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 space-time 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. (shrink)

Different approaches to special relativity (SR) are discussed. The first approach is an invariant approach, which we call the “true transformations (TT) relativity.” In this approach a physical quantity in the four-dimensional spacetime is mathematically represented either by a true tensor (when no basis has been introduced) or equivalently by a coordinate-based geometric quantity comprising both components and a basis (when some basis has been introduced). This invariant approach is compared with the usual covariant approach, which mainly deals with the (...) basis components of tensors in a specific, i.e., Einstein's coordinatization of the chosen inertial frame of reference. The third approach is the usual noncovariant approach to SR in which some quantities are not tensor quantities, but rather quantities from “3+1” space and time, e.g., the synchronously determined spatial length. This formulation is called the “apparent transformations (AT) relativity.” It is shown that the principal difference between these approaches arises from the difference in the concept of sameness of a physical quantity for different observers. This difference is investigated considering the spacetime length in the “TT relativity” and spatial and temporal distances in the “AT relativity.” It is also found that the usual transformations of the three-vectors (3-vectors) of the electric and magnetic fields E and B are the AT. Furthermore it is proved that the Maxwell equations with the electromagnetic field tensor Fab and the usual Maxwell equations with E and B are not equivalent, and that the Maxwell equations with E and B do not remain unchanged in form when the Lorentz transformations of the ordinary derivative operators and the AT of E and B are used. The Maxwell equations with Fab are written in terms of the 4-vectors of the electric Ea and magnetic Ba fields. The covariant Majorana electromagnetic field 4-vector Ψa is constructed by means of 4-vectors Ea and Ba and the covariant Majorana formulation of electrodynamics is presented. A Dirac like relativistic wave equation for the free photon is obtained. (shrink)