review of the application of newton's third law in physics

Review of the application of Newton’s third law in physics

Patrick Cornille

12 Rue M. Ravel, 94440 Santeny, France

Received 17 February 1998; in final form 9 June 1998

Abstract

We review the application of Newton’s third law in all branches of physics, namely: special relativity, electromagnetism,quantum mechanics, circuit and antenna theory. Until now, there is no experimental evidence showing that Newton’s third lawhas ever been violated in classical physics. However, in both classical physics and in special relativity theory this law is violatedfor different reasons. The violation of this law implies consequences that can be tested experimentally, namely a chargedconductor at rest in the Earth reference frame can set itself in motion and accelerate its center of mass or rotate without externalhelp. We review several experiments with conductors charged with a high voltage which show these effects.q 1999 ElsevierScience Ltd. All rights reserved.

Keywords:Newton’s third law; Conservation law of energy; Special relativity theory; Superposition principle

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1622. Newton’s third law in classical mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

2.1. Case of two particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1622.2. Fluid approach of Newton’s equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1642.3. Case ofN particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3. Newton’s third law and the principle of relativity in classical mechanics. . . . . . . . . . . . . . . . . . . . . . 1664. Newton’s third law and the principle of covariance in classical mechanics. . . . . . . . . . . . . . . . . . . . 1675. Covariance and relativity principles in relativistic mechanics .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686. Newton’s third law in relativistic collision. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707. Newton’s third law and the twin paradox. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1718. Newton’s third law in quantum mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1729. Newton’s third law in electromagnetism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

9.1. The Lorentz force law and the stimulated force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1739.2. The Weber force law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1759.3. Newton’s third law between matter and radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

10. Newton’s third law and the superposition principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17710.1. Light interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17710.2. Electrostatic interference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17810.3. Carson reciprocity theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17910.4. Newton’s third law and the Aharonov–Bohm effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18210.5. Linear circuit theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18310.6. Antenna radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18510.7. Radiation reaction and conservation of energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

11. Review of several experiments which show the Earth’s motion through the ether. . . . . . . . . . . . . . . 189

Progress in Energy and Combustion Science 25 (1999) 161–210PERGAMON

0360-1285/99/$ – see front matterq 1999 Elsevier Science Ltd. All rights reserved.PII: S0360-1285(98)00019-7

11.1. Is special relativity theory a relativity theory? .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18911.2. Doppler and aberration effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19111.3. Sagnac effect, Allan’s experiment, anisotropy in the blackbody radiation. . . . . . . . . . . . . . . . . 191

12. Review of experiments on the motion of conductors fed by direct high current or voltage. . . . . . . . . 19212.1. Experiments by Graneau, Phipps and Saumont. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19212.2. Experiments by Faraday, Ducretet, Page´s, Brown, Saxl, Allais and Graham. . . . . . . . . . . . . . . 19512.3. Calculation of the stimulated force for a charged capacitor. . . . . . . . . . . . . . . . . . . . . . . . . . . 19812.4. Cornille’s pendulum experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19912.5. The Deyo and Rambaut experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20212.6. Review of the Trouton–Noble experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

13. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

1. Introduction

This article reviews the application of Newton’s third lawin all branches of physics, namely: special relativity,electromagnetism, quantum mechanics and circuit andantenna theory. Quite often this law is brushed aside as aminor law which only applies to classical mechanics. This isnot true, on the contrary, this law is the most important lawin physics. It is fundamental for the understanding ofphysics.

For example, it is well-known that there exist two forcelaws for describing electromagnetic interactions: the better-known one is the Lorentz force law describing interactionsbetween free charges in a vacuum and the older one is theAmpere force law describing interactions between currentelements confined in a metal. There is now both theoreticaland experimental evidence [1–27] that the Ampe`re forcedoes exist in the case of charges moving in conductors. Incontrast, the Lorentz force law seems to apply very well forcharges moving in the vacuum provided the magnetic fieldis generated by an external closed circuit. Therefore, wethink that both laws do exist in nature.

These two force laws are different since the Ampe`re forcefollows Newton’s third law, whereas the Lorentz force doesnot. It is well-known that Newton’s third law can be used toclassify systems as closed or opened, depending on whethera force law follows or not Newton’s third law. But the twolaws are not equivalent even when they are used for closedsystems [28]. As demonstrated in Ref. [29], the open versusclosed classification implies the existence of absolute andrelative accelerations and velocities. Violation of this lawhas consequences which can be tested experimentally inorder to prove the existence of the ether.

It is often stated in the literature that the equality of actionand reaction has no place in relativistic mechanics. Forexample, French [30] in his book states:

The equality of action and reaction has almost noplace in relativistic mechanics. It must essentiallybe a statement about the forces acting on two bodies,

as a result of their mutual interaction at a giveninstant. And, because of the relativity of simultaneity,this phrase has no meaning.

Newton’s third law is also rejected on the ground that itimplies action at a distance when we describe the mutualinteraction of two charges. These explanations are invalidbecause each charge, located in thesamereference frame,sends at thesameretarded timet0 a signal which will arriveat the position occupied by the other charge at timet0 at thesametime t � t0 1 R/c whereR is the distance between thetwo charges at thesametime t0 as shown in Fig. 1. Then,since the charges are moving, the signal will arrive at eachparticle sooner or later thant.

For identical particles, the situation is totally symmetric ifthe force law is symmetrical. The simultaneity and retarda-tion effects have nothing to do with the fact that the mutualinteraction does not follow the law of action and reaction.The failure of such an important law only resides in theexpression of the force law itself. In fact, it has beenshown by Moon and Spencer [18–21] and Wesley [31,32] that the Ampe`re force can be formulated with retarda-tion effect and Newton’s third law is nevertheless exactlyverified.

The failure of Newton’s third law for a force law of suchgreat physical importance raises a serious problem. In thisarticle, we will examine under what conditions the third lawholds and what is responsible for its failure.

2. Newton’s third law in classical mechanics

2.1. Case of two particles

It is fundamental to recall some basic definitions in clas-sical mechanics [33–35]. Newton’s second law of motionstates that the motion of two particles in a given referenceframe is described by the differential equations

dP1

dt� F12 1 F11

dP2

dt� F21 1 F22 �1�

P. Cornille / Progress in Energy and Combustion Science 25 (1999) 161–210162

with the following definitionsP1 � m1U1 andP2 � m2U2.We must distinguish between the internal forcesF12 andF21

and the external forcesF11 andF22 acting on the particlesdue to sources outside the system. We can speak of mutualinteraction between two particles only if the internal forcesfollow Newton’s third law, namelyF12� 2 F21. Therefore,an external force is by definition a force that does not followNewton’s third law. When the external forces are zero, wesay that the system is closed or isolated.

The center of mass of the system is a fictitious pointrwhere the entire massm� m1 1 m2 of the system can bethought to be concentrated. It is defined by

mr � m1r1 1 m2r2 �2�The motion of this point is only determined by the effect

of external forces since we have

ddt

mU � dP1

dt1

dP2

dt� F11 1 F22 � Fe �3�

We can now study the motion of a second fictitious parti-cle called the relative particle with a reduced massM �m1m2/(m1 1 m2). This single particle is located at theplace occupied by either the first or the second particledepending on our choice of the rest position as shown inFig. 2. The distanceR is thereforeR12� r1 2 r2 if particle 2is located at the origin of a reference frame orR21� r2 2 r1

if particle 1 is now the origin of our reference frame. Foreach choice, we have an equation of motion:

ddt

MV12 � F12 11m�m2F11 2 m1F22�

ddt

MV21 � F21 21m�m2F11 2 m1F22�

�4�

where the relative velocityV � dR/dt between the tworeference frames is reciprocal since we haveV12 � 2V21.It follows that the reciprocityV12 � 2V21 of the rest

reference frame is linked to the existence of Newton’sthird law as shown in Fig. 2 for the three possibilities.The reciprocity concept and Newton’s third law are twofaces of the same coin. Therefore, we cannot use thereciprocity of the reference frames in special relativityand at the same time state that Newton’s third law doesnot apply in special relativity. We will show hereafter thedifference concerning the reciprocity concept betweenclassical and relativistic mechanics. The equations in Eq.(4) imply both the covariance and the invariance ofNewton’s second law under a change of reference frameif the reference frames are reciprocal. This change of refer-ence frame has nothing to do with a Galilean transforma-tion which will be discussed hereafter. Moreover, we notethat the reference frame at rest is not necessarily an inertialframe.

By definition, the equation of conservation of energymust be satisfied:

ddt� 1

2 m1U21�1

ddt� 1

2 m2U22� � d

dt� 1

2 mU2�

1ddt� 1

2 MV212�

�5�

When the external forces are zeroF11 � F22 � 0, thesystem is closed or isolated, in that case, we get

ddt

mU � dP1

dt1

dP2

dt� F11 1 F22 � 0 �6�

It follows that the velocityU � dR/dt and the kineticenergyEK � mU2/2 of the center of mass are constant.Thus, Newton’s third law can be interpreted as a law ofmomentum exchange. Hence a failure of the third lawwould be a failure of momentum conservation. The law ofmomentum conservation is regarded by physicists as morefundamental than Newton’s law because it holds in quantummechanics as well as in classical mechanics with no knownexception. Any apparent violation of momentum conserva-tion has led to the discovery of new physical objects, ofwhich the elementary particle the neutrino is a spectacularexample. The above statement is not totally correct, sincethe physicists completely ignore the splitting between inter-nal forces and external forces as discussed in this article. Wemust also point out that there is a Newton’s third law forrotation as shown in Fig. 3 with a splitting between orbitaland spin rotations.

If the external forces are zero and the internal forceF12 isderivable from a potential functionEP(R), the equation ofmotion for the reduced mass becomes:

ddt

MV � 27REP�R� �7�

One can multiply the two sides of the above equation byVto obtain

ddt� 1

2 MV2 1 EP� � 0 �8�

P. Cornille / Progress in Energy and Combustion Science 25 (1999) 161–210 163

Fig. 2. Three possible reference frames to describe the mutualinteraction between two identical particles.

Fig. 1. Electromagnetic interaction between two identical charges.

Therefore, we have conservation of mechanical energyonly in the case where the internal forces are central andsatisfy Newton’s third law for translation. As an example,let us consider the case of a simple non-relativistic harmonicoscillator of massm1 and spring constantk0 fixed to a wall ofmassm2 . m1. The equation of motion for the displacementof the mass is

Md2Rdt2� 27REP �9�

where the internal force derives from a potentialEP�R�t�� k0R2

=2. Since k0 is constant, the potentialdoes not depend explicitly on time; therefore, the systemis closed and the mechanical or total energyET is alsoconstant:

ET � 12 MV2 1 1

2 k0R2 � Ct �10�

with the approximationM < m1.

2.2. Fluid approach of Newton’s equations

Newton’s laws of motion for particles can be recoveredfrom a fluid description by using the following continuityequations:

2rmi

2t1 7·�rmiUi� � 0

2

2t�rmiUi�1 7·�rmiUiUi� � f ii 1 f ij

2

2t� 1

2 rmiU2i �1 7·� 1

2 rmiU2i Ui� � �f ii 1 f ij �·Ui

�11�

The quantityUiUi is a dyadic. The density of the forcefield f is partitioned in two force fields: the proper or self-force field f ii and the mutual force fieldf ij. For example, iff ij � ri�Ej 1 Ui ∧ Bj =c� is the Lorentz force density, thenthe partition in two forces is justified in a Pinch-like systemwhere confinement in the simple cylindrical Pinch can beobtained with a mutual magnetic force arising from an exter-nal current or by the proper magnetic force which has itsorigin in the magnetic field created by the plasma currentitself.

We can now integrate the above equations on volumes

which move with the fluid velocity; we get

ddt

ZVi �t�

rmi dV � 0

ddt

ZVi �t�

rmiUi dV �Z

Vi �t��f ii 1 f ij � dV

ddt

ZVi �t�

12 rmiU

2i dV �

ZVi �t��f ii 1 f ij �·Ui dV

�12�

To these equations, one must add the equation of theconservation of charge:

ddt

ZVi �t�

ri dV � 0 �13�

If we substitute in the preceding equations the Diracdistribution densities

rmi � mid�r 2 r i�t�� ri � qid�r 2 r i�t�� 14�then we obtain

mi � Ct qi � Ctddt�m1U1� � F11 1 F12

ddt�m2U2� � F22 1 F21

ddt� 1

2 m1U21� � �F11 1 F12�·U1

ddt� 1

2 m2U22� � �F22 1 F21�·U2

�15�

where the Lorentz forces become

F ij � qi{ Ej�r i�t�; t�11c

Ui�r i�t�; t� ∧ Bj�r i�t�; t�}

F ij ·Ui � qiEj�r i�t�; t�·Ui�r i�t�; t��16�

2.3. Case ofN particles

We can generalize the above discussion to anN-particlesystem. However, the splitting of forces as usually done inthe literature is not practical since we getN(N 2 1)/2 . Nequations to solve forN variables. Therefore, we presentanother method, not well-known, where the number of equa-tions is exactly the same as the number of particles. This


Fig. 4. The Jacobi coordinatesR1, R2, R3 for four particles.

Fig. 3. Newton’s third law for translation and rotation.

method uses Jacobi coordinates [36] (p. 169) which are ageneralization of the relative and center of mass coordinatesdefined above for two particles. The particles 1 and 2 aretreated in the usual way, that is to say the difference in thecoordinates of the two particles gives the first Jacobian coor-dinate R1 � r1 2 r2. The second Jacobian coordinate isdefined as the relative vectorR2 between the center ofmass of the first two particles and the third one, as shownin Fig. 4; therefore, by definition we have

Rj � 1m0j

Xj

k�1

mkrk 2 r j11 RN � 1m0

XNk�1

mkrk �17�

whereRN � rG is the center of mass vector of the wholesystem. The total proper mass of the firstj particlesm0j hasfor value

m0j �Xj

k�1

mk ) m0N � m0 �18�

From the preceding coordinate definitions, we obtain thevelocity definitions:

Uk � drk

dtWj �

dRj

dtWN � dRN

dt� drG

dt�19�

It follows that

W1 � U1 2 U2 Wj � Ugj 2 Uj11 WN � UgN � UG �20�where we have written

Ugj � 1m0j

Xj

k�1

mkUk �21�

One can demonstrate the conservation of the kineticenergy in the coordinate transformation:

12

XNj�1

MjW2j � 1

2

XNj�1

mjU2j �22�

with the definitions

M1 � m1m2

m1 1 m2Mj �

m0j

m0j11mj11 MN � m0N

�23�To obtain the equation of motion for the Jacobian coor-

dinateRj

ddt

MjWj � Fgj �24�

we multiply the velocityWj defined below byMj

Wj � 1m0j

Xj

k�1

mkUk 2 Uj11 �25�

and derive with respect to time the resulting equation to getthe expression of the forceFgj:

Fgj �mj11

m0j11

Xj

k�1

Fk 2m0j

m0j11F j11 �26�

knowing that

ddt

mjUj � F j �27�

For j � N and j � 1, we get the definitions

ddt

m0UG �XNj�1

F j � Fe

ddt

M1W1 � 1m1 1 m2

�m2F1 2 m1F2��28�

Therefore, the Jacobi coordinates can be used to partitionanN-particle system into parts relating to the center of massmotion and the different relative motions governed by theinternal forcesFgj for j , N which follow Newton’s thirdlaw. The forceFe is the sum of the external forces as givenin the above equation.

From the preceding equation, we can recover the follow-ing equation for two particles:

ddt

M1V12 � F12 11m�m2F11 2 m1F22� �29�

if we write m� m1 1 m2 andW1 � V12, knowing that

F1 � F11 1 F12 F2 � F22 1 F21 F12 � 2F21 �30�The above discussion may seem trivial to some physi-

cists, but this article will show it is fundamental. The split-ting between internal and external forces is independent ofthe origin of the force and, therefore, this partition mustapply in all branches of physics: classical physics, plasmaphysics, special relativity, electromagnetism and quantummechanics. Therefore, special relativity and quantummechanics are both incomplete theories, since they implythe existence of internal forces associated with the reci-procity concept and the conservation of energy and ignorethe existence of external forces. We will show hereafter thereason why the Lorentz force cannot be considered as aninternal force.

The existence of external forces which do not satisfyNewton’s third law deserves special attention since onemust recognize from the above calculation that there is noenergy conservation principle for that kind of force. Most ofour technology (motors and generators) does comply withthe energy conservation principle because of Newton’s thirdlaw. It is the reason why the efficiency of motors and gener-ators can never be higher than 100% because they work asclosed systems. However, the existence of external forcesdoes imply the existence of opened systems where theenergy is provided by other particles located outside thesystem or by the medium. Therefore, classical mechanicsdoes not forbid the existence of the so-called free-energydevices or over-unity devices provided they use forces thatdo not satisfy Newton’s third law. The reader interested bythis subject can consult the numerous web sites on free-energy devices. However, we can debunk the whole subjectof over-unity devices by pointing out the existence of


opened systems, a fact which is not well-known in the litera-ture. In the case of an opened system, the efficiency can behigher than 100% because the work of the external force isnot taken into account. The only question to be answered ishow do we generate an external force? Since the Lorentzforce does not follow Newton’s third law, this force can beused for building the so-called free-energy devices. Somephysicists may disagree with this point of view, arguing thatany system can be closed by taking into account othermaterial particles in the Universe. But this is not so, sinceone can always define the center of mass of all particles inthe Universe; in that case, the energy related to the motion ofthis center of mass cannot be taken from the particles butcomes from the ether. However, as demonstrated above withthe Jacobi coordinates, the partition between internal andexternal forces need not be applied to all particles in theUniverse, and can be a local principle.

3. Newton’s third law and the principle of relativity inclassical mechanics

Let us recall the three famous Newton’s laws in the orderof importance quoted in the literature [37]:

1. every body continues its state of rest, or rectilinearuniform motion, unless it is compelled to change thatstate by forces impressed upon it;

2. the change of motion is proportional to the motive forceimpressed and is made in the direction of the right line inwhich the force is impressed;

3. to every action, there is always opposed an equalreaction or the mutual actions of two bodies upon eachother are always equal and directed to contrary parts.

Newton’s laws, 300 years after their publication, are stillfundamental to physics. The form in which Newtonpublished them has strongly influenced the subsequentdevelopment of physics. Newburgh [38] stated thatNewton’s three laws are really two, since the first law isbeing included in the second law for the special case ofzero momentum change. We disagree with this statementfor reasons that will be examined later in this article.However, later in his article, Newburgh makes some rel-evant comments which contradict his viewpoint when hesays:

It is worth noting the rather obvious although rarelystated fact that the first two laws differ markedly fromthe third…. Newton’s first two laws are a one-bodylaw…. Contrast these two laws with the third…. Thislaw is a two-body law involving two closely relatedforces that act on different bodies. The first two lawsdiffer also in that they discuss only the net force.There is no restriction on the number of forcescomprising it.

To understand the principles of relativity and covariance

in classical mechanics, we must show that the threeNewton’s laws must be deduced from one another in theinverse order quoted above [33–35]. Therefore, Newton’sthird law must be the first law since it implies the existenceof two equations of motion as discussed in the precedingparagraph, namely:

Md2Rdt2� F i 1 a�Fe� m

d2rdt2� Fe �31�

The first equation defines the motion of a particlesubmitted to an internal forceF i resulting from the mutualinteraction with a second particle and to external forcesa (Fe) produced by other particles, the particle itself or theether.

The second equation describes the motion of the center ofmass. When the external forces are zeroa (Fe) � 0, werecover Newton’s first law which only applies to the centerof mass of the two particles. As shown hereafter, the recti-linear uniform motion of the center of mass is at the heart ofthe misunderstanding concerning the relativity and co-variance principles. It is the partition of forces obtainedfrom Newton’s third law which is the key for the under-standing of what is wrong with these two principles. Notethat the above analysis can be easily generalized to a systemof N particles by using Jacobi coordinates.

The relativity principle can be best analyzed when themotion of an object is observed from different referenceframes. A well-known example is the case of a stonedropped in a moving train. We know from the precedingparagraph that the relative motion is described by theequation

Md2R12

dt2� F12 1

1m�m2F11 2 m1F22� �32�

whereR12 � r1 2 r2 is the distance between the stone ofmassm1 and the train of massm2 knowing that the reducedmass and the total mass are respectivelyM � m1m2/(m1 1m2) andm� m1 1 m2.

Since we havem1 p m2, the preceding equation becomes

m1d2R12

dt2< F12 1 F11 �33�

Because the mass of the Earth is large in comparison withthe mass of the train, we must take into account the attrac-tion of the Earth. Eq. (32) does not change form ifm2

includes the mass of the Earth. In that case, the internalforce F12 is the gravitational force andF11 is the externalforce applied to the stone by the moving train. The equationof motion for the center of mass of the train and the stone hasthe expression

md2rdt2� F11 1 F22 � Fe) m2

d2rdt2

< F22 �34�

The coordinates of the stone and the train in the Earth


reference frame are given by the relations

r1 � r 1m2

mR12 < r 1 R12 r2 � r 2

m1

mR12 < r �35�

At the initial time t � 0, we apply an interaction force inthey direction to drop the stone from the luggage rack. Wealso assume that at that time the external forces are zero andthe train has reached the uniform velocityU � Ct in the ydirection in the Earth reference frame as shown in Fig. 5.

For an observer located in the reference frame of the train,we have

m1d2Z

dt2� F12 � 2m1g) Z � 2gt2=2

m1d2Y

dt2� F11 � 0) Y � Y1

�36�

An Earth observer who follows simultaneously themotion of the train and the stone will see the same relativemotion in both the Earth and train reference frames, namelya vertical straight line for the free falling stone. However, ifthe Earth observer only follows the absolute motion of thestone relative to the Earth reference frame, he sees a par-abola given by the relations

z� Z y� Ut 1 Y �37�which is a Galilean coordinate transformation that resultsfrom the second equation of motion in Eq. (31) in theabsence of external forces.

The reciprocity of reference frames applies only to inter-nal forces which satisfy Newton’s third law. Therefore, achange of reference frame cannot cancel an internal forcesuch as the gravitational force. By contrast, the change ofreference frame for external forces is not reciprocal, sincethis kind of force does not satisfy Newton’s third law. There-fore, the value of the external force will change with thechoice of the reference frame, for example this force can bemade zero in the train reference frame. Thus, it is notsurprising to get two different paths for the stone motion

depending on the choice of the reference frame. Conse-quently, the relativity principle for inertial reference framesin relative motion is defeated by the existence of externalforces. Brillouin [39] (p. 45) reached the same conclusionwhen he says

Let us conclude: the usual statement of the relativityprinciple requires that frames of reference beextremely heavy.

Einstein’s relativity principle refers to laws of physics butinitial conditions have to be taken into account. These initialconditions are ‘‘fact-like’’ rather than ‘‘law-like’’, they arenot invariant since they depend on the external forcesapplied, as shown in the above example.

4. Newton’s third law and the principle of covariance inclassical mechanics

The covariance principle in classical mechanics impliesthe invariance of both the acceleration and the force under achange of Galilean reference frame. Newton’s second lawapplied to the particle 1 written in two different referenceframes as shown in Fig. 6 gives the following two equationsof motion:

m1d2R12

dt2� F�R12; t� m1

d2r1

dt2� F�r1; t� �38�

with the conditionF(R12, t) � F(r1, t).On the contrary, the relationr1� r 1 m2R12/m in Eq. (35)

implies the formula

m1d2r1

dt2� m1

d2rdt2

1 Md2R12

dt2�39�

If we havem1 < M for m1 , m2, in that case the precedingequation gives the relationr1 < r 1 R12. Eq. (34) can berewritten as follows:

m1

m2m

d2rdt2� m1

m2Fe) m1

d2rdt2

<m1

m2Fe! 0 �40�


Fig. 6. Galilean change of reference frame.

