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IV. History: The Electrodynamics

It can be recognized from the previous section that the only Science reflecting the constitutive laws with purely material properties is Electrodynamics. Just because here the constitutive laws have been looked upon arbitrarily, we need to insist a little on the history of concepts in the light of the above logic. In Electrodynamics, the stress tensor is constructed based on the electric and magnetic fields –  and – and is known as Maxwell stress tensor. To update the notation, we write it again here in terms of electric intensity and magnetic induction as

In this context the parameters λ and μ are the electric permittivity and magnetic permeability respectively. One of the eigenvalues and its associated eigenvector are given by

The other two eigenvalues are with eigenvectors in the plane of  and . The vector is usually known as the Poynting vector of the electromagnetic field (Poynting, 1884). This vector has come forward from the necessity of describing the energy transfer as due to the electromagnetic field in vacuum, starting from the Maxwell equations. In this situation it does, indeed represent an energy transfer, allowing one to associate a momentum to radiation. However, in general, the transfer vector is given by equation (III.13), i.e

and the continuing argument (Campos, Jimenez, 1992 and the citations therein) about the meaning of the Poynting vector does not seem to make much sense in this context. First of all, everything about electromagnetic field is referring, according to previous results, only to intrinsic (self) properties, and the space-time equations of evolutions have to obey this requirement. Thus everything external comes some other way than simply by currents, which are the expression of a space-time theory. Secondly, the transfer responsible for the conservation of energy is not given by the Poynting vector, but by the vector . The two vectors are not identical. As long as we base our reasoning on the Maxwell equations in vacuum, the Poynting vector is always a momentum vector. It is only when we get away from the Maxwell formalism that we may find the alternative idea that the transfer of energy has a component in the plane of vectors and , depending on the relative velocity of the vacuum. As far as we know, this fact has been singularly noticed by Cunningham (Cunningham, 1909). Its consequences have been studied (Cunningham, 1915, 1921), from the very same point of view as that of the present work. The work of Cunningham’s does not seem to have been drawing much attention.

One may understand from the previous paragraphs the fact that we have sufficient reason to take part against directly concluding on the nature of electromagnetic quantities starting from Maxwell equations. Not because many concepts related to this formalism are still under debate, but the interesting fact is that we can create the 4×4 Maxwell stress tensor simply by bordering the Maxwell tensor (IV.01) with the eigenvector (IV.02) and its eigenvalue:

Thus a 4D theory should be a derived theory, it cannot be fundamental. This fact generates the idea that any symmetric 4×4 tensor would do in representing the Maxwell tensor, provided it satisfies four identities establishing the above requirement that the fourth line is given by the eigenvector of the 3×3 remaining matrix and its corresponding eigenvalue (Katz, 1964). A question is now in order: if the 4D theory is a derived theory, then what is the fundamental fact in the light of above results? In this respect, we consider a tremendous intellectual achievement the fact proved recently by Coll and collaborators (Coll, Llosa, Soler, 2002) that any 3D metric can be locally generated from any other 3D metric of constant curvature by a process of deformation involving a tensor like (III.01). In other words, if the Space is in transformation, as long as its metric reflects this by a deformation (universal deformation in the terminology of Coll and collaborators), we cannot quantify it but by a tensor reflecting our concept of uncontrollability. This realization is liable to contain in their utmost generality concepts like that of displacement current or gauging, which, in a space-time theory are the only working assumptions. In the problem of gauging, our limited experience shows that the transformation (III.09) must play an essential part. That transformation has been discovered by Page and Adams (Page, Adams, 1940).

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