**Thickness
of Present and Dimensional Cleavage**

**Benoît Leroux, B.Eng.**

© 2004-2005, Ben et Fils Net, Inc. All rights reserved.

Cabinet de physique théorique Ben et Fils Net.

**Abstract**

The
present has a time thickness. The
thickness of present defines the time interval required for the realization of
the Pauli exclusion principle. Thickness of present varies with the wavelength of
particles. It can be calculated using
the thickness of present factor, which is the reciprocal of the speed of light.

The
square of the thickness of present factor equals the product of the constants
relating to each of the three dimensional orders 1D, 2D and 3D. Each of these constants refers to a specific
type of field: magnetic, electric and
gravitational.

Dimensional cleavage is an angle that
changes with the relative velocities of two referential frames. It varies between 0 and π/2. Like Lorentz
transformations, the angle of dimensional cleavage underscores the phenomenon
of relativistic effects on masses, lengths and clocks.

A dimensional cleavage of π/2 occurs
between two contiguous dimensional orders.
This type of cleavage occurs on the front of present, inducing the
future in which the quantum probability wave is spread out.

**Introduction**

Despite the enormous upheavals brought
about in physics by relativity and the quantum theory, the notion of an
entwinement of space and time has remained unchanged.

Minkowski space is the stage for relativistic
transformations between referential frames.
Lorentz transformations allow these
differences to be calculated, but they don’t help explain how these phenomena
occur at interfaces between referential frames.

There remain significant gaps in theory
when it comes to explaining the astounding phenomena uncovered by quantum
physics (wave-particle duality, discrete quantification, decoherence,
antimatter, black holes, big bang, dark matter, etc.). A quick review of relevant physical concepts
is therefore essential.

This article presents a geometrical theory
of time, space and charges. In the
following exposition, time is wittingly excluded from dimensions because it is
considered to be of a different nature than spatial dimensions. I will demonstrate that relativity is an
illusion due to a dimensional cleavage between two referential frames moving
relative to one another.

**Photons**

A photon travels through free space at a
constant velocity c known as the speed of light. This velocity can be inferred
from Maxwell’s equations [1] and expressed as follows:

c =
1/√(m_{0}ε_{0}) (1)

where m_{0} and ε_{0} are the magnetic permeability and the electric
permittivity of empty space, respectively.

**Electrons**

Classically, an electron is viewed as an
infinite, spherical and uniform electric field which may be taken as a constant
point electric charge determined by Maxwell’s equation for Gauss’ theorem:

ε_{0} § E dS = q (2)

This equation describes the integration of
an electric field E on a Gaussian surface S surrounding a point source with a
charge q. All electrons bear the same
charge. An electron has a magnetic spin
dipole moment that is perpendicular to its electric field. All electrons are identical [2]. Electrons are the lightest persistent
particles that obey the Pauli exclusion
principle.

Although electromagnetic phenomena are now
well understood, the nature of electric charges has not yet been elucidated
[3]. The problem of electron structure, if any, remains unsolved [4]. Why do two identically-charged particles repel
each other, and why do they have opposing electric fields?

To gain insight into the phenomenon of
attraction and repulsion, one must first review the principles underlying the
notion of space. By definition, the
universe contains all things. If
something is excluded, then our definition of the universe is incomplete
[5]. From this principle, it follows
that:

*Space
must be created*. (A1)

Assumption A1 implies that a space
creation process is required for the universe to be produced as it appears to
us, in three dimensions.

The electron is viewed as a circular
eccentric electric field spread out over a surface normal to its magnetic
moment (figure 1). The magnetic field is
taken not as a flux, but as a 1D axis having the same properties as a flux in a
2D referential frame.

Figure
1. The structure of an
electron.

When two electrons come into interaction,
they repel one another in accordance with Coulomb’s law. This electric repulsive force could evidently
constitute a space production mechanism.

**Time
Arrow and Antiparticles**

Symmetry is an increasingly important
notion in modern physics. Low entropy is
ascribed to high symmetry, and increasing entropy is taken to direct the time
arrow toward the future. Yet the very
ability to persist necessarily goes hand in hand with the Pauli
exclusion principle, which is fundamental in setting
the direction of the future. The
electron being the lightest persistent particle that obeys the Pauli exclusion principle, it is
the most primitive carrier of a preferred time direction on a scale of
increasing particle weight.

Since few antiparticles are observed in
the universe, a second assumption must be made (which will be discussed below):

*An antiparticle lies a certain time interval away in the past of its
corresponding particle. *(A2)

Time has a direction. In the past, events are perfectly determined,
whereas the future is ruled by uncertainty.
In 1D or 2D universes, physical phenomena are simple, and the direction
of time is therefore easy to establish.

