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Langevin Twins Paradox Proof

  1. Aug 22, 2005 #1
    Paradox was verified by Hafele-Keating in 1971. They placed atomic clocks in commercial airplanes and then compared to a reference clock. Recently some people/groups are considering their data as questionable & unreliable. Does somebody know later experiment that also verified the paradox in a similar way?
     
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  3. Aug 22, 2005 #2

    Meir Achuz

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    It is not a paradox, unless you don't understand it.
     
  4. Aug 22, 2005 #3

    russ_watters

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    It isn't a paradox, its a well-proven consequence of Relativity.

    Currently, there are 24 GPS satellites circling the globe (plus, iirc, 4 spares) at various orbital inclinations. Each contains a clock(actually, 3 clocks, iirc) superior to the ones used in the 1971 experiment and since each is going ~40x faster and flying 10x higher than the planes in the 1971 experiment, the Relativistic effects are far more pronounced. The GPS system verifies the predictions of Relativity to an extremely high precision.
     
  5. Aug 23, 2005 #4
    Almost all authors agree there is no paradox - but there is much disagreement as to why.
     
  6. Aug 23, 2005 #5

    ahrkron

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    There are different ways to explain the effect, but there is no disagreement on the fact that it comes from relativity.
     
  7. Aug 23, 2005 #6
    Some of the explanations are mutually inconsistent - what is agreed upon, in general, is the validity of time dilation. What is not explained is the physical mechanism that produced an actual difference in the Hafele-Keating clocks when they were compared, or any other experiment where one clock has lost time relative to another.
     
  8. Aug 23, 2005 #7

    pervect

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    All of the explanations arrive at the same numerical results. Any "inconsistencies" are not the result of different predictions, but of different philosophies.

    I believe that yogi is attributing properties to clocks that they simply don't have, and that his search for a "physical" explanation of their behavior is based on the faulty assumption that two clocks following a different path "should" both read the same time when they next meet. There is no reason to suppose that clocks have or should have this property, though in older Newtonian theory clocks did have this property.
     
  9. Aug 23, 2005 #8
    Pervect - not at all - I don't know how you could interpret my post(s) as postulating that two clocks which follow different space-time paths will read the same when later compared.

    Sometime back on another thread I raised the question re the mechanism that produced actual time differences - you posted a short explanation which much intrigued me - then the thread went off in a different direction. I would appreciate your reiteration of that idea - and perhaps some additional embellishment.
     
  10. Aug 23, 2005 #9

    Hurkyl

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    So what do you mean by "What is not explained is the physical mechanism that produced an actual difference"? More precisely, what do you mean by physical mechanism, and why do you think one is needed?


    Suppose I had marked two spots on the floor, and I had drawn two different paths that led from the first spot to the second, one a straight line, and the other a semicircle.

    I find two people who happen to have devices that measure the length of a path, and set them on each path. They roll their device along their path. I claim that the devices will give different readings: the device rolled along the semicircle will measure a greater distance than the device rolled along the straight line.

    Would you object that I have not given a physical mechanism that produced an actual difference in the measurements of the devices? Would you take this as a flaw of Euclidean geometry, and any physics based upon it?


    The situation with clocks is similar. Wait, that's misleading... it's virtually identical. The paths in my experiment are directly analogous to trajectories through space-time. The length measuring devices are directly analogous to clocks.
     
  11. Aug 23, 2005 #10
    Hurkyl - that is a description - I find no fault with the description, but it is not an explanation - how does a clock that is put in uniform motion wrt to another clock to which it has been synchronized know at what rate to run?
     
  12. Aug 24, 2005 #11
    "Laws of nature are the same to every inertial observer".

    This is the basic postulate of SR. The value of the interval (or the speed of light, c) has to remain constant to every observer, no matter what their relative speed is. Otherwise laws of nature would be chaotic.

