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Paradox on Special Relativity

  1. Oct 12, 2009 #1
    hi there
    I am confused by the following setting on special relativity:

    Suppose there are 4 people, say A, B, C and D, in the deepest vacuum - essentially no gravity or any force at all. Suppose further that all move in one-dimension with constant velocities - A moves towards west with 0.8c relative to D, B towards east at 0.7c relative to D and C towards east at 0.9c relative to D.

    Everyone is moving with various speeds relative to D. From D's point of view, all A, B and C have time dilation, i.e., A, B and C will be younger after some time of travelling. But from A's point of view, the aging difference will not be the same as D and this discrepancy applies to each point of view - A or B or C.

    So who in reality, has the greatest time dilation? Or who is the youngest if they started with exactly the same age? As far as I know this is a paradox. I asked one of my physics teachers and he couldn't give me a satisfactory answer.
    I have certain amount of knowledge in Special Relativity, including mathematical analysis. So any kind of mathematical reasoning is welcome. I think there is no General Relativity going on here according to my settings and assumption.
    If my assumptions and settings are impractical, just point me out.
  2. jcsd
  3. Oct 12, 2009 #2

    Doc Al

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    Staff: Mentor

    The answer is that time dilation depends on the frame doing the observing. A sees B's clocks running slow and B sees A's clocks running slow. Both are correct; No paradox.
    Even the age of each depends on the frame doing the observing--since they disagree on who is where at any given time.
    No real paradox, since defining the age of things not located at the same place depends on the frame.

    For a more interesting scenario, look up the "twin paradox". In that scenario, two clocks start out together, then one goes off at high speed, turns around, and comes back to rejoin the other clock. The two clocks can then be compared to see which one "aged" more. And there is a real, objective, physically meaningful answer that all observers will agree upon.
  4. Oct 12, 2009 #3
    Thanks for the reply.
    But in my opinion, there is one weak point - you are using only one frame that does the observing. I know twin paradox by the way.

    To make it scientifically precise, let's say A, B, C, and D have synchronized clocks. More importantly, say there is nothing to regard as stationary - as we think of earth as proper time or whatever. Let's see each observer's view.
    From D's point of view, A is moving to west and B, C moving to east with the velocities I assumed.
    Again, from C's point of view, all A, B, D are moving to west with different velocities from what D observes.
    A and B also have different observations from both C and D.
    Now let's assume further that from D's view, C clock ticks at half his clock's speed, similarly we can give certain numerals for A and B, say 0.3 times slower for A and 0.2 times for B.

    However, when we change observer from D to, say C, C will think D's clock ticks slower than his by half. And worse C's observations on A and B will not be the same as D.
    Similar things happen for both A and B. All don't agree on time dilation. (Here I think it is because of the relativistic addition of velocities.)

    But in reality, there MUST be a consensus between all observers when they meet again.
    Is my argument clear?
  5. Oct 12, 2009 #4
    Your observers will never meet again. They are all separating from each other at constant speed. Hence, no need to ever come to a "consensus."
  6. Oct 12, 2009 #5
    this clearly is not an answer.
  7. Oct 12, 2009 #6
    if we have to set things up so that they never meet again then it is not a full theory - there are limitations.
    In twins paradox the one who speeds up finally turn back and the twins meet again.
  8. Oct 12, 2009 #7
    I am sorry that you find Relativity unsatisfying.
  9. Oct 12, 2009 #8
    We don't have to do anything; it was you who set up the problem this way. The answer is that there is no absolute answer to the question of what the time dilation "really" is, since there is no absolute reference frame by which to measure it. That is why it is called "Relativity."

    And in your example, there is no one who turns back to meet the others.
  10. Oct 12, 2009 #9


    Staff: Mentor

    Hi htetaung, I found this particular exchange very amusing. You set up a scenario where you insist on agreement at an event that does not happen (their meeting) and then don't like it when ZikZak correctly points out that the event does not happen.

    Adding a bunch of observers doesn't make your proposed scenario fundamentally different from the case where two observers disagree with each other about simultaneity, time, and length. It is all just the Lorentz transform, which is perfectly self-consistent and therefore non-paradoxical.
    Last edited: Oct 12, 2009
  11. Oct 12, 2009 #10

    Doc Al

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    Staff: Mentor

    Why do you say that? Any inertial frame is as good as another.