Fig. 5. Motion of a falling stone relative to a moving train or to theEarth.

The conditionm1Fe/m2! 0 is verified if the particle 2 hasan infinite mass, which is a necessary condition for thereference frame to be an inertial frame, and if the externalforce is not too great. Brillouin has clearly discussed in hisbook [39] of the necessity for a reference frame to have aninfinite inertial mass.

Therefore, Eq. (39) becomes

m1d2r1

dt2< m1

d2R12

dt2�41�

Only in that case, do we recover the covariance principleand the equality of forcesF(R12, t)� F(r1, t) under a changeof reference frame. From the above example, we see thateven in classical mechanics we can argue about the co-variance principle because there are two equations ofmotion. Einstein did not understand that the relative motionequation (first equation of Eq. (38)) is the only equationwhich is covariant and invariant under a change of referenceframe in the absence of any external force; therefore, thisequation does not depend on the existence of the ether. Incontrast, the center of mass equation does depend on thechoice of a reference frame. This law of motion impliesthe existence of the ether which can be chosen as thepreferred frame of rest, particularly if we take into accountall particles of the Universe.

Moreover, we can also contest the covariance principle ofEq. (38) from a point of view based on the energy equations:

ddt� 1

2 m1V212� � V12·F�R12; t� d

dt� 1

2 m1U21� � U1·F�r1; t�

�42�Thus, the particle one submitted to a forceF which is the

same in two reference frames in relative motion has a kineticenergy that is different in each reference frame since thepower is different in the two reference frames. Therefore,it suffices to change our reference frame to create as muchfree energy as we want because the velocity of the movingreference frameU � U1 2 V12 can be as large as we wish.

In not differentiating between internal and externalforces, the covariance and relativity principles blendtogether in Galilean mechanics as a principle of inertia.This principle states that Newton’s laws of motion andenergy are unaltered by the Galilean transformation

r1 � r 1 R12 � Ut 1 R12 �43�between two inertial frames in relative motion which led tothe two sets of equations in Eqs. (38) and (42). From thepreceding discussion, one can understand that the covar-iance and relativity principles are radically different in Gali-lean and Newtonian mechanics. This point is sofundamental for the understanding of physics that we mustsummarize the similarities and differences between the twomechanics. The main difference concerns the reciprocityconcept between reference frames which applies to thetwo reference frames attached to particles 1 and 2 in New-tonian mechanics, whereas in Galilean mechanics the

reciprocal reference frames are attached to particles 2 andthe origin 0. However, there is a similarity between therectilinear uniform motion of particle 2 in Galileanmechanics and the same motion of the center of masswhich is almost located at the position occupied by thesecond particle if this particle is massive and provided thatthe external force is small or zero.

5. Covariance and relativity principles in relativisticmechanics

We shall now show that the above discussion does notdepend upon the existence of the relativistic gamma factor.It follows that the relativity and covariance principles inspecial relativity can be refuted for the same reasonsdiscussed above because we must recover classicalmechanics forU/c or V/c ! 0. The invariance in form orcovariance of the equations of electrodynamics underLorentz transformations was shown by Lorentz and Poin-carebefore the formulation of the special theory of relativ-ity. In the relativistic case, the covariance principleconcerning the laws of motion is expressed through therelations

dP0

dt0� F0

dP1

dt1� F1

dE0

dt0� U0·F0

dE1

dt1� U1·F1

�44�The covariance of the above equations implies that the

quantitiesP0, P1, E0, E1 that enter these equations transformunder the following Lorentz transformations:

E1 � g�E0 1 1U·P0�

P1 � P0 1 1g

c2 E0 11

U2 �g 2 1�U·P0

� �U

U1·F1 � 1D�U0 1 1U�·F0

F1 � g21

DF0 1 1

g

c2 U0·F0 11

U2 �g 2 1�U·F0

� �U

� ��45�

where 1 � ^1 is a coefficient. To compare the laws ofmotion of the particle 1 of Fig. 6 viewed in two inertialframes in uniform relative translation in both classical andrelativistic mechanics, we use the following definitions:

D � 1 1 1U·U0=c2

E0 � m1g0c2 P0 � m1g0U0

E1 � m1g1c2 P1 � m1g1U1

gi � �1 2 U2i �=c2�21=2 g � �1 2 U2

=c2�21=2

�46�

From the two equations in Eq. (35) one deduces the


composition law of velocities for two particles:

U1 � UG 1m2

mV12 < UG 1 V12

U2� UG 2m1

mV12 < UG

�47�

We see at once, that the Galilean law of addition of velo-cities in classical mechanics is satisfied only ifm1 , m2. Wecan always assume the equalityU0 � V12; therefore, we getU1 < U 1 U0 for U2 < UG < U if the reference frame whereparticle two is at rest has an infinite mass. Now, we mustrecover classical mechanics forU/c or V/c ! 0 whichimplies to substitute one instead of the gamma factor inall the preceding relations. The transformation laws of theforce and the power are now given by the relations

F1 � 1D

F0 11

c2 �U0·F0�U� �

U1·F1 � 1D�U0 1 1U�·F0

�48�

Let us now suppose that the two particles are chargedparticles, then the Coulomb law of interaction in the restframe of particle two is given by the relation

F0 � F12 � q1q2R12

R3 �49�

In classical mechanics, the Coulomb law satisfiedNewton’s third law and, therefore, the magnitude of theforce has the same value in any reference frame. Contraryto what is said in the literature, this invariance is not aconsequence of a Galilean transformation since it is verifiedwhen the two particles have the same mass. In contrast, theforce F1 in relativistic mechanics does not have the samemagnitude as the forceF0 in a change of reference frame.Worst of all, the conservation law of energy is violated, asshown by the second relation in Eq. (48). In classicalmechanics, Newton’s third law implies the conservationlaw of energy in any reference frame. Any violation ofthis law is due to the work of external forces and is attributedto the motion of the center of mass of the system. Thecovariance principle is also criticized as a principle whichhas no physical purpose as stated by Moussa and Ponsonnet[40] (p. 59):

We recall that classical forces are known in the framelinked to the Earth. Therefore if we keep that frame,the above relations have no practical use.

What are the reasons which have led to the covarianceprinciple?. The best explanation has been given by Panofskyand Phillips in their book [41] (p. 261) when they say:

If an equation has a form which is invariant to achange in inertial frame, then an experiment basedon this equation obviously could not give a resultdepending on the particular frame of reference.

They also state:

by no experiment of any kind should it be possible todetect a preferred inertial frame.

From an experimental point of view, we will contest thelast claim later in this article. But, the covariance principle isnot required by physics as already stated by Phipps [42] andCornille [29] in both special and general relativity. In areview of the foundations of general relativity, Norton[43] summarizes Kretschmannn’s objections to the covar-iance principle by stating that general covariance is physi-cally vacuous. A good example is given in hydrodynamicswhere the equations take different forms in different changesof coordinates.

For Eulerian coordinatesr, t defined in the laboratoryframe, we get

2r

2t1 7·�rU� � 0

2rU2t

1 7·�rUU 1 PyI � � 0

2E2t

1 7·��E 1 P�U� � 0

�50�

where the quantitiesr andre are respectively the mass andinternal energy density,E� r (e 1 U2/2) is the total energydensity;P and U are the pressure and the velocity of thefluid.

For Lagrangian coordinatesr0, t0, we get

2r

2t01

r

J2J2t0� 0

r2U2t0

1 �70r�21·70P� 0

r2

2t0�e1 1

2 U2�1 ��70r�21·70�·�PU� � 0

�51�

whereJ is the Jacobian of the transformation between thelaboratory frame and the Lagrangian frame. The matrix(70r)

21 is the inverse matrix of70r.Finally, the fluid equations given on a non-rigid frame

moving with the velocity fieldUe[r(t), t] with respect tothe laboratory frame have the expression

d�rJ�dt

1 J7·�r�U 2 Ue�� 0

d�rJU�dt

1 J7·�r�UU 2 UeU�1 PyI � � 0

d�EJ�dt

1 J7·�E�U 2 Ue�1 PU� � 0

�52�

whereJ is the Jacobian of the transformation between thelaboratory frame and the moving frame. This Jacobian satis-fies the identity

1J

dJdt� 7·Ue �53�


For a rigid moving frame, we verify the condition7·Ue�0 which implies to takeJ� 1 in the preceding equations. Foridentical initial and boundary conditions, the above threesets of equations will give the same numerical values. Wenote that the preceding equations are not covariant under achange of coordinates but the quantities associated with thethree sets of equations have the same numerical values. Inspecial relativity theory, we adopt the opposite viewpointthat the equations must preserve the same form after achange of coordinates, but, contrary to the classical case,the quantities involved do not necessarily have the samenumerical values. The above discussion demonstrates thatthe ignorance of the partition between internal and externalforces is at the origin of the covariance principle.

6. Newton’s third law in relativistic collision

The theory of collision between relativistic particles is ofgreat importance for nuclear physics. In the absence ofnuclear reactions between the colliding particles the col-lision between them can be considered as elastic. Externalforces produced in a particle accelerator are needed to accel-erate and direct the particles in order to produce a collisionbetween them. When these external forces are turned off, thecolliding particles obey the equations of motion

dP1

dt1

dP2

dt� F12 1 F21 � 0

dE1

dt1

dE2

dt� �U1 2 U2�·F12

�54�

where all the quantities are generally defined in the so-calledlaboratory frame. Depending on the choice of mechanics,we have the following definitions

Mechanics Momentum Energy

Classical Pi � miUi Ei � miU2i =2

Relativisic Pi � migiUi Ei � migic2

with the identity

Ui ·dPi

dt� dEi

dt�55�

The interaction forces during the collision processcertainly follow Newton’s third law if we use the aboveequations. If we want to apply the principles of conservationof linear momentum and of energy to any closed system,Newton’s third law must be satisfied, namely:

P1�t�1 P2�t� � PG � Ct

E1�t�1 E2�t�1 E12�t� � ET � Ct�56�

The preceding equalities result from Eq. (54). These equal-ities are valid in both classical and relativistic mechanics.Therefore, the conservation laws of momentum and energyimply that Newton’s third lawmustbe satisfied. There iscertainly a contradiction when relativistic physicists affirmthat Newton’s third law does not apply in relativisticdynamics and use it when dealing with the collision of rela-tivistic particles. It is important to note that both particlesare observed in the same reference frame, namely thelaboratory frame. Therefore, no change of reference framecan be invoked to explain the contradiction. As pointed outby Beckmann [44] (p. 77)

The fact remains that the Einstein theory has someexplaining to do. For a theory that does not recognizethe equality of action and reaction cannot, withoutapology, invoke the conservation of momentum.

The law of conservation of four momentum during therelativistic collision process cannot hold exactly even if theexternal forces due to other particles are zero because wehave in fact the identity

dP1

dt1

dP2

dt� 2

ddt

q1

cA2

� �2

ddt

q2

cA1

� �± 0 �57�

The terms in the right-hand side of Eq. (57) describe theradiation effect of the colliding particles. If radiation lossesare completely negligible in a linear accelerator, the circum-stances change drastically in a circular accelerator andduring the time the collision takes place. The reader isreferred to Jackson’s book [45] (p. 701) for a general discus-sion of the radiation emitted during atomic collisions. Wemust point out that the above equation implies a motion ofthe center of mass of the colliding particles. It seems thatJackson is aware of this point when he introduces thenotions of relative particle and reduced mass in the problem15.5.

A collision can be described in four different referenceframes, namely:

the reference frame whose origin is the center of mass ofthe two particles;the Earth’s reference frame, also called the laboratoryreference frame. This reference frame can be the etherframe;the two reference frames whose origin is one of the twoparticles.

When we consider the calculation of the momentum andof the energy of the colliding particles in different referenceframes, the total classical or relativistic energy may change,or not, depending on the kind of reference frame we choose.The only case when this energy changes is when we movefrom the laboratory frame or ether frame to one of the threeother reference frames defined above. The difference inenergy must be ascribed to the kinetic energy of the centerof mass which results from the effect of the external forces


in the past. Therefore, the values of the linear momentumPG

and of the energyEG of the center of mass are zero in thereference frame of the center of mass and different from zerootherwise as shown in the following table:

Reference frame Relativistic case Classical case

Laboratory PG � ETUG=c2 PG � m0UG

Center of mass PG � 0 PG � 0Laboratory EG� ET 2 m0c

2 EG � m0U2G=2

Center of mass EG � 0 EG � 0

In the above table,m0 � m1 1 m2 is the proper mass inclassical mechanics, whereasm0 � m1 1 m2 1 E12/c

2 is theproper mass in relativistic mechanics whereE12 is themutual potential energy of the system. We can point outthat potential energy has no meaning in special relativity,no wonder that textbooks avoid discussing potential energy.

There is some confusion in the calculation of energy inrelativistic mechanics which results from the ignoring of thepartition of the kinetic energy between internal motion andexternal motion related to the motion of the center of mass.As shown in Ref. [29], the violation of the conservation ofenergy when examining the relativistic collision of twoidentical particles in two different reference frames, namely,the reference frame of the center of mass and the referenceframe of one particle, is due to the covariance principle:

�P1 1 P2�2L 21c2 �E1 1 E2�2L

� �P1 1 P2�2R 21c2 �E1 1 E2�2R �58�

The covariance of a four-vector momentum-energy in achange of reference frame leads to an inconsistency whenone uses the relativistic addition law for the calculation ofthe relative velocity between two identical particles in acollision viewed in two different frames.

By definition, we havein the laboratory frame:

classical mechanics

2P1 � P2 � m0UL E1 1 E2 � m0U2L

relativistic mechanics

2P1 � P2 � m0gLUL E1 1 E2 � 2m0gLc2

�59�

in the rest frame of the second particle:

classical mechanics

P2 � 0 E2 � 0 M � 12 m0 V � 2UL

P1 � MV � m0UL E1 � 12 MV2 � m0U2

L

�60�

relativistic mechanics

P2 � 0 E2 � m0c2

P1 � PR � m0gRUR � 2m0g2LUL

E1 � ER � m0gRc2

E1 1 E2 � 2m0g2Lc2

�61�

where the velocityUR is the velocity of the first particledefined in the reference frame of the second particle. Therelativistic definitions above are obtained by using thecovariance law and the relativistic addition law for the rela-tive velocity given below; hence it follows that

gR 1 1� 2g2L ) b2

R � 4b2L

�1 1 b2L�2

�62�

In the laboratory frame, the relative velocity between thetwo particles is the classical relative velocityV�U1 2 U2�2UL with the velocitiesU1 andU2 measured in the laboratoryframe. In Newtonian mechanics, the relative velocityV �UR is invariant in a change of reference frame; therefore, thetotal kinetic energyEK of the two particles is also invariant.In contradistinction to classical mechanics, the velocityUR

in relativistic mechanics is no longer invariant since thisvelocity is given by

UR � U1 2 U2

1 2U1·U2

c2

�63�

The inconsistency of the above formula may be shown bynoting that the total kinetic energyEK of the two particles isnot maintained in the change of reference frame since wehave in the laboratory frame

EK � 2�gL 2 1�m0c2 < m0U2L �64�

whereas in the rest frame of the second particle, we haveinstead

EK � 2�g2L 2 1�m0c2 � 2�gL 2 1��gL 1 1�m0c2 < 2m0U2

L

�65�Therefore, a change of reference frame can create energy.

This is not surprising since the two preceding relations differby a factorgL which results from the covariance principle,whereas Newtonian physics in a collision only impliesinvariance of the energy and momentum in a change ofreference frame. Moreover, the correct and unique definitionof m0 has to be rest-mass in a fundamental reference framerather then ‘‘proper-mass’’ in any particle-bound frame!

7. Newton’s third law and the twin paradox

Einstein stated that an ideal clock which moves in aclosed curve with respect to a clock at rest in the laboratoryframe will indicate an elapsed proper time smaller than the


one given by the stationary clock. The paradox arisesbecause the difference in time between the two clocks isconsidered as a velocity effect; since special relativity is areciprocal theory, from the viewpoint of the traveling clockit can be argued that it is the stationary clock that goesslower. Clocks can be replaced by twins, hence the nametwin paradox; this has generated one of the longest standingcontroversies in twentieth century physics, as proved by theprolific literature published on this topic by several authors[29, 46]. The paradox is usually explained by stating that themoving clock has suffered an acceleration while the station-ary clock has not. However, it is not consistent to use aconsideration about acceleration in order to explain theparadox after ruling out any physical effect of this accelera-tion on the behavior of the traveling clock. We know that themass variation of an electron accelerated by an electromag-netic force in the laboratory frame ism� gm0. This is a realeffect measured in modern particle accelerators. Cullwick[47] pointed out that the decay rate of a meson depends onits energy in the same proportion; therefore, a physicalchange has taken place during acceleration which is asreal as the mass variation of a moving electron. Since nophysicist will contest that if we slow down a moving elec-tron we will recover its rest mass in the laboratory frame,then the preceding formulation predicts that if, after ameasurement on the moving muons has been made, weslow them down to rest, we will recover the lifetime ofmuons at rest. By analogy one can expect that the outcomeof the twin paradox is zero. One may contest the analogy;however, the experiment has been done by Frisch and Smith[48] and confirms the validity of the analogy. Let us quotethe authors:

In addition, actually simultaneously, we slowed downand stopped a sample ofm-mesons and measured thedistribution of their decay times when they were atrest relative to us. Comparison of their rate of decayat rest with their rate of decay in flight showed thatthe moving mesons decay much more slowly.

The reader is referred to Refs. [29, 46] for a full discussionof the twin paradox.

To understand why the twin paradox is connected withNewton’s third law, one may ask why reciprocity, alsocalled kinematical symmetry, which is at the heart of theparadox, is not a valid concept for the clock problem. Toclarify the concept of reciprocity one must proceed in threedifferent steps with the help of Fig. 7. We will show that thedifficulty arises from a subtle point which has escaped thenotice of most authors. In a nutshell, in spite of kineticalsymmetry, there is no dynamical symmetry since the clockat rest is bound to the Earth laboratory frame.

Case (a): apply a forceF . 0 to clock B that will moveforward in thez. 0 direction from the clock A at rest in thelaboratory frame and will indicate the timetBM � g tAR.

Case (b): apply a forceF , 0 to clock A that will move

backward in thez , 0 direction from the clock B at rest inthe laboratory frame and will indicate the timetAM � g tBR.

We remark that there is no contradiction between the twocases concerning the time behavior of the moving clock withrespect to the rest clock.

Case (c): apply a forceF . 0 to the clock B and consideran observer at rest with the moving clock B. This observersees the clock A moving away from him in thez , 0 direc-tion with a velocityU , 0 as if the clock A is submitted to aforce F , 0. This observer will therefore attribute to theclock A the timetAM � g tBR.

Thus the reciprocity appears in cases (a) and (c) andyields a contradiction if we writetBM � tBR. We see that incase (c) the observer tries to put himself in case (b), wherethe time dilatation formulas (b) and (c) are the same with thedifference that the time dilatation formula (b) cannot contra-dict the time dilatation formula (a); therefore, where is theorigin of the contradiction? The explanation resides in theimportant fact that for a system we must make a distinctionbetween internal forces and external forces. For example, abullet moves away from a gun or two particles move awayafter a collision if they are in interaction with internal forces.For this system we know from Newton’s third law of motionthat action equals reaction; in that case the origin of theframe can be either one of the two particles or the centerof mass. We can distinguish external forces from internalones by looking to the position of the center of mass in thelaboratory frame, which is moving for external forces orfixed for internal forces. Reciprocity or symmetry cannotbe invoked in case (c) because the forceF is only appliedto clock B and is therefore an external force. Clocks A and Bcan be in interaction when we synchronize them in thelaboratory frame; but, when clock B is moved by an externalforce we have no right to switch the origin of our frame aswe did in case (c), since the center of mass of the two clocksmoves with a non-uniform velocity in the laboratory framefor the case of an external force.

8. Newton’s third law in quantum mechanics

Many systems studied in quantum mechanics consist oftwo particles. Examples are the electron and the proton of ahydrogen atom, the two atoms of a diatomic molecule and


Fig. 7. Round trip of a clock and Newton’s third principle.

the probe and target particles in a scattering event or acollision. The quantum mechanical treatment of the two-body problem starts with Schro¨dinger’s equation:

1m1

D1C 11

m2D2C 1 1j

2"

2C

2t2

2

"2 EPC � 0 �66�

wherem1 andm2 are the rest masses of the two particles andEP(r1, r2, t) is the potential energy between the two particles.Substituting the particular solutionC (r1, r2, t) �C (r1, r2) ejv t in the preceding equation produces an inhomo-geneous Helmholtz equation to be solved:

1m1

D1C 11

m2D2C 1

2

"2 �ET 2 EP�C � 0 �67�

where we have used Einstein’s relationET � 2 1"v with1 � ^ 1 depending on the sign ofET. As in classicalmechanics, the Schro¨dinger equation can be partitionedinto parts relating to the center of massr and the relativepositionR:

mr � m1r1 1 m2r2 R� r1 2 r2 �68�wherem � m1 1 m2 is the total mass of the system. Bydefinition, we haveC (r1, r2) � C0(r, R); therefore, theSchrodinger equation above can be transformed as follows:

1mDrC0 1

1M

DRC0 12

"2 �ET 2 EP�C0 � 0 �69�

where the reduced mass is defined asM � m1m2/(m1 1 m2).If the potential functionEP(R) does not depend explicitly ontime, the total energy can be split into a sum of twoconstantsET � ETr 1 ETR � 2"�1rvr 1 1RvR�. Then wecan factor the wave functionC0�r; R� � Cr�r�CR�R� andsubstitute it in the preceding equation. After a division byCrCR we get a sum of two terms which only depends oneither r or R; therefore, each term must be equal to aconstant, which implies the system of equations:

DrCr 12m

"2 ETrCr � 0

DRCR 12M

"2 �ETR 2 EP�CR � 0

�70�

The first equation describes the motion of a free particlein the absence of any external force. The second equation isused to calculate the stationary states of the electron in ahydrogen atom. Therefore, we see that the existence ofNewton’s third law in quantum mechanics is fundamental.Concerning the theory of measurement in quantummechanics, Cavalleri [49] pointed out:

It is not usually emphasized that there are two kindsof observations, one perturbing and the other non-perturbing the system under examination.

Of course the perturbing measurement occurs when theobserver is inmutual interactionwith the system underexamination, which implies both reciprocity and Newton’sthird law even if external forces are present. The affirmation

that an observer always interferes in the process of measure-ment in quantum mechanics is simply not true because ofthe non-reciprocity of external forces.

ForN particles, we can use the Jacobi coordinates definedpreviously to express the Laplacian of the functionC�r1; …r j ; …rN; t� � C0�R1; …Rj ; …RN; t� in terms ofthe new coordinates [36] (p. 264); we getXNi�1

1mi

DriC �XN 2 1

j�1

1m0j

11

mj11

!DRjC0 1

1m0

DRNY0

�71�

9. Newton’s third law in electromagnetism

9.1. The Lorentz force law and the stimulated force

As stated in Section 1, there are two force laws of motionin electromagnetism. The first one is the Ampe`re force lawand the second one is the Lorentz force law which leads tothe Lorentz–Maxwell equation of motion:

dPi

dt� FLij �72�

where the Lorentz forceFLij applied to the particlei is givenby the formula

FLij � qi Ej 11c

Ui ∧ Bj

� ��73�

The electromagnetic fieldEj ; Bj is an external fieldproduced by another charged particlej. We can makethree remarks concerning the Lorentz force law above.

