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Faster velocity than the speed of light?

  1. Feb 10, 2013 #1
    Dear Forum Users,

    I am a graduate student in Mathematics and not physics, so please bare with me. Also, I know that a similar topic has been discussed before but i could not get a clear answer from reading the previous posts.

    And here is the question i have been wondering about:

    Suppose we have two particles, p1 and p2, travelling in opposite direction, each with speed 0.6c relative to point "A".

    Question 1:
    After 1 year, the distance from A to p1 will be 0.6 lightyear, and the distance from A to p2 will be 0.6 lightyear. Thus the distance between p1 and p2 will be 1.2 lightyear - right?

    Question 2:
    If so, p1 and p2 have travelled away from each other with speed greater than the speed of light. How can that be if, as i understand, no objects can have a relative speed larger than the speed of light?

    I am looking forward to see your replies :)
     
  2. jcsd
  3. Feb 10, 2013 #2

    jedishrfu

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    If you're familiar with coordinate transformations you have to ask what speed does particle p1 see you traveling away and the answer is 0.6c by symmetry and then ask what speed does particle p1 see p2 traveling away and the answer becomes something like 0.9c (actually 0.88c) because you have to use the relativistic coordinate transformation to compute both cases.

    In relativity no observer will ever see a particle or object of any type travel faster than light speed and that goes for observers on p1 and p2 thats a given borne out by experiments that have yet to prove relativity wrong.

    http://en.wikipedia.org/wiki/Special_relativity

    and specific to your question:

    http://en.wikipedia.org/wiki/Velocity-addition_formula
     
    Last edited: Feb 10, 2013
  4. Feb 10, 2013 #3
    From the point of view of a person at point A, they will have relative velocities greater than light the same way that two cars driving away from me at 60mph in opposite directions will have a relative velocity of 120mph according to me.

    The difference is that from the point of view of a person in either car (which is moving slow enough that SR need not be applied although it technically has an effect), the other one will be moving at 120mph whereas in SR, the two spaceships that you describe will NOT see the other moving at greater than the speed of light. To find the perceived velocity of the other ship, you need the Lorentz transformation of the velocity.

    Note, just as the two cars are not driving 120mph (with respect to the earth), there is no frame of reference where either ship will be moving at greater than the speed of light.
     
  5. Feb 10, 2013 #4
    OK, so in effect the particles ARE moving away from each other with speed greater than light?
     
  6. Feb 10, 2013 #5
    The first paragraph of post #3 is incorrect.

    I suspect, syd, that you may not have noted your question 1 is posed with respect
    to the 'stationary' point A;question 2 is posed with respect to a different frame, say, p1.
    In relativity, different frames observe different results. In SR, different observers are separated not by fixed time and fixed distance but by the Lorentz transforms.
     
    Last edited: Feb 10, 2013
  7. Feb 10, 2013 #6
  8. Feb 10, 2013 #7

    phinds

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    RELATIVE TO WHAT ???

    You need to get straight on the fact that speed is relative. You can't just say "they are traveling at X speed" --- that is a meaningless statement. You have to pick a frame of reference and stick with it. There IS no frame of reference in which they are moving at > c relative to each other.
     
  9. Feb 10, 2013 #8

    PeterDonis

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    This is true in the frame in which A is at rest, but not in other frames. Distance is frame-dependent. Also, "after 1 year" is frame-dependent; you are implicitly picking out events on p1's worldline and p2's worldline which are simultaneous with 1 year having elapsed at A, but which events you pick will depend on which frame you use.

    This "speed" is not the same as the "relative speed" that can't be greater than the speed of light. To determine "relative speed", you have to pick a frame in which, say, p1 is at rest, and then ask what distance p2 covers in that frame in a given time in that frame. Since distance and time are frame-dependent, as above, you can't just use distances and times in the frame in which A is at rest to do this. You have to properly transform distances and times from frame to frame, using the Lorentz transformation. You also have to pay attention to the relativity of simultaneity.

    Have you had any experience doing any of the above? If not, I would suggest consulting a basic reference on special relativity.
     
  10. Feb 10, 2013 #9
    No, it is correct. SR holds that no object can move greater than the speed of light, but the distance between two objects that are moving with respect to a frame of reference can grow at greater than the speed of light with respect to that reference frame. Neither will appear to move faster than light in their own frames.

    No, they are not moving faster than the speed of light in ANY frame. The distance between them may be growing faster than the speed of light from a separate frame, but both will be measured to be moving less than the speed of light.
     
  11. Feb 11, 2013 #10
    Just to be clear. Is it correct to say that the distance between them is growing faster then the speed of light?
     
