Relational electromagnetism and the electromagnetic force
aa r X i v : . [ phy s i c s . c l a ss - ph ] F e b Relational electromagnetism and theelectromagnetic force
M.A. Natiello † and H.G. Solari ‡ † Centre for Mathematical Sciences, Lund University; ‡ Departamento de Física, FCEN-UBA and IFIBA-CONICET
Abstract
The force exerted by an electromagnetic body on another body in rel-ative motion, and its minimal expression, the force on moving charges or
Lorentz’ force constitute the link between electromagnetism and mechan-ics. Expressions for the force were produced first by Maxwell and laterby H. A. Lorentz, but their expressions disagree. The construction pro-cess was the result, in both cases, of analogies rooted in the idea of theether. Yet, the expression of the force has remained despite its produc-tion context. We present a path to the electromagnetic force that startsfrom Ludwig Lorenz’ relational electromagnetism. The present mathem-atical abduction does not rest on analogies. Following this path we showthat relational electromagnetism, as pursued by the Göttingen school, isconsistent with Maxwell’s transformation laws and compatible with theidea that the “speed of light” takes the same value in all (inertial) framesof reference, while it cannot be conceived on the basis of analogies withmaterial motion.
ORCID numbers:
M. A. Natiello: 0000-0002-9481-7454 H. G. Solari:0000-0003-4287-1878
The standard presentation of the Lorentz force (Panofsky and Phillips, 1955;Jackson, 1962; Feynman et al, 1965; Purcell, 1965) is entirely pragmatic; theforce is simply presented as complementing Maxwell’s equations and its ultimatesupport is left to its success, after using it intuitively, as well as to its consistencewith Lorentz transformations, which lie in the core of Special Relativity. Thereis no derivation of the force from observations or first principles in the citedbooks or any other book of our knowledge. Such an approximation reflects an a-posteriori epistemic attitude that Einstein put forward in the following form:1Physics constitutes a logical system of thought which is in a state ofevolution, and whose basis cannot be obtained through distillationby any inductive method from the experiences lived through, butwhich can only be attained by free invention. The justification (truthcontent) of the system rests in the proof of usefulness of the resultingtheorems on the basis of sense experiences, where the relations of thelatter to the former can only be comprehended intuitively. Evolutionis going on in the direction of increasing simplicity of the logicalbasis". (Einstein, 1940)In turn, Einstein’s view is a continuation of Hertz’ view (Hertz, 1893), who,in an attitude in common with other believers of the ether (such as Heaviside(Heaviside, 2011)) who resent the mechanical origin of Maxwell’s derivation,detached the production of the theory from its equations and added a free in-terpretation of the equations (Hertz offered four different interpretations, beingthe preferred one his own conception of the ether).Lorentz’ place in the development of electromagnetism is at a turning pointof it. Two forms of conceiving science came in conflict by the mid XIX-th cen-tury: we will label them the Göttingen and Berlin schools. Maxwell relies onFranz Neumann’s (Neumann, 1846) mathematical formulation of the inductionlaw which is the starting point for his elaboration ([552], Maxwell, 1873). F.Neumann as well as his son Carl, Weber, Riemann and Gauss belong to Göt-tingen. Hertz, Heaviside, Clausius and others resent of Maxwell’s constructionas being “impure” noticing that the formulae are not the result of elaborationsin terms of the ether. Hertz and Clausius stem from the (Prussian) schoolof Berlin, whose approach contrasts with the attitude of the Göttingen school(guided by Gauss). The Berlin school linked understanding with images, men-tal representations of the observed, the bild theory (D’Agostino, 2004) afterHelmholtz and Hertz, whereas the Göttingen school aimed to provide a math-ematical organisation to electromagnetic knowledge where equations and theirabduction/conceptualisation/construction constitute an unity. Lorentz was fluent in both epistemic traditions and his re-elaboration ofMaxwell tries to keep a balance on the different views, while committed to theether interpretation. However, these two views cannot coexist and Lorentz’approach turned to be incompatible with both.His deduction of the electromagnetic force from Maxwell’s background ispresented in (§74-§80, Lorentz, 1892), but it is marred by the use of steps foun-ded only in the idea of the ether. Lorentz assumed a simultaneous displacementof a body with respect to the ether and to the complement of electromagneticbodies. As soon as the argumentation rests on the ether, it falls apart when the It is worth to keep in mind that non-scientific, irrational, factors may have played a roleas well since by 1868 Göttingen (Hannover) was annexed to Prussia (hosting the school ofBerlin). Clausius, a Prussian nationalist and scientist, promoted at the same time an attackon the scientific position on electromagnetism held by the Göttingen school, from the pointof view of the ether and of Berlin’s epistemic approach (See (Clausius, 1869) and (Archibald,1986; Natiello and Solari, 2019)).
