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The Geometry of Spacetime and the Unification of the Electromagnetic, Gravitational and Strong Forces

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  • Vizaport
  • Western University
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Abstract and Figures

In this paper, a spacetime structure consisting of a body-centered cubic lattice is modeled classically as a spring-mass system, where the components of each unit cell in the lattice are based on the fundamental units discovered by Max Planck, and the common forces that govern the motion of particles in spacetime is defined and unified by geometric shapes as the spacetime lattice oscillates.
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The Geometry of Spacetime and the Unification of the
Electromagnetic, Gravitational and Strong Forces
Jeff Yee & Lori Gardi
∗#jeffsyee@gmail.com/
†#lori.anne.gardi@gmail.com
August 2, 2020
(third draft)
Abstract: In this paper, a spacetime structure consisting of a cubic lattice is modeled classically as a spring-mass
system, where the components of each unit cell in the lattice are based on the fundamental units discovered by Max
Planck, and the common forces that govern the motion of particles in spacetime is defined and unified by geometric
shapes as the spacetime lattice oscillates.
Introduction
In the early 1900s, three-dimensional space and time were linked by Albert Einstein into what became known as a
single word – spacetime – to describe the mathematics of relativity. Yet Einstein didn’t describe the mechanism for
the curving of spacetime nor how it bends and contorts to cause gravity [
1
]. Years prior to Einstein’s work, Max
Planck discovered and established a number of constants that simplify the mathematics of the universe – known as
the Planck units but he was not able to describe the meaning of these constants [
2
]. Here, the Planck units are
applied as the fundamental units of length, mass and time to define the geometry of spacetime and explain the natural
forces that cause the motion of particles within its domain. There is a reason that the Planck constants fit elegantly
into equations that represent the energy and forces of the universe.
If spacetime is considered to be a structure that curves, the structure that is curving must be defined. Similarly, if
particles and photons are considered to be wave-like, the structure that is waving must be defined. Here, the structure
of spacetime at the smallest of levels the quintessence of the universe is proposed to be a material in a lattice
structure of repeating unit cells, where each of the cells may vibrate in harmonic motion such as Fig. 1.
Fig. 1Spacetime lattice no vibration (left); cascading effect of vibration (right)
2
This structure is responsible for the forces that cause the motion of particles, including the electric, magnetic,
gravitational and strong forces. It is also responsible for the energies of photons and particles. In this paper, these
forces, the energy of photons, and the energy of the electron will be derived and explained using only five total
constants and the geometry of spacetime. Three of these constants (Planck length, Planck mass and Planck charge)
are shown in the next figure to describe a unit cell of the spacetime lattice. It will also be shown that a unit cell
exhibits behavior similar to a spring-mass system, and as a result, can be calculated using classical mechanics.
Fig. 2Spacetime unit cell
The process to define the unit cell began with a wholistic view of force and energy equations, which often have either
mass or charge as variables. Using Modified Unit Analysis (MUA), these equations were first consolidated in units,
by relating mass to charge. This was reported in the first of three papers that form the baseline for this paper The
Relationship of Mass and Charge [
3
]. The key finding was the exchange of the units of charge (Coulombs) to units of
distance (meters) to relate these properties. Charge is defined here as the displacement distance of a unit cell. The
remaining two papers that form the baseline of this paper also describe relationships The Relationship of the Mole and
Charge and The Relationship of the Fine Structure Constant and Pi. These papers define the separation distance of unit cells
and the structure of the spacetime lattice respectively [
4
,
5
].
It is the geometry of spacetime that defines particles and their motion. In this paper, it will be shown that the structure
of spacetime can be modeled as a lattice structure, where the repeating structure known as unit cells contain the
properties discovered by Max Planck for the values of Planck mass, Planck length, Planck time and Planck charge. It
is why the Planck unit system fits nicely into physics equations modeling energy and forces.
The unit cells of the spacetime lattice may expand and contract in harmonic motion, creating the presence of waves
that form the basis of particles as standing waves and the forces that cause the motion of particles as traveling waves.
All of which can be traced back to a single unit cell and the motion of its components, referred to here as granules,
as they spread linearly or spherically due to the structure of the unit cell.
The purpose of this paper is to develop a framework for the underlying structure of the universe, so that it can be
modeled with classical mechanics and simulated with computer programs to describe the energy and motion of
subatomic particles. The paper first describes the basic unit system and how it is applied to the geometry of spacetime.
Then, it offers proof of the calculations and unification of forces by deriving and explaining the fundamental physical
constants that are used in force and energy equations and are known to match experimental evidence, such as the
Coulomb constant (ke) for electric forces and the gravitational constant (G) for gravity. More than a dozen
fundamental physical constants are derived throughout this paper as such proof.
3
1. The Universe in Simplified Units (kg, m, s)
All of the equations for forces and energies can be simplified to three units for mass, distance and time. This forms
the fundamental kg/m/s unit system. What Max Planck found naturally in the constants for Planck mass, Planck
length and Planck time were tiny values when compared against these units in our human reference frame one that
had already been established for kilograms (kg), meters (m) and seconds (s) in our macro world. But the smallest unit
cells of spacetime are orders of magnitude smaller than our world. These Planck units form the reference point of
an object that occupies the space at the center of the spacetime unit cell hereafter referred to as a granule. The
spacetime unit cell can be represented by a center granule that has a mass of Planck mass (mP), a radius of Planck
length (lP), and when in motion at a universal speed, it takes Planck time (tP) to travel one Planck length. The value
of Planck mass is deceiving because it is significantly larger than the mass of the electron or proton, yet it occupies a
much smaller space. It is colored blue in this paper as a center granule that represents the collective mass of granules
in motion, collectively the mass in a spring-mass system. It will be shown that the energy from this mass is only
recognized when a granule is in motion, similar to a spring-mass that has mass but has no energy unless the mass is
displaced from equilibrium.
Fig. 1.1The basic Planck units for the kg/m/s unit system
The values for each of the basic Planck units are:
(1.1)
(1.2)
(1.3)
The previous three Planck constants apply to the default units of mass, length and time. Two additional constants
are required for the foundation of five constants that can derive everything in this paper. The remaining two constants
are related to the electron: the Planck charge (qP) is the granule displacement at the center of the electron, and the
electron’s classical radius (re) is the distance from the center of the electron where standing waves transition to
traveling waves. Both will be described in further detail later in this paper.
