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Journal of High Energy Physics, Gravitation and Cosmology, 2020, 6, 340-352
https://www.scirp.org/journal/jhepgc
ISSN Online: 2380-4335
ISSN Print: 2380-4327
DOI:
10.4236/jhepgc.2020.63027 Jul. 2, 2020 340 Journal of High Energy Physics, Gravitation
and Cosmology
Electron Shape and Structure:
A New Vortex Theory
Nader Butto
Petah Tikva, Israel
Abstract
Along with all other quantum objects, an electron is partly a wave and partly
a particle. The corpuscular properties of a particle are demonstrated when it
is shown to have a localized position in space along its trajectory at any given
moment. When an electron looks more like a particle i
t has no shape, “point
particle”, according to the Standard Model, meaning that it interacts as if it is
entirely located at a single point in space and does not spread out to fill a
three-dimensional volume. Therefore, in the sense of particle-like interac-
tions, an electron has no shape.
In this paper, a new theory is proposed in
which the electron has a structure and a shape. The central idea is that an
electron is a frictionless vortex with conserved momentum made out of con-
densed vacuum generated in the Big Bang from massless virtual p
hotons that
acquire mass when moving in the vortex at the speed of light. Using Hydro-
dynamics laws and applying them on the superfluid vacuum the basic proper-
ties of the electron are described here forth. This study provides mathematical
models to calculat
e the mass, kinetic energy, density, volume, time, charge,
and particle-wave duality. Such mathematical formulations are presented to
confirm the theory. We conclude that the shape of the electron is accessible to
human imagination, knowing its shape helps
to determine its properties and
shed a light on how matter is made and to explain the interactions of sub-
atomic
particles.
Keywords
Electron Structure, Vortex, Hydrodynamics Laws, Vacuum Density
1. Introduction
The electron is a fundamental particle of nature, is the essential constituent of
electric currents, and together with protons and neutrons, is the most important
How to cite this paper:
Butto, N. (2020
)
Electron Shape and Structure: A New Vo
r-
tex Theory
.
Journal of High Energy Phy
s-
ics
,
Gravitation and Cosmology
,
6
, 340-352.
https://doi.org/10.4236/jhepgc.2020.63027
Received:
May 19, 2020
Accepted:
June 29, 2020
Published:
July 2, 2020
Copyright © 20
20 by author(s) and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
N. Butto
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ravitation and Cosmology
element of an atom’s structure. It is negatively charged and may be free or
bound to an atom in electronic orbits. The electrons are subatomic particles,
play an essential role in numerous physical phenomena, such as electricity,
magnetism, chemistry and thermal conductivity, and they also participate in
gravitational, electromagnetic and weak interactions.
According to current experiments and theories, the electron is a structure-less,
point-like object, made out of nothing [1], and its entire mass is concentrated in
its extension-less center.
An electron looks like a particle when it interacts with other objects in certain
ways (such as in high-speed collisions) and is not point-like, as stated by quan-
tum mechanics.
In quantum mechanics, depending on the observation point of an electron, it
can appear to be a particle or it can appear to be a wave. As a
wave
, one can im-
agine “clouds” of electron orbitals around an atom, which are not physical
things
but rather representations of probabilities.
Conversely, electrons display properties that normally result from an extended
structure, namely, angular momentum (spin), a magnetic moment, and some
sort of internal oscillation.
The point-like depiction of elementary particles is so unsatisfactory that it has
spawned new theories of matter known as M theories and quantum gravity.
In 1928, when Paul Dirac presented the wave function of the electron in the
Dirac equation [2], it became obvious that there must be not only an internal os-
cillation but also some type of internal motion at the speed of light. Therefore,
the seemingly empty space that surrounds electrons is made up of “virtual par-
ticles,” and electrons are inseparable from the clouds of virtual particles sur-
rounding them.
Moreover, there is no theory that quantifies particles in a meaningful way us-
ing appropriate calculations. This implies that quantum mechanics actually have
no need for a particle as a concept because all the calculations are similar wheth-
er or not hard particles exist.
Subsequently, physicists attributed this intrinsic contradiction between the
electron’s different properties to the common-sense view that the electron is
subject to quantum mechanics and, therefore, is not accessible to human imagi-
nation.