What came to be known as the Lorentz force is the outcome of the derivationin (§74-§80, Lorentz, 1892) for the force exerted on a distinguished body (theprobe) by the ether. The action of the ether is assumed to be conveyed bythe fields E and B , which satisfy Maxwell equations according to Heaviside,i.e., regarded from a reference system at rest with the ether. Lorentz takes theexpressions for kinetic and potential electromagnetic energies from Maxwell,combining them in Lagrangian form to obtain (in §77) ∇ × E = − ∂B/∂t , the“fourth” Maxwell equation (not explicitly stated by Maxwell). Subsequently,in §80 the Lorentz force is extracted from the Lagrangian, for a probe takento be a rigid solid, consisting of particles with some charge density (nonzeroonly at the location of each particle) that adds up to a smooth charge dens-ity ρ . The procedure in both §77 and §80 is not purely mathematical but itresponds to the assumption on the ether, to which fields and velocities refer.Three main differences with Maxwell are that (a) Maxwell never wrote down a3agrangian but worked with the energy contributions instead, (b) Maxwell con-siders virtual displacements of the secondary circuit while Lorentz considers thata virtual displacement corresponds at the same time (and the same amount) toa displacement relative to the electromagnetic bodies producing the fields andwith respect to the ether (in §76). One can hardly imagine something differentthan the ether being the reference space associated to the sources of the field.Finally, (c), the velocity of the charge in motion is for Lorentz the velocitywith respect to the ether, while Maxwell’s velocity in [598] could be interpretedeither as relative to the primary circuit (the source of electromagnetic fields) –asin Faraday’s original concept– or relative to the ether. The current use of theforce adheres to Faraday’s view, thus becoming inconsistent with the derivation,inasmuch the latter rests on the ether.Before we proceed to a derivation of the force without resting upon the idea ofthe ether, it is worth to consider the difference between Lorentz’ and Maxwell’sexpressions for the force, since Lorentz appears to be following Maxwell stepsup to some point. Maxwell restrained from making an early decision on thenature of electricity. He considered a piece of matter carrying a current andincluded the contribution of the “displacement current” just by analogy withgalvanic currents. The resulting force was determined up to a contributionconsisting of the gradient of a potential. Lorentz, in turn, adopted the idea ofmaterial corpuscles (Weber’s hypothesis (Weber, 1846)). He considered that nogalvanic current was present in the moving body and introduced the Lorentzcurrent which is no more than the description of a moving (charged) body withrespect to the reference frame of the ether, made in classical terms. For bothof them, Maxwell and Lorentz, the body was a rigid solid and the force wasto be determined by a variational method. The influence of the relational viewheld by the Göttingen school on Lorentz is difficult to gauge as he writes withrespect to (instantaneous) action at a distance: “The influence that was sufferedby a particle B due to the vicinity of a second one A, indeed depends on themotion of the latter, but not on its instantaneous motion. Much more relevantis the motion of A some time earlier, and the adopted law corresponds to therequirement for the theory of electrodynamics, that was presented by Gauss in1845 in his known letter to Weber (bd.5 p. 627-629, Gauss, 1870)”.
Absolute space had earned a bad reputation by the beginning of the XIX century,only the relational space appeared to matter for scientists. The difficulties ofconceiving the wave propagation of electromagnetic phenomena as well as thephilosophical belief that “matter acts where it is” emerged as the concept of anelectromagnetic ether (having different and sometimes contradictory propertiesfor different authors). Lorentz shortly addressed this problem: Reverted brilliantly by Faraday into: matter is were it acts. Indicating that matter canonly be inferred but not sensed, thus matter is a belief while action is real. absolute rest of the aether, is self-evident; this expression would not even make sense ... When I sayfor the sake of brevity, that the aether would be at rest, then thisonly means that one part of this medium does not move against theother one and that all perceptible motions are relative motions ofthe celestial bodies in relation to the aether. (p. 1, Lorentz, 1895)For Lorentz, the ether had to be material but transparent to the (ponderable)bodies. He did not fully convince other authors such as (Einstein, 1907; Ritz,1908) for whom Lorentz’ ether was just Absolute space.Years later, Einstein discussed the matter in “Concerning the ether” (Einstein,1924). He states that:When we speak here of aether, we are, of course, not referringto the corporeal aether of mechanical wave-theory that underliesNewtonian mechanics, whose individual points each have a velocityassigned to them. This theoretical construct has, in my opinion,been superseded by the special theory of relativity. Rather the dis-cussion concerns, much more generally, those things thought of asphysically real which, besides ponderable matter consisting of elec-trical elementary particles, play a role in the causal nexus of phys-ics. Instead of ‘aether’, one could equally well speak of ‘the physicalqualities of space’[...] It is usually believed that aether is foreignto Newtonian physics and that it was only the wave theory of lightwhich introduced the notion of an omnipresent medium influencing,and affected by, physical phenomena. But this is not the case. New-tonian mechanics had its ‘aether’ in the sense indicated, albeit underthe name ‘absolute space’.Einstein closes the discussion (p. 93, Einstein, 1924) stating that..., we will not be able to do without the aether in theoretical physics,that is, a continuum endowed with physical properties; for generalrelativity, to whose fundamental viewpoints physicists will alwayshold fast, rules out direct action at a distance.With this point of view Einstein re-introduces absolute space to support theempiricist belief of “local action” (as discussed for example in (Hume, 1896)).This belief is rooted in a particular form of thinking nature, contrasting theviews of other authors such as Leibniz, Ampère, Weber and Gauss, who did notfind neither “local action” nor a subjective space necessary assumptions.5 .2 Ether-free electrodynamics