The first one is to question the meaning of the velocityUi

of the chargeqi that appears in Eq. (72). As pointed out byAssis and Peixoto [50], most textbooks do not stateexplicitly what the velocityUi is relative to. Of course,according to the special theory of relativity, the velocityof the chargeqi is the velocity relative to an inertialreference frame. Therefore, this velocity will have differentvalues in different inertial reference frames.

The second remark concerns the well-known fact that theLorentz forces do not satisfy Newton’s third law since wehaveFLij ± 2 FLji. We will demonstrate again that this factimplies the existence of external forces that can performwork whose energy is provided either by the medium orby the ether.

The third remark concerns the fact that the magnetic partof the Lorentz force never works. However, it has neverbeen recognized in the literature, except maybe in thepaper by Galeczki [51], that there is a second definition ofwork used currently in electromagnetism where themagnetic part of the Lorentz force does work. The bestexample is the work of the magnetic force dW � F·dr �I �dr ∧ B�·dr=c exerted on a current element dr � U dtsustaining a currentI which moves in a different directiondr in an external magnetic fieldB. This work is not zero and


is used in calculating the cutting flux through the surfacedS� dr ∧ dr in motional electrical circuit theory. I think thepaper by Galeczki raises interesting questions concerningthe compatibility of the special relativity theory and Fara-day’s law.

We have shown that the definition of an inertial frameimplies that this frame must have for its origin a particlewith an infinite mass. However, the mass of any real particleis finite; moreover, we have demonstrated the existence ofexternal forces that can perform work whose energy isprovided by the medium, namely the ether. Therefore, toshow the existence of the ether, we must consider the inter-action between two moving charges with forces that violateNewton’s third law. Since Lorentz forces exerted by freelymoving charges upon one another are not equal and oppositein principle, it follows that a system consisting of pair ofcharged particles in relative motion can change the state ofmotion of its center of mass without external help. Considertwo charged particlesq1 andq2 moving with velocitiesU1

andU2 relative to a reference frame where the ether is at rest.We stress that all the following calculations are done in thisreference frame; therefore, no change of reference frame isimplied in the discussion. The chargeq1 exerts onq2 a forceF21 � q2�E1 1 U2 ∧ B1=c� whereE1 andB1 are the electricand magnetic fields produced byq1 at the position occupiedby q2. Conversely, the chargeq2 produces onq1 a forceF12 � q1�E2 1 U1 ∧ B2=c�. In general, these two forceshave different directions and magnitudes:

dP1

dt1

dP2

dt� F12 1 F21 ± 0 �74�

This can be shown by rewriting the Lorentz force in termsof the potentials:

F12 � q1 27F2 21c

2A2

2t1

1c

U1 ∧ 7 ∧ A2

� �for r � r1

F21 � q2 27F1 21c

2A1

2t1

1c

U2 ∧ 7 ∧ A1

� �for r � r2

�75�where the linear momenta of the particles are now definedby the relations P1 � m1g1U1 and P2 � m2g2U2. Asalready stated in the beginning of this paper, the fact thatthe equality of action and reaction is not satisfied in rela-tivistic mechanics is not due to the relativity of simul-taneity or the retardation effect. Therefore, the two chargedparticles do not constitute a closed system because of theviolation of Newton’s third law by the Lorentz forces. Italso follows that the conservation of mechanical energy ofthe system cannot be verified. Therefore, the classicaldefinition of the center of mass given by Eq. (2), which isthe only definition physically meaningful, yields the equation

md2rdt2� F12 1 F21 2 m1U1

dg1

dt2 m2U2

dg2

dt� FG ± 0

�76�

Eq. (74) can be written in a form often encountered in theliterature, namely

ddt

m1g1U1 1q1

cA2

� �1

ddt

m2g2U2 1q2

cA1

� �� 0

�77�In that case, Newton’s third law is verified for the general-

ized momentum. It follows that the total canonical momen-tum is conserved instead of the total Newtonian momentum.From the preceding equation, one can deduce that fieldtheory attributes momentum to the electromagnetic field toallow a particle to interact only with fields at the position ofthe particle. It precludes the possibility of instantaneousparticle interactions except as an approximation. Therefore,the interaction between the particles proceeds by a transferof momentum from one particle to the field, then the fieldtransports the momentum at the speed of light to the positionof the second particle where it can be transferred from thefield to the other particle. However, this transfer cannot besymmetric, since the above equation can be rewritten asfollows:

dP1

dt1

dP2

dt� F12 1 F21

� 2ddt

q1

cA2

� �2

ddt

q2

cA1

� �± 0 �78�

Therefore, to take the fields into account in the calculationdoes not change the fact that there is a stimulated motion ofthe center of mass. We stress that the conditionF12 1 F21 ±0 alone is sufficient to state that a pair of charged particles inrelative motion can change its state of motion without exter-nal help, as stressed by several other authors [52–55] whohave reviewed this problem. There is absolutely no escapingthat conclusion.

For small accelerations, we can take into account only thevelocity fields at the simultaneous positionsr1(t) andr2(t) ofthe two particles which are given by the well-knownrelations

E1 � q1

R2

1 2 b21

�1 2 b21 sin2 u�3=2 n

E2 � 2q2

R2

1 2 b22

�1 2 b22 sin2 u�3=2 n

cB1 � U1 ∧ E1 cB2 � U2 ∧ E2

�79�

with the definitionsR � R(t) � r2(t) 2 r1(t) andn � R=R.Since the velocitiesUi are small with respect to the light

speed, then the electrical fields can be written as follows:

E1 <q1

c2R2 �c2 1 12 U2

1 2 32 �U1·n�2�n

E2 < 2q2

c2R2 �c2 1 12 U2

2 2 32 �U2·n�2�n

�80�

The forcesF12 andF21 experienced respectively byq1 and


q2 in the same reference frame are calculated from theformulas

F12 � q1 E2 11c2 U1 ∧ U2 ∧ E2

� �

F21 � q2 E1 11c2 U2 ∧ U1 ∧ E1

� � �81�

Therefore, the forceF12 is not equal in magnitude andopposite in direction to the forceF21 since we haveFG �F12 1 F21 ± 0.

As pointed out by Builder [56] (p. 285), the asymmetry,between the forces experienced simultaneously by the twocharges, obviously precludes any inference that the forcesare determined solely by the relative motions of the charges.As already noted, the failure of the third law is not due to aproblem of simultaneity. Since the above forces do notfollow Newton’s third law, we have

FG <q2

c2R2 ��U·V�n 2 3�U·n��V·n�n 1 �U·n�V 2 �V·n�U��82�

with the definitionsq � q1 � 2 q2. In the above relation,V � U2 2 U1 andU � U2 are respectively the relative andabsolute velocities where we assumedV p U. We can stillsimplify this formula by takingV·R < VR, U·R < UR cosuandU·V < UV cosu in a conductor: it follows that

FG < 2q2

c2R3 ��U·V�R 1 �V·R�U� �83�

The stimulated force in the Coulomb gauge, as given intwo preceding papers [33, 57], has the expression

FG <2q2

c2R3 �U·V�R �84�

where we have also neglected the acceleration terms in thecalculation of the force.

The two force laws in Eqs. (83) and (84) are differentbecause we used two different definitions of the center ofmass in the calculation of the stimulated force. The aboveexpressions of the stimulated force depend upon an absolutevelocityU which is defined with respect to a preferred refer-ence frame. In special relativity theory, we consider that theEarth is an inertial reference frame where the velocityU iszero; consequently, no stimulated motion can be expected inthis theory. But we all know that the Earth is moving, whichimplies the existence of a stimulated rectilinear and rota-tional motion as will be discussed in the section concerningall the experiments which reveal our motion through theether (11).

We can generalize the preceding calculation by consider-ing a cluster ofN particles which are closed to one anotherand use the Jacobi coordinates in order to calculate thestimulated force for the cluster. We must obtain an expres-sion that will depend upon the absolute motionU of thecenter of mass of the cluster with respect to the ether.

9.2. The Weber force law

The violation of Newton’s third law led Builder to realizethe contradiction between the asymmetry of the Lorentzforces viewed in different inertial frames and the relativityprinciple. Consequently, Builder came to the conclusion thatthere is no alternative but to admit the ether hypothesis. Hewas particularly clear when he said:

It is only necessary to postulate that the phenomenaare caused by motions of particles and bodies relativeto an absolute inertial system in accordance with theMaxwell–Lorentz equations.

However, Builder is partially correct since he ignores theexistence of a second force law, namely the Weber forcelaw, and its associated equation of motion:

ddt

MV � FWij �85�

where M � mimj =�mi 1 mj� is the reduced mass andV �V ij � Ui 2 Uj is the relative velocity which keeps thesame value in any reference frame. The connection betweenthe Ampere force law and the Weber force law is welldescribed in Assis’s book [58].

The Weber force law is deduced from the Weberpotential:

EP �qiqj

R1 2

V·RacR

� �2" #

�86�

wherea is a parameter which takes the valuea � ��2p

in theWeber formula. Note that fora � 2, we can consider that 2cis the relative speed of two photons moving in oppositedirections; therefore, for two particles also moving inopposite directions, each one hasc as the speed limit inone direction.

The Weber force can be calculated in two different ways.First, the force is obtained by taking the time derivative ofEP and using the equality dEP/dt � 2 V·F to get

FWij � qiqj

�1R2 2

1�acR�2 �3�V·n�2 2 2V2�

12

�acR�2 R·dVdt

�nij �87�

with the definition nij � R=R. This force clearly obeysNewton’s third law since we haveFWij � 2FWji . Theabove force is the sum of three terms, namely, the Coulombforce depending on the relative position, the Ampe`remagnetic force, depending on the relative velocity, and thethird is the induction force depending on the relative accel-eration.

The Weber force can also be obtained fromF � 2 7REP


which yields another expression for the force:

FWij � qiqj

�1R2 2

3�acR�2 �V·n�2

� �n

12

�acR�2 �V·n�7R�V·R��

�88�

One can demonstrate that the two expressions of theWeber force are the same by using the identities

�V·R��V 1 �R·7R�V 1 R ∧ 7R ∧ V�

� { V2 1 R·��V·7R�V�} R

dVdt� �V·7R�V

�89�

The Weber force conserves the total energyET � EK 1EP� Ct since the Weber force satisfies Newton’s third law.On the contrary, the Lorentz force can never satisfy theconservation of energy, even if we include the so-calledradiation force, as pointed out correctly by Wesley [32]. Itis also interesting to note that Moon and Spencer [18–21]and Wesley [31] via the field theory succeeded in introdu-cing the retardation effect in Weber electrodynamics. Sincethere is no external force in the Weber approach, then theequation of motion of the center of mass is

md2rdt2� Fe � 0 �90�

The center of mass has a constant motion with respect to theether frame.

9.3. Newton’s third law between matter and radiation

One must point out that the Lorentz force cannot bededuced from Maxwell’s equations. Therefore, it appearsas a postulate. However, if we assume that matter and radia-tion form a closed system, then we can use Newton’s thirdlaw to link the continuity equations of matter and radiationas follows.

Knowing that the energy and momentum conservationlaws for radiation obtained from Maxwell’s equations are

2Er

2t1 7·Gr � 2f ·U

2Pr

2t1 7· Tr

! � 2f

�91�

with the following definitions

Energy density Er � 18p�E2 1 B2�

Poynting flux Gr � c4p�E ∧ B�

Momentum density Pr � 14pc

�E ∧ B�

Lorentz force density f � rE 11c

J ∧ B

Stress tensor Tr ! � 1

4p� 1

2 �E2 1 B2�yI 2 �EE 1 BB��

then taking into account Newton’s third law, the continuityequations for matter have the expression

2Em

2t1 7·Gm � f ·U

2Pm

2t1 7·Tm

! � f

�92�


Em � 12 rmU2 Gm � EmU Pm � rmU Tm

! � rmUU

�93�We can now integrate the above formulas on the volumes

Vr(t) andVm(t), we get

ddt

ZVr�t�

Er dV 1Z

Sr�t��Gr 2 Erc�·dS� 2

ZVr�t�

f ·U dV

ddt

ZVr�t�

Pr dV 1Z

Sr�t�dS·� Tr

!2cPr� � 2

ZVr�t�

f dV

ddt

ZVm�t�

Em dV �Z

Vm�t�f ·U dV

ddt

ZVm�t�

Pm dV �Z

Vm�t�f dV

�94�For a charged point particle, we haver�r; t� �

qd�r 2 rS�t�� andrm�r; t� � m0d�r 2 rS�t��, it follows:

dEM

dt1

dER

dt� 0

dPM

dt1

dPR

dt� 0 �95�

knowing that the surface integrals tends to zero whenSr(t)!∞.

The recoil effect in a radiating system is a manifestationof Newton’s third law which justifies the above equations.Consider a stationary atomP0 � 0 of massm0 � E0/c

2 thatemits a photon of energyÉv and momentumÉk. Conserva-tion of energy and momentum yields

P 1 "k � 0 E 1 "v � m0c2 �96�After squaring the two preceding equations, we get

c2P2 1 c2"2k2 1 2c2"P·k � 0

E2 1 "2v2 1 2E"v � m20c4

�97�

The emission of a photon by the atom modifies its restenergy which ism

00c2 in its final state. Therefore, we can

subtract the two above equations and use the identityE2 �c2P2 1 �m0

0c2�2 to obtain the equation:

�m00c2�2 1 2"�Ev 2 c2P·k� � m2

0c4 �98�


Both equations in Eq. (96) can be used to rewrite Eq. (98) inthe form

�m00c2�2 1 2m0c2"v � m2

0c4 �99�In the co-moving frame, the energy of the photon is

"v 0 � m0c2 2 m00c2 � E2 2 E1 �100�

One can now eliminate the rest energym00c2 between the

two above equations to obtain the energy of the photon inthe laboratory frame:

"v � "v 0 1 2"v 0

2m0c2

!�101�

These results have important physical implications, forexample in the Mo¨ssbauer effect [59], because they placerestrictions on the ability of atoms and nuclei to re-absorbtheir own radiation. We can also point out that the abovetheory can only explain the existence of spontaneous radi-ation. The existence of another kind of radiation, namelystimulated radiation, is a manifestation of the partitionbetween internal and external forces which do not verifyNewton’s third law.

10. Newton’s third law and the superposition principle

A medium is said to be linear if it obeys the linear super-position principle, namely the field due to several sources isthe sum of the fields produced by each source. This principleis a consequence of the linearity of the wave equations in themedium. As pointed out by Jackson [45] (p. 10), this prin-ciple is exploited so often in electromagnetism and in quan-tum mechanics that it is taken for granted. There are, ofcourse, circumstances where non-linear effects occur, buthere we are only concerned with fields in vacuum at themicroscopic level inside atoms and nuclei. However, thisprinciple does not apply to field energy and momentum.In most textbooks in physics, the non-linearity of fieldenergy and momentum is not discussed. Even in the pro-fessional literature, we have been able to find only a fewrelevant papers dealing with the subject.

For example, the total electric energy associated with thesuperimposed electric fieldE�r; t� � E1�r; t�1 E2�r; t� hasthe expression

ET � 18p

ZV

E21 dV 1

18p

ZV

E22 dV

12

8p

ZV

E1·E2 dV

�102�

Therefore, the total energy can be written in a formal wayas the sum of four termsET � E11 1 E22 1 2E12 where thetermsEii andEij are respectively the proper (or self-) energyand the interaction (or mutual) energy of the wave fields.Because of the presence of the interaction energy term 2E12,it appears that the conservation of energy is violated. Before

examining the compatibility of the principles of super-position and energy conservation, let us discuss two simpleexamples.

10.1. Light interference

A real monochromatic electric fieldEi�r; t� written incomplex form has the expression

Ei�r; t� � 12 �Ei�r� ejvt 1 Ep

i �r� e2jvt� �103�

where the vectorEi�r� is a complex quantity of the formEi�r� � E0i�r� e2j�ki ·r1ai �, knowing that the real vectorE0i isa function which slowly varies with respect to space. Thetotal energy densityEt(r, t) is

Et � E21 1 E2

2 1 2E1·E2 �104�with the definitions

4E2i �r; t� � E2

i �r� ej2vt 1 E2i 2�r� e2j2vt 1 2Ei�r�·Ep

i �r�

4E1�r; t�·E2�r; t� � E1�r�·E2�r� ej2vt 1 Ep1�r�·Ep

2�r� e2j2vt

1 E1�r�·Ep2�r�1 E2�r�·Ep

1�r��105�

Most optical detectors used (eye, photographic plate andphotoelectric detector) are sensible to the flow of lightenergy. These detectors integrate the received density ofenergy in a certain volume of space during a lapse of timeTR called the time response of the detecting device, this timeis about 1029 s for a photoelectric detector while the lightperiod has the valueT < 10214 s. Therefore, the rapidity ofthe oscillating wave motion does not allow an optical detec-tor to measure the time dependence of the energy field butrather its time average:

k2Et�r; t�l � uE1�r�u2 1 uE2�r�u2 1 E1�r�·Ep2�r�1 E2�r�·Ep

1�r��106�

where the symbolk l means a time average operation duringthe periodT, which is defined by the expression

kEt�r; t�l � 1T

ZT

0Et�r; t� dt �107�

Taking into account the preceding definition, we get

k2Et�r; t�l � E201�r�1 E2

02�r�1 2E01�r�·E02�r�cosw�r��108�

with the definitionw�r� � �k1 2 k2�·r 1 a1 2 a2 where thequantitya1 2 a2 represents the difference between the twooptical paths of the two sources. The oscillatory term coswis at the origin of the fringe effect. Therefore, the non-linearity of energy results from the mutual interferenceterm since we have

E12 � 116p

ZV

E01�r�·E02�r� cos�w�r�� dr3 �109�


Some authors [60, 61] doing a one-dimensional study ofthe problem state incorrectly that the above integral is zerodue to the oscillatory behavior of the functions inside theintegral. In that case, the conservation of energy is satisfied.To show that this is not the case, consider two punctualsourcesS1 andS2 located at a distance 2L, where the originO is a hall-way between the sources. Although no isotropicsource of transverse light wave does exist, this does notchange the demonstration if we use such a source. More-over, the following demonstration does rigorously apply tothe case of sound waves. Therefore, a spherical wave has theexpression

E0i�r� � Ei

rie2�a2ri 1b2

=ri � �110�

The presence of the parametersa and b is necessary toobtain a finite energy solution in the following calculation:

E12 <E1·E2

32p

Z1 ∞

0

Z4p

0

1r2 e22�a2r1b2

=r��ejw

1 e2jw�r2 dr dV �111�Knowing that r1 � 2L 1 r and r2 � L 1 r, it follows

the phase relationw�r� � k1·r1 2 k2·r2 < 24kL·r=r withki � kr i =ri . After integration on the solid angle, we get

E12 < 14 E1·E2

sin�kL�kL

Z1 ∞

0e22�a2r1b2

=r� dr �112�

If we now integrate overr, we obtain

E12 < E1·E2sin�kL�

kLb2a

K1�4ab� �113�

where K1 is the modified Bessel function of the secondspecies.

Finally, the total energy has the value

ET < E21 1 E2

2 1 2E1·E2sin�kL�

kL

� �b2a

K1�4ab� �114�

The mutual energy is not zero, contrary to the statementsof the literature. This energy will be zero only in the casewhere the distanceL between the two sources tends toinfinity. We must note that the magnetic field must betaken into account in the preceding calculation; however,it turns out that its contribution is exactly the same as thecontribution of the electric field.

10.2. Electrostatic interference

The experimentally observed linear superposition offorces due to several charges means that we can write thetotal electric fieldEt�r� at r due to two chargesq1 and q2

located atr i as the vector sum of the two fields:

E1�r� � q1R1

R31

E2�r� � q2R2

R32

�115�

with the definitionsR1 � r 2 r1 andR2 � r 2 r2. The totalenergy can be calculated by using Eq. (102). As is well-

known, the proper energies become infinite if the particlesare reduced to points; however, the mutual energy calcu-lated from the relation

EM � 14p

ZV

E1·E2 dV �116�

is finite since we get

EM � q1q2

4p

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞R1·R2

R31R3

2

dr3 � q1q2

R12�117�

with the definitionR12� ur1 2 r2u.By reading the literature on the subject discussed above,

one gets the impression that the mutual energy term is thecause of the violation of the conservation of energy. Theproblem becomes even more acute when dealing withspecial relativity theory, since potential energy is notincluded in any relativistic definition of energy. To ourknowledge, with the exception of the relativistic potentialenergyEP� m0c

2(1 2 b 2)1/2 introduced by Kundu [62, 63],Brillouin is the only physicist who discussed the meaning ofpotential energy in relativistic theories in several remarkablepapers [64–66]. If the rest mass of a particle is totally ofelectromagnetic energy, then Brillouin concluded that themass of potential energy can be considered as localized inthe interacting chargesq1q2 and split 50/50 between theparticles as follows:

m01 � m11 11

2c2 EM m02 � m22 11

2c2 EM �118�

Brillouin correctly stated that the definitions of the thirdlaw of Newton and the notion of center of mass are introuble in special relativity theory. However, Brillouin didnot solve the problem because, contrary to his statement,special relativity theory does not join smoothly with Newto-nian mechanics, as shown in this paper. In fact, we knowthat the mutual energy terms are the only terms whichsatisfy the conservation of energy since, from Newtoniandynamics, one can write

ddt� 1

2 mU2� � U·Fe � P11 1 P22

ddt� 1

2 MV212� � V12·F i1 � 2P12

�119�

knowing that the external and internal forces have thefollowing expressions:

Fe � F11 1 F22 F i1 � F12 11m�m2F11 2 m1F22�

F i2 � F21 21m�m2F11 2 m1F22�

�120�

Therefore, the equalityP12� P21 is satisfied in Newtoniandynamics because of the reciprocity conceptV12 � 2 V21

and Newton’s third lawF i1 � 2 F i2. It follows that fortwo charged particles in the absence of any external forces,


we have

ddt

MV12 � F12 � 27R12EM �121�

where we have writtenEM � 2E12.Therefore, in analyzing the compatibility of the principles

of conservation of energy and superposition of fields, onemust conclude that the proper energy terms are the onlyterms which do not satisfy the conservation of energy.Chen [67] also examined the non-linearity of energy andmomentum densities of electromagnetic waves generatedby non-interacting sources. He resolved the problem bynoting the existence of an unsuspected degree of freedomof the energy–momentum four electromagnetic tensorwhich can undergo a four gauge transformation with aconstraint that allowed the situation to be physically iden-tical. However, the Lorentz covariance has nothing to dowith the problem, since we know from experiments thatenergy and momentum of photons can have physicallydistinct effects.

