# A Full Review of the Theory of Electromagnetism

###### Abstract

We will provide detailed arguments showing that the set of Maxwell equations, and the corresponding wave equations, do not properly describe the evolution of electromagnetic wave-fronts. We propose a nonlinear corrected version that is proven to be far more appropriate for the modellization of electromagnetic phenomena. The suitability of this approach will soon be evident to the reader, through a sequence of astonishing congruences, making the model as elegant as Maxwell’s, but with increased chances of development. Actually, the new set of equations will allow us to explain many open questions, and find links between electromagnetism and other theories that have been searched for a long time, or not even imagined.

## 1 Short introduction

The theory of electromagnetism, in the form conceived by J.C. Maxwell, can boast 130 years of honored service. It survived the severest tests, proving itself to be, for completeness and elegance, among the most solid theories. Very few would doubt its validity, to the extent that they may be more inclined to modify the point of view of other theories, rather than question the Maxwell equations. The trust in the model has been strong enough to obscure a certain number of “minor” incongruities and to incite the search for justifications in the development of other theories.

Nevertheless, even if the time-honored equations excellently solve complex problems, they are not able to simulate the simplest things. They are not capable for instance of describing what a solitary signal-packet is, one of the most elementary electromagnetic phenomena. Alternative models have been proposed with the aim of including solitons, but they did not succeed in gaining a long-lasting relevance, because they were based on deliberate adjustments, that, accommodating specific aspects on one hand, were causing the model to lose general properties on the other.

The development of modern field theory, which was very prosperous in the years 1930-1960, has magnified the role of the equations, giving them a universal validation in the relativistic framework. This progress came to a stop, leaving however the impression of being not too far from the goal of compenetrating electromagnetism and gravitation theory.

We are going to make some statements that many readers will certainly consider heretic. We think that the various anomalies, which are present in the model, are not incidental, but consequences of a still insufficient theoretical description of electromagnetic phenomena. Actually, it is our opinion that the flaws are more severe than expected, and therefore, such a fundamental “brick” of Physics needs extensive revision. The review process must be so deep that the entire setting necessitates re-planning from the beginning. On the other hand, if it were just a matter of small adaptations, this revision would have already been made a long time ago.

We shall start to analyse some substantial facts, that at a practical level may be considered marginal, with the aim to evidentiate contradictions. We solve these problems by suitably redesigning the Maxwell equations. This will allow for the construction of a new model, solving all the inconsistencies and achieving the scope of a better understanding of electromagnetic phenomena. In a very natural way, the new approach also leaves the door more than open, to those links and generalizations that were expected to come from the Maxwell equations, but which, although vaguely insinuated, could never be realized in practice.

None of the gracefulness that characterizes the Maxwell model will be lost. The reader who has the patience to follow our arguments through to the end, will discover that all the pieces find their exact place in a global scheme, with due elegance and harmony. We do not wish to say more in this short introduction. The model will be developed step by step, up to its final form, in order to let the reader appreciate the phases of its maturation. The mathematical tools used are classical, and maybe dated. On the other hand, our intention is to examine what would have happened to the evolution of Physics, if our model was taken into consideration, in place of the Maxwell equations. We will elaborate and clarify many important concepts, leaving the path well clear for future developments, not considered here due to lack of time.

## 2 Criticism of the theory of electromagnetism

In this section, we make some fine considerations regarding the evolution of electromagnetic waves, and the way they are modelled by the Maxwell equations. We start by pointing out deficiencies mainly at the level of mathematical elegance. These will reveal other more severe incoherences. In the end, even taking into account the correctness, up to a certain degree of approximation, of the physical approach, our judgement will be rather negative. As a matter of fact, in section 3, with the aim of finding a remedy to the problems that have emerged, substantial revision will be proposed.

From now on, until section 11, we assume that we are in void three-dimensional space. As usual, the constant indicates the speed of light. In this case, the classical Maxwell equations are:

(2.1) |

(2.2) |

(2.3) |

(2.4) |

where the vector field is dimensionally equivalent to an acceleration multiplied by a mass and divided by an electrical charge; while is a frequency multiplied by a mass and divided by a charge.

