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

1. Dec 5, 2014

VVS

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:
$p_1=\hbar k=\hbar \left(\omega,k,0,0\right)$
$p_2=\hbar k=\hbar \left(\omega,\alpha k,0,0\right)$
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
$\vec{d}=\left(0,0,d\right)$
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 $\vec{v}$ 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. Dec 5, 2014

MathematicalPhysicist

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

Parallel would be multiplying by a positive factor the expression k.

3. Dec 5, 2014

VVS

Thanks you're right. I edited it.