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Need help resolving this paradox with time dilation

  1. Jul 26, 2011 #1
    Before I post this, I want to assure everyone that I am not a conspiracy theorist who doubts Einstein or relativity theory in any aspect. I just don't understand special relativity as well as Id like to and I think having this paradox resolved in my mind will help.

    The paradoxes that Ive seen before have usually been resolved with either "One inertial frame undergoes acceleration while the other doesn't" or "events that appear simultaneous aren't actually simultaneous". I cant seem to get either of those to apply here though.

    Ok so heres the setup. A train is moving at 0.6c on a track relative to the ground. Hypothetically, an infinite number of synchronized clocks at rest on the ground line the entire length of the track. As the train passes a point A on the track, a clock at rest with the ground reads that the front of the train crosses at 4:55PM. As the train passes point A, a switch is tripped inside the train which causes a bomb in the train to activate with a timer of 5 minutes.

    Each one of the infinite clocks lining the track is equipped with a blade that will deploy off the side of the track when the clock reaches 5:00PM. This blade will slice a wire off the side of the train which will deactivate the bomb.

    Heres where the paradox arises (at least according to my understanding). From the perspective of the man on the ground, the train is moving at 0.6c and therefore the bomb timer will appear to be moving slower due to time dilation. In this case, when the clocks on the ground reach 5:00PM the slower train bomb clock will read that it still has 1 minute left since its moving at 80% speed. Therefore the blades will deploy in time and the bomb will be deactivated.

    Now from the perspective of the guy on the train. He understands that at 5:00PM a blade will cut his wire, but he observes the clocks on the ground as moving slower than the timer on the bomb. Therefore from his perspective, the bomb will tick all the way down to 0 and the clocks outside will still only read 4:59PM. Therefore the bomb will go off before any blades have a chance to cut the wire.

    In summary:

    A guy on the ground sees his clock as moving at regular speed and the bomb timer on the train moving slowly. 5 minutes of rest clock time results in only 4 minutes of bomb timer time. In this scenario the wire is cut in time and the bomb DOES NOT go off.

    A guy in the train sees the bomb timer as running at regular speed and the clock on the ground moving slowly. 5 minutes of bomb timer time results in only 4 minutes of rest clock time. In this scenario the wire is not cut in time and the bomb DOES go off.

    Clearly there is a conflict here and I cant seem to resolve it. What am I missing here?
     
  2. jcsd
  3. Jul 27, 2011 #2

    ghwellsjr

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    I don't see any paradox here at all--just a lack of understanding of what the guy on the train will see in this scenario.

    You have described a situation where there are a whole bunch of previously synchronized clocks along the track and on the train moving at 0.6c is a 5-minute timer that starts when the clocks on the ground read 4:55PM and the question is, how much time will be left on the timer when the clocks on the ground read 5:00PM? You gave the correct answer as 1 minute.

    But then you go on to make some erroneous statements regarding what the guy on the train will see. He does see each of the clocks on the ground ticking at 80% of the rate of his clock/timer but he does not see them as synchronized. Instead, if he is always looking at the clock that is closest to him, he will see the times as advancing faster than his clock/timer so that when his timer has 1 minute left on it, the closest clock on the ground will read 5:00PM. However, the clock that read 4:55PM that was closest to him when his timer started is now far away from him and will have a time earlier than 5:00PM on it, but it has no more relevance to your scenario.

    Maybe it would help to realize that he sees the clocks (and everything else on the ground) as contracted together to 80% of what it should be.

    Ask yourself the question: how far down the track in the ground frame, will the train have progressed in 5 minutes? Well, that's easy, it's 5 minutes times 0.6c (which is 0.6 light-minutes per minute) or 3 light-minutes.

    Now ask yourself the question: how does this distance appear to the guy on the train? Well, since he sees the ground as being length contracted to 80%, he will see this distance as 2.4 light-minutes.

    And finally, ask yourself the question: how long will it take the train to travel 2.4 light-seconds at 0.6 c according to the guy on the train? The answer is 2.4/0.6 = 4 minutes.

    So the train analyzes the situation exactly the same as the ground does and there is no paradox.
     
    Last edited: Jul 27, 2011
  4. Jul 27, 2011 #3
    EricD, I've sketched a space-time diagram for your thought experiment. If you have a hard time following the diagram you might google on the topic to see how to interpret it. But it's basically a picture of the 4-dimensional space picture of the observers and their respective cross-section views of the 4-D universe (X2 and X3 are supressed for simplicity). Here, instead of showing the ground observer at rest, I've chosen to establish a black rest system with a red guy moving to the left with respect to black and a blue guy moving to the right with the same speed. I've assumed blue and red clocks are synchronized at the origin of the coordinate systems. This symmetric diagram allows you to compare distances and times directly for blue and red (Otherwise you have to be bothered with hyperbolic curves, etc.).

