On the law of motion in Special Relativity

Abstract

Newton's law of motion relative to an inertial frame ("the laboratory") for a particle subject to a force acting at a certain time may be interpreted in either of two ways: (1) The force acting on the particle during an infinitesimal time imparts to the laboratory a boost (impulse divided by the mass) while the particle maintains the original velocity relative to the new frame and (2) The force acting on the particle during an infinitesimal time imparts to the particle the same boost relative to the proper frame of the particle which moves with the original velocity with respect to the laboratory. We show that the relativistic law of motion admits both interpretations, the first of which is in fact equivalent to the law of motion. As a consequence, we show that the relativistic law of motion may also be reformulated as "force equals mass times acceleration" in analogy with Newton's law, but with a relativistic mass and a relativistic acceleration defined in terms of the relativistic addition law of velocities, rather than ordinary mass and ordinary vectorial addition of velocities that lead to the classical acceleration and to Newton's law.