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Is the group velocity necessarily slower than light (STL)?

  1. Jul 16, 2009 #1
    It is well known that phase velocities can be FTL (faster than light).
    It is often said that group velocities cannot be FTL.
    But I doubt this last one.

    Starting from the Lienart-Wiechert delayed potential, what can be said for sure?
    For sure a response to a step signal cannot be FTL.
    But what about group velocity?
    And what about "signals" generally speaking?
    From classical (relativistic) physics, what can be precisely said about "signals" speed limitations?
    And what is usually meant by a "signal"?

    If, as I guess, group velocities can be FTL, I would appreciate some examples.
    If, in some circumstances, group velocities cannot be FTL, I like to know more about the conditions.

    I would like to become more precise about all these things.
    Thanks for your suggestions.
     
    Last edited: Jul 16, 2009
  2. jcsd
  3. Jul 16, 2009 #2
    GV can't be FTL. It is the velocity of a constructive interference from a group of waves, each wave traveling at c so the fastest possible GV is c. This holds for the observable universe.

    If you would like to make up a system of waves that are faster than c, say 2c, then you could have a GV that is FTL in your made up system.
     
  4. Jul 16, 2009 #3

    clem

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    The standard formula for the group velocity is v_G=dw/dk.
    This formula is derived as the first term in a Taylor series for w(k).
    If the index of refraction is changing rapidly with frequency, this formula can give a v_G that is greater than c. But then the approximation that gave v_G breaks down, so that
    dw/dk is not the velocity of a wave packet. Instead the wave packet breaks up.
    It can be shown that the velocity of the front of the wave packet still cannot exceed
    c in any case.
     
  5. Jul 20, 2009 #4
    Do you mean that the concept of group velocity isn't well defined for rapidly changing refractive index?
    How? (Is the front just defined as the first point where the amplitude isn't exactly zero?)
     
    Last edited: Jul 20, 2009
  6. Jul 20, 2009 #5

    clem

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    Yes, and paradoxes arise if this is forgotten.
    That is, the formula v_g=dw/dk still exists but it is not related to the speed of the packet.
    The next derivative relates to the spread of the packet, and if the higher terms in the Taylor series are important, the packet distorts and breaks up.
    Yes.
     
    Last edited: Jul 20, 2009
  7. Jul 20, 2009 #6

    ZapperZ

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    Note that in the NEC experiment from several years ago using an anomalous medium, the group velocity exceeded c considerably. Still, no part of the wave travels faster than c.

    Zz.
     
  8. Jul 20, 2009 #7

    George Jones

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    Greg Egan has written an applet that nicely illustrates this,

    http://gregegan.customer.netspace.net.au/APPLETS/20/20.html.

    Egan's pulse propagates without distortion because Egan has used a frequency dependence for the index of refraction that "is unlikely to hold true over a broad range of frequencies in any real medium, but it can be approximately correct for a limited range."

    In Egan's example, speed of information is not the same as the group velocity. Interestingly, in real media, speed of information transfer is usually close to the speed of light even when the group velocity is less than the speed light.

    From Jackson (page 319 in the second edition):

    "The general usage is to take the group velocity of the dominant frequency component as the signal velocity and velocity of transport. This suffices in most circumstances, but with sensitive enough detectors the signal velocity can evidently be pushed close to the velocity of light in vacuum, independent of the medium."
     
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