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Electromagnetic and Kinematic Mechanics
Synchronized in their Common Vector Field: A
Mathematical Relation (Extended Republication PI)
André Michaud
Service de Recherche Pédagogique
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Abstract: The purpose of this chapter is to establish the clear mathematical relations that
exist between kinematic mechanics and electromagnetic mechanics, in accordance with Wilhelm
Wien's project formulated in 1901. This harmonization was made possible by the integration in
kinematic mechanics of the mass increase of the electron as a function of its velocity, as
measured by Walter Kaufmann by means of his bubble-chamber experiments, which was
confirmed by H. A. Lorentz and all the leading edge physicists who analyzed his data; and the
establishment the electromagnetic structures and mutual interactions of the restricted set of stable
elementary particles within the framework of trispatial vector geometry, which emerges naturally
from the triply orthogonal relationship that Maxwell discovered between the magnetic field, the
electric field and the direction of motion of light in a vacuum. Description of the local trispatial
vector complexes of the restricted set of stable elementary particles, of their stable combinations
up to the atomic level and finally, of the four stable stationary resonance levels of the trispatial
vector field. Analysis of the experimental confirmation of the magnetic nature of the electron
spin; and establishment of its relation with the concept of magnetic monopole, of covalent
molecular bounding, of the filling of electronic orbitals by electron pairs, of the generation of
Cooper pairs, and of the related interpretation of the Stern-Gerlach experiment.
Keywords: Kinematic mechanics; Electromagnetic mechanics; Electrostatic recall
constant; Restoration force; Gravitation.
This article was initially published in the Journal of Modern Physics in May 2023:
Michaud, A. (2023) Introduction to synchronized kinematic and electromagnetic
mechanics. Journal of Modern Physics, 14, 876-932.
https://doi.org/10.4236/jmp.2023.146051
https://www.scirp.org/pdf/jmp_2023053016192489.pdf
A final expanded version of this article was republished upon invitation in November
2023 as Chapter 3 under the title “Electromagnetic and Kinematic Mechanics Synchronized in
their Common Vector Field: A Mathematical Relation” in book titled "Current Perspective to
Physical Science Research Vol. 3" which is part of a collection that pre-selects papers deemed
worthy of attention in the global offer, to make them more immediately available to the
community.
Michaud, A. (2023) Electromagnetic and Kinematic Mechanics Synchronized in
their Common Vector Field: A Mathematical Relation. In: Dr. Madogni Vianou
Irenee, Editor. Current Perspective to Physical Science Research Vol. 3.
November 23, 2023, Page 55-131.
https://doi.org/10.9734/bpi/cppsr/v3
https://doi.org/10.9734/bpi/cppsr/v3/6575B
(PI Video Promotion)
Other articles in the same project:
INDEX -Electromagnetic mechanics – The 3-Spaces Model
________________________________________________________________________
a Service de Recherche Pédagogique, Québec, Canada.
*Corresponding author: E-mail: srp2@srpinc.org;
Chapter 3
Print ISBN: 978-81-967636-7-1, eBook ISBN: 978-81-967636-5-7
Electromagnetic and Kinematic
Mechanics Synchronized in Their
Common Vector Field: A Mathematical
Relation
André Michaud a*
DOI: 10.9734/bpi/cppsr/v3/6575B
Peer-Review History:
This chapter was reviewed by following the Advanced Open Peer Review policy. This chapter was thoroughly checked to
prevent plagiarism. As per editorial policy, a minimum of two peer-reviewers reviewed the manuscript. After review and
revision of the manuscript, the Book Editor approved the manuscript for final publication. Peer review comments,
comments of the editor(s), etc. are available here: https://peerreviewarchive.com/review-history/6575B
ABSTRACT
The purpose of this chapter is to establish the clear mathematical relations that
exist between kinematic mechanics and electromagnetic mechanics, in
accordance with Wilhelm Wien's project formulated in 1901. This harmonization
was made possible by the inclusion in kinematic mechanics of Walter
Kaufmann's measurements of the electron's mass increase as a function of its
velocity, which H. A. Lorentz and all of the most cutting-edge physicists who
examined his data confirmed; and the establishment of the electromagnetic
structures and mutual interactions of the restricted set of stable elementary
particles within the framework of the general theory of relativity. Description of
the local trispatial vector complexes of the restricted set of stable elementary
particles, of their stable combinations up to the atomic level and finally, of the
four stable stationary resonance levels of the trispatial vector field. Analysis of
the experimental confirmation of the magnetic nature of the electron spin; and
establishment of its relation with the concept of magnetic monopole, of covalent
molecular bounding, of the filling of electronic orbitals by electron pairs, of the
generation of Cooper pairs, and of the related interpretation of the Stern-Gerlach
experiment.
Keywords: Kinematic mechanics; electromagnetic mechanics; electrostatic recall
constant; restoration force; gravitation.
1. INTRODUCTION
The interaction between particles with electric charge and electromagnetic fields
is known as electromagnetism in physics. One of the four basic forces of nature
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is the electromagnetic force. In interactions between atoms and molecules, it is
the dominating force. It is possible to think about electromagnetism as a mixture
of electrostatics and magnetism, two separate yet interwoven phenomena.
Magnetism is an interaction that only happens between charged particles in
relative motion, whereas electromagnetic forces happen between any two
charged particles, generating an attraction between particles with opposing
charges and a repulsion between particles with the same charge. A branch of
physics called kinematics, which was evolved from classical mechanics, defines
how points, bodies, and systems of bodies (groups of objects) move without
taking into account the forces that propel them. The discipline of kinematics is
frequently referred to as the "geometry of motion" and is occasionally considered
to be a subfield of mathematics. Any known values of the location, velocity,
and/or acceleration of points inside the system are declared as initial conditions
for a kinematics issue, together with the geometry of the system [1a-1d].
In the early 1900s, there was ongoing discussion about whether the mass of
bodies is mechanical in nature, as established by experiments with macroscopic
masses, or electromagnetic in nature, as revealed by recent findings from data
gathered about the electromagnetic behavior of electrons in Walter Kaufmann's
bubble chamber [1] [2] [3] [4], using electron beams that are accelerated and
guided on curved paths [5]. Coinciding with the beginning of Kaufmann's
experiments but in an unrelated research project, Wilhelm Wien, the famous
experimentalist who first experimentally confirmed the quantized nature of light
with his black-body experiments [6], published in 1901 an article analyzing the
possibility of harmonizing kinematic mechanics with electromagnetic mechanics
from a common basis, which is an issue that had been under discussion in the
physics community ever since Maxwell formulated his electromagnetic theory 40
years earlier [7]:
"Es ist zweifellos eine der wichtigsten Aufgaben der theoretischen Physik,
die beiden zunächst vollständig isolierten Gebiete der mechanischen und
elektromagnetischen Erscheinungen miteinander zu verknüpfen und die für
jedes geltenden Differentialgleichungen aus einer gemeinsamen Grundlage
abzuleiten." Wilhelm Wien (1901)[7]
"It is undoubtedly one of the most important tasks of theoretical physics to
link the two domains of mechanical and electromagnetic phenomena, which
are currently completely separated, and to derive differential equations that
would be applicable to each from a common basis."
According to his analysis, the dominant trend in the last quarter of the nineteenth
century, supported by Maxwell, Thompson, Boltzmann, and Hertz, was to give
priority to kinematic mechanics as a common foundation, because Maxwell had
succeeded in establishing his electromagnetic equations by adapting the
classical wave equation to account for the propagation of light in a vacuum,
which allowed him to predict the existence of the entire spectrum of non-visible
electromagnetic frequencies, which was later confirmed by Hertz. According to
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the arguments presented in his paper, Wien was rather of the opinion that
electromagnetic mechanics would be a more appropriate common basis for this
harmonization:
"Diese Untersuchungen haben zweifellos das Größe Verdienst,
nachgewiesen zu haben, dass beiden Gebieten etwas Gemeinschaftliches
zu Grunde liegen muss, und dass die gegenwärtige Trennung nicht in der
Natur der Sache begründet ist. Andererseits aber scheint mir aus diesen
Betrachtungen mit Sicherheit hervorzugehen, dass das System unserer
bisherigen Mechanik zur Darstellung der elektromagnetischen Vorgänge
ungeeignet ist."
Wilhelm Wien (1900)[7]
"These investigations have undoubtedly the great merit of having
demonstrated that both domains must be grounded on something common
and that the present separation is not rooted in their nature. On the other
hand, however, from these considerations it seems to me with certainty that
the system of our kinematic mechanics up to now is unsuitable for the
representation of electromagnetic processes."
Unbeknownst to him at this point in time, because although Planck had already
identified and calculated constant h that bears his name from Wien's black body
data, it was only in 1924 that this constant was derived from a classical
mechanics equation by Louis de Broglie, and much later yet, in 2013, that it was
also derived from an electromagnetic equation, as will be put in perspective in
Section 6, which clearly confirmed that both domains were effectively related by
this constant that could be shown to be emergent from both mechanics.
His most important argument in favor of using electromagnetism as a common
foundation for both mechanics was that the calculations made by Searle [8] using
the electromagnetic force equations developed by Heaviside [9] revealed that the
energy and mass of localized charged particles in motion should increase with
velocity, whereas the calculations made using the equations of kinematic
mechanics did not allow such an increase.
In the years following the publication of Wien's analysis, Kaufmann carried out
numerous experiments involving electrons accelerated to relativistic velocities on
curved trajectories, that allowed measuring separately their longitudinal and
transverse inertia [1] [2] [3] [4]. Extensive analysis of the Kaufmann data
successively carried out by Abraham, Lorentz, Planck, Poincaré, Bucherer,
Neumann and Einstein [10] [11] [12] [13] [14] [15] confirmed in conformity with
Searle's calculations [8], that the inertia of electrons moving at relativistic
velocities on curved trajectories, indeed increases with velocity, both
longitudinally and transversely, as particularly well analyzed and explained by
Lorentz in his 1904 article [10], even discounting the longitudinally oriented
momentum energy of the electrons, which brought solid support to Wien's
conclusion that both mechanics should be grounded on electromagnetism.
