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Questions on Relativity

  1. Jul 25, 2011 #1
    Hi! I've just began studying Special Relativity, so I'm naturally having trouble understanding some topics. I just need an opinion on whether my understanding of the topic is right or not.

    I've been wondering how time dilation is connected to space contraction. For a relativistic moving frame, space contracts in the direction of motion and at the same time, light travels a greater distance and time accordingly slows down to account for the invariant value of c?

    I'm just wondering if my understanding of the topic is correct. I'd really appreciate any input. :smile:
     
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  3. Jul 25, 2011 #2

    BruceW

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    That's right. Length contraction and time dilation are given by the more general Lorentz transform equations. These equations have the property that the speed of light is the same as measured by any reference frame
     
  4. Jul 25, 2011 #3

    BruceW

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    To explain length contraction: lets say we have a ruler that we measure to be 1 meter when stationary, then if we measure its length while it is moving, it will be less than 1 meter.
    and for time dilation: If we measure a clock to tick once every minute when stationary, then if we measure it while its moving, it will tick once every minute and a half (for example).
     
  5. Jul 25, 2011 #4
    Thanks for the reply! :)

    So could you say that time dilation and length contraction go hand-in-hand for a relativistic reference frame?
     
  6. Jul 25, 2011 #5
    I also understand how the speed of light accounts for time dilation, but I don't understand how it leads to length contraction.
     
  7. Jul 25, 2011 #6

    BruceW

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    In what way have you learned about how the speed of light accounts for time dilation?
    When I first did relativity, my lecture explained to us a geometrical example, where a combination of time dilation and length contraction led to the speed of light being the same as measured by two different observers.
     
  8. Jul 25, 2011 #7
    The way I understood it was that light appears to travel a longer path for an observer in a fixed reference frame and since c has to be constant, time appears to be lengthened as well.
     
  9. Jul 25, 2011 #8

    BruceW

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    Your terminology isn't quite right, but I think you've got the right idea. At the heart of it all, the speed of light must be the same, so clocks and rulers will go slower/get shorter than if you measured them when they were stationary.
    If you only allowed time dilation, and not length contraction, then it would not be possible to have the speed of light the same relative to all observers. This is why length contraction must happen.
     
  10. Jul 25, 2011 #9
    Would it be correct to say that length contraction causes c to be constant while time dilation happens as a consequence of the invariance of c?
     
  11. Jul 25, 2011 #10
    Blueberrynerd, I don't know if the space-time sketches will just make things more confusing or not, but some people find it easier to resolve a question like this by visualizing the relationships between different observers and between observers and objects in 4 dimensions. If you think of the four dimensions as X1, X2, X3, and X4, then you can suppress X2 and X3 in the diagrams so as to focus on how things relate in the X1-X4 coordinates. Here are some basic examples.

    Looked at in this way you might think more about the consequence of a very mysterious and fundamental aspect of the relativistic 4-D space. If an observer is in motion with respect to a rest system, his X4 axis is rotated. Further, his X1 axis is rotated also such that a 4-D photon world line will always bisect the angle between the X4 and X1 axis. All observers move along their own X4 axis (their world line) at the speed of light. And since the photon world line is always at a 45 degree angle and bisects X4 and X1 for all observers (no matter their speed), then all observers will measure a ratio of 1:1 between the photon's distance traveled along X4 and distance traveled along X1. And of course the speed of light results from our convention for calibrating time along X4 (remember all observers move along X4 at light speed--although what aspect of the observer is actually doing the moving is a very protacted philosophical discussion not appropriate here).

    So, I wouldn't put the cause for the relativistic phenomena squarely on the speed of light. It's more a consequence of the strange rotating of all observer's X4 and X1 axes. You see right away in the pictures below that different observers have different 3-D cross-section views of the 4-dimensional universe. The different geometric views result in different observations of times and distances when observing other observers in relativistic motion. I'm just trying to make the point that it is fruitful to study relativity in the context of geometry: 4-dimensions and 3-D cross-section views. You can google space-time diagrams to dig into this in much more detail.

    4-D_Object_6-1.jpg
     
    Last edited: Jul 25, 2011
  12. Jul 26, 2011 #11

    ghwellsjr

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    You are showing indications of not understanding Special Relativity. Look at these three comments of yours concerning frames:
    You need to understand several things about reference frames in Special Relativity.

