How does length contraction apply to the distance between two photons?

[pieterdb]

I'm trying to teach myself special relativity. I use the book 'Introduction to Special Relativity' by Wolfgang Rindler. I have a question about length contraction.

We consider 2 particles traveling along the x-axis of a reference frame S with a constant distance between them. We can always go to the rest frame of the particles. If the distance between the particles as seen from their rest frame is L_0, then the distance between the particles as seen from any other inertial frame moving with a velocity v in the direction of the x-axis of S is calculated by L = L_0 / gamma. This length contraction formula originates in the relativity of simultaneity.

If we now replace the 2 particles by 2 photons, it is no longer possible to go to the rest frame of the photons (since the speed of light is c in every inertial frames). Likewise it is impossible to calculate the distance between the photons as seen from another reference frame with the length contraction formula, since gamma always leads to a division by zero.

So apparently there is a difference between the length contraction of the distance between two particles and the length contraction of the distance between 2 moving photons. I guess the 2 moving photons are the limiting case ?

I don't understand this. Is there a length contraction for the distance between 2 moving photons or is this distance the same in all inertial frames (I guess there should be a length contraction since the relativity of simultaneity) ? If yes, how can we calculate this length contraction ?

 

[Fredrik]

If we now replace the 2 particles by 2 photons, it is no longer possible to go to the rest frame of the photons (since the speed of light is c in every inertial frames). Likewise it is impossible to calculate the distance between the photons as seen from another reference frame with the length contraction formula, since gamma always leads to a division by zero.

When you calculate how something is seen from another (inertial) reference frame, the gamma factor is the one defined by the velocity difference between the two frames. In this case however, since the two objects that have constant velocity in the first frame aren't at rest in the other frame, you can't just use the Length contraction formula. You should start by drawing the world lines of the photons in a spacetime diagram, and then draw a simultaneity line for the frame that you going to transform to. Check where it intersects the world lines of the photons. You need to determine the coordinates of these two events in both frames. When you've done that, it shouldn't be hard to find the distance between the photons in the other frame.

 

[pieterdb]

Thx, I understand it now.The motion of a photon seen from S is given by :

x = x₀ + ctUsing the standard Lorentz Transformation formulas, I can express the motion of a photon seen from S' by :

x' = $$\gamma$$x₀ + $$\gamma$$(c-v)(t'/$$\gamma$$(1-v/c) + vx₀/c²(1-v/c))This eventually leads to the distance between the 2 photons seen from S' :

L' = L (c+v)^1/2(c-v)^-1/2