The above equations are supposed to be satisfied point-wise at any instant of time. Their solutions are assumed to be smooth enough to allow differential calculus. Therefore, discontinuous or singular solutions are not allowed. The equations (2.2) and (2.4) could be considered unnecessary, since they are easily deduced from (2.1) and (2.3) respectively, after applying the divergence operator. Later on, for the reasons that we are going to explain, we will question the validity of (2.2) and (2.4). As a consequence, the entire formulation will lose its credibility.

As far as the evolution of an electromagnetic plane wave (with infinite extent and linearly polarized) is concerned, we have no objections to make. In Cartesian coordinates, a monocromatic wave of this type, moving along the direction of the -axis, is written as:

(2.5) |

In this case, the Maxwell equations are all satisfied point-wise.

The next step is to examine the case of a spherical wave, which is far more delicate. The wave could be generated by an oscillating dipole of negligible size. However, the way the wave is produced and supplied is not of interest to us at the moment, being more concerned with analyzing the geometrical aspects of its evolution at a distance from the source.

Let us denote by the Poynting vector. It is customary to assume that and are orthogonal, and that the wave-front propagates at constant speed , through spherical concentric surfaces. One may argue that perfect spherical waves do not exist in nature. Nevertheless, for the sake of simplicity, we maintain this hypothesis which can be removed later, without modifying the essence of our reasoning.

We are basically confronted with two possibilities. In the first one, the Poynting vector follows exactly the radial direction. This means that and locally belong to the tangent plane to the wave-front. In such a circumstance, as detailed below, we are able to show that (2.1) and (2.3) cannot be both satisfied everywhere. More precisely, it is known that (2.1) and (2.3) are true up to an error that decays quadratically with the distance from the source. Since the intensity of a spherical electromagnetic wave only decays linearly in amplitude, the above mentioned inaccuracy has no influence on practical applications. However, we record a first negative mark.

The second possibility is that, in order to satisfy all the set of Maxwell equations, we loose the orthogonality of the Poynting vector with respect to the wave-front surface. This is a more unpleasant situation, considering that the Poynting vector represents the direction of propagation of the energy flow. The lack of orthogonality between the wave-front tangent plane and the direction of propagation violates the Huygens principle (recall that we are in vacuum), leading to a deformation of the front itself. As we will check later, this results in relevant defects in the development of the wave-shape.

Let us study the problem more in detail, by taking into account the transformation in spherical coordinates:

(2.6) |

with , and large enough. We look for vector fields having the following form:

(2.7) |

where , , are functions of the variables , and (no dependency on is assumed). In (2.7), the first component of the vectors is referred to the variable , the second one to , and the third one to . The unknowns in the system of Maxwell equations reduce from six to three. Choosing a more general form for the fields only complicates the computations, without adding anything to the substance.

We start by observing that equation (2.4) is immediately satisfied. Moreover:

(2.8) |

(2.9) |

(2.10) |

Therefore, the equations in spherical coordinates become:

(2.11) |

(2.12) |

(2.13) |

To avoid discontinuities, we must introduce the following boundary constraints:

(2.14) |

(2.15) |

In the case of the pure radiation field of an oscillating dipole, when is sufficiently large, one usually sets and . Within this hypothesis, the two equations (2.11) and (2.13) are equivalent. They bring us to the general solution:

(2.16) |

where (with ) and are arbitrary functions (the only restrictions apply to their regularity). Among these solutions there is the one corresponding to , which is often present in classical texts (see for instance [2], p.284), being the one with more physical relevance. Nevertheless, we unfortunately note that equation (2.12) is compatible with only when:

(2.17) |

(2.18) |

which manifest singularities at the points corresponding to and . In general, we have the following statement:

(2.19) |

Note that such a strong singularity at the poles cannot be removed only by requiring the wave-front not to be perfectly spherical.

We observe that can be taken in such a way that is vanishing at the poles (for example ), but not in the proximity of them. In addition, we observe that, if is regular with , for any fixed , the points in which the divergence of does not vanish belong to a bidimensional set whose measure is different from zero. For instance, if , we find out that is proportional to , so that this set consists of all points of the sphere of radius , with the exception of the equator. It is certainly true that even if the divergence is not zero, it is negligible when designing, for instance, a device like an antenna. This argument, however, is not going to be valid here, since we would like to carry out an in depth analysis of what is really happening in the evolution of an electromagnetic wave, compared to what the Maxwell theory is able to predict.