4
Fig. 1.2Electron wave amplitude and wave transition radius
The values for both of these are measured as a distance, in units of meters. Eqs. 1.1 to 1.5 are the only five constants
required. Everything else in this paper will be derived from these five constants.
(1.4)
(1.5)
Unit Relationships
In the simplified kg/m/s unit system, there are only three units required for all equations. But there are also
fundamental constants that relate each of these units. Distance and time can be linked together as meters per second
(m/s), otherwise known as speed. And mass and distance can be linked together as kilograms per meter (kg/m),
otherwise known as a linear density.
The fundamental speed of the universe is the speed of light (c), which is the relationship of the Planck length to
Planck time as shown in Eq. 1.6.
(1.6)
The fundamental linear density is the magnetic constant (µ0), shown in Eq. 1.7 as the relationship between Planck
mass and Planck length, including a geometric ratio (x) that will be explained in Section 3. For the magnetic constant
to be recognized correctly in kilograms per meter units, the unit of charge (Coulombs) is replaced with the unit of
distance (meters).
(1.7)
2. The Geometry of Spacetime
Nature often repeats itself and nature often finds a way to optimize. The proposed structure of the spacetime lattice
includes repeating cells, called a unit cell in the study of molecules. Nature is likely repeating itself from the smallest
of structures as it builds. The structure of a unit cell with a separation length (a) is described in Fig. 2.1. In this
structure, granules exists at the vertices of the cube. Each of these granules has a radius of Planck length, and a
diameter of twice this length (2 * lP). Euler’s number (e) is the base of the natural logarithm and often found in nature.
The total separation length of the unit cell is a granule diameter times Euler’s number (2 * lP * e). It is nature’s way
of optimization and it is found in the separation length of granules in the unit cell.
5
Fig. 2.1Unit cell granule separation length (e is Euler’s number)
In Fig. 2.1, the center granule is color coded in blue. It is meant to signify that it is the center of vibration, such as
Fig. 2.2. If it vibrates and the total displacement is a distance of Planck charge, it collides with and has a cascading
effect on other granules. Collectively, they form a spherical wavefront with a longitudinal wavelength (l) that is also
based on Euler’s number (2 * qP * e2), but it is now the square of Euler’s number as it will be calculated based on the
surface area of a sphere.
Fig. 2.2Granule harmonic displacement and wavelength
The relationships of separation distance of granules in a unit cell (a) and the separation of granules when traveling as
longitudinal waves (l) are expressed in Eqs. 2.1 and 2.2. They will be proven later in the derivation of Avogadro’s
constant and the fundamental frequency respectively. Euler’s number (e) is 2.71828
(2.1)
(2.2)
2.1 Rectangle-to-Sphere (Surface Area Ratio)
The surface area relationship of a rectangle compared to a sphere is the first key geometric shape ratio, describing
plane waves versus spherical waves. Plane waves converging upon a center equally from all directions will be
reflected outwards and will appear to be spherical - this will be later described as particles. Further from the center,
the motion in any given direction can be described as plane waves. These two areas describe the surface penetration
of granule wave motion. A granule in motion may be displaced from equilibrium as it vibrates, affecting nearby
granules as it transfers its energy. Yet the collective energy of all granules measured at these surface areas will be
shown to be equal and conserved. A plane wave can be represented by the surface area of a rectangle (Sr) with width
(x) and length (y), and the spherical wave with the surface area of a sphere (Ss) with radius (l). Both are described in
the next figure.
6
Fig. 2.1.1Surface areas of a rectangle (Sr) and a sphere (Ss) representing plane and spherical waves
The surface areas of a rectangle (Sr) and a sphere (Ss) are as follows:
(2.1.1)
(2.1.2)
The first of two key geometric ratios is the ratio of the rectangle-to-sphere surface area (Eq. 2.1.3). It is labelled with
the alpha character (a1) due its relationship with the coupling constants of forces, as will be explained in this section.
(2.1.3)
Example
The electron’s classical radius (re) is used for the values of all three variables in Eq. 2.1.3. The notation of the function
which is used later follows this format to assign three variables to the equation:
(2.1.4)
When setting all variables to the electron’s radius, the result is the inverse of 4p, which often appears in electromagnetic
equations:
(2.1.5)
2.2 Rectangle-to-Sphere+Cone (Surface Area Ratio)
The second key geometric relationship is a slight variation of the first - the surface area of a cone is added to the
sphere. This geometry remains the same across forces but the amplitude displacement and direction will be the
difference and cause of the strong, electric, magnetic and gravitational forces. In physics equations, the relationship
of these forces relative to the electric force is described by coupling constants. For example, the fine structure
7
constant (a or ae) expresses the dimensionless ratio between the electric force and strong force [
6
]. These coupling
constants can be derived mathematically from a single geometric relationship.
The surface area of a rectangle represents the penetration of a shape such as the unit cell by a granule in (1) in Fig.
2.2.1. In a unit cell, the center granule colored blue is illustrated moving in (2), hitting maximum displacement and
reversing in (3) and finally returning to equilibrium in (4). This motion has a direct effect on the granules at the
vertices of the unit cell which spread spherically, having a cascading effect on other unit cells. At the same time, the
motion of the center granule may also affect the motion of the center granule of the next unit cell, possibly introducing
spin if not in complete alignment. This motion also cascades and the geometry can be represented by a cone. The
surface areas of the sphere and cone are added together as any granule in touch with any point on these surface areas
may cause a change in amplitude.
Fig. 2.2.1 Vibration of center granule
In this harmonic motion, the displacement and return to equilibrium is a p cycle and can be represented by a sine
wave with a half wavelength. The ratio of a rectangular surface area (Sr) will be compared to the surface area of a
sphere plus cone (Ss+ Sc). The width and height of the rectangle is x and y. The radius of the cone is d, and the slant
length of both the cone and the radius of the sphere is l. Furthermore, slant length (l) is related to cone radius by p *
d. This was described in detail in The Relationship of the Fine Structure Constant and Pi paper due to the difference between
the time that it takes for a granule to vibrate and return to equilibrium and the time that it takes for a granule that is
traveling in one direction at constant speed. It is also described visually later in Fig. 3.2.1.
Fig. 2.2.2Surface areas of a rectangle (Sr) and a sphere+cone (Ss+Sc)
The sphere and cone geometries are found in the formation of magnetic lines, which is likely replicating what is
occurring at a micro-level when atom configurations have the right alignment of electron spin (such as magnets).