10.3. Carson reciprocity theorem

The question concerning the compatibility of the energyconservation principle with the superposition principlecannot be correctly answered if we do not take into accountthe energies given by the sources. Therefore, the discussionmust start from the continuity equations of the electromag-netic fields:

2Er

2t1 7·G� 2J·E

2Pr

2t1 7·yT � 2 rE 1

1c

J ∧ B� � �122�


Energy density Er � 18p�E2 1 B2�

Momentum density Pr � 14pc

�E ∧ B�

Poynting vector G� c4p�E ∧ B�

Stress tensor yT � 14p� 1

2 �E2 1 B2�yI 2 �EE 1 BB��

The above conservation equations are non-linear fluidequations depending on Eulerian coordinates. The non-linearity results from the superposition principle since fortwo sourcesr1�r; t�, r2�r; t�, J1�r; t�, J2�r; t� the fields and

the source terms can be written as follows:

r�r; t� � r1�r; t�1 r2�r; t�J�r; t� � J1�r; t�1 J2�r; t�E�r; t� � E1�r; t�1 E2�r; t�B�r; t� � B1�r; t�1 B2�r; t�

�123�

The preceding relations imply that the continuity equationscan be partitioned in four sets of equations:

2Eij

2t1 7·Gij � 2Ji ·Ej

2Pij

2t1 7· T ij

! � 2 riEj 11c

Ji ∧ Bj

� � �124�

where the indicesij have the values 11, 22, 12, 21.In the above equations, the following definitions are used:

Eij � 18p�Ei ·Ej 1 Bi ·Bj�

Pij � 14pc

�Ei ∧ Bj�

Gij � c4p�Ei ∧ Bj�

Tij ! � 1

4p� 1

2 �Ei ·Ej 1 Bi ·Bj�yI 2 �EiEj 1 BiBj�� 125�

The preceding conservation equations satisfy certain reci-procity relations first discovered by Carson [68, 69]. Theserelations are of particular importance to antenna theory indescribing the relationship between the receiving and trans-mitting properties of an antenna. Consider now a volumeVbounded by a surfaceS and containing two sourcesr1, J1

andr2, J2 and subtract the mutual conservation equations ofenergy and momentum, then the integrals of the continuityequations throughout the volumeV giveZ

V

2

2t�E12 2 E21� dV 1

ZS�G12 2 G21�·dS

� 2Z

V�J1·E2 2 J2·E1� dV

ZV

2

2t�P12 2 P21� dV 1

ZSdS·�T12

!2 T21 !�

� 2Z

V�f 12 2 f 21� dV

�126�

wheref ij � riEj 1 Ji ∧ Bj =c is the Lorentz density force.In the above relations, we have converted the volume

integrals of the divergence terms to surface integrals bymeans of the divergence theorem. Since all sources are offinite extent, we can extend the surfaceS of the volume toinfinity, then the surface integral in the energy equationvanishes identically at infinity because the electromagnetic


fields away from the sources can be approximated asspherical waves:

Ei�r; t� � Ei�t� e2jkr

rBi�r; t� � Ei�t� ∧ r

re2jkr

r�127�

with the conditionsEi�t�·r � Bi�t�·r � 0. As a consequence,the vectorG12 2 G21 is proportional to the vector

E1�t� ∧ �E2�t� ∧ r�2 E2�t� ∧ �E1�t� ∧ r� � 0 �128�By definition, we haveE12� E21; therefore, from the energyequation, one obtains the generalized Carson form of thereciprocity theorem:Z

V�J1·E2 2 J2·E1� dV � 0 �129�

The reciprocity theorem relating two different electromag-netic fields has also been discussed by Rumsey [70] andWelch [71]. Since the preceding equation is satisfied for aninfinite volume, it is also verified for the volumesV1 andV2

containing the sources. It follows from the equalityZV1

J1·E2 dV �Z

V2

J2·E1 dV �130�

The integrals in the above equation characterize theinteraction of one source with the field produced by theother source. By definition, we haveJi � riUi then it followsJi ·Ej � f ij ·Ui and the above relation becomesZ

V1

f 12·U1 dV �Z

V2

f 21·U2 dV �131�

If the mutual density of force satisfies Newton’s third law,namely f 12 � 2f 21, then the above equation implies theequalityU1 � 2U2. For separated sources, the above equal-ity is a manifestation of the reciprocity concept associatedwith Newton’s third law. To prove this assertion, let usrecall the power equations in the case of two particles inter-acting with external forces:ddt� 1

2 MV212� � F i1·V12 � F i2·V21 � d

dt� 1

2 MV221� �132�

From a physical point of view, the reciprocity concept fortwo identical antennas means that the receiving and radia-tion properties of each antenna must be the same.

For the momentum equation, we getZV

2

2t�P12 2 P21� dV

11

4p

ZSdS∧ �E1 ∧ E2 1 B1 ∧ B2�

� 2Z

V�f 12 2 f 21� dV �133�

The equalityG12 � G21 at infinity implies P12 � P21 atinfinity. Since the mutual density of force does not satisfyNewton’s third lawf 12 ± 2f 21, it follows that the conditionP12 � P21 is possible for f 12 � f 21 if the two electro-magnetic fields are parallel overS.

We can also consider the sum of the mutual continuityequations; we obtainZ

V

2

2t�E12 1 E21� dV 1

ZS�G12 1 G21�· dS

� 2Z

V�J1·E2 1 J2·E1� dV

ZV

2

2t�P12 1 P21� dV 1

ZSdS·�T12

!1 T21 !�

� 2Z

V�f 12 1 f 21� dV

�134�

If the surfaceSgoes to infinity, then the surface integral inthe energy equation does not vanish identically at infinity,unlike the preceding case, because Carson’s theoremimplies that the radiation terms are of the same sign.However, the mutual power integral term can be zero inthe particular case where one of the two following relationsis verified:Z

SG12·dS� 2

ZV

f 12·U1 dV

ZS

G21·dS � 2Z

Vf 21·U2 dV

�135�

Therefore, in the general case, the sum of the mutualpowers is not zero and cannot be used to explain the viol-ation of the conservation of energy in the superpositionprinciple. Let us now assume that Newton’s third law issatisfied; then, it follows that the conditionU1 � 2 U2

since the right-hand side terms in the above equations areequal. In that case, the two sources must move in oppositedirections, as expected from Newton’s third law.

Two sets of conservation laws of energy and momentumcan be written in the following form:Z

Vi �t�2

2t��Pii 1 Pij �1 7·�T ii

!1 Tij !�� dV

� 2Z


ZVi �t�

2

2t�Eii 1 Eij �1 7·�Gii 1 Gij �

� �dV

� 2Z

Vi �t��f ii 1 f ij �·Ui dV

�136�

where the volumeVi containing the source of the fieldsmoves with the velocityUi with respect to a given referenceframe. Jime´nez and Campos [72] in their paper concerningthe equations of energy and momentum balance in classicalelectrodynamics did a similar analysis by stressing thedistinction between the external fields and the proper fields.However, they did not realize that the partition of the fieldsis a necessity imposed by the superposition principle andNewton’s third law. On the contrary, Cray et al. [73] pointedout the existence of three contributions to the total intensityof the electromagnetic field: the intensity of the incident


wave, the intensity of the field radiated by the atom and theinterference between the incident wave and the radiatedwave. The standard textbook treatments of spontaneousand stimulated emission make no mention of interference;to stress this omission the above authors quote Lamb:

When stimulated emission by an excited atom istreated, either quantum mechanically or by a suitableclassical model, one finds that the numbers of photonsin those modes of the radiation field which wereinitially excited are increased by the interaction. Onthe other hand, the electromagnetic field radiated bysuch an atom is found to have the appropriate multi-pole character and shows no trace of the aboveaugmentation of the incident wave. In order to getamplification of the incident wave it is necessary toconsider the interference of the incident and radiatedwaves.

We note that Lamb clearly speaks of an increase ofenergy due to the superposition principle, a very importantsubject that will be discussed again in antenna and circuittheory. However, the point of view followed by the authorsis opposite from our approach since the wave radiated by theatom is considered as a spontaneous emission, whereas theinterference contribution is due to stimulated emissionaccording to them. We will show later that an interferenceterm is a mutual term related to Newton’s third law andconservation of energy and is better considered as a termassociated with spontaneous energy.

One can show that the preceding equations can also beexpressed as laws of motion:

ddt

ZVi �t��Pii 1 Pij � dV 1

ZSi �t�

dS·��T ii !

1 T ij !�

2 �UiPii 1 UiPij �� 2Z


ddt

ZVi �t��Eii 1 Eij � dV 1

ZSi �t��Gii 1 Gij �

2 �Eii 1 Eij �Ui�·dS� 2Z

Vi �t��f ii 1 f ij �·Ui dV

�137�

We have experimental proofs concerning the generation ofthe Poynting vectorGij � c�Ei ∧ Bj�=4p by an antenna sinceKabbary et al. [74] successfully developed revolutionaryantenna systems called crossed-field-antennas whichsynthesize directly the mutual Poynting vector from sep-arately stimulatedEi andBj fields. There is also the observa-tion by Graham and Lahoz [75] of electromagnetic angularmomentum in the vacuum gap of a cylindrical capacitorcreated by quasi-static electromagnetic fields where theEi

andBj fields arise from independent sources.The laws of motion for a material fluid are given by the

relations

ddt

ZVi �t�

rmiUi dV �Z


ddt

ZVi �t�

12 rmiU

2i dV �

ZVi �t��f ii 1 f ij �·Ui dV

�138�

If we compare the right-hand sides of the field and fluidlaws of motion, we see that they have an opposite sign whichis again a manifestation of Newton’s third law in a general-ized form. If the emitted radiation can be considered as aphoton particle with momentum, then, during the radiationprocess, the material medium must recoil. To verify thisaffirmation, one must add the laws of motion of the fieldsand the fluids; we get

ddt

ZVi �t��rmiUi 1 Pii 1 Pij � dV

� 2Z

Si �t�dS·��T ii

!1 Tij !�2 �UiPii 1 UiPij ��

ddt

ZVi �t�� 1

2 rmiU2i 1 Eii 1 Eij � dV

� 2Z

Si �t��Gii 1 Gij �2 �Eii 1 Eij �Ui�·dS

�139�

The energy equation above can be found in Ginzburg’sbook [76] (p. 44, formula 3.16). When the surfacessurrounding the sources tend to infinity, all the surfaceterms depending onUi vanish while the electromagneticterms decrease asR22, but since the surface elementsdS�R2 dV increase asR2, the surface integrals depending on theelectromagnetic fields tend to finite values which representthe radiated fields. It follows that the surface integrals arenot zero, as stated by several physicists, such as Cohen-Tannoudji et al. [77] (p. 64), Landau and Lifchitz [78] (p.105) and Ginzburg [76] (p. 53). The surface integrals arecorrectly calculated in the books of Plonsey and Collin [79](p. 396) and Becker [80] (p. 285). We have no right toassume that the radiation has not yet reached the surfaceS, as stated by Ginzburg, since the radiation emitted by adipole of momentump�t� � qr�t� has a finite value given byPR � 2�d2p=dt2�2=3c3 in the non-relativistic case. Even if weneglect radiation, we can see that the recoil effect and theconservation of energy cannot be satisfied without thepresence of a mutual field defined by the subscriptsij .This justifies the sentence quoted in the literature that aparticle cannot radiate in a conservative manner withoutthe presence of an external field.

For two material fluids, the preceding relations for the


energy become

ddt

ZV1�t�� 1

2 rm1U21 1 �E11 1 E12�� dV

� 2Z

S1�t��G11 1 G12�2 �E11 1 E12�U1�·dS

ddt

ZV2�t�� 1

2 rm2U22 1 �E22 1 E21�� dV

� 2Z

S2�t��G22 1 G21�2 �E22 1 E21�U2�·dS

�140�

By adding the above equations, we obtain a relationshipdescribing an energy conservation law for both matter andradiation of a two-fluid system. However, the system is notclosed because the sum of the surface integrals is not zero inthe general case and also because the proper radiation termsin these integrals cannot compensate one another. There-fore, if there is a violation of the conservation of energydue to the superposition principle, the difference of energymust be ascribed to the center of mass of the materialsystem. This is corroborated by the fact that the materialsystem must have a stimulated motion since the properand mutual densities of the Lorentz force do not satisfyNewton’s third law:

ddt

ZV1�t�

12 rm1U

21 dV 1

ddt

ZV2�t�

12 rm2U

22 dV

�Z

V1�t��f 11 1 f 12�·U1 dV 1

ZV2�t��f 22 1 f 21�·U2 dV

�141�Therefore, one cannot expect that the sum of the right-

hand side terms in the above equation is zero in the generalcase. It is important to stress that the preceding analysis doesnot depend on the presence or not of the relativistic gammafactor, since one can replaceU2

i =2 by (g i 2 1)c2 in the aboveequation.

10.4. Newton’s third law and the Aharonov–Bohm effect

Aharonov and Bohm (A.B.) [81, 82] proposed that elec-tromagnetic potentials have a physical role in quantumtheory in contrast to their auxiliary role in classical electro-magnetism. One of the experiments purporting to illustratethis role involved the shift in an electron interference patternformed as electrons passed around opposite sides of a whis-ker or a microscopic solenoid. Experiments [83–88] haveconfirmed the existence of the A.B. effect. However, theinterpretation of the effect has been discussed in the litera-ture for more than 30 years and the effect is still regarded assomething of a mystery. One of the aspects of the A.B. effectthat generates the most scepticism is that shifts in the inter-ference pattern occur with no Lorentz force acting upon theelectrons passing around the solenoid.

Boyer, in a not well-known paper [89], was the first

physicist in 1973 to recognize the origin of the difficultyby making the statement:

Although the passing particle exerts forces on thesolenoid, it is not true that the solenoid exerts aforce on the passing particle.

Therefore, Boyer’s assertion implies the violation ofNewton’s third law for the magnetic interaction betweenthe electrons outside and inside the solenoid. This is notsurprising, since the magnetic Lorentz force is the part ofthe Lorentz force which does not obey Newton’s third law;this fact was later fully recognized by O’Raifeartaigh et al.[90] when they say:

The analysis in terms of energy shows quite clearlyand simply where the effect originates, namely not inthe interaction of the external magnetic field with theelectrons from which it is shielded but with themagnetic field of the electron, from which it is notshielded.

Since the first paper by Boyer concerning this problem,several papers [90–95] have now be published containingthe same analysis, namely that an electron moving around asolenoid produces a magnetic field which penetrates into thesolenoid. In the case in which the interior of the solenoid isshielded, the magnetic field of the electron interacts withcurrents in the shielding material. A complete analysis ofthe A.B. effect requires the calculation of all the forcesinvolved between the electrons in the beams outside thesolenoid and the electrons inside the solenoid, as well asall the electrons in the photographic plate where the inter-ference pattern takes place.

A simpler analysis used in the literature consists of calcu-lating the interaction only between the electrons outside andinside the solenoid, which is described by the relations

ddt

ZVb�t��rmbUeb 1 Pbs� dV

� 2Z

Vb�t�7·�Tbs !

2UebPbs� dV � 0

ddt

ZVs�t��rmsUes 1 Psb� dV

� 2Z

Vs�t�7·�Tsb !

2UesPsb� dV ± 0

�142�

where we used the subscript b for the beam electrons and thesubscript s for the solenoid electrons. We also have thedefinitions

Pij � 14pc

�Ei ∧ Bj�

f ij � riEj 11c

Ji ∧ Bj

Tij ! � 1

4p� 1

2 �Ei ·Ej 1 Bi ·Bj�yI 2 �EiEj 1 BiBj��

�143�


Since the electromagnetic field produced by the solenoidelectrons is zero, it follows that the conditions

f bs� 0 Pbs� 0 Tbs ! � 0 �144�

whereas all quantities inside the solenoid are different fromzero. The violation of Newton’s third law is obvious sincewe have f bs� 0 ± 2f sb ± 0. Therefore, owing to theaction of the magnetic field of the outside electrons on theinside electrons, the velocityUes of the electrons insidethe solenoid must change. However, for a good conductor,the current densityJs of the solenoid is given by the relationJs � resUes 1 risUis � resV 1 rUis < resV. Since thecurrent densityJs must remain constant if the electronsoutside the solenoid are to experience precisely zero Lorentzforce, then the relative velocityV � Uis 2 Ues must beconstant. Therefore, the solenoid must move, a point ofview shared by Herman [94].

However, there is a difficulty with the above interpreta-tion if one realizes that the electrons in the beams arediscrete and uncorrelated particles that pass through theinterferometer in a very short time of around 10 ns. There-fore, the mutual magnetic interaction occurs during a shorttime with respect to the longer time interval required tocreate the interference pattern by the arrival of a largenumber of independent electrons. Therefore, one maythink that the shift is induced rather by the mutual interac-tion between the electrons of the solenoid and the electronsin the photographic plate, since the presence of a magneticfield inside the solenoid modifies the topology of space. Thisfact is particularly clear in the Bay’s experiment [96] wherethe A.B. phase shift was demonstrated by fastening the filmto a small electric motor and advancing the film at a rateproportional to the rate of increase of current through thecoil. The film showed a continuous lateral displacement ofthe fringes within the enveloping pattern. However, as soonas the current through the solenoid becomes constant, theshifted pattern is frozen and shifts back to its originalposition when the current is stopped.

10.5. Linear circuit theory

Sinusoidal time variation at a given frequencyv is ofpractical interest in electronic circuit theory, as many ofour sources generate sinusoidal outputs. Even if the outputis not sinusoidal, it can be represented as a summation ofsinusoidal components of different amplitudes, phases andfrequencies. For sinusoidal time variation, it is convenient touse the complex form of Maxwell’s equations:

14p

E·2Ep

2t1 B p ·

2B2t

� �1

c4p

7·�E ∧ Bp� � 2J p ·E

�145�and represent the instantaneous values of the electromag-netic fields as the real parts of the complex exponentialsE�r; t� � E0�r� ejvt and B�r; t� � B0�r� ejvt where E0�r�andB0�r� are also complex vector functions of position.

Integrating the above equation over the volumeV andapplying the divergence theorem, we find

jv4p

ZV�uB0u2 2 uE0u2� dV 1

c4p

ZS�E0 ∧ Bp

0�·dS

� 2Z

VJp

0·E0 dV �146�

If we multiply the preceding equation by12 , we obtain thetime-averaged complex power provided to the fields by thesourceJ0 located insideV. The above Poynting theorem nowmay be separated into real and imaginary parts and written as

v

8p

ZV�uB0u2 2 uE0u2� dV 1

c8p

ImZ

S�E0 ∧ Bp

0�·dS� �

� 2 12 Im

ZV

Jp0·E0 dV

� �

12 Re

c4p

ZS�E0 ∧ Bp

0�·dS� �

� 2 12 Re

ZV

Jp0·E0 dV

� ��147�

The real part of the flux of the time-averaged Poyntingvector through the surfaceS is equal to the rate of energydelivered by the source terms and cannot be zero even if thesurface goes to infinity as already explained.

The imaginary equation represents the difference betweentime-averaged magnetic and electric energies stored withinV. As stated by De Broglie in his book [97] (p. 56) about thephoton, one can question the physical meaning of theimaginary part of the above equation when we work withreal physical sources. No satisfactory explanation is given atthe present time in the literature.

A close relationship between circuit theory and fieldtheory can be found in the above equations. We know thatthe impedance of a series combination of a resistanceR, aninductanceL and a capacitanceC as shown in the Fig. 8 hasthe expression

Z � R1 jv L 21

Cv2

� ��148�

This impedance is a complex quantity which can be writ-ten asZ � uZu(cosu 1 j sin u ) with the definitions

cosu � R

uZusinu � 1

uZuLv 2

1Cv

� ��149�

The resistanceR is a positive quantity by definition; there-fore, it follows that 2 p/2 < u < p/2.

For a complex currentI � I0 ejv t, the complex power is bydefinition the quantity

P� 12 ZuI u2 � Pr 1 jPi �150�

Taking into account the relations in Eq. (149), the activepowerPr� uZu cosu is always positive, whereas the reactivepowerPi � uZu sinu can be either positive or negative. By


definition, we write

P� 12 ZI2

0 � 2 12

ZV

Jp0·E0 dV �151�

If we consider that the imaginary part of the surface inte-gral in Eq. (146) is small in comparison with the other termsto a first approximation, then the classical circuit elementsare given by the relations

12 RI20 � 1

2 Rec

4p

ZS�E0 ∧ Bp

0�·dS� �

12 LI2

0 � 18p

ZV

uB0u2 dV

12

I20

Cv2 �1

8p

ZV

uE0u2 dV

�152�

In the same way, the admittance of a parallel combinationof a conductanceG � 1/R, an inductanceL and a capaci-tanceC as shown in Fig. 9 has the expression

Y � G 1 jv C 21

Lv2

� ��153�

For a complex voltageV� V0 ejv t, we can also define the

circuit elements by the relations

12 GV2

0 � 12 Re

c4p

ZS�E0 ∧ Bp

0�· dS� �

12

V20

Lv2 �1

8p

ZV

uB0u2 dV

12 CV2

0 � 18p

ZV

uE0u2 dV

�154�

knowing that

P� 12 YV2

0 � 2 12

ZV

Jp0·E0 dV �155�

It is the topology of the electromagnetic system that indi-cates which formulation must be chosen for a givenproblem. We can also define the circuit elements from thescalar and vector potentials by using the identities

uE0u2 � 4prp0F0 2 7·�F0Ep

0�2 jv

cEp

0·A0

uB0u2 � 4pc

Jp0·A0 1 7·�A0 ∧ Bp

0�2 jv

cEp

0·A0

�156�

It follows the relation

18p

ZV�uB0u2 2 uE0u2� dV � 1

2

ZV

1c

Jp0·A0 2 rp

0F0

� �dV

11

8p

ZS�A0 ∧ Bp

0 1 F0Ep0�·dS

�157�We note that left-hand side of the preceding equation is areal quantity, whereas the right-hand side is a sum ofcomplex quantities; therefore, the sum of the imaginaryparts must be zero.

For complex source terms of the formr�r; t� � r0�r� ejvt

andJ�r; t� � J0�r� ejvt, the potentialsF0 andA0 are relatedto the charge and current densities of the system through:

F0�r� �Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞e2jkR0

R0r0�r 0� dr 03

A0�r� � 1c

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞e2jkR0

R0J0�r 0� dr 03 �158�

with the definitionsk � v /c andR0 � ur 2 r 0u.After substituting the above potentials into the first term

of the right-hand side of Eq. (157), we obtain

12

ZV

1c

Jp0·A0 2 rp

0F0

� �dV � 1

2

ZV

ZV 0

e2jkR0

R0

� 1c2 Jp

0�r�·J0�r 0�2 rp0�r�·r0�r 0�

� �dV dV 0 �159�

By expanding the right-hand side of Eq. (159) into its real


Fig. 8. Series combination of a resistanceR, an inductanceL and acapacitanceC.

Fig. 9. Parallel combination of a resistanceR, an inductanceL and acapacitanceC.

and imaginary parts, we can define the relations

12 LI2

0 � 12c2

ZV

ZV 0

cos�kR0�R0

Jp0�r�·J0�r 0� dV dV 0

12 CV2

0 � 12

ZV

ZV 0

cos�kR0�R0

rp0�r�·r0�r 0� dV dV 0

12 RI20 � v

2

ZV

ZV 0

× sin�kR0�R0

1c2 Jp

0�r�·J0�r 0�2 rp0�r�·r0�r 0�

� �dV dV 0

�160�where we used the relationI2

0 � uI2u � C2v2V20 which

expresses the current as a function of the voltage at thecapacitor plates.