Let us now follow a different path and try to find other solutions, of the form given in (2.7), satisfying the set of all Maxwell equations (including ). If we do not want to be singular somewhere, we have to accept that is different from zero. This means that has a radial component, so that the Poynting vector cannot be perfectly radial. We have to better check what happens in this last case.

It is well-known that the the Maxwell equations lead to:

(2.20) |

The above are usually called “wave equations”, but, shortly, we will see that this name is not appropriate. The terminology is correct only if the fields involved are scalar. By deriving (2.11) with respect to time and using (2.12) and (2.13), we arrive at the equation:

(2.21) |

corresponding to the third component of the second equation in (2.20) in spherical coordinates.

It is worthwhile noting that (2.21) is not the wave equation for the scalar field in spherical coordinates, due to the fact that in this framework the Laplacian of a vector field is not the Laplacian of its coordinates (even if only one of them is different from zero). The wave equation for reads as follows:

(2.22) |

This is not a trivial warning, since many texts in electromagnetism erroneously confuse (2.22) with (2.21). Implicitly, we made the same mistake before, when looking for d’Alembert type solutions of the form (2.16), generating, for this reason, solutions not compatible with all the Maxwell equations.

By separation of variables, for any and any , we discover that (2.21) admits the following basis of solutions:

(2.23) |

where and are Bessel functions of first and second kind respectively, while is the -th Legendre polynomial.

A classical reference for Bessel functions is [11]. It is important to note that the solutions given in [11] at page 127, for the scalar wave equation in spherical coordinates, differ from the ones shown in (2.23). The reason is that the functions in [11] (having in place of ) are those solving (2.22), which is not the vector version of the wave equation, as we already mentioned.

For example, if we have (see [11], p.54):

(2.24) |

In order to understand what the solutions in (2.24) look like, it is standard to introduce some approximation. Thus, for and large, by taking the combination , up to multiplicative constants, it is possible to get asymptotically the monocromatic solution (compare to (2.16)), up to an error which decays quadratically with . Once again, one ends up with something similar to a travelling wave, although some cheating has been necessary (that is equivalent, in the end, to replacing once again (2.22) by (2.21)).

On the other hand, suppose that is evaluated exactly as linear combination of the functions in (2.23). Then, one recovers e by (2.12) and (2.13), through time integration. Successively, it is possible to compute the Poynting vector:

(2.25) |

which has, as expected, a non radial component. Now, let us fix and study the behavior, by varying , of the two components of . In particular, we are interested to see what happens near the poles ( or ). We start by noting that, for any , the term tends towards a finite limit for or (recall that ). Therefore, according to (2.23), the first component in (2.25) behaves as near the poles. It is a matter of using known properties of Legendre polynomials, in particular the differential equation:

(2.26) |

to check that the second component in (2.25) behaves as near the poles.

We are ready to draw some preliminary conclusions. Let us note that finally , hence all the Maxwell equations are satisfied. As already remarked, it has been necessary to keep the nonradial component of . Surprisingly, for any fixed , such a nonradial component prevails on the radial one, when approaching the poles. This implies that the shape of the wave-fronts does not resemble a sphere, but rather a kind of doughnut with the central hole reduced to a single point. The parts of the fronts corresponding to the internal side of the doughnut, progressively stratify along the -axis. We do not see a chance of recognising any sort of Hyugens principle here. This is not what we would call a travelling wave. It may be argued that this behavior is due to the influence of the source located at . But, if we stop the source, the wave-fronts already produced continue to develop. If their motion is ruled by the Huygens principle, the hole should fill up quickly, and each front should transform to something rounded which is almost a perfect sphere. The problem is that, during this smoothing process, the vector fields and are not compatible with both the constraints and . The clue is that “wave equations” in vector form have nothing to do with real waves.

Some mild analogy between the Maxwell equations and the eikonal equation, governing the movement of the fronts, was devised a long time ago. The equivalence is valid within the limits of geometrical optics (see [5], p.110). In spite of this, examining the behavior of the fronts, our impression is that their natural evolution is in conflict with all restrictions imposed by Maxwell equations. This statement will be clearer as we proceed with our study. The right connections with the eikonal equation will be defined in section 10.