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However, disregarding the momentum energy of macroscopic masses, due to
the too low velocities that can be achieved with such masses, which forever
relegates any possible measurement of any velocity related increase in
macroscopic mass far below any detectable level, as analyzed in References
[16] [17], no longitudinal or transverse mass increase was ever measured in
experiments with macroscopic masses, which entertained doubts as to the
possibility that macroscopic masses could also be subject to the observed and
confirmed mass increase of moving electrons.
In the same 1904 article in which Lorentz analyzed in depth the electromagnetic
behavior of electrons in Kaufmann's bubble chamber, he also defined, on a
completely separate issue, the set of transformations that immediately drew the
attention of the whole community, by establishing a neat foundation to the
Special Relativity Theory (SRT) proposed by Einstein in his third 1905 article
[18], that is, a solution to the then apparent impossibility at the time of identifying
a stable absolute reference in the universe with respect to which the motion of
ponderable masses could be defined and calculated, which is a conclusion that
was drawn as an outcome of the failure of the Michelson experiments to reveal
such a reference.
The interest of the Lorentz transformations lay in their ability to allow
mathematically describing and calculating the motion of macroscopic masses in
relation to each other from the point of view of kinematic mechanics, but this
unfortunately conceptually excluded the very possibility that absolute motion
could be possible in the universe, possibly from an unexpected reference that
was yet to be discovered, a question that was eventually resolved from the point
of view of electromagnetic mechanics, as we will see further on.
Interestingly, the Lorentz force equation F=q(E + v×B) whose validity for
calculating the motion of free moving electrons propelled and guided by electric
and magnetic fields – confirmed in the same 1904 article [10] from the analysis of
the data collected by Kaufmann – has been used ever since to guide free moving
electrons with the highest degree of precision in cathode ray tubes (CRT), and
other free moving charged particles in high energy accelerators, on the utterly
precise trajectories that can be established only by taking into account their
velocity related transverse increase in inertia as observed in Kaufmann's bubble
chamber:
"Hence, in phenomena in which there is an acceleration in the direction of
motion, the electron behaves as if it had a mass m1, those in which the
acceleration is normal to the path, as if the mass were m2. These quantities
m1 and m2 may therefore properly be called the "longitudinal" and
"transverse" electromagnetic masses of the electron. I shall suppose that
there is no other, no 'true' or 'material' mass."
H.A. Lorentz (1904)[10]
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On his side, Poincaré made this comment:
"Abraham's calculations and Kaufmann's experiments have shown that
mechanical mass itself is zero and that the mass of electrons, or at least of
negative electrons, is exclusively of electrodynamic origin. This forces us to
change the definition of mass; we can no longer distinguish mechanical
mass from electrodynamic mass, because then the former would disappear;
there is no other mass than electrodynamic inertia; but in this case the mass
can no longer be constant, it increases with the velocity; and even, it
depends on the direction, and a body animated by a notable velocity will not
oppose the same inertia to the forces which tend to deviate it from its
course, and to those which tend to accelerate or to delay its forward motion."
Henri Poincaré(1905)[11]
But despite Searle's calculations [8], Wien's conclusion [7] and the confirmation
brought by Kaufmann's data as analyzed by Lorentz, Poincaré, Bucherer,
Neumann, Planck and Einstein himself [12] [13] [14] [15], the confirmed
electromagnetic behavior of electrons was deemed not to apply to macroscopic
masses for which no such variation was ever measured, which ended up causing
these characteristics to be ignored in the establishment of the Special Relativity
Theory (SRT), according to a decision taken in 1907 by the leading researchers
in the community, in agreement with Einstein's opinion, that it should not be
taken account of when dealing with macroscopic masses:
"Herr Kaufmann has determined the relation between [electric and magnetic
deflection] of
𝛽
-rays with admirable care. ... Using an independent method,
Herr Planck obtained results which fully agree with Kaufmann. ... It is further
to be noted that the theories of Abraham and Bucherer yield curves which fit
the observed curve considerably better than the curve obtained from
relativity theory. However, in my opinion, these theories should be ascribed
a rather small probability because their basic postulates concerning the
mass of the moving electron are not made plausible by theoretical systems
which encompass wider complexes and phenomena."
Albert Einstein (1907) ([15], p. 159)
With these remarks, the kinematic mechanics approach was thus chosen as the
common basis from which differential equations applicable to both the kinematic
and electromagnetic domains should emerge. Abraham Pais' 1982 conclusion
regarding these remarks of Einstein and the agreement of the community clearly
hints at the problems that this decision failed to resolve with respect to the
electromagnetic properties of the electron:
"Special Relativity killed the classical dream of using the energy-momentum-
velocity relations of a particle as a means of probing the dynamic origin of its
mass. The relations are purely kinematic. The classical picture of a particle
as a finite little sphere is also gone for good. Quantum field theory has
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taught us that particles nevertheless have structure, arising from quantum
fluctuations. Recently, unified field theories have taught us that the mass of
the electron is certainly not purely electromagnetic in nature. But we still do
not know what causes the electron to weigh."
Abraham Pais (1982) ([15], p. 159)
In reality, the level of knowledge about the electromagnetic nature of the charged
and massive electron and other stable elementary electromagnetic particles of
which the atoms constituting all macroscopic masses are made was not
sufficiently advanced at the beginning of the 1900s for better conclusions to be
drawn.
When speaking of quantum fluctuations in his comment written in 1982, Pais was
of course referring to the Quantum Field Theory (QFT) developed by Paul Dirac,
grounded on the Lorenz gauge, that postulated a stable conservative zero-point
energy level in all of vacuum, about which neutral level would spontaneously and
stochastically expand and retract pairs of oppositely charged elementary
particles, such as electron-positron pairs, that could then interact to constitute all
matter in the universe.
Let us note at this point that QFT was conceived before it was discovered by
direct observation in bubble chambers in the early 1930's that such electron-
positron pairs actually can come into being only by means of the destabilization
of electromagnetic photons that possess sufficient energy to completely account
for the energy of which the invariant rest masses of both particles are made –
see Section 3.3 further on –, that is, electromagnetic photons exceeding ever so
slightly the threshold energy of 1.022 MeV, which is twice the 0.511 MeV energy
known to constitute each of their invariant rest masses, when such photons'
trajectories run close enough to charged and massive particles, such as atomic
nuclei, for them to destabilize and convert to such pairs [19] [20], and even when
coming close enough to other photons at a single point in space, as
experimentally confirmed at the Stanford Linear Accelerator (SLAC) in 1997 [21].
The difference between QFT, defined before these discoveries, and the trispatial
model of electromagnetic mechanics (EMM), that takes into account these
experimentally confirmed processes of generation of charged and massive
electron-positron pairs, made of the electromagnetic energy of localized photons
interacting with charged and massive particles or with other photons is analyzed
in Reference [22].
Of course, the classical naive image of elementary particles as small, clearly
defined spheres has definitely disappeared, as mentioned by Pais. But in light of
the more extensive knowledge base now available, it is the conclusion that the
relations between masses could be purely kinematic that proves to have been
quite illusory, given the discovery made later that the energy of which the masses
of all charged elementary particles are made, constituting the atoms whose local
accumulations establish all macroscopic masses, is purely electromagnetic in
nature.
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The same electromagnetic nature also characterizes their carrying energy, which
is permanently and adiabatically induced for each of them by the Coulomb
restoring force as a function of the inverse square of the distances separating
them, and which is made of the unidirectional energy of their ∆K momentum, that
ensures their motion or alternatively the pressure that they exert on other
particles, and by the transversely oscillating energy of the simultaneously
induced local ∆E and ∆B fields that mutually induce each other and that guide
them locally in straight line when no external influence interferes.
It is the electromagnetic properties of this energy, of which the invariant mass of
elementary particles is made, as well as their carrying energy, that we will
analyze in this article, and then put in perspective how the electromagnetic
mechanics that emerges from these properties harmonizes with traditional
kinematic mechanics.
2. THE ESTABLISHMENT OF THE SPECIAL RELATIVITY THEORY
Before proceeding to this analysis, let us proceed to a historical review of the
events that surrounded the choice of the kinematic perspective as a common
foundation for the two domains and of the consequences of this choice.
As revealed by Einstein's previously quoted remarks, the kinematic approach
was favored in 1907, which led to the adoption of the theory of Special Relativity
(SR) without taking into account the increase in the transverse mass of electrons
with velocity observed from Kaufmann's data, and that formalized the basis of
mechanics strictly on the relative motion of bodies with respect to each other
according to the Lorentz transformations [10]. Even the simple possibility that
light could move in the universe at the absolute invariant speed of light,
irrespective of the speed of the source and of that of the absorbing destination,
quickly became inconceivable to many, even though the speed of compression
sound waves in a homogeneous medium, for example, is well understood to be
absolutely independent of the speed of the source and of that of the receiver.
To compensate for the absence of the electromagnetic increase in mass with
velocity observed with Kaufmann's data, the Special Relativity theory varied the
time and length of bodies according to velocity as a function of the γ-factor, with
the lengths of masses considered as contracting and time considered as slowing
down with increasing velocity and with increasing intensity of the gravitational
gradient, while maintaining the conservative concept of potential energy
converting to momentum kinetic energy during velocity increases, and
reconversion to potential energy during decelerations, which does not imply a
continuous physical existence of this kinetic energy.