    First off, you should think in terms of a single, stationary reference frame which we use to define the positions of observers and objects as a function of time. Observers and objects can be moving (or stationary), but not the frame. If an observer/object is stationary with respect to the reference frame, then its clocks tick at a normal rate and its rulers are all a normal length, no matter their orientations. If an observer/object is moving with respect to the reference frame, then its clocks tick at a slower rate (with a longer time interval) and its rulers are shortened along the direction of motion.

    The speed of light is defined to be c in this single, stationary reference frame. It's fairly obvious that a stationary observer/object would be able to measure the speed of light to be c because its rulers and clocks are normal.

    However, you need to know that in order to measure the speed of light, an observer can only measure the round trip speed of light. He needs to have a timing device located at the source of a flash of light and a mirrror some fixed distance away. He starts his timer when the flash is emitted and stops it when he sees the reflected light arrive back at his location. Then to calculate the speed of light, he takes double the distance divided by the time interval.

    A moving observer will carry with him a moving light source, a moving timing device, and a moving mirror. Everything moves with respect to him so that for him, everything is stationary.

    If he places his mirror in a direction that is at right angles to his direction of motion, then it will take longer for the light to leave the source, travel to the mirror, and reflect back to him. In this case, if his mirror is the same distance away as for the stationary observer, then all it takes is for his timing device to take a longer time per tick so that his measurement will come out the same as for the stationary observer. However, the stationary observer will observe him as taking longer than his own measurement.

    If he places his mirror in a direction that is along his direction of motion, then it will also take longer for the light to leave the source, travel to the mirror, and reflect back to him, but if the distance to his mirror is the same as for the previous case, it will take even longer and he will not get the correct answer. For this reason, the distance has to be shortened by just the right amount so that he calculates c for the measured speed of light.

    Now, it should also be understood that observers do not have to diliberately move their mirror closer or adjust the tick rate of their clock longer in order for them to measure the speed of light to be c, it happens automatically.
     
  13. Jul 26, 2011 #12

    BruceW

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    If we assume the invariance of c and the principle of relativity are correct, then: length contraction and time dilation must both be allowed. (So they are both a consequence of the invariance of c).
    You could say it the other way round, and say that time dilation and length contraction can be used in special relativity to keep c the same as measured by all observers.

    Edit: To make it clear, time and space are put on equal footing in relativity. The reason the lengths contract and time dilates is simply because of the way we define the 'proper length' and 'proper time'.
     
  14. Aug 5, 2011 #13

    JDoolin

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    I'd say you are missing the vital third issue commonly known as the "relativity of simultaneity"

    Imagine that you have an open-ceiling circular room ,filled with smoke (to reveal where a flash of light is), and walled with mirrors (to reflect the flash of light), and there is a bright flash of light that emits from the center, passes through the smoke in an expanding circle, bounces off walls (simultaneously), and arrives again simultaneously at the center.

    From the point of view of someone hovering directly above the room, it appears that the light hits every part of the mirror simultaneously. However, to someone traveling past at 30% of the speed of light, it should appear as in the animation below.

    attachment.gif

    There are three main differences from the hovering viewpoint and the .3c viewpoint:
    (1) The light takes longer to make its outbound and return trip. (time dilation)
    (2) The room no longer appears to be circular but slightly oval. (length contraction)
    (3) The lignt no longer reaches all parts of the outer circle simultaneously, but instead hits the back end first. (relativity of simultaneity.)
     
  15. Aug 6, 2011 #14

    ghwellsjr

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    JDoolin, your animation is fantastic. I wish I knew how to create such animations directly on a webpage. The only way I have known to do aminations is to capture them on youtube and point to them indirectly.

    Your animation successfully illustrates the point you are making with regard to the relativity of simultaneity.

    However, you should state that the views are not those of a person residing within the scenario but are rather for us outside the scenario as we observe what happens according to a defined FoR in which we do not have to worry about how long the image of an event that happens in the scenario takes to reach our eyes outside the scenario.

    It's kind of like when we watch ripples on the surface of the water that travel at a very slow speed compared to the speed of light. If we were blind and had to rely on waiting until the water waves reflect off of objects and propagate back to us, we would then be in the same situation as a person in your scenario who won't be able to see the animation as you presented it.

