HyperGeometrical Universe Theory

Here I reproduce the Grand Unification derivation. The model is very simple, although conceptually difficult to understand…J

Cheer,

MP

3. Force Unification

3.1 Quantum Gravity and Electrostatic Interaction

Let’s consider a body and a particle interacting through their four-dimensional waves. The body will always have a kilogram (of mass or charge) and the particle will always be a one a.m.u. (atomic mass unit) particle (~neutron or hydrogen atom). For the gravitational interaction, this particle will have zero spin, while it will have spin half for the electrostatic interaction. Although the four-dimensional wave interaction is taking place on the hypersurface of a four-dimensional expanding hypersphere, one will make use of cross-sections to calculate interference patterns. Interference is considered on each de Broglie expansion of the hyperspherical universe. One can briefly describe the source of waving as a four-dimensional particle (four-dimensional ellipsoid of revolution or particle X for simplicity). The X particles are characterized by four axes lengths. Three axes lengths correlate with the quarks composition of matter. The fourth-axis always points in the radial time direction. Needless to say, different quarks (axis lengths) and different rotational states around the four axis will be sufficient to maps all known particles (photons, mesons, neutrinos, etc). Volume (mass) tunnels in an out of the three-dimensional space for spinning particles (particles with non-zero spin) and out and in towards the radial time dimension. Spin is considered to be a special rotation, since the rotation axis is perpendicular to radial time and one of the spatial coordinates. That gives spinning a different effect; it brings the particles in and out of the fabric of space, thus allowing for a realignment of the k-vector of associated spacetime waves. Let’s consider the interaction through a two-dimensional cross-section (X x ). Particle one (one a.m.u “zero spin neutron” or fat electron) sits on x=0, while particle two (the body of 1 Kg) sits on x=R₀. The four-dimensional dilatons are embedded in a fifth dimension (cosmological time). A position in this space is defined by the following vector:

using director cosines  and .

is the radial dimension on the RXYZ cross-section.     (3.1)

At time zero, the positions for particles 1 and 2 are given by:

and     (3.2)

After a de Broglie cycle, one has these three vectors:

and and     (3.3)

()
is the unperturbed crest of the four-dimensional wave of particle 1 after a de Broglie cycle. is the position of the same crest under the influence of particle 2.

The k-vector is given by:

     (3.4)

Where is the local metric of the five-dimensional space. Again, cosmological distances would require a further refinement and the usage of a non-local metric. This is not required in the calculation of near-proximity forces. In the derivation of the Biot-Savart law, will be rewritten with regard the corresponding non-zero relative speeds. Notice the ½ phase dependence on k-vector, corresponding to the fifth dimension for a half-spin fat electron.    

And

    (3.5)

for a “static zero spin neutron ” forward time traveling wave. Notice that the dilaton and the dilator are treated as one due to the QLP. Where

• N=1Kg of Matter @1000 Avogrado’s Number=6.0221367360E+26 particles of type 1.
• ₁
  @ h*1000*Avogrado/(1 Kg x c)= 1.3310E-15 meters ( in the MKS system).
• ₂_1Kg@ h/(1Kg x c)= 2.2102E-42 meters ( in the MKS system).
• G_{Gravitational} is the gravitational constant = 6.6720E-11 m³.Kg^-1.s^-2
• Single electric charge (Coulomb).
• q_e is the effective value of the single electric charge= charge divided by a corrective factor of 1.004145342= 1.59556231E-19 Coulomb
  
• ε₀=permittivity of the vacuum = 8.8542E-12 C².N^-1.m^-2 (MKS)

Starting with the standard MKS equation for electrostatic force between two one Kg bodies of electrons (one a.m.u. “electrons” or “protons”) = x Coulombs, one obtains:

    (3.5)

    (3.6)

The dilaton for a single particle can be represented by:

    (3.7)    

where

• || means absolute value
• 
• P (absolute value of the phase volume) is 3.5 for a particle with spin half and 3 for neutral matter. The meaning of P is that for one de Broglie wavelength traversed path by the hyperspherical universe, a propagating spacetime wave spread along by a factor of P2π (7 for charged particles and 6 for neutral-zero spin matter).

Similarly, for a 1 Kg body located at position:

        (3.8)

where the effect of the 1 kg mass is implicit in the k₂-vector and expressed by the factor N. The wave intensity scales up with the number of particles (N). One kilogram of mass has 1000 moles of 1 a.m.u. “zero-spin neutrons”, or |k₂| = 1000.Avogadro. |k₁|=N. |k₁|, where

• M=1 for neutral matter-matter, matter-antimatter or antimatter-antimatter interactions or opposite charge interactions
• M=-1 for same charge interactions

To calculate the effect of gravitational/electrostatic attraction, one needs to calculate the displacement on the crest of each particle or body wave due to interaction with the dilatons generated by the other body.