However, because subatomic particles can’t yet be directly observed, scientists
learn about the objects through indirect evidence. By observing what happens in
the vacuum around negatively charged electrons—thought to be swarming with
clouds of as-yet-unseen particles—researchers can create models of particle be-
havior.
The Standard Model predicts that particles surrounding electrons do affect an
electron’s shape, and according to this framework, the electron should be close
to perfectly spherical. But at such an infinitesimal scale as to be pretty much un-
detectable using existing technology. One framework to explain physics beyond
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the Standard Model is known as supersymmetry. However, this theory predicts
that the electron has a more distorted shape than that suggested by the Standard
Model. According to this idea, the electron is predicted to be slightly aspheric
[3].
In particle physics, the fundamental blocks of matter are continuous fluid-like
substances known as “quantum fields” that permeate the whole space around us.
With this article, the vacuum is proposed to behave as a superfluid and a vor-
tex shape of the electron is proposed as a condensation of the vacuum.
Hydrodynamics laws are applied to the superfluid vacuum to describe the ba-
sic properties of the electron. Hydrodynamic formulas are provided to calculate
its mass, density, volume, time, constant angular momentum (spin), and electric
charge.
2. Nature of the Vacuum
Both dark matter and new subatomic particles that were not predicted by the
Standard Model are yet to be directly spotted; still, a growing amount of compel-
ling evidence suggests that these phenomena do exist.
Vacuum density is generally viewed as a fundamental property of the cosmos
whose magnitude should not depend on whether we choose subatomic, astro-
nomical, or cosmological methods to assess its value.
According to quantum field theory, even in the absence of real particles, the
vacuum is always filled by pairs of created and annihilated virtual particles.
Therefore, the physical vacuum is assumed to be a non-trivial medium to which
one can associate a certain energy and density. Therefore, quantum theory re-
quires additional attributes to the vacuum. For instance, vacuum is not empty as
previously considered but rather filled with quantum mechanical zero point
energy.
The simulation of gravity (as far as Newton’s universal law of gravitation is
concerned) run through a computational fluid dynamic (CFD) approach is proven
successful [4] and it seems that such an approach (space quanta absorption af-
fected by massive particles, described as vortices of the same quanta [4]) is also
able to describe any other effect related to general relativity’s space topology.
In superfluid vacuum theory, the physical vacuum is described as a quantum
superfluid and is characterized to behave like a frictionless fluid with density and
extremely high thermal conductivity. The vacuum extends everywhere, has no
size, shape, center, direction, time, or extent, and is immovable.
Superfluid vacuum theory proposes a mass generation mechanism that may
replace or supplement the electroweak Higgs theory. It has been shown that the
masses of elementary particles could be a result of interactions with a superfluid
vacuum, similar to the gap generation mechanism in superconductors [5] [6].
Therefore, vacuum energy has real physically observable consequences, and its
properties can be observed as real physical effects [7] [8].
The quantization of gravity indicates that there is an
elementary quantum of
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matter
, indivisible, whose virtual imaginary mass is
()
73
0 min 3.9 10 kg
i
m−
=±×
[9].
These elementary quanta of matter should fill all the space in the Universe,
forming a continuous and stationary quantum fluid. The density of the universal
quantum fluid is clearly not uniform
throughout the Universe because it can be
strongly compressed in several regions (e.g., galaxies, stars, black holes, and pla-
nets). In the normal state (free space), the above-mentioned fluid is invisible.
In the
super-compressed
state, it can become visible in the form of known
matter, since matter, as we have seen, is
quantized
and is consequently formed
by an integer number
of imaginary elementary quanta of matter with imaginary
mass
( )
0 mini
m
. This means that there are no particles in the Universe with masses
smaller than the minimum mass and that all bodies are formed by a whole
number of these particles [10].
3. Internal Structure of the Electron
The Standard Model describes most of the interactions between all of matter’s
building blocks, as well as the forces that act on those particles. For decades, this
theory has successfully predicted how matter behaves, however it does not pre-
dict the structure of the electron.
The angular momentum (spin) of the electron indicates that there is an inter-
nal rotation that confers upon it its rest mass. According to Higgs theory, the
interaction between particles and the Higgs field is continuously maintained and
renewed, converting the amorphous potential energy of the field into individual
structures. The seemingly empty space that surrounds the electron is teeming
with pairs of particles and antiparticles that fleet in and out of existence; these
are called “virtual particles”.