    Once you understand that, you will see that you don't need smoke in the room to reveal where the flash of light is because we know where it is based on our defined FoR in which we define the speed of light to be c. In fact, if you think about all the propagations of light within a smoke filled room, you will see that it just presents a lot of confusion because all the smoke lights up with reflections continuing to go in all directions. Nobody could actually see the scenario as you presented it.

    But, like I say, take away the smoke and change the perspective from someone in the scenario to us outside the scenario who can observe instantly what is going on at each location within the scenario and you will have a great explanation to go with your great animation.
     
    Last edited: Aug 6, 2011
  16. Aug 6, 2011 #15

    JDoolin

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    Thank you very much. I made this animation about 10 years ago, I think, using gwbasic. I've long since forgotten how. There is a MUCH easier method now, if you have access to Mathematica:

    http://academic2.american.edu/~jpnolan/Misc/MathematicaAnimation.html
    Simply Export a Table of Graphics objects into a gif file.

    And you're right. In the smoke example, "What you see" is not necessarily what you see in the animation. In fact, as the object is approaching from your left, it would appear elongated, and as it recedes to the right, it would appear even more contracted.

    I suspect that if you are looking straight at it, though, as its path-of-motion crosses your line-of-sight at a 90° angle, a little small-angle-approximation could yield a proof that what you would see is at least "approximately" what is shown in the animation. The main concern is whether the speed of light delay is significantly different on the edges of the animation than in the center of the animation. If the object is passing at a distance far enough away, the difference between the path lengths is not significant, so the light from simultaneous events in that region would arrive (almost) simultaneously.


    (P.S. Maybe we should want to eliminate the smoke and mirrors anyway, since "Smoke and mirrors is a metaphor for a deceptive, fraudulent or insubstantial explanation or description" http://en.wikipedia.org/wiki/Smoke_and_mirrors :rofl:)
     
    Last edited: Aug 6, 2011
  17. Aug 6, 2011 #16

    ghwellsjr

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    Yes, I doubt that a proof exists because it's not correct. The only way you could "solve" this problem is to have a whole bunch of observers above the scenario all spread out so that none of them has to do a small-angle-approximation and they each see delayed in time what is happening below them, but then you have the exact same problem of defining the time delays between them as you would for the observers in the scenario.

    Just say that it is we as super observers outside the scenario who can "see" what is going on at any particular location inside the scenario in the "real time" of your FoR and the problem is solved.
     
  18. Aug 6, 2011 #17

    ghwellsjr

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    Get rid of the smoke but you need the mirrors. You could place partial rings of mirrors of different diameters to let the observer in the center "observe" the progress of the light, but the main point is illustrating how both a stationary observer and a moving observer in any FoR both see themselves in the center of their set of mirrors.
     
  19. Aug 6, 2011 #18

    JDoolin

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    I'm not entirely sure I agree here. Do a google image search for spotlight smoke or light beam. The apparent location of the light doesn't really stray significantly outside the original region, which is controlled by the shape of the lens of the spotlight. There may be a few rare secondary reflections that would hit your eye, but very few, and far fewer still tertiary reflections.
     
  20. Aug 6, 2011 #19

    JDoolin

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    Okay. Putting your observer "at the center" of the room would certainly ruin the "small-angle-approximation" that I was talking about before. My idea is to put both obserers (both the comoving observer, and the observer at .3c) far, far above the room.

    You're right, that if you filled the room with smoke, and had the observer inside the room, imagining what that observer "sees" is going to be kind of tricky.

    Whoops, I edited my edit already, and then realized you had already replied. Sorry about that. I took out the part where I said "I didn't produce a proof", and put in:

    The main concern is whether the speed of light delay is significantly different on the edges of the animation than in the center of the animation. If the object is passing at a distance far enough away, the difference between the path lengths is not significant, so the light from simultaneous events in that region would arrive (almost) simultaneously.​

    It's may not pass for a formal proof, but it's pretty strong reasoning.

    I don't see any problem in defining the time delays. It's proportional to the distance from the events to the observer. So IF the distance to the events is approximately the same (as it would be if the observers were far above the room), THEN the time delay is the same, and the events which were simultaneous would appear approximately simultaneous... just delayed.
     
    Last edited: Aug 6, 2011
  21. Aug 6, 2011 #20

    BruceW

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    The time delay due to transmission of speed of light is a problem in a lot of thought experiments.
    Often in text books, they have to say 'measurements relative to a certain frame, when taking into consideration the delay due to transmission of light'.
     
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