8
Fig. 2.2.3 Sphere and cone geometries seen in magnetic lines (left bar magnet; right bar magnet with overlay of sphere+cone)
The surface area of a cone is:
(2.2.1)
Now, equation 2.1.3 is expanded to include the addition of the cone’s surface area in the denominator. This becomes
the second of two key geometric ratios used in this paper (a2).
(2.2.2)
Example
The electron’s classical radius (re) is used for the values of all four variables in Eq. 2.2.2 (and slant length is multiplied
by radius times p). The notation of the function which is used later follows this format to assign four variables to the
equation:
(2.2.3)
When the length and width distances are the same as the radius of the cone, such as when they are all set to the
electron’s classical radius, the value is equal to the fine structure constant (
a
).
(2.2.4)
All calculations of fundamental physical constants in this paper are shown in the format equation = value (units) and
match known CODATA values of the constants [
7
]. The fine structure constant, however, is dimensionless and thus
no units appear in this equation. This special version of this geometric ratio at the electron radius is given the label
a
e as it is related to the electron.
9
(2.2.5)
3. Forces as an Effect of Spacetime Geometry
The forces that affect the motion of particles can be mathematically derived based on the conservation of energy and
geometric ratios from the previous section. It begins with the energy of a center granule with mass of Planck mass
and radius of Planck length, as illustrated.
Fig. 3.1 – Granule of Planck mass and radius of Planck length
The energy of this center granule can be expressed with Einstein’s well-known equation in Eq. 3.1, and the force at
the surface of this granule by dividing the distance of Planck length (lP) in Eq. 3.2.
(3.1)
(3.2)
All of the forces in the upcoming sections will be based on Eq. 3.2, with the inclusion of one or both of the geometric
ratios (a1 and a2) from Section 2. They will all have the following format:
(3.3)
3.1 The Fundamental Force and the Magnetic Constant
The first geometric ratio is used to prove the fundamental force that exists as constant wave motion between particles,
captured as the magnetic constant or its inverse as Coulomb’s constant. Assuming that waves travel throughout the
universe and form wavefronts according to Huygen’s principle [
8
], the convergence of these wavefronts on a spherical
granule (radius of Planck length) is represented by a plane in-wave with a displacement distance (amplitude) of Planck
charge.
10
Fig. 3.1.1 Plane wave representation of in-waves converging on a sphere with radius of Planck length (a1A)
This geometric ratio (a1A) is applied to Eq. 3.3. As an in-wave, it is the inverse of the ratio:
(3.1.1)
The geometry ratio for a plane wave to spherical wave was established in Eq. 2.1.3. Using the function described in
Eq. 3.1.2, the variables for x and y are set to Planck charge (qP) and l is set to Planck length (lP). It is substituted into
Eq. 3.1.1 to become the fundamental force on a granule.
(3.1.2)
(3.1.3)
The previous equation can be rearranged to separate c2, showing the full derivation of the magnetic constant (µ0):
(3.1.4)
(3.1.5)
It also shows that the magnetic constant is a linear density (kg/m) when the units of charge are correctly identified as
a distance (meters). The magnetic constant is now substituted into Eq. 3.1.4 for readability. This simplified format
will be used in upcoming sections proving forces and energies.
(3.1.6)
3.2 The Strong Force
All of the forces, including the strong force, are measurements at a point in space relative to the center of a particle.
This introduces a variable (r) to measure a force at this given distance.
11
Fig. 3.2.1 Reflected spherical out-wave (a1B) measured as a plane wave at a variable distance r
This geometric ratio (a1B) is applied to the in-wave equation (Eq. 3.1.1), representing the spherical out-wave that will
be measured at a variable distance (r). Energy is perfectly conserved, but the displacement of each granule decreases
as energy spreads across a greater number of granules when spreading spherically. Thus, the force decreases and is
proportional to a1B/a1A.
(3.2.1)
The geometric ratio of the rectangle-to-sphere is used again, but now with a variable distance that represents the
radius of the sphere that is being measured. The function describing this geometry with Planck charge and a variable
distance uses Eq. 2.1.3. a1B is then inserted into the previous equation and finally simplified in Eq. 3.2.4.
(3.2.2)
(3.2.3)
(3.2.4)
It is labelled as a strong force (Fs) but it may be best described as the pure reflection of energy as either longitudinal
or transverse waves measured at a given distance from the center of a particle. In a separate paper on the Geometry of
Particles, this equation is used to determine the forces and energies of particles from the neutrino to the Higgs boson
[
9
] as longitudinal waves. Here in this paper, the equation is used to prove the strong, transverse wave force in the
nucleus of atoms.
The strong force is responsible for binding quarks together to form protons and neutrons and its residual force binds
these particles together to form the nucleus of atoms. It occurs only at short distances, explained by the possibility
that particles may be stable at the nodes of standing waves, and further that standing waves occur within the boundary
of a particle’s radius (e.g. the electron’s radius). This is further described as the energy and mass of the electron in
Section 5.
3.3 The Electric Force
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The electric force is an expansion of the previous strong force, but now adds the consideration of spin within particles
like the electron. Energy is always conserved, but now the reflected energy is not purely longitudinal or transverse in
form. It is a mix of both wave forms. The longitudinal form will be shown in this section to be the electric force;
the transverse form will be shown in the next Section 3.4 to be the magnetic force.
Fig. 3.3.1 Reflected spherical out-wave (a2B) and the spin of a particle with two wave forms: longitudinal and transverse
A geometric ratio that describes this energy split into two wave forms is appended to Eq. 3.2.1 from the strong force.
(3.3.1)
The ratio for the electron (a2B) uses the second of two key geometric ratios described in Section 2. Now, it will use
the ratio of the rectangle to sphere+cone geometry as a result of introducing particle spin. Since it is based on the
electron, the electron’s radius (re) is used for the variables in the equation for this ratio, which was earlier derived to
be the fine structure constant (ae). It is inserted into Eq. 3.3.1 above and simplified.
(3.3.2)
(3.3.3)
(3.3.4)
The constant for the elementary charge (ee) is often used as the charge of a single electron. This relationship between
the elementary charge, Planck charge and the fine structure constant is shown next and then simplified.
(3.3.5)
(3.3.6)
The electric force (Fe) is rarely measured as the interaction of a single electron or proton. It is typically measured as
a collection of particles, adding particles together for the collective charge. This mechanism for adding charges
13
together will be described as constructive wave interference in Section 4. For now, the total charge (q) can be
described as a change in amplitude as a result of a number of elementary charges, expressed in Eq. 3.3.7.