Although precisely measuring this cloud is beyond the capabilities of modern
methods, the current model predicts that electrons are slightly aspheric, with a
distortion characterized by the electric dipole moment. However, no experiment
so far has detected this deviation [11]. Thus, the electric dipole moment remains
an elusive (and unproven) phenomenon, in which an electron’s spherical shape
appears deformed—“dented on one end and bulged on the other” [12].
We propose that the electron is an irrotational circular vortex of frictionless
superfluid space with concentric streamlines that was created from the primor-
dial vacuum during the Big Bang. The rate of rotation of the fluid is greatest at
the center and decreases progressively with distance from the center until there
is no gradient pressure on the boundaries of the vortex where the flow is laminar
and the friction is null. In such a case, the absence of friction would make it im-
possible to create or destroy the vortex motion. If the negative suction point vo-
lume in the center of the vortex does not have enough energy to drag the virtual
photons to the speed of light, then a stable situation cannot occur [13].
If we take a deeper look at the spiral arms of the electron vortex, we notice
that the currents are made of smaller vortices that correspond to the Higgs par-
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ticles. Higgs particles have no mass when they are created but acquire mass
when they travel through space. The interaction between the whirlpool-shaped
particles and the Higgs field (the vacuum) is continuously maintained and re-
newed, converting the amorphous potential energy of the field into Higgs par-
ticle vortices.
The superfluid accommodates the rotation by forming a lattice of quantized
vortices in which the vortex core, typically singular, breaks the topological con-
straint against rotational motion.
This new viewpoint of particles allows us to see them as a composite web
structure and at the same time as energy motion processes (Figure 1).
A spinning system along an axis with angular momentum has a torque when
the force is directed toward the center of gravity; this is known as the Coriolis
effect.
The flow to the center of the vortex due to the Coriolis effect results in vortex
tubes, which are always composed of the same virtual particles that rotate at the
speed of light; they remain
unbroken
, so they are ring-like (Figure 2).
If the velocity of the space circulation reaches the limiting speed,
c
, which is
the speed of light in the absolute vacuum, and the velocity field gradient around
the center of the vortex becomes the postulated limiting angular rotation,
ω
, the
Figure 1. Artistic presentation of an elementary particle that has a vortex structure made
by mini vortices of Higgs bosons.
Figure 2. Two vortex tube constituents of an electron vortex.
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space breaks down, creating a spherical void, which is defined as a field-less,
energy-less, and space-less volume of vacuum at the vortex center.
These maximums occur at the point where the centrifugal force and the radial
force are equalized. The inflowing medium and the free surface dip sharply and
the inflowing medium turns at 90˚ near the axis line, with the depth and velocity
inversely proportional to
r
2, to form a concave paraboloid. This is where the
highest vortex energy exists [14].
Maxwell worked out a theory of electromagnetism assuming that every mag-
netic tube of force was a vortex with an axis of rotation coinciding with the di-
rection of the force. Several properties have been mathematically proved for a
perfect frictionless fluid [15].
The magnitude of the vorticity in a vortex line increases proportionally as the
vortex line is stretched. Consider a very thin vortex tube around the vortex line,
so thin that the vorticity is practically constant over its width. As the vortex tube
stretches, the cross-sectional area decreases by the same factor; thus, the vorticity
must increase proportionally for the flux across the cross section to remain con-
stant.
Therefore, the electron is a vortex characterized by its power and volume, with
the specific magnitude depending on the rest energy of the electron.
4. Mass of the Electron Vortex
The angular momentum (spin) indicates that there is an internal rotation that
determines the rest mass. The mass of an electron is the amount of fluid-like
virtual photons with a certain density that passes in 1 s. Therefore, the mass of
the electron is calculated to be density times volume.
In hydrodynamics, the force
F
that moves the vortex is directly related to the
pressure that creates the vortex, known as the dynamic pressure
Pd
, and the area
A
according to the formula
d
F PA=
.
(1)
The dynamic pressure (
Pd
) representing the fluid kinetic energy is expressed
as
2
1
2
d
Pv
ρ
=
, (2)
where
ρ
is the density of the fluid and
v = c
is the
velocity.
