Relativistic energy and momentum in particle collisions

[swe91]

Homework Statement

Two particle P and Q each of restmass m₀ and moving in collision course at 2/3c in the laboratory frame of reference.
In the same collision but in particle P's frame of reference, P is at rest.

Homework Equations

As the total energy of the particles depends on the frame of reference, do the observers in each frame of reference agree on the number of particles and photons formed in the collision?

The Attempt at a Solution

From
E=\gammam₀c²
p=\gammam₀v
where v=v' from relativistic velocity addition.
and
E²=p²c²+m₀²c⁴
I can conclude that the total energy and momentum of the the collision differs. Furthermore, as I do not se how the available energy for particle formation can be the same when the total energy is different, I would conclude that the number of particles would differ depending on the reference frame. However, this does not feel right. As the collision takes place at a "single point" in space, wouldn't it be measured the same from all inertial frames of reference?

Thanks in advance

 

[nickjer]

Hmmm... Well it is true that momentum and energy change between different frames. People use the invariant rest mass to solve problems like this. But "number" of particles shouldn't be affected by relativity. The energy of the final products may be different but the total number of products won't change.

 

[swe91]

Okey thank you for your answer.
Is this the equation you are referring to?
E = p^2 c^2 + m^2 c^4
So if one uses " the invariant rest mass" does it imply the center of mass frame of reference of one of the particles?
If that is the case, does that imply that the energy of the particles created is equal to the invariant rest mass plus the "minimal amount" of momentum and kinetic energy of the two particles colliding at all times? and that the additional energy from other frames of reference is only kinetic?

In summary: the minimal amount of energy is used to create particles with largest possible mass, hence, at rest. Furthermore, all other energy from other frames of reference is observed as kinetic from respective frame of reference.