The different situations analyzed up to now are summarized in figure 1, which should clarify our point of view: either we keep the singularity at the poles (manifested by infinite amplitude of the fields or strong geometrical distorsion), or we allow the divergence of the electric field to be different from zero. To sustain this proposition, let us collect other elements.

Some confusion usually arises when one tries to simulate the evolution of a “fragment” of wave. We examine the case of the plane wave given in (2.5). For any fixed , we can cut out a region in the plane determined by the variables and , and follow its evolution in time. For simplicity, can be the square . Inside we assume that the electromagnetic fields evolve following (2.5), in full agreement with Maxwell equations. Outside , the fields and are supposed to vanish. The question is understanding what happens at the boundary of . It is not difficult to realize that, on the sides and , and become singular, producing concentrated distributions. Similarly, on the two sides and , the quantities and present singularities.

Some readers may complain because discontinuities of the fields may not exist in nature. Commonly, the right way to proceed is to consider a thin layer around , where the solution given by (2.5) smoothly decays to zero. Then, one lets the width of the layer tend towards zero. This in general allows us to determine special relations to be satisfied on (in place of the Maxwell equations, which are meaningless there). Unfortunately, the procedure presents some drawbacks. Let us first assume that the wave-fronts shift along the -axis maintaining their squared shape. We also assume that the fields and are orthogonal and smoothly decaying to zero in a neighbourhood of (like for instance in figure 2). Our conjecture is that there exists at least one point where Maxwell equations are not all satisfied, because and cannot both be zero at the same time. Actually, examining figure 2, we discover that there are infinite points where either or . These points form a set whose area is different from zero. We are free to try other configurations by modifying the orientation of the vector fields at each point near , but we always arrive at the same conclusion: some rule of Physics breaks down when approaching . Now, the question is: if we do not know what the governing rules are in the layer around , how can we go to the limit for the size of the layer tending to zero?

Another possibility is that the wave-fronts, due to the strong variation of the fields near the boundary of , are forced to bend a little. The electromagnetic fields are no longer on a plane, so we could probably find out the way to enforce all the Maxwell equations. However, this implies that the Poynting vectors are not parallel to the -axis anymore. Thus, the shape of is going to be further modified during the evolution. A little diffusion is bearable, yet our impression is that the wave-fronts would rapidly change their form. The more they bend, the faster they produce other distorsion. This is in contrast for instance with the fact that neat electromagnetic signals, of arbitrary transversal shape, reach our instruments after travelling for years between galaxies. The only acceptable rule is that all the Poynting vectors must stay orthogonal to the fronts and parallel to the actual direction of movement; if this does not happen the wave quickly deteriorates, fading completely.

To prove what we claimed before, we show using very standard arguments that it is not possible to construct solutions to Maxwell equations, having finite energy and travelling unperturbed at constant speed along a straight-line. We assume that the speed is and the straight-line is the -axis. Without loss of generality, such a signal-packet is supposed to be of the following type:

(2.27) |

where is a bounded function and all the components , , , , , are zero outside a bidimensional set . It is not difficult to check that (2.1) and (2.3) only hold when and are identically zero. Actually, it is straightforward to discover that e must be constant (and the sole constant allowed is zero). Then, one finds out that , , , must be harmonic functions in . Since they have to vanish at the boundary, they must vanish everywhere.

Due to the above mentioned reasons, solitonic solutions are not described by the classical theory of electromagnetism. Efforts have been made in the past to generalize the Maxwell model, in a nonlinear way, in order to include solitons. Just to mention an example, the Born-Infeld theory (see [4]) predicts the existence of finite-energy soliton-like solutions (that have been successively called BIons). These last equations have no relation with the ones we are going to develop in this paper. However, they point out the necessity of looking for nonlinear versions of the model. We will come back to the subject of solitary waves in section 5.

In many applications, a standard approach is to reconstruct the bidimensional profile of the fields inside with the help of a truncated Fourier series. This is accomplished by a complete orthogonal set of plane waves, each one carrying a suitable eigenfunction in the variables and . We must pay attention, however, to the fact that these eigenfuntions are of the periodic type. Therefore, they reproduce the same profile, not only inside , but in a lattice of infinite contiguous domains. In this way, the represented solution turns out to have infinite energy. Considering only one of these profiles, thus forcing to zero the solution outside , unavoidably brings us again to a violation of the Maxwell equations near the boundary of . Some clarifying comments on this issue can be found in [8], p.42.

