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Length contraction

  1. Jan 23, 2007 #1

    disregardthat

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    I have some questions about the relative factor:

    1/'root'1-v^2/c^2

    If an object moves close to the speed of light, the length contraction becomes significant.

    I hear this "paradox": (that was said not to be a paradox, but I didn't understand it) A train of 100 meter moves at a speed that contract it to 30 meters. It moves into a tunnel that is 40 meters long, and with open doors at both ends.

    Let's say the train moves into this tunnel, and it is 30 meters observed by the tunnel. When the train is well inside the tunnel, the doors close instantly, and at the same time. The train is trapped here contradicted.

    This was only in the tunnels point of view, in the train's point of view the front door is locked when the front of the train is in the tunnel, and when the train smashed through the front door, and the end of the train is well inside, the back door closes.

    That I understand, but there was something this podcast wouldn't explain, (only give the very question)

    What if the train is well inside by the tunnels point of view, and both doors slam down. The train is INSIDE the tunnel. The front door makes the train stop instantly, and if it was unbreakable it would smash the back door to pieces with it's end. But how can this be? From the trains point of view, the end was never really inside the tunnel, it was only the front of the train that was inside. But in the tunnels point of view the whole train is inside.

    The smashing through the front door never occurs, so how can the train smash the back door to pieces when it doesn't fit into the tunnel?

    I am sure I got it misconceptet, and I would really like to here what the correct version is!
     
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  3. Jan 23, 2007 #2

    Fredrik

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    You're assuming that all the different parts of the train will stop at the same time in the tunnel's frame, and that the length of the train will be 100 m once it has stopped. This is a contradiction. If you somehow make all the different parts of the train stop at the same time in the tunnel's frame, then the train will be 30 meters once it has stopped. You will have squeezed the train to a much shorter proper length than it had from the beginning.
     
  4. Jan 23, 2007 #3

    disregardthat

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    But in the tunnels point of view both doors are closed and the train is inside. This is true for the tunnel observer.

    The train stops as it would if it crashed into the door.

    From the train, the tunnel is contracted, meaning only the edge of the front ever reaches the tunnel. Let's say none of the doors can be smashed, neither can the train. In the tunnels point of view this undestructable train will be 30 meters fitting inside the tunnel. (before it smashes through the door)

    From the trains point of view the tunnel have contracted to about 10 meters, only 10% of the train will be inside the tunnel, it will hit the front door and stop, but when does the second door close?

    The tunnel sees the door closes, and the the train gains it's total size again, and smashes through this (ok not entirely indestructible) door.

    Now when the train has stopped the tunnel sees the train stand outside the tunnel, with the pieces lying all over the area. In the trains point of view this door must have closed on top of the train, making it unclosable...

    What do you get from this? Do anyone know?

    My question is how can the train smash the back door into pieces when it never really was inside the tunnel.
     
  5. Jan 23, 2007 #4

    Fredrik

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    Note that you're describing two events here: A) the closing of the front door, and B) the closing of the rear door. In the tunnel's frame these events are simultaneous, but in the moving frame A happens before B. So you're not describing one instant in time in the moving frame. You're describing a time interval. It's important that you understand that. What you're saying here implies that at some time between the time of event A and the time of event B (in the train's frame), the rear of the train will have entered the tunnel.

    I don't know exactly what the acceleration of the different parts of the train would be if it crashes into the door, or what the length of the train will be once it has stopped. It would take a pretty complicated calculation to find out. It is clear however, that the different parts of the train will not stop at the same time in either frame. The "message" that the front of the train has stopped will propagate through the train at a finite speed, causing the other parts to slow down and eventually stop.

    For now, just keep in mind that this assumption makes your question very complicated (unnecessarily complicated).

    If you are describing what things look like at the time of event A, you should have said that the rear of the train hasn't entered the tunnel yet. If you really mean that the rear never enters the tunnel, you're contradicting your first assumption. You have already said that there the rear will enter the tunnel shortly before event B, when/where the rear door is closed.

    The second part of this assumption is impossible. There are no rigid bodies in SR, so the train must "break" in some way.

