Monday, December 29, 2008

Energy Conservation and the Fundamental Dilator

David Freiberg has left a new comment on your post "The Hypergeometrical Force":
Wait, wait, wait. How does a proton, with hundreds of times more mass than an electron, turn into an electron? Does the Law of Conservation of Matter and Energy still apply? If so, where does all the energy go when it does this go and where does the energy come from to add the mass? 
David was kind to ask how energy conservation resists such an outrageous transformation as proposed in the Fundamental Dilator paradigm.

The answer has to contain the circumstances where these transformations take place and the emphasis on the fact that the Fundamental Dilator replaces the construct particle.  The replacement of a particle by a coherence between deformation states of a 4D spatial metric (5D spacetime metric) is basic for the understanding of this paradigm.

Currently David thinks about particles (electron, proton..) as having mass, charge - fundamental properties assigned for the life of the particle...:)  Of course, the Fundamental Dilator makes these properties transients associated with states of a coherence.  The only way this would be consistent with observation would be if we were always observing a single face (state) of this coherence.

That is exactly the case. To achieve that one has to remember that the coherence (tunneling between deformation states) is accompanied by spinning within the 4D spatial manifold where the lightspeed expanding hyperspherical shockwave Universe (our 3D Universe) exists.

The Ball Diagram shown above displays the not only the transformation (tunneling between deformation states) but also the orientation change (with respect to our 3D Universe) as the dilator goes about his life.

The initial state is the one that is in phase with the 3D Shockwave Universe. Due to the very, very small dimension along one of the directions (R), as the spinning takes place, the overlap between our dilator displacement volume and the 3D Universe vanishes.  Only at "quantized" phases of spinning is that there is overlap. ONLY at "quantized" phases there is interaction.

This stroboscopic nature and the 4D topology are the source of Quantum Mechanics.

This diagram shows two electrons belonging to a Cooper Pair.  As you can see, despite of the non-static nature of the dilator, their interaction when the dilator is flush with the 3D shockwave universe, is always consistent with what David considers their nature to be, that is, an electron always repels another electron.

The sideways phases have little overlap with our 3D Universe due to the initial symmetry of the problem and the symmetry of the involved states.

Notice that due to the coherent form in which these changes take place, the dilaton field is generated coherently, that is, the dilaton phases are always in phase with the propagating metric disturbances (dilatons) since they travel at the same speed (the speed of light).  As an aside, only in the case of Cooper pairs, the the intermediate phases anihilate each other's contributions, reducing their interaction to an Anti-Gravitational strength!!!

The Energy conservation is related to the size (3D volume overlap) of this footprint.  Local deformations of the Fabric of Space (local tilts) would impact into the dilator Kinetic Energy or Potential Kinetic Energy (Strong Force). Depending upon what is the choice for the laws of motion (Newton's Laws or my Quantum Lagrangian Principle) mass will behave in a hyperbolic manner (e.g. Lorentz Transform or Strict Relativity - hyperbolic cosines projections) or in a linear manner (through cosine projections).

There is a principle of 4D displacement volume conservation which is applied to the whole Universe, yielding ZERO, that is, the total displacement volume of all dilators in the Universe is ZERO.

The 3D projection of this law is our energy-momentum conservation law.

Please feel free to ask questions.



Wednesday, December 17, 2008

The Hypergeometrical Force

The Hypergeometrical Force

When I DERIVED the equation for the Hypergeometrical Force – the Grand Unification Force - used a setup with two infinitesimal elements of current (or mass flow).

These infinitesimal elements of current or flow defined a direction within the four dimensional spatial manifold. This direction is shown as velocities in the formulas.  Despite of having velocities, these formulas were derived for constant velocities, where the velocities were referred from a relaxed Fabric of Space reference system.  To eliminate this constraint one has to derived the potential through adiabatic integration followed by the derivation of a generalized force according to Lagrangian formalism!!!

These means that the force equations below are constant velocity forces. Despite of this constraint, they are quite usefull for making simplied reasonings about dynamics.

Here are the equations for Electromagnetism and Gravitation:


Where C1 and C2 are the charges traveling at V1 and V2 and c is the speed of light. R12 is a vector from charge 1 to charge 2.

Similarly for Gravitation:

Notice that the only difference is the proportionality coefficient. This means that the interaction is the same – dilaton field-dilator interaction controlled by the Quantum Lagrangian Principle. The proportionality coefficient ratio is given by the ratio of these two angles a0 and a1:

R0 is the age of the Universe times the speed of light (15.82 billion light years) and λ1 is the Compton wavelength of a Hydrogen atom. To be precise, this ratio has to be multiplied by an elastic coefficient of the Fabric of Space. This coefficient is the reason why dilaton and Gravitational waves exist.

