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1
Oscillating Spacetime: The Foundation of the Universe
John A. Macken
Santa Rosa, California, USA
Journal of Modern Physics (2024) 15, 1097-1143 https://www.scirp.org/pdf/jmp2024158_27505279.pdf
In this article, spacetime is modeled as a quantum mechanical sonic
medium consisting of Planck length oscillations at Planck frequency. Planck
length-time oscillations give spacetime its physical constants of c, G and ħ.
Oscillating spacetime is proposed to be the single universal field that
generates and unifies everything in the universe. The 17 fields of quantum
field theory are modeled as lower frequency resonances of oscillating
spacetime. A model of an electron is proposed to be a rotating soliton wave
in this medium. An electron appears to have wave-particle duality even
though it is fundamentally a quantized wave. This soliton wave can
momentarily be smaller than a proton in a high energy collision or can have
a relatively large volume of an atom’s orbital wave function. Finding an
electron causes it to undergo a superluminal collapse to a smaller wave size.
This gives an electron its particle-like properties when detected. The
proposed wave-based electron model is tested and shown to have an
electron’s approximate energy, de Broglie wave properties and undetectable
volume. Most important, this electron model is shown to also generate an
electron’s electrostatic and gravitational forces. The gravitational properties
are derived from the nonlinearity of this medium. When an electron’s
gravitational and electrostatic forces are modeled as distortions of soliton
waves, the equations become very simple, and a clear connection emerges
between these forces. For example, the gravitational force between two
Planck masses equals the electrostatic force between two Planck charges.
Both force magnitudes equal ħc/r2.
Keywords: unification of forces, electron model, cosmological constant
problem, foundation of physics, aether
1. Introduction
The research described in this article began with
the realization that there is a connection between the
properties of light confined inside a laser and the
physical properties of particles. The light reflecting
between the two laser mirrors is forced to have the
mirror’s frame of reference. This confined light
forms standing waves that exhibit 5 properties we
associate with fundamental particles. For example,
in a moving frame of reference, the bidirectional
light inside a laser exhibits a wave property
analogous to the de Broglie waves of a moving
electron. The standing waves in a moving laser also
undergo relativistic length contraction, relativistic
time dilation and relativistic energy increase.
Finally, the confined light waves have inertia (rest
mass). If the laser is accelerated, the light inside the
laser exerts unequal pressure on the laser’s two
mirrors. This produces a net force that resists
acceleration. This is the inertia of the light’s energy.
These 5 effects will be explained in Sections 7 and
8 of this article. These insights prove that when light
is confined to a specific frame of reference, the light
exhibits particle-like properties.
I am an inventor with many patents related to
lasers and optics. Therefore, this surprising insight
suggested to me that it might be possible to “invent”
2
a wave-based model of an electron that would
incorporate these 5 relativistic and quantum
mechanical properties. For example, a rotating
soliton wave would be a confined wave with a
specific frame of reference. Therefore, a rotating
wave model of an electron would acquire these 5
quantum mechanical and relativistic properties.
However, a wave-based electron model would
require the existence of a wave propagating medium
that went far beyond the aether. Besides propagating
light, this sonic medium would have to be the
foundation of everything in the universe. The wave
propagating medium would need extreme
properties, capable of generating all fermions and
forces from waves. The standard model has 17
named particles derived from 17 separate fields.
Therefore, the vision was that perhaps these 17
overlapping fields could be unified into one wave
propagating universal field that generates
everything in a wave-based model of the universe.
However, this wave propagating medium would
require the contradictory properties of having a
tremendous energy density capable of supporting
the most energetic waves while also appearing to be
undetectable to us.
Quantum field theory can be interpreted such
that the quantum vacuum is a field consisting of
harmonic oscillations with zero-point energy
(ZPE) of ħω/2 [1]. If ω equals Planck frequency,
ω = ωp = (c5/ħG)1/2, then the implied energy
density of this field equals Planck energy density
(Up = c7/ħG2 ≈ 10113 J/m3). However, the Wilkinson
Microwave Anisotropy Probe observed the
universe to be close to the flat Lambda-CDM
model [2] . This implies an average energy density
of the universe is roughly 10-9 J/m3. Therefore, this
10122 discrepancy has been described as "the worst
theoretical prediction in the history of physics" [3].
This discrepancy is known as the cosmological
constant problem and has been studied extensively
[4]. Virtually all the articles written about this
subject attempt to explain away the 10113 J/m3.
However, if spacetime has an unseen property that
mathematically appears to be 1011 3 J/m3, this could
potentially be the wave propagation medium I need
to be the foundation of a wave-based model of the
universe. Therefore, while others are attempting to
explain away this enormous quantum mechanical
energy density, I was motivated to prove it actually
exists.
Support for the quantum vacuum having a
property that mathematically appears to be 10113
J/m3 came from John Archibald Wheeler. He
examined the uncertainty principle and vacuum
zero-point-energy, then concluded these effects
would be explained if the "geometry of spacetime
fluctuates" [5, 6]. He concluded that 4-dimensional
spacetime consists of Planck length (Lp) vacuum
oscillations at Planck frequency (ωp) [7]. His vision
of the quantum vacuum has become the foundation
of the model of the universe proposed here. This
description of spacetime will be analyzed in
Section 4 and shown to have Planck energy density
(10113 J/m3) when described mathematically.
However, it only becomes observable energy when
angular momentum is added (an excitation is
added). Introducing a unit of quantized angular
momentum into this medium would give a rotating
soliton wave a specific frame of reference. We can
only detect differences in energy density.
Therefore, a rotating wave would be detectable but
a homogeneous medium that supports this wave
would be more difficult to detect if it is present
everywhere.
Wheeler’s model of spacetime is commonly
designated “quantum foam” or “spacetime foam”.
However, this article will use the term “oscillating
spacetime” because the spacetime model being
discussed is literally oscillating. Wheeler presented
his concept of spacetime in the last chapter of the
authoritative reference he coauthored [7]. This
reference states, “No point is more central than
this: empty space is not empty. It is the seat of the
most violent physics…. The density of field
fluctuation energy in the vacuum ∿ 1094 g/cm3
(∿10113 J/m3) argues that elementary particles
represent a percentage-wise almost completely
negligible change in the locally violent conditions
that characterize the vacuum…” The structure of
these fluctuations is described as: “The geometry
of space is subject to quantum fluctuations in
metric coefficients of the order of:
Planck length/length extension of the region under study” [7]. There
have also been many articles about spacetime foam
and related subjects [8]. None of those articles give
details suggesting that spacetime foam is the
3
fundamental building block of everything in the
universe.
The following article starts with a discussion of
the sonic properties of oscillating spacetime
consisting of Planck length oscillations at Planck
frequency. These properties include the calculation
of its propagation speed, impedance and bulk
modulus. A wave-based model of an electron is
developed from this medium. Tests of this electron
model show it achieves an electron’s energy, de
Broglie waves, and spin. This model unexpectantly
is found to also generate an electron’s electrostatic
and gravitational forces. Treating these forces as
wave interactions in oscillating spacetime results in
proof that these forces are closely related. The
wave-based model of the universe also makes
several predictions about gravitational
relationships, photons, entanglement, and the Big
Bang. The conclusion is that this is a successful and
useful model of the universe.
2. Names and units
This article addresses the big picture of a wave-
based model of the universe. Many different
subjects will be addressed. Approximations are used
by substituting the symbol k for unknown numerical
constants near 1. Another simplification is to ignore
the vectors of forces and only deal with the
magnitude of forces. These simplifications still
convey fundamental concepts without the added
burden of specifying numerical constants and
vectors. Electrons are use in examples, but these
examples also apply to muons or tauons.
Discussions of electric fields also imply magnetic
properties.
This article proposes that spacetime itself has
the properties of a sonic medium that propagates
waves at the speed of light. Spacetime will be
modeled as John Wheeler’s “spacetime foam”
(designate oscillating spacetime here) consisting of
Planck length oscillations at Planck frequency. The
multiple fields of quantum field theory will be
modeled as resonances and distortions of this
fundamental medium. All fermions will be modeled
as soliton rotating waves (sonic quasi-particles).
This is explained in Section 5, 7, 8 and Figs. 1 - 5.
This article also elevates Planck length beyond
its usual definition. The symbol lp or ℓp is usually
used to represent Planck length. However, this
article uses the symbol Lp to elevate its importance
and imply it is the fundamental wave amplitude of
oscillating spacetime. This medium has multiple
properties that equal 1 when expressed in Planck
units. Below is a list of base Planck units that will
be used in this article:
Planck length: Lp = (ħG/c3)1/2 = 1.62×10-35 m,
Planck time: tp = (ħG/c5)1/2 = 5.39×10-44 s,
Planck mass: mp = (ħc/G)1/2 = 2.18×10-8 kg,
Planck frequency: ωp = (c5/ħG)1/2 = 1.86×1043 rad/s,
Planck force: Fp = c 4/G = 1.21×1044 N,
Planck density: ρp = c5/ħG2 = 5.16×1096 kg/m3,
Planck energy: Ep = (ħc5/G)1/2 = 1.96×109 J,
Planck energy density: Up = c7/ħG2 = 4.64×10113 J/m3,
Planck pressure: p = c7/ħG2 = 4.64×10113 N/m2,
Planck temperature: Tp = mpc2/kB = 1.42×1032K,
Planck charge: qp = (4πε0ħc)1/2 = 1.88×10-18 C,
Planck electrical potential: p = (c4/4πε0G )1/2 = 1.043×27 ,
Here is a list of the symbols and equations that will be commonly used throughout the manuscript.
ωc = mc 2/ħ Compton angular frequency,
ƛ = λ/2π = c/ω Angular wavelength (Lambda bar),
ƛc = ħ/mc Compton angular wavelength of a fermion,
rc = ƛc = ħ/mc Compton radius of a fermion (3.86×10-13 m for electron),
rG = Lp2rc = Gm/c2 Gravitational radius of a fermion (6.76×10-58 kg for electron),
rq Charge radius – For charge e,: rq = α1/2Lp = 1.38×10-36 m,
4
N ≡ r/rc = rmc/ħ Number of Compton radii between fermions with mass m,
ZS ≡ c 3/G = 4×1035 kg/s Strain impedance of spacetime,
γ = (1 – v2/c2)-1/2 Lorentz factor,
α = e 2/4πε0ħc ≈ 1/137 Fine structure constant,
e = α1/2qp ≈ 1.6×10-19 C Elementary charge e,
Fe ≡ e2/4πε0r2 Electrostatic force between 2 electrons,
Fqp ≡ qp2/4πε0r2 Electrostatic force between 2 Planck charges,
FG ≡ Gme2/r2 Gravitational force between 2 electrons,
Ae ≡ (Lp/rc) =(4.18×10-23) Electron’s core strain amplitude (dimensionless).
3. The Lorentz transformation test
In the 19th century, the aether was widely
assumed to exist. Since light appeared to be waves,
it was reasoned that the vacuum of space must
contain a wave propagation medium (the aether).
However, the aether was abandoned for three
reasons. 1) Experiments failed to detect the aether,
2) Photons exhibited particle properties and 3)
Einstein’s special relativity theory postulated no
privileged reference frame required by the aether.
(A sonic medium’s “privileged reference frame” is
defined as the rest frame of the sonic medium.)
Gravitational waves (GWs) are known to
propagate through the “fabric of spacetime”. It is
not necessary to add an aether-like wave
propagation medium to spacetime because
oscillating spacetime itself has properties that
propagate GWs at the speed of light.
This article will treat spacetime as a sonic
medium and analyze its properties. The problem
is that we do not perceive any wave propagation
medium. Fermions move effortlessly through this
medium without friction. However, something
unseen limits the maximum speed of fermions to
the speed of light. The quantum vacuum also has
non-zero permittivity (ε0), permeability (μ0), and
impedance of free space (Z0). Furthermore, a
moving fermion with rest mass/energy E0
acquires relativistic energy of E = γE0 where
γ = (1 – v2/c2)-1/2. This is the well-known Lorentz
factor.
When Einstein developed special relativity,
he made two postulates that were counter
intuitive. These postulates led to special relativity
and Lorentz transformations. The postulates
were:
1) The speed of light in a vacuum is invariant
in all inertial frames of reference.
2) All the physical laws are the same in all
inertial frames of reference.
It is impossible to logically derive relativistic
length contraction, relativistic time dilation, and
relativistic energy from the currently accepted
model of particles and spacetime. These require at
least one of Einstein’s postulates. The following
example illustrates the problem. Suppose the speed
of a bullet is measured by the time it takes the bullet
to travel between two detectors. If the detectors are
moving relative to the gun, the measured velocity
will be the sum of the bullet’s velocity relative to
the gun and the detector’s velocity relative to the
gun. This calculation is logical and requires a
Galilean transformation. Now suppose this
experiment is repeated with a pulse of light. The
same speed of light will be obtained, even when the
detectors are moving relative to the light source.
This result is not logical. It requires a physical
change in the distance between the detectors
(Lorentz contraction) and a dilation in the rate of
time. Calculating this result requires a Lorentz
transformation.
Einstein concluded that Maxwell's equations
predicted the counter intuitive property of a
constant speed of light, independent of the velocity
of the observer. Therefore, he made the above two
postulates the foundation of special relativity.
Einstein’s postulates are now such an integral part
of physics that we have ignored the fact that our
generally accepted model of the universe cannot
explain the underlying physics that produces
Lorentz transformations.
In 1954, Einstein wrote: “Relativity theory can
be summarized in one sentence. All natural laws
5
must be so conditioned that they are covariant with
respect to Lorentz transformations” [9, 10]. His
two postulates artificially give us length
contraction, time dilation and all other relativistic
effects. A classical particle moving in an empty
vacuum would not acquire relativistic energy
E = γE0. A different model of particles and
spacetime is required to structurally produce
Lorentz transformations. If Einstein had a model of
spacetime, particles and forces that logically
generated Lorentz transformations, he would not
have required postulates to develop special
relativity.
Is there any alternative model of the universe
that logically generates Lorentz transformations
rather than Galilean transformations? The answer
is yes. There is a remarkable series of technical
articles [10 - 12] that imagine a thought experiment
of a hypothetical universe based on sound waves.