    Imagine an object which is instantly accelerated to a high velocity. Suppose that all the different parts started moving at the same time in the "stationary" frame. In that case the length of the object will remain the same in that frame, and that means that it has been physically stretched by a factor that exactly cancels the Lorentz contraction. Suppose instead that the different parts started moving at the same time in the "moving" frame. Then the length of the object will be unchanged in that frame, so it must have been squeezed by a factor that cancels the Lorentz expansion.

    As you can see, it's not possible to change the velocity of any object without squeezing or stretching it.

    Only 10% will be inside the tunnel at the time of event A. But when event B occurs the rear will already have entered the tunnel. You assumed this from the start (by saying that from the tunnel's point of view, there will be a moment when the whole train is inside and both doors are being closed), so if you're making a different assumption now, you're contradicting yourself.

    And again, it seems that you're assuming that the different parts of the train will stop at the same time in the tunnel's frame. They won't. And as I said, if they would, this would squeeze the train so that its proper length will be 30 meters.

    "The tunnel sees..." OK, so you're talking about what happens in the tunnel's frame. You're assuming that the train will expand to a length of 100 meters. This contradicts the previous assumptions about how the train will stop. One way to make it expand to 100 meters is to have the different parts of the train stop simultaneously in the moving frame. But then the rear would stop moving before the front in the tunnel's frame. This contradicts your initial assumption about events A and B.

    If the different parts of the train stop simultaneously in the tunnel's frame, this will not happen. The whole train will be inside the tunnel when it stops, and its length will still be 30 m (because the train will have been squeezed by a factor that compensates for the Lorentz expansion).

    If the different parts of the train stop simultaneously in the moving frame, the rear will stop first in the tunnel's frame. It will "wait" outside the tunnel as the front continues to race towards the front door, and the rear door will indeed be "unclosable".

    What I wrote above answers that question. Also note that there's no way to stop the train so that the rear "expands" (in the tunnel's frame) in the direction opposite to the motion of the train.
     
  6. Jan 23, 2007 #5

    JesseM

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    There can be no perfectly rigid objects in relativity, because it would give you the ability to transmit information faster than light (for example, I could send a message in morse code to Alpha Centauri by wiggling one end of a perfectly rigid rod 4 light years long). According to relativity, the train hitting a wall will behave more like a spring, where the back end won't react to the front end hitting the wall until a wave of compression has traveled through it (the speed of this wave would depend on the speed of sound in the material the train is made of). Of course, you could replace the train with a string of rockets, and have them all preprogrammed to decelerate "simultaneously" at the moment the front rocket reaches the wall, but you have to decide which frame's definition of simultaneity to use; if you use the tunnel's frame, then when they all decelerate simultaneously in this frame, the distance between the front rocket and the back one will remain the same as it was when they were moving, meaning that the distance in each rocket's current rest frame is shorter than it was in their rest frame before decelerating.
     
  7. Jan 23, 2007 #6

    disregardthat

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    Hmm, thanks for your posts! :)

    This contraction is that the object are unlogically taking less space than before, and not just atoms sitting closer together (like normal compression) ?

    I was writing half a page here, but I suddenly realised something:

    If we were to look at the train in the tunnels frame, would the end of the train stop if the door was unbreakable?

    Railroad: ________
    Tunnel: U_____U
    Train: -------

    Like this is the railroadand tunnel:
    _______________________U_____U
    Here comes the train (tunnel frame):
    ____-------___________U_____U
    a fraction of a second later:
    ________-------_______U_____U

    ____________-------___U_____U

    ________________-----U-____U

    _________________-----U---__U
    It hits the front door:
    _________________------U-----U
    It is still in the tunnels frame:
    _________________------U-----U

    Will it be like this from the tunnels frame? Like he sees the back of the train stopping, but the front going on to the crash. (let's say it decelerate from little size to original size at one fraction of a second I have used here. Here the train is at it's original size, and it never went through the back door because i was "going" to hit the front door.

    I understand that it would be like this if the train passes through:

    _______________________U_____U____

    _____------____________U_____U____

    _________------________U_____U____

    _____________------____U_____U____

    __________________-----U_____U____

    _____________________--U---__U____
    It passes the front door:
    _______________________U----_U-___

    _______________________U__---U---__
    Later on:
    _______________________U_____U_____-----___

    Is this way of looking at it correct, if not, what is not correct here?

