Paul Marmet Invited paper in:Abstract. Relativity theory gives a relationship predicting the increase of mass of relativistic moving particles, but no physical model has been given to describe the fundamental physical mechanism responsible for the formation of that additional mass. We show here that this additional kinetic mass is explained by a wellknown mechanism involving electromagnetic energy. This is demonstrated taking into account the magnetic field generated by a moving electric charge, calculated using the BiotSavart equation. We show that the mass of the energy of the induced magnetic field of a moving electron is always identical to the relativistic mass M_{o}(g1) deduced in Einstein’s relativity. Therefore the relativistic parameter g can be calculated using electromagnetic theory. Also, we explain that in order to satisfy the equations of electromagnetic theory and the principle of energy and momentum conservation, toroidal vortices must be formed in the electric field of an accelerated electron. Those vortices are also simultaneously compatible with the magnetic field of the Lorentz force and the wellknown de Broglie wave equation. This leads to a physical description of the internal structure of the electron in motion, which is at the same time compatible with the Coulomb field, the de Broglie wavelength equation, massenergy conservation and with the magnetic field predicted by electromagnetic theory. That realistic description is in complete agreement with all physical data and conventional logic. The paper concludes with an application, which is a first classical model of the photon, fully compatible with physical reality, without the conflicting dualistic waveparticle hypothesis. 1  Fundamental Mechanism. Let me first express my high regard to the scientific achievement of late Professor Ilya Prigogine, in honor of which this special issue is dedicated. It is well known theoretically and observed experimentally, that when a constant electric current is flowing in a wire, there is a magnetic field surrounding this wire. The emission of bremsstrahlung electromagnetic radiation, during the time interval while the electrons are accelerated from zero to the final velocity, is irrelevant in this section. We consider only the free moving electrons at constant velocity after the initial acceleration period. The magnetic field intensity distribution around a wire carrying a constant electric current is calculated using the BiotSavart’s law. This law requires, that a constant electric current must generate a stable magnetic field around the wire. That stable magnetic field, due to the electron current, is illustrated on Figure 1. Figure 1 After the initial transient, when the flow of electron current is stabilized in the wire, no more energy is directly required to maintain that induced magnetic field. That can be verified in the case of an electron current inside a conductor, since it is well known that the current (and therefore the magnetic field) remains naturally constant in time, if the electric resistivity of the conductor is zero, as in the case of a superconductor. In the case of a conductor with a nonzero electrical resistivity, the energy required to maintain the current, is totally used to heat up the wire. However, if the electron current is formed by a free electron beam, traveling in vacuum, it is even more obvious that the current of free electrons maintains a constant velocity inside the electron beam, due to momentum conservation. Since the BiotSavart equation 1 is most suitable to calculate the magnetic field, with any electron current, we can consider either a current generated by the flow of free electrons in vacuum (as for the electron beam drifting inside a cathode ray tube), or the constant electron current inside a wire. 2  Magnetic Field Produced by Single Moving Electrons. Using the BiotSavart law, let us now calculate the magnetic field when we have an extremely small current. We consider the special case of a magnetic field produced by an electron current formed by one single electron, moving at velocity v. Therefore, the long linear distribution of electric charges in the BiotSavart equation must be substituted for one concentrated electric charge existing at one point. We know that the electric current I, is defined as defined as a number of individual electrical charges (e^{}) passing through a point per second. Since the electric charge is quantized, in the case of a single electron it is impossible to calculate an infinitesimal variation of charge of one single electron. Electrons cannot produce a continuous flow of electric charge. This is particularly obvious when the number of electrons is close to unity. In one unit of electron current, (one ampere) we have N_{(1 amp)}@6.25 x 10^{18} electrons. The electron current I, is defined as the passage of a charge Q coulombs of electric charge per second. We have: The scalar form of the BiotSavart equation is: Quantization of Charges  Equation 6 gives the component of the magnetic field B in direction q, produced by one electric charge composed of N electrons distributed along the vector. In Maxwell’s time, it was unknown that the electric charges were quantized. Since the electric charge is quantized in individual electrons, the fundamental assumption of a continuous distribution of charges along assumed a century ago is