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Lorentz Transformation of Vectors from S to S' Frame

  1. Nov 13, 2013 #1
    1. The problem statement, all variables and given/known data

    The question is quite basic; what is the Lorentz transformation of the follows 4-vectors from S to S' frame:

    A photon (P) in S frame with 4-momentum

    P = (E/c,p,0,0) and

    frequency f where

    hf = pc = E. h is the planks constant, p is the magnitude of 3-momentum and E is energy.

    S' frame travels in positive x direction with position v speed (ie. not an ANTIPARTICLE).

    ...so how's P' related to P? P' is momentum in S' frame

    My attempt:

    Lorentz boost, simple gamma factor (sqrt (1-(v/c)^2)) relationship with P to give a P' answer as

    P' = √(1-(v/c)^) * P

    Correct?

    How above the relationship of f and f'? f' is the frequency of photon in S' frame.

    My attempt:

    Same as above, gamma relationship with f to give

    f' = √(1-(v/c)^) * f

    Correct?

    Thanks everyone
     
  2. jcsd
  3. Nov 13, 2013 #2

    hilbert2

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    Gold Member

    You have to write the 4-momentum as a column vector
    ##\begin{bmatrix}E/c\\p\\0\\0\end{bmatrix}##
    and operate on it with the Lorentz boost matrix, which is
    ##\Lambda=\begin{bmatrix}\gamma&-\beta\gamma&0&0\\-\beta\gamma&\gamma&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}##
    for boosts in the ##x##-direction. Here ##\beta=v/c##, where ##v## is the relative velocity of the frames. Then you get the components of the four-momentum in the new frame. The frequency changes in the transformation by the same factor as the energy component of the 4-momentum vector.
     
    Last edited: Nov 13, 2013
  4. Nov 13, 2013 #3

    vanhees71

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    2016 Award

    In addition note that for a photon the four-momentum is
    [tex](p^{\mu})=\begin{pmatrix}
    |\vec{p}| \\ \vec{p},
    \end{pmatrix}[/tex]
    because a photon's four-momentum is light-like.
     
  5. Nov 13, 2013 #4
    Hi,

    Thank you for replying (hilbert and vanhees!). I cross multiply the matrix, factor in the fact that E=pc to get:

    γP * matrix

    [ 1, -v/c]
    [-v/c, 1 ]

    How do I solve this?
     
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