Torque or no torque?! The resolution of the paradox using 4D geometric quantities with the explanation of the Trouton-Noble experiment

Abstract

In this paper we have resolved the apparent paradox of different mechanical equations for force and torque governing the motion of a charged particle in different inertial frames. The same paradox arises in all usual ``explanations'' of the Trouton-Noble experiment. It is shown that the real cause of the paradoxes is - \emph{the use of three dimensional (3D) quantities}, e.g., $\mathbf{E}$, $\mathbf{B}$, $\mathbf{F}$, $\mathbf{L}$, $% \mathbf{N}$, \emph{their transformations and equations with them.} Instead of using 3D quantities we deal with \emph{4D geometric quantities, their Lorentz transformations and equations with them}. In such treatment the paradoxes do not appear. The explanation with \emph{4D geometric quantities}% is in a complete agreement with the principle of relativity and with the Trouton-Noble experiment.