For an electron at rest, for example, the
arrow of time naturally points in the same direction as the magnetic dipole
moment, which is normal to electric surface E_{2} (figure 2).

Figure 2. The direction of time t+ is perpendicular to
electric surface E_{2}, pointing in the same direction as the magnetic
moment μ. A positron (the
antiparticle of an electron) resides in the past of its electron, separated
from it by a time interval dt
> 0.

It follows
from [A2] and the above that a positron lies under surface E_{2}, a
time interval dt away in the
past of its electron.

When an electron is set in motion,
the axis of its electric surface tilts with its magnetic dipole moment m, which then projects a positive
time vector along E_{2} in the direction of the motion. The creation of 2D space as a result of two
electrons moving away from each other is due to an increase in the internal
surface of both electrons. The
inclination of the electron’s magnetic moment also induces a time vector
projecting along E_{2 }(figure 3).

Figure 3. A moving electron projects a
time vector t+ along electric surface E_{2 }in the direction of the
motion, while the corresponding positron projects a negative time vector t- in
the opposite direction.

**Protons**

It is probable that in certain parts of
electric surface E_{2}, overcrowding prevents the surface growth of
electrons, which then lack the space they need to move away from one
another. In these disturbance areas,
electrons tilt markedly down toward E_{2} because they are unable to distance themselves. This pronounced inclination of electrons
makes it likelier for their antiparticles to come into contact with one another
(figure 4).

Figure
4. Interacting positrons brought into contact by
the repulsive stress of electrons. An
orthogonal dimensional cleavage is also brought to light by the induction of a
perpendicular future upon creation of a new dimensional order (in this case,
3D).

Three positrons may then come into contact
under high energy and react to form a quark trio, their electrons turning into antiquarks. This new
proton is ejected from surface E_{2} into 3D space, immediately drawing
an electron into its orbit. The
geometric arrangement of electrons tilting down toward the electric surface is
conducive to the formation of a 3D structure, namely a proton. Also created is a new 2D space in 3D
corresponding to the electron’s orbital around the nucleus of this new hydrogen
atom (figure 5).

Figure
5. A hydrogen atom produced by the reaction
illustrated in figure 4. The antiquarks lie on the other side of surface E_{2}. The creation of a 3D structure has liberated
the electric tension due to electron repulsion.
Three positrons have become quarks, three electrons have become antiquarks, and one electron and its positron have escaped
surface E_{2} and entered orbit around the new proton.

The possible existence of an interaction
between quark and electron has long been considered: according to Gell-Mann, a majority of
theorists believe the nature of quarks is close to that of electrons [6]. As part of GUT, Hawking mentions the possibility
of an electron being produced as a result of the annihilation of a quark by its
antiquark [7] in a reverse reaction to the
proton-forming process described above.

At any rate, the formation of a proton
provides the 2D space needed for electron persistence in the form of an
orbital. From the point of view of the
electric surface, the proton appears as the opposite of an electron, in other
words as an empty space that attracts electrons with the same intensity of
charge as the electron itself. It may
then be argued that the electric force observed between opposing charges
results from an attraction between structures of different dimensional orders
(2D and 3D for the electron-proton couple), whereas electric repulsion occurs
between structures of the same dimensional order (2D for the electron-electron
couple, and 3D for the proton-proton couple).

From the point of view of the proton,
electric surface E_{2} appears as an envelope or, ideally, a surrounding
sphere. In 2D, the antiproton (or antiquarks) and the proton lie on either side of surface E_{2} (figure 5); in 3D, on the other hand, the antiproton (or antiquarks) lies in an indeterminate location next to the
proton, separated from it by a time interval dt
wherein lies the electron’s orbital.

When 3D space is formed, gravity appears
[8], along with the weak and strong interactions that maintain the atom’s
nucleus. Gravitational force results
from an opposition between virtual (or empty) space and its occupation by
persistent 3D structures. This force is
very weak, however, when compared with that which is opposed to the presence of
electrons in 2D space and is observed as mutual electron repulsion.

In principle, gravitational acceleration g
originates from the proton’s centre of mass and points toward surface E_{2} (figure 5). When this surface becomes a spherical
interface surrounding the proton, however, gravity naturally points from the
centre outward, in no predetermined direction.

**Thickness
of Present**

As previously discussed, a geometric arrangement
in which positrons and electrons are separated by a time interval dt is conducive to the creation of
new 3D space as a result of positrons being converted into quarks, and
electrons into antiquarks.