In this thought experiment, this “sonic universe” is
filled with a medium that propagates sound at a
sonic speed designated (cs). These articles assume
everything observable in this hypothetical universe
is made of sound waves. For example, simplified
sonic quasi-particles can be visualized as spherical
standing sound waves. The point of these articles is
not to describe these sonic quasi-particles, but to
imagine the physics of a universe based on the
foundation of a single universal sound propagating
field. Particles would be sonic quasi-particles and
forces would be transmitted through this sonic
medium. In other words, everything in this
hypothetical universe is derived from the sonic
medium. All of the properties of special relativity
can be derived from a universe based entirely on
sound [10 – 12]
The first of these articles is titled “A Real
Lorentz-FitzGerald-Contraction, [10]”. In this
article, Barcelo and Jannes propose a thought
experiment of a hypothetical universe based on a
massless scalar field with the properties of a
superfluid sonic medium that propagates sound at a
sonic speed designated (cs). Particles are visualized
as sonic quasi-particles (solitons) in this sonic field.
They ask the question “Is an inertial observer
capable of discerning whether he is at rest in the
medium or moving through it at a certain uniform
velocity?” To answer this question, they propose the
sonic equivalent of the Michelson-Morley
experiment [10]. They show that “the physical
length of a quasi-interferometer arm, as measured in
the lab (using acoustic instruments) would shrink by
an acoustic Lorentz factor γ = (1 – v2/cs2)-1/2 when
moving at a velocity v with respect to the medium.”
In other words, the acoustic universe model
achieves not only a constant speed of sound
(analogous to our constant speed of light), but it is
also impossible to detect motion relative to the
medium using an interferometer incorporating sonic
quasi-particles and the sonic transfer of forces.
The second article is “Sonic Clocks and Sonic
Relativity” by Todd and Menicucci [11]. They
describe a thought experiment in which “sonic
observers” possess devices called “sound clocks”
that are analogous to Einstein’s light clocks.
Motion, relative to chains of these sonic clocks, are
shown to undergo relativistic length contraction and
time dilation. The article states, “moving observers
perceive stationary sound clock chains to be length
contracted and time dilated exactly as one would
expect from a naïve application of the relativistic
formula with c being the speed of sound instead of
the speed of light.”[11]. The third article by D.
Shanahan [12] refutes objections to the sonic
universe model and shows that wave-based
elementary particles in the sonic universe do a good
job explaining the physical origin of the Lorentz
transformation.
These referenced articles prove that a universe
based on waves can intrinsically achieve Lorentz
transformations. John Wheeler’s spacetime foam
(oscillating spacetime) has the property of a sonic
medium. A sonic medium must have elasticity. This
is the ability to absorb energy and return energy as
a wave propagates through the medium. This article
will explore whether it is possible to start with
oscillating spacetime and achieve a plausible sonic
quasi-particle. For example, is there any
macroscopic examples of waves exhibiting wave-
particle properties similar to an electron? The
answer is yes, but first we must calculate some
properties of this medium.
4. Oscillating spacetime’s acoustic properties
6
John Wheeler concluded that vacuum zero-
point energy and the uncertainty principle both
would be explained by spacetime being a medium
that has Planck length (Lp) oscillations at Planck
frequency (ωp). Here we will take this description
and calculate a few acoustic properties of such a
medium. First, what speed should waves propagate
within this medium? For example, the speed of
sound in a gas is set by the thermal velocity of the
atoms or molecules. Similarly, the speed of sound
in oscillating spacetime is set by the speed of the
spacetime oscillations. The exact model of
oscillating spacetime is not clear, but in general it
involves a sea of harmonically oscillating volumes
of space approximately Lp in radius. These are
oscillating at Planck frequency (ωp) as measured by
the local rate of time. Planck length times Planck
frequency equals the speed of light (Lpωp = c).
While the details are not defined, this
approximately satisfies the requirement that
oscillating spacetime must propagate waves at the
speed of light. Next, we will calculate the
impedance (Z) of this medium. We start with the
standard equation for the intensity (I = kA2ω2Z) of
a wave with amplitude (A), angular frequency (ω)
and impedance is Z. The intensity of a wave is
I= Uc where U is energy density and propagation
speed is c. Combining these equations and solving
for impedance (Z), we obtain
22
Zk
Uc
A
ω
=
. (1)
We know the spectral energy density of zero-
point energy is U0(ω) = ħω3/2π2c 3 [1], and its
integral can be defined as:
( )
2
1
34
44
21
23 2 3 3
1
28
ZPE
Ud k
cc c
ω
ω
ωω
ω ωω
ππ
= = −⇒
∫
(2)
Equation (2) integrates this spectral energy
density to obtain the energy density between two
frequencies: a lower frequency ω1 and a higher
frequency ω2. Equation (2) carries this one step
further (designated by arrow ⇒) and assumes we
want all frequencies equal to or less than ω2.
Therefore, ω1 = 0, and ω2 is merely designated ω.
Also, the numerical constant is 1/8π2. However,
we are interested in understanding the
approximately 10122 discrepancy between the
observed energy density of the universe (∿ 10-9
J/m3) and Planck energy density (Up = c7/ħG2 ≈
10113 J/m3) implied by quantum field theory. We
will be addressing the big picture and ignoring
numerical constants near 1. Therefore, from Eq.
(2) U = ħω4/c 3. Also, the wave amplitude equals
Planck length (A = Lp = (ħG/c3)1/2). Making these
substitutions into Eq. (1), we obtain Eq. (3),
43 2 3
32 2
d
Z
c cc c
GG
cG
ωω
ω
=
= =
. (3)
The “displacement impedance” of oscillating
spacetime (Zd = c3/Gƛ2) has kg/s∙m2 unit. This is the
same unit as specific impedance. Notice that Zd
incorporates angular wavelength squared (ƛ2). This
means that the acoustic properties of oscillating
spacetime depend on the wavelength (frequency) of
the propagating wave. For unit compatibility, the
“displacement impedance” is required when
amplitude (A) has a unit of length (meter). For
example, Eq. (3) used Lp as the wave amplitude
(A = Lp). This has unit of length (meter).
The strain impedance of spacetime (Zs) is
calculated as
4 32 3 35
32
4.04 10 kg/s
s
Z
c cc
GG
c
ω
ω
= = = ×
. (4)
Equation (4) generates the strain impedance
Zs = c3/G with unit of kg/s. It is common to use
particle displacement (meter) to express the
amplitude of a sound wave in a gas or liquid.
However, wave amplitude can also be expressed as
a dimensionless number corresponding to the
maximum slope of a sine wave. For example, if the
wave’s displacement amplitude (Ad) is specified in
meters, then the dimensionless strain amplitude is
As = Ad/ƛ. The script symbol A is used to
designate amplitude expressed as a dimensionless
number. When an interferometer is used to detect
GWs, the observed amplitude is expressed as ΔL/L
7
where ΔL is the length change in an interferometer
caused by the GW and L is the round-trip path
length of the interferometer [13, 14] This is an
approximation that requires the GW to have a
wavelength much longer than the round-trip path
length of the interferometer. The exact slope used
in mathematical analysis requires that we define
dimensionless strain amplitude as (As ≡ Ad/ƛ).
Interferometers measure the approximation ΔL/L.
When amplitude is dimensionless slope, then
impedance with unit of (kg/s) must be used for
compatibility. Equation (3) used the substitution
A2 = Ad2 = Lp2 = ħG/c3 and that substitution
generated displacement impedance of spacetime:
Zd = c3/Gƛ2. Equation (4) calculates the strain
impedance of spacetime (Zs) using the substitution
A2 = As 2 = Lp2/ƛ2 = ħG/c 3ƛ2. This generates the
strain impedance Zs = c3/G. with unit of kg/s.
It is easy to extend these acoustic properties of
spacetime to obtain useful acoustic properties such
as density, energy density and bulk modulus.
However, it must be remembered that oscillating
spacetime is a universal field that requires the
addition of an excitation to make density and
energy density observable. Without this excitation
(discussed below plus Sections 9 and 19) all the
properties of this field are virtual. It is easy to
obtain the density encountered by a wave
propagating if we know the displacement
impedance (Zd) and propagation speed (c) because
ρ = Zd/c. The density will be designated “virtual
density ρv” and can be defined as
2
2
2
p
d
p
v
Zkk
c
L
kG
ω
ρρ
≡
= =
. (5)
This Eq. (5) gives the virtual density encountered
by a wave that propagates in oscillating spacetime.
Equation (6) converts this virtual density to virtual
energy density Uv,
2
22
2
22
pp
vp
v
v
c
ck kU
G
FL
UKk k
ω
ρ
= = = = =
. (6)
Since this is an ideal acoustic medium, the
virtual energy density equals the virtual bulk
modulus (Kv) of the medium. Note that Eqs. (5 and
6) convert to several other useful forms. In
particular, the virtual energy density (Uv)
encountered by a wave with angular wavelength ƛ
is Uv = (Lp/ƛ)2Up where Up = c 7/ħG 2 ≈ 10113 J/m3.
It should be noted that there is a similarity between
the virtual energy density (Uv = kFp/ƛ2) of the
quantum vacuum in Eq. (6) and the energy density
of a black hole (Ubh = kFp/rs2) where rs is the
Schwarzschild radius of the black hole.
Now we can test Eqs. (4 – 6) because GWs
propagate in the medium of oscillating spacetime. If
we assume that GWs are propagating in a physical
medium, then it is possible to determine the
properties of the medium from GW equations. This
was first done in books on GWs [13, 14]. The
authors of Ref. [13] mentioned, “Starting from
Einstein’s field equation … the coupling constant
c4/8πG can be considered a metrical stiffness (see
Sakharov 1968 [15]) … By analogy with acoustic
waves, we can identify the quantity c3/G with the
characteristic impedance of the medium. … The
problem of detecting gravitational wave radiation
can be understood as an impedance-matching
problem.” This same point is made in the more
recent Ref. [14] on GW detectors. Neither of these
Refs. show how the “analogy with acoustic waves”
generates the implied impedance c3/G . However,
the derivation is obvious. Both Refs. [13, 14] state
23
2
1
16
L
IL
c
G
ω
π
∆
=
kg/s3 . (7)
Equation (7) is the equation for the intensity of a GW
in the limit of a weak plane wave. The terms in Eq.
(7) have been arranged in a sequence that
corresponds to
22
I kA Z
ω
=
. (8)
Equation (8) is the general equation for the intensity
(I) of a wave with numerical constant (k), amplitude
(A), frequency (ω) and impedance (Z). Comparing
Eq. (7) to Eq. (8), it is obvious the terms match and
the impedance term is
3
35
4.04 10 kg/s
pp
s
c
Zm
G
ω
=
≡= ×
. (9)
8
Equation (9) is the impedance encountered by
GWs. This exactly matches Eq. (4), the previously
calculated impedance of oscillating spacetime.
This is a successful test! GWs encounter the same
impedance (c3/G) as the predicted model of
oscillating spacetime (Planck length oscillations at
Planck frequency).
We can do one more test. We can calculate the
energy density of Planck frequency oscillations (ω
= ω
p), impedance (Z = Zs) and dimensionless
amplitude (
A
= Ad/ƛ = Lp/Lp = 1). This
dimensionless form of amplitude must be used for
compatibility with strain impedance (Zs). The
equation to be used is (U = kA2ω2Z/c). Making
these substitutions and assuming k = 1, the energy
density is U = c7/ħG2 ≈ 10113 J/m3. This is the
correct answer, but it needs a physical
interpretation. This is proposed to be the
foundation of all fields. Fields only become
observable when an “excitation” is introduced. The
10113 J/m3 is a virtual energy density. Oscillating
spacetime has a unit of energy density when
mathematically analyzed, but it lacks the
excitations required to convert a field into
observable fermions or bosons. The missing
excitations are quantized units of angular
momentum (ħ or ħ/2). This will be shown in
Section 9 for an electron and Section 19 for a
photon.
5. Quantum vortex solitons have wave-particle
properties
When a wave has particle-like properties, it is
designated a soliton. A soliton is a nonlinear, self-
reinforcing, localized wave packet that is strongly
stable. For example, solitons are unaltered in shape
and speed by a collision with other solitons [17].
Most soliton waves propagate linearly, but there are
also rotating solitons. The closest analogy to the
wave-based electron model is a rotating quantum
vortex (a rotating soliton) in a superfluid such as
superfluid liquid helium or a superfluid Bose-
Einstein condensate. Figure 1 shows several images
of rotating soliton waves in a Bose-Einstein
condensate [16]. When angular momentum is
introduced into a superfluid using two rotating laser
beams, it creates multiple rotating quantum vortices
that are solitons. For example, if a small amount of
angular momentum is introduced into the superfluid,
the angular momentum is not distributed through the
entire mass of superfluid. Instead, the angular
momentum is quarantined into small rotating
“quantum vortices” while the bulk superfluid does
not rotate. Each quantum vortex has a quantized
angular momentum term incorporating ħ.
Notice the geometric pattern the rotating
vortices make. Immediately after stirring with two
rotating laser beams for about a tenth second, these
vortices have a random distribution [16]. However,
over about the next second, these vortices assemble
themselves into the geometric pattern shown in Fig.
1. This pattern is known as a “triangular lattice” or
Fig. 1 These are images of rotating quantum vortices in superfluid
Bose-Einstein condensate droplets. Frame A has about 16 rotating
quantum vortices and frame D has about 130 vortices. Each vortex
possesses a unit of quantized angular momentum incorporating ħ [16].
9
“hexagonal lattice”. This pattern reveals there must
be a repulsive force between each vortex. The
vortices arrange themselves in the geometric pattern
that achieves the greatest distance between vortices
but within the droplet boundary. This has an
obvious similarity to electrostatic repulsion between
electrons. Apparently, each rotating vortex distorts
the surrounding superfluid in such a way that the
lowest energy state is achieved at the maximum
separation distance between vortices.
There is an obvious connection to the electron
model. If it was possible to suspend 130 electrons
in a single X-Y plane with a circular boundary,
(analogous to Fig. D), their electrostatic repulsion
would arrange the electrons into the lowest energy
state. This would be a triangular lattice pattern like
Fig. 1D. The difference is a superfluid droplet with
a surface boundary generates rotating vortices. The
superfluid universal field with no surface boundary
generates rotating spherical waves. The repulsion
between the vortices in Fig. 1 is action at a distance
transferred by a distortion of the surrounding
superfluid. The electron model developed later will
also be shown to transfer its electrostatic and
gravitational forces through a distortion of the
surrounding medium (oscillating spacetime).
