The Proof that the Standard Transformations of E and B are not the Lorentz Transformations. Application to Motional EMF

Abstract

In this paper it is proved by using the Clifford algebra formalism that the standard transformations (ST) of the three-dimensional (3D) vectors of the electric and magnetic fields E and B are not the Lorentz transformations (LT) of well-defined quantities from the 4D spacetime. This difference between the ST and the LT is obtained regardless of the used algebraic objects (1-vectors or bivectors) for the representation of the electric and magnetic fields in the usual observer dependent decompositions of F. The LT correctly transform the whole 4D quantity, e.g., E_{f}=F\cdot γ_{0}, whereas the ST are the result of the application of the LT only to the part of E_{f}, i.e., to F, but leaving gamma_{0} unchanged. The new decompositions of F in terms of 4D quantities that are defined without reference frames, i.e., the absolute quantities, are introduced and discussed. It is shown that the LT of the 4D quantities representing electric and magnetic fields correctly describe the motional electromotive force (emf) for all relatively moving inertial observers, whereas it is not the case with the ST of the 3D E and B.