In the Appendix, we demonstrate that the above integralsare positive quantities. The inductance and the capacitancedefined above do not depend on the current or the voltagebut on the geometrical form of the source. However, unlikethe definitions in Eq. (152), the circuit elements are now afunction of frequency. At very high frequencies, where thecircuit dimensions become appreciable fractions of a wave-length, we must use the above formulas. For circuits that arenot too large in terms of wavelengths,kR0 , 1, we canexpand the cosine cos(kR0) < 1 2 (kR0)

2/2 in series anduse the first and second terms. The inductance and capaci-tance defined in Eqs. (152) and (160) are not equivalentsince the formulas in Eq. (152) are independent of thefrequency, whereas the formulas in Eq. (160) are frequencydependent. Moreover, the formulas in Eq. (160) do not takeinto account the surface integral in the right-hand side of Eq.(157) which is not zero, contrary to Levich’s statement [98](p. 505).

When there are several separate sources in the medium,then the inductance and the capacitance are given by theformulas

12 Lij Ii Ij � 1

2c2

ZV

ZV 0

1R

Jpi �r�·Jj�r 0� dV dV 0

12 Cij ViVj � 1

2

ZV

ZV 0

1R

rpi �r�rj�r 0� dV dV 0

�161�

The properties of the above relations are analyzed in theAppendix. From the above relations, we see that the mutualterms which are supposed to violate the conservation ofenergy cannot be space-averaged to zero as some authorspretend [60, 61, 99] since these terms depend only on thegeometric forms of the sources.

10.6. Antenna radiation

The above relations show that one must take into accountthe sources to answer the question about the compatibility ofthe principles of superposition of fields and the conservationof energy. We will now follow the analysis of this question

by Hoh [100] who starts from the equation

12

Rec

4p

ZS�E0 ∧ Bp

0�·dS� �

� 212

ReZ

VJp

0·E0 dV� �

�162�which expresses the balance between real power flow forone radiating source located inside volumeV.

For several sources, the impedance is now given by theformula

Zij Ipi I j � 2

ZVi

Jpi ·Ej dV �163�

with the definitionsIpi �t� � I0i e2jvt andIj�t� � I0j ejvt.

Carson’s theorem implies the following identity for twosources:

12 Re�I p

1I2Z12� � 12 Re�I p

2I1Z21� �164�

Since at first order the mutual impedances depend on thegeometrical forms of the sources, it follows that these impe-dances are independent of the currents; therefore, we canassume reciprocityZ12� Z21. The real parts ofZii define theso-called radiation resistanceRri.

The time-averaged powers delivered by two sourcesS1

andS2 when each source is excited at a time are given by

P1 � 12 Re�Ip

1I1Z11� � 2 12 Re

ZV1

Jp1·E1 dV

� �

P2 � 12 Re�Ip

2I2Z22� � 2 12 Re

ZV2

Jp2·E2 dV

� � �165�

When both sources are excited simultaneously there is amutual coupling between them which results from the super-position principle. It follows that the time-averaged powersflowing out of the closed surfaces enclosing the sources willcontain more terms since we have

P01 � 1

2 Re�Ip1I1Z11 1 I p

1I2Z12�

� 2 12 Re

ZV1

�Jp1·E1 1 Jp

1·E2� dV� �

P02 � 1

2 Re�Ip2I1Z21 1 I p

2I2Z22�

� 2 12 Re

ZV2

�Jp2·E1 1 Jp

2·E2� dV� �

�166�

Consequently, the total time-averaged power radiated atinfinity by the two sources isPT� P11 1 P22 1 2P12 with thedefinitions

Pij � 2 12 Re

ZV

Jpi ·Ej dV

Pij � 12 Re

c4p

ZS�Ei ∧ Bp

j �·dS


PT � 2 12 Re

ZV�Jp

1 1 Jp2�·�E1 1 E2� dV

PT � 12 Re

c4p

ZS��E1 1 E2� ∧ �Bp

1 1 Bp2��·dS �167�

The equalityP12 � P21 is a consequence of Carson’stheorem. The proper power termsPii are always positivequantities, whereas the mutual power termsPij are quantitieswhich can be either positive or negative. Therefore, thereexist cases where the total power radiated by the sources iszero. This case is best illustrated by the situation of twosimilar antennas that are parallel and closely spaced whichverify the condition Jp

1·E1 1 Jp2·E2 � 2�Jp

1·E2 1 Jp2·E1�.

This condition is satisfied if the current densitiesJ1 �2J2 are equal in magnitude but 1808 out of phase whichimply the equalitiesE1 � 2E2 andZ11� Z22.

We know that an electron traveling in its circular orbitaround the proton in the hydrogen atom must radiate asrequired by classical electromagnetic theory. Since thepoint electron accelerates, it should lose energy and spiralinto the proton within a fraction of a second. The first answerto the problem of atomic collapse was the quantum postulateof Niels Bohr: electrons in stationary orbits do not radiate. Itis only in the transition between stationary states that radia-tion is emitted. In this paper, we can give another answerconcerning this problem and provide a simple generalcriterion using Hoh’s analysis applied to the case of twodifferent continuous distributions of charge which move ina given reference frame. All quantities in the above relationsdepend on the coordinater and the angular frequencyvbecause we have assumed that the particles are continuousdistributions of charge. For two moving particles locatedinsideV, the total radiated powerPT is zero if the followingcondition is satisfiedJp

1·E1 1 Jp2·E2 � 2�Jp

1·E2 1 Jp2·E1�

where Jp1 and Jp

2 are the current densities associated withthe motions of the electron and the proton in a given referenceframe. This implies that both the electron and the proton mustmove with respect to the center of mass. The error in theclassical reasoning was to consider that only the electronmoves with respect to the massive proton at rest in a givenreference frame. One can think that the problem can be splitinto the overall center of mass motion and the relative motion.The latter describes circular motion of a charged particle witha reduced mass, leading to radiation in the usual way and,therefore, there is still an issue concerning the radiationproblem. However, in doing so, physicists calculate theradiation due to the proper field of the electron and neglectthe effect due to the mutual radiation fields and the properfield of the proton which contribute to the calculation of theradiated power when we use a reduced mass.

The fact of taking into account the sources terms does notcancel the mutual power term 2P12 in the expression of thetotal power, which implies the existence of a mutualradiated energy 2E12 � 2P12/v outside the volumeVconfirming the discussion of light interferences. We recall

that a source is generally an antenna which is a metallicconductor with a resistanceRc � 1/s c and a generatorwhich delivers energy to the antenna. The generator is char-acterized either by its current densityJS or by its sourceelectric fieldES. Inside the source (antenna1 generator),the corresponding Ohm’s laws are satisfiedJ � scE 1 JS

or J � sc�E 1 ES�. In that case, the source terms now havefor expression

Pii � 2 12

ZVi

uJi u2Rci dV 1 1

2 ReZ

Vi

Jpi ·JSiRci dV

Pii � 2 12

ZVi

uJi u2Rci dV 1 1

2 ReZ

Vi

Jpi ·ESi dV

�168�

The first terms in the right-hand side of the precedingequations represent the power dissipated to heat the conduc-tors of the antennas and the generators. The second termsaccount for the proper powers delivered by the antennasindependently of the fact they may or may not radiate simul-taneously, especially if the generators are shielded from anyinduced electromagnetic fields. Therefore, the questionconcerning the compatibility of the superposition principleand the conservation of energy must be answered by statingif the difference of energy is delivered by the generators orby the medium. In the first case the principle of conservationof energy is saved; in the second case the principle isviolated, but the energy is provided by the medium andmust be ascribed to the motion of the center of mass ofthe system.

We are now able to comment the answers given by Hoh[100] and Levine [101] concerning this problem. Bothauthors stress that the mutual power 2P12 results from thecoupling between the sources as demonstrated by the equal-ities in Eq. (167). Levine examines the case of two half-wave antennas which are physically parallel and very closetogether. Each antenna is driven by a generator, whereI0 isthe magnitude of the sinusoidal current. The power radiatedinto space by each antenna acting alone isP� I2

0Rr=2 withRr � 73V. If both antennas are driven by in-phase currentsof the same magnitude and radiate simultaneously, thepower P� I2

0Rr =2 is now quadruple owing to the super-position principle, which implies the equalityRr � 292V.This is the reason why the folded dipole antenna, used as themain receiving element of most television antennas, is fedby a two-wire flat cable with 300V impedance. This foldedantenna is equivalent to two in-phase antennas where thesecond current source in the second leg is induced by thecurrent in the first leg.

However, these authors do not answer the question sinceone must make the distinction between the antennas and thegenerators. When Levine compares the two cases, beforeand after the superposition of the fields, he ascribes thedifference of power to a change of the radiation resistanceRr of the medium which implies an automatic change ofthe powers delivered by the generators according to the


formulasP� I20Rr=2 or P� V2

0 =2Rr (we neglectRc and theinternal resistanceRi of the generators which are small quan-tities with respect toRr) and this change occurs without anyexternal help from an observer. This point of view iscorrectly criticized by Hoh who notes that the wave impe-danceRr is a derived quantity that is computed only after theradiated power is determined. Moreover, the radiation resis-tanceRr in antenna theory [102] (p. 46) depends only on thegeometrical form of the antenna and is independent of thecurrent; therefore, the radiation resistance cannot be deter-mined by the medium. To save the conservation of energy,Levine makes the statement that there exists no generatorgiving a constant power, which is an incorrect statementsince, for the pendulum experiment described hereafter,we bought two generators which both delivered DC voltageand current with maximum limited values and, therefore,with a fixed maximum power. If the generators use themaximum power they can provide, one cannot understandhow they deliver more power to account for the difference inradiated power. Hoh does not indicate whether or not thepowerP � V0I0/2 delivered by each generator is the samebefore and after the superposition of the fields takes place:does the e.m.f.V0 or the currentI0 given by each generatorbecome doubled or not? We do not know. Consequently, itis difficult to explain the gain or the loss in radiated power ifthe part played by the medium is not taken into account. Toshow the ambiguity concerning this problem in the liter-ature, let us examine the comparison of the power radiatedby a wave antenna with the power radiated by a folded half-wave antenna as described by Houze´ [103] (p. 36):

In the folded antenna, we recover only half of thecurrentIF � I0/2 for an identical voltageV0…. Withrespect to a half-wave antenna, we get the samepowerP� 4Z0I2

F =2� Z0I20=2 with necessary different

impedances for the same voltageV0.

If we follow Houze, when a wave antenna fed by a givengenerator with a fixed voltage is folded then the currentsuddenly decreases to half of its value to radiate the samepower as before; manifestly some thing is wrong in Houze´’sreasoning. We can make an analogy by comparing the caseof a magnetic fieldBu � 2I0/cr produced by a rectilinearwire fed by a constant currentI0 and the magnetic fieldproduced by the same wire wound as a coil. We know thatthe magnetic field is stronger when the same wire iswounded withN turns per unit length as a solenoid sincewe haveBz < 2pNI0/c.

The same question concerning the compatibility of theprinciples of superposition and energy conservation ariseswhen one wants to calculate the radiated power of theconcerted oscillations of the billions of electrons thataccount for the current in an antenna. For example, for asimple antenna of lengthl, the total radiated power is givenby P� 10v2

0I20l2=c2 [102] (p. 36); therefore, the total

radiated power for a 2l antenna is 4P. When discussingthe double slit experiment, Crawford [99] affirms that the

total average power is 2P. If this statement were true, thenno radiation from antennas would be possible since the totalradiated power according to the Larmor formula woulddepend on the proper quantityNq2, a very small quantityindeed, instead of the mutual quantityN(N 2 1)q2 ascorrectly pointed out by Apsden [104]. The proper radiatedpower depending onNq2 must be associated with the motionof the center of mass of the antenna. This power is small andgenerally not observable since all antennas are stronglyfixed on building walls.

This question has also been examined by Mathews [105]in the case of two initial wave trains propagating in oppositedirections normally incident upon the interface between twodifferent elastic media. By taking into account all energiesof four waves (two reflected and two transmitted whichresult from the scattering of the two incident wave trainsby the interface), Mathews demonstrates that the principlesof superposition and conservation of energy are fullycompatible with one another. However, to prove thecompatibility of these principles, Mathews uses twomediums and affirms without proof that his demonstrationcan be extended to a homogeneous medium since thesurroundings will behave in such a way as to guaranteeenergy conservation. This is not correct; for instance,consider two light pulses emitted simultaneously from twowidely separated stars located far away from Earth whichoverlap later in the Earth’s frame: it would be absurd topretend that the luminosity of the stars will change toaccount for the change in energy when these two pulsesoverlap.

10.7. Radiation reaction and conservation of energy

We know that there are two kinds of radiation: spon-taneous and stimulated radiations. Therefore, two kinds ofelectromagnetic force can be associated with these radia-tions. For a spontaneous radiation, the force satisfiesNewton’s third law and should be called a spontaneousforce, whereas a stimulated force violates Newton’s thirdlaw in the case of a stimulated radiation. Unfortunately,most authors called spontaneous force a force that violatesNewton’s third law and, therefore, a confusion results in theliterature when we deal with the case of spontaneous radia-tion. As already stated, these two kinds of radiation arerelated to the existence of closed and opened radiatingsystems.

For a closed system, there is conservation of energy asshown by Eq. (95). Since the spontaneous radiation emittedby a charge possesses both energy and momentum, thenthere is a transfer of momentum from charge to field andthe charge must recoil because of Newton’s third law. It alsofollows that the spontaneous radiated power is given by theequalityPR� 2 dEK/dt and the radiation must come at theexpense of kinetic energy.

For an open system, there is no longer a conservation ofenergy since the energy is provided by the medium. A


striking example of an open system is the free-electron laser,where an electron beam crosses a magnetic undulator. Sincethe magnetic Lorentz force, the part of the Lorentz forcewhich does not obey Newton’s third law, does not produceany work

ddt

m0gc2 � U·ddt

m0gU � FB·U � 0 �169�

It follows that the kinetic energyEK � (g 2 1)m0c2 is

constant. Therefore, the radiated energy for point-chargeparticles cannot come at the expense of the beam kineticenergy and is provided by the medium.

Now, it is natural to go one step further and to investigatethe possibility of introducing a supplementary forceFR, theso-called radiation-reaction force, in order to obtain aconservation law for an open system. However, thisapproach, contrary to Becker’s statement [80] (tome II, p.21), does not avoid the violation of the conservation ofenergy for stimulated emission, and this is clearly shownin Becker’s equation:

ddt� 1

2 mU2 1 12 f r2� � U·FR �170�

However, the forceFR exerted on the charge cannot becalled a radiation-reaction force contrary to Newburgh’s[106] assertion:

Newton’s third law states that the field must exert aforce distinct from that causing the acceleration andhence the emission. It is this force which is called theradiation-reaction force, the complete description andexplanation of which have remained a problem. Aswe shall see below, the radiation force is a properforce which cannot satisfy Newton’s third law.

Griffiths and Szeto [107] addressed the problem in morecomprehensible terms. They noted that a charged particleaccelerates less than a neutral particle to account for theradiated energy. Therefore, a force called the radiation-reac-tion force must be present in order to avoid violating theprinciple of conservation of energy. They correctly pointedout that this force is attributable to the breakdown ofNewton’s third law in classical electrodynamics, a factrecognized by Lorentz and before him by Thomson. Whenan extended charge accelerates, the force of one part onanother is not equal and opposite to the force of the secondpart on the first. When one integrates over the entire chargeconfiguration, the result is a net force of the charge on itself;therefore, the radiation-reaction force is a self-force orproper force. The authors in their paper developed thebasic theory of the self-force on a dumbbell in longitudinalmotion.

We will review briefly this old problem with a newinsight, since we know that the particle and its radiationdo not constitute a closed system. Therefore, there are twoforces: the first one is the ‘‘mutual’’ Lorentz forceF12� FL

between the particle and another particle which causes the

acceleration and hence the stimulated emission; the secondforce is the proper forceF11� FR or radiation-reaction forceof the particle on itself. It follows the equations

dP1

dt� F12 1 F11

dE1

dt� U1·F12 1 U1·F11 �171�

with the definitionsP1 � m1g1U1 andE1 � m1g1c2. Thereare two ways of calculating the radiation-reaction forcewhich go back to Lorentz and to Abraham. Lorentz’smethod, as described by Becker [80] (tome II, p. 23), usesthe Lorentz forcef 11 � r1�E1 1 U1 ∧ B1=c� without themagnetic force forU1/c , 1; we get

FR � 243

EQ

c2

dUdt

123

q2

c3

d2Udt2

�172�

whereEQ is the electrostatic energy. Abraham’s formula ofthe radiation-reaction force has the expression

FR � 23

q2

c3 g4 K 11c2 U ∧ �U ∧ K�

� ��173�

where the quantityK is defined as

K � d2Udt2

1 3g2

c2 U·dUdt

� �dUdt

�174�

It follows that the work of the reaction force during thetime dt has the valueU·FR � 2q2g4�U·K�=3c3. The abovereaction force can also be written in the form

FR � 23

q2

m20c5

� ddt

EdPdt

� �2

g

m0

dPdt

� �2

21c2

dEdt

� �2" #

P

( )�175�

Thus, we have

U·FR � 23

q2

m20c3

ddt

P·dPdt

� �

223

q2g2

m20c3

dPdt

� �2

21c2

dEdt

� �2" #

�176�

We recognize in the last term in the right-hand side ofEq. (176) the radiated power by the charged particle in theso-called radiation zone, namely:

PR � 23

q2g2

m20c3

dPdt

� �2

21c2

dEdt

� �2" #

�177�

The radiated power is a positive quantity because we havethe identity

PR � 23

q2

c3 g4 dUdt

� �2

1c2 dgdt

� �2" #

� 223

q2g2

m20c3 P·

d2Pdt2

2E

c2

d2Edt2

!�178�


Eqs. (171) and (176) can be used to write the variation of thekinetic energy in the form

dEK

dt2

dEL

dt� dES

dt2

dER

dt�179�

with the following definitions:

the quantity dEK/dt is a positive or negative power termaccounting for the variation of the kinetic energyEK �(g1 2 1)m1c

2 of the charged particle which produces theemission of the radiation;the quantity dEL/dt � U·FL is also a positive or negativeterm which represents the variation of the internal energygiven by the other particle;the quantity dER/dt � PR is always a positive term whichgives the radiated power by the moving particle;the quantity dES/dt � U·FR 1 PR is the so-called Schottterm and may be positive or negative. The energyES isinterpreted as an internal energy bounded to the movingparticle.

Eq. (179) is a conservation law for an opened systemowing to the existence of the reaction force as already stated.Therefore, the radiated energy is provided by the medium.One can easily check the above affirmation in the case wherethe Lorentz force derives from a potentialEL � 2 EP. Then,if ET� EK 1 EP is the total energy in the system given by theobserver at some initial time, after this time we get anincrease in the kinetic energy and correspondingly anincrease of the radiated energy emitted by the moving par-ticle if we neglect the Schott term. Therefore, the total initialenergyET � EK 1 EP is not conserved. One may think thatthe Schott term cannot be ignored so readily in any consid-eration of energy balance based on Eq. (179). However, fora uniformly accelerated particle we getFR � 0 which, inturn, requires dES/dt� PR; in that case the total energyET�EK 1 EP is conserved, but clearly the particle radiates energywhich is provided by the medium through the Schott term.The physical picture provided by the Schott term becomessatisfactory if one recognizes the existence of open systems.

Although the above analysis of the reaction force is moresatisfactory from a physical point of view, it is the introduc-tion of the reaction force which must be criticized. In fact, inthe case of stimulated emission, there is a second particleinvolved in the process that must be taken into account.Therefore, the reaction force is not useful and one mustinstead have recourse to the relations in Eqs. (139) and(140) to obtain the conservation laws concerning stimulatedemission.

Teitelboim [108, 109] reworked the point-charge theoryby introducing a different splitting in the terms of the four-vector relativistic approach of the radiation force. Theseequations are better written

dPT

dt� FL 1 FR

dET

dt� U·FL 2 PR �180�

by adding the Schott terms in the definitions of energy and

momentum as follows:

PT � P 2 PS ET � E 2 ES

PS � 23

q2

m20c5 E

dPdt� 2

3q2g2

c3

dUdt

1g2

c2 U·dUdt

� �U

" #

ES � 23

q2

m20c3 P·

dPdt� 2

3q2g4

c3 U·dUdt

2FR � 1c2 PRU � 2

3q2g4

c5

dUdt

� �2

1g2

c2 U·dUdt

� �2" #

U

2U·FR � PR�1 2 1=g2�

� 23

q2g4b2

c3

dUdt

� �2

1g2

c2 U·dUdt

� �2" #

U·dPS

dt� dES

dt2

PR

g2

�181�This is the grouping of terms first suggested by Teitelboim.

11. Review of several experiments which show theEarth’s motion through the ether

We will now challenge the claim made by the specialrelativity theory concerning the impossibility of detectingour motion through the ether by internal experiments.

11.1. Is special relativity theory a relativity theory?

The result of the Michelson–Morley experiment can beeasily explained in terms of a ballistic model of light, inwhich the speed is uniquely defined with respect to thesource, not with respect to a medium. This result is in perfectagreement with the Galilean relativity and covariance prin-ciples, namely that motions observed within a uniformlymoving inertial reference frame cannot reveal any informa-tion about the velocity with which the whole system istranslated. However, the ballistic theory of light is disqua-lified by many experiments which show that the velocity oflight is completely unaffected by the motion of its source.But, the absence of any fringe shift in the Michelson–Morley experiment is in direct conflict with the wave theoryof light since an effect was expected due to the light depen-dence on the motion of the receiver through the ether. Asdiscussed in Ref. [110], there is a flaw in the experimentsince Michelson–Morley have to adjust the optical pathlengths of their interferometer to an integer number of thewavelengths in order to obtain the fringes. This adjustmentremained constant and, therefore, the initial conditions arenever changed with respect to time. The fringe effectdepends on the intensity of the field with no explicit timedependence in the mathematical formulas if the Earth speedis constant, consequently one cannot be surprised if no


fringe shift has been observed since everything is fixed withrespect to time. This is the reason why, later, Michelson–Gale succeeded with the same interferometer to get a fringeshift related to the rotational velocity of the Earth throughthe ether. In their experiment, they switch with respect totime the trajectory of light between two loops and bycomparing the fringe displacement of the large loop withthat of the small loop, the effect of the Earth’s motionthrough the ether was thus discovered. Moreover, afrequency-locking phenomena [111] in their interferometermay also explain their null result.

As pointed out by Allais [112–116], Miller [117]performed a series of experiments extending over25 years, from 1902 to 1926, which reported non-negativefringe shifts corresponding to velocities of about 8 to9 km s21. These results were interpreted at that time asdue to measurement errors. But Allais showed by a differentanalysis that there is an unexpected coherence in Miller’sdata. Moreover, there is a fundamental difference betweenthe experiments by Michelson and Miller. The Michelson–Morley experiments were done at a given time, whereasMiller’s experiments were done during many days at differ-ent periods of the year. Therefore, it is not surprising thatMiller obtained positive results. If one wishes to make ameasurement of a quantity which is slowly varying withtime, we may either take a long time to make the measure-ment or take a shorter time by varying rapidly the initialconditions. Recently, several experiments with interferom-eters done by Kantor [118], Marinov [119–121] and Silver-tooth [122–124] have obtained positive results concerningthe motion of the Earth through the ether. However, thesepositive results have not been confirmed by other research-ers. As noted by Whitney [125], these positive results are notsurprising if one interprets them not as linear velocitymeasurements but instead as measurements of the rotationvelocity of the Earth with respect to a center of rotationlocated in the Milky Way galaxy.

