Lorentz-invariant three-vectors and alternative formulation of relativistic dynamics

Abstract

Besides the well known scalar invariants, there exist also vectorial invariants in the realm of special relativity. It is shown that the three-vector $\left(\frac{d\vec{p}}{dt}\right)_{\parallel v}+\gamma_v\left(\frac{d\vec{p}}{dt}\right)_{\perp v}$ is invariant under the Lorentz transformation. The indices $_{\parallel v}$ and $_{\perp v}$ denote the respective components established with respect to the direction of the velocity of body $\vec{v}$, and $\vec{p}$ is the relativistic momentum. We prove that this vector is equal to a force of $\vec{F}_R$ satisfying the classical Newtonian law $\vec{F}_R=m\vec{a}_R$ in the instantaneous inertial rest frame of an accelerated body. Therefore the equation $\vec{F}_R=\left(\frac{d\vec{p}}{dt}\right)_{\parallel v}+\gamma_v\left(\frac{d\vec{p}}{dt}\right)_{\perp v}$, based on the Lorentz-invariant vectors, may be used as a truly invariant (not merely a covariant) relativistic equation of motion in any inertial system of reference. An alternative approach to classical electrodynamics based on the invariant three-vectors is proposed.