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Bell’s Spaceship Paradox In Another Way

  1. Sep 23, 2013 #1
    This is another version of bell’s spaceship paradox.
    Consider two spaceships A & A’ separated by a distance L, and are tied by a non elastic thread. They are on earth. This time the two spaceships are not moving instead I'm moving.
    I’m in a distant planet and i starts to accelerate and reach at 99% speed of c and stops accelerating, now from my point of view the two spaceships are moving towards me at 99%speed of c. So they must undergo length contraction, therefore the A’ ship must increase its distance from A because of length contraction and the thread must break. But it won't break in reality, So you will answer the thread won't break, then what about length contraction, won't it happen here?
    Last edited: Sep 23, 2013
  2. jcsd
  3. Sep 23, 2013 #2


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    Hi Trojan666ru! Welcome to PF! :smile:
    If the front spaceship is AB and the rear is CD, so the string is BC,

    then you're saying that AB gets shorter, so BC must get longer, so the string will break.

    No, everything gets shorter. :wink:
  4. Sep 23, 2013 #3
    if everything gets shorter then why won't the whole ship and thread get shorter in the original paradox presented by bell?
    why don't they consider the whole system as one?
  5. Sep 23, 2013 #4
    No, because in this case only you are accelerating into another system, while the rest coordinates and rest lengths of the ships and the thread are unchanged in their rest system (no reason that the thread will break). And from your perspective, the application of length contraction to all of those lengths cannot change those proportions (again no reason that the thread will break).

    In contrast to your version, the original "bell’s spaceship paradox" is built on the assumption, that the distance between the ships remains the same from your perspective. This requires that the rest length between the ships increases in its own rest system, while the rest length of the thread remains unchanged (therefore the string breaks). Now, from you perspective you have to apply length contraction to all of those lengths:
    *You apply length contraction to the increased rest length between the ships, therefore it stays the same from you perspective.
    *Then you apply length contraction to the unchanged rest length of the thread, therefore it becomes smaller compared to the distance between the ships, thus it breaks in you perspective as well.
  6. Sep 23, 2013 #5
    Histspec: So from the acceleration ships reference frame the front shop will accelerate further, why is that so? their acceleration was uniform and equal at the start but what changes it to become unequal?
  7. Sep 23, 2013 #6
    Using identical acceleration programs requires, that those programs are activated simultaneously. In relativity, simultaneity depends on the chosen frame. Consider two frames S and S'.
    In S the programs were initiated simultaneously, thus they are accelerating synchronously and therefore their distance remains the same, until they come to rest in the new frame S'.
    In S' the acceleration was not simultaneous and thus asynchronous from the beginning. It's obvious now: identical acceleration programs that have been initiated at different times lead to an increasing length between the ships, until they stop in S'.
  8. Sep 23, 2013 #7

    Let's stay in the reference frame of one accelerating Bell's spaceship.

    Accelerations are not equal at the start. At the start clocks run faster at the front ship, and also rocket fuel is burned at faster rate there.

    How can this be explained in the reference frame in question? It must be the gravitational time dilation, in our frame that's the only explanation.

    Oh yes, let's remember that an accelerating reference frame is a reference frame only in general relativity.
  9. Sep 23, 2013 #8


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    The picture above gives an analogy to Bell's Spaceship Paradox in ordinary geometry.
    Draw two curves, A and A', that are identical except one is displaced in the vertical direction.

    If the two curves are straight lines, as shown in the upper-left, then they will be parallel. The distance between the two curves is constant.

    If the two curves are NOT straight, as shown in the lower-left, then they will not be parallel. The distance between the two curves changes as you move along one of the curves.

    How is this possible? On the right, we show a blow-up of the non-straight case. What you can see is that for the two identical curves, the vertical distance between corresponding points on the curve remains constant. But vertical distance is not the relevant notion of distance between two non-straight curves. A more appropriate notion of distance is "perpendicular distance". You take a point on one of the curves, and take a line that is perpendicular to the curve at that point, and you measure the distance between the two curves along that line. The perpendicular distance changes as you move along the curve. For a curve that is horizontal, "vertical distance" and "horizontal distance" are the same. For a curve that is straight, the two are not the same, but they are always proportional. But for a curve that is not straight, the two notions of distance can be very different.

    In the Bell Spaceship Paradox, there is a similar thing going on. Instead of "vertical" and "horizontal", you have "spatial distance" and "time" (relative to the original rest frame of the rockets). The two rockets trace out two "curves" in spacetime that are identical, except one is displaced in space relative to the other. The spatial distance between the rockets, as measured in the original rest frame, remains constant. But that is not the relevant notion of distance. A more appropriate notion of distance is the distance between the rockets as measured in the instantaneous rest frame of one of the rockets. This is the analog of "perpendicular distance" in the diagram. This notion of distance changes as the rockets travel.
    Last edited: Sep 23, 2013
  10. Sep 23, 2013 #9


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    There are two criteria that you can use to determine if the thread will break in all variants of the scenario.

    The first criterion is to compare the unstressed length of the thread to the distance between the ships. That can be done, e.g. by fixing a second identical thread to the front ship and setting it out immediately next to the first thread but not connecting it to the rear ship. If the distance between the ships is greater than the unstressed length of the second thread then the first thread snaps. If this criterion is true in one frame then it is true in all frames.

    The second criterion is to calculate the expansion tensor of the congruence representing the thread. This is mathematically more advanced but doesn't require any additional threads for comparison. Since this criterion is tensor based it is manifestly invariant.

    In this case, the unstressed length of the thread is the same as the distance between the ships in all frames, so the thread does not snap in any frame.
  11. Oct 3, 2013 #10
    I would just like to endorse the conclusion expressed in the last sentence of the previous comment by DaleSpam: Yes indeed, contrary to a lot of nonsense written on the internet & elsewhere, the thread between the spaceships in Bell's scenario does not snap in any frame. This is fairly easy to prove on the one hand by using a full Lorentz transformation of both space & time coordinate from the launch frame to the co-moving frame (or any other frame) & then calculating the 'proper distance' (or 'proper length') - or on the other hand by realising that 'length contraction' in special relativity is merely an "apparent" effect & not a physically real shortening !
  12. Oct 3, 2013 #11


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    DaleSpam is not talking about Bell's scenario, but about Trojan666ru's scenario. In Bell's scenario the string snaps in every frame.
  13. Oct 3, 2013 #12


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    Closed, pending moderation.

  14. Oct 3, 2013 #13


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    To echo what A.T. said, my comments were about the OP's scenario, not the standard Bells "paradox" scenario. In the standard scenario the string breaks in every frame because the distance between the ships in each frame is greater than the length of the string in that frame.

    With that we will leave the thread closed to avoid further hijacking.
    Last edited: Oct 3, 2013
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