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Ehrenfest / rod thought experiment.

  1. Sep 11, 2010 #1
    This is new thread on an issue that was that getting slightly off topic in the original thread.


    Let's consider a slight variation of the Ehrenfest experiment. The fairly rigid carriages are all linked together by elastic couplings and are on a suitably highly banked track. As the velocity of the train increases, the elastic couplings get progressively more stretched putting a measurable strain on the carriages. At high enough velocity, the strain on the couplings get so high that they all snap. Once the couplings have snapped the stretching strain on the carriages vanishes and we end up with essentially the rod thought experiment I first proposed with no longitudinal strain parallel to the track, but there will of course be transverse strain as the carriages/ rods will of course be compressed down on to the track by the reaction force to the centripetal force exerted by the track. The transverse strain is not an issue here because I am only considering longitudinal length contraction.

    The Thomas rotation is also not an issue here, because the orientation of the carriages/ rods is maintained by the banked track.

    The key issue here is, can you in principle fit more carriages on the track when they are moving at relativistic velocities, than the number of carriages that will fit on the same track when they are at rest wrt the track?
     
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  3. Sep 11, 2010 #2

    Dale

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    Yes. Although I am not sure why this is "key".
     
  4. Sep 11, 2010 #3

    My original claim was:

    and Starthaus's counterargument was:

    Obviously we are diametrically opposed on the issue of how many moving rods/ carriages that will comfortably fit on the track / perimeter and I saw this as the key issue that needs resolving. You seem to be agreeing with my original claim. I also think it is a demonstration of length contraction, but whether it a demonstration of the "reality" of length contraction is a mute point. It certainly, to me provides a nice way to visualize length contraction, when the carriages or rods are considered as rulers, observers at wrt the track and observers at rest wrt the train all agree that the proper length of the train (with additional carriages) is longer than the proper length of the track it is riding on.
     
    Last edited: Sep 11, 2010
  5. Sep 11, 2010 #4

    Fredrik

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    There doesn't seem to be a contradiction between your claim and starthaus's. If your train cars are linked together all the way around the circumference of the circular track, they will be forcefully stretched. If they're not linked together, and accelerate individually using their own engines for a while, you will get some extra space between them.
     
  6. Sep 11, 2010 #5

    JesseM

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    How are you defining "material strain"? I would have thought that Born rigid acceleration would involve nonzero stress throughout the object since some parts of the object are experiencing greater G-forces than others, it's just that the internal forces have reached a sort of equilibrium where the stress doesn't change with time. It's sort of like how if you have an upright spring at rest on the surface of the Earth and eventually the spring should stop oscillating and reach an equilibrium where the coils nearer the bottom are more compressed than the coils nearer the top. And couldn't the same sort of equilibrium in stresses be reached by an object rotating around a central point with constant angular velocity?
     
  7. Sep 11, 2010 #6
    My original claim was for rods that are not linked together and although I did not make it clear that the rods are not linked, I probably would have said they are linked or chained together if that was what I intended. Anyway, if with the clarification that the rods are not all inked together, I wonder if Starthaus would now agree that my original claim that there would be extra space between the rods, that you you could fit more similar moving rods into?
     
  8. Sep 11, 2010 #7
    I tend to agree with your thoughts here about the stresses. There are non-zero material stresses in the linear acceleration of a rocket and the angular acceleration of a cylinder around its long axis of symmetry. If strain is measured by mechanical devices such as rulers that are subject to the same acceleration profile, then the strain appears to be zero in both cases. On the other hand, while the length of the linearly accelerating rocket as measured by rulers is in agreement with the radar length of the rocket, the perimeter length of the rotating cylinder as measured by the sum of a series of radar measurements is longer than perimeter length as measured by the sum of a series of rulers. So if we define material strain as radar length versus ruler length, then Dalespam is correct that the rotating cylinder perimeter has measurable strain, while the linearly accelerating rocket does not. It is as you say, dependent on how you define or measure strain.
     
  9. Sep 11, 2010 #8

    Dale

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    A change in the proper distance between different particles in the material.

    No, using the above definition of strain, Born rigid acceleration is strain-free by definition.

    Sure, those are all in equilibrium but all those equilibria are strained states. Equilibrium and strain are separate and independent concepts.
     
  10. Sep 11, 2010 #9
    Are you claiming that radar distance and ruler distance between the front and back of a Born rigid rocket undergoing constant proper acceleration is equal ?

    Furthermore, do you agree or disagree that rods around the perimeter cannot be Born rigid because they are rotating?
     
    Last edited: Sep 11, 2010
  11. Sep 11, 2010 #10

    bcrowell

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    I think it actually is possible to impart an angular acceleration to a Born-rigid one-dimensional object, just not to a Born-rigid two-dimensional object. The argument that it's impossible to have an angular acceleration is given in Ø. Grøn, Relativistic description of a rotating disk, Am. J. Phys. 43 869 (1975), and I think it depends on the assumption that the object encloses some area.

    I think the notion of a Born-rigid ruler is exactly equivalent to the notion of measuring distances by radar; anything that can be done with one technique can be done with the other. The difference is that radar (a) actually exists, and (b) doesn't trap unwary people into incorrect arguments.
     
  12. Sep 11, 2010 #11

    JesseM

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    Why would this necessarily be true for an object rotating at a constant angular speed?
    Well, for the spring in the gravitational field it would also be an equilibrium in your sense, right? There'd be no change in the proper distance between particles over time? If so, why couldn't a similar equilibrium be reached by a rotating object? For example, if we have a wheel-shapes space station rotating at a constant rate, why couldn't the proper distance between any given pair of particles be constant?
     
