A new view on relativity: Part 2. Relativistic dynamics

Abstract

The Lorentz transformations are represented on the ball of relativistically admissible velocities by Einstein velocity addition and rotations. This representation is by projective maps. The relativistic dynamic equation can be derived by introducing a new principle which is analogous to the Einstein's Equivalence Principle, but can be applied for any force. By this principle, the relativistic dynamic equation is defined by an element of the Lie algebra of the above representation. If we introduce a new dynamic variable, called symmetric velocity, the above representation becomes a representation by conformal, instead of projective maps. In this variable, the relativistic dynamic equation for systems with an invariant plane, becomes a non-linear analytic equation in one complex variable. We obtain explicit solutions for the motion of a charge in uniform, mutually perpendicular electric and magnetic fields. By the above principle, we show that the relativistic dynamic equation for the four-velocity leads to an analog of the electromagnetic tensor. This indicates that force in special relativity is described by a differential two-form.