Contrariwise, the electromagnetic mass increase of the electron, according to
Kaufmann's data, implies that the energy that constitutes the kinematic mass
increment of the electron ∆mmc2, corresponding to the energy represented in
electromagnetic mechanics by the mutually inducing locally oscillating ∆E and ∆B
fields representations of its carrying energy, as well as the associated
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∆Kmomentum energy, physically exists and varies adiabatically with any change
in velocity or proximity to other charged particles, without involving any variation
in time or length of masses, in conformity with the γ-factor being intrinsically
accounted for both for the momentum energy and the related oscillating fields
energy, due to their continuous physical existence [23].
As observed by Einstein, the difference between longitudinal mass m1 and
transverse mass m2 of the moving electron – as identified by Lorentz in his 1904
article – was not observable for macroscopic masses, and this condition seemed
to him and to his colleagues not relevant in their search for the cause of
gravitation, that they assumed to apply only to macroscopic masses.
In reality, given that all macroscopic masses are made of subatomic charged
electromagnetic particles stabilized in various stationary action resonance states,
including electrons, it therefore turns out that it can only be the sum of their
interactions at the subatomic level that can establish the observable behavior of
larger accumulations of such particles at our macroscopic level. Indeed, given
the low velocities possible for such large local accumulations of particles at our
macroscopic level, all experimental evidence seems to show that all processes
involving such masses can be successfully treated using classical Newtonian
kinematic mechanics.
For processes involving very small macroscopic masses interacting with very
large macroscopic masses, however, relativistic mechanics comes into play due
to the great influence of even small changes in the intensity of the gravitational
gradient on the internal distances between the stabilized charged particles that
constitute these small macroscopic masses, as for example atomic clocks
moving away from the Earth, or the motion of Mercury on its elliptical orbit very
close to the huge mass of the Sun compared to its relatively insignificant mass,
or the very small masses of the Pioneer 10 and 11 spacecrafts moving away
from the large mass of the Sun on their trajectories leading out of the solar
system, as analyzed in Reference [23].
This summarizes about all that we can directly measure from our macroscopic
perspective, of the sum of the electromagnetic interactions that exist between all
charged particles occurring at the subatomic level, of which all macroscopic
masses are made.
As already mentioned, the fact that the energy of which the rest mass of the
electron is made really is electromagnetic in nature was discovered only later, in
the early 1930s, when it was observed that photons of energy greater than 1.022
MeV could easily be converted into massive electron-positron pairs [24] [19] [20].
However, this discovery was obviously insufficient to induce reconsideration,
because much later, in the 1980s, the opinion of Pais quoted above still was that
their relations could only be purely kinematic. However, more and more
discoveries have accumulated since then to finally confirm beyond any doubt that
the common basis of physics should be electromagnetic.
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The complete historical background of the evolution of electromagnetic theory
since James Clerk Maxwell [25] and Ludwig Lorenz [26] 160 years ago, and the
evolution of the kinematic theory from its historical regrounding on relative motion
in 1907 is analyzed in Reference [23]. Since Maxwell and Lorenz established
their apparently conflicting approaches, the community has focused strictly on
the Lorenz gauge approach involving a single electromagnetic field. This
approach is obviously correct for dealing with electromagnetic energy at our
macroscopic scale, the proof being the whole set of successful engineering
developments that we benefit from grounded on the idea of such a single
electromagnetic field, in which the vectorial differences between the E-field and
the B-field have no role to play.
Metaphorically speaking, just like dealing with water as a fluid at our macroscopic
level allows successfully dealing with all aspects of its use that does not require
involving the individual characteristic of the quantized water molecules of which it
is really made, it is well understood that it would be illusory to try establishing the
characteristics of the localized quantized water molecules and of their subatomic
components by means of the macroscopic water fluidity perspective.
It turns out that the same problematic dichotomy between the fluidity perspective
of the macroscopic level of magnitude and the quantized perspective of the
subatomic level also applies to electromagnetic energy. It is at this point that
Maxwell's interpretation brings in concepts that are absent from the Lorenz
gauge approach and that resolve this problem at the quantized level of localized
photons and other charged and massive elementary particles, namely the
different spatial orientation of the oscillation of the E-field energy with respect to
the temporal orientation of the B-field energy oscillation, the displacement current
related to the E-field oscillation, and the implicit mutual LC-induction of the E-
and B-fields that Maxwell contributed with his theory.
In summary, the two possible representations of continuous electromagnetic
waves established by Maxwell and Lorenz are illustrated with Figs. 1 and 2 as an
oscillating electromagnetic pulse of intimately related E- and B-fields, the two
fields being spacewise offset by 90°, oscillating transversely on longitudinal
planes according to the classical concept of a wave propagating by transverse
oscillation in an elastic medium.
But whereas Lorenz represents them as timewise reaching simultaneously their
maximum intensity (Fig. 2), Maxwell initially conceived them as timewise
alternately reaching their maximum intensity while being 180° out of phase (Fig.
1), by introducing the concept of displacement current linked to the E-field as the
mechanical cause of the induction of the B-field, which, when reaching its
maximum intensity, reduces the E-field to zero, as in the well-known LC relation,
at which moment the B-field, being symmetrically out of balance, will re-induce
the E-field while in turn falling to zero, thus establishing the complete loop of one
cycle of the frequency corresponding to the energy of the pulse in propagation.
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Fig. 1. Spacewise orthogonal E- and B-fields transverse oscillation
representation of an electromagnetic pulse propagating in an underlying
elastic medium – defined as the aether – spacewise dephased by 90° and
timewise dephased by 180o, and mutually inducing each other, involving
the assumed existence of a displacement current as conceived by Maxwell
Fig. 2. Standard spacewise orthogonal E and B fields transverse oscillation
representation of an electromagnetic pulse propagating in an underlying
elastic medium – defined as the aether – spacewise dephased by 90° and
simultaneously peaking timewise in phase to maximum intensity,
corresponding to the Lorenz gauge interpretation
Maxwell's concepts of a displacement current and of E- and B-fields treated as
separate entities inducing each other by means of an LC oscillation, proved to be
superfluous and even brought an unnecessary level of complexity to the
treatment of electromagnetic energy as a continuous wave, and this is what
contributed to the Lorenz gauge approach being initially preferred. But these
additional features of Maxwell's theory now prove to be the elements required to
allow the establishment of the uninterrupted sequence of energy conversion
processes that mechanically establish the known sequence of stable elementary
particles quantized resonance states, progressing in intensity from the freely
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moving photon to the more intense states of the nucleons forming atomic nuclei,
which are listed in Section 7.
Let us mention at this stage of the analysis that Maxwell's initial interpretation,
more adapted to the treatment of quantized states of electromagnetic energy at
the subatomic level, does not disqualify in any way the Lorenz gauge
perspective, that proved to be totally appropriate for the treatment of
electromagnetic energy as a single continuous electromagnetic field at our
macroscopic level of magnitude, in the same way that treating water molecules
as quantized at the molecular level does not disqualify the treatment of water as
a fluid at our macroscopic level.
Maxwell conceived the motion of light in vacuum as involving a transverse
oscillation of the E- and B-fields of light energy on two longitudinal planes,
spatially offset by 90o from each other to explain the velocity of light in the
longitudinal direction in vacuum (Figs. 1 and 2) by means of an adaptation of the
classical mechanics wave equation, by similarity with a wave propagating along
an elastic cord, as analyzed in Reference [27].
2
2
L
2
2
t
y
F
m
x
y
that becomes once the mL /F constant is resolved
2
2
22
2
t
y
v
1
x
y
(1)
This equation establishes that energy pulses propagate longitudinally by
oscillating transversely on the longitudinal plane of the motion of the wave – i.e.,
when matched to the electromagnetic E- and B-fields mutually perpendicular
vectors, the electromagnetic pulse is represented as propagating on two mutually
perpendicular planes that remain parallel to the direction of motion of the wave
(Figs. 1 and 2).
2
2
00
2
2
t
εμ
x
BB
and
2
2
00
2
2
t
εμ
x
EE
(2)
with constant ε0μ0 resolving of course to 1/c2 by similarity with classical reference
Equations (1), thus establishing the related velocity as the absolutelyinvariant
speed of light in vacuum, given that both ε0 and μ0, the only two parameters that
define the speed of light are themselves known to be absolutely invariant in the
close to zero density of the vacuum medium.
The almost immediate adoption by the community of the Lorenz gauge approach
(Fig. 2), since it was easier to use for mathematical generalization purposes, led
to the Lorenz gauge eventually becoming the foundation for all subsequent
electromagnetic developments to this day, such as QFT from which Quantum
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Electrodynamics (QED) emerged; which also led, in the absence of continued
reference to Maxwell's alternative possibility, to the disappearance from the
collective awareness that Maxwell's original conclusions involved a displacement
current and that the E- and B-fields had separate and equally important functions
in his theory, and to the assumption by most of the community that the Lorenz
gauge approach was in agreement with Maxwell's own conclusions.
3. THE EVOLUTION FROM THE 3D+1 VECTORIAL SPACE
GEOMETRY TO THE 3X3D+1 VECTORIAL SPACE GEOMETRY
The very simple and easily confirmed Biot-Savart law, used by Paul Marmet to
derive Equation (30) cited further on, to reveal for the first time the simultaneous
increase with velocity of the magnetic field and of the mass of electrons moving
in a wire [28], is the perfect example to explain the triple ontological orthogonality
of electromagnetic energy, corresponding to the well-established vector cross-
product of the E- and B-fields vectors, resulting in a third velocity vector
perpendicular to the first two (Fig. 3a). Configuration 3b and 3c will be addressed
further on.