    Train_Explosion.jpg
     
    Last edited: Jul 27, 2011
  5. Jul 27, 2011 #4
    This diagram seems to be looking at the time dimension of Minkowski spacetime as if it were a dimension of space, right? like in the Euclidean version (with positive-definite signature) of SR that uses imaginary time (Wick rotation of the Minkowski diagrams).
    What confuses me is that you put additional "rest coordinates" in black, so it would seem this diagram needs more than four dimensions?
     
  6. Jul 27, 2011 #5
    Yes. You can interpret the diagram as 4 spatial dimensions. Any observer's X4 coordinate may be scaled as time just by dividing by c. It's just like you could calibrate an interstate highway for time for people driving at 60 mph. you could put up lapsed time values on signs along the way to keep track of how much time you've driven along the highway going from point A to point B.

    I have not changed the signature of the metric.

    Actually the picture represents just 4 dimensions (with X2 and X3 suppressed). You can draw in as many different coordinate systems as you wish, a different set of coordinates for as many different observers moving at different velocities as you wish. That does not change the number of dimensions in the space.
     
  7. Jul 27, 2011 #6
    Hi TrickyDicky, Here is a short derivation of the metric with the usual signature. Also, I've shown the artificial introduction of the ict to get the ++++ you mentioned. But notice that before introducing the ict I have the spatial relations without any mention of time, and it has the normal metric signature (we can make it either -+++ or +---, depending on whether you wish to multiply both sides by -1, whichever you prefer).

    Pythorean_Lorentz.jpg
     
    Last edited: Jul 27, 2011
  8. Jul 27, 2011 #7
    Hmm..., I think if you use 4 instead of 3 spatial dimensions in SR you need to change the signature since the time dimension becomes positive when using imaginary time as a space dimension.

    Ah, I would have thought (taking into account dimensions X2,X3 are suppressed), every coordinate axis represents a dimension. So you have the black coordinate axes, plus X1, and X4 axes(represented in blue), plus the 2 suppressed ones.

    Edit: while responding you posted so this post doesn't take into account your last post.
     
  9. Jul 27, 2011 #8
    Ok, I think I understand now, if I find something else I don't get , I'll ask again.
    Thanks
     
  10. Aug 20, 2011 #9
    Hey guys, I know that its been a while since I responded to my original post so thanks for the responses.

    Following off of what ghwellsjr said, it makes sense to me that if the clocks werent simultaneous in the reference frame of the train that the clock would in fact read 5:00pm when the train crosses even though the clock is seen to run slower the entire time. It makes sense to me assuming that at any point for the train, if the clock next to the train says 4:57pm then all the clocks behind the train will show less than 4:57pm (decending as you get farther from the train) and the clocks ahead of the train will show after 4:57pm (ascending as you get farther from the train).

    The problem that I have with this picture now is this. Lets say the train was starting at rest. In this case, the person on the ground sees all clocks syncronized at 4:55pm and the train conductor sees all clocks syncronized at 4:55pm as well.

    Now lets say that the train in an infinitesimally small time increment accelerates up to its cruising speed of 0.6c. Now as we mentioned earlier it will eventually cross a clock that says 5:00pm (lets call it clock5). When the train is standing still, clock5 reads 4:55 just like every other clock. When the train jumps to its new speed, clock5 must now be at 4:56.8pm. This is needed to satisfy the condition that the train will cross clock5 when it says 5:00pm and the clock is moving at 80% speed the entire 4 minutes of train time the train takes to reach clock5. (5:00pm - (4 minutes of train time * 80% speed) = 4:56.8).

    This now makes period while the train is moving make sense, however I now have trouble with the train while it is acceleration. In this scenario, while the train is at rest clock5 is at 4:55pm, but if it accelerates in 1/100th of a second to 0.6c, the clock after 1/100th of a second has now magically jumped all the way to 4:56.8pm at blistering speed, well past the speed of light.

    If anyone was able to follow all that, can you explain what I'm missing? Thanks in advance.
     
  11. Aug 20, 2011 #10

    ghwellsjr

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    There are two ways to look at a problem like this. You can analyze according to a Frame of Reference or you can analyze it from the viewpoint of an observer and what he actually sees. But what you can't do is start with an observer in one Frame of Reference and then after he accelerates look at him in a different Frame of Reference in which he is now at rest.

    Here's the problem. When we talk about clocks being synchronized in a FoR, such as the ground frame, we are not talking about what an observer at rest in that FoR can actually see. So the train conductor will not see clock 5 reading 4:55 but some earlier time. Then as he accelerates, he will see it rapidly advance during acceleration and then run at steady fast speed until it reads 5:00 when he gets to it and he has 1 minute left on his timer. I'll let you work out the details.

    You could also analyze the whole scenario in the ground rest frame (easy) or you could transform everything into the rest frame of the train while it is traveling at a constant speed (hard). But all analyses will get the same answer.
     
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