    Just because I don't have so much to do right now, I will add the trains point of view(unbreakable door), and tell me if it is correct:

    The train is coming at it's high velocity, observing the tunnel contracted:

    ________________------------------____U___U

    __ _________________------------------_U___U

    _ _____________________----------------U-___U
    Not yet smashed into:
    __________________________-----------U------_U

    __________________________---------U---------U

    This shows how the tunnel "expands" as the train hits the door. When it has hit it, and the tunnel gets to normal size and the door is unlockable.
    The time the tunnel use to "expand" is the time the door takes to decelerate the train to 0.

    Is this correct? Tell me where I'm wrong, because I would like to explain this to my friends.
     
    Last edited: Jan 23, 2007
  8. Jan 24, 2007 #7

    Fredrik

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    If the different parts of the train are forced to stop at the same time, then the train gets squeezed to a shorter length, just like you could squeeze a banana to a shorter length using your hands. I guess this is what you call "normal" compression. Lorentz contraction is something completely different, it's a property of space-time rather than something that "happens" to the object.

    The front of the train would stop at the front door. This would send a shock wave through the train causing the other parts of the train to change their velocity. If the train really is "unbreakable", the shock wave would travel at the speed of sound in the material that the train is made of. If there had existed a material that doesn't break at this velocity, the speed of sound in that material would be close to the speed of light in vacuum. So if the train is made of such a material (ignoring the fact that no such material exists), the shock wave would propagate through the train at a ridiculously high speed, causing every particle it hits to bounce back in the opposite direction. The train would then expand backwards and crash through the rear door. But this is not just Lorentz expansion. The train is like a spring that gets compressed when it hits the front door, and once it has been compressed to a certain length (which is less than 30 m), it will expand to its full length again.


    This sequence of "pictures" shows roughly what would happen if the different parts of the train somehow are forced to stop at the same time in the moving frame. (When I talk about a "moving" frame, I'm imagining an observer moving at the velocity the train has before it begins to slow down). This is something very different from what would happen if an unbreakable train runs into an unbreakable door. So it's definitely wrong to say that the rear stops because the front is going to hit the front door. If the rear stops while the front keeps moving forward, it's because some kind of machine is making the different parts of the train stop at very specific times.

    This sequence of pictures is very confusing. You have messed up many of the details. Why would the train shrink from 6 to 5 characters? Is the thickness of the doors 0 or 1 characters? Is the velocity 3 or 4 characters per picture?

    If this is supposed to be the tunnel's point of view when the train passes through without slowing down, then you don't have to draw pictures to explain it. It's easy to imagine a 30 m train going ridicilously fast through a 40 m tunnel. The doors are closing at the same time, at a time when the whole train is inside the tunnel. A moment later the train crashes through the front door.

    The same events from the train's point of view would look different: A 100 m train is going ridicilously fast through a 10 m tunnel. The front door closes, but the train crashes through it. A moment later, when the rear of the train is inside the tunnel (and the front is more than 90 m past the tunnel), the rear door closes.


    So this is the train's point of view when the different parts of the train are forced to slow down simultaneously (but not instantaneously) in the train's frame? This is called Born rigid acceleration, and is something we haven't discussed yet. It would resemble what you tried to "draw" here, so this is essentially correct. Just keep in mind that this is not at all what happens if the only reason that the train stops is that it runs into an unbreakable door. Your "pictures" show roughly what would happen if the train uses its breaks to slow down and stop.
     
  9. Jan 24, 2007 #8

    Ich

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    Why don't you read the FAQ?
     
  10. Jan 24, 2007 #9

    disregardthat

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    Thanks for your post :)

    You said: "The train would then expand backwards and crash through the rear door. " How can this be, if in the trains frame the back never enters the back door?

    I really need to know if lorentz contraction is equal to every type of matter. It doesn't matter if it's metal or a ping pong ball, right?

