Reference frames for photon collisions (pair production)

[hartin]

Homework Statement

Hi,
My question probably has a simple answer, but I've been scratching my
head over it a little too long so I thought I would ask it here. I
have three initial photons involved in a collision with 4-momenta k1,
k2 and k3. I have two reference frames:
frame 1: the centre of mass frame of photons k1 and k2 (so that 3-momenta k1+k2=0)
frame 2: the centre of mass frame of all photons k1, k2 and k3 (so that 3-momenta k1+k2+k3=0)

I want to find out what the relativistic beta v/c and gamma 1/Sqrt(1-(v/c)^2 are.

Homework Equations

beta=v/c
gamma=1/Sqrt[1-(v/c)^2]
(observed frequency/source frequency)^2=(1+beta)/(1-beta)

The Attempt at a Solution

So there seems no way to directly find the relative velocity of the two frames is as only photons are involved (relative velocity=c ?!?). I thought to use the expression for the relativistic doppler shift, but that doesn't seem appropriate.

 

[Jhero]

Hello,

Thank you for your question. It's great that you are thinking about this problem and trying to find a solution. As you mentioned, the relativistic doppler shift formula may not be applicable in this case since it is typically used for objects with mass. However, there are other equations that can help you determine the relative velocity and gamma factor in this scenario.

One approach could be to use the conservation of energy and momentum equations in the two frames. Since the 3-momenta in frame 1 and frame 2 are equal to 0, you can equate the total energy in each frame to find the relationship between the velocities. Additionally, you can use the Lorentz transformation equations to relate the velocities and gamma factors in the two frames.

Another approach could be to use the relativistic energy-momentum equation, which takes into account the energy and momentum of photons. This equation can also help you find the relationship between the velocities in the two frames.

I hope this helps and gives you some ideas on how to approach this problem. Keep exploring and don't hesitate to ask for help if you get stuck. Good luck with your research!