According to Mazur, the importance of the
particle-antiparticle duality and the notion of determinacy over time are not
only related, but necessary to one another [9].
Mazur ascribes the direction of the time arrow to weak interactions. The electron, which is the smallest
persistent particle in terms of mass, is in itself the result of a primordial
expression of a preferred time direction toward the future.

It follows from assumption (A2) that time
is aligned with an electron’s magnetic dipole moment µ. The positron lies a time interval dt away from the electron at rest,
under electric surface E_{2}.
This time interval dt,
representing the base from which an electron leaps toward the future, is what I
call *thickness of present*.

In 2D, the inclination angle θ of an
electron’s magnetic dipole moment is what I call the *angle of
dimensional cleavage*;* *it
projects a velocity vector along surface E_{2} in the direction of the electron’s motion (figure 3).

Imagine a circle with a radius equal to the
speed of light c centered on the intersecting point of two orthogonal axes
representing two referential frames moving relative to one another; θ is
the angle of dimensional cleavage between the two referential frames, and v is
the velocity of the particle in the referential frame on the y-axis as
perceived from the referential frame on the x-axis (figure 6).

Figure
6. The velocity v of a particle as perceived
from a second referential frame in relation to the speed of light and the angle
of dimensional cleavage θ between the two frames.

Perceived velocity v can be expressed in
relation to angle θ:

v =
c (sin θ) (3)

Thus:

v/c = (sin θ) (4)

Therefore, dimensional cleavage θ is
a function of velocity v divided by the speed of light:

θ
= arcsin (v/c) (5)

Deriving angle θ with respect to
velocity v gives:

dθ/dv = d(arcsin
(v/c))/dv = 1/√(1 – v^{2}/c^{2}) (6)

The right-hand side expression of equation
(6) coincides with the mass and length conversion factor for relativistic velocities
in Lorentz transformations [4]. Applying this derivative to a mass m_{0},
for instance, gives:

m =
m_{0} dθ/dv (7)

Dimensional cleavage produces a
relativistic effect on mass m_{0}, which is perceived as m from a
referential frame that is moving relative that of m_{0}.

Relativistic phenomena can be viewed as
refraction effects due to the angle of dimensional cleavage between two
referential frames. These effects occur
on a common interface that is located halfway between the frames and whose
thickness is also related to the thickness of present factor.

The
thickness of present factor is the reciprocal of the speed of light:

e_{p} = 1/c (8)

It is
estimated at:

e_{p} = 3.3356 * 10^{-9} (s/m) (9)

Its square is:

e_{p}^{2}=
1/c^{2} (10)

Replacing c^{2} in equation (10)
by the right-hand side expression of equation (1) gives:

e_{p}^{2
}= m_{0}ε_{0} (11)

Since electric permittivity ε_{0 }relates
to gravitation-free electric permittivity ε_{1 }and gravitational
freedom x_{0} as exposed in [8]:

ε_{0
=} ε_{1}x_{0} (12)

ε_{0}_{ }can be replaced in equation (11) by the
right-hand side expression of equation (12), which gives the fundamental
relation between the field types relating to dimensional orders 1D, 2D and 3D:

e_{p}^{ 2 }= m_{0}ε_{1}x_{0} (13)

Multiplying the thickness of present
factor e_{p}_{ }by the wavelength of
a particle gives a specific thickness of present value for that particle.

When the concept of speed of light c is
replaced by that of thickness of present e_{p},
maximum velocity becomes a physical limit inherent of bodies resisting to
non-being by the very constraint of persistence. The physical limit e_{p} is that of maximum cleavage (θ =
π/2).

**Orthogonal
Cleavage of Future**

In addition to affecting how masses and
lengths are perceived, dimensional cleavage plays an important role in the
deployment of space from 1D to 3D.

This can be seen in the way the future is
induced, a process wherein dimensional cleavage occurs orthogonally: θ = π/2.

For an isolated magnetic field B
considered as a 1D structure, time is perpendicular to the axis of B, and its
direction is indeterminate (figure 7).

Figure
7. For a magnetic field B, the direction of time
is indeterminate but restricted to a surface that is perpendicular to the axis
of B.

For a photon, time is perfectly aligned
with the velocity vector, which is perpendicular to the two constituent fields
E and B (figure 8). However, since
photons travel at the speed of light, for them, time ceases to pass [10].

Figure
8. The future of a photon is determinate and
perpendicular to the two constituent fields E and B.

For an electron at rest, the arrow of time
points in the same direction as the magnetic dipole moment m (figure 9).

