every 1/γ seconds, by their clocks. The pole's clocks appear fast, and light from the pole appears blueshifted, at all times, by both revolving observers. Correspondingly, the pole observer sees the revolving clocks slow and red shifted.
    The revolving clocks would see the same time on their clocks each time they passed each other. If they ever slowed down and joined the pole clock, they would see that they were younger (their clocks were behind).

    [edit: typo noted by gwellsjr corrected.]
     
    Last edited: Jan 12, 2012
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