(3.3.7)
(3.3.8)
Eq. 3.3.8 is the electric force of multiple particles separated at distance (r). However, it is more commonly expressed
as Coulomb’s law. Coulomb’s constant is a combination of all the constants found in Eq. 3.3.8. Once all the
variables are separated to the right side, the electric force in its common form can be found in Eq. 3.3.10.
(3.3.9)
(3.3.10)
3.4 The Magnetic Force -
Monopole
In the previous section on the electric force, longitudinal wave energy is reduced as it is reflected outwards due to the
spin of the electron. Due to the conservation of energy principle, this energy must take another form. The second
form is a transverse wave due to spin, and its force becomes the magnetic force.
Fig. 3.4.1 The geometry of the magnetic monopole force is identical to the electric force (a2B) but is now the inverse
This geometric ratio (a2B) is identical to the electric force except that it is now the inverse. It appears in the
denominator of Eq. 3.4.1 to become the magnetic force for a single particle (Fm).
(3.4.1)
The magnetic force of a single particle is a monopole, which is not a common equation because monopoles are not
stable in nature. The next section will derive a more common dipole magnet. However, the magnetic moment of a
single electron can be derived and is found in the Bohr magneton. The proof of the following equation for the
magnetic force of a monopole will be shown in the derivation of the Bohr magneton later in Section 5.9.
14
(3.4.2)
(3.4.3)
3.5 The Magnetic Force -
Dipole
The magnetic force of dipole magnets, which decreases in strength at the cube of distance from the source, is a slight
variation of the previous monopole magnetic force. The previous figure for magnetism is now modified for two
particles (left of Fig. 3.5.1) that have an effect on a third particle (right of Fig. 3.5.1) at a variable distance (r).
Fig. 3.5.1 The geometry of a magnetic dipole (a2C); two particles (left) have a force on a particle (right)
This geometry may apply to two particles such as a proton and electron. But it also may apply to two quarks within a
proton. For the latter reason, this force is labeled as Fo because of the effect it has on an electron in the orbit of an
atom. This will be proven in Section 5 for the calculation of electron orbital distances and energies, including the
Bohr radius.
The strong force equation (Eq. 3.2.1) now includes this geometric ratio (a2C) that describes this transformation. Like
the magnetic monopole force, it is inverse.
(3.5.1)
The same geometric ratio used for the fine structure constant (a2) is applied again. This time, there is one difference.
The x value is now variable, which is the distance (r) to the third particle. Using the variables described in the function
in the next equation, they are inserted into Eq. 2.2.2 and solved. Eq. 3.5.2 is then substituted into Eq. 3.5.1 and
simplified.
(3.5.2)
(3.5.3)
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(3.5.4)
The equation above is the force of magnetism for a dipole, following the inverse cube law. In Section 5, it will be
used to derive the Bohr radius using the form in Eq. 3.5.5, where the Planck charge is replaced with the elementary
charge, using the relationship in Eq. 3.3.5. It is referred to later as the orbital force (Fo).
(3.5.5)
3.6 The Gravitational Force
At the core of the particle, the motion of a center granule causes the spin of a particle. It was described in Section
3.3 to cause a loss in longitudinal wave energy, which is the electric force. For each electron, the motion of the center
granule with cone radius of Planck length (lP), and its effect on the electron at its surface (radius of re), can be described
visually by the following:
Fig. 3.6.1 – Geometry of a spinning granule and its effect on an electron (a2D)
Gravity will be shown to be a shading effect of two or more particles, similar to radiation pressure that causes a slight
attraction between objects. It is a shading effect of the electric force, so the geometric ratio described in the figure
above (a2D) is appended to the equation for the electric force (from Eq. 3.3.1).
(3.6.1)
The same geometry ratio (a2) from Eq. 2.2.2 is used again but now with x and y variables set to Planck length (lP).
The result is then substituted into Eq. 3.6.1 and simplified.
(3.6.2)
(3.6.3)
16
(3.6.4)
The result from Eq. 3.6.2 is recognized as the gravitational coupling constant for the electron (
a
Ge), relative to
the electric force. It is the numerical value for the gravitational coupling constant when expressed as a ratio between
the force of gravity and the electric force of two electrons [
10
]. Note that the gravitational coupling of the proton (aGp)
can be derived by multiplying the square of the proton-to-electron mass ratio to Eq. 3.6.4.
(3.6.5)
In Eq. 3.6.4, the force of gravity for the electron (FGe) can be seen as the electric force (the terms on the left match
Eq. 3.3.6) with the gravitational coupling constant described in Eq. 3.6.5 on the right. The right term describes the
energy loss from a second electron, as gravity is a shading effect between two electrons. This will be more apparent
when described in Section 3.7. Putting Eq. 3.6.4 to the test, the force of two electrons is the following:
(3.6.6)
However, most gravitational equations rely on mass (m), not charge, as variables. Using Newton’s equation for
gravity, it is found that the mass of two electrons gives an identical result, validating the equation here for gravity.
The electron’s mass (me) is 9.109 x 10-31 kg and the gravitational constant G is 6.6741 x 10-11 m3/kg s2.
(3.6.7)
The conversion from charge to mass and the derivation of the gravitational constant (G) as further proof of this
equation will be shown in Section 5.8.
3.7 The Relationship of Forces
The conservation of energy and the transition of wave forms to be the cause of forces is easier to understand when
measuring each force at the electron’s classical radius. The relationship between all of the forces becomes clear. They
are first expressed with equations and diagrams in table format (Table 3.7.1) and then described in detail.
The equations for the forces from Sections 3.2 to 3.6 are used in the table below and then distance r is set equal to
the electron’s classical radius (re) and simplified.
(3.7.1)
To simplify everything further, the common terms that exist in all equations are set to one (1). This allows the
strong force to be proportionally set to one (F ∝"1) so that everything else is relative to this force. The following
terms are set to one for the purpose of the simplified force (F) in the far-right column of the next table."
17
(3.7.2)
Strong
Electric
Magnetic
(monopole)
Magnetic
(dipole)
Gravity
Table. 3.7.1 – The relationship of forces when measured at the electron’s classical radius.
The red dot in the figures represents the point of force measurement.
The equations and accompanying figures depict the forces and their relationships as:
Strong Force The conservation of energy of in-waves as it is reflected as out-waves. When measured at
the electron’s classical radius, this is set to F ∝"1 for readability to make other forces relative. The conservation
of energy was shown earlier in Sections 3.1 and 3.2.
