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Homework Help: Lorentz-Invariance of Photons travelling parallel to each other

  1. Dec 5, 2014 #1


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    1. The problem statement, all variables and given/known data
    Show the Lorentz-Invariance of the following spatial statement: Two photons are travelling parallel to each other. The relative position vector of the two photons is orthogonal to the velocity and has length d.

    2. Relevant equations / The attempt at a solution
    The first thing that comes to my mind is to represent the two photons with 4-Momentum Vectors.
    So this is done the following way:
    [itex]p_1=\hbar k=\hbar \left(\omega,k,0,0\right)[/itex]
    [itex]p_2=\hbar k=\hbar \left(\omega,\alpha k,0,0\right)[/itex]
    Now I need a 4-vector that connects the two photons. It is easy to see that the spatial component of that vector is just
    But I am not sure what the first (time) component of that corresponding 4-vector would be?
    From here on I guess it is simple. All I need to do is show that under a Lorentz-Transformation for arbitrary [itex]\vec{v}[/itex] the four-momentum of the photons is always orthogonal to the relative position vector. I might be wrong here.
    Last edited: Dec 5, 2014
  2. jcsd
  3. Dec 5, 2014 #2


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    Gold Member

    What you wrote is antiparallel, i.e the direction reversed.

    Parallel would be multiplying by a positive factor the expression k.
  4. Dec 5, 2014 #3


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    Thanks you're right. I edited it.
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