The humongous (1036) factor is easy to understand by this simple figure. A more detailed analysis can be found in the pdf containing the theory.

This equation can be simplified for the case where there is no rotation and one of the bodies is much larger than the other (Sun and Mercury for instance) into:

This is the first time a force is actually derived from a purely geometrical theory of the Universe, so it took me some time to understand that this equation has to be integrated upon R (distance) to yield the amount of energy that can be shared into Kinetic and Potential at any position.

Adiabatic integration upon R yields the potential


This potential is the one used in the Precession of Mercury's Perihelion problem. Applying it to Lagrangian Formalism yields the more appropriate force formula for physical bodies or charges:

Similarly for F one obtains:

Thus F becomes:

Expanding it in powers of

Neglecting terms above the second order

This means that my Gravitational Force does not contains first order dependence upon v/c and also that it perfectly describes all celestial dynamics. It produces the same formula for the Precession of Mercury or Earth or anything else Perihelion…J

In summary, my formula produces not one but TWO formulas for force. One for constructs that cannot change speed (e.g. Light) and another for bodies. They are reproduced without approximations below:

The second formula is used to derive the Gravitational Lensinng (Gerber failed to do so since he only ansatz (guessed) right the first formula).

The more complex formulation for the Force per unit mass (for double jets Black Holes or White Orifices) is given by:

To understand the rest of the Universe, one only need to add the Fundamental Dilator paradigm.

The Fundamental Dilator is a coherence between two deformation states within the 4D spatial manifold. The orientation of the letters in the figure below maps the orientation of the dilator deformation with respect to our 3D Hypersurface. A vertical lettering means that the dilator overlaps minimally with our 3DUniverse (the Fundamental Dilator is quite flat (along the fourth dimension).

The Fundamental Dilator can be represented by this diagram:

Figure 1. Balls Diagram for the Proton, represented by a phase of the Fundamental Dilator.



Monday, December 15, 2008

Mercury's Perihelion Prequel

Mercury's Perihelion Prequel

I had to figure out what was wrong with Paul Gerber's derivation. Gerber's work has been dismissed by the Wikipedia people and I couldn't figure out from reading their posting if the problem was in the derivation of the Gravitational Potential Formula or afterwards.

Here I derived the Precession Rate formula starting from mine (and Gerber's) Hypergeometrical Gravitational Potential. I was able to obtain the accepted formula.

Paul Gerber's work has been dismissed because he didn't have a great argument to support his potential. That is not the case any longer. My theory is presented here in details and contains all the logical support for the Hypergeometrical Gravitational Equation. I also provided the correct derivation of the Gravitational Lensing using this potential.

Currently, the mindset is such that even when a theory predicts the experimental results, it is still considered at fault because it did not produced non-observable differences based upon General Relativity equations.

Of course, this is a non-scientific reasoning. My theory predicts observed phenomena (e.g. jets comig from Black Holes - White Orifices) that are not predicted by ad-hoc Einstein equations using an ad-hoc Schwarzschild metric.


A little research on the subject showed me that the precession was already known by the time of the prediction and was in agreement with many models where the proposers guessed left and right the actual formula for the Gravitational Potential. Some of those models had velocity dependence, although not in the manner I proposed.

In addition, the precession of Mercury was the result of many interactions with the planets of the Solar System. The unexplained precession component was of the order of 43 arc seconds per century.

Let's see if one can solve that problem in a different paradigm, say, in the Hypergeometrical Universe paradigm.

First let's write the equation for the Hypergeometrical Gravitational Force.


Remember that this equation is not guessed (Ansatz) but derived from first principles!!!!!!!!!!!!!!!!!!!!

If we consider that the Sun is not rotating, thus V1=0 the final force sensed by Mercury is given by:


Adiabatic integration upon R yields the potential


Which is equation (1) on the Gerber Model presented in Jaume Gine paper.

The Kinetic Energy T per unit mass is given by:


Is the angular momentum per unit mass.

The Potential Energy per unit mass is given by:

The generalized momentum per unit mass is given by:

Similarly the generalized force per unit mass is given by:

Solving for P one obtains:

Similarly for F one obtains:

Thus F becomes:

Expanding it in powers of

Neglecting terms above the second order

Equating generalized momentum and forces yield the generalized equation of motion:

Since the Gravitational force is radial, there is no tangential acceleration. Hence, integrating equation (5)


Yielding the following change of variables (t->θ) and (r=1/u)


This simple equation can be solved promptly if one considers that the terms in parentheses are very close to one for astronomical conditions.