Electric Force The force of longitudinal out-waves in a particle like the electron where some energy is
used for spin. It reduces longitudinal energy by a factor of the fine structure constant (ae).
Magnetic Force (Monopole) The force of transverse out-waves in a particle like the electron, where
additional energy from spin is added to longitudinal energy in a given direction (cone). It increases energy by
a factor of the fine structure constant (ae). Multiplying the ratios of the electric force and magnetic force
returns back to one (ae/ae=1), identical to the strong force.
Magnetic Force (Dipole) – At a distance of the electron’s radius, the magnetic forces for monopoles and
dipoles are identical. The dipole force is no longer the inverse cube. Similar to above, when multiplying by
the ratio of the electric force, it returns back to one (ae/ae=1).
18
Gravity When the equation for gravity is expanded to its complete form and the electron’s radius is used
as the distance, note the two red circled terms in Table 3.7.1. They are identical to the circled term for the
electric force. Each electron has a reduction of longitudinal (electric) energy due to spin, and this missing
energy is accounted for in the magnetic force. But gravity is a shading effect that requires two or more
particles. Thus, this energy loss appears twice to represent the loss of each electron in the equation. An
illustration of the shading effect and gravitational constant (G) derivation is upcoming in Section 5.8.
4. Spacetime Modeled as a Spring-Mass System
The spacetime lattice of repeating unit cells may be described as a spring-mass system and mathematically modeled
with classical mechanics, even though it is not expected that spacetime literally includes springs. It is a representation
of a unit cell with a center granule of Planck mass, with harmonic motion, affecting and displacing nearby granules in
the lattice. Its harmonic motion produces a wave-like effect over time, where the distance from equilibrium over time
is a sinusoidal wave, and the maximum displacement becomes the wave amplitude. At the center of a particle, it will
be shown that this amplitude is Planck charge (qP).
Fig. 4.1Spacetime as a spring-mass system
Displacement from equilibrium is the measurement of charge (q) and is a cumulative effect of interference. For (i)
number of particles in the same phase, such as electrons, it is additive such that the constructive interference and
calculation of charge is the addition of each individual charge (qP), such that:
(4.1)
At a granular level, each particle contains many granules that interact to cause the interference and presence of waves.
Granules collide and transfer energy, producing waves and traveling through the spacetime lattice as wavelets
according to Huygen’s principle. The process of multiple granules in the same wave phase transferring energy can
also be represented in the spring-mass system as parallel springs, where the force is additive [
11
].
19
Fig. 4.2Constructive and destructive wave interference
The behavior of granules transferring energy will be constructive or destructive, depending on the direction of travel
of nearby granules. In the same phase (traveling in same direction), amplitude is constructive as shown in the left of
Fig. 4.2, and when in opposite phase (traveling in opposite directions), amplitude is destructive as shown on the right.
This is equivalent to a spring system with parallel springs in which the spring constant (k) is additive (k = k1 + k2).
4.1 The Fundamental Frequency
For space and time to be intertwined as spacetime, the definition of time must be inherent in the geometry of the
spacetime lattice itself. It is the harmonic motion of its components. In Fig. 4.1.1, the vibration of a granule is
illustrated in blue in the left of the diagram as it completes one wavelength. On the right of the figure, it is compared
to a granule traveling at a constant speed that travels a distance (l). This is the cone radius (d) and slant length (l) used
earlier in Section 2.2.
Fig. 4.1.1Harmonic motion of granule vibration creating a fundamental frequency
It becomes a fundamental frequency and a fundamental wavelength. The displacement of granules is in the same
direction as wave propagation, becoming a longitudinal wave. Photons, which are transverse waves where vibration
is perpendicular to wave propagation, will be addressed later in Section 5.
The fundamental frequency can be mathematically represented like the frequency in a spring-mass system. Eq. 4.1.1
represents the equation for the frequency of a spring with spring constant (k) and mass (m) [
12
].
20
(4.1.1)
A spring constant (k) can be determined by the force (F) divided by displacement (x). In the case of a unit cell, the
fundamental force is the Coulomb constant (from Eq. 3.3.9) divided by the granule separation length (a) from Eq.
2.1. Note that the spring constant (k) and Coulomb constant (ke) have different units, even though both use the letter k.
(4.1.2)
In a unit cell, the mass in the spring-mass system is a center granule of Planck mass (mP). This mass is substituted
into Eq. 4.1.1. The spring constant (k) from Eq. 4.1.2 is also substituted. The fundamental longitudinal frequency
(fl) and the fundamental longitudinal wavelength (ll) are calculated as:
(4.1.3)
(4.1.4)
The wavelength value matches the value derived a second way from Eq. 2.2 using Euler’s number at 2.7 x 10-17 m.
This value also sets a lower limit of the possible wavelengths of photons, as the transverse wave would not be able to
exceed this longitudinal wavelength.
4.2 Energy of the Spring-Mass System
In a spring-mass system, energy is a function of displacement (x). In the context of a center granule in a spacetime
unit cell, if the displacement is zero then there is no energy (despite having a mass of Planck mass). The energy (U)
of a spring-mass system with spring constant (k) and displacement (x) is illustrated in the next figure and is described
in Eq. 4.2.1.
Fig. 4.2.1Energy of a spring-mass system for one-way displacement (x)
(4.2.1)
The energy equations that follow will model the total energy (E) of a granule being displaced and returning to
equilibrium. Thus, Eq. 4.2.1 is used twice as it completes an oscillation.
21
(4.2.2)
As found in Eq. 4.1.2, the spring constant (k) is the Coulomb constant (ke) divided by the distance at which the force
is applied. Now, the distance from the center of an electron will be set to a variable (r). The Coulomb constant is
replaced in terms of the magnetic constant to complete Eq. 4.2.3. In Eq. 4.2.4, the displacement at the center of the
particle is the Planck charge.
(4.2.3)
(4.2.4)
The two equations above are inserted back into the energy equation in Eq. 4.2.2 and labelled as a strong energy (Es)
in the next equation, as this is the energy of the strong force (Fs) derived back in Section 3.2. Force is energy over
distance, so this equation could have simply been derived from the force equation. However, showing its origin from
spring-mass equations will help to model and calculate the entire spacetime grid.
(4.2.5)
Similar to the transition from the strong force to electric force, the geometric ratio is applied from strong energy to
electrical energy. The ratio (ae) is applied for the electron in the next equation and simplified to use the elementary
charge. This electrical energy is labelled (El) for longitudinal energy and will be used to calculate energies of particles,
photons and orbitals as proof in the next Section 5.