We recognize that the techniques based on Fourier expansions provide excellent results in many practical circumstances, as for example the study of diffraction. Nevertheless, in this last case and in the ones treated before, it is necessary to adapt the solutions, introducing some approximation, if we want them to correspond to the real phenomenon. Indeed, these adjustments are within the so-called limits of the model. Hence, we could just stop our analysis here, with the trivial (well-known) conclusion that the Maxwell model is not perfect. We believe instead that the discrepancies pointed out are not just imperfections, but symptoms of a more profound pathology affecting the theory of classical electromagnetism.

What we learn in these pages is that there are plenty of simple and interesting phenomenon, which are inadequately explained by the Maxwell model, because the equations impose too many restrictions. Consequently, the idea we shall follow in the next section is of weakening the equations, with the aim of widening the range of solutions.

## 3 Modified Maxwell equations

The demolition process is finished, now it is time to rebuild. To begin, we propose the following model:

(3.1) |

(3.2) |

(3.3) |

that will be further adjusted in the subsequent sections. The norm is the usual one in , i.e.: . We define . Note that, when , the direction of is not determined (see also the comments at the end of section 7). The vector is supposed to be adimentional (or, equivalently, is multiplied by a constant, equal to 1, whose dimension is the inverse of the dimension of ). Consequently, is a velocity vector.

As the reader may notice, the “awkward” relation has been eliminated. It is also evident that in all the points in which , we find again the classical Maxwell system. This states that the solutions of (2.1)-(2.2)-(2.3)-(2.4) are also solutions of (3.1)-(3.2)-(3.3). Therefore, the replacement of (2.1)-(2.2) by (3.1) brings us to the property we wanted, that is the enlargement of the range of solutions.

Afterwards, we have to understand and justify what is happening from the point of view of Physics. Let us recall that we are in empty space, and that there are no electrical charges or masses anywhere. We are only examining the behavior of waves. In spite of that, we pretend that there may be regions where . The situation is not alarming, since we checked that the condition is quite frequent in the study of waves. Anyway, such a hypothesis is acceptable, as long as it is coherent with the basic laws of Physics. First of all, we observe that the equation (3.1) has been obtained by adding a nonlinear term to (2.1). The term has strong analogy with the corresponding one of classical electromagnetism, appearing on the right-hand side of (2.1) as a consequence of the Ampère law, and due to the presence of moving charges. In fact, by setting , the vector can be assimilated, up to dimensional constants, to an electric current density. Thus, even if in our case there are no real charged particles, we have to deal with a continuous time-varying medium, consisting of infinitesimal electrical charges, living with the electromagnetic wave during its evolution. Moreover, the added term does not compromise the theoretical study of a functioning device like an antenna, since, at a certain distance, the quantity is negligible.

By taking the divergence of (3.1), we get a very important relation:

(3.4) |

which is, actually, the continuity equation for the density . The equation (3.4) testifies to the presence of a transport, at the speed of light, along the direction determined by . Hence, something is flowing together with the electromagnetic fields; something that later, in sections 9 and 10, will be compared to a true mechanical fluid. On the other hand, this was also the interpretation at the end of the 19th century, before the theory of fields was rigorously developed. The fluid changes in density, but preserves its quantity, as stated by the continuity equation. It is extremely significant to remark that this property comes directly from (3.1), so it is not an additional hypothesis. In section 12, based on the density , we will construct a mass tensor that, due to (3.1), can be perfectly combined with the standard electromagnetic energy tensor. The skilful reader has already understood that this will allow us to find the link between electromagnetic and gravitational fields.

We are now going to collect other properties about the new set of equations. Considering that is orthogonal to both and , a classical result is obtainable:

(3.5) |

where the quantity , up to a multiplicative dimensional constant, is related to the energy of the electromagnetic field. Thus, the nonlinear term in (3.1) is not disturbing at this level, and the Poynting vector preserves its meaning.

By noting that and that , we get another interesting relation:

(3.6) |

Finally, one has:

(3.7) |

from which we deduce the following second-order vector equation with a nonlinear forcing term:

(3.8) |

that generalizes the second equation in (2.20). We are sorry to announce that the “wave” equations for the fields and are no longer true. On the other hand, it has emerged in section 2 that, in vector form, they are only a source of a lot of trouble.

