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III. Arguments of Detail

The previous presentation of the basis of Electrodynamics makes it clear that the whole construction can be based upon conservation of the electric charge and magnetic flux. The mathematics of Hehl and Obukhov however, raises a delicate point upon which we feel particularly urged to insist. Personally we are partial to the transport interpretation of everything in the realm of Physics. And, when it comes to transport equations one cannot assume that the Space forms supporting electro-dynamic phenomena are frozen as the presentation of our authors seems to imply. This fact is of no consequence for the scope of their work, namely to stress the importance of the differential forms for the realm of Electrodynamics. And, as this field of Physics has been penetrated by the habit of starting from the Maxwell equations, the point is served one way or another, no matter how one starts in presenting the basics. Moreover, when one is just passing through classical realm towards the relativistic one, it is hardly the case to insist upon such things as the variability of the Space supports of physical phenomena. This was just about to happen with us here, had it not been for the fact that turning back to classical context after the adventure in the relativistic realm, the spirit finds an ambiguous situation specifically tied up with constitutive laws (Le Bellac, Lévy-Leblond, 1973). This fact determined us to undertake once more the situation from the classical point of view. And it does not seem a better starting point for this particular enterprise than the work of Hehl and Obukhov.

To start with, for instance in equation (1) we think that the complete expression for the time derivative of charge should take into consideration the variability of the volume by a Space deformation of the form , in which case one has (Betounes, 1983)

(17)

thereby involving the Lie derivative of the charge density along the vector that characterizes the variation of space domain measured by V. By definition the Lie derivative of a differential form Ω is

(18)

For the specific case of electric density in equation (17), the second term of the Lie derivative is zero, and using the more familiar language of Space vectors as presented by Betounes (Betounes, 1983), we can write (17) as

(19)

where ρ_e is the scalar electric charge density. Thus, instead of equation (4) we should have

(20)

or, in terms of vectors and scalars

(21)

This way it becomes clear that, for instance, the charge density is constant in time only in case where the current is exclusively an electric current:

(22)

Otherwise, in the general case, we are bound to redefine the axiomatically introduced current form by the contribution of the carrier. This seems just natural: in general, a current does not represent exclusively stream of electric charges, for all of the known particles carry indeed more than one physical quality. Therefore the common event of this World is that the electric charge density is not constant in time.

The very same considerations apply to the time derivative of the magnetic flux from equation (13). Here we have an equivalent of equation (17) with the Lie derivative given by the formula

(23)

so that instead of (14) we must write

(24)

or, in terms of the corresponding vectors,

(25)

Again, the conclusions with respect to the fundamental equations of Electrodynamics are a little bit different. Let us expound on this point.

First, let us restate the conclusion above: the transport approach of Electrodynamics according to Hehl-Obukhov set of basic axioms, forces a redefinition of axiomatically introduced current vector and electric field strength (Post, 1997):

(26)

Now, let us say that our electromagnetic speculative knowledge is indeed easiest to follow as starting from magnetic flux and electric charge as basic concepts. Regarding the magnetic flux though, our current experience is represented by permanent magnets or magnetic dipoles, for which the condition of closure of the magnetic flux is automatically satisfied. From this point of view, one can think of the classically viewed Electrodynamics as a mere extension of experience. Accepting this extension, the equations (21) and (25), obtained by applying transport considerations to the axioms of Electrodynamics, represent two different limits of the relativistic transformation formulas of Electrodynamics, as presented by Le Bellac and Lévy-Leblond (Le Bellac, Lévy-Leblond, 1973). Namely the first transformation (26) represents the electric limit for the 4-current transformation, while the second of (26) represents the magnetic limit for fields. Consequently, the transport theory of Electrodynamics is not consistent with its relativistic counterpart. Personally we are partial to the idea that the Special Relativity has nothing to do with the deformation of Space, as indeed historically has been recognized and asserted.

Now, as the opportunity has come our way, let us expound a little bit on the existence of the right pair of transformations corresponding to the first one from equation (26) according to Le Bellac’s and Lévy-Leblond’s analysis. This pair of transformations should be something of the form

(27)

The last transformation of these two would then require an axiom analogous to equation (13), i.e. something of the form

(28)

leading to the axiomatic acceptance of displacement current from the very beginning, as historically was indeed the case. We are not aware of an interpretation of the second transformation of (27) in terms of a possible experimental arrangement. However, we are aware that there is an ongoing debate on the displacement current issues (Roche, 1998). Our point is that the axiomatic basis that has led to the Special Relativity is much larger than the necessities of a classical transport theory of Electrodynamics. The issues are, however, obscured by the kinematical approach from Relativity and only surface at a deeper analysis, as the article of Le Bellac and Lévy-Leblond plainly prove.

In closing this discussion, notice that equations (24) or (25) tell us that the variation of magnetic flux is perceived by the charge as the circulation of a Lorentz force based on Space deformation. We can even extend this image to situations where the charge is absent, as in the case of an electromagnetic model of light. In such a case, there is an electric field created by the pure interaction between the deformation of Space and the magnetic flux. Thus, the Space is actively participating in shaping the physical phenomena, but this is perhaps nothing new in Physics, inasmuch as the idea is one of the premises of General Relativity. From this perspective, we would not need in fact an axiom for introducing the Lorentz force, at least for the electromagnetic phenomena describing the radiation. This seems, again, closer to the historical order of things. That axiom is required only from kinematical point of view, a point of view that needs itself to be reviewed as, hopefully, the next section will show.

One of the most important conclusions of this discussion is coming up as we follow the logic of Hehl and Obukhov in passing from equation (14) to equation (15). For, in this case we have from equation (24)

(29)

or in terms of vectors

(30)

It is clear here that nothing can prevent us from accepting the existence of a magnetic charge in order to extend the Electrodynamics beyond experience as defined above. Care should however be exercised, because this extension is closely related to properties of Space deformation. As a matter of fact this may not be quite so strange after all, in that the Physics dealt already with such constructions in the past, even though with Space action in a concealed fashion. We just have to recall that the idea of magnetic charge first occurred as based on wave-mechanical considerations (Dirac, 1931). At the time, this was the only possible way to introduce, even though implicitly, the deformation of Space into considerations. Indeed, Dirac used the properties of wave function at the time newly introduced by Schrödinger, and this function is in fact a mean to describe at once events of a part of Space that, at a different level of perception may not be concurrent.

Apart from matters of principle, the equation (30) might prove useful in instances where our experience cannot be straightforwardly extended, for instance in the case of atoms, of interior of stars or at the level of galaxies. By accepting the existence of a magnetic density, ρ_m say, the equation (30) can be written as

(31)

where Φ is a properly chosen function. Using again the General Relativity connection, this is a potential representation of the magnetic field, closer to the way this field is considered when solving the electro-vacuum Einstein field equations (see e.g. Israel, Wilson, 1972). As one might already expect, we do not see the existence of magnetic charge as a pure theoretical coincidence.

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