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Length Contraction Issue

  1. Sep 28, 2012 #1

    bgq

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    Hi

    Consider a space ship at rest in space, this ship is equipped with a bomb connected to two rings. Once the rings connected to each other, the bomb explodes making the ship also explodes. The proper distance between the rings is one meter. Now consider a conducting rod of proper length one meter too moving directly towards the ship with constant velocity (the rings and the rod are on the same line). The rod can fit in the rings but with some friction.

    Bomb_Frame1.JPG

    According to the frame of the ship, the rod will be contracted in the direction of motion so that the rod will pass through the two rings without connecting them, so no explosion occur.

    Bomb_Frame2.JPG

    According to the frame of the rod, the distance between the rings will be contracted, so the rod will connect the two rings, and the explosion will occur.

    Rod_Frame.JPG

    Questions: 1) Will the explosion occur?
    2) Does relativity allow that the explosion could exist in one frame, but not
    in the other ?

    Please note that this problem is different from the Barn-Pole paradox.

    Thanks for any replies.
     
  2. jcsd
  3. Sep 28, 2012 #2

    Nugatory

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    Re: Lenght Contraction Issue

    What is the mechanism that makes the explosion happen when the rings are connected?

    That is, how does one ring touching the rod "know" that the other ring is also touching the rod? If you're thinking about a flow of electrical current through the rod... Don't forget to consider the travel time for the current.
     
  4. Sep 28, 2012 #3

    bgq

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    Re: Lenght Contraction Issue

    Actually, I think of flow of electrical current, but what does the travel time have to do with the situation?
     
  5. Sep 28, 2012 #4

    ghwellsjr

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    Re: Lenght Contraction Issue

    No.
    No. Haven't you learned that from your other thread? Frames don't change what happens, they only change the values of the coordinates that are assigned to events.
    No it's not, it's identical (except for the details). Why do you think it is different?
    You're welcome.
     
  6. Sep 28, 2012 #5

    bgq

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    Re: Lenght Contraction Issue

    1) Why the explosion does not occur with respect to the frame of the rod?
    Beside this, before I post this problem, I searched the web for length contraction paradoxes, and I found myself can easily resolve them (although their authors claim that no one can resolve them and they prove that SR is false), I post this own problem because I see in it something very different from other paradoxes.
    2) Yes, I expect that frame doesn't change what happens, but I ask this question to clarify the problem of question 1.
    3) It is completely different from Barn-Pole paradox. The Barn-Pole paradox could be resolved by showing that instants of opening and closing the front and the rear door are not simultaneous, and this is enough for resolving the paradox. Here you may draw the space time graph (or do calculations) for each of the ship and the rod, and you can see clearly that there is definitely a period of time the rod is in contact with the two rings at the same time with respect of the rod (although the entering and exiting are not simultaneous). In other words, in this problem no doubt that entering the region of the rings and exiting are not simultaneous, but in this situation this solves nothing, because It is still a period of time where the rod is contact with two the two rings at the same time. I don't know if my English is not good enough to express what I think, but I see that this problem is somehow opposite to Barn-Pole paradox. In that paradox, you will look at entering and exiting, but here we will look what happens in between.
    4) I really appreciate for you and all others the time you give me.
     
  7. Sep 28, 2012 #6

    PAllen

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    Re: Lenght Contraction Issue

    It's exactly the same as the barn pole because the key issue is simultaneity. Nugatory was trying to push you to see that.

    I assume you have some image of the bomb going off if the rod goes through both rings. But, obviously, in all frames, the front and back of the rod go through each ring. What you really mean is: the front is passed one ring before the back is passed the other.

    We don't need to care about how the rings detect bar crossing them. All we need to note is that any way of doing so must be equivalent to clock rest relative to ring 1 reading the same as clock at rest at ring 2 (suitably synchronized), while the front is past one ring and the back hasn't reached the other ring. So time and simultaneity have everything to do with the resolution.

    Let's label the ring on the right ring1, the ring on the left ring 2. Define 4 events:

    f1 - front of bar passing ring 1
    b1 - back of bar passing ring 1
    f2, b2 similarly.

    In the bomb frame, you agree the order of these events is: f1,b1,f2,b2 and the bomb doesn't go off. The explosive order would be f1,f2,b1,b2. Note also, that causality requires f1 before b1, and f2 before b2 in all frames.

    Now, we note that in the rod frame you have the explosive order: f1,f2,b1,b2. Note it is still consistent with the causality requirement. So you think the bomb explodes. What you miss is that in the rod frame, the clock moving with ring 1 is behind the clock moving with ring 2, sufficiently so that using the readings on these clocks the order remains f1,b1,f2,b2. Please note that any method of detecting that the rod is through both rings at the same time in the bomb frame must be equivalent to reading suitably synchronized clocks.

    This is the same sort of misunderstanding as the other thread. The ring frame measures an explosive order, but also agrees that the bomb clocks measure a safe order. Different frames won't change what a particular clock reads for a particular event.

    [edit: The following may also be worth pointing out: Suppose your event order detector (circuit of some kind) is carried with the rod (or at rest relative to it). You can posit it detonates the bomb by radio control if it detect the problem conditions. Now, analysis exactly like the above will show the bomb does go off, whichever frame you analyze it in. The key physical point is which frame's simultaneity is being detected. It can't be both, because they are different.]
     
