# [Physics] Conservation of mass in relativistic collisions

conservation-lawsenergymassmomentumspecial-relativity

It's stated in my textbook that relativistic mass is conserved in collisions, even in inelastic ones. So if you have a particle with rest mass $$m$$ moving with speed $$u$$ (considerable fraction of the speed of light) in the lab frame and it collides with a stationary particle (as seen in lab frame) also of rest mass $$m$$ and it is given that the two particles coalesce into a new particle with rest mass $$M$$ (that moves with speed $$v$$ in the lab frame), then we can say that:

$$\gamma(u)m+m=\gamma(v)M$$

This is pretty much exactly what's written in my textbook. However, this inevitably leads to:

$$\gamma(u)mc^2+mc^2=\gamma(v)Mc^2$$

Therefore, this seems to show that the collision is elastic, as no energy is lost.

This has left me very confused, especially since it's shown here as well: http://www.feynmanlectures.caltech.edu/info/solutions/inelastic_relativistic_collision_sol_1.pdf

Elastic / inelastic refers not to conservation of energy but conservation of kinetic energy. In this case we have a incident particle with speed $$u$$ and energy $$E$$, mass $$m$$. We also have a stationary particle of mass $$m$$ and after the collision we will have a single particle of mass $$M$$. If the rest mass energy changes (i.e. $$M \neq m$$) then the kinetic energy must also have changed because total energy is always conserved.
$$M = \sqrt{2(1+\gamma)} m > 2m$$