modern helmholtzian electrodynamics as a covering ... · modern helmholtzian electrodynamics as a...

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R. Smirnov-Rueda, A.E. ChubykaloDepartamento de Matematica Aplicada, Facultad de Informatica,Universidad Complutense de Madrid28040 Madrid, Spain &Escuela de Fisica, Facultad de MatematicasUniversidad Autonoma de ZacatecasApartado Postal C-58098068 Zacatecas, Mexico

Modern Helmholtzian Electrodynamics as a Covering

Classical Electromagnetic Theory

Contents

1. Introduction 540

2. Historical background of classical electrodynamics 540

3. Conventional and alternative interpretations of Hertz’s crucial experiments 543

4. Arguments in favour of alternative Helmholtz-type theory of electromagnetism 547

5. An example of ambiguity of Faraday-Maxwell’s concept of local (contact) field in theconventional electromagnetic theory 548

6. Mathematical foundations of Helmholtz-type electrodynamics 549

7. Mathematical inconsistencies in the formulation of Maxwell-Lorentz equations for one chargesystem 552

8. Reconsidered Maxwell-Hertz theory and relativistically invariant formulation of generalizedMaxwell’s equations 556

9. Analysis of classical difficulties and the Hamiltonian form of generalized Maxwell’s equations560

10.Non-radiation condition for free electromagnetic field 563

11.Conclusions 565

Abstract

The discussion of the historical background of the 19th century electromagnetic theory has shown that from the standpoint

of modern scientific method, the Hertz experiments on propagation of electromagnetic interactions cannot be considered

as conclusive at many points as it is generally implied. It has been found that alternative Helmholtz’s electrodynamics did

not contradict Hertz’s experimental observations. Mathematical analysis of the conventional electromagnetic theory showed

that numerous ambiguities are related to the treatment of time behaviours. Those difficulties turn out to be cleared up by

distinguishing between implicit and explicit time dependencies. It provides self-consistency for mathematical description of

electromagnetic theory by advocating the explicit use of full time derivatives in the mathematical formulation of Maxwell’s

equations. This approach covers conventional electromagnetic theory based on partial time derivatives. The covering theory

is showed to possess all necessary relativistic invariance properties for inertial frames of references. The idea of non-

local interactions is enclosed into the framework of Helmholtzian electromagnetic theory as unambiguous mathematical

feature. In this work we make a point that Helmholtz’s foundations and modern Helmholtz-type electrodynamics recently

developed by the authors and reviewed here promise, in general, an altogether more logical solution to self-consistent

classical electrodynamics and its reconciliation with quantum mechanics.

539

R. Smirnov-Rueda, A.E. Chubykalo

1. Introduction

There is no need to argue that classicalelectrodynamics is one of the corner-stones ofmodern physics. At first stages the development ofelectromagnetic theory proceeded in accordance withNewtonian traditional outlook on the world basedon the instantaneous action-at-a-distance (IAAAD).Faraday’s discovery of induction highlighted limitedvalidity of IAAAD in describing electromagneticphenomena. A notion of local (contact) field wasproposed by Faraday not to incorporate but toreplace Newtonian IAAAD. As a result, the state ofelectrodynamics in the middle of the 19th centurywas characterised by opposition of a few alternatives.Hertz’s discovery of electromagnetic waves played acrucial role in choosing a definitive modern version ofclassical electrodynamics. However, nowadays only afew researchers are aware of the fact that Helmholtz’selectrodynamics was also consistent with Hertz’sexperiments as well as with all known 19th century’sexperimental data (see, for instance, [1]). Why andhow Maxwell’s electrodynamics became a favouriteneeds to be revisited in detail.

2. Historical background ofclassical electrodynamics

In order to appreciate the difficulty and theimportance of the task undertaken by Hertz inhis experimental investigations, it is worth recallingthe uncertain and highly controversial state ofelectrodynamics at that time. Hertz himself wastrained in the research tradition of the Berlin schoolheaded by Helmholtz who from the middle 1860’s, hadsought to clarify existing principles in electromagnetictheory and to reach a consensus between the twomajor directions in electromagnetic research of thattime, namely, Newton’s instantaneous action-at-a-distance concept as used by Weber, and Faraday’scontact action concept as developed by Maxwell. Bythe time of Helmholtz’s first attempt at reconciliation(1870), the theoretical schemes of Weber and Maxwellhad successfully incorporated all previously well-established descriptions and empirical facts, such asthe electric potential theory (electrostatics), Ampere’smagnetostatics and Faraday’s theory of induction.

Weber developed his theory (1848) in accordancewith the Newtonian program, which prescribedthat all forces between pairs of particles shouldbe radial, acting directly through space (i.e.along the line between the particles) without anyobservable material mediator. Restriction on thisradial description of electromagnetic forces camefrom Ampere, who understood that instantaneitymeans no delay, hence no aberration. Any aberrationattending the finite propagational velocity ofinteractions would imply non-radial forces. However,

the existence of radial forces as a basic assumption ofinstantaneous action-at-a-distance (IAAAD) theorieswas confirmed by Ampere experimentally to thedegree of accuracy available at that time. Thus,electric and magnetic interactions were thought tobe completely analogous to gravitational attractionswhich, according to astronomical observations, hadno detectable aberration and always acted along theline joining the simultaneous positions of two bodies.Bearing in mind the accuracy-limit of astronomicaldata at that time, Laplace (1799) [2] calculated,measuring possible aberration effects, that the speedof propagation of gravitational interaction had toexceed at least eight orders of magnitude the speedof light. (Recently, Van Flandern claimed to rise thatlimit two orders of magnitude [3]: ”...Perhaps contactbinary star systems place the tightest constraints ona lower limit to the speed of propagation of gravity.Unless the speed of gravity exceeds 1010 times thespeed of light, such systems would fly apart within afew hundred years...”).

Laplace’s respectable conclusion gave majorsupport to the validity of the IAAAD concept, leavingopen the question of the physical cause of gravity.On this latter subject, a conceptual arm-wrestlewas initiated between supporters and detractors ofIAAAD. Newton himself had already thought thatsome physical mediator must exist [4]. ”It is absurd,”he said, ”to suppose that gravity is innate and actswithout a medium, either material or immaterial”.Many eminent scientists like Laplace explained thephenomenon as due to an ’impulsion’ of someimmaterial fluid, but did not find it appropriate atthe time to search for a reliable physical explanationof the cause of interaction, and so they bequeathedthis task to future generations. ”There is no needat all,” declared Laplace (1796) [5], ”to posit vaguecauses, impossible to submit to analysis, and which theimagination modifies to its liking in order to explainthese phenomena”.

At that stage, it is not surprising that for manyof the adversaries of IAAAD these suggestions of’immateriality’ could not be dissociated from spiritual,religious and other non-scientific notions. This pathof reasoning led them to conclude that if theIAAAD concept did not imply any material mediator,then it did not imply any physical mediator at all(compare it, for instance, with nowadays explanationof interaction mediated by material as well as bynon-material virtual particles). Thus, in their opinionIAAAD became inconceivable from the point of viewof common-sense logic. This prepared the groundfor the reappearance of an alternative Aristotelianconcept, namely, that of action by local contact (orcontact action). Thought-provoking examination ofthat resurgence and related modern topics on thedebate between far- and local actions can be foundin a well-written book by Graneau’s [6].

540 "Электромагнитные Явления", Т.2, №4 (8), 2001 г.

Modern Helmholtzian Electrodynamics as a Covering Classical Electromagnetic Theory

In its modern form, the concept of local (contact)field was reintroduced by Faraday and it appearedto give a more realistic physical description ofthe phenomena, based on the causality of localinteractions. This way of reasoning attracted Maxwell,who tried to work out his own comprehensive fieldtheory based on Faraday’s concept. However, Maxwellhimself was aware of the provisional, ’scaffolding’status of his rationale. Moreover, he encounteredsome conceptual difficulties, since he had incorporatedall the basic IAAAD results such as electrostaticsand magnetostatics without any modification. Withregard to the lines of force treated by Faradayas the representation of a material field, Maxwell’sown position was still undefined, but he cautiouslydealt with them as if they were lines of flow of anincompressible, imaginary fluid. As Maxwell stated [7]:”The substance here treated of must not be assumedto possess any of the properties of ordinary fluidsexcept those of freedom of motion and resistance tocompression. It is not even a hypothetical fluid whichis introduced to explain actual phenomena. It is merelya collection of imaginary properties which may beemployed for establishing certain theorems in puremathematics in a way more intelligible to many mindsand more applicable to physical problems than that inwhich algebraic symbols alone are used”.

Consequently, being in a static limit,mathematically equivalent to older IAAAD theories,the status of contact field theory could not have beenconsidered free of ambiguity [8]. In other words, inthis static limit, electric and magnetic fields behavedvery much as flows of an ideal, incompressible fluid,in which case they were indistinguishable fromIAAAD. These uncertainties in Maxwell’s originaltheoretical scheme were later summarised clearly andconcisely by Hertz, who wrote [9]: ”...Maxwell’s ownrepresentation does not indicate the highest attainablegoal; it frequently wavers between the conceptionswhich Maxwell found in existence, and those at whichhe arrived. Maxwell starts with the assumption ofdirect actions-at-a-distance; he investigates the lawsaccording to which hypothetical polarisations of thedielectric ether vary under the influence of suchdistance- forces; and he ends by asserting that thesepolarisations do really vary in this way, but withoutbeing actually caused to do so by distance-forces. Thisprocedure leaves behind it the unsatisfactory feelingthat there must be something wrong about either thefinal result or the way which led to it.”

As an approach to clarifying these uncertainties,let us examine the essential distinction betweenthe conceptual foundations of IAAAD and thoseof contact-field doctrines. At the beginning ofthe nineteenth century, the possibility of a finalexplanation of IAAAD as the basis of gravitationaland electric forces had not been completely ruled out.Moreover, due to the rapid development of theoretical

hydrodynamics, the attitude towards IAAAD changedfrom the summary rejection of its unphysical statusto an awareness of a deep similarity between thepotential function and the velocity-field of a fluid. Ithad been realised that the main difference betweenIAAAD and Faraday’s ‘field’ was the fact that inIAAAD a potential need not necessarily describe amaterial property of anything, whereas for Faradayit was the property of a material substance whichcould be observed in ways familiar to ordinary mattersuch as, for instance, liquids and gases. Like anyother material substance, therefore, Faraday’s ‘field’could be regarded as something movable with positivekinetic energy and, therefore, detectable empirically.This suggests that an important criterion, as Hesseputs it [10]: ”...in deciding whether or not a fieldis to be regarded as a physically continuous mediumrather than a mere mathematical device lies in itspossession of detectable properties other than the oneproperty for which it was introduced. A condition ofthis kind is often suggested as a of the physical ‘reality’of a theoretical entity, and it led Faraday to expresshis dissatisfaction with Newtonian gravitation. Butindependent detection was not the only considerationwhich weighed with the nineteenthcentury physicists.They were prepared to regard a field as a physicallycontinuous medium on other and less stringent terms,for example, if propagation was affected by materialchanges in the intervening space, if it took time,if a mechanical model could be imagined for theaction of a medium in producing the observed effect,or if energy could be located in the space betweeninteracting bodies. Any of these three conditions mightbe regarded as sufficient and no one of them wasindividually necessary. Thus, gravitation remainedan action at a distance throughout the nineteenthcentury, in spite of its description by a potentialtheory, because it did not satisfy any of these criteria,whereas the electromagnetic field theory began to takeon the characteristics of continuous action becauseit satisfied all of them. It is sometimes suggested inmodern (post-relativity) works that a finite velocityof propagation is a necessary as well as sufficientcondition for continuous action, and that this iswhy instantaneous gravitation could not be regardedas such an action, but in classical physics thereis instantaneous transmission of pressure and oflongitudinal waves in an incompressible medium, andthis would certainly be regarded as continuous action”.

From this survey of the two opposite conceptualfoundations, we may conclude that the nineteenth-century physicists considered as sufficient conditionsfor validity of Faraday-Maxwell’s field concept:

1. propagation of fields producing material changesin the surrounding space;

2. time-delay in propagation;

"Electromagnetic Phenomena", V.2, №4 (8), 2001 541


3. experimental observation of the energy locatedbetween interacting bodies.

Nevertheless, as will be discussed later, the above-mentioned conditions should be considered only asnecessary, not sufficient for establishing the existenceof Faraday-type contact fields, since there mightbe a third alternative which would combine bothIAAAD and Faraday’s contact-field features in a singlescheme. In fact, examining the historical backgroundof nineteenth century electrodynamics, such a thirdalternative did actually exist and this was the so-calledcompromise theory of Helmholtz.

By the time Helmholtz became actively involved inresolving problems of electromagnetism, in the middleof the 1860’s, Weber’s and Maxwell’s supporters hadalready been locked in a lengthy and futile conflict.Helmholtz attempted to make a decisive choicebetween them by constructing his own mathematicalscheme and designing crucial experiments to weighin favour of either Weber’s theory or Maxwell’s.Being aware of their respective advantages anddisadvantages, Helmholtz [11] attempted to elaboratehis compromise approach aimed at combining theimportant elements of the two theories. However,he did not accept the idea of Maxwell’s light-ether and adopted, instead, the concept of thedielectric and diamagnetic medium. The condition ofinfinite polarizability, in this ether model, requiredthe charge to behave like an incompressible fluid,making all circuits closed as in Maxwell’s theory.When trying to arrive at results similar to Maxwell’swithout losing the elements of action-at-a-distance,Helmholtz assumed that the electrostatic forces areconstantly present as a field in space and that thechange in the polarisation or the displacement of thecharges signalled the change in the electrostatic field.As discussed in [12, 13], under these assumptions,Helmholtz successfully derived generalised equationsvery similar to those of Maxwell and found thatin a limited case they yield equations identical toMaxwell’s. Solving these equations for a homogeneousdielectric medium, he arrived at the wave equationsfor electric and magnetic polarisations, respectively,with undetermined constant k (see, for instance, [12,13], or [14]). Equations for electric polarisation andmagnetic polarisation have solutions for transversewaves whereas equation for electric polarisation alsodefines longitudinal waves.