    For this to be valid, time and distance can't be the same to every observer, but are relative concepts that depend on the relative speed of the observers.

    btw. hi I'm new here. Sorry if I don't write perfect english, I'm finnish :smile:
     
  13. Aug 24, 2005 #12

    Hurkyl

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    The same way that a measuring tapes "know" to give different readings when they measure different paths: the measuring tapes measure length, so when they measure paths with different lengths, they give different readings.

    In the same way, clocks measure proper time. So, if used to measure two paths along which the proper time is different, the clocks will measure different durations.
     
  14. Aug 24, 2005 #13

    Hurkyl

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    If you're going to make these sorts of objections, you had better say precisely what you mean by "description" and "explanation".
     
  15. Aug 24, 2005 #14

    DrGreg

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    I have two rulers (measuring rods) and lay one in a north-to-south direction and the other in a northeast-to-southwest direction. How does the second ruler "know" to measure distance travelled northeastwards instead of distance travelled northwards ?

    (The question is, of course, rhetorical. This is another way of saying what Hurkyl just said.)
     
  16. Aug 24, 2005 #15
    Drgreg: That is the big difference - when one observes lengths in a relatively moving reference frame, the contraction is an illusion, an apparency - the lengths do no really change - as Dr Resnick puts it: The contraction is real only in the sense that the measurements are real, There is no permanent physical change as Lorentz proposed to save the ether. Or as Eddinton once said: The contraction of lengths is true, but its not really true (whatever that means).

    As for temporal changes - there is a permanent record - relatively moving clocks log different times, and this time differential can be ascertained when the two clocks are brought to rest in the same frame as per Hafele-Keating. Time dilation is not an fleeting phenomena - this is the significance of Einsteins explanation of the physical effects described in part IV of his 1905 paper - and that is the element that distinquishes SR from all the assertions that it was Poincare or Lorentz or somebody else who should be accorded recognition for the concept - Einstein stuck his neck out and predicted actual time dilation. No one else had the courage to go that far.

    My question remains - perhaps it is a consequence of spatial motion relative to Minkowski Spacetime - the problem with this interpretation is that it smacks of a preferred global reference frame.
     
  17. Aug 24, 2005 #16

    russ_watters

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    There is a difference between a clock and a tape measure that no one has picked-up on: a clock is a recording device (actually, two separate pieces: a measuring device and a recording device) and a tape measure isn't. Once a clock ticks off a second, that second is gone, never to be seen again. The only thing left is the record of that second, which for a clock is the time output by the display. So the measurements taken by a clock are exactly as permanent as those taken by a person with a clipboard standing over a tape measure. Better yet, one of those newfangled laser tape measures could keep a "permanent" record of length contraction.
     
  18. Aug 24, 2005 #17

    Hurkyl

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    Given two different trajectories in space-time, the proper time along those paths is generally unequal. Thus, the time measured by clocks traversing each trajectory will generally be different.


    P.S. when length measurements of the same "object" are conducted in two different frames, what is actually measured is different for each frame. Thus, just like the clocks, different paths have different proper lengths, and thus the results of the measurement are different.
     
    Last edited: Aug 24, 2005
  19. Aug 25, 2005 #18
    Hurkyl - in the usual case, we consider one observer to be in the rest frame, so there is no proper distance to be attributed to the at rest clock. Of course in the general case, (as per your post 9 above) three adjacent clocks A, B and C could be sychronized, and A could travel to a distant point D measured along a straight line connecting C and D and B could follow a semicircle in the CD rest frame and arrive at D. So each clock would have a different trajectory - and a different proper path length - but does this reveal anything about why the two moving clocks do not run at the same rate.

    What is interesting about the development of the conclusions in part IV of the 1905 paper is the shift from observational symmetry with regard to length contraction, to the conclusion that temporal differences are real - and consequently they are "one way" since at the end of the experiment A cannot read more than B and at the moment and in the same frame B cannot read more than A. This all comes with very little comment.
     
  20. Aug 25, 2005 #19

    Hurkyl

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    Yes it does: the paths they took are different. The paths have different proper times. Clocks measure proper time. Therefore the clocks give different measurements.



    Let's try this from the top.


