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• W2466745311 abstract "This paper deals with the Lorentz transformations of the electromagnetic field. The goal is to achieve an approach that appears elementary enough as if it needed “ only bare hands.” This idea springs from the fact that these transformation can be cast in an exceedingly appealing form― when electric and magnetic fields are packed, as real and imaginary parts, into a complex Faraday field― proper Lorentz transformations (boosts) are literally rotations by an imaginary angle. Therefore, it is advisable to seek an appropriately appealing way of getting them. The easy path is to exploit the transformation properties of currents and their derivatives― then the fields constitute a second rank antisymmetric tensor― the Faraday tensor. However, one can hardly claim that one feels these transformations in one's bones. This procedure can be translated into a form facilitating the Faraday vector― and it turns into a brute force approach― which is not entirely satisfactory in the long run. An equally swift alternative is relying on the electromagnetic potentials the transformations of which are inferred from gauge invariance― they go as four-gradients. However, certain subtlety is involved here ; when one agrees to the potential being a four-vector it becomes difficult to think of it as a vector of polarization― the time-component becomes dubious. In the end, gauge invariance saves the situation― polarization is a four-vector up to a gauge transformation. An approach based on careful examination of plane wave solutions of free Maxwell's equations proves fruitful here. Currents need not be considered any more, but wave-vectors come to play a crucial role. They are light vectors, and somewhat better behaved than currents under Lorentz transformations. As we easily establish how light rays tilt under proper Lorentz transformations, and the time-component of a wave-vector is the frequency of its wave― we are comfortable manipulating only the familiar space-part of the wave vector. All inertial observers perceive plane waves very similarly― they are always solutions of form invariant Maxwell's equations. Therefore, each observer sees an orthogonal triad of three vectors― wave, electrical and magnetic― what varies from one image to the other is their scale and orientation. By inspecting the straightforward special cases the underlying rule easily emerges― stretch and rotate. Then, linearity allows one to assemble the general case and transcription to the Faraday field language follows as a natural extension. As a bonus, an interpretation of Wigner's little group for a light-like vector presents itself. Namely, the subgroup of Lorentz transformations leaving a wave-vector invariant is the Euclidean group of transformations in the plane― it consists of rotations and translations in two dimensions. Rotations are easy, it is rewarding to inspect the “ gears and wheels” of these translations― where do they come from." @default.
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• W2466745311 date "2002-01-01" @default.
• W2466745311 modified "2023-09-24" @default.
• W2466745311 title "Boosting the Electromagnetic Field" @default.
• W2466745311 hasPublicationYear "2002" @default.
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