    All this about the unbreakable thing makes be confused. Let's say the train gets compressed after hitting the door.
    I made some drawings in paint, and I really hope of you would specify what is incorrect here. I have written it as a thought experiment with description and conclusion. (I am not trying to tell you that the way I did it is correct! I just want to know if it is incorrect, and what that is incorrect)
    (I have too much spare time at hand :rolleyes: )

    EDIT: I read the faq, but my point is that what will happen if the compressed rod/train does not fit into the barn, but the contracted rod\train do? If it did fit in, the rod\train would hit the front door, and the force this creates will compress the rod to fit inside the door, and it will be caught in the barn, and smash out if it tried to de-compress itself.

    If the rod didn't compress so much that it would not fit into the barn (when stationary) the barn door can't close because the rod is in the way. I wonder how this happen, and not why, you can see my examples I made in the thought experiment.
     

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    Last edited: Jan 24, 2007
  11. Jan 24, 2007 #10

    Doc Al

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    Both train and tunnel observers agree that the back of the train ends up inside the tunnel. What they disagree on is where the front of the train is when this happens. According to the tunnel observers, the back of the train is well within the tunnel before the front of the train crashes into the door. But the train observers disagree--they find that the rear of the train enters the tunnel after the front has already smashed into the door.

    That's right--it doesn't matter.

    You assume that such a situation is possible, but it's not.
     
  12. Jan 24, 2007 #11

    Fredrik

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    Right. Lorentz contraction is a property of space-time, not a property of matter, so it doesn't matter what kind of matter it is.

    I will look at the picture tomorrow. Right now I need to do something else.
     
  13. Jan 24, 2007 #12

    robphy

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    That's not quite correct. It's incomplete.
    Lorentz contraction is a property that arises of an observer's decomposition of spacetime into his "space" and his "time". It depends on the choice of observer... and is not a property of spacetime alone.
     
  14. Jan 24, 2007 #13

    disregardthat

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    Thanks, Doc al, that helped.
    If you would look at the drawing, you will see I have taken that in account, now no one sees the end of the train inside the tunnel
    And thanks for that you will look at it, I put in quite some time in it :P

    EDIT: Having very little to do at the moment, I thought it would be fun to create a little animation of the train moving and smashing the door outwards. (This model was not true though as I have been explained, just for fun)
    I will create the two hopefully correct models tomorrow. :)
     

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    Last edited: Jan 24, 2007
  15. Jan 24, 2007 #14

    JesseM

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    As Doc Al says, both frames agree the train enters the back door--two objects passing right next to each other is a physical event which all frames must agree on. (Imagine if a localized explosion went off at the back of the tunnel as the back end of the train was passing through it in the tunnel frame--would different frames disagree about whether a person standing at the back end of the train was killed? This would make different frames into alternate universes!)

    Again, what you have to remember is that there can be no rigid objects in relativity, the back end of the train won't instantaneously react just because the front end hits the wall, it will keep on moving forward at the same speed until a wave of compression has travelled from the front end to the back end (you might imagine replacing the train with an accordian or a spring if it helps), by which time the back end has already passed through the entrance to the tunnel.
     
    Last edited: Jan 24, 2007
  16. Jan 25, 2007 #15

    Fredrik

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    Minkowski space has a non-trivial group of isometries (the Poincaré group). The existence of that group is a property of Minkowski space. Each isometry corresponds to an inertial observer, and because of that, I would say that all that stuff about observers is a property of space-time too.
     
  17. Jan 25, 2007 #16

    Fredrik

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    It can't of course. If the rear of the train never enters the tunnel in one frame, it never enters the tunnel in any frame. The existence of an event is frame independent. You seem to be confusing two different sets of events (two different ways to force the train to stop).

    The images on the left show roughly what would happen if the train crashes into the front door, seen from the tunnel's point of view.

    The images in the middle show a very different set of events, also from the tunnel's point of view. It is roughly what would happen if the different parts of the train were forced to stop at the same time in a moving observer's frame. A crash into a door could not possibly make that happen, since the door only affects the front of the train directly. (So the sentence "the train is stopped only by the door" is definitely wrong). But some kind of machine designed to stop the train in that particular way could. Note that this would make the train's length remain 10 times the length of the tunnel in the moving observer's frame, so the train is being stretched like an accordian.