The conciliatory aspect of Helmholtz’s approachresulted in the following peculiarity. The resultsof Maxwell’s theory can be obtained by settingk = 0 (and, of course, by setting the values ofelectromagnetic constants in correspondence with themodel of the light-ether). In addition to the ordinarytransverse electromagnetic waves already confirmedby Maxwell, Helmholtz discovered the existence oflongitudinal electric waves which turned out to be

instantaneous at the Maxwellian limit (k = 0).Interpretation of this conclusion and its consequencesbecame a hard nut to crack for all contemporaryelectrodynamicists. Maxwellian followers (Heaviside,FitzGerald, Lodge etc.) refused to accept Helmholtz’stheory because they found his conceptions entirelyforeign to Maxwell’s view of the transmission ofinteraction but ignored the fact that both theoriescould be considered mathematically equivalent.(A comprehensive discussion on this mathematicalequivalence was made by O’Rahilly [15]).

Helmholtz’s attempt at a more consistentreformulation of the contemporary electrodynamicstheories could not, however, resolve the problem ofwhich approach to favour. Therefore, the need fordecisive and reliable experimental data was urgentwhen Hertz became interested in electromagneticresearch.

3. Conventional and alternativeinterpretations of Hertz’scrucial experiments

In 1879 Helmholtz proposed a prize competition,”To establish experimentally a relation betweenelectromagnetic action and the polarisation ofdielectrics” and urged his student Hertz to take upthe challenge. At first, Hertz declined, discouraged bythe poor prospects of success at that time. Later on, hebegan investigating the problem for his own interest.In 1886–88, at Karlsruhe, he attempted to establishthe compatibility of the theories of Helmholtz andMaxwell in a new series of experiments. He designedhis measurement-procedure, taking into accountHelmholtz’s separation of the total electric force intothe electrostatic and electrodynamic parts to whichdifferent velocities of propagation were ascribed. InHertz’s words [16]: ”The total force may be split upinto the electrostatic part and electrodynamic part;there is no doubt that at short distances the former,at great distances the latter, preponderates and settlesthe direction of the total force”.

He was aware of the different rate of decreasewith distance for each of these forces. Accordingto Coulomb’s law, the electrostatic component wasthought to be proportional to the inverse squareof the distance, whereas the electrodynamic partwas only proportional to the inverse of the distance(in the conventional theory of the Lienard-Wiechertpotential it would correspond to decreasing rates ofthe bound-field, or longitudinal, component and theradiation field, or transverse component, respectively).It is not surprising, therefore, that the systematicidentification of the components of the total forceappeared to be an extremely difficult task forany reliable experiment of that time because the



magnitude and the phase of the total force asa superposition of components propagating withdifferent speeds and decreasing rates would beconstantly changing with distance. Even for modernexperimental techniques this task is not an easy onebecause in addition to the knowledge of the totalforce, one of the components should also be verifiedquantitatively.

Hertz decided to carry out experiments. Despitethe uncertainty of some of the results of his firstexperiments, Hertz [17] could already make animportant, albeit only qualitative, conclusion: ”Thereare already many reasons for believing that thetransversal waves of light are electromagnetic waves;a firm foundation for this hypothesis is furnishedby showing the actual existence in free space ofelectromagnetic transversal waves which propagatedwith a velocity akin to that of light”. This qualitativeresult was published in a paper entitled ”On theFinite Velocity of Propagation of ElectromagneticAction” (1888) which nowadays, according to theestablished historiography of physics, is considered aclassical reference in which the difficult task of provingthe finite propagation velocity of electromagneticinteractions in air has been achieved.

Nevertheless, it should be noted that the titleof Hertz’s paper is perhaps misleading nowadays,because conventional Maxwellian electrodynamicsdoes not employ the Helmholtzian ’action’terminology, nor does it split?up the total electricforce into electromagnetic and electrostatic parts.However, for Hertz’s contemporaries who supportedthe Helmholtz theory, the underlying meaning ofthe presented results was clear enough: Hertz’sexperiment could qualitatively conclude about thefinite propagation of the electromagnetic (transverse)part, but could say nothing definite about theelectrostatic (longitudinal) component. Lookingcarefully through the same paper, we find Hertzdeclaring [18]: ”From this it follows that the absolutevalue of the first of these is of the same order as thevelocity of light. Nothing can as yet be decided as tothe propagation of electrostatic actions”.

Moreover, some of Hertz’s measurements [19]tended towards the instantaneous nature of theelectrostatic mode, but he was still not convincedof this instantaneity and preferred to be cautious,since his method was unable to provide him with anyreliable quantitative results: ”Since the interferencesundoubtedly change sign after 2.8 meters in theneighbourhood of the primary oscillation, we mightconclude that the electrostatic force which herepredominates is propagated with infinite velocity. Butthis conclusion would in the main depend upon asingle change of phase... If the absolute velocityof the electrostatic force remains for the presentunknown, there may yet be adduced definite reasonsfor believing that the electrostatic and electromagnetic

forces possess different velocities”.

It should be especially stressed that Hertz atthis stage was fully aware of the need for additionalexperiments to cast some light of certainty on theelectrostatic part [18]: ”It is certainly remarkable thatthe proof of a finite rate of propagation should havebeen first brought forward in the case of a forcewhich diminishes in inverse proportion to the distance[electrodynamic part], and not to the square of thedistance [electrostatic part]. But it is worth whilepointing out that this proof must also affect such forcesas are inversely proportional to the square of thedistance”.

It is interesting briefly to follow Hertz’s gradualshift towards accepting Maxwell’s field concepts. Hestarted with Helmholtz’s theory, and his conversion toMaxwell’s viewpoint was an uneasy process, possiblynever fully completed due to his (Hertz’s) prematuredeath. He began analysing the underlying conceptsin the Maxwellian limit of Helmholtz’s theory buthis final interpretation became essentially differentin form from what had been commonly acceptedby Helmholtz and his supporters. More specifically,Hertz uncritically assumed that in Maxwell’s limitthe instantaneous longitudinal component shouldbe excluded from consideration in Helmholtz’soriginal theory. All forces then became explicitlytime dependent. This was a drastic departure fromhis mentor’s philosophical foundations becauseHelmholtz rejected time dependent forces (headmitted only implicit time dependence upon spaceposition) and was deeply convinced that ”..naturecould only be comprehended through invariable causes... Helmholtz viewed electromagnetic interactions— indeed, all interactions, — as instantaneous andbipartite...” [20] and, therefore, could attributeinteractions only to longitudinal components(electrostatic forces). In Hertz’s own words [21]:”...Helmholtz distinguishes between two formsof electrical force the electromagnetic and theelectrostatic to which, until the contrary is provedby experience, two different velocities are attributed.An interpretation of the experiments from this pointof view could certainly not be incorrect, but it mightperhaps be unnecessarily complicated. In a speciallimiting case Helmholtz’s theory becomes considerablysimplified, and its equations in this case become thesame as those of Maxwell’s theory; only one formof the force remains, and this is propagated with thevelocity of light. I had to try whether the experimentswould not agree with these much simpler assumptionsof Maxwell’s theory. The attempt was successful. Theresult of the calculation are given in the paper on‘The Forces of Electric Oscillations, treated accordingto Maxwell’s Theory.”

This paper was published in 1889, one yearafter the discussion of Hertz’s first results, whichapparently were not sufficient to conclude which of



the two theoretical descriptions was more adequate.Already discouraged by the complexity of Helmholtz’sapproach for a careful account of the experimentaldata, Hertz tried in this paper to show how theobserved singularities in the propagation of the electricforce could be described by Maxwell’s theoreticalscheme. As Hertz explained [22]: ”The results ofthe experiments on rapid electric oscillations whichI have carried out appear to me to confer uponMaxwell’s theory a position of superiority to allothers. Nevertheless, I based my first interpretationof these experiments upon the older views, seekingpartly to explain phenomena as resulting from co-operation of electrostatic and electromagnetic forces.To Maxwell’s theory in its pure development such adistinction is foreign. Hence I now wish to show thatthe phenomena can be explained in terms of Maxwell’stheory without introducing this distinction. Should thisattempt succeed, it will at the same time any questionas to a separate propagation of electrostatic force,which is meaningless in Maxwell’s theory.”

In this famous paper, Hertz wrote Maxwell’sequations in the form in which they are knowntoday (the Hertz-Heaviside form) and also derivedthe distribution of force lines for the radiatingoscillator (Hertz vibrator). In other words, thisimportant contribution to the Faraday-Maxwell fieldtheory consisted in the development of the generalsource-field relation previously unknown. (Todaythis method bears Hertz’s name and is basedon Fourier analysis of dipole and multi-dipoleradiation). Using these calculations, Hertz found anexplanation (alternative to that based exclusivelyon Helmholtz’s ideas) of the singularities he hadobserved in the distribution of radiation in the nearfield (the apparently instantaneous behaviour of theelectrostatic component) In Hetrz’s words [23]: ”Letus now investigate whether the present [Maxwell’s]theory leads to any explanation of the phenomena...At great distances the phase is smaller by the value ?than it would have been if the waves had proceededwith constant velocity from the origin; the waves,therefore, behave at great distances as if they hadtravelled through the first half wavelength with infinitevelocity.”

Interestingly, this prediction of Maxwell’s theoryconcerning the infinite phase-velocity for thenear-field zone based on straightforward Fourieranalysis (Hertz’s method) appears in the modernliterature [24]. Surprisingly, however, there is nointerpretation of it in the conventional text-books.Hertz himself paid no attention to this predictionbeyond the fact that it gave him a new interpretationof his results, different from that provided by theHelmholtzian approach. It is possible that Hertzdid not realise (or had no time to realise) all theconceptual implications of the new prediction, whichhas no clear meaning in the framework of the Faraday-

Maxwell contact-field doctrine. The prediction wouldimply the existence of a small but macroscopicregion where the notion of Faraday locality becomesinvalid. On the other hand, it also shows somepossible ’fuzziness’ in the relationship between staticand dynamic limits in Maxwell’s theory, as alreadymentioned earlier. Bearing that ambiguity in mind,it would not be surprising if Maxwell’s theoreticalpredictions for static and quasistatic phenomena werefound to be similar to the older IAAAD views. Itis obvious that phenomena in the near field zone(less than half wavelength) should be regarded asquasi-static in the Hertzian analysis and therefore,implicit time dependent that is not Maxwellian butHelmholtzian feature for longitudinal components.This reasoning should have cast doubt on Hertz’sexplanation of the experimental results as not beingcompletely in the spirit of the Faraday-Maxwell’sconceptual foundations. It is surprising, then, thatalmost no-one seemed to have been worried by thepresence of non-Faraday’s elements in Maxwell’sapproach. By the same token, it is no less surprisingthat Hertz’s explanation was so unconditionallyaccepted by Maxwell’s followers. But although Hertzwas satisfied that his calculations had accountedfor the majority of the observed phenomena, hestressed that he had not succeeded in removing allthe difficulties from his experimental verification ofMaxwell’s theory. He confessed that [25]: ”...I hopedto be able to devise some way of making observationson waves in free air, that is to say, in such a mannerthat any disturbances which might be observed couldin no wise be referred to any action-at-a- distance.This last hope was frustrated by the feebleness of theeffects produced under the circumstances.”

Although Hertz’s satisfaction with Maxwell’stheory was understandable, there were still insufficientarguments for making a truly decisive choice, bearingin mind that the Helmholtzian approach remained inqualitative agreement with the observed singularities.However, no further experiments or calculations fortesting the quantitative predictions of Helmholtz’stheory have been attempted. In our retrospective viewof these stated reservations of Hertz, as well as of somany other thinkers of the time, the unconditionalacceptance with which Hertz’s experiments and theirinterpretations were received might seem somewhatunjustified. Hertz himself did not expect such supportand attributed it to the heavy philosophical burden ofthe old and unresolved dilemma of the choice betweenthe IAAAD and contact-action doctrines.

Another detail which may also have contributed tothe unconditional approval of Hertz’s results amongthe scientific community was, as already stated, therather misleading title of his paper on the finitepropagation of electromagnetic interactions, possiblydue to unawareness of Helmholtz’s classification ofelectrostatic and electromagnetic forces. However, in



contrast to this general enthusiasm for Hertz’s results,Helmholtz’s supporters adopted rather lukewarmattitude, although there was also some strongopposition from such as P. Duhem, an eminent Frenchmathematician, physicist and philosopher of scienceat the beginning of the twentieth century. He was oneof a small group of scientists who refused to acceptHertz’s experiments as conclusive. Moreover, Duhemwas the first to raise doubts about the whole concept of’crucial’ experiments [26]. A good mathematician andoutspoken critic of the inconsistencies in Maxwell’stheory, he became one of the principle advocates ofHelmholtz’s approach [27]: ”...Physicists are caughtin this dilemma: Abandon the traditional theoryof electric and magnetic distribution (electro- andmagnetostatics), or else give up the electromagnetictheory of light. Can they not adopt a third solution?Can they not imagine a doctrine in which there wouldbe a logical reconciliation of the old electrostatics,of the old magnetism, and of the new doctrine thatelectric actions are propagated in dielectrics? Thisdoctrine exists; it is one of the finest achievements ofHelmholtz; the natural prolongation of the doctrinesof Poisson, Ampere, Weber and Neumann, it logicallyleads from the principles laid down at the beginningof the nineteenth century to the most fascinatingconsequences of Maxwell’s theories, from the laws ofCoulomb to the electromagnetic theory of light; withoutlosing any of the recent conquests of electrical science,it re-establishes the continuity of tradition.”