Fig. 3. Major and minor unit vector sets applicable to the trispatial
geometry
When electrons are set in motion in a wire by applying voltage to it, a
macroscopic magnetic B-field instantly develops about the wire that can easily be
directly detected with a very ordinary magnetic compass, and whose energy
direction of motion about the wire is oriented very precisely perpendicular to the
direction of motion of the flow of electrons in the wire. It is well established that
the flow of electrons moving from the negative end of the wire towards the
positive end occurs at the surface of the wire, each moving negative electron
remaining strongly attracted all along its progression on the outside surface of
the wire to the closest positive atomic nucleus that it happens to pass by in the
wire; that is, a direction of interaction between the electrons and the atomic
nuclei that establishes the electric E field as being oriented perpendicular to both
the direction of motion of the electrons flow at the surface of the wire on one
hand, and to the direction of motion of the energy of the B field about the wire as
revealed by the compass on the other. This triple orthogonality can now be easily
visualized as corresponding to Fig. 3a.
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In the early 1930's, about 30 years after Einstein published his first 1905 paper
on the issue of the possible permanent maintenance of the localization of
electromagnetic energy as quanta moving on specific trajectories after emission
instead of its spherical propagation from its point source as conceived by
Maxwell, Anderson [19] observed experimentally that localized photons of energy
equal to or greater than 1.022 MeV, easily converted into charged and massive
electron-positron pairs moving separately in space, the two particles being
eventually measured as identical in all respects, except for the signs of their
equal and invariant charges, to which a negative sign for the electron and a
positive sign for the positron were attributed by convention.
This drew attention to the need for a consistent mechanical explanation of this
confirmed process of conversion of the energy of a localized electromagnetic
photon in free motion, then assumed to be electrically neutral and massless, into
a pair of massive and charged electron and positron, that stabilize in stable
stationary resonance states – each with an invariant rest mass of 9.10938188E-
31 kg, an invariant unit charge of 1.602176462E-19 Coulombs, and whose
invariant energy content oscillates at the stable invariant frequency of
1.235589976E20 Hz, corresponding to the electron Compton wavelength
(λc=2.426310215E-12 m).
3.1 Calculation of the Recall Constant and of the Restoration Force
of Electrons and Positrons
We will now examine with Fig. 4 the manner in which the energy of a 1.022 MeV
photon is known to convert to a pair of charged and massive electron and
positron, first observed by Anderson in the 1930's [19] and analyzed in
Reference [29].
To illustrate the mechanics of this conversion, Fig. 4 does not represent the
magnetic field energy ∆B, since this energy will be considered at the moment
when it has completely converted to the twin oscillating charges of the photon,
represented at their maximum value in Y-space.
Fig. 4. A 1.022 MeV photon decoupling into an electron-positron pair
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Fig. 4a depicts a 1.022 MeV photon before destabilization, half of whose energy
is its momentum energy ∆K, and the other half is depicted as the instantaneous
moment when its two electrical components reach their maximum distance
αλC/2πapart from each other, while its magnetic aspect reaches zero presence,
and where λC is the electron's Compton wavelength, that represents half of the
energy of this 1.022 MeV photon.
As unexpected as this may appear, it turns out that the classical spring equation
of Hooke's law also applies to the electromagnetic oscillating motion of the elastic
energy substance of which photons and electrons are made, as established in
Section XXIII of Reference [30].
kxF
(3)
In relation with which the following classical work/energy equation was
established for the case of an element subjected to elastic stretching:
2
kA
E2
(4)
Considering that Fig. 4a reveals that in the case of the 1.022 MeV photon, two
elements are subjected to elastic stretching, Equation (4) will be multiplied by 2
to account for this double relation:
2
2kA
2
kA
2E
(5)
It was established in Reference [31] that even though the energy of localized
photons is represented with traditional equation E=hc/λ, λ being the distance that
a photon travels while one of its transverse electromagnetic cycles is completed,
the transverse amplitude A of this oscillation on the transverse plane will be, with
reference to Fig. 4a:
2
Ax
(6)
And that the energy E of any related electromagnetic quantum can be resolved
by any of the following relations, the last of which having been established in
Reference [31], then the energy related to the electron Compton wavelength λC
will be:
j14E187104139.8
2
ehc
hE
C0
2
C
CC
(7)
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This is what allowed establishing the electrostatic elastic recall constant k for this
photon from Equation (5) in the following manner via a method different from that
used in 2013 [30], using the definition of the amplitude A obtained from Equation
(6) and the rest mass energy of the electron obtained with Equation (7):
2
2
C
2
C
Cm/j16E031019177.1
2
14E187104139.8
A
E
k
(8)
Thus, as illustrated in Fig. 4a, as the oscillating half of the photon's energy begins
to move away from the neutral x=0 position to reach its maximum amplitude
x=A=αλ/2π, a force named the restoring force in Hooke's law, because it is
exerted in the direction opposite to the displacement – hence the minus sign in
Equation (8) and in the following Equation (9) – begins to apply and reaches its
maximum intensity at the maximum amplitude of the transverse oscillation, that
is, a restoring force that will inevitably tend to bring the two charged components
back toward the neutral electric amplitude x=0 in Y-space, at which point whose
energy will have momentarily completely evacuated Y-space while
simultaneously reaching its maximum magnetic presence in Z-space:
Newtons05350473.29
2
kkxF C
(9)
How can we now verify that this figure is correct? Since force F is proportional to
kx in Equation (9), and that it was calculated with an amplitude A=αλC/2πwhich is
very precisely 137.0359998 times shorter than the amplitude related to the
electron's Compton wavelength λC/2π. If we multiply Equation (9) by α once, we
will get the force applicable to the longer amplitude distance related to the
electron's Compton wavelength λC:
Newtons212013666.02kF C
2
(10)
Now, the numerical figure obtained with Equation (10) is not really familiar and
does not really provide so obvious a confirmation that the figure obtained with
Equation (9) is valid. But we know that all frequencies related the stable states in
atoms are quantized in an increasing scale of precise resonance frequencies, so
it could be expected, assuming at this point that α might precisely be the required
frequency multiplier involved, that repeating the multiplication process should
eventually cause us to hit upon a familiar force value, which would then really
confirm the validity of the starting Equation (9).
About this increasing sequence of resonance frequencies/wavelengths, we know
also that the energy induced at the Bohr atom rest orbit is equal to the electron
rest mass energy multiplied by α2. Since force is proportional to energy, we can
further find the force associated with the oscillation amplitude of the energy of a
photon of same energy as that induced at the Bohr rest orbit by further
multiplying by α2:
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Newtons5E12900148.12kF C
42
(11)
But this photon is obviously moving at velocity c. We know also that force is
proportional to velocity, and we know further that the theoretical velocity at the
mean Bohr rest orbit is equal to c multiplied by α. Consequently, a final
multiplication by α should provide the well known force associated to the distance
at which the electron stabilizes from the proton when captured in the hydrogen
atom ground state orbital, whose mean distance from the proton is precisely the
Bohr radius:
Newtons8E238721808.82kF C
52
(12)
Which tentatively confirms, as initially calculated in Reference [30], that
electrostatic recall constantk=-1.031019177E16 j/m2 calculated with Equation (8)
would apply to all charges in existence, be they the pair of varying intensity
charges oscillating within localized photons or separated stabilized pairs of
charges such as the electron, the positron, the stabilized fractionary charges of
the inner scatterable subcomponents of protons and neutrons, or the charge of
the electron and the composite charge of the proton within a hydrogen atom, and
even the pairs of varying intensity neutrinic charges oscillating within stabilized
electron and positron masses.
3.2 The Origin of the Coulomb Force
Ever since the Coulomb force was discovered in relation with the discovery of
electrostatic attraction between opposite signs charges and repulsion between
same sign charges, the question remained open as to the ontological cause of
the Coulomb force.
As analyzed in Reference [32] the repulsion between same sign charges can be
neglected at the macroscopic level between elementary particles, since the
energy induced in these particles decreases to such an extent as they move
away from each other that the effect of such repulsion becomes infinitesimal and
imperceptible at the macroscopic level between any pair of such particles.
Therefore, only the energy growing with decreasing distances between opposite
charges provided by the Coulomb restoring force established in the previous
section will be considered for the rest of our analysis.
As an example of how negligible at our macroscopic level electrostatic repulsion
between same sign charged particles really is, we only need to touch our thumb
with our index finger to become aware that this touching contact involves the
mutually repelling electrons of the outer layers of the atoms of which both fingers
are made.
If one considers that even at our macroscopic level, when stretching an elastic
cord, for example, the restoration force begins to exist only when the elastic cord
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begins to be stretched, ever so slightly, from its unstretched resting state, and
that such a moment of zero tension also exists during the constant oscillating
motion of the photon energy, one can also consider that the Coulomb restoring
force involved may also not exist during that fleeting moment in which the
transverse amplitude x is momentarily equal to zero in Y-space, as illustrated in
Fig. 4a, while the oscillating energy is simultaneously momentarily immobilized at
maximum presence in magnetostatic Z-space.
This opens the door to considering the possibility that the Coulomb force could
not even exist without the prior physical existence of the fundamental
electromagnetic energysubstance, and that its cause could be related to the
intrinsic properties of this energy substance. The 4 properties identified in
References [33] and [34] that this fundamental substance must have to allow a
mechanical explanation of the behavior of localized photons, turn out to be
essential for the existence of the Coulomb force to even be possible. They are a
property of elasticity, which allows the energy substance to stretch and contract
due to a property of fluidity, without its volume varying, due to a property of
incompressibility, and finally a property of tending-to-always-remain-in-motion
that renders it physically unable to remain immobile.
As analyzed in depth in References [35] [36] [37], the first step for a pulse of
magnetic energy ejected from a fixed length dipole antenna to set itself in motion,
can only be for half of this energy to self-orient transversely to the other half – a
half-half partitioning, for symmetry considerations – in order to provide the
required ds fulcrum for the other half to press against and propel the transverse
half in vacuum, which only an intrinsic property of the energy substance such as
a tendency-to-always-remain-in-motion can logically trigger. As mathematically
confirmed in References [36] [37] [38], this symmetric half-and-half division
between a longitudinally oriented propulsive energy component ∆K and a
transversely oriented propelled energy component already establishes the
absolute invariance of the speed of light in vacuum – see Equation (14) below.