Therefore, there is no contradiction between the Michel-son–Morley and Michelson–Gale experiments, since bothexperiments must give a null result for constant translationand rotation motions. The failure by Michelson–Morley toobserve a fringe shift is due to a flaw in the experimentwhich has never been noticed by physicists until now.However, the success of the Michelson–Gale experimentdoes not mean that one can ascribe or measure the velocityof the ether. The difference between the cases of the etherand a material medium resides essentially in the fact that onecan isolate and move parts of the medium and by doing soput in evidence the influence of this motion on the wavespeed. If the ether permeates everything including materialparticles, we cannot isolate and move specific parts of it;hence the impossibility of measuring the ether motion.When physicists speak of the ether wind, they use the reci-procity concept and Newton’s third law to a case where theconcept cannot be applied, especially if the ether is somekind of universal external force field. Therefore, the claim of

our textbooks that the Michelson–Morley experimentdisproved the existence of the ether is incorrect. Since thevelocity of a material particle is by definition a relativeconcept, one must choose a material body considered byhypothesis to be at rest in the ether which can be used asa reference frame for velocity measurements. The origin ofthis preferred frame can be the center of mass of all theparticles in the universe. Today the ether is seriouslybeing re-examined by astrophysicists and is characterizedas an energy-rich particulate, subquantic medium and issometimes called the ‘‘neutrino sea’’.

Before reviewing several experiments which refute theclaim by relativist physicists that no internal experimentcan reveal our motion through the ether, we must clarifytwo points.

The first point concerns the definition of the word relative.For example, a velocity is said to be relative because it isreferred to the choice of a reference frame which is notnecessarily an inertial reference frame. This statement isperfectly correct and not ambiguous. However, from thediscussion on the application of Newton’s third law in clas-sical mechanics, we have shown that there are two kinds ofrelative velocity depending on whether or not the magnitudeof the velocity depends on the choice of a reference frame.The distinction between these two kinds of velocity is notmade in special relativity theory. A good example whichproves this affirmation can be given concerning the relativelight speed of two photons which is zero when the twophotons travel in the same direction in a given inertialframe and which becomesc instead of 2c when they travelin opposite directions. It seems that experiments by Pappasand Obolensky [126] report measurements of this relativespeed which is twice the speed of light.

Moreover, this is our second point: we contest the claimmade by relativist physicists who pretend to do the distinc-tion. This claim is perfectly clear when French [30] (p. 65),commenting on Einstein’s work, states:

At the beginning of his wonderful paper in whichspecial relativity was brought into existence, hecomments on the fact that in such phenomena asthe mutual interaction of a magnet and a conductor,it is only therelative motion that matters, and not theseparate motion of either.

Later, French insists by saying:

But equally impressive was Einstein’s conviction thatall observable physical phenomena must depend onlyon relative motions.

Unfortunately, these statements are wrong because themagnitude of the velocity and also all the other quantitiesdo depend on the choice of an inertial frame. Therefore, onecannot pretend that Maxwell’s equations and the Lorentz’sforce are formulated in terms of the relative positions,


velocities and accelerations of the particles in a given refer-ence frame as correctly stated by Wesley [127] (p. 291).

Bartocci and Capria asserted in a paper published in 1991[128] that Maxwell’s theory gives for the force exerted on astationary charge by a translating magnetic dipole a valueonly half as large as the value that the theory gives for theforce experienced by the charge moving relative to the samestationary magnetic dipole. Thus, the authors have actuallyasserted that Maxwell’s theory is incompatible with Gali-lean relativity. Soon after the publication of this paper, Jefi-menko submitted an article [129] explaining the errors ofBartocci and Capria’s calculations (I am not sure of theseerrors because there are some assumptions in Jefimenko’spaper which can be challenged). He demonstrated that theforce exerted by the magnetic dipole on the stationarycharge isF12 � 2qU ∧ B=c which is opposite to the forceF21 � qU ∧ B=c experienced by the charge when it ismoving and the magnetic dipole stationary. Moreover, Jefi-menko made in his paper the following statement:

Observe that the two forces are different regardless ofhow small the velocity of the charge is. This is acurious result since it conflicts with the most funda-mental principle of classical physics: the principle ofGalilean relativity according to which themagnitudeof the two forces should be the same.

However, Jefimenko’s statement is not correct since theforces must also have the same direction (see Eq. (38)),which is not the case. In fact, Jefimenko’s calculation provesthe correctness of our viewpoint concerning the fact that thereciprocity in the change of reference frames does implyNewton’s third lawF12 � 2 F21. One can also criticizeJefimenko’s calculation by noting that he did not prove thathis calculation applies when both the charge and themagnetic dipole are in relative motion. However, we canaffirm that it is impossible to reconcile the Lorentz approachand the Weber approach as shown in this paper.

In fact, Weber’s electrodynamics as reviewed in Assis’book [58] is the only relativity theory. In contrast, specialrelativity theory with its infinite set of inertial frames and itsvariable quantities is almost in agreement with the existenceof the ether, a point of view defended by Builder [56].However, the viewpoint sustained in Newtonian mechanicsis totally different since one associates an infinite set ofmaterial reference frames located at the centers of mass ofthe different material systems considered in the spirit of theJacobi coordinates introduced at the beginning of this paper.The difference between the quantities can be attributed todifferent choices of the centers of mass.

11.2. Doppler and aberration effects

The stellar aberration has been discussed by Hayden[130]. By reviewing Einstein’s explanations of aberration,Hayden reached the important conclusion:

The Lorentz transformation equations of the specialrelativity theory assert that there is an aberrationalangle (Uorb 2 Ustar)/c radians due to the relativevelocity of star and Earth.

Of course, Hayden challenged this conclusion since theexperiments show that stellar aberration does not dependat all on star velocity but is only due to the Earth’s orbitalvelocity. An argument frequently used to explain stellaraberration in special relativity theory is simply to note thatthe measurements of stellar aberration depend upon obser-vations taken at different times in the Earth’s orbit aroundthe Sun. Therefore, the analysis of the effect ought toinvolve only the difference between earth velocities atthese different times. Hayden [131] and Whitney [132]pointed out the fact that this argument simply disregardsthe inconsistency between stellar aberration and Dopplershift which depends on star velocity. In fact, both aberrationand Doppler effect in special relativity theory are derivedtogether as a common consequence of the Lorentz transfor-mation applied to a four-vector.

Moreover, many physicists, like French [30] (p. 134),state that for light in vacuum there is no distinction concern-ing the Doppler effect between motion of the source or theobserver that moves with respect to the vacuum. This state-ment implies that the Doppler shift depends only on therelative velocity between the Earth and the star. As notedin Refs. [29, 110], this affirmation is refuted by experimentalfacts. Indeed, one can make the distinction between themotions of a star light source and the Earth-fixed observerwith respect to the ether as indicated by Born [133] (p. 122),who clearly states that the Doppler effect due to the emissionof the stars moving in the ether does not coincide exactlywith the corresponding lines on the Earth but shows smalldisplacements due to the Earth motion around the sun. Herealso there is no reciprocity between source and receivermotions. A reciprocity would imply a link between thesource and the receiver and the verification of Newton’sthird law.

11.3. Sagnac effect, Allan’s experiment, anisotropy in theblackbody radiation

Post [134] and Anderson [135] have given a detailedanalysis of the Sagnac effect in their papers. More recently,Hayden [136] reviewed the Sagnac effect in the context ofan analysis of the isotropy of light in the frame of therotating Earth. Consider a source and an observer locatedon a disk which rotate with a velocityU�VRand two lightsignals issued from this source constrained to propagate inthe ether with a velocityc in two opposite directions aroundthe disk. The time taken by the two light signals to reach theobserver will be differentt^� 2p(R^ DL)/c and, therefore,the observer will notice a time differenceDt � 4pDL/cbetween the arrivals of the two light pulses. Since the obser-ver has moved 2pDL � Ut during the timet � 2pR/c taken


by the light to make a turn, we have

Dt � t1 2 t2 � 2Utc� 4p

VR2

c2 �182�

Knowing thatS� pR2 is the surface encompassed by thetwo light rays, the fringe shift in an interferometer is givenby the formula

DN � cDtl� 4

VSlc

�183�

The expected fringe shift was observed and Sagnacconcluded that he had proved the existence of the ether.Later, Michelson and Gale performed a Sagnac interferenceexperiment which successfully measured the Earth’srotation. The Sagnac effect is sometimes interpreted as aneffect of acceleration which allows us to measure ourabsolute motion through the ether and as such does notcontradict the null result obtained in the Michelson–Morleyexperiment for the rectilinear propagation of light in aninertial frame. But light speed cannot be relative or absolutedepending whether it propagates along a rectilinear or acircular path in the same medium. Moreover, this depen-dency cannot be related to the concept of an inertial framesince the mass of a moving ring laser gyroscope is finite. InNewtonian mechanics, there is no such thing as an inertialframe where the light speed is defined. We have only refer-ence frames where both velocity and acceleration can berelative or absolute depending on Newton’s third law. Aspointed out by Winterberg [137], the outcome of the Sagnacexperiment was used by Sagnac as a decisive argumentagainst Einstein’s claim that physics could do without theether hypothesis. Since the special theory of relativity deniesthe existence of an ether, a Sagnac effect is impossible inthis theory because the relevant conditions are the same for aco-moving receiver and for a receiver at rest in the labora-tory frame. In order to obviate the above objection, therelativist physicists remark that the co-moving frames canbe considered as successive inertial frames attached to therotating circular light path with a synchronizing discrepancygiven by the Lorentz time transformation. This is an ad hocprocedure to save the special theory of relativity which canbe challenged, since there is no procedure of synchronizationand no physical change of frame in the Sagnac experiment.

As discussed in Ref. [110], the Sagnac effect can beapplied to the motion of clocks or to the propagation ofelectromagnetic signals emitted by satellites towards severalground stations as in the microwave experiment of Allan etal. [138]. The discontinuity in time observed in the Allan etal. experiment is not a consequence of the relativity ofsimultaneity since the observers located in the groundstations are all in the same reference frame and do notmake any change of reference frame. It is quite commonin special relativity theory to invoke a hypothetical observerat rest in some hypothetical inertial frame to explain the timediscontinuity. The effect is simply a classical Sagnac effectdue to the rotation of the Earth in the ether.

The discovery of the 2.7 K cosmic black body radiationby Penzias and Wilson [139] is presented by most astro-physicists as the strongest evidence in favor of the Big-Bang theory since this microwave radiation is assumed tohave been emitted shortly after the Big-Bang. Within theaccuracy of the first measurements this radiation appeared tobe isotropic. The isotropy of the radiation indicates that theuniverse is isotropic and homogeneous on a large scale. Thisradiation is a background in front of which all astrophysicalobjects lie. Therefore, one can expects anisotropy due to themotion of the Earth with respect to the ancient matter whichemitted the radiation. This cosmic radiation can be definedas a privileged frame determined by the rest frame for whichlight is isotropic in all directions. This is a natural conse-quence of the existence of the ether provided that light speedis independent from the motion of the source. This inde-pendence is well grounded as proved by the review paperby Fox [140] on this subject. However, an anisotropy oflight speed will result with respect to the motion of anydetector trough the ether if the light speed depends on thevelocity of the receiver. Forb � U/c , 1, the temperaturemeasured by a moving observer is given by

TO < TS�1 2 b cosu� �184�whereu is the angle between the direction of motion and thedirection of measurement. The cosine anisotropy is readilyinterpreted as being caused by the motion of the Earth relativeto the rest frame of the blackbody radiation. The^3.5 mKanisotropy measurement as measured by Smoot et al. [141]corresponds to an Earth velocity of about 400 km s21 in thedirection towards the constellation Leo. This experiment fitsthe results obtained by Marinov [119, 120].

12. Review of experiments on the motion of conductorsfed by direct high current or voltage

12.1. Experiments by Graneau, Phipps and Saumont

We will now examine the question concerning the appli-cability of Newton’s third law to the interaction of currentelements in the earliest days of classical electrodynamics.Historically, the first quantitative law of action between twocurrent elements was propounded by Ampe`re and validatedby him through a remarkable combination of experimentsand theory.

Rather than considering line-current elements, one canwrite the differential force proposed by Ampe`re in termsof volume-current elementsJ1 dV1 andJ2 dV2:

d2FA12 � nc2R2 �3�J1·n��J2·n�2 2�J1·J2�� dV1 dV2 �185�

with the definitionsR12 � r1 2 r2 andn � R12/R for R �uR12u. The differential force d2FA12 is the force exerted onany volume elementdV1 due to a second volume elementdV2 which serves as the origin of the vectorn. To get


this formula, Ampe`re had to assume the application ofNewton’s third law to individual current elements.

Eq. (185) can be rewritten in a more symmetric form asfollows:

d2FA12 � 1c

��J2·72� dA1 2 �J1·71� dA2

11c�J1·J2�71

1R

� ��dV1 dV2 �186�

where the vector potentials dA1 and dA2 are given by therelations

dA1 � 12cR

�J1 1 �J1·n�n� dA2 � 12cR

�J2 1 �J2·n�n��187�

which satisfy the Coulomb gauge.In contrast, the Biot–Savart force, which is the magnetic

term of the Lorentz force, does not satisfy Newton’s thirdlaw since it is given by the formulas

d2FL12 � 1c2R2 J1 ∧ �J2 ∧ n� dV1 dV2

d2FL21 � 21

c2R2 J2 ∧ �J1 ∧ n� dV1 dV2

�188�

The Biot–Savart expressions can be rewritten in a form firstobtained by Grassmann:

d2FL12 � 1c2R2 ��J1·n�J2 2 �J1·J2�n� dV1 dV2

d2FL21 � 21

c2R2 ��J2·n�J1 2 �J1·J2�n� dV1 dV2

�189�

We note that the last terms in the preceding equationsalways satisfy Newton’s third law, whereas the first termsdo not. Since current elements are not supposed to exist butare part of complete circuits, one must integrate the Ampe`reand the Lorentz force laws over the entire current distribu-tions. It appears that the contribution of the first terms inEqs. (186) and (189) add up to zero. Therefore, the Ampe`reand the Lorentz force laws invariably predict the same netreaction forces between two closed circuits and it is always arepulsion or attraction in compliance with Newton’s thirdlaw.

To these magnetic force laws one must add the electricCoulomb law as expressed for charge elements:

d2FE12� 1R2 r1r2n dV1 dV2 �190�

The Biot–Savart and the Ampe`re force law give identicalresults when used to calculate the interaction or mutual forcebetween complete circuits provided these circuits are sep-arate. However, the two force laws predict different internalstresses in a metallic conductor. The Ampe`re law gives alongitudinal repulsive force between proximate elements ofthe same conductor contrary to the Lorentz law. The result-ing uncertainty has led Ternan [142] and Christodoulides

[143] to claim that the two laws lead to different distribu-tions of force within a circuit because of the divergence ofthe integrals. It has been shown by Graneau [144] andCornille [28] that the mathematical difficulty arising fromthe non-physical assumption of a single filamentary currentcircuit does not disappear when the line-current elementsare replaced by volume-current elements with finite currentdensities.

The reader is referred to the Graneau and Graneau book[15] and the experiments by Phipps and coworker [145–147] and Saumont [25, 26] concerning the experimentalevidence of longitudinal forces in metallic conductors.One must point out that if a closed circuit contains a movingpart then Newton’s third law implies that another part of thiscircuit must move in the opposite direction since theAmpere law implies

dP1

dt1

dP2

dt� FA12 1 FA21 � 0 �191�

This fact is demonstrated clearly in the MIT version of theAmpere hairpin experiment described in Ref. [15] (p. 63)where the observation of the jets in the mercury confirms theexistence of longitudinal electrodynamics forces for anexperiment which was done for the first time 173 yearsago. Therefore, the Ampe`re force law implies that the centerof mass of a closed circuit is at rest in the laboratory frame.Pappas [148] and Graneau and Graneau [149] have carriedout electromagnetic pendulum experiments which are varia-tions of Ampere’s hairpin experiment. They challenged thefield energy–momentum concept of special relativity theoryby stating that the opposite momentum in their experimentsshould be taken by the emitted electromagnetic radiation.Hatzikonstantinou and Moyssides [150] correctly refutetheir claims by noting that the recoiling momentum of themoving part of the pendulum cannot be imparted to theelectromagnetic field as radiation, since the radiated energyemitted by the moving part of the pendulum is a negligiblequantity, but must go to the fixed part of the pendulum. Infact, by using the Lorentz force law for the two parts of thependulum, we can explain the way by which the reactionforce of the moving part of the pendulum is transferred tothe stationary part of the pendulum by writing the relation

dP1

dt1

dP2

dt� 2

ddt

q1

cA2

� �2

ddt

q2

cA1

� �< 0 �192�

Generalizations of the above force laws were given byWhittaker [151, 152] by adding terms to the Ampe`re forcelaw:

d2FW12=dV1 dV2 � nc2R2 �3�J1·n��J2·n�2 2�J1·J2��

1 f1�J1·n�J2 1 f2�J2·n�J1

1 f3�J1·J2�n 1 f4�J1·n��J2·n�n �193�where all quantitiesf i are arbitrary functions ofR.However, one can demonstrate that all these functions


must be of the formf i(R) � a/Rn wherea is a constant.Later, Warburton [153] proposed another expression whichincludes relative acceleration terms. More recently, Munier[154] derived a general force law that satisfies the symmetrygroup properties of space. The most general expression forthe interaction between volume current distributions accord-ing to Munier is

d2FM12=dV1 dV2 � f1�J1·n��J2·n�n 1 f2�J1·J2�n1 f3��J1·n�J2 1 �J2·n�J1�1 f4�J1 ∧ J2�1 f5��n ∧ J1� ∧ �n ∧ J2��

�194�We can make a list of several magnetic force laws with

their integrated formulation as given in the literature.Ampere force law

d2FA12 � nc2R2 �3�J1·n��J2·n�2 2�J1·J2�� dV1 dV2

dFA12 � 1c�J1 ∧ 7 ∧ A2 2 �7·A2�J1 1 J1·77C2� dV1

�195�with the definitions

A2�r1� � 1c

ZV2

J2

RdV2 C2�r1� � 1

c

ZV2

J2·RR

dV2

�196�Whittaker force law

d2FW12 � 1c2R2 ��J1·n�J2 1 �J2·n�J1 2 �J1·J2�n� dV1 dV2

dFW12 � 1c�J1 ∧ 7 ∧ A2 2 �7·A2�J1� dV1

�197�Biot–Savart force law

d2FL12 � 1c2R2 ��J1·n�J2 2 �J1·J2�n� dV1 dV2

dFL12 � 1c

J1 ∧ 7 ∧ A2 dV1

�198�

Marinov force law

d2FM12 � 12c2R2 ��J1·n�J2 1 �J2·n�J1 2 2�J1·J2�n� dV1 dV2

dFM12 � 1c�J1 ∧ 7 ∧ A2 2 1

2 �7·A2�J1 2 12 �J1·7�A2� dV1

�199�In Aspden’s books [155, 156], one can find a discussion

concerning the formulation of the interaction between twomaterial particles from the point of view of Newtonian prin-ciples. By assuming the non-existence of a stimulatedcouple, Aspden was the first physicist to present a force

law which depends on the masses of the particles:

d2FAD12 � 1c2R2

��J1·n�J2

2m1

m2�J2·n�J1 2 �J1·J2�n

�dV1 dV2 �200�

However, following our discussion on the interactionbetween two particles in Newtonian mechanics, it is betterto define the force law from Eq. (4), as follows:

d2FC12� d2F12 11

m1 1 m2�m2 d2F11 2 m1 d2F22� �201�

where we have included the electrostatic force in thedefinitions:

d2F12 � 1R2 r1r2 2

1c2 �J1·J2�

� �n dV1 dV2

d2F11 � 1c2R2 �J1·n�J2 dV1 dV2

d2F22 � 1c2R2 �J2·n�J1 dV1 dV2

�202�

For two identical particles, we get the following inte-grated force law:

dFC12=dV1 � 2r17F2 11c�J1 ∧ 7 ∧ A2 1 1

2 �7·A2�J1

1 12 �J1·7�A2�

�203�with the definition

F2�r1� �Z

V2

r2

RdV2 �204�

The inclusion of mass quantities in the expression of theabove force laws implies the existence of two kinds of forcelaw to describe the interaction between particles. In the bookby Graneau and Graneau [15] (p. 143), one can find thefollowing comment:

There seems to exist no rational grounds for the claimof conventional electromagnetic field theory that oneand the same force law should apply to both the flowof electric current in a wire and an electron beam in acathode ray tube.

We subscribe to such a statement.Recently, Phipps [157] conducted an ingenious exper-

iment in order to show that the masses of the particleshave an effect in the measurement of forces exerted betweentwo closed electrical circuits. Phipps makes the reasonableassumption that the physically effective force exerted on atest element of massm1(s1) by an external element of massm2(s2) can be represented as the product of the force and aninertial factora�s1; s2� � m2=�m1 1 m2� wheres1 ands2 arelength parameters measured along continuous electricalloops. In that case, the contributions of the first terms in


Eqs. (186) and (189) do not add up to zero in a closed loopintegral because of the presence of the factora (s1, s2).Phipps succeeded in showing that a closed external currentloop does exert a non-zero longitudinal force on a testelementm1(s1) by configuring his external current loop insuch a way that a partm2(s2) of his external current loopcould recoila (s1, s2) < 0 or nota (s1, s2) < 1. He demon-strated that the longitudinal force tends to disappear whenthe weak link in the external circuit is stiffened. We mustpoint out that Phipps did his analysis in the center of massframe where his two electrical circuits are at rest. However,the same analysis can be used with the Lorentz force as donein Eq. (201); therefore, it is difficult to discriminate in thePhipps’s experiment as to what part of the motion can beattributed to the recoil effect or to the motion of the center ofmass of the two electrical circuits.

We must point out the fact that all the force laws given inthe literature differ only by terms which either satisfy or donot satisfy Newton’s third law for translations and rotations.This fact has nothing to do with either a change of referenceframe or a problem of simultaneity. As already discussedpreviously, it is the covariance principle which is the sourceof the difficulties encountered in the application of the forcelaw for particles in motion even if Newton’s third law issatisfied. This problem can be illustrated from a simpleexample taken from the literature. Consider two chargedparticles moving initially in the same direction with thesame velocityU; then the Biot–Savart force law satisfiesNewton’s third law since we have

FL12 � 2q1q2U2

c2

RR3 �205�

By adding the electrostatic force, the Lorentz force isgiven by the formula

FL12 � q1q2 1 2U2

c2

!RR3 �206�

This force satisfies Newton’s third law and has the samevalue in the laboratory frame or in the rest frame of thecharges. In special relativity theory, physicists such asPanofsky and Phillips [41] (p. 293) and Ougarov [158]

(p. 166) use the Lorentz transformation between the refer-ence frame where the two charges are at restU0� 0 and thelaboratory frame where the velocitiesU of the two movingparticles are defined. In the rest reference frame, the chargesare submitted to a Coulomb forceF0 � q1q2R/R3 satisfyingthe conditionU·F0 � 0; therefore, by using the Lorentztransformation given by Eq. (45), the force in the laboratoryframe becomes

FL12 � q1q2 1 2U2

c2

!1=2RR3 < q1q2 1 2

12

U2

c2

!RR3

�207�We note that the two force laws do not give the same

numerical results in the laboratory frame. It is not correctto assume, as the authors did, that both charges are at rest ifthey are in mutual interaction only through Coulomb forces.Moreover, from experiments on parallel conducting wirescarrying currents, we know that only a magnetic force ofattraction or repulsion exists which is independent of theposition of the observer. It is absurd to pretend that themutual magnetic force will disappear for an observerlocated on a wire. Of course, we know that the total electro-static force between electrons and ions in a wire is assumedto be zero.