(4.2.6)
5. Explaining the Universe’s Fundamental Constants
The derivation of the upcoming energy and force equations are ultimately traced back to only five constants in Section
1 and two geometric ratios in Section 2. The magnetic constant, the fine structure constant, the Coulomb constant
and the elementary charge constants used in this section are fundamental physical constants that are naturally derived
in the process of explaining the conservation of energy as it changes in geometry and wave form.
Ten additional fundamental physical constants from physics equations are also derived from the base Planck unit
system here, explaining why they appear in equations and offering further proof of this proposed geometry of
spacetime. For example, Eq. 3.6.4 describes the gravitational force, which is well documented and known to be
proportional to the gravitational constant (G) when mass and distance are used as variables. By deriving G, it is
assumed that it is proof of the gravitational force equation found in Section 3. Or by deriving the Planck constant
(h), it is assumed that it is proof of the energy equation found in Section 4.
22
This section includes derivations based on single electron interactions. As a result, no constructive or destructive
wave interference is used or needed for the derivation of fundamental physical constants (Δ=1). However, for
calculations of energies and forces of multiple particles, constructive interference should be assumed (Δ<>1). This
is known to be the addition or subtraction of particles of charge (q) or mass (m).
5.1 Electron Energy & Mass
From the description of the energy of the spacetime lattice in Section 4 (Fig. 4.1), energy continually spreads from
the center of the particle spherically, decreasing in granule amplitude (displacement) proportional to the distance from
the center. Within a defined radius, this energy is measured as particle energy (or mass without c2). Beyond this
radius, energy continues but it is no longer considered to be within the confines of a particle. Its energy affects other
particles, which is seen as the electric force (energy over distance).
It is the wave form that defines a particle and its boundary. When incoming waves of the same frequency meet
outgoing waves of the same frequency, a phenomenon known as standing waves may occur. The outgoing waves created
by each and every particle is traveling the spacetime lattice until reaching other particles, becoming in-waves for other
particles. When this occurs, a standing wave forms near the center of the particle and spreads spherically outwards.
A standing wave contains energy, but there is no net propagation of energy [
13
]. In other words, it is stored energy,
which is why the volume contained in these standing waves appears as a spherical particle.
Fig. 5.1.1Electron energy and mass
For an electron, the transition point where standing waves become traveling waves and energy is no longer stored is
at the electron’s classical radius (re). The longitudinal energy equation from Eq. 4.2.6 is used to derive the electron’s
energy (Ee) and mass (me) in electrical constants, where the distance (r) is the edge of the standing waves - the electron’s
radius. The electron’s energy is correctly calculated using this equation at the electron’s radius in Eq. 5.1.1. And
the electron’s mass is the same equation without wave speed c2, also correctly calculated.
(5.1.1)
(5.1.2)
5.2 Bohr Radius
23
The energy of an electron continues beyond its radius, as derived in the electric force. When the force experiences
constructive wave interference between an electron and a particle of the same wave phase (e.g. another electron), the
particle motion is away from each other. When it experiences destructive wave interference with a particle of opposite
wave phase (e.g. a positron) the particle motion is towards each other. In Fig. 5.2.1, an electron is attracted via the
electric force (Fe) to a positively charged particle in the proton.
A proton is a known composite particle, containing at least three quarks (in some cases, five quarks have been found)
[
14
]. It can be shown that an orbital force (Fo), also called the magnetic dipole force in Section 3.5, keeps the electron
in orbit, forcing it out of the proton to balance the attractive force that pulls it in. The electron’s orbit in an atom is
where opposing forces are equal, illustrated in the next figure.
Fig. 5.2.1 – Single proton and electron (Bohr radius)
The electric forces and orbital forces are set to equal for the position where the forces on an electron is zero and it
will be in a stable orbit. Eqs. 3.3.6 and 3.5.5 are substituted into Eq. 5.2.1 to solve for the forces.
(5.2.1)
(5.2.2)
After solving for r in Eq. 5.2.2, the distance is found to be the Bohr radius (a0) – which is the most probable location
of an electron in an orbit around a single proton (hydrogen).
(5.2.3)
(5.2.4)
5.3 Rydberg Unit of Energy
24
The energy between a single electron and proton can be solved now, knowing the Bohr radius as the distance between
the two. This energy is known as the Rydberg unit of energy (ERy). The energy equation from Eq. 4.2.6 is used once
again the same equation used to calculate the electron’s energy and the electric force.
In Fig. 5.2.1 above, the one-dimensional orbital force (Fo) is on an axis between the proton and electron. This force
travels in both directions from the proton, eventually requiring an electron on the opposite side in an atom when
more protons are added to balance the electric force. As a result, half of the energy is on the side between the proton
and electron. A factor of ½ is applied to the longitudinal energy, as shown in Eq. 5.3.1. Then, Eq. 4.2.6 is substituted
into Eq. 5.3.1 and the Bohr radius (a0) is used as the distance r. It resolves correctly to be the Rydberg unit of
energy.
(5.3.1)
(5.3.2)
5.4 Avogadro’s Constant
The energy between a single electron and proton is constant, as found in the Rydberg unit of energy. The mass
contained in the unit cells that transfer this energy between the electron and proton is also constant. And the total
number of unit cells will be shown to be equal to Avogadro’s constant (NA) – another constant.
In 1834, Michael Faraday found that the mass of a substance altered proportional to the charge in electrolysis, in what
became known as Faraday’s law of electrolysis the Faraday constant is proportional to the elementary charge (ee)
and Avogadro’s constant (NA). It demonstrates that Avogadro’s constant represents a number of something
responsible for charge, which are unit cells between a proton and electron where the initial displacement is the
elementary charge. Furthermore, since the proton and electron form the fundamental atom (hydrogen), and since all
atoms and molecules are formed from a combination of protons and electrons, it would follow that Avogadro’s
constant would appear in other atoms with a greater number of protons, and also from molecules that are formed
from such atoms.
Fig. 5.4.1 – The number of unit cells between a single proton and electron – Avogadro’s constant
The calculation of Avogadro’s constant is found in Eq. 5.4.1. It is the probable distance between a single proton and
electron (the Bohr radius, a0, from Eq. 5.2.4), divided by the length of a spacetime unit cell (a, from Eq. 2.1). It
resolves to be the total number of unit cells between the proton and electron – Avogadro’s constant.