Figure
9. For an electron at rest, the direction of
time is perpendicular to the electric surface.

For a moving electron, the time vector
tilts down with the magnetic dipole moment m closer to the direction of the motion
(figure 3). When a proton is formed,
dimensional cleavage is nearly or perfectly orthogonal, resulting in the
ejection of quarks and antiquarks from the electric
surface. This suggests that the
direction of the future is perpendicular to the electrons’ surface of origin
(figure 4).

Finally, a proton in 3D has an eccentric,
field-shaped future, whereas its environment displays a concentric future
(figure 10).

Figure
10. In 3D, particles such as hydrogen atoms have
an indeterminate and eccentric future, whereas their environments have a
concentric future.

An analysis of the orthogonal “direction
adaptation nature” of Coulomb and gravitational forces performed by Cui [11]
shows that these forces always act perpendicularly to the four velocity vectors
of particles in 4D spacetime rather than along a line
drawn between interacting particles.
According to Cui, the direction adaptation nature of these two types of
forces might have something to do with the quantum aspect of the phenomenon. Though Cui’s
argument is not directly related to my model, his intuition concerning the role
of orthogonal angles in physics is worth pointing out as an analogy in the
present context.

**Uncertainty
of Future**

The uncertainty of future is necessarily
connected to the rules governing the quantum probability waves of
particles. The immediate future of an
electron presents an uncertainty (a cone defined by solid angle Φ)
relating to magnetic dipole moment m whose diameter where it intersects the tangent
plane to velocity v, projected on E_{2}, defines the front of present
(figure 11).

Figure
11. The cone and uncertainty line Ψ
associated with the future of a moving electron. The intersection of the cone with the plane
perpendicular to the front of present on surface E_{2} projects a line
of uncertainty that is perpendicular to the motion.

The quantum probability wave forms an
uncertainty line Ψ in 2D and an uncertainty surface Ψ in 3D
(figure 12), both velocity-dependent and lying on the front of present,
perpendicularly to the time axis.

Figure
12. The surface of uncertainty Ψ lies perpendicularly to
the motion of a particle in 3D.

**Interfaces
and Wave Function Reduction **

The uncertainty lines Ψ_{1}
in 2D become uncertainty surfaces Ψ_{s}
in 3D, forming interfaces where the futures of objects meet and the coherence
of spacetime can be adjusted. Wave function reduction, far from
representing some sort of interference between the observer’s conscience and a
physical phenomenon, simply requires that two fronts of future meet in such a
way as to become the cause of other subsequent phenomena.

Decoherence is natural in 3D if the future of a
quantum object does not meet the future of another object to produce a
cause. When an electron goes through two
slits at once, an interference pattern is produced in the surface Ψ of
probabilistic future. The very
observation of the quantum state of a particle induces the future of the
observer, which reacts with that of the particle to reduce the wave functions
of the particle and the observer simultaneously at the interface. This reaction of mutual wave function
reduction allows the coherence of spacetime to be
adjusted through an interface that forms a perpendicular surface Ψ common
to the observer and the observed.

**Maintenance
of Present**

The process of maintenance of present can
be viewed as a vibration: an instance of
an antiparticle annihilates an instance of the present of the corresponding
particle as a new instance of the particle appears with a new instance of its
antiparticle.

Without some sort of continuous dissolution
and creation process, or “rebound effect”, physical events would accumulate in
limited persistence spaces.

**Discussion**

Doubts could be raised about the validity
of the electron structure presented in this article, since the electron’s
magnetic dipole moment cannot be formally determined on the z axis based on
observations. It should be stressed that
regular observations are made from a 3D point of view. A 2D phenomenon, however, will necessarily
bear uncertainties in 3D, a higher dimensional order than that in which 2D
structures are defined.

The time location of the past and thus of
a proton’s antiquarks outside the nucleus of the atom
(figure 5) is a geometric configuration that is conducive to the maintenance of
the orbital of the hydrogen atom’s electron, due to the repulsion between the
charges of the quarks and the positron and those of the antiquarks
and the electron. It wouldn’t be
surprising if the proton’s thickness of present coincided with the electron’s –
thus ensuring the stability of the atom in 3D – and helped determine the energy
levels of the various electron orbitals in more
complex atoms.

The interactions between positrons,
electrons, quarks and antiquarks can be analysed
easily with this model. However, I save
this work – as well as a study of the formation and structure of neutrons – for
later publications.