The rotating quantum vortices shown in Fig. 1
are shown because they have similarities to the
simplified electron model that will be proposed
later. Then the simplified electron model will be
expanded to allow it to achieve an electron’s wave
functions. However, first, a more complete
description of the oscillating spacetime model will
be given. Then the electron model will be
developed, starting with the simplified model. The
electron model will build on analogies to the
quantum vortices of Fig. 1.
6. Description of oscillating spacetime
Wheeler’s concept of oscillating spacetime [7]
can be condensed into the following two postulates.
1) The quantum vacuum is a sea of Planck length
vacuum oscillations at Planck frequency.
2) These oscillations create vacuum zero-point
energy and the uncertainty principle.
In addition to these, the following postulates are
proposed.
3) These spatial and temporal oscillations make
spacetime a stiff elastic sonic medium that
propagates waves at the speed of light.
4) This medium has a privileged frame of reference
where the medium is truly at rest. This privileged
frame corresponds to the local cosmic microwave
background (CMB) rest frame.
5) This oscillating spacetime is the single medium
that generates everything in the universe – all
fermions, all forces, all other fields and all the laws
of nature.
6) All fields are lower frequency resonances and
distortions of the ωp oscillating spacetime.
7) An electron is a soliton wave, rotating in the
oscillating spacetime. This rotating wave has ħ/2
angular momentum (Z axis) with undetectable
displacement amplitude of Lp.
We will start by giving a more detailed
description of oscillating spacetime. On the scale of
a volume with a radius of approximately Lp,
spacetime has the properties of a harmonic
oscillator. This volume undergoes both spatial and
temporal oscillations at (ωp ≈ 1043 rad/s). An
adjacent volume probably oscillates out of phase so
that there is offsetting expanding and contracting
volumes. Therefore, even describing the size as
approximately Lp in radius is a problem because the
coordinate distance between points fluctuates. The
fluctuations are happening at the speed of light.
Therefore, a wave in this medium will also
propagate at the speed of light. This satisfies the first
requirement for the universe to be a sonic universe.
The speed of sound of the sonic medium must equal
the speed of light.
The Planck frequency oscillations give
oscillating spacetime its 4th dimension (its time
dimension). Every point in spacetime needs a local
clock to enforce the local rate of time. The constants
c, G and ħ all have time in their units. The rate of
time is affected by gravity and frame of reference.
The oscillations of oscillating spacetime are
required to impose a local rate of time and generate
the constants c, G and h. The oscillation frequency,
measured locally, always equals Planck frequency.
These oscillations include a fluctuation of the
rate of time on the scale of Planck length. A
comparison of two hypothetical perfect point clocks
10
separated by more than Lp would show that they
speed up and slow down relative to each other. They
will differ by ± Planck time (tp) because of the
fluctuating rates of time at each location. The
following thought experiment will help to explain
the proposed spacetime fluctuations. Imagine a
spherical mass with the density of a neutron star. If
there is a small, evacuated cavity at the center of this
mass, this vacuum volume would not have any
gravitational acceleration. However, this internal
space would have a slower rate of time and a larger
proper distance between stationary points compared
to the same space without the surrounding mass.
Next, imagine this cavity volume if a
hypothetical negative gravity (antigravity)
substance is substituted for the surrounding shell.
Surrounding a cavity with this hypothetical negative
gravity substance would produce the opposite
effects including, 1) a faster rate of time, 2) a
smaller proper distance between stationary points
(smaller volume), and 3) no gravitational
acceleration. A substance that generates negative
gravity must be made of “negative energy.” There
are no examples of negative energy, but the concept
is useful.
The spacetime model to be tested has spatial and
temporal fluctuations between positive and negative
energy distortions. Adjacent volumes probably
oscillate out of phase, so these average to zero
observable energy and zero average distortion.
Therefore, the macroscopic average appears to be a
quiet vacuum with no observable energy. These
oscillations give the quantum vacuum its physical
properties (natural laws) that include the constants
of c, G and ħ.
Spherical volumes with radius much larger than
Planck length (r > Lp) contain vast numbers of these
Lp harmonic oscillators. Collectively, they also
produce a “noise” that is a distributed Lp fluctuation
across the radius r at the lower frequency of ω = c/r.
These larger volumes also achieve the virtual
energy of E = ħω/2 of zero point energy oscillators.
As discussed later in Section 9, a few frequencies
are resonances. For example, the resonance at
ω = 7.8×1021 rad/s is associated with both real
electrons and virtual electrons. A virtual electron is
a distortion of oscillating spacetime that
momentarily achieves an electron’s properties.
However, this distortion disappears in 1/ωc ≈ 10-21
seconds. The reason proposed here is because it
lacks the quantized angular momentum required to
achieve the stability and energy of a real wave-
based electron.
This model is supported by the fact that the
distance between two points cannot be measured to
the accuracy of Lp, and a time interval cannot be
measured to the accuracy of tp. [18 - 21]. These
limits are proposed to be the result of the vacuum
“noise” associated with the Lp and tp vacuum
fluctuations over macroscopic distances limiting the
accuracy of measurements. Also, the proposed
model of oscillating spacetime gives spacetime its
time dimension. The Planck length/time
fluctuations just described are not a field that
occupies empty spacetime. Instead, spacetime is an
oscillating sonic medium. All fermions will be
shown to be sonic soliton waves in this medium.
This meets the requirement described in [10 – 12] to
achieve Lorentz transformations. All the fields of
quantum field theory are proposed to be resonances
or distortions of oscillating spacetime. This is the
ultimate simplification and unification.
7. De Broglie waves
Richard Feynman famously said that “The
double-slit experiment has in it the heart of quantum
mechanics. In reality, it contains the only mystery.”
[22]. He was talking about the effects of a double
slit on both photons and electrons. When electrons
pass through a double slit, they exhibit wave
properties. This was first predicted by Louis de
Broglie in his 1924 PhD thesis. This prediction has
been confirmed by numerous experiments. (Google
Scholar lists over 1,000 articles on de Broglie waves
or matter waves.) Therefore, a key requirement of a
wave-based electron model is that it must exhibit
these wave-like properties. In a frame of reference
moving at velocity v, the model must achieve the
following: 1) the electron’s de Broglie angular
wavelength (ƛd = ħ/p = ħ/γmev); 2) the de Broglie
phase velocity (vphase = c2/v) and 3) the de Broglie
group velocity of (vgroup = v).
11
Fig. 2 The standing waves in a laser exhibit this modulation envelope when
the laser is translated at 5% the speed of light. This modulation envelope
wavelength (λm) would equal an electron’s de Broglie wavelength if the
laser wavelength equaled an electron’s Compton wavelength. These
standing waves also have relativistic length contraction.
In the Introduction, it was mentioned that the
initial motivation for starting this research was the
realization that the confined waves inside a laser had
5 quantum mechanical and relativistic properties of
particles. Normally, light waves freely propagate at
the speed of light with no frame of reference.
However, the light inside a laser reflects between
two mirrors. This creates standing waves inside the
laser and gives this confined light a specific frame
of reference. When light is forced to have the
particle-like property of a specific frame of
reference, it also exhibits 5 other particle-like
properties. In a moving frame of reference, this
confined light also exhibits 1) de Broglie waves, 2)
relativistic length contraction, 3) relativistic time
dilation, 4) relativistic kinetic energy and 5) inertia
when accelerated. This is proven mathematically in
references [23 - 25]. However, computer
simulations give a conceptual understanding that
even helps understand the mathematical analysis.
A laser has an integer number of counter
propagating light wavelengths reflecting between
its two mirrors. These counter propagating waves
form standing waves between the mirrors. In a
single frequency laser in a stationary frame of
reference, the amplitude of these waves oscillates in
unison. If phase is ignored, the amplitude of these
standing waves appears to oscillate at twice the
laser’s optical frequency. However, when phase
reversal is included, one complete cycle equals the
optical frequency. Figure 2 shows a laser moving
from left to right at about 5% of the speed of light.
This figure depicts how the standing waves would
appear if they could be observed at an instant of time
in this moving frame of reference.
Waves moving in the direction of relative
motion are Doppler shifted to a higher frequency
(shorter wavelength) and waves moving in the
opposite direction are doppler shifted to a lower
frequency. The superposition of these waves creates
the modulation envelope shown in Fig. 2. The
modulation envelope pattern is moving to the right
at a phase velocity of c2/v. For the example depicted
( v = 0.05 c ), the modulation envelope would be
moving to the right at 20 times the speed of light.
This is an interference effect equivalent to a moire
pattern that can move faster than the speed of light
without violating any laws of physics. The waves
that form this pattern are real; the moire pattern
12
Fig. 3 The A and B panels are Doppler distorted spherical waves moving to the
right at 25% the speed of light: In Fig. 3A, the waves are propagating outward
from a central source and in Fig. 3B the waves are propagating inward towards
a central point. Fig. 3C is a superposition of Fig. 3A and 3B. The moving linear
interference pattern in Fig. 3C has similarities to an electron’s de Broglie wave
properties if the waves have an electron’s Compton wavelength in the rest frame.
(modulation envelope) is ethereal. The modulation
envelope wavelength (λm) in Fig. 2 matches the
electron’s de Broglie wavelength if the optical
wavelength equals the electron’s Compton angular
wavelength ƛc = ħ/mec = 3.86×10-13 m. What
happens if there are spherical standing waves rather
than linear standing waves?
The mirrors in Fig. 2 reflected the laser light
and achieved the standing waves inside the laser.
Fig. 3 assumes an unseen reflection mechanism
such as a spherical mirror which reflects the
outward propagating waves back towards the
central source of the waves. The 3 panels in Fig. 3
assume the source of these waves and the reflector
are moving to the right at 25% the speed of light.
Figure 3A shows just the outward propagating
waves and Fig. 3B shows just the inward
propagating waves. The relative motion (25% of c)
produces the Doppler distortion of these waves.
Fig. 3C is the superposition of panels 3A and 3B.
The dark bands in Fig. 3C are regions of
destructive interference. These are equivalent to
the amplitude minimums (≈ 0 amplitude) in Fig. 2.
Notice that there is a 180-degree phase reversal at
these dark bands. For example, notice that a yellow
standing wave segment changes to a blue standing
wave segment going across a dark band. This
indicates a 180-degree phase reversal. In Fig. 2,
there also is a phase reversal going through a
minimum amplitude, but the difference in phase is
not as noticeable as in Fig. 3. The point of these
figures is to graphically show how spherical
standing waves with a wavelength equal to an
electron’s Compton angular wavelength achieve an
electron’s de Broglie wave properties. References
[23 and 24] prove that confined light in a moving
frame of reference not only exhibits de Broglie
waves, but it also exhibits relativistic length
contraction, relativistic time dilation and
relativistic energy increase. Reference [25] shows
confined light exhibits inertia. For example, if a
laser is accelerated along its optical axis, there is
unequal light pressure on the two reflectors. The
acceleration means that higher frequency light
strikes the rear reflector compared to the front
reflector. This produces a net force that resists
acceleration. This is the inertia of the confined light
energy. No Higgs field is required to give inertia to
confined waves.
8. The simplified electron and muon model
13
The proposed wave-based model of an electron
and other fermions is fundamentally a rotating
soliton wave in the medium of oscillating
spacetime. The following description will focus on
the simplest wave-based fermion, an electron. The
wave that forms an electron can have many
different shapes, sizes and characteristics
depending on the interactions with other fermions
and boundaries. Therefore, the description of an
electron will start with a simplified model of an
isolated electron in its most compact form. This
compact model allows calculations to be made
such as an electron’s energy, wave properties and
forces. Later in Section 18 there will be a brief
discussion of how this quantized wave can expand
and contract to other sizes and shapes.
There is an analogy between the proposed wave-
based model of an electron and a rotating quantum
vortex previously described in Fig. 1. It was shown
that when angular momentum is introduced to the
superfluid Bose-Einstein condensate, it forms
rotating vortices. Each vortex in this superfluid is a
rotating soliton wave possessing quantized angular
momentum incorporating ħ. These quantum
vortices arrange themselves in a pattern indicating
each vortex repels its neighbors by distorting the
surrounding superfluid.
Oscillating spacetime is a perfect superfluid.
When quantized angular momentum is introduced
into oscillating spacetime, it can form a rotating
soliton wave with similarities to the quantum
vortices of Fig. 1. Oscillating spacetime has
resonances that form stable or semi-stable rotating
waves at specific frequencies. We will only be
discussing electrons and muons. The rotating wave
that forms an electron must form standing waves in
the surrounding oscillating spacetime to achieve an
electron’s de Broglie wave characteristics. The
previous analysis indicated an electron’s de Broglie
wave characteristics imply the existence of
spherical standing waves with a frequency equal to
an electron’s ωc = 7.76×1020 rad/s. Therefore, the
indication is that these standing waves probably are
generated by a wave rotating at an electron’s ωc. The
simplest electron model would be a rotating wave,
approximately one Compton wavelength in
circumference. This would have a radius equal to an
electron’s Compton radius (rc = ħ/mec = 3.86×10-13
m). This radius is also equal to an electron’s
Compton angular wavelength (rc = ƛc = ħ/mec) This
simplified wave-based model of an electron is
illustrated below.
The proposed wave-based model of an electron
can be broken into two parts: 1) A core wave,
rotating at an electron’s ωc and 2) the standing,
rotating waves that surround this core wave. The
rotating wave that forms an electron’s core does not
have a sharp edge. However, it has a “mathematical
radius” equal to rc. This mathematical radius is used
Fig. 4A and 4B are two different ways of representing the rotating distortion of
oscillating spacetime that is the core of the wave-based model of an electron. Figure
4A represents the distortions of space and time as a distorted membrane. Figure 4B
uses blue and yellow to represent spatial and temporal distortions. The arrows in Fig.
