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In this paper a geometric approach to the special relativity (SR) is usedthat is called the "invariant special relativity" (ISR). In the ISR it isconsidered that in the four-dimensional (4D) spacetime physical laws aregeometric, coordinate-free relationships between the 4D geometric,coordinate-free quantities. It is mathematicaly proved that in the ISR theelectric and magnetic fields are properly defined vectors on the 4D spacetime.According to the first proof the dimension of a vector field is mathematicalydetermined by the dimension of its domain. Since the electric and magneticfields are defined on the 4D spacetime they are properly defined 4D vectors,the 4D geometric quantities (GQs). As shown in an axiomatic geometricformulation of electromagnetism with only one axiom, the field equation for thebivector field F [33], [T. Ivezi\'c, Found. Phys. Lett. 18, 401 (2005), arXiv:physics/0412167], the primary quantity for the whole electromagnetism is thebivector field F. The electric and magnetic fields 4D vectors E and B aredetermined in a mathematically correct way in terms of F and the 4D velocityvector v of the observer who measures E and B fields. Furthermore, the proofsare presented that under the mathematicaly correct Lorentz transformations,which are first derived by Minkowski and reinvented and generalized in terms of4D GQs, e.g., in [23], [T. Ivezi\'c, Phys. Scr. 82, 055007 (2010)], theelectric field 4D vector transforms as any other 4D vector transforms, i.e.,again to the electric field 4D vector; there is no mixing with the magneticfield 4D vector B, as in the usual transformations (UT) of the 3D fields. Thisformulation with the 4D GQs is in a true agreement with experiments inelectromagnetism, e.g., the motional emf.

journal of physics. conference series

Electric and magnetic fields as explicitly observer dependent  four-dimensional vectors and their Lorentz transformations according to  Minkowski-Ivezi\'c