However, it appears that this call of Duhem’s fora ’third solution’ fell mainly on deaf ears. So, whilstappreciating the difficulties of Hertz’s pioneeringinvestigations, and taking into consideration hisstruggle through the uncertainties and controversiesof the electrodynamics of his time, the fact remainsthat his final opting for Maxwell’s theory was notbased on strict scientific logic. What needs to bedone, therefore, is clearly to identify the criteria foracceptance and apply these to the existing alternativetheories. The detailed examination of how and why acertain theory is confirmed or refuted by experimentaltests is, of course, a matter of the methodologyand philosophy of science. Hertz was apparently notfully aware of the need to test his choice fromthis systematic, methodological and philosophicalstandpoint.

As did the majority of his contemporaries, Hertzintuitively applied a criterion of empirical verification,in the hypothetico-deductive manner prevailing in19-th century science. This method consisted ofcreating hypotheses in the form of postulates and thenmaking deductions from these which could be eitherconfirmed or rejected by experiment. However, fromthe modern methodological standpoint, as it is nowwell recognised, empirical verification is a conditionthat is necessary but not sufficient for establishingthe truth of a theory. The fact is that an empirical

observation may ’verify’ any number of different, yetequally valid theories sharing that same prediction.The only reliable way, therefore, of deciding betweentheories is not to ‘verify’ any one of them in particular,which might just as well verify any number of thembut, if possible, to eliminate all but one of thecontenders. The only way of doing this is, of course, bythe method of refutation. This method may be carriedout logically, by pure ratiocination (as by pointingout some logical or mathematical contradiction) orby demonstrating that empirical predictions made bysome logically consistent theory are false. However,even if a theory is logically sound, if it makes nofalsifiable predictions, it cannot be refuted. Such atheory, according to Popper, cannot qualify as ascientific theory. This criterion, introduced by Popperin the late 1920s, thus provided a reliable criterionfor separating genuine scientific theories from pseudo-scientific or ‘metaphysical’ theories by which, Poppermeant theories which make no predictions that can,even in principle, be empirically falsified (it appearsto be the case of some of the modern quantum fieldtheories such as quantum gravity, string theory etc.).

It may be of interest, then, to consider how thismore modern criterion of falsifiability might have beenapplied to Hertz’s decision, had it been availableat the time. Qualitatively, as we have seen, bothMaxwell’s and Helmholtz’s theories fit equally well allthe observations made by Hertz, namely:

1. material changes in the surrounding space;

2. finite propagation of transverse componentswith the speed of light;

3. empirical observation of energy betweeninteracting bodies etc.

The present-day scientific method suggests thatthe next step is to explore the difference between theexperimental predictions of the two theories, beyondthose that are already known, and to separate-out thedifferent, non-compatible but empirically verifiablepredictions that is, to determine which of them, ifany, are sufficient to explain all the known facts, asdistinct from those that are merely necessary. Thus,in the case of alternative electrodynamics theories,the core of the new ‘crucial’ experiment shouldbe to test, experimentally, statements specificallydescribing the character of the longitudinal electriccomponents which distinguish the Helmholzian fromthe Maxwellian theory. With regard to the latter,no decisive, unambiguous information of the kindnecessary to refute Helmholtz’s theory has yet beenfound in Hertz’s experimental papers. As a result,by this same reasoning Helmholtz’s theory couldnot be conclusively ruled out, since it remainedperfectly falsifiable and did not contradict any knownobservational fact.



4. Arguments in favour ofalternative Helmholtz-typetheory of electromagnetism

As already mentioned, the criterion of falsifiabilityrequires any truly scientific alternative theory to belogically self-consistent. Logical inconsistencies leadto bogus predictions. In this way it is interesting toremind that several aspects of standard conventionalelectrodynamics are found to be unsatisfactory,despite all the advances claimed by relativity andquantum mechanics. Conventional electrodynamics isthus still not free from untreatable inconsistencies,as in its implications regarding self-interaction,infinite contribution of self-energy, the concept ofelectromagnetic mass, indefiniteness in the flux ofelectromagnetic energy, etc. These internal difficultiesexplain why, from the beginning to the middle ofthe 20-th century, there were unceasing efforts tomodify either Maxwell’s equations or the underlyingconceptual premises of electromagnetism. The presentstatus of classical electrodynamics can by expressed bywords of R. Feynman [28]: ”...this tremendous edifice[classical electrodynamics], which is such a beautifulsuccess in explaining so many phenomena, ultimatelyfalls on its face. When you follow any of our physicstoo far, you find that it always gets into some kind oftrouble. ...the failure of the classical electromagnetictheory. ...Classical mechanics is a mathematicallyconsistent theory; it just doesn’t agree with experience.It is interesting though, that the classical theory ofelectromagnetism is an unsatisfactory theory all byitself. There are difficulties associated with the ideasof Maxwell’s theory which are not solved by and notdirectly associated with quantum mechanics...”.

One of the latest systematic accounts of thesedifficulties has been made recently in [29–31].In particular, the old problem of Maxwell’selectrodynamics, concerning the uncertainrelationship between static and dynamic limits, hasbeen brought into greater relief. Pure mathematicalanalysis [29] has shown that the conventional theorydoes not ensure a continuous transition betweenstatic and dynamic limits, which is, surely, in itself,strange to contemplate in the context of the Faraday-Maxwell continuous-field concept. Interestingly, it hasalso been found that if the condition of continuoustransition between static and dynamic limits isimposed explicitly in mathematical terms, thenconventional boundary condition should be replacedby generalised boundary conditions. As a result, thestructure of general solutions to Maxwell’s equationshave to be modified to include non-local longitudinalcomponents. It is interesting to note that the possibledifficulty with conventional boundary conditions inthe classical field theory was realised by A. Einsteinhimself a few month before his death in 1955. In the

last edition of The Meaning of Relativity he added thefollowing [32]: ”...A field theory is not yet completelydetermined by the system of field equations: Shouldone postulate boundary conditions?: Without sucha postulate, the theory is much too vague. In myopinion the answer to the question is that postulationof boundary condition is indispensable.”

Thus, a different choice of a boundary conditioncan result in a different approach (Maxwellian orHelmholtzian) in the framework of the same systemof Maxwell’s equations (a limit case of Helmholtz’sequations). It, therefore, results in a different timedependence of forces: explicit time dependence forlongitudinal and transverse components in Maxwell’stheory and explicit/implicit time dependence fortransverse/longitudinal components in Helmholtz’stheory. As far as it is reflected in the historyof electromagnetism literature, Hertz did notattribute any importance to the choice of boundaryconditions and believed that the same differentialequations should have only one basic interpretationalbackground.

In respect to the longitudinal components ofelectromagnetic field, it is noteworthy that theyturn out to be in agreement with the requirementsof relativistic invariance in the neo-Helmholtzianapproach [29–31]. Moreover, they behave in thesame manner as it is accepted by conventionalrelativistic theory for fields and potentials of oneuniformly moving charge under Lorentz relativistictransformations. With regard to the latter, it is well-known that field lines of one uniformly moving chargeare radial, i.e. exhibit IAAAD features. However, inthe traditional interpretation, this radial characterof field-lines is considered fictitious (as a class oflegitimate singularities), whereas in the Helmholtz-type approach developed in [29–31] it is a perfectlyrealistic representation. Thus, contrary to whatcould have been expected from the point of viewof the conventional electrodynamics, instantaneouslongitudinal fields can behave in accordance withrequired Lorentz relativistic symmetry.

In the approach reviewed below, longitudinalcomponents do not take part in a local energy transferavoiding possible conflict with the special relativitytheory. A local energy transfer is an exclusiveprerogative of transversal components associated withradiation. In fact, it is well-known that Einstein’stheory does not limit phase velocities, if there is nolocal energy transfer. Thus, a purely mathematicalanalysis of a self-consistency of electromagnetictheory with required relativistic properties leadsautomatically to the alternative approach very similarby spirit to the Helmholtzian doctrine based on asuperposition of local transversal and instantaneouslongitudinal forces.

In summarising this discussion it is interesting tonote another possible attractiveness of Helmholtz’s



conceptual foundations. As well as having theseabove-mentioned difficulties, Maxwell’s theory alsocould not provide any reliable model of the atom.This left a theoretical gap which had to be filledindependently. This, of course, is what led to thecurrent quantum mechanical theory of the atom,with its implications of essential non-locality. Thuswas created the other serious dilemma of present-dayphysics: significant incommensurability between theclassical relativistic theory with its basic concept oflocal field, and quantum mechanics with its essentialnon-locality. This fundamental conflict is such that,for some people, the only way of resolving it seemsto be to combine or superimpose these incompatiblerequirements for locality and non-locality in somepurely expedient and compromising way. In viewof these reasoning it also can be suggested thatany physical theory of interaction might not bepure conventional field theory (in the sense of localfield) but be complemented by non-local longitudinalcomponents (in the sense of non-local potential field).

In modern physics, as it is well-known, longitudinalcomponents are responsible for the appearance ofinfinities that in special cases and with enormousdifficulties can be eliminated by mathematicalmeans, leaving nevertheless some kind of conceptualinsatisfaction. In some cases such as quantum gravitythe infinities resist to be removed indicating thatthere is no unified conceptual approach to theinfinities problems and it appears to have moremathematical convenience than physical justification.In Helmholtz’s and neo-Helmholtz’s theory [29–31] with the clear distinction between transverseand longitudinal component, the latter one doesnot contribute infinite amount of self-energy as itdoes not in every classical (newtonian) non-localpotential theory with bipartite type of interactions.This property of longitudinal components alloweda formulation of non- radiation condition in [29]which states that ”...a limited class of motionexists when accelerated charged particles do notproduce electromagnetic radiation.” It means thatwithin Helmholtz’s foundations it is even possible toformulate a classical model of the atom contrary toMaxwell’s electrodynamics where all such attemptsfailed. All these developments will be reviewed in thiswork.

Thus, as the relevant historical literature shows,Helmholtz’s foundations promise an altogether morelogical solution to fundamental problems of modernphysics. This may suggest a way of reconciling classicalelectrodynamics and quantum mechanics in a lessad hoc and altogether more rational way than has,up till now, seemed obligatory. Recent experimentalconfirmations of the violation of Bell’s inequalities inquantum mechanical measurements and entanglementin quantum optics shed some light on a possiblealternative foundations of classical electrodynamics

incorporating IAAAD longitudinal components in thesame framework of Maxwell’s equations.

5. An example of ambiguity ofFaraday-Maxwell’s conceptof local (contact) field in theconventional electromagnetictheory

At the beginning of this Section we show one of theconfusions of classical electrodynamics in describingelectromagnetic field of an accelerated charge. Theattractiveness of this example consists in the way itlightens the main conventional theory difficulties andthe way it leads to the Helmholtzian-type foundationsof classical electrodynamics. Let us consider a charge emoving in a laboratory reference system with constantacceleration a along the positive direction of the X-axis. An electric field created by an arbitrarily movingcharge is given by the following expression obtaineddirectly from Lienard-Wiechert potentials [33]:

E(R, t) = e

(

R−Rv

c

)

(

1−v2

c2

)

(

R−Rv

c

)3 +

e

[

R,[

R−Rv

c

]

,v]

(

R−Rv

c

)3 . (1)

All values in the right-hand are taken in themoment of time t0 = t − τ , where τ is the retardedtime. Since all vectors are collinear, the second termin (1) is zero. In the conventional theory, the Poyntingvector represents electromagnetic field energy flow perunit area per unit time across a given surface,

S =c

4π[E,H] , P =

1

c2S, (2)

where S is the Poynting vector, P is momentumdensity, and E and B are electric and magnetic fieldstrengths, respectively. Analysing (2), one can easilynote that S and P (and, therefore, all electromagneticenergy flow) are exactly zero (S = 0) along the X-axis.On the other hand, from the energy conservation law,

W =E2 +B2

8π,

∂W

∂t= −(∇,S) (3)

we conclude that W and ∂W/∂t should differ fromzero everywhere along X-axis because there is a linearrelationship betweenW and E2 changing in time alongX-axis. An ambiguity takes place if, for instance, thecharge is moving in some arbitrary way along the X-axis. As a result the energy density W should alsoalter as a function of changing electric field E. Thenthe question logically arises: what is the mechanism



that changes electric and magnetic field components atsome fixed distance from the charge on X-axis if thereis apparently no electromagnetic field energy transferin that direction (S = 0)? This ambiguity is due tothe fact that in the conventional theory based on theconcept of local (contact) field, which energy has to bestored locally in space, any change of field componentsis indispensable without field energy flux. This isobviously violated in the above-mentioned examplethat brings into question an assumed sufficiency oftransverse solutions alone to describe all properties ofelectromagnetic field. At least, the resolution of thisambiguity cannot be based on transverse solutions toMaxwell’s equations because it is well-established thatany moving charge does not radiate electromagneticwaves along the direction of motion. Only longitudinalcomponents, if they exist, can be useful in thatrespect.

Let us make several qualitative observations onthe possible role of longitudinal fields components.The solution (1) indicates the existence of longitudinalperturbations along the X-axis. The energy transfer(the Poynting vector) S is a product of the energydensity and its spreading velocity v,

S =Wv (4)

then either the spreading velocity v or the energydensity W must be zero along the X-axis. The firstassumption neglects any possibility of interactiontransfer. The second one (W = 0) is uncomprehensingin the framework of Faraday-Maxwell local field whichshould be locally stored in space with non-zero energy.We can assume that longitudinal modes possess nolocal energy (in other words, cannot be stored locallyin space (W = 0) contrary to what their transversecounterpart do) but arrive to the conclusion that theirspreading velocity v may be any positive value eveninfinite. This is, of course, only our hypothesis hereto resolve above-mentioned ambiguity but as we willshow later on it has solid mathematical foundations.

