In every reference frame energy and momentum are conserved in collisions between free particles.
For each component pm of the 4-vector (p0,p1,p2,p3) we have
,
where i denotes the particles going into the collision and j denotes the particles emerging from the collision.
For transformations between reference frames we have
.
This is a consequence of the invariance of the dot product under a Lorentz transformation.
Assume we have a collection of initially free particles. They interact with each other and possibly change into different particles. After the interaction the new particles are free again. The following rules always apply:
(a) The "length" of the 4-vector (P0,P) is
invariant when transforming
from one one reference frame to another.
(b) P0 and P are conserved in any reference frame.