Therefore,
2
1
2
F cA
ρ
=
. (3)
The area of the vortex is approximately a circle,
2
2Ar= π
, that interacts with
the adjacent vacuum from both sides; therefore,
22
F cr
ρ
= π
. (4)
If we multiply and divide the right side of the above equation by time
t
, we
obtain
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2
F ct r c t
ρ
π=
.
(5)
However,
v/t
is equivalent to the acceleration
a
,
and
vt
is equivalent to the
length
L
.
According to Newtonian theory, the force
F
divided by the acceleration equals
the mass. Dividing both sides by the acceleration, we obtain
2
Fa m rL
ρ
π= =
. (6)
However, the area times the length is equal to the volume
Q.
Therefore,
2
Fa m rL
ρ
π= =
.
(7)
which corresponds to the inertial mass of the electron.
5. Minimum Time of the Electron Vortex
Because the electron has no shape in quantum physics, the time of the electron
has never been reported before. The electron as a vortex has a minimum time
less than which the electron converts into a virtual particle.
If
2
m rL
ρ
= π
and the length is the product of velocity times time,
L ct
=
,
then
2
m r tc
ρ
=π
.
(8)
The mass of the electron depends on the spin and the time. If the relative rota-
tion velocity of the vortex or time is zero, the mass will be zero; accordingly, the
mass will disappear and convert into an amorphous vacuum. The minimum
time of the electron, that needed to complete one rotation cycle 2π
re
, is
21
2 8.08586 10 s
e
t rc
−
= ×π=
, (9)
where
13
3.86 10 m
e
r
−
= ×
and
8
2.99792458 10 m sc= ×
.
6. Density and Volume of the Electron
Normally, electron density is a measure of the probability of an electron being
present at a specific location. However, what we mean by the density of an elec-
tron is the density of virtual photons of which the electron is made. Hydrody-
namically, vortex formation depends on the density of the medium in which it
occurs. In our case, the vacuum’s “energy density” is related to the number of
virtual particles that are continually created and annihilated in the vacuum;
however, the density of the electron is in a supercompressed state and is higher
than that of the surrounding vacuum field.
Knowing the time of the electron, the density of the electron can be calculated:
2 63
2.519470 10 kg mm r tc
ρ
π= = ×
,
(10)
where
2
r tcπ
is the volume. Knowing the density of the electron, the volume
Ve
can be calculated:
37 3
3.6118 10 kg m
e
Vm
ρ
−
= = ×
. (11)
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7. Particle-Wave Relationship
Particle-wave duality is a central tenet of quantum physics, and an electron has
wave-like properties. The essence and physical relationship between the particle
and the wave remains an unresolved problem in physics. The vortex model of
the electron gives three different solutions:
1) Louis de Broglie developed a hypothesis [16] to relate the dual wave and
particle behavior that can be applied to electrons. The de Broglie wavelength
formula relates the wavelength
λ
to the momentum
mv
of a wave/particle: [17]
h p h mc
λ
= =
and
h mc
λ
=
.
In hydrodynamics, the velocity of the fluid element instantaneously passing
through a given point in space in the vortex with radius r is constant in time;
therefore, the circulation or the vorticity in the core of the vortex
2rc
π= Γ
is
constant. Γm is a conserved momentum; therefore,
2rcm
π
is constant, which
corresponds to the Planck constant.
If
2h rcm= π
, then
2r
λ
= π
, which is the circumference of the core of the
vortex.
If the radius of the core of the vortex electron,
13
2 3.86 10 mr h mc
−
= = ×π
where
m
is the mass of the electron, the value of the circumference is
12
2 2.42408 10 mr
−
= ×π
, which is similar to the CODATA 2014 [18] value for
the Compton wavelength of the electron, 2.4263102367 (11) × 10−12 m.
Vortices also have frequencies described in units of time (rotations per second). If
the time required to complete a cycle around the vortex is
21
2 8.08586 10 s
e
t rc
−
= ×π=
, then the frequency of the electron related to the
cycles of rotation of the vortex is
20
1 1.2367 10 Hz
e
ft= = ×
. In fact, the rota-
tional velocity of the vortex
2f cr
ω
= π =
; therefore,
2fc r= π
,
which gives
us the same frequency as Planck’s theory. If the invariant mass of an electron is
approximately 9.109 × 10−31 kilograms, then its energy would be
2 14
8.1981 10 JE mc
−
= = ×
And if
E hf=
where
h
is Planck constant = 6.62607004 × 10−34 m2∙kg/s
then
20
1.2372492 10 Hzf Eh
= = ×
.