At the end of this Section, we stress that inthe conventional electrodynamics longitudinal fieldcomponents in vacuo do not play any role at alland, in fact, they are eliminated from considerationby means of appropriate gauge. In Dirac’s ownwords [34]: ”...As long as we are dealing only withtransverse waves, we cannot bring in the Coulombinteractions between particles. To bring them in, wehave to introduce longitudinal electromagnetic waves:The longitudinal waves can be eliminated by meansof mathematical transformation. Now, when we domake this transformation which results in eliminatingthe longitudinal electromagnetic waves, we get a newterm appearing in the Hamiltonian. This new term isjust the Coulomb energy of interaction between all thecharged particles,

∑

1,2

e1e2

r1,2, (5)

...This term appears automatically when wemake the transformation of the elimination ofthe longitudinal waves.” As we know from theclassical physics, (5) means the existence of bipartiteinstantaneous longitudinal interaction with nopotential energy stored locally in the interparticlespace. What is then the meaning of the eliminationof longitudinal components in the conventionaltheory? In the following we will try to show that theproblem of longitudinal components is unreasonablyunderestimated in classical electrodynamics (perhapsby historical reasons). There should be a changeof attitude towards its status. Mathematical andphysical reasons in favour of Helmholtz-typefoundations will be given to show a paramountimportance of longitudinal components to build upa self-consistent classical electrodynamics and itspossible reconciliation with quantum mechanics.

6. Mathematical foundations ofHelmholtz-typeelectrodynamics

Let us recall that a complete set of Maxwell’sequations in vacuum is

(∇,E) = 4πρ, (6)

(∇,B) = 0, (7)

[∇,H] =4π

cj+

1

c

∂E

∂t, (8)

[∇,E] = −1

c

∂B

∂t. (9)

If this system of equations is really complete andboundary conditions are adequate, it should describeall electromagnetic phenomena without exceptionsand ambiguities. It is often convenient to introducepotentials, satisfying the Lorentz condition

(∇,A) +1

c

∂φ

∂t= 0. (10)

As a result, the set of coupled first-order partialdifferential equations (6)–(9) can be reduced tothe equivalent pair of uncoupled inhomogeneousD’Alembert’s equations:

∆ϕ−1

c2∂2ϕ

∂t2= −4πρ(r, t), (11)

∆A−1

c2∂2A

∂t2= −

4π

cj(r, t). (12)

Differential equations have, generally speaking, aninfinite number of possible solutions. An uniquelydetermined solution is selected by laying downsufficient additional conditions. Different forms ofadditional conditions are possible for the secondorder partial differential equations: initial value andboundary-value conditions. A general solution of the



D’Alembert equation is considered as an explicit time-dependent function of the type g(R, t). Let us discuss avery subtle point related to the use and interpretationof implicit and explicit time dependencies in theconventional electrodynamics. We think that as far asthis problem is not cleared up, the classical theory willremain beset of ambiguities. Helmholtz-type approachreviewed below makes that distinction very clear.

Special relativity well established that in thestationary approximation (charge moving with aconstant velocity) all fields components are implicittime-dependent functions of the type f(R(t)). Fieldlines remain radial in all inertial frames of referencesand, hence, depend on the instant position of thecharge. As a consequence, time t is not an independentvariable any more in this case and enters as aparameter through space position of the charge R(t).Hence, the use of partial time derivatives ∂/∂t,∂2/∂t2 etc. (according to their formal mathematicaldefinition) is inadequate if a function has not twoor more independent variables. Nevertheless, in basictexts on classical electromagnetic theory partial timederivatives are indiscriminately applied even forimplicit time dependent functions in the proper senseof total time derivatives. (Some clear examples willbe done in the next Section discussing the use ofcontinuity equation).

Looking back at D’Alembert’s equations (11),(12),space variable R should be fixed under the actionof partial time derivative ∂2/∂t2. Fixing R(t), meansthat there is no change with time t playing the role of aparameter. Thus, partial time derivatives vanish fromD’Alembert equation in the case of uniformly movingcharge. Poisson’s equation for four-vector (φ,A) withimplicit time dependence appears to be appropriateone. We especially made a detailed analysis becauseof a confusion in conventional texts on classicalelectromagnetism about explicit use of Poisson’sequations for uniformly moving charge (but as wehave seen, they do it tacitly). It is commonly thoughtthat only D’Alembert’s equation (i.e. that onlyD’Alembert’s operator ∆ − ∂2/∂t2) is relativisticallyinvariant under Lorentz’s transformations. As wewill discuss later in the Section VII in connectionwith gauge invariance and summarising it up inthe Section VIII, Poisson’s equation in four- vectorrepresentation (φ,A) (as well as Poisson’s differentialoperator ∆) can also be considered relativisticallyinvariant when applied to implicit time-dependentpotentials, reproducing all results of special relativityfor inertial frames of reference. Poisson’s differentialoperator ∆ is not covariant but invariant underLorentz’s transformations. Time variable is not anymore independent in this case and cannot be used forcovariant representation of D’Alembert’s differentialoperator. It is endorsed by the well-known fact thatcovariance is not necessary, it is only sufficient forrelativistic invariance.

Thus, we can conclude that D’Alembert equationshave general solutions in form of explicit time-dependent functions whereas Poisson’s equations haveonly implicit time dependent solutions. The followingquestion becomes obvious: how any transition fromD’Alembert and Poisson’s equations is described inthe conventional formalism? As a matter of fact,this question has not even been asked becausePoisson’s equation has not been recognised as coveringimplicit time-dependent phenomena (it was appliedexclusively in electro- and magnetostatics with notime dependence at all). This question, unexploredby the conventional approach, contains a very seriousdifficulty.

As we shall demonstrate below, a continuoustransition between solutions of D’Alembert’sand Poisson’s equations, respectively, is notmathematically ensured in classical electromagnetism.Based on the premises of a continuous nature ofelectromagnetic phenomena, one can assume that anygeneral implicit time solution of Poisson’s equationshould be continuously transformed into explicittime solutions of D’Alembert’s equations (and viceversa). This requirement can also be formulatedas a mathematical condition on the continuity ofgeneral solutions of Maxwell’s equations at everymoment of time. By force of the uniqueness theoremfor the second order partial differential equations,only one solution exists satisfying given initial andboundary conditions. Consequently, the continuoustransition from solutions of D’Alembert’s equationinto solutions of Poisson’s equation (and vice versa)should be ensured by the continuous transitionbetween respective initial and boundary conditions.This is the point where the conventional approachfails again. Only implicit time-dependent functionf(R(t)) can be unique solution of Poisson’s equationsand boundary conditions for external problem areto be formulated in the infinity. On the other hand,the solution of D’Alembert’s equation is an explicittime-dependent function g(R(t), t) since only it fitsrequirements of Faraday- Maxwell’s electrodynamicsas a physically sound solution for the notion of local(contact) field. The boundary conditions in this caseare given in a finite region. It makes no sense toestablish them at the infinity if it cannot be reachedby any perturbations with finite spread velocity. Asfar as one deals with large external region, effects ofboundaries are still insignificant over a small intervalof time, and, therefore, it is convenient to considerthe limiting problem with initial conditions for aninfinite region (initial Cauchy’s problem). This is howin mathematical physics areas of infinite dimensionsare introduced into consideration.

Let us look carefully at the standard formulationof respective boundary-value problems in a regionextending to infinity. There are three externalboundary-value problems for Poisson’s equation.



They are known as the Dirichlet problem,Neumann problem and their combination. Themathematical formulation, for instance, for Dirichlet’sboundary conditions requires to find a function u(r)satisfying [35]:

(i) Laplace’s equation ∆u = 0 everywhere outsidethe given system of charges (currents).

(ii) Solution u(r) is continuous everywhere in thegiven region and takes the given value G on theinternal surface S: u

∣

∣

S= G.

(iii) Solution u(r) converges uniformly to 0 atinfinity: u(r)→ 0 as |r| → ∞.

The final condition (iii) is essential for a uniquesolution! In the case of D’Alembert’s equationthe standard mathematical formulation is different.Obviously, we are interested only in the problem foran infinite region (initial Cauchy’s problem). So it isrequired to find the function u(r(t), t) satisfying [35]:

(j) homogeneous D’Alembert’s equationeverywhere outside the given system of charges(currents) for every moment of time t ≥ 0

(jj) initial conditions in all infinite regions as follows:u(r, t)

∣

∣

t=0= G1(r); ut(r, t)

∣

∣

t=0= G2(r).

The condition (iii) about the uniform convergenceat infinity is not mentioned. Recall here thatCauchy’s problem is considered when one of theboundaries is insignificant over all time of a process.In conventional electrodynamics it means that anyperturbation with finite spread velocity will neverreach the limits of the region under considerationduring the time of observation. From the conventionalpoint of view, condition (iii) formally included intoCauchy’s problem can never affect the solution and,hence, might not be taken into account seriouslyfor selecting of adequate solutions. In fact in thecontext of local field, the inclusion of the condition(iii) becomes meaningless since only explicit time-dependent solutions (retarded waves with finite spreadvelocity) are allowed by conventional electrodynamicsto solutions of D’Alembert’s equation.

On the other hand, we underline here that theabsence of the condition (iii) for every momentof time in the standard mathematical formulationof Cauchy’s initial problem does not ensure thecontinuous transition into external boundary-valueproblem for Poisson’s equation and, as a result, mutualcontinuity between the corresponding solutions cannotbe expected by force of the uniqueness theorem. Thisunambiguous mathematical fact should be consideredas one of the most warning signals of possibleflaws in the mathematical formalism of contemporaryMaxwell’s electrodynamics. The only way that seemsto be obligatory to satisfy the property of continuityof electromagnetic field (in other words, to keep

the continuity in transition between solutions ofD’Alembert and Poisson’s equations), is the inclusionof the condition (iii) for every moment of time inthe standard mathematical formulation of Cauchy’sinitial problem. It obviously ensures the continuoustransition into external boundary-value problem forPoisson’s equation (and vice versa) and implies astructure of a general solution as a superposition ofseparate non-reducible to each other functions of thetype

f(R(r)) + g(r, t). (13)

When applied to potentials, this statement takes aform:

ϕ(r, t) = ϕ0(R(t)) + ϕ∗(r, t), (14)

A(r, t) = A0(R(t)) +A∗(r, t), (15)

where for one charge system R(t) = r − rq(t); r isa fixed distance from the point of observation to theorigin of the reference system and rq(t) is the positionof the charge at the instant t.

The presence of the condition (iii) in theformulation of Cauchy’s problem turns out to bemeaningful for any moment of time, and thecorresponding boundary conditions keep continuity inrespect of mutual transformation. That makes thecondition (iii) irremovable from the formulation ofinitial Cauchy’s problem resulting in fundamental(irremovable) nature of implicit time-dependent (orlongitudinal) components φ0, A0 responsible for theinterparticle interaction. Potentials with explicit time-dependence φ∗ and A∗ vanish in the steady-state case,leaving only implicit time-dependent functions φ0 andA0 in the total potential (left-hand side of (14) and(15)). Now, contrary to the conventional approach,it is clear how the total solution φ (or A) in left-hand side of (14),(15) with explicit time dependenceundergoes transformations into solution with implicittime dependence (and vice versa).

Faraday-Maxwell’s approach does not allow totake into account the first term in right-hand sideof (14),(15) as full-value part of any general solution.Turning to the above-mentioned ambiguity at thebeginning of the previous section, we see now thatthe novel solution in form of (14),(15) can describethe change of electric field component along theX-axis at any distance and at any time. It castsdoubts on the general belief that Lienard- Wiechertpotentials (as only explicit time-dependent solutionsof D’Alembert’s equations for Cauchy’s problem)should be considered as unique general solutionsto Maxwell’s equations regardless the context ofboundary conditions. In connection to the latterit is worth reminding Einstein’s words (see theSection III) that field theory is not only determinedby the system of field equations but also bypostulation of boundary conditions. In fact, Lienardand Wiechert formulated the initial Cauchy problem



for electromagnetic components several years beforethe appearance of Einstein’s principle of relativity.Thus, a priori imposed boundary conditions werenot assumed to have adequate relativistic properties.This is another open question in the conventionalapproach whether relativistic requirements should bereflected in the mathematical formulation of the initialboundary problem. In this respect, we only stress thatadditional condition (iii) is such an invariant becauseit is irremovable and unchangeable in every frame ofreference.

Let us consider again a pair of uncoupledinhomogeneous D’Alembert’s equations (11),(12) withinitial conditions (j), (jj) and (iii). For some purposes,it is convenient to decompose (11),(12) into twopairs of second order differential equations for eachcomponent of general solution (14),(15):

∆ϕ0 = −4πρ(r, t), (16)

∆A0 = −4π

cj(r, t) (17)

and

∆ϕ∗ −1

c2∂2ϕ∗

∂t2= 0, (18)

∆A∗ −1

c2∂2A∗

∂t2= 0, (19)

with initial and boundary conditions given, forinstance, in the case of electric potential. Theequation (16), apart from (iii), is supplemented by

ϕ0(r)∣

∣

S= G (20)

whereas (18) has to be added with

ϕ∗(r, t)∣

∣

t=0= G1 − ϕ0(r)

∣

∣

t=0, (21)

ϕ∗τ (r, t)∣

∣

t=0= G2 −

d

dtϕ0(r)

∣

∣

t=0. (22)

In the theory of differential equations anycomplete solution of (11),(12) consists of a generalsolution of homogeneous D’Alembert’s equation plussome particular solution of the inhomogeneous one.Thus, we can assume that the same procedurecan be applied to its equivalent formulationin form (16)–(19). On one hand, a completesolution should be formed by two independentgeneral solutions satisfying homogeneous Poisson’sand homogeneous wave equations, respectively,and, on the other hand, it has to includeone particular solution (as a linear combinationof non-reducible components (14),(15), satisfyinginhomogeneous D’Alembert’s equations (11),(12).Relationship between both components (longitudinaland transverse) of electromagnetic field is guidedby (21) and (22) and is contained in the particularsolution of inhomogeneous D’Alembert’s equations. Amore comprehensive study of the matter will be doneelsewhere.