As the magnetic pulse began to self-distribute in two equal parts by causing
some of its substance to move in a transverse direction, the motion of this
transversely oriented energy could proceed only by self-distributing as two
quantities moving in opposite directions – again due to symmetry considerations
– thus initiating the elastic stretching represented in vectorial Y-space as
illustrated with Fig. 4a.
This distribution as two quantities elastically moving away from each other
immediately triggers the coming into being of a restoring force related to an
increasing elastic return intensity that will reach the constancy of the k level when
the maximum amplitude of the oscillation is reached, an intensity that universally
stabilizes at the maximum level of exactly e=1.602176462E-19 Coulomb for a
separating pair, which is the maximum charge intensity reached by electrons and
positrons in the universe as each pair decouples, physically separating into equal
parts the entire amount of 1.022+ MeV energy of the photon from which they
originated, and which then remains at this maximum return intensity in all
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electrons and positrons in the universe, as illustrated with Fig. 4d. It could even
be considered that the unit charge of electrons and positrons is nothing other
than the fundamental elastic recall intensity constant of the universe.
As they reached maximum transverse separation, due to the incompressibility
property of its substance, and the requirement for motion now having been
satisfied for the ∆K momentum energy half, and given that longitudinal motion is
now forbidden for the two energy components now in motion of the transverse
energy half, the only avenue for it to obey its tendency-to-always-remain-in-
motion after reaching maximum extension turn out to start symmetrically moving
backwards towards the common center-of-presence that they share with the now
fully extended momentum energy component, and the only mechanical way that
the incompressible volume of the returning energy substance will allow it to
continue moving will be for it to start moving symmetrically in a third direction by
expanding omnidirectionally as an energy sphere in what is represented by the
vectorial Z-space of Fig. 6c shown further on.
Having then completely evacuated Y-space as its amplitude reaches zero in this
space and as its volume reaches maximum in Z-space, to continue moving, the
energy will then start moving back to Y-space as the two separate elements
moving away from each other illustrated with Fig. 6e, initiating the second cycle
of the now established LC oscillation of the electromagnetic energy quantum,
moving at speed c in the vacuum of normal X-space, whose set of centers-of-
presence of all existing photons establishes a level 0trispatial vector field weakly
interacting with each other, all moving at the speed of light in all directions in the
universe.
Any trispatial photon of level 0 reaching the threshold intensity of 1.022 MeV is
then likely to decouple into an electron-positron pair as represented in Fig. 4,
whose set of 1.602176462E-19 Coulomb intensity level centers-of-presence
would establish two opposite signs trispatial vector fields, one on either side of
the zero level field, each element of which constantly seeking to join any
1.602176462E-19 Coulomb intensity level element of the opposite signs level 1
field with the recall intensity established with Equation (8), thus defining a
trispatial gravitational field of level 1 trispatial vector complexes, and each
element of which is accompanied by a level 0 photon induced by the Coulomb
interaction that allows it to move or apply pressure depending on the energy level
of this photon and of the local electromagnetic equilibrium. The level 2 trispatial
vector complexes of the stable trispatial gravitational field will be described later.
It could thus be tentatively concluded that the very existence of the Coulomb
restoring force could be ultimately due to the ontological existence of this
property of the fundamental energy substance of always-tending-to-remain-in-
motion.
3.3 The Decoupling of 1.022 MeV Photons
Fig. 4b illustrates the beginning of the destabilization process of the return motion
of the two charged elements towards each other that prevents them from directly
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returning toward each other, as illustrated in Fig. 4a, that initiates the usual
transfer process to Z-space. This deflection of their return trajectories initiates an
orbital motion around the photon center-of-presence that will inexorably drive
them to reach a circular escape orbit, as analyzed in Reference [29], i.e., a
motion sustained by the momentum energy ∆K of the photon that gradually
transfers into Y-space to provide the increased energy needed to establish this
circular orbit, which will be established when both elements simultaneously reach
the speed of light on that orbit, as shown in Fig. 4c, at which point they will
separate to move separately into X-space, sharing the remaining energy of the
initial photon, as shown in Fig. 4d. The complete mechanical decoupling cycle of
the initial photon into an electron-positron pair is analyzed in depth in reference
[29].
This confirmed conversion of freely moving electromagnetic photons into
massive charged elementary particles, first observed by Anderson in 1933 [19],
is what confirmed the electromagnetic nature of the energy of which their mass is
made, a confirmation that directly invalidated the conclusion reached by the
community in 1907, according to which the electron could only be a mass in the
kinematic sense of the term as defined in classical mechanics, and brought to
light the fact that this invariant rest mass energy was then also likely to be
represented as half of it corresponding to an invariant E-field, given the
invariance of its unit charge, while the other half could only correspond to an
oscillating B-field – oscillating, given that no other portion of the total amount of
energy of the invariant rest mass of the electron remains available to explain the
oscillation frequency related to the known Compton wavelength of the electron
rest mass energy (λc) –, i.e., a magnetic B-field as discovered by Marmet in 2003
[28], whose oscillation, already determined by Maxwell to mandatorily apply in
the case of free moving electromagnetic energy, was experimentally confirmed
by similarity with the experiment published in 2013 [38], and was directly
confirmed to apply to electrons with the 2014 Kotler et al. experiment [39]. See
Section 10 regarding this experimental confirmation issue.
A first telltale as to which direction should be investigated to allow establishing
this mechanical conversion illustrated by Fig. 4 was provided by Louis de Broglie
when he concluded in 1937 that 3D/4D space geometry was too restrictive to
allow exactly describing and explaining the existence of elementary particles:
"… la non-individualité des particules, le principe d'exclusion et l'énergie
d'échange sont trois mystères intimement reliés : ils se rattachent tous
trois à l'impossibilité de représenter exactement les entités physiques
élémentaires dans le cadre de l'espace continu à trois dimensions (ou plus
généralement de l'espace-temps continu à quatre dimensions). Peut-être
un jour, en nous évadant hors de ce cadre, parviendrons-nous à mieux
pénétrer le sens, encore bien obscur aujourd'hui, de ces grands principes
directeurs de la nouvelle physique."
Louis de Broglie 1937 ([40], p. 273).
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"... the non-individuality of particles, the exclusion principle and exchange
energy are three intimately related enigmas; all three are tied to the im-
possibility of exactly representing elementary physical entities within the
frame of continuous three dimensional space (or more generally of
continuous four dimensional space-time). Some day maybe, by escaping
from this frame, will we better grasp the meaning, still quite cryptic today,
of these major guiding principles of the new physics".
It so happens that the conditions established by de Broglie in the 1930's for all
symmetry requirements to be respected and for Maxwell's equations to be
complied with – analyzed in depth in [33] [34] – can be satisfied for localized
electromagnetic quanta if the self-sustaining oscillation occurs on a
planeperpendicular to the direction of motionof the energy in space, a plane
already hinted at by the traditional plane wave treatment of energy (Fig. 5a), and
that did not require an elastic medium in which to propagate, if related to an
amount of momentum energy that would provide for the propagation of the
transversely oriented energy quantum that would be oscillating in standing mode.
Fig. 5. Comparison between traditional plane wave treatment of the energy
of an electromagnetic energy pulse that would be spherically expanding in
an underlying medium (the aether) from its point of emission (Fig. 5a), and
treatment of the same energy pulse remaining localized as it propagates
without spherically expanding, requiring no underlying medium according
to Einstein's conclusion [41] and de Broglie's conditions [40] (Fig. 5b)
Let us recall that plane wave treatment in the traditional spherical expanding
wave perspective involves treating an infinitesimally small ds surface section of
the wavefront, assumed flat due to the infinitesimal curvature of such a small
portion of the surface of a sphere, to calculate the same amount of energy
emitted at the point-like source of the wave (Fig. 5a), as if it was not spherically
distributed. This method mathematically provides the same amount of energy
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emitted at the source and measured at its point of absorption as if the emitted
energy quantum had remained localized all the way to its point of absorption (Fig.
5b).
"Es scheint mir nun in der Tat, daß die Beobachtungen über die 'schwarze
Strahlung', Photolumineszenz, die Erzeugung von Kathodenstrahlen durch
ultraviolettes Licht und andere die Erzeugung bez. Verwandlung des Lichtes
betreffende Erscheinungsgruppen besser verständlich erscheinen unter der
Annahme, daß die Energie des Lichtes diskontinuierlich im Raume verteilt
sei. Nach der hier ins Auge zu fassenden Annahme ist bei Ausbreitung eines
von einem Punkte ausgehenden Lichtstrahles die Energie nicht
kontinuierlich auf größer und größer werdende Räume verteilt, sondern es
besteht dieselbe aus einer endlichen Zahl von in Raumpunkten lokalisierten
Energiequanten, welche sich bewegen, ohne sich zu teilen und nur als
Ganze absorbiert und erzeugt werden können."
Albert Einstein, 1905 ([41], p. 133)
"In fact, it seems to me that the observations on 'black-body radiation',
photoluminescence, the production of cathode rays by ultraviolet light and
other phenomena involving the emission or conversion of light can be better
understood on the assumption that the energy of light is distributed
discontinuously in space. According to the assumption considered here,
when a light ray starting from a point is propagated, the energy is not
continuously distributed over an ever increasing volume, but it consists of a
finite number of energy quanta, localized in space, which move without
being divided and which can be absorbed or emitted only as a whole."
In Fig. 5b, the vector representation is a freeze of the motion of the oscillating
energy at step 6d of Fig. 6 halfway crossed over into Y-space coming from Z-
space. In this case the condition
∇
∙B=0always applies by structure since all of the
photon energy remains contained within its local oscillating volume, its source
always remaining local to the center-of-presence of the photon all along its
trajectory.
