**Gravitational Freedom of Empty Space**

**Ben****oît**** Leroux, B.Eng.**

Cabinet de physique
théorique de Ben et Fils Net.

**Abstract**

The constant of gravitational freedom of
empty space is inferred through a completion of the symmetry between Coulomb
force and gravitational force equations.

I suggest that gravitational freedom of
empty space is the 3D equivalent of 2D electric permittivity of empty space and
1D magnetic permeability of empty space.
The present value of electric permittivity must be corrected to account
for gravitational freedom of empty space, which fits naturally into the speed
of light equation.

**Introduction**

The
scientific community has been struggling for almost a century to relate the
various universal constants, which are precisely set on seemingly arbitrary
values, though their effects on nature and the fate of
the universe are so obvious today.

Of these
universal constants, the constant of gravitation is surely the most
isolated. Not only is it devoid of any
relation to other constants, but the very phenomenon of gravity eludes the
principles of quantum physics. Through
general relativity, however, Einstein demonstrated that gravity plays a
significant role in nature, interacting with other forms of energy to distort Minkowski spacetime.

In this
article, Minkowski spacetime
– a geometry comprising three spatial dimensions and one time dimension – is
set aside intentionally, because my discovery challenges the notion that a
complete description of space can be made with Minkowski
space alone. This way, space is
expressed by itself, with no mention of time.

A property
inherent to each dimensional order (1D, 2D and 3D) is identified, making it
necessary to modify the general conception of space, and thus spacetime.

Gravitational
freedom is a surprisingly elegant and fertile new concept that enhances
theoretical symmetry without invalidating any of the calculations or
experiments performed over the last hundred years based on classical or quantum
physics.

**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 magnetic permeability and 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 (see discussion
below). All electrons are identical
[2]. Electrons are the lightest
persistent particles that obey the Pauli exclusion principle.

The electric repulsive force between
identically-charged particles is defined by the Coulomb force:

F_{e} = k q^{2}/d^{2} (3)

where q is the charge, d is the distance
between the charges and k is Coulomb’s constant. Comlomb’s constant
can be expressed as a function of electric permittivity of empty space [3]:

k = 1/(4πε_{0}) (4)

where ε_{0} is the electric
permittivity of empty space, which is estimated at:

ε_{0} = 8.854187818 × 10^{-12} C^{2}/(N×m^{2}) (5)

Although electromagnetic phenomena are now
well understood, the nature of electric charges has not yet been elucidated
[4]. The problem of electron structure, if any, remains unsolved [5]. 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 [6]. 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 (see discussion).

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.

In spite of the Bohr theory, the radius of
an electron is not clearly determined and depends on the electron’s kinetic
energy. One can thus consider that the
2D space created by the mutual repulsion of two electrons is internal to each
electron rather than external and independent.
This may be stated mathematically using the Coulomb force equation by
replacing the distance d between electrons by the equivalent radii r of both
electrons, assuming that d = 2 r:

F_{e} = q^{2}/ (4πε_{0}(2r)^{2
}) (6)

When denominator whole numbers 4 and 2^{2} are multiplied, equation (6) becomes:

F_{e} = q^{2}/(16πε_{0}r^{2}) (7)

A modified version k_{l} of Coulomb’s constant k**
**can then be formulated:

k_{1} = k/4 = 1/(16πε_{0}) (8)

It follows from assumption (A1) that the
space in which the first electrons are confined is 2D, since no 3D space has
yet been created. This fundamental
electric surface is called E_{2}.
Tiwari provides convincing arguments for a 2D
electron structure [4].

**Gravitational
Freedom of Empty Space**

The gravitational force between two
protons is expressed as follows:

F_{g} = Gm_{p}^{2}/d^{2} (9)

where G is the universal constant of
gravitation from Newton’s law, m_{p} is the mass of each proton and d
is the distance between the two protons.

Unlike Coulomb’s constant, the
gravitational constant is not related to a property of empty space. To correct this discrepancy and obtain
perfect symmetry, the gravitational constant must be expressed in the same form
as the modified Coulomb’s constant in equation (8):

G =
1/(16πx_{1}) (10)

where x_{1} is a property equivalent to electric
permittivity of empty space for gravitation.
Equation (10) is solved for x_{1} by introducing the value of G
(6.67×10^{‑11} N*m^{2}/kg^{2}):

x_{1} = 1/(16πG) = 298266385.1 (11)

Since the properties of empty space are
characterized by ε_{0} and m_{0}, the two of which determine the speed of
light c in accordance with equation (1), and the result of equation (11)
is very close to the value of c (c = 2.99792458 *10^{8} m/s), x_{1}
is obviously related to the speed of light.
A proportionality factor x_{0} between x_{1} and c may thus be formulated:

x_{0} = x_{1}/c = 0.994909569 (12)

Replacing x_{1} by x_{0}c in equation (10) gives:

G =
1/(16πx_{0}c) (13)

Equation (9) may then be stated in a final
form:

F_{g} = m_{p}^{2}/(16 πx_{0}cr^{2})
(14)

It should be noted that in equation (14),
the distance between protons (d in equation (9)) is replaced by the
half-distance (r = d/2) so as to follow the reasoning used in the Coulomb force
equation (7), that is to say that this value is already doubled by the factor
of 16 appearing in the denominator.