    The images on the left and the images in the middle both describe possible sets of events. The difference is in how the train is forced to stop.

    I don't know what to make of the images on the right. It seems that you're confusing the two (different) sets of events that were depicted on the left and in the middle, or maybe describing a third possibility (the train using it's breaks to slow down). You may also be confusing two different frames. (You say "the train's point of view". Does that mean you're using an accelerating frame?)

    If I understand the question correctly, the rod/train will either be trapped in a compressed state, or smash one or both of the doors (because of "spring force" expansion, not because of Lorentz expansion), depending on what breaks first.

    Edit: I clearly didn't understand the question. What I said describes what happens when all the different parts of the train are stopped simultaneously in the tunnel's frame. The train will then be compressed (like a spring) by exactly the same factor as it was Lorentz contracted when it was moving.

    You were asking what happens if the different parts of the train slow down in a different way, one we haven't discussed yet. It is possible (in principle) to slow down and stop the different parts of the train in a way that compresses the train by a smaller amount than it was Lorentz contracted when it was moving, so that the rear never enters the tunnel. The rear door would fail to close, and then the train would expand backwards (because of the "spring force", not because of Lorentz expansion).

    Is there a negation too many in the first sentence? I assume you mean, "if the rod/train doesn't compress enough to fit in the barn/tunnel". Then the rear of the train would never enter the tunnel, in any frame. (It's actually redundant to mention that this is "in any frame", since the existence of an event is frame independent).
     
    Last edited: Jan 25, 2007
  18. Jan 25, 2007 #17

    disregardthat

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    Thanks for a long and informative post :)

    My pictures were merely suggestions of what I might think it would work like, and from what I got from your post was:

    'The reason the door gets smashed out is not because the object is going slower so the lorentz contraction cancels, but because the object itself works like a spring to smahs the door out.'

    Well, this is all very confusing, and I wonder how excactly the train would look like picture by picture, in reality.

    If the train really didn't fit into the tunnel when compressed, then how excactly would it work out, picture by picture? In the tunnels frame the end of the train can never enter the tunnel. As you say, if the train is only stopped by a door, the Situation 2 is not correct.

    You say that there is no rigid object in relativity? Does this mean that when the train stops, the atoms are too close to eachother, and will press themself from eachother to the right position, and by that working like a ordinary spring? Or is the atoms really just being compressed like any normal football would when it hits the ground, because of the force of the door.

    Do you have a link to an easy understanding of the relation between molecular compression and the lorentz compression (contraction).

    Ok, if my questions was confusing I can try to define the really question here:
    Is the force that smashes the door out, the atoms tension between eachother, so they try to stabilize into the correct position, and by that expanding the train after it has stopped, (the atoms in the back of the train is having a velocity backwards towards the back door of the tunnel, and smashing)

    or

    The reason because the door smashes is: The train in same frame of referance as the tunnel, makes it expand, (contraction cancels) and by that the train smashes the door out.

    Hmmm, I can't seem to make this clear: What direction would the lorentz contraction cancel? If an object is moving faster and faster, would the object squeeze it's sides towards the center, would the front squeeze towards the back of the object, or would the back squeeze towards the front? Front = the front of the object in the way it is moving

    And by my first picture series, is it the atoms pushing on eachother making the train expand I am drawing, or is it the lorentz contraction canceling I am drawing... It's hard to sort out any difference between these as we see it. Isn't there any equation for this?
     
    Last edited: Jan 25, 2007
  19. Jan 25, 2007 #18

    Fredrik

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    Yes, that's correct. (What I'm about to say is from the tunnel's point of view). If the train stops because it runs into the front door, the rear will already have entered the tunnel when this happens. So a very short time after the train has crashed into the door, it won't be Lorentz contracted anymore, but it's still shorter than it was when it was Lorentz contracted. This means that it has been compressed like a spring, and that means that we can expect it to expand like a spring.

    I hope you understand now that the following are completely different sets of events:

    a) The train runs into the door, which causes it to stop.
    b) The train uses its breaks to slow down and stop.
    c) The different parts of the train are forced to stop simultaneously in the moving observer's frame.
    d) The different parts of the train are forced to stop simultaneously in the tunnel's frame.
    e) The different parts of the train are forced to decelerate in a way that compresses the train by a factor that's less than the Lorentz contraction factor associated with the train's initial speed, so the train won't fit in the tunnel once it has stopped.

