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Observed speed of light [actually about relativistic velocity addition]

  1. Apr 23, 2010 #1
    I apologize if this is a repost, my original post didn't seem to take.

    If you accelerate a particle to 99% the speed of light and accelerate another particle to 99% the speed of light directly into the path of the first would this not create the observed effect from either particle that the opposite particle is traveling at 198% the speed of light?

    If everything is relative to the observer this would seem to break the speed of light limitation in the observation of either particle.

    Thanks!
     
  2. jcsd
  3. Apr 23, 2010 #2
    Re: Observed speed of light

    Velocites just don't add like that. What you are talking about is Galilean relativity which is the low velocity limit of Einstein's Relativity.
     
  4. Apr 23, 2010 #3
    Re: Observed speed of light

    ...for the more you change the relative velocity of a particle, the more you change the relative time and velocity and time are intertwined.
     
  5. Apr 23, 2010 #4

    jtbell

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  6. Apr 23, 2010 #5
    Re: Observed speed of light

    Would this also apply to two objects moving away from each other? I ask because I'm also interested in the rate of acceleration of distant objects away from each other.

    Does this mean that even though I can observe two objects on distant opposite ends of the universe traveling away from me at nearly the speed of light, they themselves view each other at traveling less than the speed of light?
     
  7. Apr 24, 2010 #6

    Doc Al

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    Re: Observed speed of light

    That's correct.
     
  8. Apr 24, 2010 #7

    jtbell

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    Re: Observed speed of light

    Yes. The equation on the page that I linked to applies in this case also. Substitute a positive number for a velocity to the right, and a negative number for a velocity to the left.
     
  9. Apr 25, 2010 #8
    Re: Observed speed of light

    This poses a bit of a challenge for me.

    I always assumed that distant objects from us were accelerating away from us faster than nearer objects because space itself was expanding. (Place two objects at different distances from you then double "space". The more distant object will appear to have accelerated at a greater rate even though it's "velocity" was constant).

    Reason I ask is because one theory of multiple universes implies that anything moving away from us faster than the speed of light exists in another universe since nothing from it can effect us (forces withstanding, because aren't they instant?).

    I assumed that since space itself was expanding that distant objects would eventually become outside of our observable universe. Was my assumption wrong? Has this ever been tested?

    Thanks for the replies so far... I've always pondered ideas and I'm always happy to hear why they are wrong! Things are more interesting when they aren't simple....
     
    Last edited: Apr 25, 2010
  10. Apr 25, 2010 #9

    jtbell

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    Re: Observed speed of light

    The equation I linked to is for special relativity (flat spacetime). If you want to get into general relativity (curved spacetime), I'm outta here. :uhh:
     
  11. Apr 25, 2010 #10
    Re: Observed speed of light


    I've never taken "higher" math (beyond high-school), so that I can ask a challenging idea gives me a giggle though you most likely already know why it(the thought) doesn't work.

    How does my question of objects moving away, then towards, each other put a chink in the special/general relativity armor. Where does the weak link arise?

    I can visualize that two object approaching each other will "compress" spacetime and therefore not break any laws. My question throws in "dark energy", which expands space, which implies observed velocity when in fact there is none.

    Srry. I'm no math whiz, I ride in a truck all day, I have time to ponder the crazy stuff.
     
    Last edited: Apr 25, 2010
  12. Apr 25, 2010 #11

    Dale

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    No chink in the armor. It is just that the math becomes quite a bit more complicated. Also, the question itself becomes rather ill-posed in GR since in a curved spacetime there is no unique way to compare the relative velocity of two distant objects.
     
  13. Apr 25, 2010 #12
    Re: Observed speed of light

    Thanks for the answers so far, I think I've got my head wrapped around it. Here is what my thought process has put together from everything... tell me if I'm wrong.

    Galilean Relativity explains why two objects moving at .9c in opposite directions from a center point are in fact not moving at 1.8c away from each other. I'm assuming this due to a curve in spacetime?

    So, how can distant objects be moving away from each at faster than c? The reason is because the space between them is expanding. The objects are not moving through space faster than c.

    Imagine two ballplayers on the opposite side of the field. One ballplayer throws a ball to the other. If you somehow expand the playing field faster than the ball is traveling it would appear as if the ballplayers are moving away faster than the ball can travel. This is not the case because the ballplayers never moved, the playing field just got bigger.
     