  13. Sep 11, 2010 #12
    Yes, I am. They could calibrate a short rulers using a radar measurement and when they lay these rulers end to end to measure the length of the rocket, the ruler length would agree with radar length of the rocket. If you are asking if the ruler length of the accelerating rocket is the same as the ruler length and radar length of the rocket when it was not accelerating, that is a more technical question, because the transfer from not accelerating to accelerating is not constant acceleration and Born rigid motion requires constant acceleration. A rocket with constant Born rigid acceleration was never not accelerating by definition.

    If the ruler length and radar length length of a rocket is measured while it is initially at rest and not accelerating, it will find that its radar length is shorter after it starts accelerating, even with Born rigid acceleration. What it finds its ruler length to be depends on how and where the rulers are mounted on the rocket and how the rulers are accelerated. For Born rigid motion, every atom of the rocket has to accelerated by the right amount, so each atom technically needs its own rocket motor. If the rulers are accelerated in the same manner they will obviously not notice any change in the length of the rocket. However, if they check the calibration of the Born rigid accelerating rulers with radar, they will find they are shorter than when the rocket was not accelerating. Consider one ruler mounted only at the nose of the rocket and one ruler mounted only at the tail of the rocket. They overlap at the middle.When the rocket is not accelerating, they put a aligned marks on the two rulers where they overlap and another mark on the wall aligned with the first two on the rulers. If the rockets takes off and goes to Born rigid acceleration, and if no no special acceleration is applied to the rulers, the nose mounted one will stretch under the acceleration forces and tail mounted ruler will compress. Now while the two aligned marks on the rulers may still be aligned they will not still be aligned with the mark on the wall. This means strain and stress can be detected in a rocket that goes from a state of not being accelerated to a state of being accelerated even if the acceleration phase is Born rigid.


    I never claimed that the rods around the perimeter are Born rigid and never claimed Born rigidity is an important part of the experiment. Quite the opposite. I claimed you could fit more moving rods around the perimeter than you can fit stationary rods around the perimeter. The moving rods are not squashed together because they are not touching. If you slowed down the moving rods they would expand and eventually when they came to a stop they would be squashed up together, with no gaps and under serious longitudinal compression.
     
    Last edited: Sep 11, 2010
  14. Sep 11, 2010 #13

    bcrowell

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    OK, here's a more formal argument that it is possible to give an angular acceleration to a one-dimensional Born-rigid ruler.

    Let a one-dimensional ruler be initially rotating at a certain angular velocity with one of its end-points fixed at the origin. At times judged to be simultaneous in the nonrotating lab frame, apply impulses at evenly spaced points P1, ... Pn along the ruler, in such a way that the lab observer sees the result as an impulsive angular acceleration about the origin. To prove that this is consistent with Born rigidity, we have to prove that the result of this is to preserve the distance between Pk and Pk+1 as measured using radar by an observer comoving with Pk and Pk+1. (If n is made large enough, it makes sense to have an observer who is approximately comoving with both of these points.) But this is certainly true, because the effect on Pk and Pk+1 is simply a Lorentz boost in the direction perpendicular to the line connecting them. The Lorentz boost does not affect the simultaneity of the two impulses, because the two events are separated in space along a line perpendicular to the boost. Nor does the boost affect the radar distance between the two points, because Lorentz contraction doesn't apply along a line perpendicular to the boost.
     
  15. Sep 11, 2010 #14

    Dale

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    Sorry if I caused any confusion. You can have Born rigid constant angular velocity. You cannot have Born rigid angular acceleration.
     
  16. Sep 11, 2010 #15

    bcrowell

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    ...of an object that encloses a finite area.
     
  17. Sep 11, 2010 #16

    Dale

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    I don't know about that. The derivations I have seen have been for cylindrically symmetric objects, but I didn't find the above very convincing. I don't want to make a claim either way on a rod.
     
  18. Sep 11, 2010 #17

    bcrowell

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    Which part of it did you find unconvincing?
     
  19. Sep 12, 2010 #18
    Whether they are linked together or not, to keep them in place around the perimeter would require restraint of some kind yes??
    They would be subjected to the same outward inertial forces in effect on the disk of the Ehrenfest conditions wouldn't they?.
    SO would those inertial forces cancel or mitigate the contraction or not??
    starthoaus seems to be saying they would negate the contraction entirely.
    From what I have read of the resolutions to the original Ehrenfest scenario it appears they would be a significant factor in all cases. Is this wrong?
     
    Last edited: Sep 12, 2010
  20. Sep 12, 2010 #19
    Hi
    I have questions
    As I understand the Born hypotheses it proposes to eliminate the strain of contraction by controlled distributed acceleration, at the same time countering an assumed force of disruptive expansion.
    If this is the case then, even given an ideal [impossible] perfect control of force, wouldn't the best result acheivable be an equilibrium??
    But can a balance of counter forces ever be strain free??
    What is the meaning of proper distance in this context?
    In the context of inertial frames the meaning is clear but here it seems to be without a reference of any kind.
    But isn't an accelerated state equialent to the exact conditions JesseM is refering to [in gravity]??
     
    Last edited: Sep 12, 2010
  21. Sep 12, 2010 #20
    WHy would the radar length with rotation be longer than the ruler length?
    WHy would the radar length agree with rulers with linear acceleration?
    Aren't the clock rates different at the front and the back in a Born rigid rocket???
    Radar distance longer from the Front to Back to Front , than from B->F->B ??
     
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