Necessity of many-particle forces in relativistic Newtonian mechanics for massive and massless particles

Abstract

The Lorentz-invariance conditions for Newtonian equations of motion for three particles are assumed to be satisfied by sums of two-particle forces that satisfy the Lorentz-invariance conditions for two particles. Then it is shown that a particle can be accelerated only by forces from particles that do not accelerate, provided every particle has positive mass. There are exceptional cases when one or more of the particles has zero mass. Relativistic Newtonian mechanics for zero-mass particles is formulated two different ways. When the equations of motion specify the accelerations as functions of the positions and velocities, the result is the same as for positive-mass particles. When the time derivatives of the momenta are specified as functions of the positions and momenta, the result is that a particle can be accelerated only by forces from particles that do not accelerate continuously. However, there are forces that change the magnitude of the momentum without changing the velocity, for a particle with zero mass. They produce discontinuous acceleration when the velocity abruptly changes direction as the momentum reaches zero and changes sign. For a particle accelerated by a force from a massless particle that accelerates in this discontinuous way, there are two-particle forces with acceleration of both particles.