12.2. Experiments by Faraday, Ducretet, Page´s, Brown,Saxl, Allais and Graham

It seems that the first experiment described in the litera-ture concerning discs charged with a high voltage was donein Faraday’s time around 1870, where a mica disc movingon a point, as shown in Fig. 10, takes a rapid rotation whenconnected with a Wimshurst machine. This fact is reportedin a communication presented by Ducretet to The Academyof Sciences around 1898 concerning a similar experiment.We give below the text in French published in Page´s’ book[159] (p. 87):

Nous avons e´galement vu dans la maison Ducretet unancien appareil oublie´ depuis longtemps et qui me´rited’etre remis en honneur. C’est, comme on le voit, undisque de mica qui est mobile sur une pointe et quiprend un mouvement de rotation tre`s rapide lorsqu’onle presente a` une machine e´lectrique tres puissantecomme les machines de Wimshurst. La rotation estalors si energetique que la pesanteur paraıˆt supprimeepar la force centrifuge, quoique celle-ci ne sembledevoir donner que des composantes horizontales, etle disque s’envole…

Ne pourrait-on pas invoquer cette expe´rience pourrendre compte du retard a` la chute des plans horizon-taux que Monsieur Langley a constate´ lorsque le planest lance´ dans l’air avec une vitesse horizontale plusou moins conside´rable?


Fig. 10. Rotating mica disk polarized with a high voltage.

La marche du fluide e´lectrique est facile a`comprendre, l’effluve sort de la machine et se preći-pite vers la pointe. Il monte le long de la premie`recolonne verticale, passe de la` sur le disque, puis il sortdu disque par la second colonne verticale, et de la` serend ala terre; mais le pheńomene de la rotation estbeaucoup plus difficile a` expliquer.

J’ai vu le disque tourner, pour la premie`re fois aLondres, quelques temps apre`s le coup d’e´tat, lorsqueje suivais les cours de Faraday. Quelques temps apre`setre revenu d’exil, Ruhmkorff me montra de nouveaul’experience, et nous discutaˆmes sur les causes duphenomene que Faraday n’avait pas indiqueés, maisni l’un ni l’autre nous ne puˆmes arriver a` une ideeacceptable.

Cette circonstance me revint a` l’esprit vingt anneésplus tard, lorsque j’imaginai d’employer un disque enfer qui ne tourne pas avec une rapidite´ moins grande,et que l’on met en mouvement d’une foule demanieres differentes, comme nous aurons l’occasionde l’expliquer plus en long. Alors je dećouvris pour lemouvement du disque de fer une explication, qui jecrois est la bonne, et que j’ai l’espoir de voir accepterpar la science officielle. Je me re´serve de voir si ellene s’applique pas par hasarde au disque de mica,mutatis mutandis.

Le mouvement du disque de fer produit par l’e´lectro-magnetisme a de´ja ete employedans l’industrie sousla forme que j’ai imagineé et par les proce´des que j’aiindiques. Des modifications plus ou moins heureuses,ont permis d’en e´tendre conside´rablement l’usage etnous pensons qu’il est loin d’avoir dit son dernier motdans la grande question du transport de la force a`distance.

Quel est l’inventeur du disque en mica, qui me paraıˆtun complement obligatoire de toute machine e´lectri-que qui se respecte, a` un moment ou` il est tant ques-tion de champs magne´tiques tournants et de rotationsdirectes auxquelles, par une se´rie de circonstancesbizarres, il a indirectement donne´ naissance?

Monsieur Ducretet qui a construit le mode`le que nouspresentons m’a appris que Ruhmkorff pre´tendaitl’avoir invente, et que l’invention lui e´tait disputeepar l’AbbeLaborde; mais la description inse´ree dansLes Mondes N823 ne date que de 1870, a` une datebien poste´rieure al’experience alaquelle j’ai assiste´.

Reste la question de la priorite´ de Faraday que jereserve.

Ce qui est certain, c’est qu’un disque analogue setrouve decrit sous le nom de tourniquet de Franklina la page 271 du traite´ de Sigaud de la Fond, mais cedisque est pourvu d’une bande d’e´tain qui n’existepas dans la machine dont nous parlons. Place´ entreles deux boules d’une machine de Wimshurst ou deHoltz, le disque de Franklin prend une vitesse tre`sgrande, sans que l’on ait besoin d’employer depointes.

Cette expe´rience oublieé pendant plus d’un sie`cle, estevidemment analogue aux deux autres et leur sert depreface.

The mica disk experiment was also done by Ruhmkorff asquoted above in the French text and described in 1876 byMascart [160]. This experiment was also studied in Jefimen-ko’s book [161] on electrostatic motors. Below the disk, asshown in Fig. 10, there are two vertical corona-producingneedles mounted on a hard rubber base. One of the needles isconnected to earth while the other is connected to a long,stiff, horizontal wire terminating in a sharp point. To set thedisk in rotation, a high voltage terminal is brought intoproximity with the sharp point of the horizontal wire. It iscorrect to state that, by a corona discharge, one needlesprays charges onto the disk while the other one dischargesthem to the ground. However, these corona discharges areperpendicular to the disk; therefore, the rotation cannot beattributed to the electrostatic forces which are also perpen-dicular to the disk. In fact, this charging process produces asmall current and a polarization of matter necessary toinduce a stimulated rotation of the mica disk. As discussedbelow, the corona effect cannot be the direct cause of anystimulated motion.

Pages, who was a physician by education, reproducedsuch an experiment with similar results in 1921. He alsoquotes an experiment with a capacitor [159] (p. 56) whichshows a 5 g weight decrease for an applied voltage 200 kV.Taking account Eq. (208), one obtains a forceFG � 4.9 ×103 dynes which involves 5.5× 1018 electrons in the calcu-lation of this force. From this date up to 1960, Page´s didmany experiments with disks charged with high voltageswhich led him to the theory of the electromagnetic Magnuseffect as described in his book.

In 1923, Biefield, a physicist at the California Institute forAdvanced Studies, discovered that a heavily charged elec-trical capacitor moved towards its positive pole. He assignedBrown to study the effect as a research project. AlthoughBrown carefully conducted experiments for 30 years withcharged bodies in air, oil and in a high vacuum, he wasnevertheless unable to have the results published in the


scientific media of America. The author of this article hasviewed a movie taken in Brown’s laboratory which doesshow these effects. We have now to rely on secondarysources, such as the books by Schaffranke [162], LaViolette[163], Valone [164], the report made for the Air Force byCravens [165] and the patents taken by Brown to get apartial report of his experimental achievements.

The results have usually been discounted because theywere attributed to ion wind [166] and corona discharge.The criticisms formulated concerning the results of theseexperiments can be easily refuted because the ion windeffect is too small as we shall show later. Moreover,Brown performed experiments in a high vacuum andobserved that the effect remained, as explained inLaViolette’s book [163]:

The earlier misconception that Brown’s discs werepowered by an ion wind was finally cleared up towardthe end of 1955. During laboratory experimentsconducted in Paris under the auspices of a Frenchaeronautics corporation, Brown was able to observehow his devices performed under vacuum conditions.He attached two aluminum plate gravitors to each endof a rotor arm so that their combined thrust wouldrotate the assembly somewhat like a fireworkspinwheel. He placed this device inside a vacuumchamber and brought the air pressure down to lessthan one billionth of an atmosphere. This dropped theion wind contribution to a negligibly small value.But, rather than slowing down, as skeptics hadexpected, his rotor sped up. His gravitor apparentlyoperated far more efficiently in a vacuum. In fact,when he increased the voltage to 200 kV, the rotor’s

speed began to increase unchecked, reaching such ahigh rate of rotation that they had to reduce thevoltage to keep the rotor from flying apart.

In fact, we will show that the stimulated force is propor-tional to the current flowing through the conductor. There-fore, the effect of the corona discharge in air is just toproduce the necessary current inside the conducting metalto show the effect; one can use instead a capacitor with adielectric with a high leakage current. Moreover, therotation also can produce a lag effect between the motionof the ions and the electrons. There is one more factinvolved in these experiments, namely only a non-uniformelectric field can produce a current. But, non-uniformelectric fields can also set uncharged bodies in motion, afact which has been recognized only in 1960 by Pohl[167]. One has also to take into account electrostatic forcesproduced by the environment. Brown made specific experi-ments in order to eliminate any electrostatic effect, whichconfirmed the existence of the force. The maximum effectwas observed in 1928 for a body weighing approximately10 kg charged at 150 kV which results in a 105 dynes thrust.Several experiments over many years were done by Brownat different laboratories throughout the world with thefollowing findings:

a charged body is found to be acted on equally well inevery direction;the acceleration effect exhibits a forward thrust towardsthe positive pole of the charged body, as demonstratedabove;the effect seems to be more pronounced when thesymmetry axis of the affecting body and the chargedistribution coincide;the higher the voltage, the greater was the force observed.Brown claimed that the force can rise exponentially withincreasing voltage.

One of Brown’s experiment described in his US patent5 949 550 has been successfully duplicated recently byDeavenport [168].

A similar effect has recently been reported by Deyo [169],who used a standard, classroom-style Van de Graff gener-ator to produce the high voltage, which is about 250 kV, onthe top of the copper cone of the high tension terminal; seeFig. 11. The surface of the copper cone had been coated toreduce any premature coronal discharging before thecharges crowded to the focal point where the small ball isplaced. A coronal discharge composed of a thin, blue, flameabout 2.5 cm in height with an inverted funnel of lavenderhaze above the flame. A vigorous buzzing sound wasemanating from the coronal discharge. A small paper cylin-der about 5 cm high and 6 mm in diameter is released abovethe cone, then let us quote Deyo:

For a split second it just floated above the cone [Fig.11] and then it accelerated into a spin of about


Fig. 11. Copper cone charged with a 250 kV high voltage with aspinning paper cylinder above it.

400 rpm about its vertical axis! After it had reachedits top speed, it began a secondary oscillation aboutits centre of mass. The loci of the gyrating cylinderformed a surface similar to an hourglass. In thedarkened room, the ends of the cylinder were givingoff a lavender glow which sprayed behind the issuingedge.

We must point out that the coronal wind cannot produceby itself the two stimulated rotations observed here. As wewill show hereafter with the examination of the Trouton–Noble experiment, the stimulated rotations result from theviolation of Newton’s third law for rotation. Therefore, thecoronal wind is necessary only to polarize the neutral matterof the cylinder in order to produce the rotation effect.

Saxl, a post-doctoral student with Einstein, made thou-sands of careful observations and records for more than10 years with electrically charged torque pendulums. Saxl[170] shows that the voltage versus the pendulum periodfollows a square law, as expected from the above theory.Unusual variations of the pendulum period were noted parti-cularly during solar and lunar eclipses [171]. A retrospectivediscussion concerning the experimental results of Saxl canbe found in the review by Maccabee [172]. We can alsoquote the pendulum experiments by Allais, the FrenchNobel Prize winner in Economics, who is a physicist byeducation. A complete report concerning the researchwork by Allais can now be found in his recently publishedbook [116].

All the authors quoted above explain their experimentalresults by invoking some physical phenomenon whichcancels or modifies the gravitation field. The idea that thegravitational and electromagnetic fields might be induc-tively coupled is not new, since it was first proposed byFaraday. The reader interested by the subject can consultthe experimental work of Woodward [173, 174]. However,we refute their interpretations concerning the existence ofsuch an effect. On the contrary, the existence of externalforces due to the violation of Newton’s third law showsthat classical electromagnetic forces can partially counter-balance the existing gravitational force.

Ball lightning is another phenomena which has a connec-tion with the subject of stimulated forces. Several eye-witness reports refer to the direct developments of long-lived fireballs in air. Ball lightning has been the subject ofinvestigations in science since the early nineteenth century[175]. The nature and origins of ball lightning remain asubject of controversy [175–178]. Ball lightning is consid-ered as a spherical standing wave of electromagnetic radia-tions trapped in a plasma shell [176]. Recently, Ohtsuki andOfuruton [178] reported the production of plasma fireballsin a natural atmosphere by microwave interference. Balllightning is an interesting phenomena for two reasons: therelatively long lifetime poses a major problem for anytheory dealing with the existence of a stable soliton objectand their stimulated motion cannot be easily explained.

Although several witnesses report that fireballs move inan apparently capricious manner, certain general featureshave been observed. Fireballs created in the upperatmosphere have relatively high velocities and traveltowards the Earth’s surface in a near vertical direction,whereas fireballs which occur close to the Earth’s surfacetend to have low velocities. The question of whether themotion of fireballs is guided by local air motion is still acontroversial subject. However, there are some reports thatfireballs did not follow the wind direction and took an inde-pendent path. This independence is a problem for severaltheoretical models where the fireballs are supposed to beguided by external fields. The independence of motion isnot a problem if we consider that fireballs are self-propelledby stimulated forces which result from the violation ofNewton’s third law inside the plasma.

In 1980, Graham and Lahoz [75] made the first directobservation of quasi-static electromagnetic angular momen-tum in vacuum. The experiment consists of the measure-ment of the axial torque on a cylindrical capacitor locatedin an axial magnetic field. Thus the Poynting vectorGij �c�Ei ∧ Bj�=4p is azimuthal inside the vacuum gap of thecapacitor. They observed a stimulated torque which is ingood agreement with the calculation done with Maxwell’sequations. However, if we take into account both the elec-trons and the ions particles, this experiment can be inter-preted as a consequence of the violation of Newton’s thirdlaw for rotation.

Finally, we can quote plasma experiments with thetapinches [179] which prove that a cylindrically symmetricplasma column, globally but not necessarily locally chargeneutral, without external currents and subject to a radialmagnetic compression starts to rotate ‘‘spontaneously’’. Intwo papers, Witalis [180, 181] explains the origin of thisstimulated rotation as the consequence of the violation ofNewton’s third law about central force interaction, which isthe same effect that we have considered for the case ofrectilinear motion.

All the different experiments reported by many physiciststhroughout the world do show that one can detect our motionthrough the ether by internal experiments, contrary to theclaims of special relativity theory. Today, there are so manyexperiments proving this fact, more than 10 experiments arequoted in this paper, that negating them is an attitude whichbelongs more to religious faith than to science.

12.3. Calculation of the stimulated force for a chargedcapacitor

The violation of Newton’s third law by the Lorentz forcesimplies that a charged capacitor must accelerate its center ofmass or rotate without external help if the capacitor has anabsolute motion with respect to the ether. Moreover, theexistence of an external force must also result in the viol-ation of energy conservation. An experiment showing thelinear stimulated motion of the capacitor through the ether


has been reported elsewhere [34, 182, 183]. The experiment,as shown in Fig. 12, consists of two heavy metallic ballssuspended by fine cotton wires to the ceiling of the labora-tory. In order to keep the balls at a fixed distanceD, aninsulating rod is used between the balls. Therefore, the bi-filar pendulum with the two balls make a capacitor thatmoves as a solid with the Earth’s velocityU � Ui whereUi is also the ionic velocity defined with respect to the etherframe. We know that an electric current circulating througha metal consists in the motion of free electrons contained inthe body of the metal. If the capacitor moves through theether, the free electrons will be accelerated differently,lagging behind with a relative drift velocityV � Ui 2 Ue.This effect has been shown to exist in the Earth’s frame byTolmann and Stewart [184] and Barnett [185] and must alsoexist in the ether frame sinceV is the same quantity in anyframe. At all times, there is an electron drift motion in eachplate trying to follow the motion of the ions in the twoplates. Therefore, the stimulated force will exist if there isalways a drift velocity between the electrons and the ions.This drift velocity can be increased if the capacitor is notperfect; in that case, there is a leakage current which willenhance the magnitude of the stimulated force since thevelocity V depends on the current (J � rV).

In a first step, let us assume that only the presence ofelectrons and ions on the surface of the balls and the wiresaffect the calculation; this implies that all the internal forcescancel one another. In that case, the stimulated force is thenproportional to the square of the applied voltageV0 by virtueof the formulaQ � CV0, whereC is the capacity of thecapacitor; therefore, we get

FG < 2Q2 U·Vc2

DD3 �208�

whereD can be taken as the distance between the center ofthe two balls. Ifd � 6.7 cm is the diameter of a ball, thecapacitance can be calculated from the formulaC � d(1 2d2/4D2)/(4 2 2d/D), knowing thatD < 2d, we getC < 15d/48� 2.09 cm or 2.33 pF. The capacitance of the bi-filar linewhich brings the voltage to the balls is given by the formulaC � (L × 1029/36) log(2D/d) F, where d� 0.05 cm is thediameter of the wires andD� 14 cm is the distance between

the two wires. Therefore, we getC � 8.8 pF for a line oflengthL � 2 m. The relative drift velocity of the free elec-trons circulating in the wires and the balls is small, aboutV�1021 cm s21. Knowing thatU/c < 1023 and V0 � (5/3) ×102 statvolt or 50 kV, we get a small forceFG < 1 ×10210 cosu dynes forNe < 3.5 × 1012 surface electrons.Eq. (208) gives the minimum force one can expect from acalculation that neglects volume forces. The value of thestimulated force will be greatly enhanced if the chargesinside the balls participate in the calculation of this forceand if the total resulting force does not cancel. The numberof conducting electrons in a body of massM is given by theformula Ne � neM/(mpA), wherene is the number of freeelectrons per atom. Therefore, we realize the importanceof the quantityQ� Neq since the stimulated force is roughlyproportional either toQ2 or to N2

eq2. This effect has beentested in the experiment described below.

In the Coulomb gauge, the direction of the stimulatedforce and the resultant motion thereby produced are towardthe positive electrode of the capacitor, as shown by theexperiment described hereafter. Eq. (208) shows that it isthe magnitude and not the direction of the stimulated forcethat will vary with the direction of the Earth’s absolutemotion through the ether. To show this effect, one mustregister the amplitude of this force over months, which isnot possible at the present time. However, we tested thedirection effect by switching the poles of the generator inthe experiment. We verify that the direction of the stimu-lated force has indeed changed 1808 since the stimulatedforce depends on the direction of the current given byVin Eq. (208).

12.4. Cornille’s pendulum experiments

Fig. 12 shows a pendulum of massnM whereM � 500 gis the mass of one ball,n the number of balls andL is thelength of the string which makes an angleu with thevertical. The forces acting on the system are the gravita-tional force nMg, the tensionT in the string and thestimulated forceFG. The horizontal component of thetension force is balanced by the stimulated forceFG whenthe stationary state is reached; therefore, we haveFG � nMgtgu < nMgx=L, wherex < Ltgu is the displace-ment of the ball.

The bi-filar pendulum was placed in the middle of thelaboratory room empty of any metallic object. We usedhigh quality power supplies which were grounded and layon a table which was parallel to the axis of symmetry of thecapacitor. The distances between the laboratory walls andthe balls or the wires were abouth � 2.25 m. The wallsurface was about 1.3× 105 cm2; therefore, the inducedforce [80] (p. 122) between one ball and the insulatingwall of the laboratory room is less thanF � QE/2 �2pQ2/S< 5.7 dynes which is 102 smaller than the forceF �CV2

0 =2h� 6:2 × 102 dynes between one wire and the wall,


Fig. 12. Various forces acting on a double solid pendulum (n� 2).

knowing thatC� (L × 1029/18) log(4h/d) F is the capacityof the wire with respect to the wall.

First question: is the stimulated force produced by theelectrostatic forces resulting from the induced charges inthe surroundings?

Four tests were done to reject the hypothesis that the forceis induced by the surroundings.

First test: we switch the polarity of the power supplies andobserve that the direction of the force has also changed 1808with no observable effect on the magnitude of the stimulatedforce. We recall that the direction of the force must notchange if the stimulated force is produced by asymmetriesof the electrostatic forces. From the above calculation, anyperturbation of the symmetry of the laboratory geometrywill induce a force which will be smaller than 6.2×102 dynes.

Second test: we put a wooden plate of about 5 mm nearthe negative ball. When the voltage is increased, one can seethe pendulum attracted by the induced charges in thewooden plate, at about 30 kV; when we increase the voltageabove this value, then the stimulated force takes over andone can see the pendulum moving away from the wood platein the opposite direction.

Third test: the induced forces in the surrounding environ-ment are tested by replacing the bare wires by coated wires.In this case, the leakage current drops to 0.003 mA and notranslation motion of the pendulum, even with oscillatingthe DC voltage, was observed in spite of the fact that theelectrostatic forces are rigorously the same in both cases.We recall that the leakage currentI � 1.5 mA is mainly dueto the thin bare wires,d� 0.5 mm, which induce the ioniza-tion of the air surrounding the wires and to the balls and thatthe stimulated force is proportional to the currentI �NquVuS, whereSis the cross-sectional surface of the metallicconductors: no current, no stimulated force!

Fourth test: when the ionization current is present, weoscillate the DC voltage and obtain a huge increase of thekinetic energy of the pendulum, which proves that we areusing an external force to produce work. We recall that aninternal force can be distinguished from an external force byoscillating the potential function, since for an internal forcethe oscillating kinetic energy does not increase with respectto time (think of the case of the harmonic oscillator),whereas for an external force the kinetic energy doesincrease (think of the case of a swing pushed by an externalobserver).

We can conclude that no induced forces in the surround-ing environment can explain the existence of the hugestimulated force. As soon as the voltage is increased and acurrent is flowing through the capacitors, one can see athrust of the two balls in the direction of the positive ball.

Second question: is the stimulated force produced byelectrical wind or corona discharge?

We must point out again that the stimulated force does notexist if there is no current circulating inside the conductors,as proved by Eq. (208). Let us recall that a current flows

inside a conductor if there is a permanent non-uniform elec-trical field (at least inside the generator) inside the conduc-tor. There are several ways to produce this non-uniformfield; for example, it can be produced by the presence ofthin wires near to the balls, as explained in paragraph 6-11high-voltage breakdown in the Feynman lecture book; seealso the very important and not well-known paper by Pohl[167].

The current can also be produced by ionization of the airor by the leakage current in a dielectric. In the experimentdescribed in the report by Talley [186], the pendulum was invacuum, no effect was observed because there was nocurrent involved in the experiment except in one experimentwhere a dielectric was placed between the two electrodes. Inthat case, the observed effect can be explained by the leak-age current in the dielectric. I think that this positive resultjustifies our viewpoint that a current is needed in order toproduce a stimulated force through the violation ofNewton’s third law. One can also criticize the experimentby noting that the voltage of 19 kV is too small to show theeffect. Brown used voltages up to 150 kV and obtained hugeforces; see the book by LaViolette [163].

There are several possible mechanisms to explain themotion of the pendulum by an electrical wind effect.

The first is that the observed thrust can result from theejection of charged particles by the wires due to the coronaeffect and ionization in the air surrounding the wires. But thecorona discharges around each thin cylindrical wire musthave the same cylindrical symmetry and, therefore, the reac-tion forces must cancel to zero on the average.

The second is that electrons are attracted by the positiveball and the positive ions by the negative ball; due to thedifference of mass of the two species of particles, the motionshould be in the direction of the negative ball and cannotexplain the thrust in the positive direction.