    You would have to draw a different set of pictures for each one of these cases that you find interesting. Option e) is a lot less interesting than the others. Options a) and b) are somewhat realistic, and c) and d) at least have a pedagogical value, but e) adds nothing new. (The train won't fit in the tunnel in b) either).

    No, that's not what it means. The material will behave like an ordinary spring, but that doesn't really have anything to do with relativity.

    If you want to understand what "no rigid object" means, go back to post #4 in this thread and try to understand the section that starts with "Imagine an object which is instantly accelerated to a high velocity". You should also read #5 (JesseM's post).

    I'm confused. Aren't you just saying the same thing in a different way? I don't see why you use the word "or". The answer is "yes". The train will behave like a football.

    I don't. I also don't think it's meaningful to say that there's a relation between the two. We can surmise that there's been a forceful compression of the train in scenario a) because we know that the rear of the train is inside the tunnel when the front hits the door.

    The first alternative. The second is wrong, because an object slowing down (and therefore being Lorentz expanded) will never expand backwards. Consider scenario b). From the tunnel's point of view, the rear of the train will simply begin to slow down before the front, but it will keep moving forward until it has come to a complete stop. When this happens, the front is still moving, but it will of course also come to a complete stop very soon.

    There isn't a single answer to this question. It depends on when and how you accelerate the different parts of the object. This is what I talked about in #4. I suggest you try to figure out what happens in scenarios c) and d).

    Edit: I think I read your question too fast and misunderstood it. I thought you were asking for when there will be a "squeeze" accompanying a boost and when there will be a "stretch". But you're just asking about what a Lorentz contraction looks like while it's happening. What I said is the correct answer to that question too, but if I had understood exactly what you were asking I would also had said that in the typical scenario (when the acceleration is small, so that shock waves propagating through the material are negligible compared to the overall motion of the object) the rear is starting to accelerate a bit earlier than the front, so that it has a higher velocity than the front. This will make the object shorter. The front will be the last part of the object to reach the final velocity.

    When the train reaches its shortest length (in the tunnel's frame), the "Lorentz contraction cancellation" has already taken place. That's how we know that the train must have been forcefully compressed.
     
    Last edited: Jan 25, 2007
  20. Jan 26, 2007 #19

    disregardthat

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    Ah, thanks! I think I understand it now, I think I have thought of this lorentz contraction as a big deal here, but it is actually just observer dependent, and the object have no different properties other than the "slow" reaction between the particles in the train moving is respect to eachother. (because they cannot intereact one by one faster than light.

    I have read that for a object in high (any) velocity the planckmeter shrinks compared to the "stationary" frame. So I thought that it was not
    the atoms electromagnetic field being slow to interact, but the object was taking less space in the stationary "dimension".

    So lorentz contraction isn't in that way equal to time dialation, where planck seconds are different from the "stationary" frame. Is this correct? I have always thought of this to have a close relation.

    Actually, I did not mean the relation between the two phenomenas, but rather the difference between these two in the train crashing into door situation.

    Ah, thanks for that, it pretty much is the core of much the problem. How excactly would the lorentz contraction cancel? if the train was smashing into the door, and not slowing down by braking. would the back and front stop at simultaneously, and THEN compress physically because of the force in a tunnel frame? (And then eventualy being "shot" back again, because of the molecular structure restabilizing (trian situation))

    (quote] The front will be the last part of the object to reach the final velocity.[/quote]
    But the front will slow down first at the crash, of course. (?)
     
  21. Jan 26, 2007 #20

    disregardthat

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    Just to add the other consequences of relativistic speeds, the time the train uses will be slower for the train, but faster for the tunnel, right?

    But the relativistic mass? Does it have any effect on the impact, will the damage on the train increase as the relativistic mass increase? I understand that it is not the rest-mass, or invariant mass that is increasing. But if the damage does not increase, what effect does the relalativistic mass have other than making the object harder to accelerate(more Newtons in "stationary frame") (if any)
     
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