The starting point of our task are Maxwell equations (Maxwell, 1873) that inmodern notation, with C = ( µ ǫ ) − , can be stated as B = ∇ × A (1) E = − ∂A∂t − ∇ V (2) ǫ ∇ · E = ρ (3) µ j + 1 C ∂E∂t = ∇ × B (4)All quantities are evaluated at a reference point, x , relative to the referenceframe in which the distribution of charge is given by ρ ( x, t ) and satisfies thecontinuity equation ∂ρ ( x, t ) ∂t + ∇ · j ( x, t ) = 0 ([295], Maxwell, 1873). In otherwords, it is the frame where the fields E, B have been determined through ρ and j . The equations correspond to the electromagnetic momentum, A, (todaycalled vector potential), the magnetic induction, B ([619] eq. A, Maxwell, 1873),the electric field intensity E ([619], eq. B, Maxwell, 1873) (being the sum of aninduction contribution ([598], Maxwell, 1873) and a gradient generalising theelectrostatic potential) and the galvanic current, j . The third equation, takenfrom ([612], Maxwell, 1873) and inspired in Coulomb and Faraday, generaliseshere Poisson’s equation from electrostatics, while the fourth equation ([619], eq.E, Maxwell, 1873) refers to ([610], eq. H*, Maxwell, 1873) .Maxwell obtained eq. 4 through considerations about the ether. He ar-gued that the propagation of electromagnetic waves asks for a propagation me-dium, by analogy with “the flight of a material substance through space” ([866]Maxwell, 1873). However, Ludwig Lorenz (Lorenz, 1867) expresses discomfortwith the ether hypothesis, which had only been useful to “furnish a basis for ourimagination” (p. 287). Consequently, he offered an ether-free derivation of thefield equations that took into account Gauss’ suggestion of delayed action at adistance (bd.5 p. 627-629, Gauss, 1870). . For static (time-independent) situ-ations, the electromagnetic potentials are defined through Poissons’s equationstarting from charges and currents, i.e., ( A, VC )( x, t ) = µ π Z (cid:18) ( j, ρC )( y, t ) | x − y | (cid:19) d y. Inspired in Gauss’ proposal, Lorenz considered that the charges and currentscontributing to the potentials at time t act with delay, after being originated ina previous time s = t − | x − y | C : The same equations can be found in (Maxwell, 1865) Section III under the equation labels:B, 35, G, C (taking into account A) and H. Other authors arrived to similar equations, also without involving the ether (Riemann,1867; Betti, 1867; Neumann, 1868). These authors (but not Lorenz) were in turn criticisedfor not involving the ether in their conceptualisation (Clausius, 1869). A, VC )( x, t ) = µ π Z (cid:18) ( j, ρC )( y, t − C | x − y | ) | x − y | (cid:19) d y, (5)where only galvanic currents and actual charges are involved. After standardoperations of vector calculus (see Appendix A) it is found that ✷ A = − µ j (6) ✷ V = − ǫ ρ where ✷ = ∆ − ∂ ∂t is D’Alembert’s wave operator. In Appendix A we de-rive eq. 4 from Lorenz’ setup, taking advantage of the Lorenz gauge . Theidea of electromagnetic waves was circulating at least since 1857 after work byKirchhoff Kirchhoff (1857) and Weber (Poggendorff, 1857; Weber, 1864). Thesewaves, however, were present inside material media (e.g., in wires). Lorenz’work (Lorenz, 1867) combines elements from both Weber and Kirchhoff.
Following Lorentz, charges are assumed to be rigid bodies (§75c, Lorentz, 1892)or “corpuscles” and all charges are considered to be accounted for directly, sothere is no need to introduce polarisation/induction fields. With this setup, theforce on a probe charge will be identical to the electromotive force.While the idea that a charge in movement corresponds to a current goesback (at least) to Weber (Weber, 1846), the form of constructing the currenthad been the subject of controversy. For Maxwell the velocities that matter wererelative velocities between primary and secondary circuits in full agreement withFaraday Maxwell ([568-583] 1873). For the sake of the argument we considerboth circuits as rigid solids. Lorentz in turn considered that the velocities hadto be considered relative to the ether, an idea that has to be disregarded sinceno evidence of the ether can be found. Hence, only the Faraday-Weber-Maxwell(and others) idea of relative velocity appears to be tenable. We let ˙¯ x ( t ) bethe relative velocity between two reference points in the primary and secondarybodies. We hereby define current according to ( x, t ) = ( y + ¯ x ( t ) , t )¯ ρ ( x, t ) = ρ ( x − ¯ x ( t ) , t )¯ j ( x, t ) = j ( x − ¯ x ( t ) , t ) + ˙¯ xρ ( x − ¯ x ( t ) , t ) (7)where y is a “local” coordinate of the secondary body (labelled with the index , the body where we aim to compute the acting force) and x is the coordinaterelative to the reference frame above in which the potentials are A and V .Hence, neither ether nor any inertial frame is considered, but only the relativecoordinate between the electromagnetic bodies undergoing mutual interaction.The charge density and internal current ( ρ ( y, t ) , j ( y, t )) are described in a7rame at rest with respect to the rigid body , while (¯ ρ ( x, t ) , ¯ j ( x, t )) are theirexpressions with reference to the primary circuit. Also, it is assumed that thebody does not rotate. In either reference frame, charge and current for body also satisfy the continuity equation.This definition of current (eq. 7) reduces to Lorentz’ current for the partic-ular case j = 0 , only that the velocity is now relational, while the first term ineq. 7 allows for other sources of current, as it was entertained by Faraday andMaxwell. Maxwell’s energy is introduced through a process in which matter acquires itselectromagnetic state (Maxwell, 1873). The electrostatic energy is obtained in[630-631] bringing charges from infinity. Next, the magnetostatic energy is ob-tained in similar form in [632-633], based upon magnetostatic results previouslyobtained in [389]. Maxwell proceeds to add an electrokinetic energy due to thecurrents [634-635]. In the even numbered articles he presents the physical ideaand in the odd numbered articles he transforms the expression using integrationby parts (Gauss’ law).Thus, the energy required to create a given electromagnetic state (a distribu-tion of charges and currents) can be regarded as the time-integral of the power,first bringing charges from a condition of zero energy (from “infinity”) workingagainst −∇ V and bringing also the current distribution, now working againstthe electromagnetic momentum A : P = (cid:18) ∇ V + ∂A∂t (cid:19) · j = − E · j the time-integral from a situation in which E (0) = 0 , leads to E = − Z t dt Z d x ( E · j ) = Z t dt Z d x (cid:18)(cid:18) ∇ V + ∂A∂t (cid:19) · j (cid:19) . (8)Assuming that all of | B | , | E | , A · j, V ρ decrease faster than r at infinity (ahypothesis needed for most manipulations performed by Maxwell and Lorentz)and applying Gauss theorem to convert volume integrals of a divergence intosurface integrals (vanishing at infinity by the assumption), a straightforwardcomputation (see Appendix A) shows that the energy provided by the electri-fication of the body is E = 12 Z d x (cid:18) µ | B | + ǫ | E | (cid:19) . (9)From the electrostatic and magetostatic situations it is clear that the individualterms correspond to the electrokinetic, T = R d x (cid:16) µ | B | (cid:17) , and potentialenergies, U = R d x (cid:0) ǫ | E | (cid:1) , thus suggesting that the electromagnetic Lag-rangian reads L = T − U and the action A = R dt L . Another straightforward8omputation and application of Gauss theorem (see Appendix A) leads to A = 12 Z dt Z (cid:18) µ | B | − ǫ | E | (cid:19) d x = 12 Z A · j d x − Z ρV d x + f ( t ) − f ( t ) (10)The dynamic equations are independent of f , so we can proceed from hereby taking f ≡ . While the intuition of terms can be taken as a suggestion, wemust ask: What kind of dynamical situation is reflected by this action? Theorem 1.