    Last edited: Sep 28, 2012
  8. Sep 28, 2012 #7

    bgq

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    Re: Lenght Contraction Issue

    OK, thank you PAllen. But Let me ask you another question related to this situation:

    With respect to the rod frame of reference, if the proper distance between the rings is smaller than the proper length of the rod, what will be different in the previous analysis?
     
  9. Sep 28, 2012 #8

    PAllen

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    Re: Lenght Contraction Issue

    I'm confused. I thought we were assuming that. Perhaps a terminology issue?

    - proper length of an object is length as measured in its rest frame.
    - proper distance is more general, but assumed, from your picture, that we had distance between rings in bomb frame less than proper length of rod. That is, that the bomb would go off if the rod was moving slowly.

    So nothing would change, because that is what I assumed (plus, that the relative speed was such that contracted length of the rod was smaller than ring distance in the bomb frame).
     
  10. Sep 28, 2012 #9
    Re: Lenght Contraction Issue

    bgq, the reason this problem is confusing you is that you haven't specified exactly how the rod-detector works, and under exactly what circumstances it causes the bomb to go off. Specifying a mechanism for this, and analyzing it in both frames, should clear your confusion by letting you examine exactly what happens in both frames.

    Let's say the bomb operates by the following simple mechanism: Whenever a ring is in contact with a rod, it is sending a signal toward some electronics at the center of the bomb saying, "I am currently in contact with the rod!" Let's say this signal propagates at the speed of light. If the electronic controller at the center of the bomb detect signals from both rings at once, the bomb explodes.

    What happens in the bomb frame? First the rod comes into contact with the right-hand, ring. While the rod is in contact with this ring it is sending signals to the central controller. Then there's a brief interval when no ring is touching the rod, and no signals are being sent. Then the left-hand ring contacts the rod for a while, and sends signals to the central controller. What does the central controller see? It sees what I just described, but delayed by the time the signals take to travel from the rings to the electronics. The central controller never sees signals from both rings at once, so the bomb doesn't explode.

    Now, what happens in the rod's frame? At first the rod is touching the right-hand ring, and the right-hand ring is sending signals to the central controller. Then there's an interval of time when the rod is touching both rings, and both rings are sending signals to the central controller. Then there's a time when the rod is touching only the left-hand ring, and only the left-hand ring is sending signals to the central controller.

    What does the central controller see, then? Does it ever see signals from both rings at once? No--even though there was a time when signals were being simultaneously from both rings, those signals do not arrive at the central controller at the same time. This is because the central controller is moving rapidly to the right, fleeing any signals emitted from the left-hand ring. This causes it to see the right-hand signals earlier than it otherwise would, and the left-hand signals later than it otherwise would, because the left-hand signals have to go farther before they catch up with the fleeing controller. The controller ends up seeing all the signals from the right-hand ring before any signals from the left-hand ring start to arrive. So the central controller never sees simultaneous signals from both rings, and never explodes the bomb.

    The deep lesson here is that you can't really talk about whether two events happen "simultaneously" if they happen at different points in space. You have to specify a frame of reference before you can talk about simultaneity. If you construct a mechanism that decides whether two events happen simultaneously, what it will actually do is decide whether the events happened simultaneously *in a certain frame* (the rest frame of the mechanism). In other frames, events that your mechanism decides are "simultaneous" will not look simultaneous.
     
    Last edited: Sep 28, 2012
  11. Sep 29, 2012 #10

    bgq

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    Re: Lenght Contraction Issue

    That's really great! It seems very consistent, and I am satisfied by this, thank you very much.
     
  12. Oct 1, 2012 #11
    Re: Lenght Contraction Issue

    Excuse my ignorance, but doesn't that imply that the flow of current must also be according to the synchronized clocks? assuming the rod completes a circuit.
     
  13. Oct 1, 2012 #12
    Re: Lenght Contraction Issue

    Excellent illustration.
    Likewise you could , given the same setup of rings and rod , adjust the detection parameters slightly to create a situation where the bomb DOES go off in both frames even though in the ring frame there is still NEVER a moment of of actual connection between the rings by the rod.
     
  14. Oct 1, 2012 #13

    PAllen

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    Re: Lenght Contraction Issue

    Yes, current carries a signal, and closure of a circuit at two points measures simultaneity. A circuit being briefly closed (in this scenario) is exactly equivalent to clocks at rest relative to the circuit showing simultaneous connection at the closure points. If the circuit is attached to the rod, the bomb will explode. If the circuit is attached to the bomb, it won't explode. In either case, the same will be true analyzed in any frame.
     
  15. Oct 1, 2012 #14

    zonde

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    Re: Lenght Contraction Issue

    As rod is touching rings very briefly you can't really talk about flow of current but rather about short electrical pulse. And then we have to find out in what direction this pulse is supposed to travel. If it's from first ring to second then the rod might carry this pulse along as it travels to second ring (bomb explodes). If it's in other direction certainly nothing happens. Nothing much to do with SR actually.
     
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