Fig. 6. Representation of the stationary transverse oscillation cycle of the
oscillating electromagnetic half-quantum of a free moving photon or of the
carrier-photon of an electron
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This solution emerged from the long established invariant triple vectorial
orthogonality of the vector cross product of the E and B vectors which is so
fundamental in electromagnetism (Fig. 3a). When the j and k minor unit vectors
of normal space representing the E and B fields are expanded into becoming
fully developedmajor 3D vectorial spaces represented by J and Kmajor unit
vectors, each possessing its own internal ijk minor unit vectors set, a fully
developed major 3D vectorial normal space, represented by a major unit vectorI
emerges by vectorial cross product of the majors vectors J and K, that also
maintains its usual internal set of minor ijk unit vectors (Figs. 3b and 3c).
Thus emerged, for visualization purposes, the 3x3D+1 augmented vector space
that underpins the trispatial model, with the +1 element of course representing
the time dimension. The difficulty for us of mentally visualizing more than 3
perpendicular dimensions at once is solved by treating each of the major 3D
vector spaces I J Kas if they were folded 3-ribbed umbrellas meeting
perpendicularly at their three tips, and which, once folded, reduce the set of
major 3x3D vector spaces to the basic 3D cross product vector representation of
Fig. 3a. Simply opening the umbrellas one by one, allows visualization in
sequence of the movement of the energetic substance as it circulates within each
3D vector space of the set.
The common punctual origin of the three orthogonal vector spaces then becomes
an infinitesimal dV volume through which the energy of the quantum, now
perceived as a physically existing local amount of substance, can now transit
between the three spaces as if they were communicating vessels, to establish
the equilibrium state required by symmetry, and whose infinitesimal ds cross
section serves as a fulcrum against which the momentum energy of the quantum
can apply its pressure to cause motion of the transversely oscillating half when
the local electromagnetic environment allows it.
This entirely new vectorial space geometry effectively allowed logically
representing not only free moving photons, but also to mechanically explain how
such photons of sufficient energy can decouple into pairs of electron-positron as
illustrated with Fig. 4 [29], and also to mechanically explain how triads of
sufficiently thermal electrons and positrons can accelerate to stabilize as the
most energetic triads of elementary electromagnetic particles configurations that
can exist in the universe, that is, protons and neutrons [42], represented as level
2 vectorial complexes in the universal trispatial vector field. See Figs. 14 and 15
further on.
The development of the trispatial vector complex is what allowed the
development in References [33] [34], of the first LC equation of internal
electromagnetic mechanics of the photon (13) in conformity with the conditions
identified by Louis de Broglie as being required for localized photons to satisfy
both the Bose-Einstein statistic and Planck's law, and perfectly explain the
photoelectric effect while respecting Maxwell's equations and remaining
consistent with the properties of Dirac's theory of complementary corpuscle
symmetry ([40], p. 277):
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t)(ωsin K
2
iL
t) (ωcos)jJ,jJ(
C4
e
2iI
λ2
hc
iIE 2
Z
2
2
Y
2
X
(13)
4. THE ESTABLISHMENT OF THE ELECTROMAGNETIC
MECHANICS OF ELEMENTARY PARTICLES
The first step in preparation for the harmonization of kinematic and
electromagnetic mechanics according to Wien's project [7] consisted of course in
reversing the consequences of the 1907 decision that led to the adoption of the
incomplete kinematic theory of Special Relativity, and in finally taking into
account the electromagnetic behavior of the electron observed and measured
during Kaufmann's experiments.
The particularity of this observed electromagnetic behavior of the electron
compared to its previously accepted kinematic behavior is that its transversely
measurable mass increases with velocity, an increase that becomes measurable
only when velocities reach more than 2000 km/s, velocities that were largely
exceeded in the Kaufmann bubble chamber.
What allowed reversing this long established perspective was the publication,
shortly after the trispatial geometry was presented at Congress-2000 [43], of Paul
Marmet's article in 2003 [28] in which he derived an equation from the Biot-
Savart equation, that confirmed that the energy of the magnetic field of an
accelerating electron, known to increase with velocity, was in fact the same
energy that was measured as the increasing electron mass as measured from
the data collected by Kaufmann [1] [2] [3] [4].
This discovery allowed separating for the first time the velocity related ∆B
magnetic field increment of the accelerating electron from the invariant Be field of
its invariant rest mass in an article published in 2007 [31] and observing that the
electron carrying energy had the very same electromagnetic structure as
Equation (1) for free moving photons. The only difference being that in the case
of the electron carrying energy, its momentum component was propelling the
inert rest mass of the electron in additionto also propel its own ∆B field related
∆mm complement inert mass, which is what forever prevents the electron from
reaching the speed of light, because the energy ratio ∆K / (∆mmc2 + m0c2) can
never reach unity as in the case of Equation (1), in which the energy ratio ∆ K /
∆mmc2 is invariably equal to 1/1, which is what sets the velocity of light as an
asymptotic velocity limit for all massive elementary particles, as established with
Equation (14) defined in Reference [37]:
c
x
x
c
x
x
c
x0
x0
c
xa2
xax4
cv 222
(14)
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In which a represents the energy in joules of the electron's rest mass
(E=m0c2=8.18710414E-14 j) and x represents the energy in joules of its carrying
energy. This equation provides the relativistic velocity of the electron on the full
scale of relativistic velocities without any need to use Lorentz's γ factor, and in
which, if the energy of the electron's rest mass a is set to zero, then it will provide
the light-invariant velocity of its carrier-photon as if now moving freely as an
isolated electromagnetic photon. The first step of the kinematic-electromagnetic
harmonization thus involved the incorporation into the equations of kinematic
mechanics of the energy that contributes to the transverse increase of the mass
of the moving electron.
This first step was accomplished by incorporating this magnetic energy into
Newton's kinetic energy equation ∆K= ½m0v2 in order to account for the entire
energy induced by the Coulomb interaction in Kaufmann's experiments in an
article published in 2013 [37], i.e., the electron momentum energy plus the
transverse magnetic energy induced simultaneously.
Equation (14) was established precisely as an outcome of this conversion, that
first involved converting Newton's kinetic energy equation to its electromagnetic
version in Reference [37], using as a confirming numerical example the well
known wavelength of the energy induced at the Bohr radius of the hydrogen
atom as a reference, that happens to provide the mean energy induced in the
ground state electronic orbital of the hydrogen atom:
Joules182E2.17987190
2
mv
2λ
hc
K2
B
(15)
The missing magnetic energy component induced in the electron at the Bohr
radius distance from the proton was then added to the electromagnetic version of
the kinetic energy equation:
2
iL
2λ
hc
cm
2
mv
E2
λλ
2
m
2
(16)
And finally, by combining LC Equation (1) for the photon developed in Reference
[30] with LC Equation (31) – shown further on – for the rest mass of the electron,
developed in Reference [29], Equation (17) was obtained, providing both the
trispatial kinematic energy momentum equation and its electromagnetic version:
2
iL
2λ
hc
2
iL
2λ
hc
cmcm
2
mv
E
2
λλ
C
2
λλ
2
0
2
m
2
Cc
(17)
The process of integrating the electromagnetic versions of all three kinematic
components of Equation (17) into a single ratio of unidirectional energies over
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magnetic energies, to isolate a squared velocities ratio in line with Marmet's
Equation (30) led to the following form in Reference [37]:
2
2
2
2
λλ
2
CC
2
C
C
2
c
v
)i(L)i(2Lλλ
λ4λ(hc)
(18)
from which Equation (14) was derived, as well as Equation (19), from which the
Lorentz γ-factor was derived for the first time in history from an electromagnetic
equation in Reference [37],thus demonstrating that the gamma factor is naturally
embedded into all electromagnetic equations, and is related to the non-rectilinear
variation of the energy adiabatically induced in all elementary charged particles
by the Coulomb interaction and has consequently no relation whatsoever with the
time dilation and/or masses length contraction assumed as a premise in the SR
theory.
2
2
2
C
2
CC c
v
λ2λ
λ4λλ
(19)
These developments then allowed the establishment of the series of interaction
sequences between elementary charged particles that provide an uninterrupted
sequence of causality between the two sets of kinematic and electromagnetic
equations for all mechanical energy conversion processes:
1) from the quantities of unidirectional kinetic energy that constitute the
momentum of elementary charged and massive particles and their
electromagnetic complement, both induced simultaneously and
adiabatically in each charged particle by the Coulomb interaction, whose
mechanics is analyzed in References [16] and [17],
2) to the release as a free-moving electromagnetic photon of any quantity of
this energy that becomes in excess of the precise amount allowed by
some stable or metastable electromagnetic equilibrium state in atoms, for
example, when an electron is suddenly stopped in its forward motion as it
becomes captive of the resonance state of an atom's available orbital after
having accelerated to reach this equilibrium state, due to the resistance of
its forward moving momentum energy to have its state of motion modified
in accordance with Newton's first law of motion, and whose resulting
emission mechanics is analyzed in References [35] and [36],
3) to the creation of electron-positron pairs from the destabilization of free
moving photons of energy 1.022 MeV or more, whose mechanics is
analyzed in Reference [29],
4) to the creation of protons and neutrons from the interaction of thermal
triads of electrons and positrons in volumes of space sufficiently small and
with insufficient energy to escape mutual capture, whose mechanics of
stabilization is analyzed in Reference [42],
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5) to the final shedding in the form of neutrino energy of momentary
metastable excess mass – different from velocity related momentary
relativistic mass increment – as overexcited newly created massive
elementary particles are forced by local electromagnetic equilibrium states
into reaching their lowest possible and henceforth stable and invariant
electron or positron rest mass, whose electromagnetic mechanics of
emission is analyzed in Reference [44].