2) The third solution is related to the energy of the particle. A particle of mass
m
has a rest energy of
2
E mc=
. The non-reduced Compton wavelength for this
particle is the wavelength of a photon of the same energy. According to Planck’s
theory, for photons of frequency
f
, the energy is given by
e
E hf=
.
(12)
The frequency of the electron is
e
f Eh=
,
(13)
where
7
0.511 MeV 8.1866 10 ergs
e
f Eh h h
−
= = = ×
. Therefore,
20
1.2355 10 cycles s
e
f= ×
.
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where
h
is 6.626176 × 10−27 erg-seconds.
How is
E hf=
related to the vortex model?
The force that rotates the vortex is
22
F cr
ρ
= π
. (14)
If we multiply and divide the right side of Equation (3
.
5
.
3)
by time,
t
, we ob-
tain
2
F ct r c t
ρ
π=
,
(15)
where
ct
is equivalent to the distance
L
,
L
π
r
2 is equivalent to the volume
Q
,
ρQ
is
equivalent to the mass, and 1/
t
is equivalent to the frequency
f.
Therefore,
F mcf=
.
(16)
If
E
= force × distance and the distance that an electron moves in one cycle is
2rπ
, then
2E rmcf= π
, where
2rmcπ
is equivalent to
h.
Therefore,
E hf=
and
e
f Eh=
.
8. Electron Charge
Charge is a fundamental physical property of matter that is responsible for its
interactions with electromagnetic fields. An electron is a particle that possesses
this property, and experiments show that it possesses a negative charge.
The real nature and the essence of charge are unknown. In this section, a new
theory to describe the nature of electric charge is formulated based on the vortex
model of the electron.
In hydrodynamics, the rotation of a vortex creates a drag force that attracts
the vacuum to the center of the vortex. This force is directly related to the densi-
ty of the vacuum, the speed of rotation, and the area is inversely related to the
distance from the vortex center according to the equation
2
1
2
F cAr
ρ
=
.
(17)
Multiplying and dividing the right side of the above equation by time, we obtain
11
22
F ctAc rt Vc rt
ρρ
= =
. (18)
However, this force is reduced owing to interactions with the adjacent va-
cuum. If the density of the vacuum is
ρv
and the rotation speed of the vortex is
c
,
dividing the momentum by the length of the circumference of the vortex will
give the diminished momentum
P
for a unit of length:
v
Pc
ρλ
=
.
(19)
Therefore, the momentum of the vortex is reduced for every unit of length
according to the equation
2
1
2
v
F V c rt
ρρ λ
=
.
(20)
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If
2r
λ
= π
, then
2 22
12 42
2
vv
F Vcrtr Vc tr
ρρ ρρ
π= π=
. (21)
In hydrodynamics,
ρvc
2
is the elasticity of the vacuum. In fact, the velocity of a
particle in an elastic medium can be expressed by the formula
( )
12
c Ed=
, (22)
where
c
, the speed of light,
E
is the elasticity, and
d
=
ρv
, the density of the me-
dium.
Therefore, the elasticity,
E
, can be written as
2
v
Ec
ρ
=
, (23)
which is the inverse of the stiffness of the vacuum and has the same value as the
electric permittivity,
i.e.
,
21
0v
v
ρε
−
=
.
This makes sense in terms of dimensions because the elasticity modulus is
Newton × m−2 whereas the permittivity is Newton−1 × m−2 (C2/Nm2).
Then, the Equation (3
.
6
.
5)
becomes
2
0
4
v
F V tr
ρε
π
=
. (24)
The force density at a point in a fluid divided by the density is the acceleration
of the fluid at that point:
2
0
4F f Vt r
ρε
= π
=
. (25)
In fluid mechanics, the force density [19] is the negative gradient of the pres-
sure. It has physical dimensions of force per unit volume. The force density is a
vector field representing the flux density of the hydrostatic force within the bulk
of a fluid.