Let ( A , V ) be the known values of the electromagnetic potentialsin a piece of matter supported on a region of space with characteristic function χ . Then, under the assumptions of Lemma 5, Hamilton’s principle of leastaction (Ch 3, 13 A p. 59, Arnold, 1989), δ A = 0 , subject to the constraintsgiven by ( A , V ) the constraints implies that the manifestation of the potentialsoutside matter obeys the wave equation.Proof. The result follows from the computation of the minimal action under ( V − V ) χ = 0( A − A ) χ = 0 Multiplying the constraints by the Lagrange multipliers λ and κ (the latter avector), while we use the shorthand notations B = ∇× A and E = (cid:0) − ∂A∂t − ∇ V (cid:1) ,we need to variate A = 12 Z dt (cid:18)Z (cid:18) µ | B | − ǫ | E | − κ · ( A − A ) χ + λ ( V − V ) χ (cid:19) d x (cid:19) . After variation and applying Gauss theorem as usual leads to ✷ A = − χκ ✷ V = − χλ which allows us to identify j = χκ (the density of current inside the materialresponsible for A and ρ = χλ the density of charge responsible for V .This theorem deserves to be called Lorenz’ theorem since he wrote aboutthese relations (p.300, Lorenz, 1867): “This result is a new proof of the identityof the vibrations of light with electrical currents; for it is clear now that not onlythe laws of light can be deduced from those of electrical currents, but that theconverse way may be pursued, provided the same limiting conditions are addedwhich the theory of light requires.”Thus, electromagnetism has been summarised in terms of the potential V and vector potential (or electrokinetic momentum, according to Maxwell-Faraday) A (defined via eq. 5), the principle of least action and the continuity The continuity equation, eq.(5) and ∇ · A + 1 C ∂V∂t = 0 , see Appendix. B = ∇ × A and E = − ( ∂A∂t + ∇ V ) , or stating consequences of thesedefinitions, such as ∂∂t ∇ × A = ∇ × ∂A∂t (which leads immediately to Faraday’sinduction law), or ∇ · E = ρ (that follows after the use of the continuity equa-tion).The connection of this theoretical structure with observations in nature isgiven by the phenomenological map (Solari and Natiello, 2020) providing obser-vational content to charges and currents in matter. We are now ready to showhow Lorentz’ force belongs to this context and is already inscribed in the formerequations. When we consider two pieces of electrified matter in interaction we can envisagea different form of constructing the system. In the first step, the bodies arefar apart, so that we can assume that they do not interact, and are electrifiedto reach their actual state. Next, they will be brought together to their cor-responding mechanical positions in terms of a thought process called a virtualdisplacement . The formalisation of this idea already present in Maxwell is calleda virtual variation (Ch 4, 21 B p. 92, Arnold, 1989). The force associated tothis virtual displacement will then result from the variation of the interactionterms in the Lagrangian. Given the electromagnetic contribution to the ac-tion, determining the contribution to the force amounts to applying Hamilton’sprinciple using a virtual displacement of the probe (which we indicate with su-bindex 2) with respect to the primary circuit producing the fields (subindex 1).In formulae, we must request δ ¯ x ( t ) A = 0 with δ ¯ x ( t ) = 0 and δ ¯ x ( t ) = 0 , where ¯ x ( t ) denotes the relative distance betweenprobe and primary circuit. In so doing, we must take into account that there isa corresponding variation of the velocity δ ˙¯ x ( t ) . Recall that according to 6 and7, ¯ x occurs in ( ρ , j ) and ˙¯ x occurs only in j . For the sake of convenience, wemay express the action using Lemma 8 before proceeding: δ ¯ x ( t ) A = δ ¯ x ( t ) Z dt Z d x [ A · j − V ρ ] = 0 (11)The required variation reads (dropping the index ¯ x ( t ) ), δ A = Z dt Z d x [ A · δj − V δρ ] The variation of the current and of the charge distribution due to the motionof the probe relative to the primary circuit are, after eq. 7, δ ¯ j = − ( δ ¯ x · ∇ )¯ j + ¯ ρ δ ˙¯ xδ ¯ ρ = − ( δ ¯ x · ∇ )¯ ρ