5. ESTABLISHMENT OF THE RELATION BETWEEN THE ENERGY
OF THE MAGNETIC FIELD AND THE ENERGY OF THE
ELECTRON MASS
It was only after Paul Marmet established the relation between the variable
magnetic field of the moving electron and its variable mass in 2003 that
Maxwell's original interpretation was again brought to the fore as being required
to mechanically explain this relation [28], because it implies by structure that the
E and B fields must induce each other in alternance as hinted at by the 1998
experiment (see Section 10), since it is not physically possible for the ∆B field,
revealed by Marmet's derivation as being induced simultaneously with the ∆K
momentum energy, not to be accompanied by a ∆E field with which it would
alternate, to account for the oscillating frequency of this carrying energy, a
process which must involve by structure the displacement current that Maxwell
conceived of as being involved on the E side of the relationship, that would
induce the magnetic field B increasing to its maximum intensity while the E field
reduces to zero, followed by the re-establishment of the displacement current of
the E field as the B field reduces to zero in turn, thus initiating the individual LC
electromagnetic cycle of the corresponding frequency.
This means, since the Coulomb interaction – linked to the first Maxwell equation
by the relation eE –, which is known to induce in each charged particle twice the
energy of its momentum, that:
0
2
0
21
2
0
21
2
0
2
12
e
d4
qq
d4
qq
d
d4
q
qdqdFdE E
(20)
in which d=x=a=αλ/2π (ref: Equation (6))
That is, the ∆K momentum energy provided for by the traditional relativistic
equation for calculating momentum energy:
1
2
0
cmK
(21)
plus the ∆mmc2 magnetic mass increment revealed by Marmet's derivation, which
is also equal by structure to the same relation:
1γcmcΔm 2
0
2
m
(22)
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Which means, as observed in Reference [37], that in line with equation (20), the
total amount of energy induced in an electron at any velocity will be equal to:
1cm2cmKE 2
0
2
m
(23)
This involves that half of the energy induced in each elementary charged particle
by the Coulomb interaction self-transposes by structure transversely to the
direction of application of its momentum energy, a transversely oriented half that
will then start oscillating on its own between an electric ∆E state and a magnetic
∆B state that provides the relativistic mass increment – i.e. the sum of the
instantaneous energies represented by ∆E + ∆B, or the energy of ∆E or ∆B at
maximum intensity, that is, a mass increment ∆mm –, which is added to the rest
mass m0 of the particle, a sum that turns out to be the total mass propelled at a
given relativistic velocity determined by the simultaneously induced relativistic
momentum energy ∆K.
Marmet's discovery that the B-field of the electron's rest mass energy is only half
of the energy of its rest mass then led to further derivations that allowed
understanding that the B-field of the second term of the Lorentz force equation is
the sum of the invariant Be-field of the electron's rest mass energy, plus the
variable ∆B field of its carrying energy, that oscillates in alternating motion with
the associated ∆E field on planes transverse to the direction of motion of the
electron, this associated ∆E field itself being in vector cross product relation with
the invariant Ee field of the electron's rest mass energy, which means that the
Lorentz force equation:
BvE qF
(24)
can be amended to the following form to describe a straight line motion of particle
q:
)()(qF ee BBvEE
(25)
Let us note at this point, that it is the equal density by structure of the ∆E and the
∆B carrying-energy components – since it is the same amount of energy that
oscillates between the two states – as they reach in alternance their maximum
intensity that causes the default straight line motion of charged particles. What
causes curved trajectories of elementary particle beams that can be calculated
with the Lorentz force equation is the addition of external B fields established in
the environment of the moving charged particle beams, that add their energy to
the ∆B field component of the carrier-photon of the particle induced by the
Coulomb interaction, thus causing the default 1/1 equal ∆E / ∆B energy density
ratio to drift in favor of the density ratio ∆E/(∆B +Bexternal) that applies a transverse
force favoring the magnetic force at the expense of the force exerted by the
electric force, which is what causes these curved trajectories. See Section 9
about this specific issue.
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Let us also note that the Ee and Be fields accounting for both halves of the
invariant rest mass of the electron that are also part of the Lorentz force Equation
(25), for calculation requirement, play no role in guiding the electron, since they
represent the omnidirectionally inert transverse energy of the invariant rest mass
of the electron.
This led to the establishment and publication in 2007 [31] of the first level
equations of these two separate magnetic fields from the specific wavelengths of
the separate energy quanta involved:
2
3
0
e
C
λα
πecμ
B
23
0
λα
πecμ
B
(26)
whose sum provides the first level composite B-field usable in the Lorentz force
Equation (24) to guide electrons on straight line trajectories (The establishment
of the composite B field defining curved trajectories will be addressed in Section
9):
2
23
2
2
0
C
C
e
ec
BBB
(27)
Similarly, the corresponding first level invariant Eefield of the other half of the rest
mass energy of the electron and the variable ∆Efield of its carrying energy could
be separated in the same reference [31]:
2
3
0
e
C
λαε
πe
E
23
0λαε
πe
E
(28)
whose vectorial cross product – given that their energies are oriented
perpendicular to each other within electrostatic Y-space, the Ee-field energy
being oriented in the Y-x direction and the ∆E-field energy being oriented in the
Y-y direction – provides the first level composite E-field component used in the
Lorentz force Equation (24) in relation with the same density B-field to guide
electrons on straight line trajectories:
CC
CCC
e
2
4
2
2
2
2
3
0
eΔEEE
(29)
Then, from the electromagnetic definition of the invariant magnetic rest mass M0
of the electron, amounting to exactly half the invariant m0 mass of the electron,
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that emerged from Marmet's so critically important equation (30), numbered
"Equation 23" in his article [28]:
2
2
e
2
e
22
0c
v
2
m
cr8π
veμ
M
, leading to
2
m
r8π
eμ
Me
e
2
0
0
(30)
an LC equation could be derived in Reference [29] to describe both the invariant
electric energy corresponding to the electric charge localized in Y-space and the
invariant energy transversely oscillating between spaces X and Z corresponding
to the magnetic field of the invariant rest mass of the electron:
t)(ωsin
2
iL
t)(ωcos
4C
e'
2
2λ
hc
cmE 2
Z
2
2
X
2
Y
C
2
eCC
C
(31)
And for its carrying energy, an LC equation identical to Equation (13) previously
derived for free moving photons could also be derived, representing its
momentum energy residing in X-space while its magnetic energy oscillates
between spaces Y and Z:
t)(ωsin
2
iL
t)(ωcos
4C
e
2
2λ
hc
E2
Z
2
2
Y
2
X
(32)
This is what allowed understanding that the varying carrying-energy of the
electron in motion had the exact same electromagnetic structure as localized free
moving photons, whose internal electromagnetic structure was hypothesized by
Louis de Broglie in the 1930's in Reference [40], hence the name of carrier-
photon given afterwards to the carrying energy of the electron in numerous other
articles of the electromagnetic mechanics project.
As previously mentioned, this development in turn allowed logically converting
Newton's non-relativistic momentum kinetic energy equation:
2
0vm
2
1
K
0
m
K2
v
(33)
to its electromagnetic equivalent by integrating the missing magnetic energy
component revealed by Marmet's revolutionary Equation (30) in the
establishment of LC equations (13) and (31):
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2
λλ
2
2
c
C
2
iLi2Lλλ
λ4λ
hcv
CC
(34)
From which, two equations were derived in Reference [37] that provide the full
range of relativistic electron velocities from the theoretical zero m/s velocity to
near the asymptotic limit for massive particles of the speed of light, either from
the wavelengths of the energy of the rest mass of the particle and of its carrier-
photon, or directly from the corresponding energy quanta in joules:
C
CC
λ2λ
λ4λλ
cv
K2E
K4EK
cv 2
(35)
In the same reference, was established the fact that the magnetic ∆B-field
energy increment that accounts for the velocity related mass increment ∆mm
observed in Kaufmann's data, is always equal in quantity to its ∆K accompanying
relativistic momentum energy amount, that can be calculated with the traditional
relativistic kinetic energy Equation (21), which means that the instantaneous
relativistic mass of the moving electron can easily be calculated without using the
γ-factor, simply by dividing by 2 the ∆E amount of carrying energy calculated with
Equation (20), or setting it as equal to ∆K or calculating it with Equation (21) by
direct identity with the energy calculated for ∆K:
2
0m0 2c
E
mmmm
(36)
and that the corrected energy-momentum equation that accounts for both the
relativistic momentum energy and the related relativistic mass increment to be
added to the rest mass of the electron can be represented by the following
equation, by simply adding the total amount of energy ∆E induced in the electron
calculated with Equation (23) to the invariant rest mass energy m0c2 of the
electron, to establish the simplified trispatial energy-momentum equation:
2
0
2
m
2
0e cmcΔmΔKcmΔEE
(37)
6. THE RELATIONSHIP BETWEEN PLANCK'S CONSTANT AND THE
RESONANCE FREQUENCIES OF ELECTRONIC ORBITALS
At the beginning of the 20th century, Planck discovered a major relationship
between the different frequencies of the blackbody radiation that Wien had
recently discovered to be quantized. He observed that the correct energy of any
frequency recorded was systematically obtained by the product of the frequency
by a smallest common multiplier that he had calculated to have the very precise
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value of 6.62606876E-34 that was then symbolized by the letter h and given the
name of Planck's constant to honor his discovery:
hE
and since
c
then
hc
E
(38)
In his 1924 doctoral thesis, Louis de Broglie then succeeded by a brilliant
deduction to relate Planck's constant to the theoretical ground orbit of the
electron in the Bohr atom and to bring to light the fact that all frequencies emitted
by the hydrogen atom are integer multiples the energy frequency related to the
Bohr ground orbit. The fundamental reference that he established involved the
length of the Bohr orbit – λ=2πR, R being the Bohr radius – and the classical
momentum equation p=mv applied to the electron rest mass m0 on the idealized
circular Bohr orbit:
pvmh 0
(39)
And rearranging:
h
p
(40)
Substituting p of Equation (40) for h/λ in Equation (38), the well known
momentum energy equation E=pc applicable to localized photons was obtained:
pcc
h
E
(41)
These well known equations are generally mentioned in popular textbooks, such
as References [45] and [46], without explaining how de Broglie related Planck's
constant to the resonance frequencies of the electron in the hydrogen atom
orbitals in his doctoral thesis published in 1924 by the French Académie des
sciences. Due to its radical ideas, the peer-reviewers Jean Perrin, Paul Langevin,
Elie Cartan and Charles Maugin sought Einstein's advice during the review
process to obtain his opinion, which resulted in Einstein bringing it to
Schrödinger's attention:
"The examining board, perplexed by apparently radical ideas of de Broglie,
asked Albert Einstein (1879-1955) whether the thesis deserved a doctoral
degree. Einstein responded quickly by saying that the thesis deserved a
Nobel Prize rather than a doctoral degree. Einstein recommended the thesis
to Schrödinger, which resulted in celebrated Schrödinger equation."
