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Question concerning wave 4-vector

  1. Feb 9, 2010 #1
    Can someone show me where I'm going wrong in the following?

    From Rindler's Relativity book, page 103, the wave 4-vector is defined as L = f (n/w, 1/c ) where f is the frequency and n the unit vector parallel to the velocity of propagation of a plane wave, w.

    To get a feel for how it transforms, I'm starting with the y, z components since they should remain unchanged for a frame s' travelling along the x-axis of the stationary frame with velocity v:

    f n_y/w = f' n'_y'/w' - (1)

    Where cos theta is the direction cosine of the velocity of propagation, it is easily shown that:

    f' = gamma (1 - v/w cos theta ) - (2)

    n'_y' is parallel to w'_y' and so:

    n'_y' = w'_y'/w' - (3)

    So, substituting for f' and n'_y' in (1) should = f n_y hence verifying that the y and z components transform as the components of a 4 vector:

    f' n'_y'/w'

    = gamma (1 - v/w cos theta ) w'_y'/w'^2

    = gamma (1 - v/w cos theta ) (u_y/gamma(1 - vw_x/c^2) ) / w'^2

    But w'^2 = ( gamma^2 (w_x - v)^2 + w_y^2 + w_z^2)/ gamma^2(1 - vw_x/c^2)^2

    So, it looks as if it won't simplify the way I want to show (1)

    Thanks for your comments in advance.
     
  2. jcsd
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