Relativistic momentum

Introductory treatments of relativistic dynamics rely on the invariance of momentum conservation (i.e., on the assumption that momentum is conserved in all inertial frames if it is conserved in one) to establish the relationship for the momentum of a particle in terms of its mass and velocity. By contrast, more advanced treatments rely on the transformation properties of the four-velocity and/or proper time to obtain the same result and then show that momentum conservation is invariant. Here, we will outline a derivation of that relationship that, in the spirit of the more advanced treatments, relies on an elemental feature of the transformation of momentum rather than on its conservation but does not have as a prerequisite the introduction of four-vectors and invariants. The steps in the derivation are no more involved than in the usual introductory treatments; indeed, the arithmetic is almost identical.