10e have then δ A = δ Z dt L = Z dt Z d x [ A · δ ¯ j − V δ ¯ ρ ]= Z dt Z d x (cid:20) ¯ j · ( δ ¯ x · ∇ ) A − ¯ ρ ( δ ¯ x · ∇ ) V − δ ¯ x · ∂∂t ( A ¯ ρ ) (cid:21) (12)after some integrations by parts and using the wave equation for the vectorpotential. Further transformation with mathematical identities allows us towrite Z dt Z d x δx · [¯ j × B + ¯ ρ E ] . This is, following the standard use of Hamilton’s principle in mechanics we arriveto an electromagnetic contribution to the force on the probe F em = ¯ j × B + ¯ ρ E Lorentz considered only the case j = 0 , hence ¯ j = ¯ ρ ˙¯ x . Maxwell consideredthe general case in Maxwell ([602] 1873) but using his “total current” instead,consistent with his belief in the ether-based displacement current as a materialcurrent. However, his elaboration is based on galvanic currents, finally modify-ing the final result by analogy, eq. (C) [619]. The present ether-free approachdiffers with Lorentz’ in the broader concept of current and in that the particip-ating quantities are fully relational and describe only the interaction betweenprobe and primary circuit. Unlike Maxwell, we have no use for the hypothesisof a displacement current. Since the Lagrangian is expressed by an integral, we are free to change theintegration variable by a fixed translation leaving the integral unchanged. Ac-tually, we can use a different translation for each time in (10). We propose tochange integration variable from x to z, with x = z + ¯ x ( t ) . Instead of performingthe change in eq.(11) we will save effort and perform it in eq.(12), prior to thepartial integration in time, namely δ A = Z dt Z d x [¯ j · ( δ ¯ x · ∇ ) A − ¯ ρ ( δ ¯ x · ∇ ) V + δ ˙¯ x · ( A ¯ ρ )] (with ¯ j ( x, t ) , ¯ ρ ( x, t ) given by eq.7). We introduce the following notation z = x − ¯ x ( t ) V ( z, t ) = V ( z + ¯ x ( t ) , t ) (13) A ( z, t ) = A ( z + ¯ x ( t ) , t ) Hence, the variation reads now 11 Z L dt = Z dt Z d z [( j ( z, t ) + ˙¯ xρ ( z, t )) · ( δ ¯ x · ∇ ) A − ρ ( z, t ) ( δ ¯ x · ∇ ) V ]+ Z dt Z d z [ δ ˙¯ x · ( Aρ ( z, t ))] integrating by parts in time the last term and using the relations Z dt [ δ ˙¯ x · ( Aρ )] = Z dt (cid:20) − ρ δ ¯ x · (cid:20) ∂A∂t (cid:21) − δ ¯ x · A ∂ρ ∂t (cid:21)(cid:20) ∂A∂t (cid:21) ≡ ∂∂t A ( z + ¯ x ( t ) , t ) = ∂∂t A ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) x = z +¯ x ( t ) + ( ˙¯ x · ∇ ) A ( z + ¯ x ( t ) , t ) −∇ V − (cid:20) ∂A∂t (cid:21) = −∇ V − ∂∂t A ( x, t ) (cid:12)(cid:12)(cid:12)(cid:12) x = z +¯ x ( t ) − ( ˙¯ x · ∇ ) A (along with the continuity equation) we arrive after some algebra and furtheruse of Gauss’ theorem to: δ Z L dt = Z dt Z d z [ j · ( δ ¯ x · ∇ ) A − j · ∇ ( δ ¯ x · A )]+ Z dt Z d z (cid:20) ( ρ ˙¯ x ) · ( δ ¯ x · ∇ ) A + ρ δ ¯ x · ( −∇ V − ∂A∂t ) (cid:21) Finally, the following relations (the second one valid for any sufficiently dif-ferentiable scalar function Φ ) ˙ x · ( δ ¯ x · ∇ ) A = δ ¯ x · ∇ ( ˙ x · A )( δ ¯ x · ∇ )Φ( x, t ) = − δ ¯ x × ( ∇ × Φ) + ∇ ( δ ¯ x · Φ) j · ( δ ¯ x · ∇ ) A − j · ∇ ( δ ¯ x · A ) = j · ( − δ ¯ x × ( ∇ × A ) lead us to the next result: δ Z L dt = Z dt Z d z [ j · ( − δ ¯ x × ( ∇ × A ))]+ Z dt Z d z (cid:20) δ ¯ x · ρ (cid:18) − (cid:20) ∂A∂t (cid:21) − ∇ ( V − ˙¯ x · A ) (cid:19)(cid:21) = Z dt Z d z δ ¯ x · (cid:20) j × B + ρ (cid:18) − (cid:20) ∂A∂t (cid:21) − ∇ ( V − ˙¯ x · A ) (cid:19)(cid:21) Hence we have two expressions for the mechanical contribution of the elec-tromagnetic force: The one obtained from eq.(12) above and the present one,i.e., F em = Z d x [¯ j × B + ¯ ρ E ] = Z d z (cid:20) j × B + ρ (cid:18) − (cid:20) ∂A∂t (cid:21) − ∇ ( V − ˙¯ x · A ) (cid:19)(cid:21) : Theorem 2. (Maxwell’s invariance theorem ) : Let x = x ′ + ¯ x ( t ) , andcorrespondingly v = v ′ + ˙¯ x . Define A ′ ( x ′ , t ) ≡ A ( x, t ) , then the value of theelectromotive force at a point x does not depend on the choice of reference systemif and only if ψ ( x, t ) transforms as ψ ′ ( x ′ , t ) ≡ ψ ( x ′ − ¯ x, t ) − ˙¯ x · A ′ ( x ′ , t ) . In for-mulae, E ′ ( x ′ , t ) = E ( x, t ) , where ψ is an undetermined electrodynamic potential(introduced for the sake of generality): E ′ ( x ′ , t ) = v ′ × ( ∇ × A ′ ( x ′ , t )) − ∂A ′ ( x ′ , t ) ∂t − ∇ ψ ′ ( x ′ , t ) . Proof.