4B imply rotation. The blue and yellow areas indicate the wave maximum and
minimum of the soliton wave that forms an electron’s core. The rotation rate is equal
to an electron’s Compton frequency ≈ 1020 Hz
14
in calculations. The energy in the electron’s electric
field, external to rc, is Eext = αħc/2rc ≈ 3×10-16 J. This
is roughly 0.4% of an electron’s total energy. There
is also a roughly comparable energy in the electron’s
magnetic field. Therefore, it can be said that in this
model, more than 99% of an electron’s energy is in
the rotating core and less than 1% of an electron’s
energy is in its electric/magnetic field.
Figure 4 shows two ways of attempting to
illustrate the core wave of the electron model. This
wave is completely unlike anything we have
previously encountered. This is a 4-dimensional
soliton wave rotating in oscillating spacetime. The
enormous impedance of this medium (c3/G =
4×1035 kg/s) makes it possible for this wave with
undetectable displacement amplitude of Planck
length/time to generate a testable electron model.
The spatial aspect of the electron’s core wave is
represented by Figure 4A, while Fig. 4B represents
the temporal component. Both components are
simultaneously present and form a single 4-
dimensional rotating wave The representation in
Fig. 4A has three inaccuracies. First, the Lp spatial
amplitude is about 1022 times smaller than an
electron’s Compton radius. Therefore, the wave’s
height depicted in Fig. 4A is vastly exaggerated.
Second, there appears to be an effect incorporating
the fine structure constant that distorts the electron’s
core. This distortion is only partly understood and
not incorporated into Fig. 4. This effect will be
discussed later in Section 17, but this α1/2 reduction
may reduce the size of the core somewhat. The third
inaccuracy is that Figure 4 implies the rotation is in
a single direction and plane. This is completely
wrong. The rotating wave that forms an electron
only distorts oscillating spacetime by Planck length.
It exists in a turbulent sea of Lp spatial and Planck
time temporal fluctuations that cause distortions.
Furthermore, this wave is at the limit of causality. It
has an expectation rotational direction and axis, but
this rotation is chaotic. It also rotates around all
other axes with a lower probability amplitude than
the expectation axis. Only the reverse of the
expectation rotational direction has a probability of
zero.
Now we will switch to attempt to explain how
the electron’s core wave also modulates the rate of
time (the 4th dimension). To illustrate this, we will
utilize blue and yellow colors in Fig. 4B. Imagine
the blue lobe of Fig. 4B represents a volume of
space where the rate of time is faster than the local
norm and the yellow lobe has a slower rate of time
than the local norm. The temporal distortion is
± Planck time (± 5×10-44 s) displacement of the
temporal dimension. For every radian of rotation of
the wave, a hypothetical clock in the blue lobe
would gain one unit of Planck time (∿10-43 s) and a
clock in the yellow lobe would lose one unit of
Planck time. This imperfect example is compared to
the local norm. The wave is rotating at
ωc = 7.76×1020 rad/s, therefore the blue lobe would
have a faster rate of time of tpωc = 4.18×10-23
seconds per second compared to the local norm and
the yellow lobe loses time at this rate. This is an
extremely small difference in the rate of time. For
example, if two perfect clocks differed by this
amount, they would only differ by about 40
microseconds over the age of the universe.
The rotating wave in the core of an electron is
technically a “dipole wave in spacetime”.
Macroscopic dipole waves are forbidden by general
relativity. For example, a standard text on general
relativity [7] states that there can be no mass dipole
radiation because the second time derivative of mass
dipole is zero. For example, if dipole waves existed
in spacetime on the macroscopic scale, they would
violate the conservation of momentum. However, it
is proposed here that the uncertainty principle
permits dipole waves to exist in spacetime if they are
undetectable. It has been theoretically proven that it
is impossible to make measurements accurate to Lp
or tp, [18 - 21]. Therefore, quantum mechanics
permits the dipole waves in spacetime required for
the wave-based model of fermions.
A sound wave in a sonic medium has an
amplitude that can be quantified as the maximum
displacement of the vibrating particles from the mean
position. This has been designated “displacement
amplitude (Ad)” and it has unit of length (meter). The
previously discussed alternative way of expressing
the amplitude of a sine wave is its strain amplitude
(As = Ad/ƛ). This is the dimensionless maximum
slope of the sine wave. The maximum slope of the
rotating wave that forms an electron will be
designated the “electron’s strain amplitude
(Ae = 4.185×10-23)”. It is a dimensionless number
15
Fig. 5 Both Fig. 5A and 5B represent clockwise rotating sinusoidal Archimedean
spirals. Figure 5A is outward propagating Archimedean spiral waves emanating from
a central rotating wave. Figure 5B is the inward propagating (reflected) spiral waves.
Figure 5C is the superposition of the outward and inward traveling waves to form
standing waves in Fig. 5A and 5B. This shows a central rotating core wave surrounded
by rotating standing waves. Figure 5C is a cross-section of the model of an electron.
obtained by dividing electron’s displacement
amplitude (Lp) by the electron’s Compton angular
wavelength (ƛc = ħ/mec = 3.86×10-13). Therefore
Ae = Lp/ƛc = 4.18×10-23. This dimensionless number
will be shown to also represent all the electron’s
properties (expressed in dimensionless Planck units)
except for the electron’s EM properties.
9. The source of an electron’s fields
Now, we are going to move on to describe
another part of the simplified electron model. If we
actually had a rotating hill and valley on a
membrane, such as shown in Fig. 4A, then an
extended elastic membrane would propagate a
sinusoidal Archimedean spiral wave, radiating away
from the source. Figure 5A shows an outward
radiating sinusoidal Archimedean spiral wave. The
clockwise rotating source of these waves is not
shown. For example, the yellow spiral waves in Fig.
5A can be visualized as hills and the blue spiral
waves can be visualized as valleys. These waves
would move outward from the central source at the
speed of surface wave propagation (c for an
electron).
If the elastic membrane has a circular boundary
that reflects these waves back towards the source,
the reflected waves will also form a sinusoidal
Archimedean spiral wave propagating back towards
the source. The inward propagating spiral waves in
Fig. 5B appear to imply a reversal in rotational
direction, but this is a wrong interpretation. Figure
5B also has a clockwise rotation but the inward
propagation produces this spiral. Figure 5C is the
superposition of the outward and inward
propagating waves. This figure should be rotating at
an electron’s Compton frequency, ∿ 1020 Hz. The
central core is surrounded by the rotating standing
waves that create an electron’s electric and
magnetic field. The distortion that creates an
electron’s gravitational field is not shown. This is a
cross section of the simplified wave-based model of
an electron.
As previously justified, about 99% of an
electron’s energy is in the electron’s central core
and less than 1% of the electron’s energy is in the
rotating standing waves. Beyond the core, these
waves decrease in amplitude with inverse radius
(1/r) and never go to zero. Therefore, an electron’s
“cloud” of standing waves extends indefinitely.
There is no mystery about action at a distance in this
model. A first electron’s standing waves physically
overlap a distant second electron. Each electron
distorts the oscillating spacetime medium used by
the other electron. When a rotating wave such as an
electron rotates in a distorted medium, the distortion
16
causes the rotating wave to migrate either away
from or towards the other rotating wave. This results
in the transfer of a force through a distortion of
oscillating spacetime. Two electrons migrate away
from each other (repel each other), but an electron
and a positron migrate towards each other. The
difference appears to be a difference in phase,
perhaps at the level of ωp. This is currently not
understood. However, the key point is that the force
is transferred through the medium. Recall the
repulsion between quantum vortices in a Bose-
Einstein condensate previously discussed in relation
to Fig. 1. They appear to also repel each other
through a distortion of the medium. There is no need
to postulate the exchange of virtual messenger
particles.
Figures (5A) to (5C) were made using a
sinusoidal Archimedean spiral. The equation for an
Archimedean spiral in polar coordinates is r = aθ.
In this equation, θ is the angle in radians, and “a ”
is a scaling factor with unit of length. The electron
model used for illustrations is based on an
Archimedean spiral with a = rc. Therefore, the
electron’s Archimedean spiral equation is r = rc θ .
Figure 5C is believed to be a fairly accurate
depiction of the electron’s rotating standing waves
beyond about one Compton wavelength. However,
there is an effect discussed in Section 15, involving
vacuum polarization that reduces the amplitude of
the waves beyond the core by a factor of
α1/2 ≈ 0.085. The electron model shown in Fig. 5C
depicts the electron model without the effect of
vacuum polarization. Without this effect, an
electron would generate Planck charge (qp) rather
than charge e = α1/2qp. This reduction in amplitude
probably also reduces the size of the core somewhat.
The (1/r) reduction in wave amplitude is also not
shown to depict the electron’s external rotating
standing waves more clearly.
The computer simulations assumed the inward
propagating waves were created by an unseen
spherical reflector. However, rather than a single
external reflector, the reflection in the model of an
electron is assumed to be the result of resonance
with oscillating spacetime in which rotating
standing waves in Fig. 5C become their own
reflectors. Waves attempting to escape are returned
to the core. There are only a few combinations of
frequencies, amplitudes, angular momentum, etc.
that achieve stability (this resonance). These are the
stable fundamental fermions and baryons.
In Bragg reflection, electromagnetic (EM)
waves in a transparent medium reflect off acoustic
waves [26]. In stimulated Brillouin scattering, an
intense laser beam creates the acoustic waves in a
medium that then reflects the laser beam in the
opposite direction.[26] The simplest type of
resonant reflection would be for a wave to create a
density variation (like a multilayer dielectric
reflector) that reflects the wavelength. An ideal gas
cannot achieve this type of acoustic wave resonance
because it has a single speed of sound and has no
nonlinearities. However, the electron model has two
different communication speeds. Some internal
communication happens at the same speed as
entanglement communication (instantly). Other
communication happens at the speed of light. For
now, it is only necessary to postulate the existence
of standing waves and calculate the results of these
standing waves.
10. Zitterbewegung
In 1930, Erwin Schrodinger [27] analyzed the
Dirac equation and derived a prediction that an
electron should exhibit a fluctuating interference
between positive and negative energy states. An
electron should appear to have a jittery motion,
which he designated “zitterbewegung” in German.
Paul Dirac explained in his 1933 Nobel Prize lecture
[28], “As a result of this oscillatory motion, the
velocity of the electron at any time equals the
velocity of light.” In the article On the
Zitterbewegung of the Dirac Electron [29], Kerson
Huang states, “Zitterbewegung may be looked upon
as a circular motion about the direction of the
electron spin, with a radius equal to the Compton
wavelength (divided by 2π) of the electron.” This
describes the rotating wave that forms an electron’s
core depicted in Fig. 4 and 5. The chaotically
rotating wave is moving at the speed of light with a
mathematical radius equal to the electron’s
Compton radius (rc).
The frequency of an electron’s zitterbewegung
is twice the electron model’s Compton frequency
(ωc). However, this is compatible with the proposed
17
electron model because the waves in Fig. 5 are the
electron’s wave function and the electron’s
zitterbewegung is analogous to the electron’s
probability function (intensity) that scales with the
square of the wave function (sin2θ = ½(1 – cos 2θ).
This squaring doubles the frequency to 2θ.
Therefore, the zitterbewegung should be twice the
wave function frequency. The quantized rotating
wave is moving at the speed of light. However, the
interference effects move at both less than and
greater than the speed of light. The electron’s
zitterbewegung oscillations extend far from the core
and are a component of the electron’s EM and
gravitational fields. The quantized wave-based
model is a natural fit to the Dirac equation and de
Broglie waves. Electron models incorporating a
point particle, or point excitation, are an arduous fit.
11. Energy calculation
We are next going to test the wave-based electron
model by calculating the energy of the electron
model. This is believed to be the first time an
analytical electron model has been proposed with
sufficient detail that it is possible to calculate the
implied energy. Recall that this wave-based model
of an electron is simplified to permit calculations.
There are several ways of doing this calculation.
The most direct uses the equation E = kA2ω2ZV/c.
This equation gives the energy (E) of a wave with
amplitude A, frequency ω, in volume V propagating
at speed c and encountering impedance Z . If we
substitute A = Ae = Lp/rc= (Gme2/ħc)1/2, ω = ωc =
c/rc = mec2/ħ, Z = c3/G, rc = ħ/mec and V = krc3,
then, ignoring numerical constants near 1, the
answer equals an electron’s E = mec2 energy
3
2
22
22 3
2
1
ee
e
e
Gm m c
A ZV c
c c G mc c
E k k mc
ω
= = =
(10)
Equation (10) shows this is a successful test because
the wave-based electron model generates electron’s
approximate energy (approximate because we
ignored k). If we were calculating a muon’s energy,
there would be different values for a muon’s
amplitude, frequency, and volume. This
combination yields a muon’s energy in Eq. (10).
The point particle models of electrons and muons
have no internal structure. These are primitive
models that make no structural difference between
an electron, a muon or any other fermion.
I want to put this energy calculation into
perspective. This model’s electron’s energy density
is about 3.4×1023 J/m3. This is about 167 times the
energy density of osmium (ρ ≈ 23 gm/cm3). The
reason that a wave with Lp displacement amplitude
can achieve this large energy density is because
impedance Zs = c 3/G makes oscillating spacetime
an extremely stiff wave propagation medium. A
high frequency wave with a very small amplitude
can have this energy density with this impedance.
This enormous impedance makes it possible for a
wave with an electron’s Compton frequency
(∿ 1020 Hz) and Lp displacement amplitude to form
an easily detectable electron. This is the “excitation”
of the universal field that creates an electron.
However, this enormous impedance also means
that GWs are hard to detect. Even GWs with
substantial intensity produce a very small
displacement of oscillating spacetime. For example,
the first GW detected (GW150914) had an intensity
of about 0.02 w/m2 at about 200 Hz. This intensity
would be a very loud tonal noise if it was a 200 Hz
sound wave. However, this GW produced a nearly
undetectable strain amplitude of only
ΔL/L ≈ 1.25×10-21 in oscillating spacetime [30, 31].
Detecting this infinitesimal effect required the
LIGO experiment; an interferometer with arms 4
km long.
12. Quantized angular momentum
Now that we have confirmed that the wave-
based electron model achieves an electron’s energy,
we can calculate the approximate angular
momentum of this rotating wave. A wave
propagating at the speed of light with energy E, has
momentum p = E/c. The electron model is a
distributed wave. However, for an approximation,
we can assume this wave is confined to only
propagate in a narrow circular channel (hoop) with
radius equal to an electron’s Compton radius
rc = ħc/E. The angular momentum of a rotating
hoop with radius r and momentum p is ℒ = pr . With
18
these postulates, we have ℒ = pr = (E/c)(ħc/E) = ħ.