Furthermore, in fluid dynamics, the volume of a fluid that passes per unit of
time,
V/t
, is the volume flow rate, which is usually represented by the symbol
Q
.
Its SI unit is m3/s.
The force between two vortices is directly proportional to the magnitude of
the flow rate of the elementary density in each vortex
q
1
q
2 and inversely propor-
tional to the distance of the separation between their centers, r, diminished by
the stiffness of the vacuum represented by the electric permittivity per unit of
length:
2
12 0
4f qq r
ε
π=
. (26)
The origin and essence of the electric permittivity will be discussed in detail
separately in another paper.
9. Discussion
The nature and the essence of the electron was central to the development of
quantum theory early in the twentieth century, and remains at the frontier of
physics today. A century after Danish physicist Niels Bohr conceived of the elec-
tron as the proton’s satellite [20], our perception of the electron continues to
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evolve and expand.
In particle physics, the fundamental blocks of matter are continuous fluid-like
substances known as “quantum fields” that permeate the whole space around us.
In this article, the electron is proposed to have frictionless vortex shape and
the hydrodynamic laws are applied.
The central idea is that an electron is a frictionless vortex with conserved
momentum made out of condensed vacuum generated in the Big Bang from
massless virtual photons that acquire mass when moving in the vortex at the
speed of light, as described by Higgs theory. Considering the vacuum density
and applying classical hydrodynamics, analytical formulations are applied to
calculate the mass, volume, density, time, and frequency of the vortex. We ob-
tained the properties of an electron and unified particle-wave duality using the
same vortex model.
We summarize the findings of this article in the following points:
1) The electron has a vortex shape with density, mass, radius, area, circumfe-
rence, volume, rotational velocity, minimal time, frequency and flow rate.
2) The force that rotates the vortex is directly proportional to the density of
the vortex, times the square of its rotation speed and to its area according to the
Equation (4)
22
F cr
ρ
= π
.
3) The electron vortex mass is directly proportional to its density times its
area and its length according to the Equation (6)
2
m rL
ρ
= π
.
4) The density of the electron is directly proportional to the vortex mass and
inversely proportional to its area, and to the time needed to complete one cycle
and rotation speed of the vortex according to the Equation (10)
2
m r tc
ρ
= π
.
5) The minimum time of the electron is the time needed to complete on rota-
tion cycle of the vortex which directly proportional to the vortex circumference
and inversely to the rotation speed according to the Equation (9)
2
e
t rc
= π
.
6) Therefore, the frequency of the electron is directly proportional to its rota-
tion speed and inversely related to its circumference according to the equation in
section (6.1)
2
e
t rc= π
.
7) Furthermore, the frequency of the electron vortex is directly related to its
energy and inversely related to its mass, circumference and rotation speed ac-
cording to the equation in section (6.2)
2f E rmc= π
.
8) The electric charge q is equal to flow rate of the elementary density V/t
from the periphery to the center of the electron vortex divided by the area in
both sides of the vortex diminished by the stiffness of the vacuum according to
the Equation (25)
22
00
44f Vt r q r
εε
ππ= =
.
9) The electric force between two charges is the force between two vortices
which is directly proportional to the magnitude of the flow rate of the elementa-
ry density in each vortex and inversely proportional to the distance of the sepa-
ration between their centers, diminished by the stiffness of the vacuum
represented by the electric permittivity per unit of length according to the Equa-
tion (26)
2
12 0
4f qq r
ε
π=
.
N. Butto
DOI:
10.4236/jhepgc.2020.63027 351 Journal of High Energy Physics, G
ravitation and Cosmology
We conclude that the shape of the electron is accessible to human imagina-
tion.
The electron properties can be accurately described using classical laws of hy-
drodynamics. Knowing its shape helps to determine its properties and shed a
light on how matter is made and to explain the interactions of sub-atomic par-
ticles.
This theory could overturn several alternative physics theories that attempt to
fill in the blanks about phenomena that the Standard Model can’t explain.
The electron wave-particle duality, electron charge and the origin of electron
spin will be discussed in detail in separate papers.
Future experimental studies are needed to confirm the vortex structure of the
electron.
Acknowledgements
The author would like to thank Enago (https://www.enago.com/) for the English
language review.
This research did not receive any specific grant from funding agencies in the
public, commercial, or not-for-profit sectors.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this pa-
per.
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