incorrect. However, since there are about 10^{19} electrons in one unit of electron current, there is generally no appreciable difference, whether we consider that large number of individual charges or a continuous flow of charges. Yet, in equation 6, we wish to consider a number of electrons as small as N=1. Equation 6 must be reconsidered in order to look for that necessary adjustment, due to the disappearance of the linear distribution of electric charges given by vector as a consequence of the quantization of the electron charge. It is interesting to note that H. Poincaré ^{(2)} in 1906, was the first to recognize that another force, called the “Poincaré stress” has to be present to prevent the electric charge of an electron from flying apart due to the Coulomb repulsion. Since we now consider the magnetic field produced by one single quantized electron, additional transformations are also required. For example, the BiotSavart equation defines the magnetic field in a direction with respect to the element . In the BiotSavart equation, the electron charge distribution is not isotropic. From the explanations above, in the BiotSavart equation, we have seen that the Sin (q) function (equation 6) determines the direction of the calculated magnetic field with respect to the continuous charge distribution. However, when we have a single electron, it becomes impossible to define the direction of a nolongerexistent continuous distribution of electric charges. Since the axis of distribution of the electric charges no longer exists, we have to find the new geometry. However, since we now have an isotropic electric field around an individual electron, let us assume that the magnetic field generated is also isotropic. Equation 7 becomes: 7 We can easily visualize qualitatively that the magnetic field produced by a row of electric charges is, as calculated by the BiotSavart equation, not isotropic, because the magnetic field generated backward by the particles in the first part of the row of charged particles cancels out the magnetic field generated forward by the ones in the last part of that row of electric charges. Of course, that cannot exist for a single electron. It is not surprising that the passage from a geometry of linearly distributed electron charges (in an electron current) to the geometry of a point source (single moving electron) produces a similar change of geometry in the resulting magnetic field. We will see below that the hypothesis above is valid because the isotropic distribution of the magnetic field around single electrons is compatible with equation 1, (which is applicable to a smooth linear distribution of electrons). Experimentally, due to its smallness, it has never been possible to measure the faint magnetic field induced around one single moving electron. Nevertheless, the validity of that relationship is verified indirectly by the correctness of the BiotSavart equation. We remember that the BiotSavart equation was planned only to calculate the component of the magnetic field appearing in a specific direction, in the case of a large number of electrons distributed linearly. It was not destined to calculate the total magnetic field around one single isolated electron. The existence of quantized electron charge in an electron was still unknown in Maxwell’s time. Only the effect produced by a continuous flow of electric charges could be considered then. Consequently, this calculation here, which involves independent electrons, is more fundamental than the BiotSavart equation, since the fundamental corpuscular nature of the electric charge, which is more realistic in physics, is now taken into account. 3  Induced Magnetic Energy around a Single Moving Electron. We have seen that equation 7 gives the amplitude and the distribution of the total induced magnetic “field” around individual moving electrons. Corrections have been made in order to take into account that we have now point like electric charges so that the electric charge of the electron is no longer distributed along a line. Also the electric field around one electron is isotropic and is not distributed along a line as in the BiotSavart problem. These new considerations are such that the BiotSavart equation is still valid when the electric charge is distributed as a continuous flow. From that induced magnetic field, let us calculate now, the magnetic “energy” around one single (N=1) electron. The density of magnetic energy u_{m} which is defined as the magnetic energy U_{m} per unit volume V, is given by the relationship ^{(3)}: 4  Total Mass of Magnetic Energy in One Single Moving Electron. Equation 10 gives the total energy of the induced magnetic field around one moving electron in volume dV. Let us calculate the total mass M of that magnetic field surrounding the moving electron, using the BiotSavart’s equation. We know that the constant of proportionality between energy and mass is c^{2} (in the relationship E=mc^{2}). Since equation 10 gives the energy per unit volume dV, it must be divided by c^{2} in order to get the mass of the magnetic field. We have: On figure 2, we see that the differential surface element of the surface of a sphere, at a distance r, is equal to a thin square, having sides respectively equal to r dq along meridians, multiplied by the element of longitude j of the circle 2prSin(q). Figure 2 Magnetic fields are well measured