The third is that both negative and positive ions areattracted by the balls of opposite polarity. The transfer ofmomentum in the positive direction can be explained by acollision process and is due to the difference of massbetween the two kinds of ion. Let us assume that the stimu-lated motion is due to a direct collision of both positive andnegative ions with the metallic conductors. Then the transferof momentum must be attributed to the difference of massesbetween the two kinds of ion, namely the masses of theemitted electrons. In Moore’s book [187] (p. 84), it is statedthat 6× 1012 electrons s21 leave the negative electrode for acorona amounting to 1026 A. For a 1.5 mA leakage current,we obtain 9× 1015 electrons s21, which amounts to a masstransfer 8× 10212 g s21 which is several orders smaller thanthe3.5 gstimulated forceobserved.Therefore, thecalculationtaking into account the mass of the electrons shows that thistransfer isquitesmall incomparisonwith thestimulated force.

We also recall that the stimulated rotation in the exper-iment of Graham and Lahoz [75] is obtained in vacuumwithout any possible ionization effect. Moreover, asshown in Fig. 14, the stimulated force is roughly


proportional to the mass; this fact cannot be explained by anionization effect, since both the voltage and the leakagecurrent are the same for different values of the mass. There-fore, the thrust observed cannot be caused by ambient ionmomentum transfer when the experiment is conducted in theair. Moreover, Deyo’s experiments [169] at low voltage andhigh current show the same stimulated force with no poss-ible ionization effect. It is interesting to point out that in thehigh voltage experiment we use 75 W of power whereas inthe low voltage experiment we use almost the same power,i.e. 50 W.

Third question: is the stimulated force produced by theEarth’s magnetic field?

Owing to the leakage current, the capacitor (wires1balls) can be considered as a linear conductor of lengthL � 2 m located in the Earth’s magnetic field. In thatcase, there is a force applied to the conductor given by therelation F � IL ∧ B=c. Knowing that the Earth’s magneticfield is aboutB� 0.5 G, the magnitude of this force is 1.5×1022 dynes. Therefore, the thrust observed cannot resultfrom the Earth’s magnetic field since the Laplace forcedue to the leakage current is smaller than the stimulatedforce by several orders of magnitude and, moreover, is notapplied in the right direction.

To measure the displacementx of the pendulum when itreaches the stationary position, we used two wooden platesplaced near each ball at the same distance, which wasmeasured before the balls were charged. This distance ismeasured when the positive ball almost touches the positivewooden plate. We also used the wooden plates to test thespace-charge effects in the surrounding environment, sincethese effects are greater than the corresponding effectsinduced by the laboratory walls. We know that an insulatingplate always exerts less influence than a conducting plate.Knowing the chargeQ < 3.48 × 102 statcoulomb, we canestimate that the force exerted by the wooden plate on a ballwill be less thanF � QE/2 � 2pQ2/S < 3 × 102 dynes,

whereS< L2� 2.5× 103 cm2 is the surface of the woodenplate. The magnitude of this force must be compared withthe magnitude of the stimulated force, which is about 3.5×103 dynes for two balls charged at 50 kV. To test the factthat the force exerted by a wooden plate on a ball is smallerthan the stimulated force, we put only one wooden plate nearto the negative ball. When the voltage is increased, one cansee the pendulum attracted by the wooden plate; at about30 kV, when the stimulated force takes over, one can see thependulum moving away from the wooden plate in the direc-tion of the positive ball. However, when we used twowooden plates, the forces of attraction on the two ballscancel one another since the insulating plates attract theballs in opposite directions. Therefore, the induced forcedue to a lack of symmetry in the wooden plates is quitesmall.

The distancesx measured for two balls weighingM �500 g are respectively 3 mm, 5 mm, 8 mm for the appliedvoltages 30 kV, 40 kV, 50 kV and 8 mm, 5 mm, 5 mm fornballs 2, 4, 6 charged with a potential of 50 kV. Thesedistances are used to plot the magnitude of the stimulatedforce as a function of the voltage in Fig. 13 and of the massin Fig. 14. The accuracy of the measurements is roughly^1 mm; however, we think that the increase of thestimulated force versus the mass must be confirmed bymore accurate measurements. The experiment is perfectlyreproducible and we did it several times for many physicistswho witnessed this experiment. Owing to theV2

0 depen-dence of Eq. (208), the stimulated force is too small to bemeasured with a good accuracy below 30 kV. However, theaccuracy with two different kinds of measurement is suffi-cient to prove the existence of the effect, which is, of course,the main point of this experiment. The results are reliable,since we can increase the effect to 5 cm by oscillating thehigh voltage. In that case, no measurement is indeed neces-sary to prove the existence of this effect. We also usedanother method to measure the displacement of the balls


Fig. 13. Force versus voltage. Fig. 14. Force versus mass.

which consisted of placing a wooden measuring rod parallelto the pendulum and taking a video movie of the experimentwhen the voltage was switched on and off. By measuring theamplitude of the oscillation one can determine the displace-ment. We did check that the displacement measured with thewooden plates was almost the same as the displacementobtained when using the video recorder without the woodenplates.

In classical circuit theory, the generator, the wires and thecapacitor form a closed system where the conservation lawof energy for internal forces applies. Therefore, the energyES of the generator is converted into energy storedEP in thecharged capacitor and into heatER dissipated during thecharging process; it follows the conservation law

ES � EP 1 ER

with

EP � 12 CV2

0 � ER �Z∞

0RI2�t� dt �209�

If the capacitor is not perfect, there is a leaking currentand a corresponding dissipated energy which is provided bythe generator. However, the above law will be violated whenthere is a stimulated forceFG since we have now

ddt� 1

2 nMU2� � U·FT �210�

where FT is the sum of the gravitational forcenMg, thetensionT in the string and the stimulated forceFG, knowingthat U is the velocity of the pendulum in the Earth’s refer-ence frame. When the generator is switched off, a kineticenergyEK � nMU2/2 � nMLg(1 2 cosu ) < nMgx2/2L isrecovered. This energy cannot be given by the generator butis taken from the ether. For two balls charged at 50 kV, thekinetic energy of the pendulum isEK � Mgx2/L < 1.4 ×103 ergs, whereas the electrostatic potential energy isEP � CV2

0 =2 < 1:4 × 105 ergs. The kinetic energy due tothe stimulated force is not a small quantity and cannot betaken from the generator since in classical circuit theory nomotion of the capacitor is taken into account during thecharging process. We also applied the high voltage in anoscillatory manner in synchronism with the oscillatorymotion of the pendulum. It results in an amplification ofthe displacement of the pendulum which reaches a magni-tude of ^ 5 cm. This implies the existence of the stimulatedforce and an increase of the kinetic energy almost 25 timesthe above kinetic energy. We also measured the existence ofthe stationary state where the stimulated force is present formore than 6 min; during this time both the voltage and theleakage current are constant.

12.5. The Deyo and Rambaut experiments

Deyo points out in his book [169] (p. 171) that when anarc has been struck between a welding machine, typicallyusing 20–100 A of direct-current electricity at 20–60 V andthe aluminum stock, one notices the sudden jerking motion

of the power supply wires connecting the hand-piece to thetransformer. This phenomena can be explained either byusing the Ampe`re force, in which case another part of theequipment must also move in the opposite direction tocounterbalance the momentum of the wires, or by thegeneration of a stimulated force applied to the center ofmass of the equipment.

Deyo duplicated this phenomena by hanging a loop of0.08 mm diameter wire across the laboratory room, eachend of a portion of the wire was fixed to the walls. A carbattery was used to supply the high-current to the test wire.When the power was applied to this experimental setting,the portion of the wire which drooped between the twoanchor posts deflected toward one wall for a given polarityand toward the other wall when the polarity of the batterywas reversed. In both cases, as the circuit was closed thewire deflection was momentarily exaggerated beforecoming to rest slightly off its unenergized position.

Deyo also tested the interaction of the Earth’s magneticfield with the magnetic field of the wire by hanging fourloops from the ceiling, each loop facing a cardinal directionof the compass. Whenever, he applied power to the system,the loops would all deflect equally either toward or awayfrom the center of the loop arrangement, as shown in the Fig.15. We know that two extended parallel conductors in closeproximity mutually repel one another when carrying currentin opposite directions. Although the portions of the wirewhich face one another are not in close proximity, theoutside deflections can be easily explained by the actionof the mutual repulsive Ampe`re forces. However, the inside


Fig. 15. Top view above the ceiling of hanging loops.

deflections corresponding to a change of the polarity of thebattery contradict this interpretation since the currents arealways circulating in opposite directions for the portions ofthe wire facing one another, whatever the polarity of thebattery.

Finally, Deyo repeated the first experiment with this timea portion of the wire tightly stretched between the anchorposts. Along the length of the portion of the wire were gluedsmall strips of paper. All of them were in an upright positionalong the wire. When the power was applied to this config-uration, the pieces of paper were seen to twist about thelength of the wire and then to return to their upright, restposition after power was removed.

Deyo’s experiments confirm the findings of the pendulumexperiments described above. The bare wires in the pendu-lum experiments are fixed on the nylon wires supporting thetwo metallic balls. When we loosened the copper wires fromthe nylon wires, we can see an interesting phenomenon,namely the oscillation of both wires bringing the highvoltage to the balls. This oscillation can be explained asfollows: the attractive electrostatic forces between the twowires bring them closer to each other and, therefore,increase the ionization current to a value where the repulsivemagnetostatic forces take over. It then follows that the twowires move in the opposite direction from one another untilthe ionization current decreases to such a value that theelectrostatic forces take over again and the whole cyclerepeats. However, one can see that the oscillations of thetwo wires are not symmetric, since the magnitude of theoscillation of the positive wire is far greater and is subjectto a torque which was so violent at one time that the wireloosened itself from the ceiling.

Rambaut, a retired scientist from the French AtomicEnergy Commission, published several papers on theAmpere force [188–190] and the cold fusion problem[191–196]. He participated in the pendulum experimentsdescribed above. After witnessing the translation motionof the electrostatic bi-filar pendulum, not knowing ofDeyo’s experiments, Rambaut duplicated these experimentsin a different manner since he used a one-ball pendulumconnected to a low-voltage generator (12 V) with a highcurrent (4 A) crossing the metallic ball. The experiment isso simple that any reader can repeat it in his garage andconvince himself of the veracity of our assertion concerningthe existence of the stimulated force. It suffices to connect acar battery with wires to the metallic ball. One must use verythin wires in order to provide the necessary resistance toavoid short-circuiting the battery. Moreover, the finenessand the flexibility of the wires which must be hung fromthe ceiling prevent any mechanical coupling through a heat-ing process between the wires and the ball (m� 0.5 kg). Assoon as the current is turned on, one can see, if the experi-ment is properly done, a rotation and a small translation ofthe ball. These effects certainly do not result from any windeffect or induction effects. To increase the translation effect,one can oscillate the DC voltage in phase with the

oscillatory motion of the pendulum. The fact that the ampli-tude of the oscillatory motion increases is proof that thestimulated force is an external force whose work increasesthe kinetic energy of the pendulum, as anybody who hasplayed with a swing in his youth knows very well.

12.6. Review of the Trouton–Noble experiment

The Trouton–Noble [197] experiment is generallyregarded as the electrostatic equivalent of the Michelson–Morley optical experiment: it looks for an effect predicted tobe caused by the absolute motion of the Earth through theether. This experiment has recently been reviewed in aPh.D. thesis by Janssen [198]. The Trouton–Noble exper-iment was performed with a parallel plate capacitor, but it isgenerally described in textbooks with the much simplerconfiguration of two chargesQ1 and Q2 fixed to the endsof a stick of lengthR. The stick is free to pivot about itscenter of mass, which is presumed to be moving with respectto the ether with a constant velocityU. The Lorentz forcesbetween the charges are given by the relation

F12 � Q1Q2 R 11c2 U ∧ �U ∧ R�

� �1R3 � 2F21 �211�

which can be rewritten in the form

F12 � Q1Q2 1 2U2

c2

!R 1

1c2 �U·R�U

" #1R3 �212�

The force is not along the direction of the vectorR;consequently, there is an electromagnetic torque whichtends to orient the stick perpendicular to the velocity if thecharges have opposite sign. Therefore, if a parallel platecapacitor is suspended by means of a fine torsion fiber andcharged, an electromagnetic torque is expected due tomagnetic forces since the capacitor is moving through theether. Specifically, the torqueG should beG � (Q2U2/2c2D) sin(2u ) sin2 w , whereQ is the charge of the capacitor,D the distance between the capacitor plates,U the velocityof the capacitor carried along by the Earth in its motionaround the Sun,u the angle between the velocity vectorand the normal to the capacitor plates andw the anglebetween the velocity vector and the fiber. The originalTrouton–Noble experiment looked for the effect due to theorbital velocityU � 3 × 106 cm s21 of the Earth about theSun. It found a null result. However, Chase [199] identifiedexperimental problems making the original Trouton–Noblenull result inconclusive. Chase repeated the experimentwithout the identified sources of error and found the samenull result. More recently, Hayden [200] designed an exper-iment which is 105 times more sensitive than the originalTrouton–Noble experiment, and that also yielded a nullresult.

As pointed out by Page and Adams [201] and morerecently by Singal [202], there is a fallacy in the usualreasoning about the Trouton–Noble experiment. It iswrong to neglect the torque caused by the forces exerted by


the insulating separators necessary to keep the plates fromapproaching each other under their mutual electric attraction.Therefore, the torque must not be obtained by calculating theinteraction between the charges of the plates which face oneanother, but instead by calculating the interaction betweenthe charges of the plates symmetric with respect to the axis ofsymmetry of the capacitor, as shown in Fig. 16. For asymmetric and homogeneous distribution of charges on theplates, the total torque will be zero at low voltage. Thus thecapacitor is in equilibrium under both the electromagneticand the mechanical forces of constraint exerted by the rodsfor all orientations of the capacitor in all reference frameswithout the need to invoke the relativity principle. Since thetorque is proportional to the square of the voltage we mustincrease the applied voltage to show the effect.

Hayden [203] recently offered an analysis of the Trou-ton–Noble experiment in which he concluded that it is notcompetent to decide about the existence of the luminiferousether. But I will argue here that the Trouton–Noble effectcan indeed be observed, under the proper conditions. Asdiscussed in recent papers [34, 57, 182, 183, 204], anystimulated motion, either rectilinear or circular, of a chargedcapacitor located in the Earth’s reference frame is a conse-quence of the violation of Newton’s third law. We showpreviously that the effect is small if the charges on thesurfaces of the plates are the only charges which participatein the effect. However, the charges inside the plates can givea more important contribution to the predicted effectprovided one uses a voltage higher than 40 kV. Therefore,the Trouton–Noble experiment previously failed becausethe voltages of 2 or 0.6 kV used in the experiments weretoo small to show any stimulated rotation effect in the caseof a small homogeneous distribution of charges on theplates. However, we would like to report that the stimulatedrotation of a parallel plate capacitor [204] has been observedand reproduced recently. The parallel plate capacitor ofabout 505 pF was obtained by sticking two thin aluminumfoils 150× 190 mm2 on an insulating Plexiglas plate (1 r �4) with a thickness of 2 mm and suspended by a thin wire tothe ceiling of the laboratory. The wires bringing the voltagewere coated; therefore, the leakage current was below

0.003 mA. The plan of the capacitor was initially orientedin the north–south direction; as soon as the voltage wasincreased to about 70 kV using a Wimshurst machine, onecould see the torque effect bringing the plates of thecapacitor to the east–west direction where they rested inan equilibrium position. On the contrary, nothing happenedwhen the capacitor plates were initially parallel to the east–west direction parallel to the Earth’s rotational velocity.Therefore, we did observe the expected motion or rest ofthe capacitor as a function of its initial position with respect tothe north–south and east–west directions. The experiment iseasy to do and suffers no ambiguity concerning its interpreta-tion. This positive result alone justifies our interpretationconcerning the violation of Newton’s third law and provesthat the special relativity theory cannot be correct.

A recent study by Szames [205] sheds new light on thehistorical aspects of the Trouton–Noble experiment. Thisauthor reviewed the original papers of Trouton and Nobleand their subsequent, inadequate analysis by physicists.Szames demonstrated that Trouton and Noble observedthe long-sought effect but amazingly reported a negativeresult. Additionally, he showed how physicists quotedthese ‘‘null’’ results in the literature, without ever consult-ing the original papers. Among others, the Trouton–Nobleexperiment was replicated in the 1920s by Tomashek andgave a null result. This experiment is often quoted toconfirm the Trouton–Noble fiasco. However, Tomashek’sreplication was experimentally disproved by Kennard, seeRef. [205]. In the early 1920s, positive results of Trouton–Noble-like experiments were achieved by Thomas Town-send Brown and his professor, Dr. Paul A. Biefeld, givingbirth to the Biefeld–Brown effect. The connection betweenthe two experiments was not established at that time. Thefull details of this story and some of its possible applicationshave been explored in depth by Szames.

13. Conclusion

By taking into account Newton’s third law, we haveshown at the beginning of this paper that Newtonianmechanics is not equivalent to Galilean mechanics. Thenon-existence of Newton’s third law in both Galilean andspecial relativity mechanics necessarily leads to the falseconcepts of covariance and inertial frames. Moreover, wedemonstrated that the negation of Newton’s third law andthe existence of the ether in special relativity theory contra-dicts one another. Most physicists believe erroneously thatthe luminiferous ether of the nineteenth century has beenruled out by the Michelson–Morley experiment and by thedevelopment of the theory of special relativity. As pointedout in Ref. [110], there is a flaw in the Michelson experimentwhich can explain the negative result. Moreover, the posi-tive result of the Michelson–Gale experiment has laterproved the existence of the ether. All the experimentsreviewed in this paper and the experiments done with


Fig. 16. Torque on a charged capacitor in motion.

conductors supplied with direct, high-voltage, low-currentor high-current, low-voltage prove the existence of stimu-lated forces which are external forces violating Newton’sthird law. The experimental evidence concerning theseforces can no longer seriously be denied and should leadto important technical applications in the near future. Agrowing minority of physicists working today on thefoundations of special relativity seem now to be etheroriented. A new physics of the ether is emerging that inour opinion will explain better the constitution of matterand radiation. In this paper, we have shown that Newton’sthird law is the key for a better understanding of physics.Moreover, the experimental proofs of violation of Newton’sthird law reviewed in this paper prove the existence of theluminiferous ether. A serious revision of our understandingof the physical laws which govern the universe seems nowunavoidable.

Appendix

The inductanceLij and the capacitanceCij of a system ofconductors where currentsI i, I j and voltagesVi, Vj have beendefined can be calculated from the equations of energy:

12 Lij I i I j � 1

2c2

Z Z 1R

Ji�r�·Jj�r 0� dr3 dr 03

12 Cij ViVj � 1

2

Z Z 1R

ri�r�rj�r 0� dr3 dr 03�A1�

knowing thatR� uR � r 2 r’ u.The coefficientsLij andCij defined above have the follow-

ing properties:

Reciprocity Lij � Cji Cij � Cji

Proper inductances and capacitancesLii . 0 Cii . 0

Mutual inductances and capacitancesLij ., 0 Cij , 0

Condition for positive definiteness of energy:

Lii Ljj > L2ij Cjj >

Xnj±i�1

uCij u �A2�

The above definitions can be applied to a system of twoconductors. In that case, we obtain the positive definitequadratic forms I � a2L11 1 2abL12 1 b2L22 and I �a2C11 1 2abC12 1 b2C22 for any real a and b constants.This quadratic form results from the superposition principleand since the inductances and the capacitances are definedfrom the energy relations in Eq. (A1), then we may again askthe question of the compatibility of the superposition prin-ciple and the energy conservation principle. However, themutual inductances and capacitances in circuit theory arequantities which can be measured in experiments and itwould be absurd to pretend that these quantities can inany manner be averaged to zero.

The positive definiteness of energy can be demonstratedin three separate proofs, as shown by Power [206].

However, the proof based on the Fourier transform of theintegral

I �Z Z 1

RJ�r�·J�r 0� dr3 dr 03 �A3�

is the most interesting one as we will see hereafter. In thepreceding integral, we have writtenJ�r� � aJ1�r�1 bJ2�r�.We can also make the similar study for the quantityr�r� � ar1�r�1 br2�r�. The three quantities inside theabove integral admit the following inverse transforms:

1R� 2�2p�2

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞e2jK·R

K2 dK3

J�r� � 1�2p�3

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞e2jk·rJ�k� dk3

J�r 0� � 1�2p�3

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞e2jk 0·r 0J�k 0� dk 03

�A4�

Inserting these relations into the integral in Eq. (A3), wefind that

I � 2�2p�8

Z Z Z 1K2 J�k�·J�k 0� d�K 1 k�

� d�K 2 k 0� dk3 dk 03 dK3 �A5�Then it follows that

I � 2�2p�2

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞1

K2 J�2K�·J�K� dK3 �A6�

Since the functionJ(r) is a real quantity, we haveJ�2K� � J p �K� and therefore Eq. (A6) becomes

I � 2�2p�2

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞1

K2 uJ�K�u2 dK3 > 0 �A7�

The same demonstration can be applied when the currentdensities are complex quantities:

I �Z Z 1

RJ p �r�·J�r 0� dr3 dr 03 �A8�

In that case, we obtain instead

I � 2�2p�8

Z Z Z 1K2 J p �k�·J�k 0� d�K 2 k�

� d�K 2 k 0� dk3 dk 03 dK3 �A9�and the same relation follows:

I � 2�2p�2

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞1

K2 uJ�K�u2 dK3 > 0 �A10�

We can generalize Power’s proof to the following integral

I �Z Z e2jk0R

RJ p �r�·J�r 0� dr3 dr 03 �A11�

and apply the same analysis as above, knowing that thespectral component of the Green functionG�R� � e2jk0R

=R


has the expression

G�K� � 1K2 2 k2

0

2 1�k0� j2d�K2 2 k2

0� �A12�

where1 (k0) � 1 1 for k0 . 0 and1 (k0) � 2 1 for k0 , 0.The integralI is now a complex quantity:

I � 2�2p�2

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞uJ�K�u2K2 2 k2

0

dK3

2 j1�k0��2p�2

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞uJ�K�u2d�K2 2 k2

0� dK3

�A13�The theorem of residues cannot be used directly to calcu-

late the real part of the integralI in Eq. (A13) since Jordan’slemma is generally not satisfied. However, if we split thedenominator in the real part ofI by the well-known identity

2K2 2 k2

0

� 2K2 1 F�K� �A14�

with the definition

F�K� � k0

K2

1K 2 k0

21

K 1 k0

� ��A15�

then the residue theorem can be applied to the first termI1

given by Eq. (A10), whereas the second term has the expres-sion

I2 � 1�2p�2

Z1 ∞

2 ∞

Z1 ∞

2 ∞

Z1 ∞

2 ∞F�K�uJ�K�u2 dK3 �A16�

with the definition

I2 �Z Z cos�k0R�2 1

RJ p �r�·J�r 0� dr3 dr 03 �A17�

The operation Re(I) � I1 1 I2 corresponds exactly to theone made in quantum field theory for the renormalization ofenergy in the Lamb-shift calculation [207]. The partition ofthe real part of the integral also has a physical meaning,since the integralI1 defines the stationary part of themagnetic energy. An advantage of this model which wasfirst developed by Barut and Huele [208] and Boudet andcoworker [209–212] is that no infinite quantities and nophotons are to be considered, in contrast with quantum elec-trodynamics where the theory is plagued with infinitenumbers. Therefore, the integralsI1 and I2 converge andthe proper inductances calculated fromI1 and I2 give finitenumbers.

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