Nishimura, H. (2021) ([50], p. iii)
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His thesis was finally translated to English only in 2021 by the Minkowski Institute
[50]. It may seem surprising that such an important historical document remained
available only in French for almost one century, but it was not unusual in the first
half of the 20th century for scientific articles published in Europe not to be
translated to a common language, given that most European scientists were
generally multilingual.
This could not be better illustrated than by a thank you note found in the
introduction page of a 1900 major Dutch archive of exact and natural sciences
publication, to the authors who had accepted to write their contribution either in
French, in German "or" in English ([48], p.10), which clearly suggests that most
researchers and potential readers of the archives of this era were expected to be
familiar with at least these three languages.
When English became the standard formal publication language in the mid 20th
century, Quantum Mechanics was already a well established science that now
drew more attention to the complementary statistical developments of
Heisenberg and the recent addition of Feynman's path integral rather than to
Schrödinger's wave equation and its underlying de Broglie hypothesis, both
having by now become a part of history, and which were no longer attracting
sufficient attention to be translated for further study. Such historical scientific
documents are now progressively being translated to English by institutions such
as the Minkowski Institute to become available to the international scientific
community.
Let us now examine this equation from de Broglie that so revolutionized
fundamental physics [49] [50]. Here is how he introduced his equation:
"Dans le cas particulier des trajectoires circulaires dans l'atome de Bohr, on
obtient:"
"In the particular case of circular trajectories in the Bohr atom, we obtain:"
hnvmR2πdlvm 00
(42)
Before analyzing in detail this simple classical kinematic mechanics equation that
so deeply revolutionized physics, let us review how he conceived of this relation,
also explained in introduction to References [51] [52]. Here is de Broglie's
description in his own words of the observation that he published in 1923 that led
him to this major conclusion:
"L'apparition, dans les lois du mouvement quantifié des électrons dans les
atomes, de nombres entiers, me semblait indiquer l'existence pour ces
mouvements d'interférences analogues à celles que l'on rencontre dans
toutes les branches de la théorie des ondes et où interviennent tout
naturellement des nombres entiers." ([53], p. 461).
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"The occurrence of integers, in the laws of quantified motion of electrons
in atoms seemed to me indicative of the existence for these motions of
interferences analogous to those met in all branches of wave theory,
where integers naturally occur".
Shortly after, he published another note in Les Comptes rendus de l'Académie
des Sciences in which he was proposing a preliminary interpretation of the
conditions that might explain the stability of the electron within atomic structures
[54].
The critical conclusion of this note is the following:
"l'onde de fréquence ν et de vitesse c/β doit être en résonance sur la
longueur de la trajectoire. Ceci conduit à la condition:"
"The wave of frequency ν and velocity c/β must be in resonance on the
whole length of the trajectory. This leads to condition:"
nhT
β-1
cβmr
2
22
o
n being an integer (43)
which is the stability condition determined by Bohr and Sommerfeld for a
trajectory being run at constant velocity as noted by de Broglie [54].
The following year, de Broglie published two more notes [55] [56], to which he
refers in Reference ([53], p. 462), in one of which he mentioned that from this
viewpoint, Bohr's famous frequency condition law could be interpreted as
involving some sort of beat or pulsation (un battement in the original French text),
that is, a resonance state associating the frequency of the emitted wave to the
initial electron stationary state and to its final stationary state. And then he
submitted his doctoral thesis to the examination board.
We observe that the electron momentum energy p=m0v is part of the resolved
integral in Equation (42). It was already understood at the time that when an
electron is captured by a proton to form a hydrogen atom, its momentum kinetic
energy is liberated in the environment just as when a macroscopic mass is
suddenly stopped in its motion. In the case of the electron, this sudden capture
causes it to stabilize in the ground state of the hydrogen atom at the Bohr radius
mean distance from the proton, well established to be R1=5.291772083E-11 m,
corresponding to integer value n=1 in Equation (42).
The related emitted bremsstrahlung photon is well established to have an
amount of energy equal to 13.60569162 eV, which when converted to joules
gives:
j18E179871902.219E602176462.160569162.13K
(44)
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The classical velocity of the electron on the theoretical Bohr orbit at the Bohr
radius distance can then be established with Equation (33):
m252.2187691
31E10938188.9
18E179871902.22
m
K2
v
0
(45)
The length of the Bohr orbit is then:
m10E32491846.311E291772083.52R2 1R1
(46)
Having now on hand the numerical values of every element of the resolved
integral of Equation (42) for the first orbital, we can only imagine the surprise that
de Broglie must have felt in obtaining Planck's constant by numerically resolving
his equation, as also explained in References [57] [58]:
sj34E626068757.6vmh 0R1
(47)
That is, an equation which is at the origin of the introduction by Heisenberg of his
uncertainty relation ∆x
≅
h/m∙∆vx in relation with de Broglie's previously proposed
intuition in Reference [54] that the electron had to be in resonance about its
ground state trajectory in the hydrogen atom, and moreover, which provided
confirmation that all allowed orbitals of the electron in the hydrogen atom had to
be integer multiples of the ground state orbital constant, a condition that he had
previously suspected as mentioned in Reference [53], with the now added
clarification that their energies could only be multiples of Planck's constant.
Is it surprising then that Einstein would have immediately told the Sorbonne
board of reviewers upon having been consulted as to the value of de Broglie's
discovery, that he deserved a Nobel Prize rather than a doctoral degree [50], for
having resolved an issue that had mystified the community ever since Planck had
calculated this constant from the blackbody radiation, that is, discovering a way
to derive this constant from an already established equation on top of having
demonstrated that the energy spectrum emitted by the hydrogen atom involved
an integer related resonance sequence!
Incidentally, despite the fact that de Broglie derived Planck's constant as far back
as a hundred years ago from a simple classical kinematic mechanics equation,
and that it was also derived from an electromagnetic equation in 2013 [30], it is
put in perspective in References [57] [58] that it is even definable by a precise set
of well-established fundamental constants (h=e2/2ε0αc), despite still being
considered in the community as a calculated constant, not derivable from first
principles.
Planck's constant is then directly related to the length of one orbit that the
electron would theoretically travel at Bohr radius distance from the proton in a
hydrogen atom while moving at classical velocity 2187691.252 m/s, each orbit
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taking 1.59186E-16 second to be complete, and that add up to a number of orbits
traveled per second exactly equal to the frequency of the energy induced in the
electron at Bohr radius distance from the proton of:
Hz15E580495968.6
10E32491846.3
252.2187691v
1
1R
R
(48)
When multiplied by Planck's constant, this frequency provides the exact amount
of energy induced by the Coulomb interaction at Bohr ground state R1distance
from the proton in the hydrogen atom:
j18E3602818768.4hE 1
RB
(49)
Equation (49) is also directly confirmed by applying the Coulomb Equation (20) to
this distance, both the electron and the proton having the unit charge
e=1.602176462E-19 Coulombs:
j18E359743805.4
R4
e
cmKhE
10
2
2
mR1
(50)
Of course, when the true relativistic velocity 2187647.561 m/s related to the total
amount of energy calculated with the Coulomb equation is used in calculating the
frequency with Equation (48), the exact value of 4.359743805E-18 J is obtained
with Equation (49), which when divided by 2 and converted to eV, confirms the
whole sequence by recovering the energy of the 13.6059162 eV bremsstrahlung
photon that initiated the whole sequence of reasoning that de Broglie followed to
ultimately establish Equation (42).
This is how de Broglie could relate the momentum of the electron on the Bohr
orbit to Planck's constant and then to the frequency and wavelength of the
electromagnetic energy that causes the electron to theoretically move at the
rated velocity by adapting Equation (47), which is only a simplified form of his
historical Equation (42), That is, in summary, the well-known equations (38), (39),
(40) and (41) cited at the beginning of this Section:
vmh 0
=>
c
hh
mvp
=>
hpcE
(51)
This is how the 1924 equation allowed the establishment of the following
equivalence E=pc=hν=∆K+∆mmc2= 4.359743805E-18 jfrom the kinematic and
electromagnetic parameters of the stabilized electron in the ground state of the
hydrogen atom, thus proving that the pc term in Equation (41) really provides the
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total amount of energy induced adiabatically in the electron by the Coulomb
interaction.
So we observe that not only could Planck's constant be calculated from the
experimental data collected by means of the black body experiments, but that it
could also be derived from the very structure of the classical Bohr atom, thus
establishing a clear relation between classical kinematic mechanics and
electromagnetic radiation frequencies. Moreover, Planck's constant could also be
derived from strict electromagnetic considerations in Reference [30] – see
comment following Equation (17) of this Reference – thus relating
electromagnetic mechanics and kinematic mechanics at the most fundamental
level by means of these derivations of Planck's constant from equations
emerging from both mechanics.
7. GROUNDING THE MASS OF THE ELECTRON ON AN
ELECTROMAGNETIC FOUNDATION
A clear demonstration that a common basis can be established to eventually