experimentally at any large distances, but such a measurement is no longer possible directly at an infinitesimal distance, very near the center of the electron. At r=0, equation 15 shows that it would require an infinite amount of energy to generate magnetic fields right down to the center of the electron. Therefore, the total electron energy of 511. KeV, gives the information of how close to the geometrical center, the field can exist. As a result, a hollow structure of electric and magnetic fields of the electron is absolutely necessary, due to the finite energy of the electron (511 Kev). More explanation is given below in section 7. There is a minimum radius expected due to the fact that the electron energy is finite (511 Kev), which is called the classical electron radius (r_{e}) ^{(3)}. There cannot exist any electromagnetic energy within the classical electron radius (r_{e}), because the velocity of that empty part of the electron (hollow cavity) cannot induce a magnetic field as calculated by the BiotSavart equation. Let us integrate the magnetic energy from the wellknown (hollow) classical electron radius r_{e} up to infinity, where an electric field can exist. From equation 15, we have: 5  The “Relativistic Increase of Mass”. We wish to compare the magnetic mass of a moving electron given in equation 17, with the kinetic mass (which is the increase of the socalled “relativistic mass”) of the same electron. When we apply the principle of massenergy conservation, we have found ^{(4)}, that the mass of a moving particle in motion M_{v} is given by the same relationship as in Einstein’s relativity. The mass of a moving particle is given by the relationship 6  The Magnetic Mass Versus the Relativistic Mass. Let us compare the increase of magnetic mass calculated above in equation 17, with the increase of electron mass, using relativity theory as given in equation 22. In fact, we are testing whether the relativistic mass is the same thing as the magnetic mass. Equations 17 and 22 give: In fact, the real fundamental nature of the kinetic mass, which increases with velocity, is nothing else than the magnetic energy, as given by the BiotSavart equation. From equations 17 and 23, we can conclude that for an electron, the physical nature of the parameter g in relativity is:Therefore the increase of the socalled relativistic mass is in fact nothing more that the mass of the magnetic field generated due to the electron velocity. 7  Physical Meaning of the Classical Electron Radius. Let us examine a natural interpretation to the Classical Electron Radius. Following the above calculation (r_{e} in eq. 17), we see that the “Classical Electron Radius” can be described, as the size of a central cavity with radius r_{e}, in which there is no field, because this would require an amount of energy (and mass) which is not compatible with the electron mass. An infinite amount of energy is required if we assume that there is a field at the center of the electron, inside that cavity. This is physically unrealistic and therefore impossible. The fact that there is always only 511. KeV of energy available in an electron is a natural physical constraint, which prevents any electrical field inside that classical radius. It has been shown that electrons at rest are pure electromagnetic fields ^{(3)}. Since the electron is pure electromagnetic energy, the total electromagnetic field surrounding the electron has an energy equal to U=m_{e}c^{2}=511. KeV. Using that relationship ^{(3)}, the entire energy of an electron is: 
Index
Preface
Preface

Figure 3 illustrates the cross section of an electron at rest, representing the electric field density by a dark area. The electric field of an electron at rest is isotropic (in 3D) around its hollow core and extends to infinity.Figure 3.
Figure 6 illustrates the internal structure of the first three vortices drawn on figure 5. Figure 6 illustrates the vortices, assuming a perfect conservation of energy and momentum, when the electron has been accelerated by a force “F” applied (downward) on the central part of the electric field of the electron.
However, as mentioned above, this is the “sole” property, which is needed, because the principle of massenergy conservation is fully satisfied without any energy belonging to that assumed medium. Then again, it is a geometrical property. Other phenomena have also been observed, which shows that an absolute frame of reference is required without the interaction of any physical medium (ether). The GPS ^{(16)}, which requires a nonrelativist correction (because it requires the Sagnac effect), provides a proof of a need of an “absolute frame of reference” for light propagation without involving any interaction with the media. It is shown that the velocity of light is actually (cv) in a frame moving at velocity v, ^{(17, 18)} even if the moving observer always measures c in his own frame. In fact, the velocity of light is an absolute constant in an “absolute frame” at rest, but due to the different clock rate in the moving frame, there is an apparent velocity of light equal to c in all frames ^{(16)}.“ For the moment, the sole property of that assumed ether is to establish an absolute origin to the velocityframe of light and physical matter, because this frame of reference is absolutely needed to comply with the principle of energy and momentum conservation.”