Thus, the initial set of Maxwell’s equations hasbeen decomposed into two pairs of equations withindependent general solutions for each pair that arecoupled only through the partial solution of the wholeset of equations (16)–(19) or (11),(12). The first pair(16),(17) manifests the instantaneous and longitudinalaspect of electromagnetic interactions (action-at-a-distance) while the second one (18),(19) characterisesexplicit time-dependent phenomena related to thepropagation of transverse waves (light, radiation etc.).It is obvious thus that Helmholtz’s basic ideas arefundamentally compatible with Maxwell’s equations.

The potential separation (14),(15) implies thesame procedure with respect to the field strengths,

E(r, t) = E0(R(t)) +E∗(r, t), (23)

B(r, t) = B0(R(t)) +B∗(r, t), (24)

where E0 and B0 are instantaneous longitudinal fields.To finish this Section we would like to mention

that Villecco’s independent analysis endorsed ourclaims on discontinuity problem in the classicalelectromagnetic theory. He found that [36]: ”...thetransition between two different states of uniformvelocity via an intermediate state of accelerationresults in a type of discontinuity in functional form:Though no known law is violated in this processes,there is a sense of intrinsic continuity which isnevertheless violated...”.

7. Mathematical inconsistenciesin the formulation ofMaxwell-Lorentz equationsfor one charge system

Let us come back again to the original set ofMaxwell’s equations (6)–(9) for the reference systemat rest supplemented by the continuity equation

∂ρ

∂t+ (∇, j) = 0. (25)

In the phenomenological theory of electromagnetismthe hypothesis about the continuous nature of themedium was one of the foundations of Maxwell’stheoretical scheme. This point of view succeeded inuniting so many electromagnetic phenomena withoutthe necessity to consider a specific structure ofmatter. Nevertheless, a macroscopic character of thecharge conception defines all well-known limitationson Maxwell’s theory. For instance, the system ofequations (6)–(9) in a steady state approximationcorresponds to a quite particular case of continuousand closed conduction currents (motionless as awhole).

In 1895, the theory was extended by Lorentz for asystem of charged particles moving in vacuum. Since



then it has been widely assumed that the same basiclaws are valid microscopically as it is macroscopicallyin the case of original Maxwell’s equations. This meansthat in Lorentz form all macroscopic values of chargeand current densities have to be substituted by theirmicroscopic values. Let us write explicitly the Lorentzfield equations for one point-charged particle movingin vacuum [33]:

(∇,E) = 4πqδ(r− rq(t)), (26)

(∇,B) = 0, (27)

[∇,H] =4π

cqvδ(r− rq(t)) +

1

c

∂E

∂t, (28)

[∇,E] = −1

c

∂B

∂t, (29)

where rq(t) is the coordinate of the charge at themoment of time t.

In order to achieve a complete descriptionof a system consisting of fields and charges inthe framework of electromagnetic theory, Lorentzsupplemented (26)–(29) by the equation of motion:

dp

dt= eE+

e

c[v,B], (30)

where p is the momentum of the particle.The equation of motion (30) introduces an

expression for the mechanical force known as Lorentzforce which in the electron theory formulated byLorentz has a clear axiomatic and empirical status.Later on we shall discuss some disadvantages relatedwith the adopted status of the Lorentz forceconception.

Macroscopic Maxwell’s equations (6)–(9) may beobtained now from Lorentz’s equations (26)–(29) bysome statistical averaging process, using the structureof material media. The mathematical language forequations (26)–(29) is nowadays widely accepted inthe conventional classical electrodynamics. However,there is an ambiguity in the application of theseequations to the case of one uniformly movingcharge. A simple charge translation in space producesalterations of field components. Nevertheless, theycannot be treated in terms of Maxwell’s displacementcurrents. Strictly speaking, in this case all Maxwell’sdisplacement currents proportional to ∂E/∂t and∂B/∂t vanish from (28)–(29). This statement can bereasoned in two different ways:

1. ∂E/∂t = 0 and ∂B/∂t = 0, since all fieldcomponents of one uniformly moving chargeare implicit time-dependent functions (timedoes not enter as an independent parameterbut only through space variable) so thatfrom the mathematical standpoint only totaltime derivative makes sense in this casewhereas partial time derivative turns out tobe not adequate (time and distance are notindependent variables);

2. a non-zero value of ∂E/∂t and ∂B/∂t wouldimply a local variation of fields in time regardlessany change in the position of the charge(space co-ordinate is fixed when partial timederivative is taken) and, hence, would imply thepropagation of those local variations in form oftransverse electromagnetic waves.

This would strongly contradict the well-establishedin special relativity fact that one uniformly movingcharge does not produce any electromagnetic radiationat all.

Thus, a mathematically rigorous interpretationof (28),(29) in the case of a charge moving with aconstant velocity leads to the following conclusion:in a charge-free space the value of ∂E/∂t = 0 and,therefore, the value of rotH is also equal to zero infree space

[∇,H] =4π

cqvδ(r− rq(t)). (31)

On the other hand, field components of oneuniformly moving charge can be treated exactly in theframework of Lorentz’s transformations. Therefore,for any purpose exact relativistic expressions forelectric and magnetic fields and potentials should beapplied [33]

E = q(1− β2)(R−R~β)

(R−R~β)3, (32)

H =1

c[v,E], (33)

where ~β = v/c.Thus, we arrive here at the important conclusion:

generally speaking, according to special relativitytheory the value of rotH is not equal to zero in anypoint out of moving charge and takes a well-definedvalue

[∇,H] =1

c[∇, [v,E]]. (34)

For instance, this gives immediately a non-zerovalue of rotH along the direction of motion (X-axis):

[∇,H](x, x > x0) = q2β(1− β2)

(1− β)3(x− x0)3. (35)

The conflict with the previous statement ofthe equation (31) is inevitable. In order to obtainadequacy between the set of field equations (26)–(29) and their relativistic solutions in the case ofuniformly moving charge, it is necessary to consideran additional term like that considered in (34).As will be shown in continuation, this assumptionfor static and quasi-static fields is a supplementof Maxwell’s displacement currents introduced forexplicitly time varying fields (explanation of thelight as the propagation of transverse electromagneticwaves).



As it is well-known, the necessity of Maxwell’sdisplacement current was realised on the basis ofthe following formal reasoning. In order to makeequation (8) consistent with the electric chargeconservation law in form of continuity equation (25),Maxwell supplemented (8) with an additional term.However, for stationary processes, as we already haveseen, this term disappears and equation (8) becomesconsistent only with closed (or continuous going off toinfinity) currents

(∇, j) = 0. (36)

It is also a direct consequence of continuityequation (25) in any stationary state when allmagnitudes have to be treated as implicit time-dependent functions. Thereby, we meet here anotherdifficulty of Lorentz’s equations: uniform movementof a single charged particle (as an example of opensteady current), generally speaking, does not satisfythe limitations imposed by (36). It implies someadditional term to be taken into account in (36) tofulfil Maxwell’s hypothesis on the circuital characterof total currents (conduction plus displacementcurrents).

Let us have a close look on the continuityequation and its conventional interpretation. Indeveloping the mathematical formalism of his theoryMaxwell adopted Faraday’s idea of field tubes forelectric and magnetic fields as well as for electriccharge flow (conduction currents). As a consequence,in accordance with hydrodynamics language, thecontinuity equation was accepted as valid to expressthe hypothesis that a net sum of electric chargescould not be annihilated. In this case, the continuityequation reproduces the charge conservation law inthe given fixed volume V

dQ

dt=

∫∫∫

V

{

∂ρ

∂t+ (∇, j)

}

dV = 0 (37)

or in the form of a differential equation

∂ρ

∂t+ (∇, j) = 0,

(

dQ

dt= 0

)

. (38)

It should be remarked that equation (38) describesexclusively the conservation but not the change of theamount of charge (or matter) in the given volume V .In many scientific writings on electromagnetic theorythere is no clear distinction between these two aspects.If one wants to describe the change of something in thegiven volume V , the equation (37) should be replacedby a balance equation (see, for instance, [37])

dQ

dt=

d

dt

∫∫∫

V

ρdV = −

∫∫

S

(∇, j)dS, (39)

here j is a total current of electric charges througha surface S that bounds the given volume V . In

the mathematical language common to all physicaltheories it means that the rate of increase in thetotal quantity of electrostatic charge within any fixedvolume V is equal to the excess of the influx overthe efflux of current through a closed surface S.On contracting the surface to an infinitesimal spherearound a point one can arrive at the differentialequation [37]

dρ

dt+ (∇, j) = 0,

(

dQ

dt6= 0

)

. (40)

The balance equation (40) covers the continuityequation (38) as a particular case in which the amountof something (charge or matter) is kept constant inV during the course of time. Earlier we mentionedthat a single charge in motion, generally speaking,could not be treated in terms of the continuityequation (38). When the particle leaves the givenvolume, it violates locally the charge conservation,invalidating the continuity equation (38). Instead ofit the balance equation (40) has to be used. Onesimple method to prove that is to consider againthe example of point-charge moving with a constantvelocity. In particular, the charge density is assumedto have implicit time dependence as follows

ρ(r, rq(t)) = qδ(r− rq(t)), (41)

where r is a fixed distance from the point ofobservation to the origin of the reference system atrest; rq(t) and vq = drq/dt are the distance and thevelocity of the charge at the instant.

It is easy to show that the total density derivativewith respect to time consist of the convection termonly, since time enters in equation (41) as a parameter(∂ρ/∂t = 0):

dρ

dt=∂ρ

∂t+

{

d

dt(r− rq(t))

}

∇ρ = −(vq,∇ρ). (42)

Thus, the balance equation for a single chargedparticle is fulfilled directly:

− (vq,∇ρ) +∇(ρvq) = −(vq,∇ρ)+

(vq,∇ρ) = 0. (43)

The next step is to analyse equation (43) in termsof Maxwell’s hypothesis in respect to the circuitalcharacter of the total electric current (includingdisplacement current). In other words, the totalcurrent of one uniformly moving charge has to beformed by two contributions: the motion of the chargeitself (conduction current) and displacement currentin outer space:

(∇, (jcond + jdispl )) = 0, (44)

where jcond and jdispl are conduction anddisplacement currents, respectively.



Thus, we can rewrite (40) in the form ofequation (44)

(∇, jdispl ) =dρ

dt=

d

dt

(

1

4π(∇,E)

)

=

(

∇,1

4π

dE

dt

)

. (45)

It may be easily verified that two field operations∇ and d/dt are completely interchangeable in (45).Thus, for general motion of the charge when onecan disregard its size, Maxwell’s condition on a totalcurrent takes the following form (see for the sake ofcomparison the formula (42)) taking into account thestandard expansion of the total time derivative:

(∇, jdispl ) =1

4π

(

∇,

{

∂E

∂t−

(v,∇r)E− (a,∇v)E− . . . }) , (46)

here a is acceleration and further terms correspond toderivatives of non-uniform acceleration.

So far we have made use of the formalmathematical approach without any physicalinterpretation. More specifically, in calculatingthe full time derivative of E, the convectiveterm (second right-hand term in (46)) should beconsidered as implicit time-dependent (time variableis fixed when space partial derivative is taken)in agreement with the mathematical definitionof partial derivatives. In mathematical languageit means that all field alterations produced bya simple charge translation (convective part ofthe total derivative) take place at the same timein every space point (i.e. instantaneously). Thisinterpretation has no precedents in conventionalclassical electrodynamics for the case of arbitrarymotion whereas for uniformly moving charge thisdescription is the only possible formalism (in specialrelativity field lines of uniformly moving chargeremain radial, i.e. exhibit no retardation in respectto the space position of the charge). Turning backto (46), it is clear that the first right-hand termwith partial time derivative describes explicit time-dependent phenomena. Thus, in the same wayas it was independently concluded in the SectionVI, all field components can be split up into twoindependent classes with explicit E∗ and implicit E0

time dependencies, respectively:

dE

dt=∂E∗

∂t− (v,∇r)E0 − (a,∇v)E0 − . . . (47)

A general expression of full displacement current isthen taken by the formula:

jdispl =1

4π

∂E∗

∂t−1

4π(v,∇r)E0−

1

4π(a,∇v)E0 − . . . (48)

Let us stress here one subtle point which will beindispensable in the following discussion of relativisticinvariance properties of the present Helmholtz-typeapproach. The derivation of (47) has consideredthe partial time derivative to be independentfrom the space derivative in full agreement withthe mathematical formalism of partial derivatives.Thus, the time parameter of implicit time-dependentcomponents (let us call it t) comes into considerationas an afterthought through the space variable R(t)and, therefore, can be, in principle, considered asindependent from the time variable of explicit time-dependent components (in special relativity this isthe so-called proper time τ). As we will discuss later,special relativity does not distinguish these two timedependences and tacitly implies t = τ that leadsto the Lorentz invariance of electromagnetic fieldcomponents.

In order to come back to the previous discussionof the displacement current concept, let us remindthat our initial aim was to find a reasonable formfor Maxwell’s circuital condition (44). It would allowto relate field alterations in free space producedby one moving charge with the Maxwell conceptionof displacement current. From the standpoint ofconventional classical electrodynamics, the first termrepresents the well-known Maxwell displacementcurrent coming up only in non-steady processeswhereas the second term can be interpreted only asquasistationary due to its dependence on a chargetranslation in space (with time as implicit parameter).Further, we will call that term as ”convectiondisplacement current”. By the same token, the thirdright-hand term is due to uniform acceleration andcould be called ”uniform acceleration displacementcurrent” etc.