  14. Apr 25, 2010 #13

    diazona

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    Re: Observed speed of light

    Actually, you got the terminology mixed up a little. Galilean relativity would say that two objects moving at .9c in opposite directions from a center point are moving at 1.8c away from each other. The corresponding formula is
    [tex]v_{13} = v_{12} + v_{23}[/tex]
    and this is what everyone believed prior to about 1900. But eventually it was discovered that that is not really correct. Lorentzian relativity is the name we attach to the correct formula (at least, correct as far as we can tell), which, if I remember correctly, is
    [tex]v_{13} = \frac{v_{12} + v_{23}}{1 + v_{12}v_{23}/c^2}[/tex]
    This formula would tell you that two objects moving at .9c in opposite directions away from a center point are moving at .994c away from each other.

    This has nothing to do with curving of spacetime. Curved spacetime is related to gravity and general relativity, and this Lorentzian relativity stuff is true even when there is no gravity.
    Yep, that's exactly right. (At least, I have heard similar analogies used many times to describe the expansion of the universe)
     
  15. Apr 25, 2010 #14
    Re: Observed speed of light

    Has anyone got a link to something that explains why that formula is true?
     
  16. Apr 26, 2010 #15
    Re: Observed speed of light

    Something odd has to be going on. I can see the mathmatical proof but logic says there is more to it. If object A and B have both traveled for one second at .9c, then their combined distances from center C (according to logic) should be roughly 539 million meters. Measured from object A to B their distance from each other will only be roughly 298 million meters.

    If you freeze the system at this one second mark, how would you account for the different measurements?
     
  17. Apr 26, 2010 #16
    Re: Observed speed of light

    Let us say an observer is standing at C holding a clock and measures the distance between A and B after one second on his clock. Observer A measures a different distance between A and B after one second on his clock, partly because A's clock is ticking at a different rate to C's clock. That is basically the "something odd" that is going on.
     
  18. Apr 26, 2010 #17

    JesseM

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    Re: Observed speed of light

    In the frame that sees them both moving at 0.9c in opposite directions, the distance between them is increasing at 1.8c. But that's the "closing speed" between two different objects (how fast one is closing in on the other, or how fast the distance between them is growing) in the frame of a third observer who's not at rest relative to either object, this is a distinct notion from the speed of one object in the other object's own rest frame--relativity just says that no individual object or signal can move faster than light in any inertial frame. And keep in mind, each inertial frame uses rulers and clocks at rest in that frame to define "speed" in terms of distance/time...since rulers at rest in one frame are measured to be length contracted in another frame, and clocks at rest in one frame are measured to be time dilated in another frame, so hopefully you can see it makes a kind of sense that the relativistic velocity addition formula (which relates measured speed in one frame to measured speed in a different frame) would be different from the Galilean formula in Newtonian mechanics, and that different frames could disagree about the rate that the distance between two objectss was increasing (unlike in Newtonian physics where everyone agrees on this).
    But "one second mark" in what frame? Again, you have to keep in mind time dilation, which means clocks in different frames measure time differently (not to mention something called the relativity of simultaneity which says different frames disagree on whether a pair of events at different locations happened at the same time or not)
     
    Last edited: Apr 26, 2010
  19. Apr 26, 2010 #18
    In coordinate system of the 'ground' observer, A is moving at 0.9c, and B is moving just slightly faster. So relative velocity B-C (in the 'ground' frame) will be really small (<0.1c)

    In the frame of A, B is going away at 0.9c, and ground observer is receding at 0.9c in opposite directions.

    Note that observers A,B and groudn observer do not agree on distances.

    And finally, you can't freeze the system at some mark because A, B and ground observer are in different spacial locations. So, 'freeze at some moment of time' is observer dependent. Some moment of time for A is not the same as for B.
     
  20. Apr 26, 2010 #19
    Gotcha!

    How then do you calculate the time dilation between object A and C on say an 8.74 lightyear round trip (Alpha Centauri)?

    What I'm looking for exactly is the time passed for both observers.
     
  21. Apr 26, 2010 #20

    JesseM

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    In any inertial frame, if an observer travels for some time T (in terms of that frame's time coordinate) at speed v, the time elapsed for that observer is [tex]T\sqrt{1 - v^2/c^2}[/tex]
     
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