Relativistic scattering Lab and CM frames

[trelek2]

Hi!

I have the following problem:
Example: collision of 2 electrons
For non-relativistic scattering it is easy to show that the speed of the CM frame with respect to the lab frame is equal to the speed of the electrons in the CM frame, expoloiting the fact that in the lab frame, one of the electrons is at rest.

Now this also holds in the relativistic regime. I'm not really sure where does this follow from. Is it valid to take the lorentz velocity addition formula, and taking the speed of the particle in the lab frame to be 0 we see that for x-coordinate the speed of the electron in CM frame must then be uqual to to the speed of CM frame in lab frame. Saying that the lab and CM frames are in standard configuration makes this proof general enough?

 

[bcrowell]

I think your proof is correct. What's different between the relativistic and nonrelativistic cases is that in the relativistic case, the velocities in the c.m. frame are not half the velocity of the projectile in the lab frame.

 

[Entropia]

Hello!

The relationship between the lab and CM frames in relativistic scattering can be derived using the Lorentz transformation equations. In the lab frame, one of the electrons is at rest and the other has a velocity v. In the CM frame, both electrons have the same velocity v' (since the CM frame is moving with the same speed as the electrons). Using the Lorentz transformation equations, we can relate the velocities in the two frames, and we get that v' = (v + u)/(1 + vu/c^2), where u is the velocity of the CM frame with respect to the lab frame. Since v=0 in the lab frame, we get v' = u, which means that the speed of the electrons in the CM frame is equal to the speed of the CM frame in the lab frame. This holds for any standard configuration of the lab and CM frames, making the proof general enough. I hope this helps!