The above results motivate an important extensionof displacement current concept. First, it postulatesthe circuital character of the total electric current as itwas originally assumed by Maxwell. Second, it permitsto fulfil the circuital condition for non-steady as wellas for steady processes (static and quasistatic fields),contrary to the conventional approach. Let us give anequivalent mathematical expression of the convectiondisplacement current (in the case of single chargedparticle):

1

c(v,∇)E =

1

cv(∇,E)−

1

c[∇, [v,E]]. (49)

Accordingly, for our purpose we need to remindthat in the right-hand side of equation (28) the totalcurrent (jtot = jcond + jdispl ) must be considered as:

[∇,H] =4π

cqvδ(r− rq(t)) +

1

c

∂E

∂t−

1

cv(∇,E) +

1

c[∇, [v,B] + . . . (50)



For the sake of simplicity we omit acceleration andother expansion terms in this general formula but theyare tacitly implied.

This approach allows the treatment ofequation (29) in the same way as (28):

[∇,E] = −∂B

∂t+1

c(v,∇)B =

−∂B

∂t+1

cv(∇,B)−

1

c[∇, [v,B]] + . . . (51)

Turning back to the beginning of this Section wenote now that for uniform motion curlH is definedby (50) in every space point out of the charge in theexpected way (see (34)). As a final remark, the set ofequations (26),(27) and (50),(51) can be regarded asa generalized form of Maxwell-Lorentz system of fieldequations. In the next section they will be comparedwith modified Maxwell-Hertz equations extended onone charge system.

8. Reconsidered Maxwell-Hertztheory and relativisticallyinvariant formulation ofgeneralized Maxwell’sequations

Independently of Heaviside, the problem of themodification of Maxwell’s equations for bodies inmotion was posed by Hertz in his attempts to build upa comprehensive and consistent electrodynamics [38,39]. A starting point of that approach was thefundamental character of Faraday’s law of inductionrepresented for the first time by Maxwell in the formof integral equations,

∮

C

Hdl =4π

c

∫∫

S

jdS +1

c

d

dt

∫∫

S

EdS, (52)

∮

C

Edl = −d

dt

∫∫

S

BdS (53)

where C is a contour, S is a surface bounded by C.In qualitative physical language Faraday’s

observations had been expressed in form of thefollowing statement: the effect of magnetic inductionin the circuit C takes place always with the changeof the magnetic flux through the surface S regardlesswhether it relates to the change of intensity ofadjacent magnet or occurs due to the relative motion.More over, Faraday established that the same effectwas detected in a circuit at rest as well as in thatin motion. The latter fact provided the principalbasis of Hertz’s relativity principle based on Galileoinvariance. In order to avoid details of Hertz’s originalinvestigations [38, 39], let us only note its similarity

with the traditional non-relativistic treatment ofthe integral form of Faraday’s law [40]. Namely, ifthe circuit C is moving with a velocity v in somedirection, the total time derivative in (52),(53) musttake into account this motion (convection derivative)as well as the flux changes with time at a point(partial time derivative) [40],

∮

C

Edl = −1

c

d

dt

∫∫

S

BdS =

−1

c

{

∂

∂t+ (v,∇)

}∫∫

S

BdS, (54)

where S is any surface bounded by circuit C, movingtogether with a medium.

This approach is valid only for non-relativisticconsideration and leads to Galilean fieldtransformation 43. In Hertz’s theory any motionof the ether relative to the material particles had notbeen taken into account, so that the moving bodieswere regarded simply as homogeneous portions of themedium distinguished only by special values of electricand magnetic constants. Among the consequences ofsuch assumption, Hertz saw the necessity to movethe surface of integration in equations (52),(53) atthe same time with the moving medium. Thus thegeneration of a magnetic (or electric) force within amoving dielectric was calculated with implicit use ofGalilean invariance in equation (54) unless one makesany additional assumptions on the special characterof transformations in a moving frame of reference.

Recently, T. Phipps Jr. again drew attention tothe failure of Maxwell’s equations in partial timederivative to describe first-order effects related toconvective terms of total time derivatives [41, 42]. Heproposed to revive Hertz’s Galilean-invariant versionof Maxwell’s theory written in total time derivatives.He only differs from Hertz’s own interpretation of thevelocity parameter. However, in this review we shallshow how total time derivatives can be compatiblewith the requirements of special relativity in inertialframes of reference.

Let us now examine the case of a point sourceof electric and magnetic fields. In order to abstainfrom the use of moving contour C and surface S thatimplies a priori application of some relativity principle(Galileo’s or Einstein’s), we limit our considerationto a fixed region (C and S are at rest) whereas thesource is moving through a free space. According toFaraday’s law, there must be an electromotive forcein the contour C due to the flux changes with timeand convection derivatives simultaneously. Using themathematical language for total time derivatives, wearrive at the expression analogous to the differentialform (47),

dΦ

dt=∂Φ∗

∂t− (vs,∇r)Φ0 − (as,∇v)Φ0 − . . . (55)



making use of the definitions:

ΦE0 =

∫∫

S

E0(r− rs(t))dS, or,

ΦB0 =

∫∫

S

B0(r− rs(t))dS

(56)

and

Φ∗(E) =

∫∫

S

E∗(r, t)dS, or,

Φ∗(B) =

∫∫

S

B∗(r, t)dS,

(57)

where r is a fixed distance from the point ofobservation to the origin of the reference systems atrest; rs(t), vs = drs/dt, as = dvs/dt are the distance,instant velocity and instant acceleration of the electric(or magnetic) field source.

For the sake of simplicity, we can conserve for thepresent the same denomination of field flux in twoindependent parts of total time derivative (56), takinginto account additional (fixed space and fixed time)conditions, respectively, in the following expression:

d

dtΦ =

{

∂

∂t− (vs,∇r)− (as,∇v)− . . .

}

Φ. (58)

Using a well-known representation for the convectionpart in equation (56),

(v,∇)

∫∫

S

EdS =

∫∫

S

v(∇,E)dS+

∫∫

S

[∇[E,v]]dS, (59)

we obtain an alternative form of Maxwell’s integralequations (52),(53) for a moving electric charge in thereference system at rest,

∮

C

Hdl =4π

c

∫∫

S

jdS +1

c

∫∫

S

{

∂E

∂t−

v(∇,E)− [∇, [E,v]]− . . . } dS, (60)

∮

C

Edl = −1

c

∫∫

S

{

∂E

∂t+ [∇, [v,B]] + . . .

}

dS. (61)

Here we omit, for the sake of simplicity, accelerationand other expansion terms in general formula but they,of course, are tacitly implied.

Before going on to a more general consideration ofa large number of sources, it is worth to draw attentionthat we arrived to the most compact differential formof Maxwell- Hertz equations in the reference system

at rest [30],

(∇,E) = 4πρ, (62)

(∇,B) = 0, (63)

[∇,H] =4π

cρv +

1

c

dE

dt, (64)

[∇,H] =4π

cj+

1

c

∂E

∂t, (65)

where the total time derivative of any vector field valueE (or B) is,

dE

dt=∂E

∂t− (v,∇)E− (a,∇v)E− . . . (66)

The above-mentioned form (62)–(65) was for thefirst time admitted by Hertz for electrodynamics ofbodies in motion [38,39]. It was the covering theory forMaxwell’s original approach which became the limitcase of motionless medium (a reference system at rest)when values of instant velocity v, instant accelerationa etc. tend to zero in (66) leaving only partial timederivatives in agreement with (6)–(9). The differenceof the present approach [30] with Hertz’s coveringtheory (and with Phipps’ neo-Hertzian approach [41,42]) consists in the definition of the total timederivative (66) for a medium at rest (not in motionwith the possible implication of Galilean invariance).Below we shall demonstrate that the set (62)–(65)possesses invariance properties in any inertial frameof reference.

There is no difficulty in extending this approachto a many particle system, assuming the validityof the electrodynamics superposition principle. Thisextension is important in order to find out whetherthe generalised microscopic field equations cover theoriginal (macroscopic) Maxwell’s theory as a limitingcase. To do so one ought to take into account allprincipal restrictions of Maxwell’s equations (6)–(9)which deal only with a continuous and closed (or goingoff to infinity) conduction currents. They also have tobe motionless as a whole (static tubes of charge flow),admitting only the variation of current intensity.

Under these assumptions, it is quite easy toshow that the total (macroscopic) convection andothers displacement currents are cancelled by itself bysumming up all microscopic contributions,

∑

i

(vi,∇)Ei + (ai,∇v)Ei + · · · = 0,

∑

i

(vi,∇)Bi + (ai,∇v)Bi + · · · = 0.(67)

In other words, every additional terms in (50),(51)(as well as in (60),(61) disappears and we obtain theoriginal set of Maxwell macroscopic equations (6)–(9)for continuous and closed (or going off to infinity)conduction currents as a valid approximation.

To conclude this part we would like to notethat the set of equations (60),(61) can be called as



modified Maxwell-Hertz’s equations extended to onecharge system. It is easy to see that in this formthey are completely equivalent to modified Maxwell-Lorentz equations (50),(51) obtained with the help ofthe balance equation. Thus, differential and integralapproaches to extend the original Maxwell theory leadto the same result.

Let us write once again the generalised formof Maxwell-Lorentz equations explicitly for a singlemoving particle that is a source of electric andmagnetic fields simultaneously,

(∇,E) = 4πρ, (68)

(∇,B) = 0, (69)

[∇,H] =4π

cρv +

1

c

{

∂E

∂t−

(v,∇)E− (a,∇v)E− . . .

}

, (70)

[∇,E] = −1

c

∂B

∂t−1

c[∇, [v,B]]−

1

c(a,∇v)B− . . . , (71)

at the same time with the balance equation,

dρ

dt+ (∇, ρv) = 0. (72)

Splitting up field components into explicit and implicittime-dependent contributions E∗ (B∗) and E0 (B0),respectively, the basic field equations (70),(71) can berewritten as follows:

[∇,H] =4π

cρv +

1

c

{

∂E∗

∂t− (v,∇)E0−

(a,∇v)E0 − . . .

}

, (73)

[∇,E] = −1

c

∂B∗

∂t−1

c[∇, [v,B0]]−

1

c(a,∇v)B0 − . . . , (74)

where the total field values have two independentparts,

E = E0 +E∗ = E0(r− rq(t)) +E∗(r, t), (75)

B = B0 +B∗ = B0(r− rq(t)) +B∗(r, t). (76)

Here we note that implicit time-dependent fieldcomponents E0 and B0 depend only on the pointof observation and on the source position at aninstant whereas time varying-fields E∗ and B∗ dependexplicitly on time at a fixed point. The separationprocedure may be similarly extended to the electricand magnetic potentials introduced as

E = ∇ϕ, B = [∇,A], (77)

where

ϕ = ϕ0 + ϕ∗, A = A0 +A∗. (78)

Let us establish invariance of field equations intotal time derivatives. As far as in special relativitythe invariance is looking for inertial frames ofreference moving with a constant velocity v, thenin total time derivative expansion we should omitall acceleration and higher order terms. Thus, usingdefinitions (77),(78) we obtain from equation (74) that

E = −∇ϕ−1

c

∂A∗

∂t−1

c[v,B0]. (79)

Separation of implicit time-dependent fromexplicit time-dependent components in (79) isstraightforward

E0 = −∇ϕ0 −1

c[v,B0],

E∗ = −∇ϕ∗ −1

c

∂A∗

∂t.

(80)

Using this separation we obtain two second orderdifferential equations for total potentials (78)

∆A = −4π

cρv + F1, (81)

∆ϕ = −4πρ+ F2, (82)

where

F1 = ∇(∇,A0 +A∗)−1

c(v,∇)∇ϕ0+

1

c

∂∇ϕ∗

∂t+1

c2∂2A∗

∂t2, (83)

F2 = −1

c

(

∇,∂A∗

∂t

)

, .

(84)The second term in (83) can be easily transformedusing mathematical operations of field theory,

(v,∇)∇ϕ0 = ∇(v,∇ϕ0)− [v, [∇,∇ϕ0]]. (85)

Since [∇,∇(. . . )] is always equal to zero, we canrewrite F1 in a new form,

F1 = ∇

[{

(∇,A0)−1

c(v,∇ϕ0)

}

+

{

(∇,A∗) +1

c

∂ϕ∗

∂t

}]

+1

c2∂2A∗

∂t2. (86)

The principal feature of (86) consists in the fact thatall implicit and explicit time-dependent componentsof total electric and magnetic potentials enterindependently and, therefore, can be characterized byrespective gauge conditions,

∇ ·A0 −1

c(v,∇ϕ0) = 0, (87)

∇ ·A∗ +1

c

∂ϕ∗

∂t= 0. (88)

Lorentz’s gauge (88) is applicable now only for explicittime-dependent potentials and is invariant under



Lorentz’s transformations. It suggests that the propertime τ (let us call here τ the time variable of explicittime-dependent components in the entire spirit ofthe special relativity theory) for two inertial framesmoving with respect to each other are related byan imaginary rotation in space-time. The amount ofrotation depends on the relative velocity.

Implicit time-dependent potentials turn out to berelated through the novel gauge (87) which coversa well-known relationship between the componentsof electric and magnetic field potentials of uniformlymoving charge [33],

A0 =v

cϕ0. (89)

Strictly speaking, this relationship is true for Galileanas well as for Lorentz’s transformations. The differenceis attributed to a mathematical formulation ofpotentials in a new frame of reference. For instance,the Lorentz transformation corresponds to a rotationin the space-time plane whereas the Galilean oneleaves A0 and φ0 unchanged, for it is assumed thatno operation can rotate the time axis into the spaceaxis or vice versa. For Galilean invariance, the timedirection is supposed to be the same for all inertialframes of reference.