First, according to the definition, we have A ′ ( x ′ , t ) = A ( x ′ + ¯ x, t ) .Next, we note that by straightforward vector calculus identities, Maxwell’selectromotive force (eq. B in ([598], Maxwell, 1873)) can be restated as E ( x, t ) = − ∂A ( x, t ) ∂t − ( v · ∇ ) A ( x, t ) − ∇ ( ψ ( x, t ) − v · A ( x, t )) In the new coordinate system we compute: E ′ ( x ′ , t ) = v ′ × ( ∇ × A ′ ( x ′ , t )) − ∂A ′ ( x ′ , t ) ∂t − ∇ ψ ′ ( x ′ , t )= − ∂A ′ ∂t ( x ′ , t ) − ( v ′ · ∇ ) A ′ − ∇ ( ψ ′ ( x ′ , t ) − v ′ · A ′ ( x ′ , t )) . Subsequently, under the present assumption A ′ ( x ′ , t ) ≡ A ( x, t ) we may rewrite ∂A ′ ( x ′ , t ) ∂t = ∂A ( x, t ) ∂t (cid:12)(cid:12)(cid:12)(cid:12) x ′ +¯ x + ( ˙¯ x · ∇ ) A leading to E ′ ( x ′ , t ) = − ∂A ( x, t ) ∂t − ( ˙¯ x · ∇ ) A − ( v ′ · ∇ ) A ′ − ∇ ( ψ ′ ( x ′ , t ) − v ′ · A ′ ( x ′ , t ))= − ∂A ( x, t ) ∂t − ( v · ∇ ) A − ∇ ( ψ ′ ( x ′ , t ) − v ′ · A ′ ( x ′ , t ))= − ∂A ( x, t ) ∂t − ( v · ∇ ) A − ∇ ( ψ ( x, t ) − ˙¯ x · A ( x, t ) − v ′ · A ′ ( x ′ , t )) − ∂A ( x, t ) ∂t − ( v · ∇ ) A − ∇ ( ψ ( x, t ) − v · A ( x, t )) = E ( x, t ) Maxwell’s result covers also rotating reference systems, but we skip this case to keep theargumentation simple. Maxwell refers to this expression as: “the theory of the motion of a body of invariableform“. For any property of matter, this relation is immediate. ψ ′ ( x ′ , t ) ≡ ψ ( x, t ) − ˙¯ x · A ( x, t ) . Note that ψ is Maxwell’s undetermined po-tential that once determined in one system of reference transforms according tothe theorem to other systems.Maxwell’s Art. [601] of the Treatise states “It appears from this that theelectromotive intensity is expressed by a formula of the same type, whether themotions of the conductors be referred to fixed axes or to axes moving in space,the only difference between the formulae being that in the case of moving axesthe electric potential ψ must be changed into ψ + ψ ′ .“ The invariance of the force as determined from the point of view of the source(primary circuit) or of the target (probe, secondary circuit) in the previoussubsection follows from the general invariance of integrals in front of coordinatechanges, supported in Maxwell’s invariance theorem. The result is a special caseof the No Arbitrariness Principle (NAP) (Solari and Natiello, 2018), stating thatthe description of natural processes cannot depend on arbitrary choices (in thiscase the choice of subjective reference frame). Indeed, we can attain a fullyrelational description of electromagnetic phenomena.The consideration of three electrified bodies will allow us to inspect thisproblem. Let x ij ( t ) the relative position between the i and the j body, with i, j ∈ { , , } , clearly, x ii = 0 . The relative positions satisfy x + x + x = 0 and correspondingly, the relative velocities satisfy v + v + v = 0 . Let ζ ii be each of the components of the four dimensional vector ( j i , ρ i ) describingthe density of currents and the density of charges in body i , as perceived froma frame fixed to itself, and ζ ji the same components as described from theframe of body j . We will denote by W the operator that produces the delayedpropagation of the EM situation in the body and, finally, T ( x ) the operator thatapplies a time-dependent translation to current+charge vector as in 7. Clearly T (0) = Id . Then, given the potentials of body i in its own frame, we get thepotentials in the body j frame as ( A ji , V ji ) = ( W T ( x ji ) ✷ ) ( A ii , V ii ) The operators T ( x ij ( t )) form a group of transformations. It is easy to verifythat the product law is: T ( x ( t )) T ( y ( t )) = T (( x + y )( t ) Letting ˜ T ( x ij ) = W T ( x ji ) ✷ we notice that the operators ˜ T ( x ij ) are conjugated to T ( x ij ) since the relation ✷ W = Id and W ✷ = Id produce the conjugation relation ✷ ˜ T ( x ij ) = T ( x ij ) ✷ ˜ T ( x ij ) W = W T ( x ij ) T ( x ij ) acts witha conjugate representation on the wave representatives. This is the abstractcontent of Maxwell’s theorem.We can finally write all the fields in terms of one reference body, say, thebody with i = 1 as ( A , V ) = W (cid:0) ζ + T ( x ) ζ + T ( x ) ζ (cid:1) We next apply ˜ T ( x ) to this expression, we get ˜ T ( x )( A , V ) = ˜ T ( x ) W (cid:0) ζ + T ( x ) ζ + T ( x ) ζ (cid:1) = ( W T ( x ) ✷ ) W (cid:0) ζ + T ( x ) ζ + T ( x ) ζ (cid:1) = ( W T ( x )) (cid:0) ζ + T ( x ) ζ + T ( x ) ζ (cid:1) = W (cid:0) T ( x ) ζ + T ( x ) T ( x ) ζ + T ( x ) T ( x ) ζ (cid:1) = W (cid:0) T ( x ) ζ + ζ + T ( x ) ζ (cid:1) = ( A , V ) Which shows how the subjective representation of fields transforms consistentlywith the time-dependent-translations group. When the admitted reference isrestricted to inertial bodies (Natiello and Solari, 2019), the group of transform-ations is the Galilean group.Let us specify the above construction for the electromagnetic force thatbodies and exert on a moving charge q . Expressing the fields as computedby bodies and respectively, this force reads, F = q (cid:0) v × B + E (cid:1) + q (cid:0) v × B + E (cid:1) From Maxwell’s invariance theorem, the relation ( A , V ) = T ( A , V ) is justa recasting of the vector potential A in the coordinates of body . Hence,both potentials take the same value on any given point of space. Consequently B = B . The transformation of the electric field and scalar potential reads(also by Theorem 2), E = −∇ V − ∂∂t A = −∇ (cid:0) V − v · A (cid:1) − ∂∂t A − ( v · ∇ ) A = v × (cid:0) ∇ × A (cid:1) − ∇ V − ∂∂t A = v × B + E Hence, F = q (cid:0) v × B + E (cid:1) + q (cid:0) v × B + (cid:0) v × B + E (cid:1)(cid:1) = q (cid:0) v × (cid:0) B + B (cid:1) + (cid:0) E + E (cid:1)(cid:1) i.e., we have proven Corollary 3.
