Free-space Maxwell equations

Up to now, we have examined how the Beltrami vector field relation surfaces in many electromagnetic contexts, featuring predominantly plane-wave solutions (PWSs) to the free-space Maxwell equations in conjunction with biisotropic media (Lakhtakia-Bohren), in homogeneous isotropic vacua (Hillion/Quinnez), or in the magnetostatic context exemplified by FFMFs associated with plasmas (Bostick, etc.). 

This is clearly a Beltrami equation, but what is more amazing is that the field result (88) describes a solution to the free-space Maxwell equations that, in contrast to standard PWS, the electric (E0) and magnetic (Bo) vectors are parallel [e.g., Eo x Bo = 0, where Eo x Bo = i(E0 A Bo)], the signal (group) velocity of the wave is subluminal (v < c), the field invariants are non-null, and as (91) clearly shows, this wave is not transverse but possesses longitudinal components. Moreover, Rodrigues and Vaz found similar solutions to the free-space Maxwell equations that describe a superluminal (v > c) situation [71]. 

Consequently, much like the Beltrami vortex filaments discussed earlier in conjunction with the magnetostatic FFMF, the Beltrami vector relations associated with nonluminal solutions to the free space Maxwell equations, are directly related to physical classical field phenomena currently unexplainable by accepted scientific paradigms. For instance, such non-PWS of the free-space Maxwell equations are direct violations of the sacrosanct principle of special relativity [72], as well as exhibit other counterintuitive properties. Yet, even more extraordinary, these non-PWS are not only theoretical possibilities, but have been demonstrated to exist empirically in the form of the so-called evanescent mode propagation of electromagnetic energy [72-76]. 

The central equations of electromagnetic theory are elegantly written in the fonn of four coupled equations for the electric and magnetic fields. These are known as Maxwell s equations. In free space, these equations take the fonn ... 

To prove the invariance of the cyclic theorem [11-20], it is necessary only to prove the invariance of the free-space Maxwell-Heaviside equations ... 

The basic laws of electromagnetism were summarized in 1865 by Scottish physicist James Clerk Maxwell in a set of four differential equations that yielded a number of practical results almost immediately. For free space, these equations had wavelike solutions that traveled at the speed of light, which was immediately seen to be a form of electromagnetic radiation. Further, it turned out that visible light covered only a small frequency range. Applied scientists soon discovered how to transmit messages by radio waves electromagnetic waves of much lower frequency. 

In order to rmderstand how light can be controlled, we must first review some of tire basic properties of tire electromagnetic field [8], The electromagnetic tlieory of light is governed by tire equations of James Clerk Maxwell. The field phenomena in free space with no sources are described by tire basic set of relationships below ... 

Above we described tire nature of Maxwell s equations in free space in a medium, two more vector fields need to be... 

Maxwell s equations can be combined (61) to describe the propagation of light ia free space, yielding the following scalar wave equation ... 

The polarization vectors vanish in free space, so that in the absence of charge and matter D = e0E, H = —B and the Maxwell equations are ... 

The electromagnetic field in free space is described by the electric field vector E and the magnetic field vector H, which in the absence of charges satisfy Maxwell s equations... 

In U(l) electrodynamics in free space, there are only transverse components of the vector potential, so the integral (158) vanishes. It follows that the area integral in Eq. (157) also vanishes, and so the U(l) phase factor cannot be used to describe interferometry. For example, it cannot be used to describe the Sagnac effect. The latter result is consistent with the fact that the Maxwell-Heaviside and d Alembert equations are invariant under T, which generates the clockwise... 

Maxwell s and wave equations in free space. As discussed elsewhere in the book, one can also question the validity of the conventional representation itself in this chapter, however, we will keep as close as possible to Maxwell s equations. Without any pretension for completeness, some of the issues are as follows. 

The situation with photon propagation in free space is quite diferent. If vacuum is equated to absence of a fluid, what is the support for the waves Of course, particle-like propagation solves the problem, but it (strictly) invalidates Maxwell s equations in vacuum. There is a positive aspect. Since vacuum is nondispersive, all velocities have the same magnitude. 

A second question concerns the existence of longitudinal components of the magnetic field. Maxwell s equations in free space are (completely ) equivalent to two homogeneous uncoupled wave equations for the vector fields E and B. The uncoupled wave equations admit longitudinal components for both fields E and B. However, longitudinal components are prohibited in the conventional interpretation of Maxwell s equations. 

As noted elsewhere [67], Eq. (14) means that the continuity condition does not prohibit the existence of an electromagnetic current density J in free space. It is stressed that Eq. (14) is a mathematical prediction of Maxwell s equations, completely independent of any interpretation. 

For the time being, let us consider the conventional view wave-particle duality. Then, propogation of photon is the same as propagation of electromagnetic field E, B. In free space the charge density is null everywhere, except possibly at the source. The photon is chargeless hence, if Maxwell s equations are applicable to a photon in vacuum, pe = 0 everywhere. This leads to some contradiction. 

In Section III we reviewed our own work on the solutions of Maxwell s equations, which hint to the existence of non-conventional magnetic scalar potentials in free space. The symmetrized set of Maxwell s equations [87] suggests the existence of two novel electromagnetic fields P, N, that lead to the conventional fields E, B. 

In utilizing a complex three-vector (self-dual tensor) rather than a real antisymmetric tensor to describe the electromagnetic field, Hillion and Quinnez discussed the equivalence between the 2-spinor field and the complex electromagnetic field [63]. Using a Hertz potential [64] instead of the standard 4-vector potential in this model, they derived an energy momentum tensor out of which Beltrami-type field relations emerged. This development proceeded from the Maxwell equations in free homogeneous isotropic space... 

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