The electron’s single axis angular momentum is ħ/2
and the 3-axis angular momentum is (3/4)1/2ħ ≈
0.87ħ. Therefore, this simple calculation is an
approximate success. Notice that this calculation
gives the same angular momentum for muons.
The rotating center of Fig. 5c is an interference
effect. Parts of this are moving slower than c and
parts are moving faster than c. The faster than c
component does not violate the laws of physics
because an interference effect is the sum of two or
more waves. The interference effect does not
transfer information or energy faster than light.
Only the underlying Archimedean spiral waves
possess energy and momentum. Only the very small
tangential component of these spiral waves carries
angular momentum. The electron’s rotation is also
chaotic. This means that it has an expectation
rotational axis, but all other rotation axes are present
at reduced probability. This substantially reduces
the single axis net angular momentum to something
less than ħ. We are only looking for
approximations, so ħ/2 is plausible. A point particle
has no radial dimension. Therefore, a point particle
must have zero angular momentum.
13. Electron’s gravity
When sound is transmitted through an acoustic
medium, nonlinearities are introduced by the finite
properties of the medium. For example, when sound
waves propagate in air, nonlinearities occur that
slightly modify the original waveform. The finite air
pressure sets a boundary condition that creates a
very weak nonlinearity at conversational levels.
If oscillating spacetime could propagate waves
without any frequency/wavelength limitation, there
would be no nonlinearities. However, oscillating
spacetime has finite properties that create a
boundary condition. For example, oscillating
spacetime cannot propagate a wave with a
frequency exceeding ωp or a wavelength less than
Lp. This single boundary condition means that a
sinusoidal wave in this medium will undergo a
nonlinear distortion even for frequencies much less
than ωp (wavelengths much longer than Lp). This
implies that even a wave-based electron with
frequency ωc = 7.8×1020 rad/s should produce a
small but quantifiable nonlinear distortion in
oscillating spacetime. We will test this hypothesis
and see if it explains the mechanics of how an
electron creates curved spacetime and gravitational
forces. Einstein just postulated that mass caused the
curvature of space without explaining the
underlying mechanism. The model proposed in this
work also aims to provide the foundational
explanation of how a fermion creates the
gravitational curvature of spacetime.
The gravitational curvature of spacetime can be
thought of as a slowing in the coordinate speed of
light. From my laser background, I postulated there
might be a similarity between gravitational slowing
of the coordinate speed of light in oscillating
spacetime and the optical Kerr effect that increases
the index of refraction (slows light) in transparent
materials. All transparent materials (all solids,
liquids, and gases) exhibit a nonlinear effect known
as the Kerr effect. The atoms that form a transparent
material are bound by electric fields with a finite
strength. This finite electric field strength is a
boundary condition. Applying an electric field to a
transparent material produces a nonlinear optical
distortion that results in a change in the index of
refraction (Δn = λKℇ 2) where λ is wavelength, K is
the Kerr constant of the transparent material and ℇ
is the electric field [32]. Note that the magnitude of
this nonlinear effect scales with electric field
strength squared (ℇ 2). The nonlinearity reaches
maximum when the imposed electric field equals
the boundary (the binding electric field).
In the optical Kerr effect, the oscillating electric
field of the light itself is the imposed electric field
that produces a change in the index of refraction.
Even low intensity light generates a weak nonlinear
effect, but this nonlinearity went unnoticed until
high intensity laser beams produced large nonlinear
effects. This effect becomes obvious when the
intensity of a focused laser beam is roughly about 1
GW/cm2 [33]. In glass, the increase in index of
refraction can be so great that the nonlinear index of
refraction gradient can cause the beam to self-focus
and prevents a focused laser beam from expanding
beyond the radius of the high intensity focus. This
is known as electrostrictive self-focusing. [33]
This hypothesis implies that squaring the
electron’s strain amplitude (Ae2 = (Lp/rc)2 =
19
1.75×10-45) should be analogous to squaring the
electric field (ℇ2) in the optical Kerr effect. The
prediction is that squaring the electron’s strain
amplitude should be the dominant first term in a
nonlinear expansion that defines the nonlinear
effect produced in an electron’s core wave. A
prediction is that multiplying electron’s strain
amplitude squared Ae2 by the electron’s Compton
radius (Ae2rc = Lp2/rc ) should give the
gravitational distortion produced by an electron.
This predicted gravitational distortion will be
designated the electron’s “gravitational radius”
rG, which is equivalent to
2 58
6.76 10
Gc
p
r Lr
−
= = ×
m. (11)
Equation (11) quantifies this prediction. It can also
be written as rG = As2rc. This predicted rG is half of
an electron’s Schwarzschild radius (rs = 2 Gme/c2 =
1.35×10-57). General relativity was generated from
a top-down approach that started with the
assumption that acceleration and gravity were
equivalent. “Einstein's happiest thought was his leap
from the observation that a falling person feels no
gravity to the realization that gravity might be
equivalent to acceleration.” [34]. His successful
general relativity addressed just the macroscopic
properties of gravity and acceleration.
Equation (11) was generated by a bottom-up
approach that started with oscillating spacetime
being a nonlinear medium and the wave-based
electron model causing a nonlinear distortion in this
medium. This bottom-up approach reveals the
underlying physics of how fermions create curved
spacetime. It is possible to judge the validity of this
approach by analyzing Eq. (11). The left side of Eq.
(11) is a gravitational concept that conforms to the
weak gravity approximation from general relativity
because rG = Gm/c2. The right side of Eq. (11)
contains only quantum mechanical terms.
(Lp2 = ħG/c3) and (rc = ħ/mc). Equation (11) is
unique since it is the first equation to derive a
gravitational property from a quantum mechanical
structural model of a fundamental particle. The gold
1 In general relativity, the exact equation is dt/dτ =
(1 – 2Gm/c2R)-1/2 and the weak gravity approximation is
dt/dτ = (1 – Gm/c2r). For an electron’s mass and Compton
standard for validating a new theory is whether it is
capable of making a prediction that can be proven
correct.
The essential nature of a particle’s gravitational
radius is highlighted by the straightforward Eqs.
(12) and (13)
22
G
g rc r=
, (12)
2
11
Ge
r Gm
dt
d r cr
τ
=+=+
. (13)
An electron will be used in the explanation, but this
applies to any fermion or even any hadron. Equation
(12) incorporates rG to generate the gravitational
acceleration (g) produced by an electron at a
distance r > rc. Equation 13 incorporates rG in an
equation equivalent to the weak gravity
approximation for the gravitational effect on the rate
of time. The left side of Eq. (13) is the gravitational
rate of time gradient (dt/dτ) produced by an
electron. The right side of Eq. (13) contains an
electron’s gravitational radius. The ratio rG/r is the
dimensionless slope of the gravitational curvature
produced by an electron. This slope can also be
written as Lp2/rcr or Gme/c2r . Technically, Eq. (13)
is the weak gravity approximation. However, for an
electron beyond its Compton radius (rc = 3.86×10-13
m), this approximation is accurate to better than 1
part in 1040 and is being considered exact.1
As illustrated below, Eq. (14) reveals there is a
symmetrical relationship between rG : Lp : rc
because
p
G
pc
L
r
Lr
=
. (14)
To understand this, imagine these three lengths are
designated on a logarithmic length scale. The
shortest length is an electron’s rG (= 6.76×10-58 m)
and the longest length is an electron’s Compton
radius (rc = 3.86×10-13 m). Planck length,
(Lp = 1.62×10-35 m) is exactly midway between
these two radii on the logarithmic length scale. This
symmetrical relationship also applies to any
fermion or even any hadron. A particle more
radius, the weak gravity approximation is accurate to better
than 1 part in 1040.
20
massive than an electron has a larger gravitational
radius and a smaller Compton radius. These two
radii scale in a way that keeps Lp exactly in the
middle of the logarithmic length scale. A
hypothetical particle with Planck mass would be the
limit of wave-based particles because a particle with
Planck mass would have its gravitational radius and
its Compton radius both equal to Lp.
This article proposes that an electron’s
gravitational radius (∿ 10-57 m) distortion is not
measurable for an individual electron. However,
this is the contribution each electron makes to the
total gravitational curvature (distortion) produced
by a large mass. For example, the mass of all the
electrons in the sun equals about 150 times the mass
of the Earth. It is possible to have a distortion of
oscillating spacetime that is smaller than Lp (
∿
10-35
m), even though it is not possible to have a
wavelength smaller than Lp (frequency > ωp).
14. Electron’s gravitational force test
Next, we will develop further the concept that
gravity is a wave-based nonlinear effect. The
previous derivation of the electron’s gravitational
radius used only the strain amplitude (slope) of the
rotating wave that forms the electron’s core. The
distortions of oscillating spacetime that create an
electron’s electromagnetic (EM) and gravitational
fields are outside the electron’s Compton radius
(outside the mathematical radius). To analyze an
electron’s gravity, we will now assume two wave-
based electrons separated by a distance larger than
an electron’s Compton radius (r > rc). The test will
be whether the wave-based model generates the
correct gravitational force magnitude between two
electrons.
Beyond the core (r > rc), the model has rotating
standing waves that decrease in amplitude with
(1/r). The dimensionless nonlinear strain
amplitude external to the core (designated AG)
must equal the square of the nonlinear strain
amplitude (AG = Lp2/rc2 = Ae2) at distance r = rc.
Outside the core, there is a 1/r decrease in
amplitude. We accomplish both goals if we
designate the distance between the two electrons
using the dimensionless number (N) of Compton
radii (N ≡ r/rc = r mec/ħ). The gravitational strain
Fig. 6 This shows the center electron’s standing
waves overlapping a second electron’s core (on
right). The white dots are separated by one complete
Compton wavelength or 2π Compton radii (2πrc).
Therefore, the separation between the two electron
cores in Fig. 6 is N = 8π ≈ 25.
amplitude (slope) then is (AG = Ae2/N) at a
distance of N Compton radii from the electron’s
center.
Figure 6 shows an electron core centered on
zero. This rotating wave structure includes a
rotating standing wave “cloud” that extends
indefinitely. Figure 6 shows a second rotating
electron core on the right. This also has rotating
standing waves, but these are not shown. Figure 6
illustrates the wave scale number N = r/rc. An
electron’s natural unit of distance measurement is
the number of Compton radii (rc) separating two
electrons. The second electron core in Fig. 6 is
rotating in the distorted volume of oscillating
spacetime created by the central electron’s standing
waves. The separation between the two electron
cores in Fig. 6 is N = 8π ≈ 25.
Moving forward, we will proceed with testing
this hypothesis. Can we generate the gravitational
force magnitude between two electrons using these
properties and the properties of oscillating
spacetime? The equation F = k A 2ω2Za/c gives the
force (F) exerted on area (a) by a wave with
amplitude (A) and frequency ω in a medium with
impedance (Z) and propagation speed c. The
unknown numerical constant near 1 is (k).
21
The simplified calculation assumes waves being
continuously emitted by the central electron. A
small part of these waves would strike the second
electron’s core area of (a = krc2). This would
produce a repulsive force (FGe). We will calculate
the force magnitude using this model. If it gives the
correct gravitational force magnitude, then the
refined model will be discussed. For this initial
calculation, the gravitational strain amplitude at
separation distance of N Compton radii is:
AG = Ae2/N. Other substitutions into
F = kA2ω2Za/c are: ω = ωc = c/rc , area = a = krc2,
Zs = c3/G, and N = r/rc results in
42
2
22 2
23
22
cs
G ec p
e
e
G
c
NN
kr
cc
F k kF
c Gc
r
ω
= = =
a
A
AA
(15)
If we set k = 1, then Eq. (15) becomes:
FGe = (Ae4/N 2)Fp = Gme2/r 2. Success! No
experiment is required. The starting point was the
insight that oscillating spacetime should be a
nonlinear medium that is unable to propagate waves
with frequency greater than ωp. This boundary
creates the nonlinear amplitude AG in Eq. (15). This
converts to Newton’s gravitational equation for the
gravitational force magnitude between two particles
with an electron’s mass (me). This is believed to be
the first time a gravitational force magnitude has
been derived in a bottom-up calculation of the
quantum mechanical properties of particles. Before
commenting further on this significant advance, we
will first develop the electrostatic force between two
electrons using the same electron model and the
properties of oscillating spacetime.
15. Electron’s electrostatic force test
We will now proceed to use similar logic in
an attempt to derive the magnitude of the
electrostatic force between two electrons. The
same force equation (
F
=
k A
2ω2
Za
/
c
) will be
used
.
The only difference compared to the
gravitational calculation Eq. (15) is the following
calculations will test two slightly different first
order (linear) strain amplitudes. First:
A
= (Ae/N) and second A = α
1/2(Ae/N). The
symbol Fqp is the electrostatic force (Coulomb’s
law) between two Planck charges and Fqe is the
electrostatic force magnitude between two electrons
with charge e. Other substitutions are ω = ωc = c/rc;
Z = Zs = c3/G and area = a = krc2 which leads to
22
2
2
22
23
2
ee
p
c
qp
c
N
kF
N
r
kA c c
Fk
c Gc
r
ω
= = =
a
AA
, (16)
22
2
2
22
23
2
e
e
p
c
qe
c
N
kF
N
r
kA c c
Fk
c Gc
r
αα
ω
= = =
aA
A
(17)
If we set k = 1, Eq. (16) becomes:
Fqp = (Ae2/N2)Fp. This converts to Coulomb’s law
for the force between two Planck charges
(Fqp = qp2/4πεor2). It was hoped that Eq. (16) would
generate the electrostatic force between two
electrons. Instead, it generates the force between the
two most fundamental units of charge (Planck
charge qp). The electron model is attempting to
generate Planck charge. If 100% of the core energy
is radiated, it would create Planck charge. However,
vacuum polarization apparently reduces qp by a
factor of α1/2 ≈ 0.085 to charge e. The nonlinear
wave component that creates gravity is unaffected
by vacuum polarization. Therefore, vacuum
polarization creates a difference of factor of α1/2
between the first order waves that create an
electron’s EM properties and the second order
waves that create an electron’s gravity.