The expression (87) and all physically possibletransformations based on it, do not involve explicitlyany time dimension. The time t here can be addedas an afterthought (a parameter describing thespace coordinate R(t)). In above discussion of fulltime derivative we noted that time variable τ (forexplicit) and time parameter t (for implicit timebehaviours) are, generally speaking, independent. Ifwe assume, as they do it tacitly in special relativitywith no distinction of time behaviours, that bothtime variables are identical t = τ then we arriveto the implication of Lorentz’s invariance for A0

and φ0. Without additional hypothesis, the presentHelmholtzian approach cannot rule in favour ofGalilean or Lorentz’s transformations for implicittime dependences. The novel gauge (87) as wellas (89) are compatible with both of them. Theonly way of resolving this dilemma now seems tobe to suggest experimental verification of electricfield transformation in a moving frame. In fact, Leusrecently proposed such experiment [51]. A uniformbeam of electons moving with the velocity close toc has to produce electric field strenght which differsfor Galilean and Lorentz transformations.

Two gauge conditions (87) and (88) can be writtenjointly in a more compact formula that we can call thegeneralized Lorentz condition,

∇ ·A+1

c

dϕ

dt= 0, (90)

where A0 and φ0 are defined by (78) and the totaltime derivative is taken as in (66) up to the convectionterm.

We have not done it yet and will do it elsewherebut, perhaps, it is possible to prove that generalisedLorentz gauge (90) is valid also for non-inertial frames(acceleration and higher order terms in the total timederivative expansion). It would have a very attractiveconsequence that the field equations (62)–(65) writtenin total time derivatives could be considered invariantregardless a frame of reference (inertial or non-inertial).

Recall that in special relativity, electric andmagnetic potentials of uniformly moving charge A0

and ϕ0 are interrelated through the relationship (89)under application of Lorentz transformation. Here wefound that relativistic potentials (or components ofpotential four-vector) are connected in a more generalway (87). Another important aspect of the presentapproach can be attributed to the verification of someambiguity in the use of Lorentz gauge since it isapplicable only to explicit time-dependent potentials.In fact, there are some difficulties in the conventionalelectrodynamics concerning the inconsistency of thisgauge with implicit time-dependent functions. Thestandard Lorentz gauge condition

∇ ·A+1

c

∂ϕ

∂t= 0 (91)

is assumed to be valid for total electric andmagnetic potentials (transverse plus longitudinal) andis considered suffice to hold Maxwell’s equationsinvariant under Lorentz transformation. In the quasi-stationary approximation, the Lorentz condition inevery frame of references takes the form of the so-called radiation gauge [43],

∇ ·A = 0. (92)

It contradicts the expected relation (89) (or in ourapproach (87)) between electric and magnetic implicittime-dependent potentials. To make (92) consistentwith (89) in the given frame, they used to putan additional condition on the electric potentialsatisfying the so-called Coulomb gauge [43],

∇ ·A = 0, ϕ = 0. (93)

In mathematical language the invariance of implicittime-dependent fields in the conventional approachinvolves more strong limitations than those imposedpreviously by the Lorentz gauge. Generally speaking,the conventional classical electrodynamics has toadmit more than one invariance principle sinceevery time the Lorentz transformation is done, oneneeds also simultaneously to transform all physicalquantities in accordance with the Coulomb gauge (93).This problem was widely discussed and in thelanguage adopted in the general Lorentz group theory,is known as gauge dependent representation (or jointrepresentation) of the Lorentz group [43]. In fact, itmeans an additional non-relativistic adjustment of



electric potential, every time we change the frame ofreference. This difficulty vanishes when the relativisticgauge (87) for implicit time-dependent potentials isintroduced.

A rigorous consideration of (81),(82) gives anotherimportant conclusion: simultaneous application oftwo independent gauge transformations (87),(88)discomposes the initial set (68)–(71) into two pairsof differential equations, namely,

∆A0 = −4π

cρv, (94)

∆ϕ0 = −4πρ (95)

at the same time with the homogeneous waveequations,

∆A∗ −4π

c

∂2A∗

∂t2= 0, (96)

∆ϕ∗ −4π

c

∂2ϕ∗

∂t2= 0. (97)

Likewise (89), Poisson’s second order differentialequations (94),(95) for electric and magneticpotentials covers the conventional approach in thesteady-state approximation and can be considered asvalid extension to implicit time-dependent potentials.A general solution, as one would expect, satisfiesa pair of uncoupled inhomogeneous D’Alembert’sequations. It can be verified by summing up (94),(95)and (96),(97) (here we omit premeditatedly allboundary conditions for the sake of simplicity),

∆A−1

c2∂2A

∂t2= −

4π

cρv, (98)

∆ϕ−1

c2∂2ϕ

∂t2= −4πρ, (99)

where the total values A and φ are defined by (78).The same result has been derived in the Section

VI independently, starting from the analysis ofboundary conditions for inhomogeneous D’Alembert’sequations [29]. It has been shown mathematically thatany general solution of Maxwell’s equations has tobe obligatory written as a superposition of implicitand explicit time-dependent functions. The aboveanalysis endorsed that conclusion by demonstratingrelativistic invariance of (98),(99) and, therefore, (68)–(71), if and only if the relativistic gauge condition (90)is satisfied by respective components of the totalfield. Thus, the covering theory based on the totaltime derivatives possesses all necessary relativisticsymmetry properties.

To conclude this section, some remarks worthto be done concerning the empirical and axiomaticstatus of the Lorentz force concept in the electrontheory formulated by Lorentz. In the first versionof Maxwell’s theory published under the name ”OnPhysical Lines of Force” (1861–1862) there wasalready admitted an unified character of a full

electromotive force in the conductor in motion bydescribing it as [44,45]

E = −∇ϕ−1

c

∂A

∂t+1

c[v,B], (100)

where (1) the first term is the electrostatic force, (2)the second is the force of magnetic induction and (3)the third one is the force of electromagnetic inductiondue to the conductor motion. Later investigationsbegan to distinguish between the electric force ina moving body and the electric force in the etherthrough which the body was moving and as a result,did not consider 1/c[v,B] as a full-value part of theelectric field, as afterwards was argued by Hertz. Thisdistinction was one of the basic premises in Lorentz’selectron theory and was closely related to the specialstatus of the Lorentz force conception. It also can benoted in the way how it forms part the formalism ofthe conventional field theory. The equation of motionwith total time derivative (30) should be contrastedfrom the form of partial differential equations (26)–(29). It does not correspond to the mathematicalstructure of a consistent system.

In special relativity the Lorentz force, is theresult of the transformation of the componentsof Minkowski’s force. Thus, the expression forthe Lorentz force can be obtained in a purelymathematical way from the general relativisticrelationships [33]. In the present Helmholtz-typeapproach the Lorentz force is one of the terms in thetotal time derivative expansion. This has advantageto be consistent by itself with the set of generalisedfield equations. There is no need to supplementMaxwell’s theory with equation of motion. Givensuch interpretation of Lorentz’s force, we remindthat in our approach it can be related only toimplicit time-dependent components whereas in theconventional electrodynamics it was the product ofthe total magnetic field leading to some ambiguities.In this respect it is interesting to mention very recentworks by Wesley [46] and Phipps [47] challengingthe sufficiency of the Lorentz force law to describeexperimental observations. They advocated the use oftotal time derivatives (in the above-mentioned neo-Hertzian sense) and their data roughly agreed withtheoretical predictions, while the conventional theorydoes not predict any effect at all.

9. Analysis of classicaldifficulties and theHamiltonian form ofgeneralized Maxwell’sequations

Maxwell’s equation in the form of D‘Alembert’sequations lend themselves to the covariant descriptionand are in agreement with the requirements of special



relativity mathematical formalism. For four-vectors ofseparated potentials, the standard four-vector form ofbasic equations can be used. We immediately have thefollowing expressions:

(

∆−1

c2∂2

∂t2

)

[A0µ +A∗µ] = −

4π

cjµ,

µ = 0, 1, 2, 3, (101)

where

A0µ +A∗µ = (ϕ0 + ϕ∗,A0 +A∗),

jµ = (cρ, j). (102)

The first Poisson’s operator ∆ acts only on thefour-vector of implicit time-dependent componentsA0µ whereas ∆ and ∂2/∂t2 act together on explicittime-dependent components A∗

µ. The equation (101)is relativistically invariant under the generalizedrelativistic Lorentz gauge condition (90). To give somesubstance to the above formality we exhibit explicitlyPoisson’s equation for implicit time-dependent four-vector A0µ

∆A0µ = −4π

cjµ, (103)

whereA0µ = (ϕ0,A0). (104)

As we demonstrated in the previous Section,equation (103) is relativistically invariant underthe Lorentz gauge (83) if the time parameter there is considered identintical to the time variableτ for explicit time components A∗

µ. Under thiscondition, Poisson’s differential operator ∆ acting onimplicit time-dependent potentials becomes invariantin every inertial frame of reference under Lorentz’stransformations. This is due to the fact that timevariable t is not any more independent from τ asit is assumed for partial derivatives in full timederivative formalism. Non-covariant representation ofD’Alembert differential operator ∆ − ∂2/∂t2 or, inother words, non-covariance of equation (103) is nota stumbling block here for relativistic invarianceand endorses the well-known fact that covarianceis not necessary, it is only sufficient for relativisticinvariance.

More over, it is tacitly implied in the conventionalapproach and corresponds to the relativistic invarianceof field components of an uniformly moving charge(implicit time-dependent functions) that remain radiallines of electric field regardless the choice of inertialframe. This fact is odd to contemplate in the Faraday-Maxwell electrodynamics based on the concept of local(contact) field which mathematically fits explicit time-dependent behaviour.

Actually, electric field lines of an unmoving chargeare radial. Under Lorentz’s transformation into theinertial frame of reference moving with the velocity v

explicit time- dependence does not appear and fieldlines remain radial. Without any approximation, theinfluence of a possible retarded effect cancels itself atany distance from the moving charge.

On the other hand, the conventional theoryis unable to give any reasonable interpretationdescribing a transition from an uniform movementof a charge into an arbitrary one and then againinto uniform over a limited interval of time. In thiscase, the first and the latter solutions can be givenexactly by the Lorentz transformation as implicittime- dependent functions. What mechanism changesthem at a distance unreachable for retarded Lienard-Wiechert fields? The lack of continuity between thecorresponding solutions is obvious. It has the samenature as discussed in the Section VI.

The Helmholtz-type approach based on separationof implicit and explicit time behaviours, alsohighlights serious ambiguities associated withthe self-energy concept in the framework of theconventional electrodynamics. Let us confine ourprevious qualitative reasoning to the example ofelectrostatics. A rigorous analysis will be done laterapplying Hamiltonian formalism.

In electrostatics the total energy of N interactingcharges is

W =1

2

N∑

i=1

∑

j 6=1

eiej

|ri − rj |. (105)

Here, the infinite self-energy terms (i = j) are omittedin the double sum. The expression obtained byMaxwell for the energy in an electric field, expressedas a volume integral over the field, is [45]

W =1

2

∫

V

E2dV (106)

This corresponds to Maxwell’s idea that thesystem energy must be stored somewhere in space.The expression (106) includes self-energy terms andin the case of point charges they make infinitecontributions to the integral.

In a relativistically covariant formulation theconservation of energy and the conservation ofmomentum are not independent principles. Inparticular, the local form of energy-momentumconservation can be written in a covariant form, usingthe energy- momentum tensor,

∂Tµν

∂xµ= 0. (107)

For an electromagnetic field, it is well-knownthat (107) can be strictly satisfied only for a freefield (when a charge is not taken into account),whereas, for the total field of a charge this is nottrue, since (107) is not satisfied mathematically (four-dimensional analogy of Gauss’s theorem). As everyoneknows in classical electrodynamics, this fact gives



rise to the ”electromagnetic mass” concept, whichviolates the exact relativistic mass-energy relationship(E = mc2). Let us examine this problem in aless formal manner. The equivalent three-dimensionalform of (107) is the formula (3). The amount ofelectrostatic self-energy of an unmoving charge in agiven volume V is proportional to E2 (see (106)).According to (107) (or (??)), in a new inertial frame,energy density W as well as electric field E mustbe, generally speaking, an explicit time-dependentfunction (∂W/∂t 6= 0 and ∂E/∂t 6= 0). On the otherhand, the electric field strength of an unmoving chargekeeps its implicit time behaviour under Lorentz’stransformation (∂E/∂t = 0). It contradicts thecommonly accepted view that electrostatic self-energyis stored locally in space.

In the framework of Helmholtian approach theseambiguities can be cleared up. Actually, lookingback at the general solution (23) with explicitlyexposed longitudinal and transverse components,the term E0 is responsible for bipartite interactionbetween charges. No local energy conservation lawin the form (107) or (3) is adequate for implicittime-dependent field E0. We suggest that theoriginal mathematical form (105) should be used.Nevertheless, the local form (107) or (3) is perfectlyadequate for explicitly time-dependent free fieldE∗. Clear separation on implicit and explicit timedependencies in Helmhotz-type electrodynamics leadsto the correspondent separation in the total electricfield energy expression,

W =1

2

N∑

i=1

∑

j 6=1

eiej

|ri − rj |+1

2

∫

V

E2dV. (108)

This is a logical conclusion of our qualitativereasoning that will be mathematically verified belowin Hamiltonian formulation.

Let us discuss generalised field equations in totaltime derivatives (62)–(65) for arbitrary fields from thestandpoint of the principle of least action. Applyingexplicitly separation of field components we have notdone any modifications in the general four- vectorrepresentation of Maxwell equations (101),(102). Weonly noted that in this case the set of field equationscan be split up for equations of implicit and explicittime-dependent potentials such as (16)–(19) or (94)–(97). A relativistic action for implicit time potentialA0µ can be written in the conventional form [33],

Sm + Smf =

2∫

1

(−N∑

a=1

macdsa−

N∑

a=1

eac

3∑

µ=0

A0(ma)dxµa). (109)

This expression is sufficient to derive the first couple ofequations (16),(17) (or (94),(95)) from the least action

principle. It can be directly verified by rewriting thesecond term in (109) as

Smf = −1

c

∫

∑

µ

A0µjµdV dt (110)

and using Dirac’s expression for four-current,

jµ(r, t) =∑

a

[

−ea4π∆

(

a

|r− ra|

)]

Uµa, (111)

where Uµa is the four-velocity of the charged particlea.

