The Proof that the Standard Transformations of E and B and the Maxwell Equations with E and B are not Relativistically Correct

TL;DR

This paper proves that the conventional 3D transformations of electric and magnetic fields are not relativistically correct, and introduces well-defined 4D quantities to replace them, challenging the compatibility of Maxwell's equations with special relativity.

Contribution

It demonstrates that standard 3D E and B transformations are invalid in relativity and proposes 4D E^a and B^a as proper quantities, providing a new formulation of Maxwell's equations.

Findings

Standard 3D transformations are not relativistically correct.

Maxwell's equations with 3D E and B are incompatible with special relativity.

4D E^a and B^a are well-defined relativistic quantities.

Abstract

In this paper it is exactly proved that the standard transformations of the three-dimensional (3D) vectors of the electric and magnetic fields E and B are not relativistically correct transformations. Thence the 3D vectors E and B are not well-defined quantities in the 4D spacetime and, contrary to the general belief, the usual Maxwell equations with the 3D E and B are not in agreement with the special relativity. The 4-vectors E^a and B^a, as well-defined 4D quantities, are introduced instead of ill-defined 3D E and B. The proof is given in the tensor and the Clifford algebra formalisms.