Planck charge is the most fundamental unit of
charge. It not only is derived by setting fundamental
constants equal to 1, but two Planck charges have
the maximum possible coupling constant equal to 1
because qp2/4πε0ħc = 1. For comparison, charge e
has a coupling constant of α because e2/4πε0ħc = α.
The wave-based model of an electron is
attempting to generate Planck charge because an
initial wave amplitude of Lp at the base of the
rotating standing waves would create Planck
charge. However, if an electron had Planck charge,
all of an electron’s energy would be in the
electric/magnetic field external to the electron’s rc.
This would be unstable because there would be no
energy in the electron’s essential rotating core.
Without vacuum polarization reducing the first
order wave amplitude at the edge of the core from
22
Lp to α1/2Lp, the stable electron would not exist.
Mathematically, it is easy to accommodate this α1/2
reduction in standing wave amplitude. This is done
by manually inserting (α1/2) into Eq. (16) to generate
Eq. (17).
The force between two electrons, each
possessing a charge of e, is denoted as Fe in Eq. (17).
To achieve this charge e force, Eq. (16) uses the
substitution A = (α1/2Ae/N). This generates
Fe = (αAe2/N2)Fp = e2/4πεor2. This is Coulomb’s
law for the electrostatic force between two
electrons. Planck charge is: qp = (4πεoħc)1/2 =
1.87×10-18 coulomb. An electron’s charge e is about
8.5% of Planck charge (e = α
1/2qp). This addition
(α1/2) reflects the structural modification nature
requires to achieve the stable resonance condition
that forms an electron. This α1/2 modification should
have an unknown effect on the size and distribution
of the electron’s core.
In Table 1, the information in Eqs. (15 – 17) is
presented in different ways that makes it easier to
see that the electrostatic and gravitational forces
between two electrons are closely related.
Column 1
Introductory force
equations using terms
FG, Fe and Fqp
Column 2
Force equations using
Planck force Fp and
Planck distance r/Lp
Column 3
Force equations using
the wave properties Ae
and N
Row 1
FGe
2
2
e
Ge
Gm
r
F
≡
(18)
2
2
2
p
ep
Ge
LF
Fr
=
A
(21)
4
2
e
e
p
Ge F
FN
=
A
(24)
Row 2
Fqe
2
2
4
o
qe
r
e
F
πε
≡
(19)
2
2
p
p
qe
LF
r
F
α
=
(22)
2
2
e
e
p
qe
F
N
F
α
=
A
(25)
Row 3
Fqp
2
2
4
p
o
qp
q
r
F
πε
≡
(20)
2
2
p
p
qp
LF
r
F=
(23)
2
2
e
e
p
qp
F
FN
=
A
(26)
Table 1 The objective of this table is to prove the gravitational and electrostatic force
magnitudes between two electrons are mathematically closely related. All the equations
in Row 1 are equivalent to Newton’s gravitational equation for an electron’s mass me.
All the equations in Row 2 are Coulomb’s law for two electrons. Row 3 is Coulomb’s
law for two Planck charges (qp). A brief definition of some terms is in the footnote2.
Equation (18) is the Newtonian gravitational
equation for two electron masses (me). Equation
(19) is Coulomb’s law for two electrons with charge
e. Equation (20) is Coulomb’s law for two
hypothetical particles with Planck charge (qp).
These standard equations are included for easy
comparison to other equations in Table 1.
2 Ae ≡ Lp/rc = me/mp = Ee/Ep = ω
e/ωp = (Gme2/ħc)1/2 =
4.18×10-23 This dimensionless number is an electron’s
“structural constant”. All of an electron’s non-EM structural
properties can be expressed in natural units with this number.
F Ge = the gravitational force magnitude between two electrons
Column 2 converts the three standard force
equations in Column 1 to force equations
incorporating Planck force Fp = c4/G = 1.2×1044 N
and dimensionless distance r/Lp measured in Planck
units.
The equations in column 2 made a partial
conversion to the wave properties of the electron by
incorporating the wave’s strain amplitude Lp/r at
Fqe = the electrostatic force magnitude between two electrons
(charge e)
Fqp = the electrostatic force magnitude between two Planck
charges (qp)
Other terms are defined in Section 3
23
distance r. However, distance r is designated in
meters, and meters are a length standard devised by
humans. The model is predicting the electron’s
natural standard of length is its Compton radius
(rc = 3.86×10-13 m). If we want to fully convert the
force equations to wave properties, (illustrated in
Fig. 6), we should designate the distance between
two electrons as N Compton wavelengths (N ≡ r/rc).
The other wave property used in column 3 is an
electron’s strain amplitude Ae. It specifies the strain
amplitude of the core wave Ae = Lp/rc = 4.18×10-23.
Column 3 shows the maximum mathematical
unification between the gravitational and
electrostatic force magnitudes because this column
incorporates the most wave properties.
We will start by comparing the gravitational
force magnitude Eq. (24) to the Planck charge
electrostatic force magnitude, Eq. (26). The only
difference between these two equations is the
gravitational equation has Ae4 and the Planck charge
electrostatic force equation has Ae2. Therefore, the
difference (ratio) between these two forces is
FGe/Fqp =Ae2 = 1.75×10-45. Equation (25) gives the
electrostatic force between two electrons with charge
e rather than charge qp. Equation (25) requires the
additional term α to accommodate the reduced
electric field energy. Therefore, the force ratio is
FGe/Fqe = Ae2/α = 2.4×10-43. This is a simplification
compared to the conventional ratio of
FGe/Fqe = Gme24πεo/e2 ≈ 2.4×10-43. The electron’s
constant Ae can be used to quantify all of a wave-
based electron’s non-EM structural properties.
16. What is electric charge?
From this analysis, it is now possible to
understand another mystery of physics. Electrons
and muons are currently visualized as charged point
particles that are excitations in the electron and
muon fields respectively. Electrostatic forces are
currently visualized as virtual photon fluctuations in
the EM field. According to this commonly held
model, an electron, and a muon both have charge e
because they both emit the same virtual photon
fluctuations in the EM field. The EM force between
two electrons is currently understood to be the
exchange of virtual photons between charged
particles. A muon’s mass is about 207 times larger
than an electron’s mass. The gravitational force
between two muons is (207)2 ≈ 43,000 times larger
than the gravitational force between two electrons.
However, the electrostatic force is the same between
two electrons or two muons. Therefore, charge is
treated as an unexplained fundamental property,
independent of mass. This results in the concept that
the EM and gravitational forces are completely
separate.
In the simplified model proposed here, the cores
of an electron and a muon are both rotating soliton
waves with the same displacement amplitude (Lp).
The obvious differences are that a muon has more
energy because it has 207 times higher rotational
frequency and 207 times smaller Compton radius
than an electron. However, there is also a more
subtle difference between an electron and a muon.
The model predicts a muon, and an electron should
both have the same percentage of their energy in
their respective electric fields (external to their
cores). This implies a muon should have 207 times
more energy in its electric field than an electron.
This last statement seems impossible because they
both have the charge e. Is this a fatal flaw in the
model? No! This will be shown to be a reasonable
prediction that actually supports the model.
The electric field energy is defined as the first
order rotating standing wave energy external to their
respective Compton radii. A muon’s Compton
radius is 1.86×10-15 m. An electron’s Compton
radius is 207 times larger (3.86×10-13 m). The
muon’s extra electric field energy is all packed into
the small spherical shell volume with an inner radius
of 1.86×10-15 m and outer radius of 3.86×10-13 m.
Therefore, the muon does have 207 times more
energy in its electric field than an electron. We only
measure charge and electrostatic force at distance
greater than 3.86×10-13 m. Therefore, we do not
detect any difference between the electric field of a
muon and an electron. The prediction is reasonable.
This is a partial answer, but it does not give a
conceptually understandable answer to how two
electrons and two muons can exert the same
electrostatic force. The deeper answer is that the
property we call “charge” is a first order distortion
of oscillating spacetime. The rotating waves that
form an electron and the muon have different
24
frequencies and radii, but these waves both have the
same displacement amplitude equal to Planck
length (Ad = Lp). This wave amplitude produces a
volumetric distortion of oscillating spacetime.
Inside the cores this distributed distortion is
equivalent to a total volume approximately equaling
the volume of a sphere with radius of Lp. However,
the vacuum polarization that takes place at the edge
of the core reduces the charge from Planck charge
to charge e. (from Lp to: α1/2Lp ≈ 0.085Lp).
Therefore, all fermions and hadrons with
elementary charge e produce the same magnitude of
first order distortion. This type of distortion will be
designated “charge radius rq” For charge e, the
charge radius is: rq = α
1/2Lp = 1.38×10-36 m and
Planck charge qp has rq = Lp = 1.62×10-35 m.
The concept of charge radius can be extended to
improve our understanding of the unit of coulomb
in equations. If oscillating spacetime is the universal
field, coulomb must be able to be converted to a
property of spacetime. The model predicts that
Planck charge produces a Lp polarized distortion of
the oscillating spacetime field. Therefore, (Lp/qp) is
the proposed “charge conversion constant” that
converts electrical charge, with unit of coulomb
(C), to a distortion of spacetime with unit of meter.
This is actually a unit of “polarized length” that
incorporates a vector (discussed later). The
following Eq. (27) is the proposed “charge
conversion constant Lp/qp” that will be used to
convert a unit of coulomb to a unit of polarized
length
18
4
8.617 10 m/C
4
p
po
LG
qc
πε
−
= = ×
. (27)
The way Lp/qp is used will be demonstrated using
the Coulomb force constant 1/4πεo. This has
dimensional analysis units of ML3/T2Q2. Therefore,
we want to eliminate the Q-2 term (inverse coulomb
squared). This is accomplished as follows:
(1/4πεo)(qp/Lp)2 = c4/G. Converting coulomb to
meter using Lp/qp, the Coulomb force constant
converts to Planck force (c4/G). This is reasonable
and gives a new insight.
Electrical current with unit of coulomb/second
converts to m/s. Therefore, 1 ampere converts to
3.33×10-9 m/s and Planck current (Ip = 3.48×1025
amp) converts to the speed of light. Electrical
potential (voltage) converts to a force. For example,
one volt converts to 1.16×1017 N. An electron,
passing through a 1 volt drop, does not experience a
force of 1.16×1017 N. Instead, an electron’s
distortion of spacetime (α1/2Lp = 1.38×10-36 m)
encounters a force of 1.16×1017 N. Force times
distance equals energy. Therefore, 1.16×1017 N x
1.38×10-36 m = 1.6×10-19 J = 1eV. This indeed is the
energy gained by an electron passing through a 1-
volt electrical potential. Equation (28) shows that
the impedance of free space (Zo = 1/ε0c ≈ 377 Ω)
encountered by photons converts to the impedance
of spacetime c3/G because
243
4
14
po
o
po
qcc
ZL cG G
πε π
ε
= =
. (28)
The 4π can be ignored in Eq. (28). Proving that Z0
converts to c3/G is important because this says that
photons propagate in the medium of oscillating
spacetime. Recall that c3/G was the calculated
impedance of both oscillating spacetime, and the
impedance encountered by GWs. Therefore, Eq.
(28) completes this concept and adds photons as
quantized waves propagating in the universal field
of oscillating spacetime. Terminal collapse gives the
photon its particle-like properties.
The term “polarized length" implies it is a unit
of length that has a vector. The charge conversion
constant can only be used in electrical equations
where the radial terms are aligned with the electrical
field gradient. The charge conversion constant
cannot be used to convert the magnetic field
strength H (unit: C/sm) and the magnetic flux
density B (unit: kg/sC). The reason for this appears
to be that magnetic equations incorporate cross
products that create vector conflicts with the
polarized length property of the electrical charge.
The distortion of oscillating spacetime produced
by electrical charge appears to be more complex
than the distortion produced by gravity. For
example, this model has not yet explained the
structural difference between an electron and a
positron. For example, the difference between
positive and negative charge perhaps requires a
phase distortion at the level of ωp. This is one of
25
many challenges remaining before this wave-based
model of the universe matures.
17. Unification of forces at the Planck limit
There is a thought experiment that proves that the
gravitational and EM forces converge and produce
the same force magnitude at the Planck particle
limit. It will be shown that the gravitational force
between two particles with Planck mass (mp) equals
the electrostatic force between two particles with
Planck charge (qp). This equality at the Planck
particle limit clearly establishes the close
connection between these forces. Before describing
this thought experiment, it is useful to briefly
describe Planck charge and Planck mass.
Planck charge (qp = (4πεoħc)1/2 = 1.88×10-18 C)
is the natural unit of charge that results from setting
c = 1, ħ = 1 and setting the coulomb constant also
equal to one (1/4πε0 = 1). However, Planck charge
is more than just a unit of charge derived from
fundamental constants. Planck charge implies the
theoretical maximum charge radius equal to Planck
length (rq = Lp). In the wave-based model, Planck
charge would result if 100% of the particle’s energy
was in its electric/magnetic field. This limiting case
would be unstable because no energy would be in
the essential core. Charge e is a factor of α1/2 ≈ 8.5%
of Planck charge (e = α
1/2qp). Two hypothetical
particles with Planck charge would repel each other
with a force about 137 times larger than the
electrostatic repulsive force between two electrons.
Planck mass (mp = (ħc/G)1/2 = 2.18×10-8 kg) is
the maximum possible mass/energy a single
fermion can have. A wave-based particle with
Planck mass/energy would have a Compton radius
equal to Lp. Therefore, a Planck mass particle
would have the maximum possible nonlinear strain
amplitude of nonlinear strain amplitude
(AG = Lp2/rc2 = Lp/Lp = 1). A hypothetical particle
with Planck charge would have the maximum
possible displacement amplitude external to the
core. Therefore, a Planck mass particle has
similarities to a Planck charge particle because they
both represent the limits of particle properties. The
related force equations are
2
22 2
p
Gp
Gm Gc
Fr rG
c
r
= = =
, (29)
2
222
4
44
po
qp
oo
r
c
r
qc
Fr
πε
πε πε
= = =
, (30)
2
Gp qp
F F cr= =
. (31)
Equation (29) calculates the gravitational force
(FGp) between two hypothetical particles with
Planck mass (mp) and Eq. (30) calculates the
electrostatic force (Fqp) between two hypothetical
particles with Planck charge (qp). The amazing
result is that at arbitrary separation distance (r), they
both generate the same force magnitude. These
forces have opposite vectors, but the same force
magnitude. This is summarized in Eq. (31). The
equality of forces at the Planck limit of mass and
charge was predicted by the wave-based model
because the distinction between first and second
order effects disappear at this Planck limit. This
concept will be explained by comparing the
electrostatic Eq. (23) for two Planck charges (qp), to
gravitational Eq. (21) for arbitrary mass. The only
difference is that gravitational equation (21)
contains strain amplitude squared (As2 = Lp2/rc2)
and Eq. (23) does not. For Planck mass, the
Compton radius equals Planck length (rc = Lp).
