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I'm having difficulty understanding 4-momentum in relativity. The problem I have is that two particles of equal mass are colliding, with one particle moving and one at rest. I know the three-momentum of the moving particle, and I need to find the total energy in the COM frame.

What's confusing me is that it appears I can completely ignore the initial frame. Using the Energy-momentum relation, I can say that the total momentum will be zero (since both particles will have opposite momentum), then I end up with:

$E^2=m^2$

Which I can solve because I know the mass of the two particles. This doesn't seem right since it would mean that that the energy total in the COM frame doesn't depend on the velocity of either of the particles.

I also tried a different approach where I tried to velocity of the COM frame. The velocity of the center of mass in the initial frame should be $v/2$, but when I use the Lorentz Velocity transformation I end up with the velocity of the rest particle being $v/2$, and the velocity of the moving particle being:

$\ v'=\frac{v-v/2}{1-v*v/2}$

Which does not simplify to $v/2$ as far as I can tell. Can someone please help me understand what is going on?

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1 Answer 1

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Your calculation is incorrect because the total energy in the COM frame is not equal to the sum of the masses. In addition to the sum of the masses of the two particles you must also include the kinetic energy. The system of particles has kinetic energy even though it does not have momentum.

The easiest way to do this problem is to recognize that the total energy in the center of momentum frame is equal to the Minkowski norm of the four-momentum. Since this is a relativistic invariant you can calculate it in any frame. So you can do the calculation in the lab frame where you have the information and then simply report the result without further transformations.

Regarding the velocities. They will not be equal to $v/2$, but they must be equal.

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  • $\begingroup$ I think I understand. So I can't just add the two masses, but can I still add the energies and momentums? $\endgroup$
    – Kevin
    Feb 9, 2020 at 2:53
  • $\begingroup$ Think vectors.... you can’t just add magnitudes. But you can add corresponding components together. $\endgroup$
    – robphy
    Feb 9, 2020 at 3:06
  • $\begingroup$ @Kevin yes. You can add components of four-vectors, just like you can add components of three-vectors. You cannot add the norms of four-vectors just like you cannot add the norms of three-vectors. $\endgroup$
    – Dale
    Feb 9, 2020 at 3:24
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    $\begingroup$ I think I get it now. Thanks for the help guys! $\endgroup$
    – Kevin
    Feb 9, 2020 at 3:39

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