The Proof that Maxwell Equations with the 3D E and B are not Covariant upon the Lorentz Transformations but upon the Standard Transformations. The New Lorentz Invariant Field Equations

摘要

In this paper the Lorentz transformations (LT) and the standardtransformations (ST) of the usual Maxwell equations (ME) with thethree-dimensional (3D) vectors of the electric and magnetic fields, E and Brespectively, are examined using both the geometric algebra and tensorformalisms. Different 4D algebric objects are used to represent the usualobserver dependent and the new observer independent electric and magneticfields. It is found that the ST of the ME differ from their LT and consequentlythat the ME with the 3D E and B are not covariant upon the LT but upon the ST.The obtained results do not depend on the character of the 4D algebric objectsused to represent the electric and magnetic fields. The Lorentz invariant fieldequations are presented with 1-vectors E and B, bivectors E_{Hv} and B_{Hv} andthe abstract tensors, the 4-vectors E^{a} and B^{a}. All these quantities aredefined without reference frames, i.e., as absolute quantities. When some basishas been introduced, they are represented as coordinate-based geometricquantities comprising both components and a basis. It is explicitly shown thatthis geometric approach agrees with experiments, e.g., the Faraday disk, in allrelatively moving inertial frames of reference, which is not the case with theusual approach with the 3D E and B and their ST.