Therefore, for Planck mass the strain amplitude
equals 1 because (As2 = Lp2/Lp2 = 1). Therefore, the
enormous difference between the gravitational and
electrostatic forces disappears at the Planck limit.
Below, Eq. (32) extends Eq. (29) to cover two
particles with the same arbitrary mass (m) and Eq.
(33) extends Eq. (30) to cover two particles with the
same arbitrary charge (q)
2
2
G
p
mc
Fmr
=
, (32)
2
2
q
p
qc
Fqr
=
. (33)
By dividing arbitrary mass (m) by Planck mass (mp),
we have a scaling factor (m/mp) that references mp.
Equation (32) converts to (FG = Gm2/r2), the
Newtonian gravitational law equation for two of the
26
same masses (m). Equation (33) is similar, but it
gives the electrostatic force (Fq) between two
particles with the same charge q. Equation (33)
converts to (Fq = q 2/4πεor2). This is the Coulomb law
equation for the electrostatic force between two
particles with charge (q).
There is another type of unification of forces that
occurs when two particles are assumed to be
separated by a distance equal to their Compton
radius. We will compare the gravitational and
electrostatic forces between two hypothetical
particles with an electron’s mass and Planck charge.
The force magnitude between these particles will be
expressed in dimensionless Planck units. Therefore,
we will define FGe ≡ FGe/Fp and Fqp ≡ Fqp/Fp. The
relationship is
()
4
2 4 23
4.18 10
Ge qp e
−
= = = ×FFA
. ( 34 )
At this separation, the square of the electrostatic
force equals the gravitational force. These
dimensionless numbers also equal the electron’s
strain amplitude raised to the 4th power (Ae4). These
relationships are easily derived by setting N = 1 in
Eqs. (24 and 26).
These connections between the gravitational and
EM forces do not conflict with the mathematical
equations of general relativity or electromagnetism.
The connections only conflict with the common
physical interpretation that the EM force is conveyed
by the exchange of messenger particles and gravity
is a pseudo force created by the geometric curvature.
These physical interpretations cannot explain the
various unifications of forces described in this
article.
However, in the wave-based model, gravity and
the EM force are both conveyed through related
distortions of the universal field (oscillating
spacetime). Therefore, the FG/Fe ratio of these two
forces between two electrons can be expressed
simply as the ratio of distortions. FG/Fe = As2/α =
2.40×10-43. The gravitational force is an obvious
curvature of spacetime. The other three forces are
less obvious distortions of oscillating spacetime. For
example, electromagnetism might be a subtle phase
shift of the Planck frequency oscillations present in
oscillating spacetime. This is a subject for future
study.
18. How big is an electron?
The simplified electron model previously
discussed is suitable for the various tests associated
with calculating the properties of the wave-based
electron model. However, an electron is a quantized
wave that can easily change its size and shape.
Therefore, the electron model needs to be expanded
to accommodate the ability of an electron to change
its size and shape.
This question about the size of an electron can be
answered on three different levels. The first level is
to visualize the traditional concept of an electron
that has a central particle surrounded by an
electric/magnetic field. Even with this description,
the wave-based model differs with the traditional
concept of an electron’s size because the wave-
based model includes the electron’s electric field as
part of the electron’s structure. Figure 5C shows a
rotating wave surrounding the central core. This
external rotating standing wave is the part of the
electron’s structure that creates an electron’s
electric/magnetic field. The energy in the electron’s
electric field contributes to the electron’s total
energy. For example, the energy in an electron’s
electric field external to radial distance r is:
Eext = e2/4πεor = αħc/r. Even with a large value of r,
this energy never goes to zero. For example, the
energy in the electron’s electric field beyond the
Bohr radius of a hydrogen atom (∿ 5×10-11 m) is
13.6 eV. The wave-based model of an electron
shown in Fig. 5C shows the extended electron
model.
An electron’s mass/energy has been
experimentally measured to an accuracy of
1.1×10-38 kg ≈ 10-21 J [35]. This means using
Eext = αħc/r , the electron’s experimentally
measurable energy includes the contribution of an
electron’s electric field energy out to a radius of
about 2×10-7 m. This is more than 1011 times larger
radius than 10-18 m. However, there is no limit to
the extent of an isolated electron’s electric field. We
should say an electron is the opposite of a point
particle – it is indefinitely large. We ignore the
27
electron’s distributed energy and wave properties
when we search for a nonexistent central particle.
The second level of answer to this question
considers an electron bound in an atom. The
positive charge of the nucleus limits the extent of an
electron’s negative electric field. For example, an
electron in a hydrogen atom has different wave
functions associated with different orbitals. These
wave functions can exceed 10-10 m in radius. This is
much larger than the electron’s Compton radius
(3.86×10-13 m) previously discussed in the
simplified electron model. If an electron is confined
in some way, then the electron’s wave structure
adjusts to achieve the distribution of the electron’s
wave function determined by the Schrodinger
equation.
An electron in a hydrogen atom encounters the
positive electric field of the nucleus. This creates a
distributed, soft boundary for the electron. The
single Compton frequency of the simplified electron
model is replaced by the multiple frequency wave
components of an electron confined to a hydrogen
atom. These different frequencies are close to the
electron’s ωc but constructive and destructive
interference between the slightly different
frequencies create the moving modulation
envelopes that result in the orbital shapes described
by the Schrodinger equation. These are distributed
wave effects rather than the probability distribution
of a moving point excitation.
The third way of describing an electron’s size
ultimately explains how an electron acquires its
particle-like properties. We will start this
explanation with the wave function of a particle-in-
a-box. The Schrodinger equation allows the
calculation of the wave function of a hypothetical
particle in a one-dimensional box with impenetrable
boundaries. The wave function is usually
considered to be a mathematical description of the
quantum state of an isolated particle. Squaring the
wave function turns the complex probability
amplitude into an actual probability. The wave-
based model considers the wave function to be a
representation of an actual quantum mechanical
wave distortion of oscillating spacetime. For
example, an electron, confined in a Penning trap,
has a quantifiable wave function within this device.
In the wave-based model, the trapped electron is a
physical quantized soliton wave in oscillating
spacetime. Squaring the wave function gives the
intensity of the wave. This intensity corresponds to
the probability of the location where the distributed
wave is likely to collapse into the compact form that
can be observed (an observable electron).
Now we come to the important question. How
does the wave-based model convert the widely
distributed quantized wave represented by the
electron’s wave function, into an observable
electron particle? The key to this transition is the
electron’s quantized angular momentum. All
quantization in the universe is proposed to be
ultimately traceable to angular momentum being
quantized. Fermions have ħ/2 angular momentum
when measured along a single axis and photons
have ħ angular momentum. In the wave-based
model, when a widely distributed quantized wave is
observed, there is a superluminal collapse from the
distributed wave structure to a much smaller wave
structure that has the properties of a particle with
quantized angular momentum.
We know that two entangled particles can
undergo superluminal adjustment when one of these
is observed. Within a single quantized wave such as
an electron, superluminal adjustments must also
happen. The electron possesses a quantized unit of
angular momentum. This quantized angular
momentum requires that 100% of the quantized
angular momentum in the widely distributed
quantized wave must be preserved as a unit. The
distributed wave, described by the wave function,
undergoes a superluminal collapse to a much
smaller size wave that retains 100% of the quantized
angular momentum. The superluminal “collapse of
the electron’s distributed wave function” really
happens when an electron is ”observed”. This will
be designated as a superluminal “terminal collapse”.
To give an extreme example, we will switch
from an electron to a single photon confined in an
optical resonator with 100% reflecting mirrors.
Imagine we introduce several absorbing atoms into
this optical resonator volume. It takes time for a
photon to be absorbed [36]. Therefore, it is possible
for several atoms to start to absorb the single photon
simultaneously. However, the photon’s ħ quantized
angular momentum cannot be divided. This
example requires that a partial absorption can be
28
reversed so that 100% of the quantized angular
momentum can be deposited into a single atom. The
idea of reversal of a partial absorption seems
unrealistic. However, this has already been
experimentally observed! In the article in Nature
titled To catch and reverse a quantum jump mid-
flight, [37] Minev et al experimentally demonstrate
that the jump from the ground to an excited state of
a superconducting artificial three-level atom can be
tracked and reversed. They say, “using real-time
monitoring and feedback, we catch and reverse a
quantum jump mid-flight, thus deterministically
preventing its completion.”[37] This is a
demonstration of the power of quantization.
The ability to achieve a superluminal collapse is
required to achieve the complete transfer of a
photon’s quantized angular momentum into a single
atom. Even if a photon was starting to be absorbed
by multiple other atoms, these partial interactions
must be reversed. Quantized angular momentum
demands that the photon’s entire ħ angular
momentum must be transferred into a single lucky
atom. Usually, this requires a superluminal collapse.
All the photon’s energy is also transferred as a
byproduct of the complete transfer of angular
momentum.
There is one more question about an electron’s
size that needs to be addressed. How is it possible
for an electron to probe the quark structure of a
proton if the electron’s mathematical radius
(Compton radius) is rc = 3.86×10-13 m? A wave-
based electron’s size can not only get larger when
unbounded, but the size will also momentarily
decrease to less than rc in a collision. For example,
in an ultra-relativistic collision, the kinetic energy
of the collision is momentarily converted to an
electron’s internal energy. This additional energy
increases the electron’s rotational frequency,
reduces its radius, and keeps the electron’s angular
momentum constant. In a collision with a 200 GeV
proton, the electron’s mathematical radius can
momentarily be reduced to about 10-18 m. This
explains how an electron can probe the much
smaller proton’s internal quark structure yet have a
relatively large wave function volume in other
experiments.
The electron has a wave structure that is
undetectable. It is impossible to detect a Lp
displacement of oscillating spacetime. Experiments
such as the double slit experiment indicate that an
electron has wave properties, but the waves
themselves are undetectable. This quantized wave
can shrink to less than 10-18 m or increase to
indefinitely large. Therefore, visualizing this as a
corpuscular particle or even a point-like excitation
of a field creates numerous mysteries. The wave-
based model solves most of these mysteries.
19. Photon Model
In the late 19th century and early 20th century,
there was nearly universal acceptance that light was
a wave that propagates in the luminiferous aether.
The Michelson-Morley experiment, the
photo-electric effect and Compton scattering
experiment all seemed to require that a photon was
a quantum mechanical particle that did not require a
propagation medium. Since the luminiferous aether
was not observable, the aether was declared
“superfluous” and abandoned. This article proposes
that oscillating spacetime fulfills the functions of all
the 17 fields of quantum field theory. Therefore, all
17 separate fields are proposed to be superfluous.
Eliminating 17 overlapping fields and replacing
them with a single universal field (oscillating
spacetime) achieves the simplification supported by
Occam’s razor.
A photon is modeled as a quantized wave in
oscillating spacetime. The quantization occurs
because each photon has ħ quantized angular
momentum. Even a large wave with ħ angular
momentum is quarantined as it propagates through
the superfluid oscillating spacetime. The photon’s
particle-like properties only appear when the photon
undergoes a terminal collapse. There is no particle
component or particle-like excitation in this model
of a photon. The following two thought experiments
demonstrate that a photon cannot be a point
excitation of the EM field or have any other particle-
like property.
The first thought experiment incorporates lasers
that are extremely monochromatic. Lasers are
routinely stabilized to achieve a bandwidth of about
1 Hz. The most stable lasers have achieved a
bandwidth of 0.01 Hz. [38]. A laser with a spectral
bandwidth of 0.01 Hz is almost perfectly
29
monochromatic. Photons with 0.01 Hz bandwidth
must be a perfect sine wave over a time period of at
least 100 seconds. Therefore, each photon with 0.01
Hz bandwidth must be at least 3×1010 m long. This
is more than 75 times the 3.84×108 m distance
between the Earth and the moon! This
monochromatic wave can also be spherically
diverging and be spread over an enormous volume.
The idea of a particle-like single excitation making
discontinuous jumps (superluminal jumps) over a
diverging beam with 3×1010 m length to create a
photon’s wave properties is ridiculous. The
alternative wave-based model of a photon has no
particle component (no point excitation). All the
photon’s particle-like properties are achieved by a
superluminal terminal collapse when the widely
distributed wave-based photon is absorbed. If the
photon’s widely distributed waves encounter a non-
absorbing object, the modified wave pattern
undergoes a superluminal adjustment of the
distributed wave that preserves all the photon’s
angular momentum and energy.
The second thought experiment uses a
Michelson interferometer with the two arms that are
unequal lengths. For example, suppose that one arm
is 1,500 meters longer than the other. The light that
makes the round trip in this arm is delayed by 10
microseconds and 3,000 meters compared to the
light that goes down the shorter arm. The two
reflected beams are then combined in the usual 50%
reflecting beam splitter used in a Michelson
interferometer. Therefore, if the laser providing the
input beam to this interferometer has a coherence
length less than 3,000 meters, (bandwidth greater
than 100 kHz), then the two beams will combine
incoherently at the output beam splitter. However, if
a stable laser with a bandwidth much less than 100
kHz is used (e.g. 1 Hz bandwidth), the beams will
combine coherently at the output beam splitter. It
will be possible to achieve destructive interference
in one output direction and constructive interference
in the other output direction even with a large
unequal path length. If there is a large optical path
length difference between the two arms, the particle
or point excitation model of a photon becomes
unworkable.