The modified Coulomb’s constant k_{1}
in equation (8) can now be related to the gravitational constant G by isolating
the term 1/(16π) in equations (8) and (13):

k_{1}ε_{0} = 1/(16π) (15)

Gx_{0}c
= 1/(16π) (16)

and then combining equations (15) and
(16):

Gx_{0}c
= k_{1}ε_{0} (17)

Therefore, x_{0} is to gravitation in 3D what permittivity
of empty space ε_{0} is to electricity in 2D and permeability of empty
space m_{0} is to magnetism in 1D.
Since the constant x_{0} has a value that is close to 1, it is given the name *gravitational
freedom of empty space*.

The SI units of gravitational freedom of
empty space can be inferred from equation (13):
kg×s^{3}/m^{4}.

**Integrating
Gravitational Freedom in the Speed of Light Constant**

Since permittivity of empty space ε_{0} is measured indirectly through
observations and calculations on electric displacement currents in condensers
[3], that is to say in a 3D
configuration, the gravitational freedom of empty
space x_{0} must be taken into account.
This is done using the following relation:

ε_{0} = x_{0}ε_{1} (18)

To calculate gravitation-free permittivity
ε_{1}, the values of ε_{0} (5) and x_{0}
(12) are entered into equation (18):

ε_{1} = 8.8091161837×10^{-12} (C^{2}×m)/(s×kg^{2}) (19)

Finally, x_{0} can be inserted in the speed of light
equation (1) by replacing ε_{0 }by
x_{0}ε_{1}, in accordance with equation (18):

c =
1/√(m_{0}x_{0}ε_{1}) (20)

Factors m_{0}, ε_{1} and x_{0} are
properties of empty space related to dimensional orders 1D, 2D and 3D and
corresponding to magnetic, electric and gravitational fields, respectively.

Combining equations (17) and (18) gives:

G x_{0}c
= k_{1} x_{0} ε_{1} (21)

**Discussion**

My assumption that an electron’s magnetic
moment is perpendicular to its electric field is not supported by current
observations. In fact, the direction of
the electron’s magnetic moment is not determined, nor is the electric field
considered as a planar phenomenon. Until
now, all known observations of the electron have been made from a 3D point of
view. Thus, as I suggested, if the
electron is indeed a 2D structure, the directions of an electron’s magnetic
moment and electric field cannot be fully determined from a 3D point of view.
My description is based on an analysis of the electron structure from the point
of view of 2D space, in which the magnetic moment is inherited from 1D
space.

Introducing the hidden variable x_{0}
to link the universal constants G, c and k may seem mathematically weak. However, since the values of G, ε_{0}
and c are all derived from observation, a line of reasoning that sheds new
light on well-known and understood physical phenomena cannot be rejected
straightaway.

It seems quite natural to assign a
specific dimensional order to each type of field (magnetic, electric and
gravitational) and assume that lower dimensional orders are included in higher
ones.

**Conclusion**

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 past and future
components of the time vector of an electron set in motion in 2D space (an
electric surface) as a result of the mutual repulsion of electrons.

By completing the symmetry between the
equations of electric and gravitational forces, I linked the universal
gravitational constant to the speed of light and inferred the concept of
gravitational freedom of empty space.

I demonstrated that gravitational fields
bear the same specific relation to 3D space as do electric fields to 2D space
and magnetic fields to 1D space.

**References**

1. R. Resnick,
D. Hallyday, *Ondes**,
optique et physique moderne,
physique 3* (translation of *Physics*, John Wiley & Sons), Éd. du renouveau
pédagogique, Montreal, 1979. p. 55

2.
B. Greene, *The Fabric Of The Cosmos*, Alfred A. Knopf, NY, 2004. p. 439

3. R. Resnick, D. Hallyday, *Électricité et Magnétisme physique
2* (translation of *Physics*, John Wiley & Sons), Éd. du renouveau
pédagogique, Montreal, 1979. pp. 5, 89

4. S. C. Tiwari,
« The Nature Of Electronic Charge » (2004)

http://xxx.lanl.gov/abs/physics/0408053

5. *Oxford
Dictionary of Physics*, Oxford University Press Inc, NY, 2003. p. 141

6. L. Smolin, *Three
Roads To Quantum Gravity*, Basic Books, NY, 2001. p. 17