What is possible intuitive explanation of inelastic relativistic collsion? In classical mechanics, we say an inelastic collision happens when some energy is transferred to heat and noise without changing the total sum of momentum. However, in special relativity, every component of 4 momentum is preserved, but not the sum of masses. How can we explain it intuitively like we did in classical mechanics?
 A: In relativistic collisions, the lost kinetic energy appears as rest masses of products. Or the kinetic energy can increase in, say, a $1 \to 2$ decay process, where rest mass energies are converted into kinetic energy. 
A: In a collision, the total relativistic energy is conserved.
$$E_{rel,total,f} = \gamma_{Ai} m_A  + \gamma_{Bi}m_B$$
If, in addition, the particles retain their rest masses in the interaction, 
we have for the left-hand-side:
$$E_{rel,total,f} = \gamma_{Af} m_A  + \gamma_{Bf}m_B$$
and (without using the "final" labels on the left hand side)
$$ m_A + m_B = m_A +m_B.$$ 
By subtraction,
$$(\gamma_{Af}-1) m_A  + (\gamma_{Bf}-1) m_B = (\gamma_{Ai}-1) m_A  + (\gamma_{Bi}-1)m_B$$
so the "total relativistic-kinetic-energy is conserved"... in an elastic collision,
similar to what we say in classical mechanics...
the "total kinetic-energy is conserved"... in an elastic collision.
By having the particles change mass in the collision, 
this last equation will no longer hold... and you now have an inelastic interaction.