    Clocks that move with respect to each other cannot be synchronized. Perhaps you mean they start from the same point reading the same time?
    OK. So?

    Ah... so if you change the scenario and have the clocks accelerate and switch frames so they can turn around and meet again, then yes you will find consensus. (But that depends on the details of how they made the round trip.) Note that this is just a variation of the twin paradox! (Quadruplet paradox, I guess.)
    What argument?

    In order to predict what each clock will read when then all come back together, you'll need to describe their round trips in precise detail.
  12. Oct 12, 2009 #11


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    Gold Member

    You don't have to set up things so that they never meet again. They can, of course. But if they do (as in the twins paradox) there'll be no discrepancy as to whose clock marks less proper time, i.e. who has aged less. And if they don't, their discrepancy as to whose clock runs more slowly does not entail discrepancy as to the resolution of practical problems: it is an essential element of the theory that all observers will agree on which events happen and which do not and, what is more, by knowing the time and space coordinates that one observer assigns to an event, all other observers can figure out their own coordinates for the same event and thus guess whether it happened or will happen. In practical terms, I have not hit on any situation or problem where the theory does not provide a frame-independent solution, in spite of discrepancies among observers about frame-dependent magnitudes. If you think of one, let me know.
  13. Oct 12, 2009 #12
    Unfortunately, it is the correct answer. While the observers continue to separate at constant speed, they will all have separate points of view and it is not possible for the observers to come to any concensus about who is correct about the relative clock rates in the scenario you described.

    In a simpler example consider an object moving at 0.7c relative to observer A but stationary with respect to observer B who is comoving with the object. Can A and B reach a consensus about the true absolute velocity of the object? Not in relativity. Both have there own opinion of the velocity of the object and neither can prove the other wrong. Since time dilation is a function of velocity, how can A and B reach a consensus on the true clock rate of the object if they can not even agree on the velocity of the object?
    Last edited: Oct 12, 2009
  14. Oct 12, 2009 #13
    Where is the cut-and-past answer sheet on commonly misconstructed statements of special relativity?
  15. Jan 20, 2010 #14
    If you subscribe to all of the consequences of the Lorentz transformations there is always somthing akin to a paradox. One thing that is never discussed is the fact that when you use the form that gives you time dilatation you can't also have foreshortening and vice versa. Only one variable is changed by each permutation of the transformations and that does indeed seem paradoxical.

    The problem for all of the arguments is that there is no validity to any of the Lorentz transformations. Lorentz devised the transformations to explain why it was impossible to detect the aether that he and many others refused to believe did not exist. Poincare still believed in the aether in 1908 when he published his book "Science and Method". he was still lauding Lorentz transformations in the context of their original purpose, "the forshortening of each body of light" as it moved within the aether current caused by the Earths movement in its orbit. (That rate is about 29.68 meters per second). Einstein adapted the equations to a different purpose and that was to resolve conflicts he saw when you put c as an invariable into light thought problems that concern addition of velocity problems. His reasoning seems to be that since light can't vary every other function of an addition of velocities problem must be able to. Thus his conclusions about foreshortening and dialatation. It is his use of the Lorentz transformations that suggested these ideas to him not the other way around.

    None of this means that either of Einstein's postulates of Special Relativity is untrue. It just means that what they indicate is the relativity of perceptions one reference frame to another which is a concept that people are forever trying to mix in with Lorentz Transformation results which seem to predict absolute results whenever you actually use them. The fact that you can't derive all of the phenomena at the same time renders them suspect. The fact that their use with light thought problems that appear in one version in texts and encyclopedias and claime to demonstrate time dilatation require the use of another misconception about the propagation of light is reason to scrap their use and the things that derive from their use altogether.

    The misconception about light I mention is that all of these thought problems presume that the beam of light continues to follow along with the forward momentum of the moving train or rocket or Galileos Ship and is observed by the stationary observer to angle forward as he watches. The light actually moves in a straight line from its point of propagation and will look like a straight up and down beam from the stationary point of view and will angle backward for the observer travelling with the source. This is very different from what Einstein thought and from what all science has gone along with ever since.

    I have posted a thourough dicussion of this at my blog: http://drmphysics.blogspot.com
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