Past exam of the physics 1 paper of the University of Oxford / 2014 resit / 3 / a

In the centre-of-mass frame, the two equal incoming rest masses have opposite momenta and total energy $2E$. The squared total four-momentum fixes the composite invariant mass $\mathcal{M}$:A change of inertial frame, including going to either particle's rest frame, cannot change this invariant mass. It includes the internal kinetic energy and is generally larger than $2M_1$.

There is a terminology convention worth making explicit. The older “relativistic mass” means frame-dependent total energy divided by $c^2$. To stop particle 1 requires a boost with Lorentz factor $\gamma =E/(M_1{c}^2)$, so the system has total energy $2\gamma E=2E^2/(M_1{c}^2)$ in that frame. Under that alternative convention, its relativistic mass would be $2E^2/(M_1{c}^4)$. This is not its invariant mass; both quantities coincide only when $E=M_1{c}^2$ and the incoming particles already have zero relative motion.