25
(5.4.1)
5.5 Planck Constant
The Planck constant (h) is used in equations to describe photon energies, where energy is proportional to frequency
(E=hf). A photon is a transverse wave, which is when the vibrating element moves perpendicular to wave propagation
(unlike a longitudinal wave that is in the same direction of wave propagation).
Referring back to Fig. 5.2.1, the electron’s position in a stable orbit is when the attractive electric force (Fe) and the
repelling orbital force (Fo) are the same. When the electron is closer or further than this range, they are unequal,
causing motion of the electron. While in motion, the granules at the center of the electron likely avoid the one-
dimensional Fo force, since it repels along a defined axis. Yet, they should continue to be attracted to the Fe force,
which is spherical, but the attraction is strongest when minimizing the distance between the particles. This causes a
vibration of the granules within the center of the electron in the direction perpendicular to the line of the Fo force
from the proton. Fig. 5.5.1 describes this vibration (illustrated as up-down in the figure), creating a transverse wave.
It now has a two-dimensional vibration while the electron is in motion as it continues to respond to incoming waves,
oscillating a distance of Planck charge (qP).
Fig. 5.5.1Generation of a photon as a transverse wave
The transverse wave energy (Et) is a conservation of energy. It can be described mathematically as the electrical force
(Fe) at a distance (r) between the proton and electron. As there are two transverse waves on each side of the particle,
the energy measured on one side is half the value, expressed in the next equation.
(5.5.1)
Substitute for the electric force (Fe) from Eq. 3.3.6 and then rearrange terms.
(5.5.2)
(5.5.3)
26
The Planck constant (h) is a proportionality constant representing the constants in the left, outside of the
parentheses, in Eq. 5.5.3. The remainder is used in the next section on frequency.
(5.5.4)
5.6 Rydberg Constant
Eq. 5.5.2 expresses the Planck relation (E=hf) in terms all derived from the original four Planck units and the
electron’s classical radius from Section 1. While the constants that constitute the Planck proportionality constant are
on the left, the remaining variables and constants within parentheses are frequency. For a single proton and electron,
the distance (r) is the Bohr radius (a0). Inserting this distance into Eq. 5.5.3 yields the transverse wave frequency for
a photon at ground state hydrogen.
(5.6.1)
This frequency is more commonly expressed as the Rydberg constant (R), which is found by removing wave speed
(c) from Eq. 5.6.1 such that it becomes an inverse of wavelength:
(5.6.2)
This section derives fundamental physical constants with only single particle interaction (one electron or one proton),
ignoring constructive wave interference. Calculating the frequency of atoms with multiple protons does indeed
require constructive wave interference (D), which results in two variables in the frequency equation (wave interference
and distance). The complete equation for frequency is found below, and has been validated by calculating energies
and frequencies of photons from hydrogen to calcium in a separate paper [
15
].
(5.6.3)
5.7 Compton Wavelength
The Compton wavelength (lC) occurs when photon energy is equal to a particle’s rest energy, found in annihilation
when an electron and positron annihilate. Eq. 5.6.3 can be used to find this frequency and wavelength. It describes
the interaction between single particles (D=1). The only remaining variable is the distance at which the particles rest
and are stable (r).
From the description of the electron in Section 5.1, standing waves occur within the electron’s radius (re). From the
description of destructive waves in Section 4, it is assumed that the position halfway between the electron’s standing
wave radius is the position in which the net amplitude of all granules within the particles is zero. In other words, it is
completely destructive at half the electron’s radius (re/2), providing a position for stability of the particles where no
27
longitudinal waves emerge because of their destructive properties. This distance is substituted into Eq. 5.6.3 as shown
below to resolve for frequency in Eq. 5.7.1.
Then, the Compton wavelength is solved by using the relationship of frequency and wavelength by taking the inverse
and removing wave speed (c). This results in the calculation of the Compton wavelength in Eq. 5.7.2.
(5.7.1)
(5.7.2)
5.8 Gravitational Constant
The gravitational constant (G) is used in the calculation of gravitational forces when the masses of two objects are
used as variables. Mass is a collection of a number of particles, including the electron, all of which experience gravity.
Thus, at the lowest level, the gravitational constant should appear in single particle interaction even if other forces
dominate at this level (the electric force is significantly stronger for two electrons than gravity, repelling them instead
of attracting them).
From the explanation of the geometry of gravity in Section 3, a single electron particle is now expanded to illustrate
two particles and a shading effect between the particles. In the figure, each electron has incoming wave amplitude at
the surface of the sphere that is greater than the outgoing wave amplitude at the surface of the sphere as a result of
transferring some energy to motion at the electron’s core (causing spin). Ignoring other forces (i.e. the electric force),
it would cause an attraction of the particles due to a net force as a result of unequal energy on the opposite sides of
the particles. This causes a shading effect between the particles.
Fig. 5.8.1 – Two particles and a net force forcing particles together as a result of an energy shading effect
The equation to model two electrons begins with the gravitational force equation from Eq. 3.6.4, shown again in the
next equation labelled as Fg.
28
(5.8.1)
The issue is that the gravitational constant (G) assumes a measurement in mass (m), not the elementary charge (ee).
However, mass and charge are related and were described earlier in Section 5.1. Total mass will eventually be
described in equations as a summation of the constructive wave interference of many particles. For single particles,
the mass of one electron in terms of electrical constants comes from Eq. 5.1.2 and is shown again below. Then,
one
elementary charge is isolated in Eq. 5.8.3.
(5.8.2)
(5.8.3)
Substitute Eq. 5.8.3 into 5.8.1 and simplify.
(5.8.4)
(5.8.5)
The elementary charge in the denominator and the fine structure constant in the numerator can be replaced to convert
back to Planck charge, per the relationship in Eq. 3.3.5. It can be further reduced by converting the magnetic constant
back to original Planck constants, per Eq. 3.1.5. These substitutions greatly reduce the equation for gravitation.
(5.8.6)
(5.8.7)
29
The constants on the left of Eq. 5.8.7 are the gravitational constant (G), as calculated correctly in both value and
units. It demonstrates that gravity occurs at a single particle, and its force will be proportional to mass due to
constructive wave interference, as mass is the additive interference of multiple particles.
(5.8.8)
5.9 Bohr Magneton
The Bohr magneton (µB) represents the electron’s magnetic moment, described earlier in Section 3 on the magnetic
force of a monopole. In the gravitational explanation of two electrons in Fig. 5.8.1, the wave amplitude and energy
coming into the surface of the electron’s sphere is not equal to its outgoing amplitude and energy. The difference is
very slight as a result of the motion of a center granule being displaced a Planck length. Due to the conservation of
energy principle, there should be energy transferred as a result of this center granule motion. This is illustrated in the
next figure.