In the classical Maxwell equations the role of the field can be interchanged with that of field . This is not true for the new formulation. We will later see, in section 9, how to solve this problem. For the moment, we keep working with (3.1)-(3.2)-(3.3), just because the theory will be more easy. In the coming sections 4 and 5, we will see how elegantly it is possible to solve the problems raised in section 2.

## 4 Perfect spherical waves

In the case of a plane wave of infinite extension, for both the Maxwell model and the new one, we are able to enforce the condition and realize the orthogonality of the Poynting vectors with respect to the propagation fronts. Concerning a “fragment” of plane wave, the classical method runs into problems. However, with the new approach the situation radically improves. Let us see why.

With the same assumptions of section 2, let be the square . We already noted that, on the sides and , the quantities and become infinite. Nevertheless, when substituted into equation (3.1), they come to a difference of the type . The two singular terms reciprocally cancel out, leaving a finite quantity, so that the equation has a chance to be satisfied. To show this, we can create a layer around the boundary of . Then, without developing singularities, we pass to the limit for the width of the layer tending to zero. The trick now works, because, in contrast to the classical Maxwell case, equation (3.1) can be satisfied exactly in all the points, since it is compatible with the condition . In the limit process we can also guarantee that the Poynting vectors remain parallel to the -axis. Therefore, the fragment does not change its transversal shape. Explict computations will be carried out in section 5, in the case in which is a circle.

For example, the situation represented in figure 2 is perfectly allowed for by our equations, except near the lower and the upper sides. Actually, on the sides and , given that and are singular, we still have problems (clearly equation (3.2) and (3.3) are not true). Similar problems are encountered by modifying the polarization of the wave. These troubles will be solved in section 9, by unifying (3.2) and (3.3) in a single equation similar to (3.1), in such a way that the roles of and are interchangeable.

The case of a spherical wave is very interesting. Let us consider the transformation of coordinates given in (2.6). Let us also suppose that the fields are given as in (2.7), with , , not depending on . We have:

(4.1) |

(4.2) |

where is the sign of .

The new equations in spherical coordinates become:

(4.3) |

(4.4) |

(4.5) |

These expressions may seem rather complicated, but there is nothing to be afraid of.

To avoid discontinuities, we also impose the boundary conditions (2.14) and (2.15). Now, by choosing and , one obtains:

(4.6) |

Therefore, one gets:

(4.7) |

(4.8) |

(4.9) |

Once again, the first and the last equations are equivalent, providing the general solution (2.16). Anyway, this time, thanks to the nonlinear corrective term of (3.1), the second equation is compatible with . We are not obliged to choose , in order to enforce the condition , because this constraint is no longer required. The conclusion is that perfect spherical waves are admissible with the new model. The functions e may be truly arbitrary (the only restriction is ). Continuing with our analysis, we will construct later infinite other solutions which are unobtainable with the classical Maxwell model.

In the perfect spherical case, the Poynting vector only has the radial component different from zero. As expected, this component has a constant sign (even if and oscillate). Since the set of equations is of a hyperbolic type, we can introduce the characteristic curves. In the example of the spherical wave, such curves are semi straight-lines emanating from the point , and the vector is aligned with them. The nonlinear term introduced in (3.1) does not adversely affect the behavior of the wave, because, with and , the corresponding equations (4.7) and (4.9) are linear. Therefore, the superposition principle is still valid. Any piece of information, present at the boundary of the sphere of radius , propagates radially at the speed of light, without being disturbed (except by the natural decay in intensity). The nonlinear effects of the model are latent. They show up when we try to force, with some external solicitations, the Poynting vector not to follow the characteristic lines. This circumstance will be taken into account in sections 7 and 8.

In a very mild form, we can state that the divergence vanishes, by observing that, for any :

(4.10) |

that is has zero average when integrated over a period of time. Nevertheless, in section 13, we will get an astonishing result. We will see that an electromagnetic wave produces, during its passage, a modification of space-time. In the new geometry, the 4-divergence of the electric field is zero. This could make it difficult, or even impossible, to set up an experiment that emphasizes the condition at some point. The measure could be affected by the modified space-time geometry in such a way the condition cannot be revealed.