Under the assumptions of Theorem 2, the interaction force betweenelectromagnetic bodies is invariant in front of arbitrary (subjective) translationsof the reference system. Discussion
The attitude towards error defines our science. When our observations are in-compatible with a hypothesis, i.e., they refute it, we can only think of suppress-ing and/or replacing the indicted hypothesis as well as all of its consequences,going back to the point where we had the wrong idea, and restarting our progressfrom that point.Maxwell worked out his electromagnetism following the path of the Göttin-gen school, but he needed to persuade himself of analogies with matter (Maxwell,1856), hence his introduction of the ether. Lorentz as well worked out from thismixed epistemological position. More than a hundred years later we can saythat their success comes from the mathematisation while the problems withtheir approaches come from their analogical thoughts, as we have observed forexample in Maxwell’s force and Lorentz deduction of his force.In this work we have shown that a relational electromagnetism that reachesa higher level of consistency and harmony than currently accepted electromag-netism is possible. Within this approach, Lorentz’ force emerges from the sameprinciples that produce Maxwell equations, this is, they form a consistent the-ory a-priori rather than a-posteriori. The resulting relational electromagnet-ism admits a subjective form that complies with the no-arbitrariness-principle(Solari and Natiello, 2018), a principle that is more general and demanding thanPoincaré’s principle-of-relativity. The construction begins with one concept ofspace and ends within the same concept.Electrodynamics was built in part resting on the hypothesis of the ether.This construction was never revised, despite the claims of having disposed ofthe ether after Einstein’s famous contributions (Einstein, 1905, 1907). Currentelectrodynamics still rests on the ether. It offers only an alternative interpret-ation of its symbols, thus dissociating the construction of the theory from theresults. Probably the only argument raised in favour of modern electrodynamicsis that “it works”, it gives “the right answer”. The pragmatic approach (Fixa-tion of belief, Peirce, 1955) regards human actions as the outcome of a strugglebetween doubt and belief. The former provokes tension and an urge for resol-ution, the latter is a state of peace of mind towards which we strive. Vulgarpragmatism (which is not a proper current of thought but pervades modern sci-ence) admits justifications foreign to reason in order to achieve “peace of mind”:If something fits a given purpose (it “works”, in the given sense), then it must beright or “close” to it. As much as such an attitude may be of help when explor-ing new ideas, it becomes an obstacle when the ideas are organised by reason.Can the conclusions of a theory contradict its own hypotheses? Clearly not.In this work we show how to construct the basic elements of electrodynamicscompletely ether-free.Not surprisingly, the elements of our construction can be found in Gauss,Maxwell, Lorenz, and the Göttingen school. The correspondence between Lorentz’current and Maxwell’s invariance theorem is a key element for this state ofharmony. Lorentz’ and Maxwell’s transformations, when restricted to inertialsystems reduce to not-so-obvious presentations of Galileo’s group of transform-16tions.In this relational approach, electromagnetic interactions manifest themselvesin any reference system as waves of speed C , refuting the idea that this isincompatible with classical space-time. But then, what do we need to drop toaccept this theory? The answer is clear: analogy must be trusted no more thatMaxwell did:“...It appears to me, however, that while we derive great advantagefrom the recognition of the many analogies between the electric cur-rent and a current of material fluid, we must carefully avoid makingany assumption not warranted by experimental evidence, and thatthere is, as yet, no experimental evidence to shew whether the elec-tric current is really a current of a material substance, or a doublecurrent, or whether its velocity is great or small as measured in feetper second.” ([574], Maxwell, 1873)The electromagnetic interaction does not support an analogy with matter withoutthe resource of introducing absolute space and attributing properties to thespace or, alternatively, to propose material-like carriers of interactions. Thealternative goes back to Maxwell in his closing of his treatise, objecting theGöttingen school of thought:“But in all of these theories the question naturally occurs: – If some-thing is transmitted from one particle to another at a distance, whatis its condition after it has left the one particle and before it hasreached the other ? If this something is the potential energy of thetwo particles, as in Neumann’s theory, how are we to conceive thisenergy as existing in a point of space, coinciding neither with theone particle nor with the other ?” ([866] Maxwell, 1873)The paragraph shows Maxwell’s need to represent interactions in terms of ana-logies with matter. Since bodies have a place in the (always subjective) space,the paragraph assumes, without saying it, that we should consider interactions(an abstract concept) in terms of the more accessible (intuitive) idea of bodies.Because bodies, not interactions, have a place in space. References
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A Lemmas and Proofs
Lemma 4. A ( x, t ) = µ π Z U (cid:18) j ( y, t − C | x − y | ) | x − y | (cid:19) d y ⇒ ✷ A = − µ j , and sim-ilarly for ǫ ✷ V = − ρ , where ✷ ≡ ∆ − C ∂ ∂t .Proof. We perform the calculation in detail only for A , since the other one issimilar. We use the shorthand r = | x − y | . ∇ x A i = µ π Z d y j i ∇ x r − ∂∂t j i ∇ x rC r ! ∆ A i = ∇ x · ∇ x A i = µ π Z d y j i ∆ 1 r − (cid:18) ∇ x r (cid:19) · (cid:18) ∂∂t j i ∇ x rC (cid:19) − ∂∂t j i ∆ rC r + ∂ ∂t j i r |∇ x rC | ! Moreover, standard vector calculus identities give ∂∂t j i (cid:18) ∇ r · ∇ rC + ∆ rC r (cid:19) = 0 |∇ rC | = 1 C ∆ A i ( x, t ) = µ π Z d y j i ( y, t − rC )∆ (cid:18) r (cid:19) + (cid:18) C (cid:19) µ π Z d y ∂ ∂t j i ( y, t − rC ) r The time derivative in the last term can be extracted outside the integral, thusyielding, ✷ A i ( x, t ) = ∆ A i ( x, t ) − (cid:18) C (cid:19) µ π Z d y ∂ ∂t j i ( y, t − rC ) r = ∆ A i ( x, t ) − (cid:18) C (cid:19) ∂ ∂t A i ( x, t )= µ π Z d y j i ( y, t − | x − y | C )∆ (cid:18) r (cid:19) = − µ j i ( x, t ) Lemma 5.
The continuity equation along with eq.5 imply ∇ · A + 1 C ∂V∂t = 0 (theLorenz gauge).Proof. Still using the shorthand r = | x − y | , and applying Gauss theorem overvolume integrals of total divergences of functions vanishing sufficiently fast at21nfinity, ∇ · A + 1 C ∂V∂t == Z d y ∇ x · j ( y, t − rC ) | x − y | + ∂∂t ρ ( y, t − rC ) | x − y | = Z d y ∇ x | x − y | · j ( y, t − rC ) − | x − y | (cid:20) ∂∂s j ( y, s ) (cid:21) s = t − rC · ∇ x rC + ∂∂t ρ ( y, t − rC ) | x − y | = Z d y −∇ y | x − y | · j ( y, t − rC ) − | x − y | (cid:20) ∂∂s j ( y, s ) (cid:21) s = t − rC · ∇ x rC + ∂∂t ρ ( y, t − rC ) | x − y | = Z d y ∇ y · j ( y, t − rC ) | x − y | − | x − y | (cid:20) ∂∂s j ( y, s ) (cid:21) s = t − rC · ∇ x rC + ∂∂t ρ ( y, t − rC ) | x − y | = Z d y ∇ y · [ j ( y, s )] s = t − rC | x − y | − | x − y | (cid:20) ∂∂s j ( y, s ) (cid:21) s = t − rC · ( ∇ y + ∇ x ) rC + ∂∂t ρ ( y, t − rC ) | x − y | = Z d y (cid:2) ∂∂s ρ ( y, s ) + ∇ y · j ( y, s ) (cid:3) s = t − rC | x − y | ! since ( ∇ y + ∇ x ) | x − y | = 0 . Lemma 6.
Lemma 5, together with eqs. 1, 2 and 5 imply eq. 4.Proof.
The proof requires standard applications of vector calculus. From eq. 5we derive µ j = − (cid:18) ∇ ( ∇ · A ) − ∇ × ( ∇ × A ) − C ∂ A∂t (cid:19) = 1 C (cid:18) ∇ ∂V∂t + ∂ A∂t (cid:19) + ∇ × B = − C ∂E∂t + ∇ × B Lemma 7.
Eq. 8 is equivalent to eq. 9 (the total electromagnetic energy), upto the volume integral of the gradient of a function that vanishes at infinity.Proof.
Under the general assumption that R ∇ · F ( x, t ) d x vanishes at infinity,being F a vector function that decays sufficiently fast (i.e., faster than r − ), we22btain Z t dt Z d x (cid:18)(cid:18) ∇ V + ∂A∂t (cid:19) · j (cid:19) == Z t dt Z d x (cid:18) ∇ · ( V j ) − V ∇ · j + ∂A∂t · (cid:18) − ǫ ∂E∂t + 1 µ ∇ × B (cid:19)(cid:19) = Z t dt Z d x (cid:18) V ∂ρ∂t + ∂A∂t · (cid:18) − ǫ ∂E∂t + 1 µ ∇ × B (cid:19)(cid:19) = Z t dt Z d x (cid:18) ǫ V ∇ · ∂E∂t − ǫ ∂A∂t · ∂E∂t + 1 µ ∂A∂t · ∇ × B (cid:19) = Z t dt Z d x (cid:18) ǫ ∇ · ( V ∂E∂t ) − ǫ ∇ V · ∂E∂t − ǫ ∂A∂t · ∂E∂t + 1 µ (cid:18) ∂B∂t · B (cid:19) − µ ∇ · (cid:18) ∂A∂t × B (cid:19)(cid:19) = Z t dt Z d x (cid:18) ǫ E · ∂E∂t + 1 µ (cid:18) ∂B∂t · B (cid:19)(cid:19) = 12 Z t dt ∂∂t Z d x (cid:18) ǫ | E | + 1 µ | B | (cid:19) = 12 Z d x (cid:18) ǫ | E | + 1 µ | B | (cid:19) where all integrals involving total divergences have been set to zero by Gauss’theorem. We have also used the continuity equation. Lemma 8.
Up to an overall function of time and the divergence of a functionvanishing sufficiently fast at infinity, the electromagnetic action satisfies A = 12 Z dt Z (cid:18) µ | B | − ǫ | E | (cid:19) d x = 12 Z ( A · j − ρV ) d x Proof.
The proof requires standard vector calculus operations on the integrand,namely A · j − ρV = A · (cid:18) µ ∇ × B − ǫ ∂E∂t (cid:19) − ǫ V ∇ · E = 1 µ (cid:0) | B | − ∇ · ( A × B ) (cid:1) − ǫ A · ∂E∂t − ǫ ∇ · ( V E ) + ǫ E · ∇ V = 1 µ (cid:0) | B | − ∇ · ( A × B ) (cid:1) − ǫ A · ∂E∂t − ǫ ∇ · ( V E ) + ǫ E · (cid:18) − E − ∂A∂t (cid:19) = 1 µ | B | − ǫ | E | − ∇ · (cid:18) µ A × B + ǫ V E (cid:19) − ǫ ∂∂t ( A · E ) emma 9. The result of Lemma 8 is independent of the choice of gauge.Proof.