Fig. 5.9.1Flow rate of granules at two poles of a particle
Fig 5.9.1 shows the energy from this vibration flowing through the electron at two poles. But the Bohr magneton is
not expressed in terms of either energy nor force. In the simplified kg/m/s unit system from Section 1, the Bohr
magneton’s units resolve to cubic meters per second (m3/s) when the units of charge (Coulombs) are replaced with
the units of distance (meters). These units make the Bohr magneton the equivalent of a flow rate, calculating the flow
from one pole of the electron.
The derivation begins with the magnetic force (monopole) from Eq. 3.4.3. This equation is slightly modified to insert
the square of the electron’s classical radius in both the numerator and denominator, which is the original derivation
of the fine structure constant back in Eq. 2.2.4. Next, in Eq. 5.9.2, Planck charge is replaced with the elementary
charge, by inserting another fine structure constant in the denominator (the relationship per Eq. 3.3.5).
(5.9.1)
30
(5.9.2)
The Bohr magneton volume flow is a measurement at the electron’s surface, so the radius (r) is set to the electron’s
classical radius (re). This is simplified and rearranged in the next equation.
(5.9.3)
Rearranging in this format allows the force to be expressed in terms of the flow rate (squared), through the electron’s
cross section (surface area – Se) of a given density (r). This simplified format is expressed as:
(5.9.4)
The surface area of the electron’s cross section is:
(5.9.5)
The magnetic constant was described earlier in Section 3.1 as a linear density. To make it a volumetric density for the
electron, it is the magnetic constant divided by the electron’s radius squared (re2) to give it complete density units of
kg/m3.
(5.9.6)
Eq. 5.9.4 was arranged to show the density and surface area properties on the right side of the equation. The remaining
constants in parentheses in that equation make up the flow rate. This is the square of the Bohr magneton. Taking
the square root of this leads to the derivation of the Bohr magneton derived from the magnetic force equation.
(5.9.7)
(5.9.8)
31
Conclusion
The structure of spacetime was found to be similar in equation to a spring-mass system, allowing all calculations of
energy and forces to be based on classical mechanics and properties where the components of the spacetime unit cell
have a known position in time and space.
A ratio of geometric shapes, including a rectangle, sphere and cone were found to be important in this paper. The
strong force used a ratio of the surface area of a rectangle compared to the surface area of a sphere. Then, adding
the surface area of a cone to the sphere provided the unification of forces as the coupling constants for the electric
force, gravitational force and magnetic forces – all of which were the same geometry yet differed in variables for the
lengths and widths of the geometries.
The simplified unit system with three basic units (kg, m, s) and only five constants (Planck length, Planck mass, Planck
time, Planck charge and the electron’s classical radius) were used along with the geometry ratios established for forces
and correctly derived and calculated more than a dozen fundamental physical constants associated with the electron.
It includes proportionality constants such as the gravitational constant (G), the Planck constant (h) and the Rydberg
constant (R) which have been validated in physics equations across a wide range of measurements for gravitational
forces, photon energies and photon wavelengths respectively, thus validating the simplified equations in this paper.
32
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Book
Over the last forty years, scientists have uncovered evidence that if the Universe had been forged with even slightly different properties, life as we know it - and life as we can imagine it - would be impossible. Join us on a journey through how we understand the Universe, from its most basic particles and forces, to planets, stars and galaxies, and back through cosmic history to the birth of the cosmos. Conflicting notions about our place in the Universe are defined, defended and critiqued from scientific, philosophical and religious viewpoints. The authors' engaging and witty style addresses what fine-tuning might mean for the future of physics and the search for the ultimate laws of nature. Tackling difficult questions and providing thought-provoking answers, this volumes challenges us to consider our place in the cosmos, regardless of our initial convictions.
Article
Presenting the history of space-time physics, from Newton to Einstein, as a philosophical development DiSalle reflects our increasing understanding of the connections between ideas of space and time and our physical knowledge. He suggests that philosophy's greatest impact on physics has come about, less by the influence of philosophical hypotheses, than by the philosophical analysis of concepts of space, time and motion, and the roles they play in our assumptions about physical objects and physical measurements. This way of thinking leads to interpretations of the work of Newton and Einstein and the connections between them. it also offers ways of looking at old questions about a priori knowledge, the physical interpretation of mathematics, and the nature of conceptual change. Understanding Space-Time will interest readers in philosophy, history and philosophy of science, and physics, as well as readers interested in the relations between physics and philosophy.
Article
This paper gives the 2006 self-consistent set of values of the basic constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA) for international use. Further, it describes in detail the adjustment of the values of the constants, including the selection of the final set of input data based on the results of least-squares analyses. The 2006 adjustment takes into account the data considered in the 2002 adjustment as well as the data that became available between 31 December 2002, the closing date of that adjustment, and 31 December 2006, the closing date of the new adjustment. The new data have led to a significant reduction in the uncertainties of many recommended values. The 2006 set replaces the previously recommended 2002 CODATA set and may also be found on the World Wide Web at physics.nist.gov/constants.
The Relationship of the Fine Structure Constant and Pi. ResearchGate
  • J Yee
Yee, J., 2019. The Relationship of the Fine Structure Constant and Pi. ResearchGate. Online: https://www.researchgate.net/publication/333224718.
Physics Questions People Ask Fermilab
  • Fermilab
Fermilab, 2019. Physics Questions People Ask Fermilab. Online: https://www.fnal.gov/pub/science/inquiring/questions/strong_force.html.
Simple Harmonic Motion Frequency
  • Hyperphysics
Hyperphysics, 2019. Simple Harmonic Motion Frequency. Online: http://hyperphysics.phy-astr.gsu.edu/hbase/shm2.html.
CERN's LHCb experiment reports observation of exotic pentaquark particle
CERN, 2015. CERN's LHCb experiment reports observation of exotic pentaquark particle. Cern.com. Online: http://press.cern/pressreleases/2015/07/cerns-lhcb-experiment-reports-observation-exotic-pentaquark-particles.
Atomic Orbitals: Explained and Derived with Energy Wave Equations
  • J Yee
Yee, J., 2018. Atomic Orbitals: Explained and Derived with Energy Wave Equations, Vixra.org, 1708.0146.