Laser radar systems that utilize heterodyne
detection are essentially interferometers with
unequal arm lengths. A beam splitter generates two
beams. One beam is sent to a distant target and the
very weak reflection is returned to the
interferometer where it is mixed in a beam splitter
with the other beam (the local oscillator beam) that
has traveled a much shorter distance. I was a co-
inventor and part of a small team that
experimentally demonstrated the feasibility of such
a heterodyne detection laser radar system. The laser
radar apparatus is described in the patent [39]. The
difference between the two optical path lengths in
the laser radar system exceeded 3 km. The CO2 laser
used had a narrow bandwidth that met the
theoretical stability requirement.
The usual answer is to merely say that the
photon’s particle properties only occur when the
photon is observed. However, that describes the
properties of the proposed wave-based model of a
photon. This quantized photon wave only gains its
particle-like properties when it undergoes a
superluminal terminal collapse when it is detected.
There is no point excitation in a hypothetical
background EM field. The description of the photon
model is further clarified in the next section dealing
with entanglement.
20. Wave-based explanation of entanglement
We can also test the concept of terminal collapse
to see if it explains the many counterintuitive
properties of entanglement. Here is the concept.
Two photons that share a single quantum state are
said to be entangled. Entanglement experiments that
use linear polarizers will be used in this explanation.
If one entangled photon is detected after it passes
through a linear polarizer with a particular
orientation, the other entangled photon instantly
achieves the orthogonal liner polarization. This
presents the following three mysteries: 1) How do
the two entangled photons achieve superluminal
communication? 2) How are the two separated
photons able to keep track of each other?
and 3) How does one entangled photon convey the
exact polarization angle information to its entangled
partner?
This explanation will use the annihilation of an
electron-positron pair to generate two entangled
photons. This exotic example is used because it
30
avoids the complexities of the more conventional
ways of generating two entangled photons. Figure 7
shows a central point that was the location of the
electron-positron annihilation. The wave-based
model does not produce two, point particle photons
propagating in opposite directions. Instead, Fig.7
shows that the two entangled photons are
represented by a spherical shell of waves
propagating away from the annihilation point. In
this form, these two entangled waves have not yet
chosen their momentum vectors and polarization
directions. Figure 7 shows the concentric waves just
Fig. 7 Two entangled photons created by the annihilation of an electron-
positron pair are represented by concentric spherical waves that propagate
in oscillating spacetime.
Fig. 8 The shades of gray represent the amplitude distribution of the
remaining waves after the first wave-based photon has been extracted.
These remaining waves form the second of the two entangled photons.
31
prior to the first of the two entangled photons
undergoing a superluminal terminal collapse into
the absorbing particle at the bottom of Fig. 7. This
collapse extracts the wave amplitude distribution of
the first photon. This distribution is given by the
Kirchhoff obliquity factor K(θ) = cos2(θ/2),
explained in [40]. Reference [41] gives more details
about a photon’s amplitude and predicted intensity
limitations.
Figure 8 shows the amplitude distribution of the
remaining concentric waves after the first photon’s
waves have been extracted. The collapse of the first
photon’s waves deposited a specific momentum
vector and polarization into the absorbing particle
near the bottom of Fig. 8. The amplitude of the
remaining waves is represented by the gray scale in
Fig 8. These waves have zero amplitude (white
color) near the absorbing particle because the first
photon extracted the maximum wave amplitude
there. The remaining waves have maximum
amplitude (black color) at the top of Fig. 8 because
the remaining photon has maximum amplitude in
this direction. This distribution creates the
momentum direction and uncertainty angle for the
remaining waves. This wave distribution will also
undergo a super luminal collapse when it is finally
absorbed by a particle within the “momentum
uncertainty angle”.
This model conceptually explains the three
mysteries of entanglement previously enumerated.
1) There is no superluminal communication
between the two entangled photons. The first of the
two entangled photons to be observed merely
undergoes a superluminal collapse into the
absorbing particle. This withdraws the waves
needed to achieve the first photon’s polarization and
momentum. 2) The two entangled photons do not
need a mysterious way of keeping track of each
other. The distributed cloud of waves can generate
any polarization for the first photon. Once this
polarization and momentum is extracted from the
wave function, the remaining waves automatically
have the orthogonal polarization and orthogonal
momentum vector. 3) There is no need to have
superluminal communication of information
between entangled partners, which is also explained
in point #2. The wave-based model of entanglement
is conceptually understandable.
21. Falsifiable Predictions
This section gives two additional falsifiable
predictions that are logical extensions of the
oscillating spacetime model. Previous predictions
such as the electron’s gravitational radius
(rG = Lp2/rc) and the merging of the gravitational and
electrostatic force magnitudes at the Lp-qp limit are
mathematically proven (FGp = Fqp = ħc/r2).
Therefore, they do not qualify as falsifiable
predictions that need further proof. However, the
following predictions are uncertain and therefore
qualify as falsifiable.
The first falsifiable prediction is: The physical
laws are not the same in all inertial frames of
reference. One of the starting postulates used by
Einstein to develop special relativity was: The
physical laws are the same in all inertial frames of
reference. This postulate is universally accepted.
There have been no confirmed experiments that
disprove this postulate. However, this postulate has
not been tested in ultra relativistic frames of
reference addressed here. The articles that examined
the sonic universe hypothesis [10 - 12]
mathematically proved that it is not possible to
observe motion relative to the privileged rest frame
of the assumed sonic medium. However, the sonic
medium assumed by these references was overly
simplified. There was no mention that a real sonic
medium would have a boundary condition set by the
finite maximum frequency the sonic medium can
propagate. Oscillating spacetime cannot propagate
waves with a frequency higher than ωp or
wavelength shorter than Lp in the privileged frame
where the medium is stationary (the CMB rest
frame). This is a boundary condition that leads to
the following prediction: The physical laws are not
the same in all inertial frames of reference.
To explain this shocking prediction, we will start
with Fig. 3A. This figure shows an electron’s
outward propagating waves viewed with relative
motion equal to 25% the speed of light. For this
explanation, we will assume the “stationary” frame
of reference is the CMB rest frame. An electron
moving relative to this privileged rest frame
requires the medium to be capable of propagating
frequencies both higher and lower than an electron’s
32
ωc. For example, Fig. 3A depicts a computer
simulation, using Mathematica, of spherical waves
viewed from a frame of reference with relative
motion equal to 25% of the speed of light. These
outward propagating waves, moving in the direction
of travel, undergo a relativistic Doppler shift to
about 28% higher than the Electron’s Compton
frequency. As shown in Fig. 3A, this higher
frequency produces a shorter wavelength in the
direction of motion. Waves propagating in the
opposite direction are redshifted to a longer
wavelength. In the electron’s rest frame, the
standing waves have an electron’s ƛc. However, an
electron moving relative to the privileged frame
requires the privileged frame medium to be able to
support a shorter wavelength (higher frequency)
than an electron’s Compton wavelength/frequency.
This becomes impossible for an electron in the most
extreme, ultra relativistic frames of reference.
For an ultra-relativistic frame of reference,
relative to the privileged CMB rest frame, the
following approximation can be used: ƛd ≈ ƛc/γ. For
example, when γ > ∿10, an electron’s de Broglie
wavelength (ƛd) is approximately equal to an
electron’s Compton wavelength ( ƛc) divided by the
Lorentz factor γ. To achieve an ultra-relativistic de
Broglie wavelength ƛd, the privileged frame must be
capable of propagating a wave with wavelength
ƛ = ƛd.
Therefore, there are ultra relativistic frames of
reference where wave-based electrons cannot exist
because they would require wavelengths shorter
than Lp in the privileged frame. For example, a
frame of reference with a Lorentz factor of
γ > 2.4×1022 would require waves shorter than Lp
(frequencies higher than ωp) in the privileged frame
(ƛc/2.4×1022 = Lp). Therefore, there are no electrons
in frames of reference with γ > 2.4×1022 relative to
the privileged frame (relative to the CMB rest
frame). Protons and neutrons have de Broglie wave
properties [42]. These more energetic particles have
a lower cutoff γ than electrons. For example, the
wave-based model predicts there are no protons or
neutrons in frames of reference with γ > 1.3×1019
relative to the privileged frame (relative to the CMB
rest frame). The physical laws would clearly be
different in frames of reference that do not allow
protons or neutrons. Therefore, the wave-based
model predicts that the physical laws are not the
same in all inertial frames of reference.
The second falsifiable prediction is: The Big
Bang (BB) started as oscillating spacetime with
Planck energy density Up ≈ 10113 J/m3. The wave-
based model of the universe starts with the BB with
the highest energy density oscillating spacetime can
generate – Up. This highest energy density
(∿10113 J/m3) of observable energy density is
modeled as Planck energy density of Planck energy
photons. This starting condition is perfectly
homogeneous. This starting condition will be
designated “Planck spacetime” because all the
properties of this spacetime equal 1 in natural Panck
units. At 1 unit of Planck time (tp = 1 ≈ 10-43 s), this
medium has the following observable properties:
Planck energy density (Up ≈ 10113 J/m3), Planck
temperature (Tp ≈ 1032 K) and Planck pressure
(Pp ≈ 10113 N/m2). The only force is Planck force
(Fp ≈ 1044 N). Each photon has Planck energy
(Ep ≈ 2×109 J). These starting conditions all equal 1
in dimensionless Planck units.
This starting condition is perfectly
homogeneous because it is the limiting maximum
condition of spacetime. In the first unit of Planck
time, each Planck energy photon is isolated. There
is no gravity at tp = 1 because there was no prior
time for gravity to spread. The gravitational
influence of each Planck volume begins to
uniformly spread at tp = 2 (the first running time unit
in the age of the universe). This produces a
nonlinear gravitational effect that results in an
increase in volume and a decrease in energy density.
This perfectly homogeneous starting condition is
maintained as this model of the universe evolves to
lower density and lower temperature. No
inflationary phase is required to achieve
homogeneity. The starting homogeneity is merely
maintained as the universe changes with age.
One of the mysteries of the universe is
designated as the “flatness problem” [43 – 45]. The
Wilkinson Microwave Anisotropy Probe (WMAP)
[2] determined the current density of the universe is
within 1% of the critical density required to achieve
flat space. Extrapolating back in time to the “Planck
era” at the start of the BB (the first ∿10-43 s), the
homogeneity of the universe must have matched the
critical density to better than 1 part in about 1060 at
33
the Planck era [43 – 45]. The currently accepted BB
theory is a physical theory that describes how the
universe started with an infinitely hot and dense
single point that inflated and stretched. [43]. A
high-density entity does not expand uniformly. It
cannot achieve today’s homogeneity and critical
density. The inflationary BB explains neither the
origin of the 17 fields required by quantum field
theory nor a model of a sonic universe required to
achieve Lorentz transformations (see Section 3).
The alternative suggested here is to start the BB
with Planck spacetime with all its properties
equaling 1 in natural units. This is the ultimate
simplicity. Furthermore, Planck spacetime evolves
into the sonic universe we have today. Lorentz
transformations, a universal field, and the constant
speed of light are all logical results of starting the
BB with Planck spacetime. An attempt to describe
in more detail how Planck spacetime evolved into
today’s the wave-based universe is given in chapters
13 and 14 of reference [46]. This alternative model
of the BB is in its infancy.
22. Summary and Conclusion
This article describes a model of the universe
based on John Wheeler’s proposal that spacetime is
a medium that has Planck length oscillations,
predominantly at Planck frequency. This article
quantifies the properties of this medium and
proposes that oscillating spacetime is the foundation
of everything in the universe. Therefore, this single
“universal field” holds the key to unifying all
particles and forces. The multiple fields of quantum
field theory are characterized as lower frequency
resonances and properties of this Planck frequency
oscillating medium.
There are no particles or point-like excitations
in this model of the universe. Electrons and other
fermions are modeled as rotating quantized waves
in the medium of oscillating spacetime. An electron
appears to have wave-particle duality even though it
is fundamentally a quantized wave because each
wave-based electron possesses a quantized unit of
angular momentum. The article explains how a
distributed quantized wave undergoes a
superluminal “terminal collapse” when “observed”
and achieves its particle-like properties.
This wave propagation medium has enormous
impedance of Zs = c3/G = 4×1035 kg/s. This gives the
spacetime medium enormous stiffness and makes it
possible for a rotating wave with undetectable Lp
amplitude to generate the energy and gravity of any
fermion or boson. This model of oscillating
spacetime is successfully tested by seeing if it is
theoretically possible to generate a plausible model
of an electron from this medium. Tests of the
wave-based electron model are shown to
approximately generate an electron’s energy, de
Broglie waves, angular momentum,
zitterbewegung, particle-like properties and
variable sizes wave distributions. This soliton wave
can momentarily be smaller than a proton in a high
energy collision or can have a relatively large
volume of an atom’s orbital wave function.
This model predicts that oscillating spacetime is
a nonlinear medium. Waves in this medium have
both a linear component and a much weaker
nonlinear component. Analysis of the electron
model shows that the electron’s EM properties are
derived from the linear component and the
electron’s gravitational properties are derived from
the nonlinear component. These forces are predicted
to be related because they are similar distortions of
oscillating spacetime. This insight leads to the
prediction that the gravitational force between two
Planck masses should equal the electrostatic force
between two Planck charges. This prediction is
shown to be correct. Both force magnitudes equal
ħc/r2. Furthermore, the electron’s quantum
mechanical wave properties are shown to generate
an electron’s gravitational radius (rG = Lp2/rc =
6.76×10-58 m) and an electron’s charge radius
(rq = α1/2Lp = 1.38×10-36 m).
Conclusion: The proposed wave-based model of
the universe is both plausible and useful. This model
simplifies all of physics because everything in the
universe is derived from a single wave propagation
medium (oscillating spacetime). It is easy to
calculate the predicted properties of this model
because the model incorporates quantifiable waves
in a quantifiable medium. This model is also useful
for education because quantum mechanical
properties that are counter intuitive when visualized
as particles are conceptually understandable when
visualized as quantized waves.
34
Acknowledgment
The author acknowledges Jim Macken made the
computer-generated figures and simulations. Ron
Macken provided useful suggestions and Donald
Macken made the video. Most important, the author
acknowledges 14 years of invaluable discussions
and education from Nikolai Yatsenko (deceased).
Conflicts of Interest
The author declares no conflicts of interest
regarding the publication of this paper.
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