Assuming the antiparticle lies in the past
of its particle, the effect of the antiparticle’s mass on the particle must be
examined. According to quantum theory,
the mass of an antiparticle is equivalent to that of the corresponding
particle. In order for the antiparticle
to remain invisible to the particle (to my knowledge, no gravitational effect
revealing the presence of an antiparticle near a particle has ever been recorded),
its gravitational wave would never have to be able to reach the present of the
particle. The fact that the thickness of
present is the reciprocal of the speed of light, which Einstein assumed to
coincide with the velocity of gravitational waves, certainly has something to
do with it, but the paradox remains.

My description of protons as being formed
through the mutual repulsion of electrons provides an interesting lead in the
search for an explanation to Feynman’s ratios of gravitational attraction to
electric repulsion, and proton diameter to universe diameter, both of which
have an order of magnitude of ~10^{-42} [12].

This proton formation mechanism could be
responsible for the unexplained jets of matter observed at the centre of
disk-shaped celestial structures; apparently, the impressive focusing of these
jets can only be explained by the presence of intense magnetic fields [13]. The centre of a rotating 2D charged surface
is the most likely place for a high concentration of electrons to be found.

Unlike the speed of light, which is the
highest speed at which information can travel through space, the thickness of
present not only serves as a foundation for the persistence of fermions, making
them obey the Pauli exclusion principle,
but also acts as a local resistance to or friction against the spread of
information through space.

The present model explains and reconciles
certain contradictory aspects of the general relativity and quantum mechanics
theories. General relativity is entirely
based on a continuous universe in 3+1D, while quantum mechanics rest on
discrete mathematics relating to 2D binary phenomena that strictly exclude
gravity, which is confined to a 3D geometry [8]. I eliminate this discrepancy by introducing
the notion of cleavage between dimensional orders 1, 2 and 3 and referring to
my concept of gravitational freedom of empty space [8], which provides new
insight into both the separation and connectedness of the different dimensional
orders.

Having established that the electron has a
2D structure and that gravitation only acts in 3D, I must ascribe a meaning to
the electron’s rest mass. This mass can
simply be taken as a vibration of surface E_{2} in 3D, which would explain the considerable mass
difference between protons and electrons (m_{p} = 1836 m_{e}).

Interface reactions between futures are
consistent with Smolin’s conception of the universe
as a network of evolving relationships [5].
Smolin’s analysis of Penrose spin networks
should help understand how spacetime coherence is
maintained through the interaction of magnetic fields on the front of present
where the wave function is spread out.

These discoveries open a whole new field
of thought with regard to the role of interfaces between referential frames. This issue must be related to the Beckenstein concept of surfaces, entropy and the Planck
constant. The occurrence of wave
function reduction on 2D surfaces in 3D and the resolution of spacetime coherence governed by this process constitute
substantial clues as to what underlies the illusion to which three-dimensional
nature has confined us.

At present, the general trend is to
consider space and time at the microscopic level as discrete phenomena
[5]. In my theory, time acts discretely
through the thickness of present and the instantiation of particles obeying the
Pauli exclusion principle, a
process wherein resulting space is fragmented into quanta.

**Conclusion**

This article presents a geometric theory
of time, space and charges.

I suggested that space does not exist a
priori and assumed that the fundamental principle underlying electric repulsive
forces is the provision of the space required for electron persistence. I then postulated that this 2D space is produced
by the mutual repulsion of electrons.

I established relevant conditions and
proposed a proton formation process wherein three-dimensional space is
produced.

By assuming that antiparticles lie in the
past of their corresponding particles, I derived a mechanism for the deployment
of new spatial dimensions involving, in each instance, an orthogonal induction
of time on the front of present.

I assumed and then demonstrated that it is
relevant for the present to possess a time thickness that is related to the
wavelength of particles. I established
that the thickness of present factor is the reciprocal of the speed of
light.

I established the fundamental relation
defining the thickness of present factor:
e_{p}^{ 2 }= m_{0}ε_{1}x_{0}. This equation describes a model of the
universe based on three dimensional orders, each of which is characterised by a
constant related respectively to magnetic, electric and gravitational fields. There should be a fourth dimensional order,
of order zero, which is time but in the form of frequency. This prime dimensional order can only be
perceived through other dimensional orders.

Finally, I linked quantum probabilities
with the uncertainty of future using a line or surface – depending on the dimensional
order in which a given phenomenon occurs – extending on the front of present
perpendicularly to the arrow of time.

I identified an angular phenomenon,
dimensional cleavage, to which I ascribed relativistic effects on the
perception of masses and lengths from different referential frames in motion
relative to one another.

I discussed interface wave function
reduction phenomena.

I associated dimensional cleavage with an
orthogonal induction of future inherent in the deployment of dimensional orders.

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