This is done for the lighter particle, by calculating the derivative of the waveform and considering the extremely fast varying gravitational wave from the macroscopic body always equal to one, since the maxima of these oscillations are too close to each other and can be considered a continuum.

The total waveform is given by:

        (3.9)

The term contains the treatment for retarded potentials, but for simplicity we will neglect differences in dimensional time between and .

Why is the lightspeed c the limiting speed in this Universe?

The reason can be seem from equation (3.9)

        (3.9)

Taking the derivative and equating it to zero, yields:

    (3.9.1)

As interaction increases, r shifting asymptotically reaches some , that is, is achieved at saturation.

    (3.9.2)

This means that the limiting angle of acceleration is 45 degrees or the speed of light. Notice that there was no use of any postulate as in the Theory of Relativity. This speed limit is the direct result of the choice of interaction and the proposed topology.

Equation (3.9) is the one and only unification equation, that is, it is the four-dimensional wave equation that yields all the forces, when one consider four-dimensional wave constructive interference. It shows that anti-matter will have gravitational repulsion or anti-gravity with respect to normal matter. The derivative for y₁ is given by:

        (3.10)

due to << 2.

Similarly

        (3.11)

Solving for x:

        (3.12)

There are two regimen of spacetime travel and they are depicted in Figure 9 below:

Figure 9. This figure shows the geometry of a surface bound particle. This is a X versus R cross-section of the hyperspherical expanding universe. Notice that the two circles represent a one de Broglie expansion of the hyperspherical universe.

At each de Broglie step both types of particles (zero and non-zero) change position by the same amount x and that defines a change in k-vector direction. The difference is with which referential that change in angle occurs. In the case of volumetric waves (non-zero spin particles), the k-vector is allowed to change by the angle ₁, while in the case of superficial waves (zero spin particles), the k-vector changes just by the amount given by ₀ since its k-vector has to remain perpendicular to the fabric of space. Tan() is given by tan(_) =x/₁ or by tan(_) =x/₁*( ₁/R₀) depending upon if the interaction is such that the particle k-vector shifts as in ₁ or it just acquires the radial pointing direction as in ₀. A further refinement introduced by equation (3.13) below introduces a level of local deformation of the de Broglie hypersurface or fabric of space. A change in angle ₀ corresponds to a much smaller angle change between the radial directions (by a factor ₁/R₀ = 9.385E-42, with R₀ (circa 15 billion light-years) as the dimensional age of the Universe). The experimental spacetime torsion due to gravitational interaction lies someplace in between 1 and 10^-41, thus showcasing a level of local deformation of the fabric of space. From figure 7, one calculates tan() as:

        (3.13)

Where and M=1. It will be shown that the upper limit is valid for charged particle interaction, while the lower limit modified by a slight deformation of the fabric of space will be associated with gravitational interaction. For the case of light, one has the following equation:

        (3.14)

That is, light propagates with proper time projection/propagation direction  at 45⁰ with respect to the radial time/direction. To calculate the derivative of tan () with respect to , one can use the following relationship:

        (3.15)

Acceleration is given by:

        (3.16)

To calculate the force between two 1 Kg masses (1000 moles of 1 a.m.u. particles) separated by one meter distance one needs to multiply equation (3.15) by 1Kg (N particles/Kg* 1Kg):

        (3.17)

For d=1 and P=3.5 one obtains the G_{Electrostatic} (3.5).

=     (3.18)

Where one made use of ₁=N₂ and considered the absolute value. It is important to notice that the derivation of the G_Calculated never made use of any electrostatic property of vacuum, charge etc. It only mattered the mass (spacetime volumetric deformation) and spin. Of course, one used the Planck constant and the speed of light and Avogadro’s number. By setting d=1 one recovers the electrostatic value of G!

To analyze gravitational interaction, let’s consider that Hubble coefficient measurements estimate the universe as being around 15 Billion Years old or 1.418E26 meters radius. To obtain the elasticity coefficient of spacetime, let’s rewrite d=₁/R₀)x on equation (3.17) and equate the G_Calculated to G_{Gravitational} for
two bodies of 1 Kg separated by 1 meter.