13 –
References.
1  A. Serway, Électricité et
Magnétisme, Les éditions HRW Ltée Montréal, 1989, page 195 éq. 8.4
(1983). Also:
http://hyperphysics.phyastr.gsu.edu/hbase/magnetic/biosav.html
2  H.
Poincaré ”Sur la dynamique de l’électron” Rend. Circ. Matem. Palermo 21,
129, (1906)],
3  John David Jackson, “Classical Electrodynamics“,
John Wiley & Sons, New York, (1963).
4 – P. Marmet, “Einstein’s Theory of
Relativity versus Classical Mechanics”, Newton Physics Books, 2401
Ogilvie Road, Ottawa, Ontario, Canada, K1J 7N4, (1997)
5  M.
Abraham, “Prinzipien der Dynamik des Elektrons” Ann. Der Phys. 10,
105, (1903).
6  Classical Electron Radius, Web:
http://scienceworld.wolfram.com/physics/ElectronRadius.html
and
http://www.tcaep.co.uk/science/constant/detail/classicalelectronradius.htm
7
 R. P. Feyman R. B. Leighton and M. sands, ”The Feynman Lectures on
Physics” Vol II, Chap. 28 (AddisonWesley, Reading 1964)
8  H. A.
Lorenz, “The Theory of Electrons” (Teubner, Leipzig 1916, also Dover, New
York (1952)
9  F. Rohlich “SelfEnergy and Stability of the
Classical Electron” Am. J. Phys. 28, 639, (1982)
10  T. H. Boyer,
“Classical Model of the Electron and the Definition of Electromagnetic Field
Momentum”, Phys. Rev. D. 25, 3246 (1982).
11  A. K. Singal,
“EnergyMomentum of the SelfFields of a Moving Charge in Classical
Electromagnetism”, J. Phys. A 25, 1605, (1992)
12  J. G.
Williamson, M. B. van der Mark, “Is the Electron a Photon with Toroidal
Topology” Annales de la Fondation Louis de Broglie 22, 133, (1997)
13 
Louis de Broglie, “Ondes et quanta”, Comptes rendus de l'Académie
des Sciences, Vol. 177, pp. 517519 (1923).
14  P. Marmet, book:
"Absurdities in Modern Physics: A Solution", ISBN 0921272154,
Les Éditions du Nordir, (1993). Also on the Web at: http://www.newtonphysics.on.ca/HEISENBERG/index.html
15  P. Marmet, "Explaining the Illusion of the Constant Velocity of
Light", Meeting "Physical Interpretations of Relativity Theory VII",
University of Sunderland, London U.K., 1518, September 2000. Conference
Proceedings "Physical Interpretations of Relativity Theory VII" p. 250260 (Ed.
M. C. Duffy, University of Sunderland). Also in "Acta Scientiarum" (2000)
as: "The GPS and the Constant Velocity of Light". Also: "GPS and
the Illusion of Constant Light Speed" Galilean Electrodynamics", Vol. 14,
No:2, p. 2330. March/April 2003. Web address: http://www.newtonphysics.on.ca/Illusion/index.html
)
16  P. Marmet, “Natural Length Contraction Mechanism Due to
Kinetic Energy” Web: http://www.newtonphysics.on.ca/kinetic/length.html).
17  P. Marmet, “Experimental Tests Invalidating Einstein's Relativity”
Web address: http://www.newtonphysics.on.ca/faq/invalidation.html)
18  P. Marmet, “Simultaneity and Absolute Velocity of Light”.
Ch. 9 of the Book “Einstein’s Theory of Relativity versus Classical Mechanics”,
Newton Physics Books, 2401 Ogilvie Rd. Ottawa, Canada, K1J 7N4, ISBN
0921272189 (1997). Web address: http://www.newtonphysics.on.ca/EINSTEIN/Chapter9.html)
Ottawa, Canada,
Revised October 14, 2003.
Updated Nov. 4, 2003

<><><><><><><><><><><><><>