Among the stationary solutions we find:

(4.11) |

as well as:

(4.12) |

with , , arbitrary constants. In particular, we recognize the classical stationary electric field:

whose divergence is zero for any . Unfortunately, most of these solutions show singularities.

Due to the nonlinearity of the equations, the study of the interference of waves looks quite complicated. As long as the waves are such that and (as in the plane case) there are no problems, since the nonlinear terms do not actually activate. For waves of different shape, the situation may be truly intricate. In first approximation, the nonlinear effects should attenuate faster than the amplitude of the waves. Thus, at a certain distance, these anomalies may not normally be observed. Although we do not wish to discuss it here, the subject is of crucial relevance and deserves to be studied in more detail.

## 5 Travelling signal-packets

In this section, it is convenient for us to express our new set of equations in cylindrical coordinates. After taking , we assume that the fields are of the form , , where the first component is referred to the variable , the second one to the variable and the third one to the variable . Moreover, for simplicity, the functions , and will not depend on . In cylindrical coordinates, the counterparts of equations (4.3)-(4.4)-(4.5) are:

(5.1) |

(5.2) |

(5.3) |

By setting , we can easily find solutions when and do not depend on . In this case one has and . From (5.1) and (5.3) it is easy to get the equations:

(5.4) |

whose solutions are related to Bessel functions.

Anyhow, extremely interesting solutions in cylindrical coordinates turn out to be the following ones:

(5.5) |

Note that the divergence of is equal to , so that it is different from zero, unless is proportional to . The functions and can be arbitrary (to guarantee the continuity of the vector fields, we only impose ). The relations in (5.5) give raise to electromagnetic waves shifting at the speed of light along the -axis. If and vanish outside a finite measure interval, for any fixed time the wave is constrained inside a bounded cylinder. The packet travels unperturbed for an indefinite amount of time. The corresponding field is perfecly radial and the vector is parallel to the -axis.

Given , suppose that is zero for . Suppose also that , in a neighborhood of has a sharp gradient. It is evident that the vector remains bounded even if we let the derivative of tend to at . Therefore, as we anticipated at the beginning of section 4, we can give a meaning to equation (3.1), even if is a discontinuous function in .

We can get a transport equation for the unknown by using the equation (3.4), i.e.:

(5.6) |

which, thanks to the fact that is a constant field, takes the simplified form of:

(5.7) |

with the sign depending on the orientation of .

We recall that the Maxwell equations do not allow for the existence of solitary waves, as the ones we have introduced right now. Therefore, here we obtained another important result.

The energy of these solitary waves is obtained by integrating the energy density, given by: . Thus, one gets:

(5.8) |

Suppose that, at an initial time , the electromagnetic fields are assigned compatibly with (5.5). The vector turns out to be automatically determined, then the wave is forced to move in the direction of at speed . There are no stationary solutions, unless is constant. But, in this last case, due to (5.8), the energy is not going to be finite. The wave-packets take their energy far away, with no dissipation, until they react with other waves or more complicated structures (like, for instance, particles).

Let us study more closely the expressions given in (5.5). Assume that, for , the function is non negative, and that . If the function has a constant sign, we distinguish between two cases, depending if the sign is positive or negative (see figure 3). The sign determines the “orientation” of the vector (note however that is not exactly parallel to the -axis, despite what is shown in figure 3). Then, we have subcases, depending whether is directed toward the -axis or not. In conclusion, two possible types of solitary waves can occur, depending on the orientation of the electric field (external or internal). These will be denoted by and , respectively. In figure 3, indicates the direction of motion. Of course, could also have a non-constant sign. In this case, the corresponding wave can be seen as a sequence of waves of type and , shifting one after the other.

In vacuum, the electromagnetic fields at rest, are assumed to be identically zero. During the passage of a soliton, the calm is momentarily broken only at the points “touched” by the wave. The information shifts, but does not irradiate. Thus, also if the wave-packet displays a negative or positive sign, depending on the orientation of , this is not in relation to what is usually called electric charge. Hence, as long as the cylinders containing two different solitons do not collide, they can cross very near without influencing each other. On the contrary, we expect some scattering phenomena, through a mechanism that will be studied in section 8.