    (3.19)

Where P =3 since we are considering a spin-zero interaction. Solving for x:

    (3.20)

If we consider that the force is given by mass times acceleration:

    (3.21)

    (3.22)

The natural frequency of spacetime oscillations is:

    (3.23)

Notice that this is not dependent upon any masses. That should be the best frequency to look for or to create gravitational waves. Of course, Hubble red shift considerations should be used to determine the precise frequency from a specific region of the universe. At last one can calculate the value of the vacuum permittivity from equations (3.5) and (3.18) as:

        (3.24)

Not surprisingly, there is a perfect match between theoretical and experimental (8.85418782E-12 C².N^-1.m^-2 ) values. The correction factor used to calculate the effective charge per particle is due to the effect of non-zero spin on matter, thus related to the particle gyromagnetic ratio. It is important to notice that this derivation don’t use any parameterization. The “gyromagnetic ratio” and the “FS elasticity” are predictions of the theory, which uses only electron charge, speed of light, Avogadro’s number and Planck’s constant to relate it to non-hypergeometrical physics.

The complete gravitation equation is given by:

    (3.25)

Quantum aspects can be recovered by not using fast oscillation approximations. It is also important to notice that equations (3.8) and (3.9) can be used to calculate the interaction between any particles (matter or anti-matter) or to perform quantum mechanical calculations in a manner similar to molecular dynamic simulations. The quantum character is implicit in the de Broglie wavelength stepwise quantization. It is also relativistic in essence, as it will become clear when one analyzes magnetism next.

3.2 Magnetic Interaction

The Derivation of the Biot-Savart Law

Let’s consider two wires with currents i₁ and i₂ separated by a distance R. Let’s consider i₂ on the element of length dl₂ as the result of a moving charge of mass of 1Kg of fat electrons (one a.m.u. electrons). This is done to obtain the correct scaling factor.

Without loss of generality, let’s consider that the distance between the two elements of current is given by:

and     (3.26)

The velocities are:

and     (3.27)

Due to the spin half, one has after a two de Broglie cycles:

and and     (3.28)

Since one expects that the motion of particle 2 will produce a drag on the particle 1 along particle 2 direction of motion.

The figure below showcase the geometry associated with these two currents.

Figure 10. Derivation of Biot-Savart law using spacetime waves.

Notice also that the effect of the ½ spin is to slow down the rate of phase variation along the dimensional time  in half.

In the case of currents, the velocities are not relativistic and one can make the following approximations to the five-dimensional rotation matrix or metric: cosh() @1 and sinh(_) @v_i/c where v_i is the velocity along the axis i.

The k-vectors for the two electrons on the static reference frame are given by:

    (3.29)

Similarly:

    (3.30)

The wave intensities at are:

    (3.31)

    (3.32)

Where N= 1000 Avogadro, ₁=de Broglie wavelength of a one a.m.u (atomic mass unit) particle, ₂=de Broglie wavelength of a 1Kg particle= ₁/N.

Now one can calculate:

    (3.33)

    (3.34)

    (3.35)

due to << 2.

Similarly:

    (3.36)

    (3.37)

Hence:

    (3.38)

    (3.39)

And

    (3.40)

Thus,

    (3.41)

    (3.42a)

since v2=0     (3.42b)

since v1=0    (3.42c)

since v1=v2=0    (3.42d)

    (3.42f)

Where p stands for proton and e for electron.

    (3.43)

    (3.44)

Where non-velocity dependent and single velocity dependent contributions where neglected due to the counterbalancing wave contributions from static positively charged centers.

The force is given by:

        (3.45)

Where one took into consideration that a particle with spin half has a cycle of 2₁ instead of ₁.

The Biot-Savart law can be written as:

    (3.46)

Comparing the two equations one obtains:

    (3.47)

Thus

    (3.48)

From equation (3.24)

Thus

    (3.49)

Thus one recovers the relationship between ₀ and ₀.

3.3 Grand Unification Supersymmetry

As the dimensional age of the universe becomes smaller, the relative strength of gravitation interaction increases. Conversely, one expects that as the universe expands gravity will become weaker and weaker. This and the four-dimensional light speed expanding hyperspherical universe topology explain the acceleration of expansion without the need of anti-gravitational dark matter.

For gravitation the spring coefficient is given by:

    (3.50)

Similarly for electrostatic interaction, one has:

    (3.51)

Thus     (3.52)

Thus when R₀ was smaller than times ₁ (3.8E-19s), gravitational and electromagnetic interactions had equal strength. They were certainly indistinguishable when the radius of the universe was one de Broglie wavelength long. This section is called Grand unification supersymmetry, because condition (3.52) plays the role of the envisioned group theoretical supersymmetry of the grand unification force in future theories. Of course, it has a geometrical interpretation. At that exact radius, an elastic spring constant of the fabric of space allows for a change in the local normal such that it is parallel to the redirection of k-vector of a freely moving dilator. This is not what most scientists in this field expected but science is not about expectations.