  12. Feb 11, 2013 #11
    In your frame of reference 'A', yes. The distance is increasing with a rate of 1.2 lightyears/year, that is 1.2c. I said rate and not speed since it's not the speed of anything. There is nothing going faster than light here.
     
  13. Feb 11, 2013 #12
    Thank you all for the replies. I feel that i understand the basic concept now.

    By the way, can anyone recommend a good introductory physics book explaining time dilation and related topics
     
  14. Feb 11, 2013 #13

    Fredrik

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    It's not an "introductory physics" book, but I really like the presentation of SR in "A first course in general relativity" by Schutz. It doesn't go very deep into it, but it covers all the basics very well.

    I don't like that book's presentation of GR however. It's considered the easiest intro to GR, but it's easy because it does everything it can to avoid explaining differential geometry.

    Your question in the OP seems to have been answered, but I'll offer my thoughts anyway, since you got the answer in bits and pieces. Perhaps this will make it easier for the next person who asks this question.

    The distance between the position coordinates of the two objects in the inertial coordinate system comoving with A is increasing at a rate faster than c. Some people even call that rate the "relative speed" of the two objects. Another term is "separation speed". The velocity of object p2 relative to p1 is however something completely different. This is the ##dx^i/dt## of the line in ##\mathbb R^4## (or the dx/dt of the line in ##\mathbb R^2##) that p2's world line is mapped to by the inertial coordinate system that's comoving with p1. Since you specified that the coordinate velocities of p1 and p2 in the inertial coordinate system comoving with A are in opposite directions, you can calculate the velocity of p2 in the inertial coordinate system comoving with p1 by using the velocity addition formula. In units such that c=1, it takes the form
    $$w=\frac{u+v}{1+uv}.$$ In this case, u and w are known, and we're looking for v, so we solve for v.
    $$v=\frac{w-u}{1-wu}.$$ Now you can just plug in the values w=0.6 and u=-0.6 to get the result v=1.2/1.36≈0.88.

    I think the easiest way to prove that the right-hand side of the velocity addition formula is in the interval (-1,1) for all u,v<1 is to define ##\theta(r)=\tanh^{-1}(r)## for all ##r\in\mathbb R##, and use the identity
    $$\frac{\tanh x+\tanh y}{1+\tanh x\tanh y}=\tanh(x+y)$$ and the fact that ##|\tanh x|<1## for all ##x\in \mathbb R##.
    $$|w| =\left|\frac{\tanh\theta(u)+\tanh\theta(v)}{1 +\tanh\theta(u)\tanh\theta(v)}\right| =\left|\tanh(\theta(u)+\theta(v))\right|<1.$$
     
    Last edited: Feb 11, 2013
  15. Feb 11, 2013 #14

    jedishrfu

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  16. Feb 11, 2013 #15
    New question: Will the light from p1 ever reach p2?
     
  17. Feb 11, 2013 #16

    jedishrfu

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    Yes it will.
     
  18. Feb 11, 2013 #17
    Earlier in the thread it said that in A's FoR the distance is increasing at a rate of 1.2 ly.

    In FoR P1 & P2 the rate is 0.88c


    in A's FoR does the light from P1 reach P2? if so how?

    Never mind I see the 1.2 ly distance increase from FoR A is useless, 0.6 isn't.
     
    Last edited: Feb 11, 2013
  19. Feb 11, 2013 #18
    A related topic that I have a difficulty understand is the expansion of the universy. I read that the universe is expanding such that some galaxies move away from each other faster than the speed of light. If this is the case, does it not mean that some galaxies are moving away from us faster than c?
     
  20. Feb 11, 2013 #19
    It is difficult to understand. Our everyday notion of distance doesn’t work on cosmological scales, so everyday intuition is useless.
    Yes, that is possible in curved spacetime. But it is a different concept than in your original post which is in flat space-time. At the Hubble radius, the recession velocity is c; beyond the Hubble radius galaxies move away from us at greater than c. [We also move away from such observers at greater than c.]

    The 'distance' measure used in FRW cosmology model of our universe is most commonly the proper distance, is based on the FRW metric with a rate of change as measured by co-moving observers; this velocity exceeds c for sufficiently large distances.

    All this ‘superluminal’ velocity at great distances tells us is how one of many different possible definitions of distance changes; Other metrics that use different co-ordinates may not contain any apparent superluminal recession.

    For more, checkout the Balloon Analogy by pHinds from these forums: http://www.phinds.com/balloonanalogy/
     
    Last edited: Feb 11, 2013
  21. Feb 11, 2013 #20

    phinds

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    Not in the sense you mean. They are RECEEDING from us at FTL but that is not the same as MOVING. Seems weird, I know. Google "metric expansion". Nothing is moving, it's just that the distance is increasing because space is expanding.
     
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