Let us consider the second pair ofequations (18),(19) or (96),(97) defining explicitlytime-dependent potentials (ϕ∗, A∗) or A∗

µ inrepresentation (101). It is easy to see that theconventional Hamiltonian form can be adopted todescribe transverse components of electromagneticfield [33],

St = −1

16π

∫

∑

µ,ν

FµνFµνdV dt, (112)

where

Fµν =∂A∗

ν

∂xµ−∂A∗

µ

∂xν. (113)

Finally, it remains to be proved that the variationalderivative,

δSf = −

∫

∑

µ

(

1

4π

∑

ν

∂Fµν

∂xν

)

δA∗µdV dt (114)

can be used to obtain the covariant analogueof (18),(19) (or (95)–(97)) in the following form:

∑

ν

∂

∂xνFµν =

∑

ν

∂

∂xν

[

∂A∗ν

∂xµ−∂A∗µ

∂xν

]

= 0. (115)

The difference with the conventional interpretationconsists in the way electromagnetic potentials A0µ

and A∗µ take part in this Hamiltonian formulation.

In the light of the Helmholtzian approach, theelectromagnetic energy-momentum tensor demandssome corrections in the interpretation of itsmathematical formulation [33],

Tµν = −1

4π

∑

p

Fµp F

νp +1

16πgµν

∑

β,γ

FβγFβγ . (116)

As a consequence of the definition (113), it candescribe the energy-momentum conservation law for,exclusively, free electromagnetic field as follows,

∑

ν

∂Tµν

∂xν= 0. (117)

Consequently, contrary to the traditionalinterpretation, the quantity F µν can be defined



as a transverse electromagnetic field tensor becauseit contains only transverse field components but nottotal as in the conventional approach. There is nomore violation of (117) even if the charge is taken intoaccount, contrary to the situation in the conventionaltheory (see above discussion of equation (107)).

Strictly speaking, the total field energy W shouldbe split up into two parts:

1. energy Wmf longitudinal implicit time-dependent fields responsible for electro- andmagnetostatic interaction between charges(non-local term) and

2. energy Wf of transverse explicitly time-dependent electromagnetic field (local term),

W =Wmf +Wf . (118)

Following these results we suggest that the conceptof potential (non-local) energy and potential forcesmust be re-established in classical electrodynamics.So, the system of charges and currents in absenceof free electromagnetic field Wf must be consideredas a conservative system without any idealization.Introduction of interaction energy Wmf in theform (109) equivalent to (105) definitely eliminates theproblem of infinities of self-energy terms.

The physical meaning of the Poynting vectorhas been changed notably. So far the conventionaltheory dealt with it as a quantity describing dynamicproperties of the total electromagnetic field. Now itis adequate only for conservation law in the form ofequation (117) and, therefore, makes sense only fortransverse components of electromagnetic field. Long-time well-known ambiguities related to the definitionof the field energy location in space, do not take placein Helmholtz-type electrodynamics. In particular,there should be no flux of electromagnetic energyfor stationary currents. Contrary, the conventionalapproach predicts senseless flux of energy coming frominfinity towards the current [28].

At the end of this Section we would like topresent a valuable mechanical analogy of Maxwell’sequations in the form of (16)–(19) (or (94)–(97)).It helps to understand why general solutions mustbe split up into (orthogonal) potentials (14),(15)(or (78)) with explicit and implicit time-dependence,respectively. The set of differential equations forelastic waves in an isotropic media (see [48]) canbe considered as mechanical analogy of Maxwell’sequations to endorse Helmholtzinan foundations ofclassical electrodynamics

∂2ul

∂t2− c2l∆ul = 0, (119)

∂2ut

∂t2− c2t∆ut = 0, (120)

here cl and ct are spreading velocities of longitudinaland transverse waves, respectively.

The general solution of (110),(111) is the sum oftwo independent and orthogonal terms correspondingto longitudinal ul and transversal ut waves,

u = ul + ut. (121)

If the longitudinal spreading velocity approachesformally to infinity (cl → ∞) then (119) transformsinto Laplace’s equation whereas the general solutionturns out to have an implicit time dependence.Solution (121) takes the form of separated potentialsolution (14),(15) (or (78)). Longitudinal componentdoes not vanish in this limit from mathematicalconsideration, though the time behaviour undergoes afundamental transformation. Thus, longitudinal wavesul have to be considered as full-value solution ofthe total system of differential equations (119),(120).It allows to understand why Hertz had no right toeliminate longitudinal components from mathematicalsolutions of Helmholtz’s theory in Maxwellian limit(see corresponding discussion in the Section II,quote [21]).

To end the Section we conclude that theidea of non-local interactions is enclosed into theframework of Helmholtzian electromagnetic theoryas unambiguous mathematical feature. On theother hand, some of the quantum mechanicaleffects like Aharonov-Bohm effect, violation ofthe Bell’s inequalities etc. point out indirectlyon the possibility of non-local interactions inelectromagnetism. During the last century modernphysics had faced fundamental difficulties in unifyingrelativistic classical physics elaborated mainly in theframework of the locality concept of relativistic theoryand quantum physics characterized essentially by theemergence of non-locality. Regretfully, nowadays thereis no rigorous mutual correspondence between thesetwo fundamental areas of physical science. Helmholtz-type approach offers an altogether more promisingsolution.

10. Non-radiation condition forfree electromagnetic field

In this section we make a qualitative discussion ofthe energy balance between the system of interactingcharged particles and free electromagnetic field,namely, energy and momentum lost by radiation. Wemust examine carefully one essential difference in theelectromagnetic energy interpretation. Let us writethe total relativistic action as

S = Sm + Smf + Sf . (122)

Although we adopt denominations used in theconventional theory, the physical essence of the lasttwo terms has changed significantly. Usually, theinteraction between particles and electromagnetic



field was attributed to Smf whereas the propertiesof electromagnetic field manifested itself by theadditional term Sf .

In the new approach, no concept of local (contact)field as intermediary is suffice to describe theinteraction between charges (currents). Hence, Smf

is understood in terms of non-local conservativefield. Local field (transverse components attributed toradiation or free field) is represented by the last termSf . The possible free field interaction with the systemof charges (currents) depends entirely on its locationin space.

The internal structure of the relativisticaction (122), where local and non-local contributionsare separated in different terms, allow us to considerthe isolated system of charged particles and free fieldas consisting of two corresponding subsystems. Eachof the subsystems may be completely independentif there is no mutual interaction (for instance,free electromagnetic field is located far from thegiven region of charges and currents). In static andsteady approximations the subsystem of charges andcurrents can be considered as conservative. The totalHamiltonian of our isolated system can be split upinto two corresponding parts,

H = H1 +H2, (123)

where H1 is the Hamiltonian of the conservativesystem of charges and currents. It involves apart fromelectro- and magnetostatic energy also mechanicalenergy of particles (corresponding to the actionSm + Smf ). H2 is the Hamiltonian of the freeelectromagnetic field (corresponds to the action Sf ).

It is important to note that the separation intotwo subsystems is possible only in this new approach.In the conventional interpretation of Sf described theproperties of the total electromagnetic field withoutmaking any distinction between longitudinal andtransverse components. In static approximation theterm Sf should vanish in our approach but in theconventional approach it is not zero, and correspondsto the field self-energy [33],

Sf =

t2∫

t1

Lfdt, (124)

where

Lf =1

8π

∫

V

(E2 −B2)dV. (125)

Here E and B are the total electric and magnetic fieldstrengths, respectively, admitted in the conventionalapproach.

In relativistic theory, energy is zero componentof the four-momentum. For an isolated systemthe total Hamiltonian is not time-dependentand, therefore, the energy conservation as well

as the momentum conservation may be treatedindependently. Conservation laws are useful tools toget some general insight on the problem even whencomplete solutions are very difficult to find.

Let us consider the case where charges (currents)and free electromagnetic field are located in thesame region and become interacting. Internal forces ofmutual reaction between two subsystems are usuallynamed as internal dissipative forces. They carry outthe energy exchange inside the total isolated system.In terms of the Hamiltonian formalism [49] it can beexpressed as a corresponding Hamiltonian evolution,

dH1,2

dt=∂H1,2

∂t+ P ex

1,2 + P in1,2, (126)

where P ex1,2 (P in

1,2) is the power of the external (internal)forces acting on two subsystems, respectively. In ourcase P ex

1 and P ex2 appear as a result of mutual

interaction. On the other hand, any internal non-potential force in the first subsystem can also causeenergy dissipation (P in

1 ). Even in the absence of areal mechanical friction, other internal non-potentialsforces (for example, inhomogeneous gyroscopic forces)can still act in this subsystem and dissipate energy. Inother words, if initially there is no free electromagneticfield (H2 = 0), it can be created by internal non-potential forces (P in

1 ) acting in the first subsystem(H2 is no more zero). It means that energy is lost byradiation in the subsystem of charges and currents.In mathematical language the corresponding energybalance can be written as follows:

d

dt(H1 +H2) = 0 (127)

where d/dt(H1) and d/dt(H2) are energy change ratesfor the first and the second subsystems, respectively. Itmight be easily noted that the energy balance (127) issymmetrical in respect to time reversion. This featureis in accordance with the time symmetry of generalizedMaxwell’s equations in total time derivatives.

Here it should be specially noted that the energybalance in the conventional electrodynamics is notalways reversible in time. For instance, the dipolarradiation energy is proportional to the square of acharge acceleration. Thus, the total dipolar radiationenergy is always positive regardless the mathematicaloperation on the reversion of time (negative signof time variable). It indicates (not proves) thatconventional Maxwell’s equations may not be entirelytime symmetrical. Perhaps, the failure to build upclassical Rutherford’s model of the atom is due to thatdeficiency.

To end this Section we formulate the previousstatement about the energy conservation as thecondition of non-radiation of the free electromagneticfield: if in an isolated system of charges (currents) inthe absence of free electromagnetic field (d/dt(H1) =0), all internal non-potential forces are compensated



or do not exist then this system will not produce(radiate) free electromagnetic field (remains zero H2 =0) and will keep the conservative system itself.

This implies not only an equilibrium betweenradiation and absorption but no radiation at all.This possibility would be of particular interest inthe attempt to understand the quantum mechanicalprinciples and may be explored elsewhere.

11. Conclusions

From the modern philosophy of science standpoint,then, it is plain that Helmholtz’s theory could not bedefinitely ruled out, since it was perfectly falsifiable byPopper’s criterion, did not contradict any observablefact and predicted all well-known electromagneticphenomena in Maxwellian limit. As the relevanthistory of physics literature shows, Hertz himselfindicated in his last experimental work that hedefinitely could not disprove any reference to actionat a distance. Moreover, Hertz’s crucial experimentsprovided no explicit information on longitudinalcomponents which were such an essential featureof Helmholtz’s theory. From this point they couldnot be considered conclusive to refute alternativeapproach. This may indicate that there was aneed for further experimental investigations of Hertzinto the possibility that Helmholtz’s theoreticallypredicted longitudinal electromagnetic componentsdid in fact exist. More explicit and direct experimentalinformation on the existence of longitudinal forcesshould have determined the Hertz choice betweenMaxwell’s and Helmholtz’s theories. As far as thisexperimental part was not fulfilled, the problemof the completeness of Hertz’s experiments onpropagation of electromagnetic interactions could notbe considered as fully resolved. Thus, according to themodern criteria of scientific method, philosophy andmethodology of science say that Hertz’s experimentscannot be considered as conclusive at some points asit is generally implied.

Mathematical analysis of Maxwell-Lorentzequations for one charge system shows ambiguousconventional treatment of implicit and explicit timedependencies. It was found that all conventionalapproach is beset with the same ambiguity leadingto many mathematical inconsistencies and paradoxes.We suggested that it is possible to solve thosedifficulties by clear distinguishing between functionswith implicit and explicit time dependencies.This consideration provided self-consistency formathematical description of electromagnetic theory.Maxwell’s equations resulted to be written in fulltime derivatives that consistently covers conventionalapproach. We showed that the covering theorypossesses all necessary relativistic invarianceproperties for inertial frames of references. UsualLorentz’s gauge condition is covered by generalised

gauge condition. It promises to keep generalisedMaxwell’s equations invariant also in non-inertialframes but this issue will be studied elsewhere.

Consistent mathematical interpretation ofgeneralised field equations gives a solid ground forHelmholtzian foundations of classical electrodynamicsbased on the superposition of implicit time dependentlongitudinal and explicit time dependent transversecomponents. This approach demonstrates advantagesover the conventional field description in eliminatingthe large number of internal inconsistencies fromclassical electrodynamics and promises more adequatesolution to fundamental problems of modern physics.Recent experimental data [46, 47] highlightedcertain limitations of the conventional approach.Graneau’s monograph on modern Newtonianelectrodynamics [50] reviewed numerous researchdata in exploding wires, railguns, differentelectromagnetic accelerators, jet propulsion inliquid metals, arc plasma explosions, capillary fusionetc. as unambiguous indication on the existence ofnon-local longitudinal forces. Thus, a new area ofelectromagnetic research emerges that is interested inthe study of longitudinal components by experimentalas well as by theoretical means.

As a final conclusion of this review, we would liketo quote P. Duhem’s significant words [27]: ”... Anexcessive admiration for Maxwell’s work has led manyphysicists to the view that it does not matter whethera theory is logical or absurd, all it is required to do issuggest experiments: A day will come, I am certain,when it will be recognised: that above all the objectsof a theory is to bring classification and order intothe chaos of facts shown by experience. Then it willbe acknowledged that Helmholtz’s electrodynamics is afine work and that I did well to adhere to it. Logiccan be patient, for it is eternal”.

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