If we place a mirror parallel to the -axis, at some distance from it, the reflected image of the travelling wave-packet will be the same wave-packet shifting in the opposite direction (because changes sign, while maintains its orientation). This disagrees with our common sense. In other words, equation (3.1) does not preserve plane symmetries. The same is true for the Maxwell equations. In both cases we have no elements to decide the correct sign of the vector product (left-hand or right-hand). As a matter of fact, without modifying the equations, a change of the sign of can be compensated for by a change of the sign of the electric (or the magnetic) field. To learn more about this problem we need to study the interactions between waves and matter. Hence, for the moment, we have no sufficient information to understand from which part of the mirror is our universe. An answer to these crucial questions will be given in section 8.

In cylindrical coordinates, we can find many other solitonic solutions. Here is another example:

(5.9) |

with and . In order to fulfill the condition , it is necessary to impose:

(5.10) |

Note that and . Except for the condition (5.10), the functions and are arbitrary, so that the new solutions are very general. Actually, they include the previous ones (take and ). In section 9, we will remove the condition , allowing for the existence of even more solutions. We can force and to vanish outside a bidimensional domain . Again, this plane front, modulated by the function , travels along the -axis at the speed of light.

We may reasonably expect that these solitary solutions are modified, or even destroyed, when they encounter another external electromagnetic field. In fact, the equations being nonlinear, the superposition principle does not hold, in particular if the motion is disturbed in a way that is in contrast to the natural evolution along the characteristic curves. As far as we know, there are no documented experiments evidencing these facts. In section 11, we instead examine the behavior of solitary waves under the action of gravitational fields.

An electromagnetic radiation can be suitably considered as an envelope of solitary waves, travelling in the same direction. If, transversally, these solitons are of infinitesimal size, they can be compared to “light rays”. This observation clarifies how a wave can be viewed at the same time as a whole electromagnetic phenomenon and as a bundle of infinite microscopic rays.

And then there are photons. They are also pure electromagnetic manifestations, but, unfortunately, they are not modelled by the Maxwell equations. Modern atomic and subatomic physics would not exist without photons, yet they find no place in the classical theory of electromagnetism. This is an unpleasant gap. Although physicists are acquainted with this dualism, the general framework remains blurred. From our new standpoint, we contend that the photons observed in nature have very good chances to be modelled by the equations introduced here. As a matter of fact, we have enough freedom to be able to build solutions (no matter how complicated) resembling real photons. We can assign a frequency to them, longitudinally or transversally. Then, we know that they must move at the speed of light and can have finite energy, given by the energy of their electromagnetic vector fields. They can be “positive” or “negative” without being electrical charges. Even with no mass at rest, their motion can be distorted by gravitational fields (see section 11). A concept of spin can be also introduced (see section 15). If what we are proposing here is a new functioning model for electromagnetism (and we will collect other evidence supporting this hypothesis), then it explains why photons can be self-contained elementary entities and electromagnetic emissions at the same time. In this case, a first link between classical and quantum physics is set forth.

## 6 Lagrangian formulation

In order to recover the equations (3.1)-(3.2)-(3.3) from the principle of least action, we follow the same path bringing to the classical Maxwell equations. Thus, we introduce the scalar potential and the vector potential , such that:

(6.1) |

From the above definitions we easily get the equations: and . The third equation (3.1) is going to be deduced from the minimization of a suitable action function.

Let us first note that, by taking the pontential equal to zero and setting , one obtains:

(6.2) |

This allows us to infer that is orthogonal to and that (see also [10], p.126).

Successively, for and between 0 and 3, we introduce the electromagnetic tensor:

(6.3) |

where and . Explicitly, we have:

(6.4) |

with and . Replacing by , one gets instead the contravariant tensor :

(6.5) |

Therefore, up to multiplicative constants, the action turns out to be (see for instance [9], p.596):

(6.6) |

where, summing-up over repeated indices, the Lagrangian is . As customary, the variations are functions , having compact support both in space and time (between two fixed instants).

With well-known results, one obtains:

(6.7) |

Imposing , the corresponding Euler equations are exactly the standard Maxwell equations. As a matter of fact, due to the arbitrariness of the variations , one recovers: (for ), that is equivalent to write and .

Let us now introduce a novelty. We require that the variations are subjected to a certain constraint, so that the conclusions are going to be different. Actually, we impose the condition: