Mechanical model of the inhomogeneous Maxwell's equations and of Lorentz transformations
aa r X i v : . [ phy s i c s . h i s t - ph ] J u l Mechanical model of the inhomogeneous Maxwell’s equations and of Lorentztransformations
Lachezar S. Simeonov
Department of Physics, Sofia University, James Bourchier 5 blvd, 1164 Sofia, Bulgaria (Dated: July 23, 2020)A mechanical model of a quasi-elastic body (aether) which reproduces Maxwell’s equations withcharges and currents is presented. In MacCullagh’s model [31], a quasi-elastic body with anti-symmetrical stress tensor was considered but it did not include any charges or currents. Indeed,any presence of charges or currents appeared to contradict the continuity equation of the aetherand it remained a mystery as to where the aether goes and whence it comes. We present herea solution to this mystery. In the model here presented, the amount of aether is conserved evenin the presence of charges and currents. It turns out that the charges themselves appear to bemoving with the same local aethereal velocity. In other words, the charges appear to be part ofthe aether itself, a kind of singularities in the aether. The positive (negative) charge seems to blastaway (draw in) aether with a rate equal to the amount of charge, the electric field is interpreted asflux of the aether and the magnetic field as a torque per unit volume. In addition it is shown thatthe model is consistent with the theory of relativity, provided that Lorentz-Poincare interpretationof relativity theory is used instead of the relativistic or Minkowskian interpretation (three verydifferent but empirically equivalent interpretations). Within the framework of the Lorentz-Poincareinterpretation a statistical-mechanical interpretation of the Lorentz transformations is provided. Itturns out that the length of a body is contracted by the electromagnetic field acting back on thebody produced by the constituent particles of this same body. This self-interaction causes also delayof all the processes and clock-dilation results. This analysis can be extended even to muons andother elementary particles.
I. INTRODUCTION
Contrary to customary views, a special reference frameand superluminal velocities are quite consistent with thetheory of relativity. There are three very different empir-ically equivalent interpretations of relativity theory [1]and one of them - Lorentz-Poincare interpretation canquite easily accommodate Lorentz transformations witha single special reference frame and superluminal veloci-ties. Lorentz-Poincare interpretation means simply thatthe physical clocks and rods have been distorted by theforce fields, not time and space, and Lorentz transfor-mations connect reference frames which measure spaceand time with such distorted clocks and rods. Accord-ing to Bell [5] this is the ’cheapest solution’ in order toreconcile EPR experiments and relativity theory (oth-erwise the very notion of spin of the particles will beframe dependent, for more details see below). Indeed,Ives [2], Builder [3] and Prokhovnik [4] have developedLorentz-Poincare interpretation and reduced it to as fewa number of postulates as the customary relativistic andMinkowskian interpretations. In this manner Lorentz-Poincare interpretation has become as much elegant andsimple as the other two interpretations. Ives [2] derivesLorentz transformations from the laws of conservation ofenergy and momentum of charges interacting with elec-tromagnetic field as well as the laws of transmission ofradiant energy. Thus, he derives the velocity dependenceof the mass and from thence, Lorentz transformations.Prokhovnik [4] derives the distortions of the rods andclocks and thus Lorentz transformations as a retardedpotential effect. Lorentz-Poincare interpretation can have multiple ad-vantages. First, it allows nonlocal deterministic, i.e. non-local classical approaches to quantum theory to be consis-tent with Lorentz transformations. Examples of such ap-proaches to quantum mechanics are Bohmian mechanics[6], and the stochastic quantization procedures of Nelson[7], Guerra and Ruggiero [8], and Parisi and Wu [9]. Suchapproaches are finding increasing interest today. Thismay be due to the fact that these approaches can ex-plain some quantum phenomena that cannot be easilyexplained via the standard approach. Examples can begiven as multiple tunneling [10], mesoscopic physics andquantum Brownian oscillators [11], critical phenomenaat zero temperature [12], quantum-field theoretical reg-ularization procedures which preserve all symmetries ofthe bare theory such as gauge symmetry and chiral sym-metry [13]. This allows to quantize gauge fields, bothAbelian and non-Abelian, without gauge fixing and thefollowing cumbersome Faddeev-Popov ghosts [14].Second, Lorentz-Poincare interpretation does not usean arbitrary convention of the simultaneity of distantevents (putting the famous ǫ = 1 /
2, [1]) as in the rel-ativistic interpretation and is not based on defunct pos-itivistic principles. It can more easily explain the Bell’sspaceship paradox [15] as well as numerous others, for in-stance the length-contraction paradox [16], shown in Fig.2 than the relativistic interpretation. At the same timeit does not unite time and space (as in Minkowskian in-terpretation) and treats space-time diagrams at the levelof pressure-volume diagrams, i.e. instrumentally not re-alistically [1]. This allows us to treat time as flowing (A-theory of time) and not static (B-theory of time, for moredetails, see below) as is done in Minkowskian interpreta-tion. In addition since space-time is treated merely in-strumentally, we do not face the difficulty of interpretingrealistically space-like intervals which are complex num-bers. Finally, within the framework of Lorentz-Poincareinterpretation one can easily incorporate gravity as a bi-metric theory and not a curvature in space-time becausespace-time is merely a notation in this interpretation.This can resolve multiple problems - like the existenceof energy-momentum tensor, uniting gravity field in thesame field-theoretical formalism as the other fields bytreating gravity as spin 2 field, instead of geometry, etc.[17].Not only that but the presence of a single special ref-erence frame is in fact well grounded from the observa-tional point of view [18]. Indeed, it is well known thatif the universe is homogeneous and isotropic, as assumedby the standard ΛCDM cosmological model, there exista special reference frame in rest with the average mo-tion of the cosmic matter. In this special foliation ofspace-time there is a global time. Quoting Craig [1]:”To return to Eddington’s analogy of the paper block,suppose that only by foliating the block into a stack ofsheets do we discover that on each sheet is a drawing ofa cartoon figure and that by flipping through the sheetssuccessively, we can see this figure, thus animated, pro-ceed to pursue some action. Any other slicing of theblock would result merely in a scrambled series of inkmarks. In such a case it would be silly to insist thatany arbitrary foliation is just as good as the foliationwhich regards the block as a stack of sheets. But anal-ogously the Robertson-Walker metric discloses to us thenatural foliation of spacetime for our universe.” In otherwords, special foliation (i.e. a special reference frame)exists in an isotropic and homogeneous universe which isin rest relative to the average cosmic matter. However,our universe is in fact statistically isotropic to one partin 10 [21] as can be seen from the statistical isotropyof the CMB, cf. Planck 2018 results - [18]. And unlessone adopts an anti-Copernican view (i.e. one accepts theimplausible hypothesis that our place in the universe isspecial) the universe is homogeneous. Thus, there exist aspecial reference frame in rest to the cosmic matter witha global time coordinate. In fact, our velocity (i.e. thatof the Sun) with respect to this special reference framecan be measured from the amplitude of the Solar dipoleand Planck 2018 results show that it is known to about0.025 % and v = (369 . ± .
11) km s − . What Michel-son and Morley failed to achieve using light in the visiblespectrum [22], we have succeeded using microwave radi-ation.Therefore Lorentz-Poincare interpretation has manyreasons to be recommended - philosophical, observationaland theoretical. Not only that but Milonni [20] arguesquite consistently for the reality of the quantum vacuum,i.e. the quantum vacuum is not nothing but has multipleproperties and causes such diverse phenomena as Casimireffect, spontaneous decay, etc. To quote Whittaker [24]: ”But with the development of quantum electrodynam-ics, the vacuum has come to be regarded as the seat ofthe ’zero-point oscillations of the electromagnetic field,of the ’zero-point’ fluctuations of electric charge and cur-rent, and of a ’polarization’ corresponding to a dielec-tric constant different from unity. It seems absurd toretain the name ’vacuum’ for an entity so rich in physi-cal properties, and the historical word ’aether’ may fitlybe retained.” Therefore, from the modern perspective weare quite justified in accepting a kind of aether, whichpossesses rich properties and is not nothing, if by ’noth-ing’ we understand the classical philosophical definition -lack of properties [23]. This shows that we could attempta mechanical explanation of the physical fields. The his-tory of aether as a mechanical explanation of optics, grav-ity and electromagnetism is quite long and distinguished[24]. It can be traced back to Descartes [24] who in-troudced the aether as a mechanical substance necessaryto explain gravitation and optics. However Huygens at-tempted more concrete aether explanation of light assum-ing that light was a wave in the aether and he introducedhis famous prinicple which bears his name [25]. Thisstarted a long debate whether light was corpuscules or awave. It was unitl Young’s famous experiments [26] thatstarted to settle the matter in favour of the wave theory oflight. About that time, Fresnel made a mechanical modelof the aether, which gave a theory of double refraction,solved the problem of biaxial crystals, showing that Huy-gens’ construction for uniaxial crystals was a special caseof his own more general wave surface [27]. Other impor-tant contributions in that period were the two aberrationtheories - that of Fresnel [28] who assumed partial aetherdrag and of Stokes [29] who assumed complete aetherdrag. Nineteenth century produced a plethora of athermodels. Green [30] made the first complete mechanicalmodel of aether in terms of elastic body with symmetricstress-tensor. He removed the longitudinal waves of thisbody by assuming that their speed was infinite. Perhapsthe most famous and successful model was that of Mac-Cullagh, who assumed an anti-symmetric stress tensor[31]. The equations MacCullagh produced are equivalentto the modern Maxwell’s equations empty of matter. In1888 Lord Kelvin (W. Thomson) proved that Green’selastic theory (with symmetric stress tensor) is stable,contrary to what Green thought even when one puts thespeed of the longitudinal waves zero [32]. His theoryhowever is equivalent to that of MacCullagh because theequations of motion of both theories are the same. LordKelvin even produced his famous mechanical gyroscopicmodels which resist rotations and not distortions andthus provided a real system that behaves like the aetherand is equivalent to Maxwell’s equations in vacuum [33].Larmor introduces charges and currents in the MacCul-lagh’s model [34] by simply postulating charge density tobe equal to the divergence of the electric field while at thesame time the electric field is interpreted as the curl ofthe displacement of the aether in vacuum. This seemingcontradiction is explained by saying that the electric fieldin presence of charges is no longer equal to the curl of thedisplacement. In this manner the meaning of the electricfield and the charge is confounded. Also, the electric fieldis a polar vector and not axial and thus Larmor modelthough formally equivalent to Maxwell’s equations withcharges suffers certain limitations. If in the Larmor’smodel the electric field is interpreted as the velocity (orthe flux) of the aehter, it turns out that the presence ofcharges is again difficult to interpret, since then the con-tinuity equation of the aether will be violated [35] and itremains a mystery where the aether goes and whence itcomes.In the model here presented, unlike Larmor’s model,we do not assume an aether with a constant density. Thecharge and the electric field have simple interpretations.At the same time, the amount of aether is conserved. Notonly that but it turns out that the charges move with thesame aethereal velocity, i.e. they are singularities in theaether.The paper is organized as follows. In Section II wepresent the basic equations of the model and show itsequivalence with the customary Maxwell’s equations. Weshow a path to extend the theory and answer possi-ble objections. In Section III we present a Lagrangiandescription of the model and by introducing velocity-dependent mass we derive Newton’s equations of motionof the charges in electromagnetic field. In Section IV weexamine in great detail the many hidden assumptions inthe three different interpretations of relativity theory andargue that Lorentz-Poincare interpretation have certainadvantages over the other two. We show a possible routetowards a mechanical model of Einstein’s gravity equa-tion and we even make an aether model of the linearizedEinstein’s equations. In Section V we give a statisticalinterpretation of Lorentz transformations and show howself interacting gas of constituent particles, interactingvia electromagnetic field is delayed and contracted by itsown electric field. Finally in Section VI we give conclu-sions. II. THE MODELA. Basic equations
Let us examine a quasi-elastic continuous body, bywhich we mean a body with anti-symmetrical stress ten-sor, σ ik = c ( ∂ i A k − ∂ k A i ) i, k = x, y, z, (1)where σ ik are the components of the anti-symmetricalstress tensor, A i are the components of a vector field A ( r , t ) yet to be defined and r = ( x, y, z ) are the coor-dinates in a Cartesian coordinate system with an arbi-trarily chosen origin. The vector field A ( r , t ) is definedthus: ∂∂t A ( r , t ) = ρ ( r , t ) v ( r , t ) + ∇ φ ( r , t ) , (2) where ρ ( r , t ) is the aether density, v ( r , t ) is the aetherlocal velocity field and the scalar field φ ( r , t ) is an ar-bitrary sufficiently smooth field. That A is defined upto such a gradient is obvious from Eq. (1). The stresstensor is not changed by adding ∇ φ to A . It is for thisreason that gauge freedom arises. Writing down New-ton’s equations ρ∂ t v i = ∂ k σ ki (summation by repeatedindexes is implied), we obtain, ρ ∂ v ∂t = − c ∇ × ( ∇ × A ) . (3)We have neglected the convective acceleration ρ ( v · ∇ ) v which is quite customary in any elastic theory. We as-sume in our model that the displacement of the aetheris quite small and the convective acceleration is entirelynegligible.The last equation necessary to complete our model isthe continuity equation ∂ρ∂t + ∇ · ( ρ v ) = 0 . (4)Equations (2), (3) and (4) complete our model. In thenext subsection we shall show that these equations areequivalent to Maxwell’s equations with charges. Notethat continuity equation (4) is obeyed.Finally, we should take notice that we could have de-rived these model equations by assuming the stress tensorto be symmetrical and of the form σ ik = c ( ∂ i A k + ∂ k A i ) − c δ ik ∇ · A , (5)where δ ik is Kronecker’s delta. The relation between Eqs.(1) and (5) is similar to the relation that holds betweenMacCullagh’s model [31] and Lord Kelvin’s model [32]without charges. However in our model we have charges,the density of the aether is not constant and the aether is conserved in the presence of charges, as we shall seebelow. We shall treat both models as equivalent. Thegauge freedom remains even in the model Eq. (5). B. Equivalence of the proposed model withMaxwell’s equations
In order to show that the above equations reproduceMaxwell’s equations with currents and charges we shallmake several definitions.
Definition 1 : There exist independent structures inthe aether which blast away (or draw in) aether with aconstant rate. We call this rate the charge Q , Q ≡ − ∂∂t Z V ρdV, (6)where we have integrated along some fixed control volume V where the charge is placed at some moment of time t .It is obvious that if Q >
0, aether is blasted away fromthe volume V and if Q < (cid:2026) (cid:3051)(cid:3052) (cid:2026) (cid:3052)(cid:3051) (cid:1856)(cid:1876) (cid:1856)(cid:1877) (cid:1856)(cid:1878)(cid:2026) (cid:3051)(cid:3052) (cid:2026) (cid:3052)(cid:3051) (cid:1876)(cid:1877) (cid:1878) (cid:1841) (cid:1842)(cid:4666)(cid:1876)(cid:481) (cid:1877)(cid:481) (cid:1878)(cid:4667)
FIG. 1: (Color online) Torque produced by anti-symmetricalstresses on an elementary volume dV = dxdydz relative tothe center P ( x, y, z ) of the volume. On the right positive x surface, the force σ xy dydz has an arm dx/
2. Summing up allfour torques one obtains the torque dM z in z direction. Thenapplication of Eq. (1) leads to Eq. (10). the charge. Thus, positive charges resemble an electricfan, which blasts away air but does not produce it. Now,if the elementary charges (electrons say) are stable andmove, then while they travel in the aether, they drawin (since electrons have negative charge) aether with aconstant rate, equal to the total elementary charge e ofthe electron.If we examine a contiunous distribution of charges andcall ̺ the charge density, then Eq. (6) implies ̺ = − ∂ρ∂t . (7) Definition 2 : The electric field E ( r , t ) is defined as theflux of the aether, i.e. E ≡ ρ v . (8) Definition 3 : The magnetic field B ( r , t ) is defined thus B = − c ∇ × A . (9)In order to interpret this result one should consider aninfinitesimal volume dV of the aether and calculate thetorque using Eq. (1) acting on this volume, cf. Fig.1.The torque is, d M = 2 c ( ∇ × A ) dV. (10)Therefore the magnetic field is B ( r , t ) = − c d M dV . (11)Thus, apart from some numerical factor, the magneticfield at a point r is a torque acting per unit volume atthis point. Having made these definitions one readily ob-tains the Maxwell’s equations in Lorentz-Heaviside sys-tem. Indeed, ∇ · E = ∇ · ( ρ v ) = − ∂ t ρ = ̺. (12) We have used Eqs. (4), (7), and (8). This is the firstMaxwell’s equation. Next, − c ∂∂t B = ∇ × ∂ A ∂t = ∇ × ( ρ v + ∇ φ ) = ∇ × E . (13)This is the second Maxwell’s equation. We have usedEqs. (2), (8), and (9) as well as the fact that ∇ × ∇ φ = 0for any sufficiently smooth scalar field φ . Next, ∇ · B = − c ∇ · ( ∇ × A ) = 0 . (14)This is the third Maxwell’s equation. And lastly, ∇ × B = − c ∇ × ( ∇ × A ) = ρc ∂ v ∂t . (15)We have used Eq. (9). Next, we complete the full deriva-tive and we obtain ∇ × B = 1 c ∂∂t ( ρ v ) − c ∂ρ∂t v = 1 c ∂ E ∂t + 1 c ̺ v . (16)We have used Eqs. (7) and (8). Amazingly, in orderto obtain the customary Maxwell’s equations we haveto postulate that the charge moves with velocity of theaether , i.e. v charge ( r , t ) = v ( r , t ) , (17)and with the additional definition of charge current J = ̺ v charge , (18)we finally have ∇ × B = 1 c J + 1 c ∂ E ∂t . (19)Thus, in order to obtain the customary Maxwell’s equa-tions, the charge should have the same local velocity ofthe aether. But why should this be so, unless the chargeis part of the aether, a kind of singularity in it, whichblasts away or draws in aether with a constant rate. Wecan speak therefore of a constitutive aether. C. Possible objections answered
A possible objection to the proposed theory is that atany point r the theory seems to suggest that the electricfield E and the current J should be parallel since bothare proportional to the velocity field v . This objectionis easily dealt with. Indeed, we have made a tacit spatialaverage over a volume dV . The volume dV is such thatit is much larger than the dimensions of the elementarycharges and at the same time may be considered as aninfinitesimal volume. More concretely, let us considerthe volume dV centered around the point r . Let us alsodivide this infinitesimal volume dV into N equal cells.We assume that N ≫
1. The electric field is thereforedefined as E ( r, t ) = 1 N X k ρ k v k ≡ h ρ v i , (20)where the sum is spread through all cells. Here ρ k and v k are the aether density and the aether velocity fieldat cell with number k and the brackets hi mean spatialaverage. All other quantities are defined as such spatialaverages, i.e. J = h ̺ v i , B = − c h∇× A i , etc. Now clearly J = N P k ̺ k v k and the sum is spread only where thereare charges in dV , i.e. where ̺ k = − ∂ t ρ k = 0. Thenobviously E is not parallel to J .Another possible objection is that given the model’sbasic equations (2), (3) and (4), it seems that the wholemotion of the aether, including the motion of the charges ,can be calculated. Therefore Maxwell’s equations seem tobe able to produce the equations of motion of the chargesindependently from Newton’s equations. However this isnot the case. Indeed, the aether equations (2), (3) and(4) are not assumed to be valid inside the charges or inthe vicinity of the charges. Some other equations arenecessary to explain the motion of the charges, their sta-bility and their properties which in our model are postu-lated. Therefore the motion of the charges cannot be cal-culated from the insufficient information provided by theaether’s equations. The aether’s equations only displaythe connection between the charges and the electromag-netic field. Thus, the current J and the charge density ̺ are to be considered as independent field sources.In order to see that more clearly, we have J = N P a ̺ a v a = − N P a ( ∂ t ρ a ) v a . The index a showsthat the sum is spread through all cells where there arecharges. Here ̺ a and v a are considered as postulated, notderived, i.e. J and ̺ are external sources of the electricand magnetic field. We do not know what is the equa-tion of motion for the charges, i.e. we do not claim thatwe know ρ a ∂ t v a at the cell a , where there is a charge.However we assume that at the cell where the charge ispresent | ∂ t ρ a /ρ a | ≫ | ∂ t v a | / | v a | , i.e. there are two char-acteristic times, the time for which the charge blasts away(or draws in) aether τ and the characteristic time of mo-tion of the charge τ . We assume that τ ≪ τ . Withthat assumption in hand we have J = − N X a ( ∂ t ρ a ) v a ≈ − N X a ( ∂ t ρ a ) v a − N X a ρ a ∂ t v a . (21)The above equation can be simplified J ≈ − N ∂ t X a ρ a v a . (22)Next, we add the term − N P b ( ∂ t ρ b ) v b which is 0.Indeed, here the index b indicates that the summationis through all cells where there are no charges, and thus ∂ t ρ b = 0. Thus we have J ≈ − N ∂ t X a ρ a v a − N X b ( ∂ t ρ b ) v b . (23) We complete the full derivative in the second term, andwe have J ≈ − N ∂ t X k ρ k v k + 1 N X b ρ b ∂ t v b , (24)the index k shows that the summation is spread throughall N cells. Therefore the first sum is the electric field E The second sum of the above equations is spread throughall cells b , where there are no charges and then we canapply the aether equation (3). Thus, J ≈ − ∂ t E + cN X b ( ∇ × B ) b . (25)But there are much more cells b where there are nocharges than cells a . Therefore we have1 N X b ( ∇ × B ) b ≈ N X k ( ∇ × B ) k ≡ h∇ × B i . (26)Note that if we chose to imply the spatial average andnot write it, the right hand side is just ∇ × B . With this,we finally have J ≈ − ∂ t E + c ∇ × B . (27)By neglecting the term ρ a ∂ t v a , we have been able toshow a connection between J on the one hand, and B and E on the other. However we do not know the equationsof motion of the charges (only how they are connectedwith the electromagnetic field) and thus the current J isto be considered as a source in the equations of Maxwellwhich cannot be calculated from these equations. Similarreasoning can be applied to the rest of the equations ofMaxwell. Not only that but we see that the spatial aver-age is essential in order to consider J and ̺ as sources.These considerations show that our derivation of Eq. (19)was somewhat simplistic but we still have v charge = v , i.e.the charge is a singularity in the aether. Henceforth weshall imply the spatial average but we shall not write it. D. Electromagnetic field in aether devoid ofcharges
In the absence of any charges, ρ is time-independent,the equations are quite simplified. Indeed, Eq. (2) couldbe integrated and we have A = ρ ξ , (28)where the vector field ξ is the displacement of the aetherand we have chosen φ = 0 for simplicity. Then, thestress-tensor becomes σ ik = c ρ ( ∂ i ξ k − ∂ k ξ i ) , i, k = x, y, z. (29)Therefore in the absence of charges, the theory is re-duced to the old MacCullagh’s quasi-elastic aether. Thisaether resists rotations in the same manner as elasticbody resists displacements according to Hooke’s law.More concretely the torque necessary to rotate an in-finitesimal aether particle with volume dV is proportionalto the angle of rotation δ ϕ = ∇× ξ . Indeed, it is readilyobtained that d M = 4 ρc δ ϕ dV. (30)This equation should reminds us of Hooke’s law. InHooke’s law the force is proportional to the displacement ξ valid for elastic bodies. In our case however, it is the torque d M which is proportional to the angle of rotationof the aether particles. The aether thus resists rotations,not displacements as in the elastic theory. Lord Kelvin’sgyroscopic machines can simulate such a behavior [33].In addition, since ρ = const., the aether is incompress-ible and the continuity equation becomes ∇ · ξ = 0 . (31)Newton’s equation is simplified to ρ ∂ ξ ∂t = − c ρ ∇ × ( ∇ × ξ ) = c ρ ∇ ξ . (32)Cancelling ρ we obtain the homogeneous wave equationwith travelling transverse waves with speed c , the speedof light. The light becomes thus a kind of ’sound’ wavein the aether. E. Nonlinear theory
It is interesting to ponder upon the following possibil-ity. Let us imagine the full Newton’s equation for theaether, including the convective acceleration as well assome higher nonlinear terms ρ ∂ v ∂t + ρ ( v · ∇ ) v = − c ∇ × ( ∇ × A ) + .... (33)Where ... means some higher order nonlinear terms, i.e.terms which depend upon the velocity of the aether v in a nonlinear manner. Is it possible to choose somesuch terms and to derive soliton solutions of the proposedtheory which give all the properties of the charges whichin the present model are simply postulated? This seems aformidable mathematical problem. It is quite intriguingthough if one can obtain them. The convective term willtend to keep the soliton stable as in the famous Korteweg-de Vries equation. III. LAGRANGIAN DESCRIPTION ANDNEWTON’S EQUATION.
In this section we shall examine the Lagrangian of thefield equations. We start with considering the motion ofthe charges as given. The charges are thus conceived of as external sources of the field. First, let us choose forsimplicity the Weyl gauge φ = 0 in Eq. (2). Of course,this gauge is incomplete and some additional restrictionsought to be implied to fix the vector A . However theseare quite familiar and we shall not bother with them here.In addition, we note that any gauge whatsoever may beused.Now by completing the full derivative in Eq. (3) weobtain − c ∇ × ( ∇ × A ) = ∂∂t ( ρ v ) + J = ∂ A ∂t + J . (34)We have used Eqs. (2) and (7). Rearranging, c ∇ × ( ∇ × A ) + ∂ A ∂t = − J . (35)This field equation can be readily obtained from the prin-ciple of least action with Lagrangian density, L = L f + L fc , (36)where L f is the field Lagrangian density, L f = 12 (cid:18) ∂ A ∂t (cid:19) − c ∇ × A ) , (37)while L fc is the Lagrangian density of interaction be-tween field and charges L fc = − J · A . (38)Of course, if the charges are modeled as delta functions,which means we neglect their dimensions, one obtains theLagrangian L fc = Z V L fc dV = − X a q a v a · A ( r a , t ) , (39)where the sum is spread through all the charges. It iseasy to prove that the above Lagrangians readily givethe field equation (35).So far, we have merely rewritten our model in La-grangian terms. If we now include the Lagrangian forthe elementary charges L c = − X a m a c r − v a c , (40)into the total Lagrangian L = L f + L fc + L c , (41)where L f = R L f dV is the field Lagrangian, one obtainsnot only the field equation (35) but also Newton’s equa-tion for the charges ddt m a v a q − v a c = q a E ( r a , t ) + q a v a c × B ( r a , t ) . (42)The stipulation (40) is another assumption in ourmodel, that the mass of the charge is velocity dependent,i.e. m ( v ) = m q − v c . (43)For a single free charge, we have ddt m v q − v c = 0 . (44) IV. LORENTZ-POINCARE INTERPRETATION,RELATIVISTIC INTERPRETATION ANDMINKOWSKIAN INTERPRETATION OF THETHEORY OF RELATIVITY. ASSESSMENTS
So far the proposed model of electrodynamics assumesthe existence of a special reference frame - that of theaether, which at first sight might appear to contradict thetheory of relativity. This is not so however, if one takesinto account that three very different interpretations ofrelativity theory [1] exist with very different ontologies(but sadly, often mixed in textbooks), though being em-pirically equivalent. These are the Lorentz-Poincare in-terpretation, the relativistic interpretation (which wasthe original Einsteinian interpretation of his 1905 paper[36]) and the Minkowskian interpretation. In this sectionwe show that Lorentz-Poincare interpretation is consis-tent with the model. In addition we shall argue thatLorentz-Poincare interpretation has certain advantagesover the other two interpretations. We shall do this bylisting certain difficulties with the relativistic interpreta-tion and with the Minkowskian interpretation, which canbe more easily resolved with the Lorentz-Poincare inter-pretation. This will provide even further support for themodel of electrodynamics here presented.In order to understand the great differences betweenthe three interpretations of relativity theory, one needsfirst to take into account that there are two models oftime [1], called tensed theory of time (also A-theoryof time) and tense less theory of time (also B-theory oftime). The great difference between the Minkowskian in-terpretation and the other two interpretations is rootedin the fact that the Minkowskian interpretation assumesB-theory of time, while the other two interpretations as-sume A-theory of time.According to A-theory of time only the present is real(i.e. only the present exists), the future does not exist(it will exist) and the past does not exist (it no longer exists). This is the common sense notion of time. Letus imagine a staircase and let each stair represents a mo-ment of time. According to A-theory of time one par-ticular stair (present) exists, the stairs below this stair(past) no longer exist and the stairs above this stair (fu-ture) do not exist yet. When the next moment of timecomes (and it becomes present), it comes into being and the previous stair (which becomes past) ceases to exist.Such is the classical notion of the flow of time.According to B-theory of time, the whole staircase ex-ists and is real, i.e. not only the present exists but alsothe past and the future. Such a model of time allows thehypothetical possibility of going back in time (gettingdown to lower stairs), while the A-theory of time doesnot allow this possibility (for the past does not exist).The Minkowskian interpretation as we shall see belowrests on the assumption of B-theory of time. A. Lorentz-Poincare interpretation
Faced with the stellar aberration phenomenon [1](which seems to imply that no aether wind exists) andwith the null effect of the Michelson-Morley experi-ment on detecting the motion of the Earth relative tothe aether (which could be explained by aether wind),Lorentz commenced a research program to explain theseseemingly contradictory observations. He went to theexpedient of introducing contraction of the length of thearms of the Michelson-Morley interferometer along theirdirection of motion with the standard now FitzGerald-Lorentz contraction. Later it was discovered that clockdilation was necessary as well to explain the null effectof the Michelson-Morley experiment. Lorentz attemptedto derive these effects (length contraction and clock di-lation) from a microscopic model of interaction betweenthe constituent particles of the interferometer. Howeverhis research program was interrupted by Einstein’s pa-per of 1905, where he introduces his famous special the-ory of relativity, which we shall discuss in the next sub-section. Lorentz interpretation (which we should bettercall Lorentz-Poincare interpretation) is fully consistentwith Lorentz transformations and is empirically equiva-lent with both the original Einsteinian special theory ofrelativity of his 1905 paper (which Einstein later aban-doned in favor of the Minkowskian interpretation) as wellas with Minkowskian interpretation. However it is a dif-ferent theory.Lorentz-Poincare interpretation may start with the no-tions of space and time according to Newton. This meansthat there exists an absolute time and absolute space.Absolute time flows uniformly of its own nature and with-out reference to anything external. It is different than the physical time, which is the measure of absolute time by physical clocks and material bodies. The physical clocksare delayed when in motion but not the absolute time.The same with space. According to Newton there aretwo kinds of spaces - absolute and physical. Absolutespace is homogeneous and immovable, it exists withoutreference to anything external. However, physical spacemeasured by physical processes (light signals or physicalrods) is merely a measure of the absolute space. Newtondid not know that if physical bodies are in motion, theyare contracted and the physical space changes. However absolute space does not. Therefore one should distinguishbetween the absolute and the physical. This distinctionshows immediately that there exist a special referenceframe which should give physical time and physical spacein coincidence with absolute time and absolute space.However, Lorentz-Poincare interpretation does not ne-cessitate the existence of absolute time and space, butinstead it necessitates the existence of a special refer-ence frame relative to which the true time (which meansby definition the time in this special reference frame)and true space (which by definition means the physicalspace as measured in this special reference frame) aremeasured. Even if this reference frame is never to bemeasured and even if the laws of physics look exactly thesame as in all other reference frames, yet it exists. Weshould correct our measuring instruments, if it is possi-ble, to be in accordance with this special reference framein which the aether is at rest. If it is not possible then weshould imply its existence nonetheless. In addition, weshould seek physical causes as explanations as to why theclocks are delayed and rods contracted when in absolutemotion. The further point of whether absolute time andabsolute space exist cannot be established with empir-ical means but rather with philosophical or theologicalones [1]. However I shall call the true time and the truespace, absolute time and absolute space, following longtradition, even though these are different.Lorentz transformations therefore merely describe how physical rods and clocks are contracted and delayed whenin motion. They connect reference frames made by phys-ical rods and clocks amenable to alteration when in mo-tion.Lorentz-Poincare interpretation has been further de-veloped (neo-Lorentzian interpretation) to as few as pos-sible assumptions. In fact it has become as simple asthe relativistic and Minkowskian interpretations. In latersection we show that Lorentz-Poincare interpretation canbe reduced to two assumptions only - the existence of aspecial reference frame (that of the aether) and velocity-dependent mass of the particles. All else follows fromthese assumptions, including Lorentz transformations.We show that a system in motion emits electromagneticfield and it acts back on this system thus slowing downall processes and contracting the bodies in the directionof motion with the standard FitzGerald-Lorentz contrac-tion if they are to preserve their thermal equilibrium.Please note however that even the velocity dependenceof the mass of the particles has been made quite naturalby Ives [2], where he derives it from two reasons only -the laws of conservation of energy and momentum as wellas the laws of transmission of radiant energy.This interpretation gives us physical causes for theclock dilation and rod contraction, namely physical forces . These effects are not derived simply as a resultof the postulates of the theory as is done in the relativis-tic interpretation and in the Minkowskian interpretationwithout any causes. For more details, see below.The interpretation assumes A-theory of time and stan-dard notion of the flow of time.
B. Relativistic interpretation
There are two other interpretations of the theory ofrelativity. The second interpretation, which we shall callrelativistic interpretation is the original Einstein’s inter-pretation of his 1905 paper [36]. In this interpretationspacetime is merely an instrument, a helpful tool, and isnot interpreted realistically (as in Minkowskian interpre-tation). It assumes A-theory of time. Einstein droppedthis interpretation later in favor of the Minkowskian in-terpretation.It is customary to present this interpretation in termsof the postulate of the relativity of all inertial referenceframes and the postulate of the constancy of the veloc-ity of light c . However, in order to define properly themeaning of the words ’reference frame’, one needs morepreparatory work and we shall see that instead of twoaxioms, we need in fact eleven, eight of which are mereconventions (at least in this interpretation) and three areempirical. We shall follow Reichenbach [37].
1. The relativity of the geometry of space
The defenders of this interpretation reason thus. Letus start with the geometry of space. Let us imagine two-dimensional creatures living upon a two-dimensional in-finite surface which is not a plane, i.e. the geometryis not
Euclidean. How could these creatures (being to-tally unaware of a third dimension) discover that theydo not live on a plane? Perhaps they could sum the an-gles of a triangle and discover that it is not π . If theymake such a discovery they can reason that their spaceis not Euclidean. However there is an alternative ex-planation. There may be some force acting on the rodsand distorting them such that it only appears that thespace is not Euclidean, for example some heat which dis-torts the metal rods. However, if the cause is heat, thetwo-dimensional creatures can easily discover this by us-ing rods made of different material. Heat alters differentmetals in a different manner. Such a force, which acts dif-ferently on different bodies is usually called ’differentialforce’. It is easy to detect the presence or absence of suchdifferential forces by simply choosing different rods andestablishing different distortions. However, what if theforce is universal, that is a force which distorts all rods of whatsoever material in the same manner? Then, the two-dimensional creatures will not be able even in principle (according to the defenders of this interpretation) to dif-ferentiate between the two situations - a flat space with auniversal force which affects all rods or a non-Euclideanspace without such universal forces. The defenders ofsuch an interpretation claim that it is not merely a caseof technical impossibility but of impossibility in principle (we shall prove below that such a hasty presumption isfalse and there is a method to differentiate between thesetwo scenarios). Neither does transporting of rods help.For by transporting them, the universal forces may affectthem in such a manner that when the rods are united inthe same point in space they may be equal but when theyare separated in distant points they are no longer equal.Therefore, the relativist concludes that the geometry ofspace is merely a convention . One may choose flat geom-etry always but allow universal forces or choose a curvedspace and deny universal forces. One may also choose byconvention that part of the measured curviture of space isdue to non-Euclidean geometry and part - due to univer-sal forces. Such a convention, the relativist continue, isa necessary convention, not very different from choosingby convention the unit of distance - meter, foot, etc.
2. The relativity of the flow of time
The same reasoning could be applied to time. We can-not say that time flows uniformly in some particular pointin space. Neither can we say that time in different posi-tions flows the same. For if universal forces exist, theymay alter the duration of all processes. Neither does ithelp to transport the clocks for the same argument of rodtransportation is applied here. Thus, we again have nomeans, even in principle to distinguish between the twoscenarios - clocks are distorted by universal force or thereis no universal force but time flows in different manner(again, such a presumption is quite hasty and it is in factfalse, as we shall see below).
3. The relativity of the simultaneity of distant events
What of simultaneity? It is obvious that we can es-tablish whether two events are simultaneous at the samepoint in space by a simple comparison of these events.But what of simultaneity of events in different locationsin space? Let us choose two distant points A and B . Letus measure the distance between them and call it | AB | .Let us send some signal (say a sound wave) from point A to point B at a moment of time t (as measured by aclock in A ) . At what moment of time t (measured by aclock in A ) has the signal reached point B ? This questionis the same as asking what moment of time t as mea-sured by a clock in A is simultaneous with the momentat B when the signal reached B . Thus we are askinga question of simultaneity of two distant events - clockmeasurement t in A and the moment when the signalreaches B . One possible answer is this - by knowing thespeed of sound c s , the moment of time t = | AB | /c s . Butthis answer is clearly misleading. For how is the speed ofsound c s measured in the first place? One simply dividesthe distance traveled by the sound | AB | by the time in-terval t − t . But this presupposes the knowledge of t inthe first place. Therefore the measurement of speed pre-supposes the notion of simultaneity of distant events. Ifthere were infinitely fast signals, then we can send such asignal from point A at moment of time t to point B andreflect it back to A . Clearly this signal, being infinitely fast, returns back to A again at the moment t . However,the relativist would claim that there are no such signals,and thus absolute simultaneity of such a type cannot beestablished even in principle. Therefore we can use thefastest possible signal - light signal and send it from point A at a moment of time t . Then the light signal reachespoint B and is reflected back to A . It returns to A at mo-ment t . What moment of time t measured by a clockin A is simultaneous with the event when the light signalreached B ? Obviously t < t < t . But since there areno infinitely fast signals, nor there are signals faster thanthe light signal, then it is impossible even in principle(according to the relativists) to establish t . Thus, therelativist claims that the moment t is chosen by con-vention! In other words t = t + ǫ ( t − t ) and we canchoose by convention any ǫ such that 0 < ǫ <
1. Thechoice ǫ = is one such possibility. If we choose ǫ = itappears that we have assumed that the light signal trav-els in both directions with the same speed. However thisis not true. We have in fact defined it to travel in bothdirections with the same speed. The choice of any ǫ is aconvention and it defines simultaneity of distant events.Therefore the constancy of the speed of light in both di-rections (being the fastest signal) is a convention, not anempirical fact. There are other empirical facts which arenot conventions and we shall discuss them below. In factdifferent choices of ǫ lead to different Lorentz transfor-mations. The familiar Lorentz transformations are trueonly by the convention ǫ = . Please note that the no-tion of a signal being faster (or fastest) does not requirethe notion of simultaneity. We simply send two signalsfrom A simultaneously (as measured by A ) and discoverwhich signal reached B first. No notion of simultaneityis required here.The relativist thus concludes that the geometry ofspace, the flow of time, the notion of simultaneity of dis-tant events are mere conventions. Before one builds thetheory one needs to separate what is a convention andwhat an empirical fact. With this preparation in hand,the relativist can finally build the whole theory.
4. Definition of reference frames. Lorentz transformations
Let us imagine a continuum of points in the whole ofspace, each endowed with an observer. Let us consider aparticular point A . The observer at A defines his unit oftime by some periodic process and let this unit of timebe the second. Now, the observer at A cannot say thathis time flows uniformly. There may be universal forceswhich affect all periodic processes at A and thus maskingthe fact that the time does not flow uniformly. But theimpossibility to establish the uniformity of time in prin-ciple, forces us, says the relativist to define that the timeat A flows uniformly by mere convention. Axiom 1: (convention):
Time flows uniformly at allpoints in space.Next, the observer sends a light signal to some point0 B and reflects it back to A . Let us denote with ABA the time interval for the whole trip of the light signal A − B − A as measured by the clock at A . Axiom 2 (convention) : If the point B has the propertythat the time interval ABA is always the same as mea-sured by a clock at A , no matter when the light signalis sent from A , we define such a point of being at rest relative to A .Please note that this is a mere convention and in facta definition of rest. Now, the observer at A finds otherpoints C , D , etc. being at rest relative to A . We callsuch a system of points at rest relative to A . However,just because ABA = const.,
ACA = const., etc. it does not follow that
BAB = const. or
CAC = const. Inother words, the points B , C , etc. are at rest relativeto A but it does not follow that A is at rest relative to B or to C or to any other point. That such systems ofpoints exist with the special property that all points areat rest relative to each other is an empirical fact (we donot consider general relativity here). Axiom 3 (empirical fact):
There exist special systemsof points A , B , C ,..., such that all points are at restrelative to each other.Note, there is not just a single system of points butinfinite such systems. Axiom 4: (convention):
We select such a system ofpoints which are at rest relative to each other.Next, the observer at A sends his time unit (second) tothe other observers at B , C , etc. He may do so by merelysending light signals every second. Please note that theunit of time is thus transferred to the other observers,but clocks are not yet synchronized, i.e. the notion ofsimultaneity of distant events is not established yet.Let us choose three points, A , B and C of our se-lected system of points. Therefore these points are atrest relative to each other. And let us send two signals simultaneously from A . One of the signal travels the trip A − B − C − A and the other A − C − B − A . Now, generally the two signals will not return to the point A simultaneously (measured by the clock at A ) even thoughthe points A , B and C may be at rest relative to eachother. That there exist such systems of points that theround trip journey takes the same amount of time is anempirical fact (again, we exclude general relativity here). Axiom 5: (empirical fact)
There exist special systemsof points, at rest relative to each other such that theround-trip journeys
ABCA = ACBA are always thesame.We are finally ready to define the simultaneity of dis-tant events by light signal synchronization.
Axiom 6: (convention)
Distant clocks are synchronizedusing light signals. In other words if we choose two ar-bitrary points A and B of our selected system of pointswhich are at rest relative to each other, we send a lightsignal at a moment of time t measured by the clock at A . It travels the distance A − B − A and returns at A at amoment of time t by the clock at A . The moment t at A simultaneous with the moment at B when the signal reached B is defined to be t = t + ǫ ( t − t ) for ǫ = .In this manner clock B is synchronized by the clock in A . The clocks in all other points can be synchronized bythe clock at A in the same way.The above definition may seem to have chosen a specialpoint A . But it can be easily proved that the above syn-chronization procedure is symmetric. This means thatthe point A is not special in any way and in fact if wewere to choose any other point to synchronize all clocks,both synchronizations will agree, provided we choose thesame ǫ (in our case by convention ǫ = ) . In additionthis synchronization is transitive, i.e. if two clocks at dif-ferent points B and C are synchronized by A they aresynchronized by each other.Thus far we have dealt with the concept of time in ourselected system of points. Now we continue with space.The first notion is the topological notion of between . Axiom 7:(convention):
If we choose three points A , B and C in our selected system of points we define point B to be between A and C if ABC = AC . Axiom 8 (empirical fact):
If points B and B arebetween A and C , then either B is between A and B or B is between B and C .The above two axioms help us to define the notion ofstraight line. Axiom 9 (convention):
The straight line through A and B is the set of all points which among themselvessatisfy the relation between and which include the points A and B .With this preparation in hand, we can define the equal-ity of distances in our selected system of points. Axiom 10 (convention):
If the time interval
ABA = ACA for three different points A , B and C in our selectedsystem of points, then we define | AB | = | AC | .This concludes the geometry of space. The above ax-ioms are quite sufficient to prove that space becomes Eu-clidean. Axiom 11 (convention):
Let us choose two inertial sys-tems K and K ′ as defined by the above axioms in differ-ent states of motion. Let l be a rest-length in a system K and l ′ be a rest-length in K ′ . If l is measured by observesat rest in K ′ , they will not in general measure the samelength l as observers at rest in K . There will be someexpansion or contraction factor. The same principle istrue if l ′ is measured by observers at rest in K . We re-quire by convention the identity of these expansion (orcontraction) factors obtained by the observes at rest in K and K ′ .With these eleven axioms at our disposal we finallyhave a correct meaning of the notion of reference frame.Obviously the above axioms define the light signal tohave the same velocity in each reference frame. Not onlythat but the geometry is Euclidean (we are still in specialrelativity) and the distance traveled by a light signal frompoint ( x, y, z ) to point ( x + dx, y + dy, z + dz ) is c dt = dx + dy + dz , (45)where the right hand-side is the distance between two1infinitesimally close points and dt is the time requiredfor the light signal to traverse that distance. In anotherreference frame we have the same speed, thus c dt ′ = dx ′ + dy ′ + dz ′ . (46) Given our axioms , the only transformations between x, y, z, t and x ′ , y ′ , z ′ , t ′ that obey the above two equationssimultaneously are the familiar Lorentz transformations.All familiar results follow from here.Imagine a rod placed in x direction in a reference frame K and let it move with a velocity V along x directionrelative to K . How is the length of the rod measured?One simply places two observers at some moment of time t (in K ) placed at both ends of the rod and measuresthe distance between the observers. However, if one per-forms the same experiment in a reference frame K ′ whichmoves with the rod (i.e., the rod is at rest relative to K ′ )the very notion of the same moment of time t ′ in K ′ isquite different than that in K and thus different lengthis measured. Therefore the difference of the length ofan object in different reference frames is connected withthe relativity of simultaneity in different reference frames(according to the relativists).The interpretation uses A-theory of time. This con-cludes the relativistic interpretation. C. Minkowskian interpretation
Minkowskian interpretation unites time and space intoa four-dimensional manifold, called space-time. Thespace-time is not merely a helpful instrument but is in-terpreted realistically. The physical objects are four-dimensional. This interpretation assumes B-theory oftime. The four dimensional distance between two points( x, y, z, t ) and ( x + dx, y + dy, z + dz, t + dt ) in space-timeis, ds = c dt − dx − dy − dz . (47)The geometry in space-time is thus defined, as beingpseudo-Euclidean geometry. Going from one inertial ref-erence frame to another is again given by Lorentz trans-formations, but they are here interpreted as a change ofcoordinates in the space-time manifold. D. Assessment of the three interpretations.Criticism of the relativistic interpretation and theMinkowskian interpretation
Of the three interpretations of relativity theory, onlyLorentz-Poincare interpretation is consistent with ourmodel of the electromagnetic field and with a specialreference frame, that of the aether. Relativistic andMinkowskian interpretation are not consistent with aspecial reference frame. However, let us examine more (cid:1841) (cid:1876)(cid:1878) (cid:1877) (cid:2178)(cid:1864) (cid:883) (cid:3398) (cid:1848) (cid:2870) (cid:1855) (cid:2870) (cid:1864)(cid:2178) (cid:2869)
FIG. 2: (Color online) The reality of FitzGerald-Lorentz con-traction. As the rod, with rest-length l moves with velocity V in x direction it is shortened to l p − V /c . On the otherhand, the radius of the metal ring which moves with velocity V in z direction is not . Therefore the rod can pass through the ring! carefully the relativistic and the Minkowskian interpre-tations.Let us start with the relativistic interpretation. Let usimagine two objects (see [1] and the references therein) -a rod and a metal ring in a reference frame K , see Fig. 2.Let the rod has a rest-length l placed along x directionin K . Let the rod moves in the x direction with velocity V with respect to K and let initially, at t = 0 all thepoints in the rod have the same coordinate z >
0. Letus also imagine a circular metal ring with a diameter l ,equal to the length of the rod, whose plane is in the xy plane. Let the ring moves upward along z direction withvelocity V so that its plane remains parallel to the xy plane. The two bodies - the rod and the metal ring moveand let us set this system of two bodies in such a mannerthat the two objects meet at some moment of time t > l p − V /c while the diameter of the metal ring isnot changed, since the velocity V is perpendicular tothe plane of the ring. Therefore, the rod will be able topass through the metal ring! Of course, if one examineswhat happens from the reference frame of the rod, itis trivial to show that the ring will be inclined due toLorentz contraction and the rod will still pass the ring.However in K we see that the rod passes through thering and so the FitzGerald-Lorentz contraction is a realphysical phenomenon, not simply a result of the relativityof simultaneity as claimed by the relativists.Let us examine another famous example - Bell’s space-ship paradox, see [1] and [15]. Let us consider two space-ships moving with the same velocity in an inertial refer-ence frame K , say along x direction. Therefore the dis-tance between them remains constant as they move andlet us call this distance L . Let these spaceships acceleratesimultaneously (with respect to K ) with the same accel-eration. Obviously the distance between the spaceships2will remain the same, i.e. L , even after they acceleratebecause the two spaceships have the same acceleration.Now, let us consider this scenario again but this timelet us imagine a delicate string or thread that hangs be-tween the spaceships, i.e. the string has a length L . Now,let the ships accelerate again with the same acceleration(with respect to K ). However, this time the string willbe subjected to FitzGerald-Lorentz contraction, i.e. itslength will tend to be less than L . On the other hand thedistance between the ships remains L . Thus the stringwill break! That it will break can be seen from the mo-mentary inertial frame of the spaceships K ′ , where dueto the relativity of simultaneity the ships will not begintheir acceleration simultaneously even though they accel-erate simultaneously in K . Therefore FitzGerald-Lorentzcontraction can break delicate strings.Both of these scenarios can be multiplied [1] and peo-ple who are trained to think in terms of the relativisticinterpretation will be quite startled at first. The reasonfor their surprise is that the FitzGerald-Lorentz contrac-tion is quite real - as real as the contraction of metal rodswhen their temperature is decreased. Lorentz contractionis a true physical contraction. Within Lorentz-Poincareinterpretation these two examples are not difficult to ex-plain because bodies that move with a velocity relativeto the aether are indeed contracted by physical forces.There is a true physical force that causes the contractionand it may well break delicate strings and threads. InMinkowksian interpretation the bodies are not three di-mensional but four-dimensional objects. And when theobjects move it is like seeing them in the four-dimensionalspace-time from different ’angles’. Thus effects like theabove are explained also in Minkowskian interpretationbetter than the relativistic interpretation. Examples likethat show that Minkowskian interpretation has more ex-planatory power than the relativistic interpretation. Andfor that reason the practitioners of relativity theory favorthe Minkowskian interpretation rather than the relativis-tic interpretation.Therefore these examples show that the relativis-tic interpretation is explanatorily impoverished as com-pared to the Lorentz-Poincare interpretation and theMinkowskian interpretation. However there are moreproblems. Indeed, since the relativistic interpretationassumes A-theory of time only the present exists. Thepast and the future do not exist. The future will exist,and the present no longer exists. But the very notionof the present (and thus of what exists) is frame depen-dent. Imagine two inertial reference frames. In one of theframe, a person may be shot dead, while in another refer-ence frame he may still be alive (not yet shot). If the tworeference frames are to have an equal status, then eachreference frame is a new world in which different thingsare real. Going from one reference frame to another isthe same as going from one world to another. Such a plu-ralistic ontology is fantastic. Even worse, the relativisticinterpretation is based upon arbitrary conventions. Therelativist believes that he is compelled to choose ǫ by convention because one cannot establish empirically dis-tant simultaneity. However the philosophy behind thatis the old defunct philosophy of positivism (according towhich things that one cannot measure are meaningless).However, this philosophy has been abandoned (see [1]and the references therein) by the philosophers of sciencesince it is too restrictive and is contrary to the scien-tific endeavor. A scientist quite often postulates the ex-istence of many things which are not yet empirically es-tablished in order to give explanations of a phenomenon- the molecular hypothesis in statistical mechanics haseasily explained thermodynamics and chemical reactionswell before these molecules were detected directly. Manyother examples could be multiplied - the Higgs boson,great many elementary particles, chemical elements, theprediction of the existence of the planet Neptune, etc. Inaddition, positivism confuses epistemology (what we canknow) with ontology (what exists). As to the matter ofabsolute distant simultaneity one can simply imagine in-finitely fast signals and clock synchronization with thesesignals, even if such signal are not observed and even ifthey do not exist in nature. Even if one cannot establishabsolute simultaneity by empirical means (in fact we can as we shall show below) does not mean that there is nosuch absolute simultaneity. Neither does it mean thatwe have to have a recourse to such a fantastic pluralis-tic ontology which is explanatorily impoverished. Oneshould distinguish between what one can know and whatactually exists. These are different things.Neither does Minkowskian interpretation solves theabove problems satisfactorily because it is beset withother difficulties. Indeed, the first difficulty is the unionof space with time. Just because one can write spaceand time coordinates on the same coordinate system, onecannot consider them united. One can unite pressureand volume on a single coordinate system. This doesnot mean that there is such a thing as a pressure-volumespace. Neither does it help to claim that space-time isdifferent than volume-pressure space by the presence offour-dimensional metric. But how has one detected thismetric in the first place? One had to apply the clock syn-chronization procedure first, which is quite arbitrary andrests on arbitrary conventions (the choice of ǫ ) and on de-funct positivistic principle. Different conventions of ǫ willlead to different metrics (in [37] Reichenbach gives suchexamples). If one denies the doctrine of Positivism andaccepts absolute simultaneity, the metric in space-timefalls apart and space-time is thus reduced to the rank ofthe pressure-volume space. In addition, if one is to ac-cept the realism of the space-time one has to accept thatspace-like four-dimensional intervals exist and are com-plex numbers, which is quite incredible. But even worsethan that is the acceptance of B-theory of time which fliesin the face of our experience of time. B-theory assumesthat past and future exist, that there is a hypotheticalpossibility of time-travel in the past. But there is no ev-idence of such things. In fact, one can argue that theA-theory of time is a properly basic belief (see [1] as well3as [23] for the definition of properly basic belief) and theburden of proof lies upon the shoulders of the B-theorist.What is the evidence for B-theory? There is none. B-theory is simply postulated without any evidence. Thus,it is quite save to say that space-time is merely a goodinstrument, already used in Newtonian physics and is notto be accepted as the true reality.Things are aggravated greatly if quantum mechanicalconsiderations are taken into account. Quantum mechan-ics is in great deal of tension with both Minkowskian andrelativistic interpretation regardless of whether one ac-cepts Copenhagen’s interpretation of quantum mechan-ics or some non-local hidden variable theory. Indeed, letus consider a version of the EPR (for much more details,see [38]) paradox and examine a decay of the π mesoninto an electron and positron. Let the π meson decaysat the moment of time t = 0 at the origin of an iner-tial reference frame K . Obviously in K the electron andpositron have opposite momenta. Since the spin of the π meson is 0, due to the conservation of angular mo-mentum the electron-positron pair is in the singlet spinstate | i = √ [ | ↑↓i − | ↓↑i ], where | ↑i means spin in z direction and | ↓i means spin opposite to z direction. Atsufficiently great distances from the origin O of the coor-dinate system we place detectors that measure the spin ofthe electron and of the positron in directions a and b re-spectively, where | a | = | b | = 1. Simple quantum mechan-ical considerations give us that the correlation of the spinprojections upon a and b directions is h ˆ S a ˆ S b i = − ~ a · b ,where ~ is Planck’s constant. Very general considerationslead us to the conclusion that local hidden variable theoryis not consistent with this correlation as can be seen byBell’s inequalities [38]. Thus, if there is a hidden variabletheory one is compelled to choose some non-local theorywhich is inconsistent with neither the Minkowskian inter-pretation nor with the relativistic interpretation. How-ever, such a non-local theory is consistent with Lorenz-Poincare interpretation, which already assumes a specialreference frame and is compatible with both the Lorentztransformations and with superluminal velocities. Notonly that but the tension remains quite strong even inCopenhagen interpretation. Though it is true that onecannot send superluminal signals with quantum mechan-ical correlations, yet a problem remains. Indeed, uponmeasurement of the spins by the detectors the collapseof the paired wave function is simultaneous. But simul-taneous with respect to what reference frame? If it issimultaneous in some inertial reference frame K then inanother reference frame K ′ , due to the relativity of si-multaneity, one of the particles would have been with acollapsed wave-function before it is even measured by thedetector. Thus, the very spin state becomes not intrinsicproperty of the electron or the positron but frame depen-dent. If one is to avoid such highly implausible scenario,one has to hold the relativistic and the Minkowskian in-terpretations into question. Quoting Bell [5]: ”I think it’sa deep dilemma, and the resolution of it will not be triv-ial; it will require a substantial change in the way we look at things. But I would say that the cheapest resolutionis something like going back to relativity as it was beforeEinstein, when people like Lorentz and Poincare thoughtthat there was an aether - a preferred frame of reference- but that our measuring instruments were distorted bymotion in such a way that we could not detect motionthrough the aether...The reason I want to go back to theidea of an aether here is because these EPR experimentsthere is the suggestion that behind the scenes somethingis going faster than light. Now, if all Lorentz framesare equivalent, this also means that things can go back-ward in time...this introduces great problems, paradoxesof causality, and so on. And so it is precisely to avoidthese that I want to say there is a real causal sequencewhich is defined in the aether”.Therefore Bell’s inequalities are criticism of the local-ity assumption rather than the hidden variable theories.For they may close the gate for local hidden variable the-ories but also open the gate for more rational non-localhidden variable theories of the type of de Broglie-Bohminterpretation. Lorentz-Poincare interpretation is quiteconsistent with Lorentz transformations, aether and withnon-local hidden variable theories.The introduction of general relativity as a proof thatthe space-time is necessary not just as an instrument butas a reality is also implausible. Let us examine the mat-ter more carefully. Einstein had attempted to introducea principle of ’general’ relativity that will relativize allmotion, not simply the motions with constant speed asin special relativity. It is generally accepted that he failedin that regard [1] and instead he drafted a complicatedgravity theory. Therefore the name ’general relativity’is a kind of misnomer. Let us follow his reasoning. Heimagines two fluid spheres S and S at rest relative toeach other with the same radii. The sphere S begins torotate and due to the centrifugal forces, it will changeits shape into an ellipsoid of revolution as measured bythe instruments which are at rest relative to S . How-ever, from the reference frame of S , S will not changeand there is thus asymmetry between the two spheres.It seems therefore that the rotating motion is absoluteand detectable. However, Einstein applies Mach’s prin-ciple, according to which the inertia of the spheres isdetermined by the distant bodies, the distant stars. Ifthere were not distant stars and distant bodies the ro-tating motion would not have been detected. However,Einstein confuses here Mach’s principle with the exten-sion of the relativity principle. The relativity principlewould have required that from the reference frame of S ,both S and all distant stars with respect to S appeardistorted and S appears as an ellipsoid of revolution aswell. Thus, Einstein confuses relativity principle withMach’s principle. Not only that but Mach’s principlefails to eliminate absolute motion. Indeed, let us leavethe fluid spheres aside and examine another scenario. Letus imagine a rotating universe. The centrifugal effects arevisible again (see [1] and the references therein) and thistime the absolute rotation of the universe as a whole can-4not be explained as a rotation relative to other bodies.Thus Mach’s principle fails to eliminate absolute motion.Next, Einstein invites us to consider the principle ofequivalence - the local equivalence between gravitationalfields and accelerating reference frames. He then makesan error of thinking that by making the theory generallycovariant he had mathematically implanted the equiv-alence principle. However, this is false. The fact thata theory is written in generally covariant form does notmake all reference frames equivalent. Indeed, one canwrite all physical equations generally covariant. For in-stance in Newton’s equations one has to simply uniteall inertial forces with the physical forces into a singleexpression F µ , µ = 1 , , g µν ( µ, ν = 0 , ...,
3) with the gravity field. However atthis point Einstein makes a departure from the field the-ory of the type of Faradey-Maxwell in favor of a geo-metrical theory. But this physico-geometrical dualism of g µν does not allow the possibility to write a symmetricenergy-momentum tensor but pseudo-tensor. Indeed, apossible reference frame could remove the gravitationalfield in a small area of space-time and thus all compo-nents of the symmetric energy-momentum tensor wouldbe zero at this point. This is not a behavior of a ten-sor but of a pseudo-tensor. The lack of a symmetricenergy-momentum tensor separates gravity theory fromthe other field theories, while there is a perfectly con-sistent field approach to gravity [17] that leads to ex-actly the same field equations as Einstein’s equationsand at the same time it allows that a symmetric energy-momentum tensor can be written for the gravity field.Indeed, let us start from field theoretic notions. Firstwe start with a flat space-time (which in Lorentz-Poincare interpretation is merely a tool) with a flat met-ric η µν . We raise and lower indexes with η . Second,from very general considerations (long range of the grav-ity field, the attractiveness of the gravity, deflection oflight by gravity, etc.) we can reason that gravity fieldought to be described by spin 2 field h µν defined on flatspace-time. The requirements of positive definiteness ofthe energy density [17] and a conserved source of thefield equations lead to the Lagrangian for a free tensormassless field h µν , L h = 12 ∂ µ h νρ ∂ µ h νρ − ∂ µ h νρ ∂ ν h µρ + ∂ µ h∂ ρ h µρ − ∂ µ h∂ µ h, (48)where h = h µµ . Thus far the theory is quite similar to thevector theory of electrodynamics if one considers that the above Lagrangian has a gauge symmetry h µν → h µν + ∂ µ Λ ν + ∂ ν Λ µ , (49)for arbitrary Λ µ ( x ), where x is a four-dimensional coor-dinate.We follow Thirring [39]. Due to this gauge symmetryone can have the condition ∂ µ h µν = 12 ∂ ν h, (50)which is preserved if we restrict ourselves to gaugeswhich (cid:3) Λ µ = 0, similarly to the Lorentz gauge in elec-trodynamics. The free field equations become (cid:3) h µν = 0 . (51)The simplest manner to introduce interaction with par-ticles is to add on the right hand side the source term dueto the energy-momentum density of the particles, i.e, (cid:3) h µν = f (cid:18) T ( p ) µν − η µν T ( p ) (cid:19) . (52)Here f is a coupling constant and T ( p ) µν is the energy-momentum tensor for the particles. Eq. (52) leads tothe condition ∂ µ T ( p ) µν = 0. However this will violate thelaws of motion of matter as will be seen below. Indeed,this is due to the fact that the total energy conservationshould be ∂ µ (cid:16) T ( p ) µν + T ( h ) µν (cid:17) = 0, where T ( h ) µν is the energy-momentum of the field h µν . In order to try to correct thatinconsistency, we could augment the right-hand side ofEq. (52) with the energy-momentum tensor of the field h µν itself, which immediately introduces non-linearities.However, we shall first restrict ourselves to the lineartheory to lowest order in f .Let us examine for simplicity’s sake a single particle.Then the action for a free particle is, S fp = Z dsL fp , (53)where s is the proper time of the particle and L fp is theLagrangian for a single free particle. It is given by L fp = − mη µν dx µ ds dx ν ds . (54)Here x µ are the four-dimensional coordinates of the par-ticle and m is its mass.The interaction term on the right-hand side of Eq. (52)can be introduced into the action S fp for the free particleinto the form S int = Z dsL int , (55)where L int = f mh µν dx µ ds dx ν ds . (56)5Thus, the total Lagrangian for the particle in a spin 2field (considered as an external field) will be L = L fp + L int = − mg µν dx µ ds dx ν ds , (57)where g µν = η µν − f h µν , (58)is an effective renormalized metric. It is this renormalizedmetric that is ’seen’ by the particle as it interacts withthe tensor field h µν . Indeed, the above Lagrangian showsthat the particle moves in such a manner as to minimizethe four-dimensional interval ds = g µν dx µ dx ν , i.e. itmoves as if upon a geodesic in a curved space-time withmetric g µν . The true metric is however η µν not g µν inthe field-theoretic approach to gravity. In fact one caneasily derive from the Lagrangian (57) the following con-servation law g µν dx µ ds dx ν ds = const. , (59)where the above constant can always be chosen by con-vention equal to 1 and thus ds = g µν dx µ dx ν . In thismanner the gravity field h µν is concealed in such a man-ner that the true metric η will not be visible but insteadthe metric g will be observed. Obviously this is due tothe interaction of the particles of the bodies with thegravitational field.Now, one can introduce higher order powers of f to cor-rect the above mentioned inconsistency. If one includesthe energy-momentum tensor of the gravity field on theright-hand side of Eq. (52) another inconsistency ap-pears, namely the condition ∂ µ (cid:16) T ( p ) µν + T ( h ) µν (cid:17) = 0 whichis derived from the field equation is inconsistent with thematter equation. It was proved by Wyss [42] and inde-pendently by Ogietsku and Polubarinov [43] that one canconstruct iterative procedure by requiring consistency toall orders in the coupling constant f . They showed thatrequiring gauge invariance for the tensor potential h µν toall orders in f , the procedure converges to general rela-tivity. Then Deser [17] dropped this limitation by intro-ducing the second-order energy-momentum tensor thatcan be obtained by a variational principle.Such a bimetric approach to gravity makes possible toconsider gravity as a field and energy-momentum tensorcan be written. Not only that but the field approachunites all forces of nature under a single unified frame-work. Even more, according to Logunov [44] one is com-pelled to consider gravity as a field in flat space-time suchthat the gauge is organically built into the theory. Oth-erwise Einstein’s gravity equations will not give uniquepredictions.With this preparation in hand one can finally showhow a universal force can be detected and the reasoningof the relativists who say that such forces are in principleunobservable is implausible. Indeed, let us imagine that we have two-dimensional creatures that measure the an-gles in a triangle and they discover that their sum is not π . Is the geometry non-Euclidean or there is a universalforce that distorts their measuring rods? The creaturescan reason thus. They can first try the hypothesis thata universal force exists. Within this hypothesis they canobtain the properties of this force and compare it withthe other forces of nature available to them. If they dis-cover great many similarities with other forces and a sin-gle unifying formalism can unite all the forces and in factcan derive all the properties of this universal force, thenit is much more plausible that the universal force exists.Thus with the help of additional information from otherparts of physics (or knowledge in general) one can provethat universal force exists. And gravity is such a force.Not only that but even within the geometrical ap-proach to gravity as a curviture in space-time it is simplyuntrue that there are no special reference frames. Thegeodesics in the curved space-time play the same role asthe inertial reference frames played in special relativity.This can be seen easily, by recalling that infinitesimallysmall regions of curved space-time are flat and specialrelativity can be applied there. The geodesic motion inthis infinitesimal region is an inertial observer. The onlydifference between flat Minkowski space-time and curvedMinkowski space-time is how these infinitesimal regionsare connected into a single manifold. Therefore the prin-ciple of equivalence does not render all reference framesequivalent.And finally cosmological considerations (see [21] and[1]) when united with Einstein’s gravity equations caneasily show that there exist a special reference frame evenwithin the limited geometrical approach of curved space-time which deals with generally-covariant equations. In-deed, if the universe is homogeneous and isotropic thereis a special foliation of space-time (within Minkowskianinterpretation) such that ds = c dt − R ( t ) (cid:0) dr + r dθ + r sin θdφ (cid:1) , (60)i.e., there exists a global cosmic time t . This foliationof space-time corresponds to a reference frame that co-incides with the average motion of the cosmic matter.Thus, symmetries can disclose to us the existence of aspecial reference frame. One may object that this ismerely a special foliation of space-time and some otherfoliation is allowed. However, this reasoning applies real-istic understanding of space-time which we already crit-icized. Therefore what Einstein’s special theory of rel-ativity took with one hand (a special reference frame),Einstein’s gravity theory gave back with the other.And finally we know that our universe is indeed sta-tistically homogeneous and isotropic ([18] and [21]) withgreat accuracy, see the Introduction.6 E. A possible route to a mechanical model ofEinstein’s gravity equations
The same aether we have used to construct mechanicalmodel of Maxwell’s equations can reveal to us a possibleroute toward a mechanical model of gravity. We shalldeal with the liner theory devoid of matter. Then A = ρ ξ and the aether density ρ is constant. Let us substitute h µν = h νµ = ρc ρcv x ρcv y ρcv z ρcv x − σ xx − σ xy − σ xz ρcv y − σ yx − σ yy − σ yz ρcv x − σ zx − σ zy − σ zz . (61)Then Newton’s equations ρ∂ t v i = ∂ k σ ki and the conti-nuity equation ∂ t ρ + ∂ i ( ρv i ) = 0 can be written in a fourdimensional notation as ∂ µ h µν = 0 . (62)The above equation corresponds to the gauge condition.The equations for h µν are (cid:3) h µν = 0 . (63)If we wish to derive mechanical model of Einstein’s grav-ity equations we must postulate different stress-strain re-lations and include matter. Such a procedure does notseem a priori impossible. V. STATISTICAL-MECHANICALINTERPRETATION OF LORENTZTRANSFORMATIONS
In this section we shall derive
Lorentz transformationswithin the framework of Lorentz-Poincare interpretation.We do this by calculating the one-particle probability dis-tribution of a gas of molecules. We take into accountthat the molecules of the gas emit electromagnetic fieldand this field acts back upon the molecules themselvesand thus distorting the one-particle distribution func-tion. The result of this distortion is such that in orderfor the gas to remain in thermal equilibrium all processesin the gas are delayed by the standard clock dilation ( not time dilation) and the gas as a whole is contracted inthe direction of the general locomotion of the gas withthe standard FitzGerald-Lorentz contraction. In otherwords, the dilation of the processes and the contractionof the moving body relative to the aether are derived as true physical effects caused by self-interaction.To this end we shall make only two assumptions: Assumption 1:
There exist a special reference frame,that of the aether.
Assumption 2:
The mass of the charge is velocity-dependent, more concretely we assume Eq. (43).It is obvious that the molecules of the gas interact withthe electromagnetic field they created. This does not ex-clude the possibility of external electromagnetic field but for simplicity of derivation we shall exclude it. In ad-dition, it is possible that even the so called elementaryparticles (say muons) consist of more elementary con-stituent particles (the ’molecules’ of the muon) and theyinteract via some other field. I shall argue below thatso long this field propagates with the finite velocity oflight c , the muon will still be contracted and its life timeincreased if it moves relative to the aether. Thereforethe considerations below are not confined to macroscopicbodies.In order to derive how the electromagnetic field dis-torts a body, we shall consider the one-particle Boltz-mann’s equation for a gas of molecules interacting viaelectromagnetic field. Following [40] and [41] we reasonthus. Let f ( r , p , t ) is the probability distribution at mo-ment t of finding a single particle in position r and withmomentum p in the volume dV and momentum volume d p . After an infinitesimal time interval dt we have that f ( r , p , t ) dV d p = f ( r + v dt, p + F em dt, t + dt ) dV d p , (64)where r = r + v dt , F em = e ( E + v × B /c ) is the elec-tromagnetic force acting on a charge e (not necessar-ily the electron charge), v is the charge of the particle, p = p + F em dt , dV = dxdydz and dV = dxdydz . FromEq. (43) we have p = m v q − v c , (65)and thus v = p c p p c + m c . (66)These equations allow us to calculate the Jacobian andhence the change of the volume dV d p in phase space, dV d p = (cid:18) ∂∂ p · F em (cid:19) dV d p = dV d p , (67)because ∂∂ p · F em = 0 . (68)Here ∂/∂ p is the gradient operator with respect to mo-menta. The same for ∂/∂ r = ∇ . We shall use bothnotations, ∂/∂ r as well as ∇ . Using Eq. (67), (64) issimplified to ∂f∂t + v · ∂f∂ r = − e (cid:16) E + v c × B (cid:17) · ∂f∂ p . (69)We have assumed so far a system of positive charges only( e > both positiveand negative charges we have to consider two probabilitydistributions for both types of constituent particles. Weshall call them f , . Boltzmann’s equations for both are ∂f , ∂t + v · ∂f , ∂ r = ∓ e ( E + v c × B ) · ∂f , ∂ p . (70)7Please note that we do not include collision terms be-cause they are already taken into account on the right ofthe above equation. Indeed, any kind of collision is hereassumed to be of electromagnetic origin and is thus in-cluded in the electromagnetic force. In other words, thecharges do not interact directly but in directly throughthe electromagentic field. Therefore the above equationis the most general Boltzmann’s equation of interactingparticles.So far we have shown how the electromagnetic fieldproduced by the particles in the system distorts theprobability distributions f , . We need to completethe system with the equations which show how thecharges themselves create this field. Obviously these areMaxwell’s equations. In order to simplify them we applythe standard Lorentz gauge: ∇ · A − c ∂φ∂t = 0 . (71)Then, Maxwell’s equations are converted to inhomoge-neous wave equations, as is well known: (cid:3) φ = − ̺, (72) (cid:3) A = 1 c J , (73)where (cid:3) = ∇ − c − ∂ /∂t is the D’Alembert operator.Let us assume that there are N positive and negativecharges. Then the charge and current density are ̺ = eN Z f ( r , p , t ) d p − eN Z f ( r , p , t ) d p , (74) J = eN Z v f ( r , p , t ) d p − eN Z v f ( r , p , t ) d p (75)The full system of equations describing the self-governingsystem of charges is now complete. A. A system in thermal equilibrium moving withvelocity V relative to the aether along the x axis.New notation for the Maxwell’s wave equations
Let us begin by considering a system in thermal equi-librium at rest relative to the aether. Obviously then theprobability distributions are time- in dependent. Let ususe superscript (0) to denote the system at rest with re-spect to the aether. Thus, the equations describing thissystem in absolute rest and at the same time in thermalequilibrium are, ∇ φ (0) = eN Z f (0)1 ( r , p ) d p − eN Z f (0)2 ( r , p ) d p . (76)Similarly for A (0) . Boltzmann’s equations are v · ∂f (0)1 , ∂ r = ∓ e (cid:16) E (0) + v c × B (0) (cid:17) · ∂f (0)1 , ∂ p . (77) We shall assume that the solution of any system at restand in thermal equilibrium can be obtained, more pre-cisely we can obtain φ (0) , A (0) and f (0)1 , .Now, let us examine another system in thermal equilib-rium , which however moves with velocity V in x directionwith respect to the aether. We do not change the refer-ence frame. The frame is still the aether frame. We justconsider a moving system. The equation describing thepotential for this moving system is (cid:3) φ = − ̺ ( x − V t, y, z ) . (78)Similarly for A . The only time dependence is due tothe general locomotion of the system with respect to theaether. Let us substitute x = x − V t,y = y,z = z,t = t. (79)Please note that this is simply a notation . No physicalmeaning whatsoever is imparted to these substitutions(yet). The wave equation for the potential φ is changedto, (cid:18) − V c (cid:19) ∂ φ∂x + ∂ φ∂y + ∂ φ∂z + 2 Vc ∂ φ∂x ∂t = − ̺ ( x , y , z ) . (80)Similarly for A . On the right hand side, the charge istime-independent in the new notation. However the waveequation on the left is distorted. It is obvious though thatthe left hand side will be simplified if we substitute x = x q − V c = x − V t q − V c ,y = y = y,z = z = z,t = t = t. (81)Then Eq. (80) becomes (cid:3) φ + 2 Vc ∂ φ∂x ∂t = − ̺ x r − V c , y , z ! , (82) (cid:3) is the D’Alambert operator with respect to the no-tation (81). Eq. (82) however is not a wave equation,even though on the right hand side we have a time- in dependent source. It is not difficult however to guessa method to correct that. We substitute, x = x = x − V t q − V c ,y = y = y,z = z = z,t = r − V c t − x Vc = t − xV /c q − V c . (83)8In this manner we finally obtain, (cid:3) φ = − ̺ x r − V c , y , z ! . (84)This is truly an inhomogeneous wave equation with atime- in dependent source. Again we stress that Eqs. (83)are simply a notation. Nothing else. The same consider-ations are done for the vector potential, (cid:3) A = 1 c J x r − V c , y , z ! . (85)One might think that the job to convert the equations ofa moving system in an effective system at rest is done.However we are not ready because for these new equa-tions the sources on the right hand side do not obey thecontinuity equation. Indeed, if we rewrite the continuityequation ∂̺∂t + ∇ · J = 0 , (86)with the new notation (83), we have, ∂∂t ̺ − V J x /c q − V c + ∂∂x J x − V ̺ q − V c + ∂∂y J y + ∂∂z J z = 0 . (87)In order for the new sources to obey the continuity equa-tion we have to perform another substitution ̺ = ̺ − V J x /c q − V c , (88) J x = J x − V ̺ q − V c , (89)and J y = J y and J z = J z . We have to perform similarlinear combinations of φ and A such that the sources ofthe wave equations are ̺ and J . φ = φ + V A x q − V c ,A x = A x + V φ/c q − V c , (90)and A y = A y as well as A z = A z . Then we finally have (cid:3) φ = − ̺ x r − V c , y , z ! . (91)Similarly for A . The wave equations are finally reducedto stationary equations, whose sources obey the continu-ity equation ∂̺ ∂t + ∇ · J = 0 . (92) Since the sources in these wave equations in the new no-tation are effectively time-independent, we may discardthe time derivative ∇ φ = − ̺ x r − V c , y , z ! . (93)Similarly for A . We stress that we have simply made anew notation. Nothing else is done. No physical inter-pretation whatsoever is applied . Next, we consider Boltz-mann’s equations.
B. The new notation applied to Boltzmann’sequations for the moving system with velocity Valong the x-axis in thermal equilibrium
Boltzmann’s equations for the moving system in ther-mal equilibrium for both type of charges are, ∂f , ∂t + v · ∂f , ∂ r = ∓ e ( E + v c × B ) · ∂f , ∂ p . (94)However we have f , = f , ( x − V t, y, z, p x , p y , p z ). Theonly time dependence of f , which describes a thermalequilibrium of a moving system is due to general locomo-tion of the system. We use directly substitution (83). Inorder to rewrite Boltzmann’s equations we have to cor-rect the electromagnetic field using E and B derivedfrom the potentials φ and A . In addition, we haveto distort the axes in the phase space. The coordinates x, y, z, t are distorted according to Eqs. (83). But weneed to do the same for the moment. To this end wehave v x = dx dt = v x − V − v x Vc . (95)Similarly, one could derive v y and v z and from herethe momentum p which allows us to distort all axes inphase space using Eqs. (83). After a somewhat laboriouscalculation (the reader may also follow [40] for a four-dimensional notation) we obtain, v · ∂f , ∂ r = ∓ e (cid:16) E + v c × B (cid:17) · ∂f , ∂ p . (96)We see that Boltzmann’s equation is covariant with re-spect to the new notation Eq. (83). This equation, be-ing combined with the effective stationary field equation(93), and a similar Poisson’s equation for A helps us toreach the conclusion that the whole of the moving systemis effectively reduced to a system which is in absolute rest(i.e. rest relative to the aether). Of course, the system isin fact moving with velocity V along the x axis but withthe new notation, the system behaves mathematically asif it is in absolute rest. We have assumed that any systemof equations, describing a body in absolute rest, can besolved and the solution is denoted with a superscript (0).Therefore the solution is f , ( r , p , t ) = f (0)1 , ( r , p ) and9similarly for the fields. Reverting back to the originalnotation we have f , ( r , p , t ) = f (0)1 , x − V t q − V c , y, z, p x − V p p c + m c /c q − V c , p y , p z (97)The solution for the distributions f , is rewritten interms of some effective system which is now effectivelyin absolute rest. However, the above rest solution f (0)1 , is not any effective solution but is the solution of the partic-ular system which is in motion , if it were not in motionbut if it were in absolute rest. This is immediately seenby taking the limit V → In this manner fromEq. (97) we can compare the distribution functions f , of the system in thermal equilibrium, which moves withvelocity V with the distribution functions f (0)1 , in thermalequilibrium if the system were in absolute rest. Let us assume that the charge distributions f (0)1 , ( r , p )describe a body in absolute rest with finite length L alongthe x -axis. This means that f , → s → L where s is the distance from one end of the body to the other .Eq. (97) shows that the distribution is translated alongthe x -axis with amount V t and then contracted with afactor p − V /c . Therefore for the moving body inthermal equilibrium described by the distributions f , we have that f , → s → L p − V /c , i.e. thebody has a new dimension L p − V /c . This is thecustomary FitzGerald-Lorentz contraction. C. Dilation of all processes in a body not inthermal equilibrium
We could easily examine a body, which is not in ther-mal equilibrium and yet it is in absolute rest. In otherwords, the body has some internal motion but its centerof mass is not moving. For example, an oscillating macro-scopic dipole which consists of many molecules and yetits center of mass is in absolute rest. The internal motionfor this dipole is its oscillations. Of course, this may beany other system in absolute rest and not in equilibrium.Therefore the field equations for such a system are, (cid:3) φ (0) = − ̺ ( r , t ) . (98)Similarly for A . Here, despite the fact that the systemof charges is in absolute rest, yet ̺ is a function of time,due to the internal motion of the body. Boltzmann’sequations are ∂f (0)1 , ∂t + v · ∂f (0)1 , ∂ r = ∓ e (cid:16) E + v c × B (cid:17) · ∂f (0)1 , ∂ p . (99) And again, the distributions are still time- de pendent be-cause the system is not in thermal equilibrium due to theinternal motion of the body.If any body not in thermal equilibrium were to movewith velocity V along the x axis, the equations become (cid:3) φ = − ̺ ( x − V t, y, z, t ) , (100)and similarly for A . Note that the time-dependence of ̺ is not only due to the internal motion but also due to theexternal general locomotion of the system with velocity V along the x axis. Boltzmann’s equations are ∂f , ∂t + v · ∂f , ∂ r = ∓ e (cid:16) E + v c × B (cid:17) · ∂f , ∂ p , (101)where f , = f , ( x − V t, y, z, p , t ), i.e., f , also dependson time in two ways - internal motion and general loco-motion. We can follow exactly the same line of reasoningas we did for the system which was in thermal equilib-rium. We obtain f , ( r , p , t ) = f (0)1 , x − V t q − V c , y, z, t − xV /c q − V c , ... , (102)where ... is the momentum-dependence which is the sameas in Eq. (97). We observe that the body is againFitzGerald-Lorentz contracted. We see however that theinternal motion is delayed with the factor (cid:0) − V /c (cid:1) / ,i.e. all processes are delayed. This is immediately seen asfollows. Choose a point in space, say x = y = z = 0 andlet us examine the density of particles with momentum p in the range d p . Obviously it is, N f , (0 , , , p , t ) d p = N f (0)1 , − V t q − V c , , , t q − V c , ... d p . (103)The first term in the brackets is the time-dependencedue to the general locomotion of the system. The secondtime-dependence is due to internal motion. We observethat the internal processes in the point ( x, y, z ) = (0 , , (cid:0) − V /c (cid:1) / . One couldsay even more. Not only are the processes delayedwith this factor, but they are delayed with the amount xV /c (cid:0) − V /c (cid:1) / depending on x . This means thatif two processes were simultaneous in the absolute restposition of the body, they will no longer be if the bodymoves relative to the aether.By using the notation (83) we have been able to reachthe conclusion that a moving body is distorted and theprocesses in it delayed because the probability distribu-tion functions f , are slowed down and contracted. It iseasy to see that the cause of these distortions is the elec-tromagnetic field created by the molecules of the bodyand which acts back upon the body. In other words,0we claim that the cause of the distortions and the delaysis self-interaction through the medium of the electromag-netic field. Indeed, we have nowhere made any clock syn-chronization as in the relativistic interpretation. We onlyexamine a complicated self-interacting system within theframework of Newtonian time and space. Not only thatbut we know that the fields φ and A obey a wave equa-tion. But it is well known that the wave equation isalready Lorentz covariant. If the field equations were not Lorentz covariant, but Galilean covariant, then it is ele-mentary to prove that Boltzmann’s equation would havebeen Galilean covariant as well, and no distortion what-soever, neither to the length of the body, nor to the du-ration of the processes in the body would have occurred.Therefore the cause of the distortion is the peculiar wayin which the electromagnetic field is altered when thebody moves.
The field changes when the body moves andthis change in turn affects the body itself, which has emit-ted this particular field . This is obvious if one considerswhat would happen if the covariance of the electromag-netic field equations were Galilean instead of Lorentzian.Lastly, we should point out that in our model we do not require that the ’molecules’ (i.e. constituent particlesof the body) change when the body is in motion. Theyare simply considered as point particles.
D. Another reference frame moving with respectto the aether using physical clocks and physical rodsas measuring instruments
So far, the coordinates (83) were merely a notationnecessary to compare the one-particle distribution func-tions f , of the moving system with the distributionsin absolute rest f (0)1 , . Therefore Lorentz covariance isseen in our interpretation as a mathematical device forthis comparison. In customary derivations of Lorentzcovariance, people usually neglect Boltzmann’s equation(or any other equation which governs the motion of thecharges) and thus neglect a very vital part, namely howthe system is affected by its own electromagnetic field asit moves.Having done all that work, we are finally ready to ex-amine what is going to happen if a new reference system K ′ is used, which moves relative to the aether K withvelocity V in x direction. We assume that the centers ofthe two coordinate systems O and O ′ coincide at t = 0.In this new reference system however, we must take intoaccount that the clocks and rods which we use to mea-sure time and distance are physical devices, i.e. they aremade of molecules and therefore they are themselves dis-torted. All physical rods are contracted and all clocksare delayed. Note that it is not time which is delayed,it is that the clocks and the physical processes are be-ing delayed. Time is absolute and metaphysical, and isthus unaltered. The true time is measured by clocks inabsolute rest.Let an event M occurs in a point ( x, y, z ) in the aether at absolute moment of time t . What are the coordinatesin the new reference system K ′ ? Obviously, if there wasno FitzGerald-Lorentz contraction, then x ′ = x − V t .However, since the moving measuring rods are contractedthey measure greater distance. Indeed, the distance is defined as the number of unit rods placed in x ′ direc-tion. Contracted rods therefore measure greater distancebecause more rods are required. We therefore obtain x ′ = x − V t q − V c . (104)As we have established, only the rods in x direction arecontracted. Therefore y ′ = y and z ′ = z . Please take intoaccount the difference between ( x ′ , y ′ , z ′ ) and ( x , y , z )in Eqs. (83). The latter are mere notation which wasuseful to obtain Eq. (102) and this in turn allows us tocompare the distribution functions f (0)1 , of a system inabsolute rest and the distribution functions f , of thesystem in motion. However, the coordinates ( x ′ , y ′ , z ′ )are not notation but rather the false reading of the dis-torted measuring rods in a moving reference system.Next, we shall consider the time coordinate t ′ . To thisend let us first consider the clock synchronization in theaether frame. To this end, let us define the unit of secondin a clock positioned at O , the center of the aether co-ordinate system K . We neglect the physical dimensionsof the clocks. Let us synchronize two clocks at the samepoint O . This can be done by using the same definition ofsecond in both clocks or by merely comparing them. Thiscan be done because the clocks are at the same point O .We slowly move the second clock along the x axis withvelocity v . As the clock moves, it is slowed down. Thefirst clock measures moment of time t = x/v . Here t = 0is the moment when the second clock has started its mo-tion in the x direction. The second clock measures time t = x p − v /c /v because it is slowed down. If wetake the difference in measurements in the second clockand at the same time we take the limit v →
0, we obtainlim v → ( t − t ) = x lim v → − q − v c v = 0 . (105)This means that so long the second clock is being movedsufficiently slowly, we can consider it synchronized withthe clock at O . In other words, we can consider thesetwo clocks as measuring the absolute time. We can evenrepeat the procedure by moving the second clock suf-ficiently slowly along y and z directions and consider itagain synchronized with the clock at O . The same proce-dure can be now applied over and over again and we canfill the whole of space with such synchronized clocks. Letus now consider a moving frame of reference K ′ with ve-locity V along x direction. Before it starts its motion wefill this coordinate system of synchronized clocks whichmeasure the absolute time. Then we move them all withthe velocity V . We can consider all these clocks as asingle physical system made of molecules. Therefore, for1this system of molecules we can apply Eq. (102). Thus, t ′ = t − V x/c q − V c . (106)First, we see that all clocks are slowed down with thesame factor q − V c . Second, we observe that clocksat different positions x are slowed down with differentamount xV /c p − V /c with respect to the absolutetime. This means that if two events are simultaneousin K they are not so in K ′ . In our interpretation weexplain that these clocks are false and they measure thetime incorrectly due to the slowing down of the clocks.Again, note the difference between t in Eqs. (83) and t ′ .Here t is a mere notation which was helpful to establishEq. (102) while t ′ is the measure of the distorted clocksin a moving reference frame K ′ .From these Lorentz transformations for x ′ , y ′ , z ′ and t ′ we can derive all familiar results. That all inertial ref-erence frames appear to be indistinguishable and equiva-lent, that the speed of light c appears to be invariant in allinertial reference frames. However in Lorentz-Poincareinterpretation, this appearance is just that - an appear-ance which is however false and is due to the distortion ofthe measuring instruments of the moving reference sys-tem.Further, we could ask why is the Lorentz covariance soubiquitous. And why all the forces of nature obey it, notjust the electromagnetic phenomena? The answer seemsto be this. Firt, we point out that the Lorentz covarianceis already built into the wave equation for the fields φ and A . Any wave equation is Lorentz covariant. This in turnled to the distortion of the physical body explained bythe Boltzmann’s equations. If the fields obeyed equationswhich were Galilean covariant, then the body would not have been distorted when it moves with velocity V (inother words the Boltzmann’s equations would have beenGalilean covariant as well). It all boils down then to theLorentz-covariance of the wave equation. However, theLorentz covariance of the wave equation is due to the factthat the fields propagate with the finite speed of light c not with infinite speed. If the fields were to propagatewith some other finite speed c ′ say, then Lorentz trans-formations would have been changed accordingly and wewould have had to replace c with c ′ everywhere. In otherwords the very presence of c in the Lorentz transforma-tions is an artifact of the field equations. Second, it seemsthat the reason why all forces are Lorentz covariant isthat all fields in nature (that we know so far) propa-gate with the same speed of light c . If the fields were to propagate with different speeds, we would have differentLorentz transformations for each field. Thus, the ubiqui-tousness of Lorentz covariance is traced back to the factthat all fields in nature we know so far propagate with the same speed c . But why should this be so? If one acceptsthe aether point of view, the answer seems to be simplythis - the aether is the seat of all forces of nature , they arefields in the same aether. We do not need to postulatedifferent aethers for different forces. Otherwise it wouldbe very difficult to explain as to why the fields propa-gate with exactly the same speed. This shows us thata unification of all forces of nature ought to be possibleand a single aether, distorted in different fashions, shouldlead to the different forces in nature. From the ubiqui-tousness of the Lorentz covariance we can conclude thatall attempts to unify all forces of nature are quire rea-sonable and even expected. Such a conclusion cannot bemade easily in the standard relativistic or Minkowskianinterpretation.In addition, we can explain why the muon has a greaterlife time when it moves with greater speed. The explana-tion from the point of view of the aether is that the muonhas some internal structure. There are internal forces in-side the muon, which propagate with finite speed anddistort the muon and increase its life time.Lastly, even if quantum mechanical considerationswere to be applied and even if some collision terms in-cluded in the Boltzmann’s equations (such terms are notnecessary because the molecules are assumed to interactthrough the field not directly), so long as the Lorentz co-variance is applicable then the body would be distorted.Therefore the results are quite general. VI. CONCLUSION
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