Why doesn't relativistic momentum appear conserved in this frame?

Question

Suppose I have an inelastic head on collision between two idential particles of mass $m$ that come to rest in the centre of momentum frame where relativistic momentum is obviously conserved. If I now switch to the proper frame of one of the particles, the velocity of the other before the collision is $2v\gamma^2$, and after they both travel with velocity $v$.

This would appear to suggest that relativistic momentum isn't conserved in the proper frame of either particle if I write the initial relativistic momentum as $(\gamma_{2v\gamma^2}) (m)(2v\gamma^2)$ and the final as $(\gamma_v) (2m)(v)$. So where is the error in this reasoning?

Answer 1

The collision is inelastic. In either frame some of the initial kinetic energy is lost.

Where can it go?

1. It could be carried off by massless ejecta (photons).
2. It could remain as a binding energy of the combined system or in internal modes of the combined system. Either way the mass, $M$, of the combined system is greater than $2m$.
3. Or some combination of the above.

Your calculation assumes the final mass is $2m$, which is not true unless you have failed to tell us about some relevant momenta in the ejecta.

Assuming case 2, let's figure the kinematics in your second frame (one mass stopped); I'm going to use $c=1$ units and every mass I write is a rest mass:

$$ E_f = \sqrt{M^2 + P^2} = E_i = m + \sqrt{m^2 + p^2} $$

Applying conservation of momentum we can set the $P = p$, so we solve for $M$

$$ M^2 = m^2 + 2m\sqrt{m^2 + p^2} + m^2 + p^2 - p^2 $$

or

$$ M^2 = 2m^2 \left[ 1 + \sqrt{1 + \left(\frac{p}{m}\right) ^2} \right] $$

which is a bit more than $2m$.