    Space-time is four-dimensional. We have a metric on space-time: the measure of a straight line segment is given by, in any inertial frame: (using the +--- convention, and units where c = 1)

    d(P, Q)² = (Δt² - Δx² - Δy² - Δz²)

    And through the methods of calculus, we can turn this into an infinitessimal metric which can be used to measure curves through space-time.

    A bit of work shows that d(P, Q)² is independent of the inertial reference frame used to compute it, thus the metric is a "pure" geometric object -- we can speak of its value without referring to any sort of coordinate system.


    When d(P, Q)² > 0, we say that the line segment PQ is time-like, and that |d(P, Q)| is the proper time along the line segment. When d(P, Q)² < 0, we say that PQ is space-like, and that |d(P, Q)| is the proper length of the line segment.

    This terminology also applies to curves through space-time where the metric does not change sign.


    Now, if we have a point-like object travelling through space-time, its trajectory is a time-like curve, called its world-line. If it's travelling inertially, it's actually a time-like line! We can pick two points on the worldline, and ask about the proper time experienced by the object as it travelled from one point to the other.

    A clock, Special Relativistically speaking, is a device that measures the proper time along the worldline it traverses.

    So, if I have two (time-like separated) points in space-time, and two clocks follow two different worldline (segments) joining the two points, then they've measured the metric along two entirely different curves, and it should be clear that the circumstances would have to be very special for the readings to be the same.

    (Note that in this discussion of travelling points and clocks, I've not invoked a single mention of any sort of coordinates!)

    We can draw space-like curves in space-time, and also ask about the lengths of these things.


    Measuring the length of an actual object is another question all together. Suppose we have a one-dimensional object travelling through space-time. It traces out a two-dimensional shape through space-time, called its worldsheet.

    Now, it doesn't make sense to ask about the length of a two-dimensional shape! However, I can ask about the proper length of a cross-section of a worldsheet. Clearly, differently angled cross-sections will have different lengths.

    Now, I'll step back to an inertial reference frame: when we ask about the length of an actual (one-dimensional) object, what we really mean is that we want to compute the proper length of a cross-section of the object's worldsheet. We use our coordinates to specify exactly which cross-section: usually, we want something like the intersection of the worldsheet with the hyperplane of a given, constant time coordinate. (A hyperplane of simultaneity)

    Different inertial reference frames have different hyperplanes of simultaneity, and thus will use different cross-sections when measuring the length of a one-dimensional object.


    Inertial reference frames are also good for selecting cross-sections of worldlines as well (a.k.a. a point on the worldline). Given a worldline and two values of coordinate time (or hyperplanes of simultaneity, if you prefer), we can speak of the segment of the worldline between those coordinate times. Since we can compute the difference of the chosen coordinate times, and the proper time along the segment, we can form the ratio Δτ / Δt. The infinitessimal version is interpreted as the "rate change of proper time with respect to coordinate time", or time dilation.

    Note that time dilation is always with respect to coordinate time! It doesn't make sense to ask about the rate of one clock with respect to another clock directly! You have to do it indirectly by comparing both clocks to a coordinate time!

    It should be clear that we should have no expectation of clocks running at the same rate with respect to coordinate time: if two clocks are travelling along different worldlines, and we select intervals between two coordinate times, the clocks are used to measure the proper time along two entirely different worldline segments, and entirely different worldline segments will generally have entirely different proper times along them.



    This is all pure geometry: the only physics involved is the hypotheses that this geometry describes the universe, and that there is a device capable of measuring proper time. This is why I say that when you ask for an "explanation" of why two clocks run at different rates (with respect to a chosen coordinate system), that it's tantamount to asking for an explanation of why two line-segments in Euclidean geometry can have different lengths.


    edit: fixed mistake involving the metric
     
    Last edited: Aug 26, 2005
  21. Aug 25, 2005 #20

    robphy

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    typo: d(P, Q) = (Δt² - Δx² - Δy² - Δz²) [no square root]... or else use d² to distinguish the classes of 4-vectors.
     
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