The solution is given by:

Where ε is a constant of integration.


The precession of Mercury's perihelion is given by the change in angle for θ=2π

If one consider that

Considering that the precession of the perihelion or apogee are the same, let's consider the apogee. The precession is given by:

Then the formula for the precession rate is given by:

Gerber and Einstein obtained a more complex version of this formula. Let's see if one can simplify Einstein's equation into ours.

Einstein equation is given by:

ε=Eccentricity, a=Semi-major axis, τ=Orbital period.

Under standard assumptions the orbital period () of a body traveling along an elliptic orbit can be computed as:


  • is standard gravitational parameter,
  • is length of semi-major axis.
Replacing the period and the semi-major axis in terms of R0 yields:


Which is the same result!!!!!!!!!!!!!!!

This means that given that Gerber's Gravitational potential is also the Hypergeometrical Gravitational Potential and that Gerber's work has been totally dismissed, I cocluded that the cause was inconsistencies in the logic used to derive the Gravitational Potential.

In Gerber's Gravity, the author emphasized that despite Gerber's result yielded the observed precession, it does not provide the same equations as General Relativity – as if this were necessarily a problem.

The gravitational radius of the Sun is m = 1.475 km and the semilatus rectum of Mercury's orbit is L = (5.544)107 km, so the precession of Mercury's orbit due to this effect (excluding the perturbations of the other planets, etc.) is 0.1034 arc seconds per revolution. Mercury completes 414.93 revolutions per century, so its orbit precesses (due to this effect) by 42.9 arc seconds per century, in excellent agreement with what is observed. This formula also gives values for all other known bodies orbiting the Sun consistent with observation.
We should mention that although general relativity and Gerber's potential predict the same first-order precession (for weak fields), the respective equations of motion are not identical, even at the first non-Newtonian level of approximation. In terms of the parameter u = 1/r the equations of motion are

so the non-Newtonian terms are actually quite different. Of course, any non-Newtonian terms will lead to orbits that fail to close, so there will be some cumulative precession for the two-body problem. It just so happens that the term +3mu2 in the GR equation of motion and the term 6mu d2u/d2 in Gerber's equation of motion both result in a first-order precession of 6m/L in the slow weak-field limit. Thus Gerber did not in any way anticipate the two-body equation of motion predicted by general relativity, let alone the field equations from which the relativistic equation of motion is derived.
Nevertheless, we have verified that Gerber's potential (1) does indeed yield the observed non-Newtonian precession for planetary orbits, so we return to the question of the how Gerber arrived at this particular form, and whether it has any consistent representation.

Eventually, the author tries to derive Gerber's potential using an argument based upon retarded potentials.

Since we arrived at the same equation obtained by Gerber and Einstein and we know that there has been extensive testing of this equation on the calculation of precession of other planets, spacecrafts, etc. there is no need to pose any more arguments to support the Hypergeometrical Gravitational Equation.

Overzealous Wikipedia editors concluded that because Gerber's argument leading to the Hypergeometrical Gravitational Potential was flawed and thus he did not preceded Einstein in deriving the correct formula for the precession of Mercury's Perihelion. Currently this subtle details is very relevant because the editors believe that General Relativity equations and reality are the same. Somehow, they see other theories as ad-hoc but failed to see how ad-hoc General Relativity and Einstein equations are. Anything that describes the Universe with a parametrized (Unknon) metric cannot be anything other than an Ad-Hoc theory...:)

It also seems that Gerber made a mistake in deriving the Gravitational Lensing angle. Somehow "experts" did not catch his mistake.

I did it correctly and reached the appropriate result.
So Gehrcke initiated a reprint of Gerber's 1902-paper in the Annalen der Physik in 1917, where he questioned the priority of Einstein and tried to prove a possible Plagiarism by him.[A 5]However, according to Albrecht Fölsing[B 5] and Roseveare,[B 6] those claims were rejected, because soon after Gerber's paper was reprinted, scientists like Hugo von Seeliger,[A 6] Max von Laue[A 7] published some papers, where it was shown that Gerber's theory is inconsistent and his formula is not the consequence of his premises. Also Roseveare argued that Gerber's theory is inconsistent and that the value for the deflection of light in the gravitational field of the sun in Gerber's theory was too high by the factor 3/2. And Einstein wrote in 1920:[A 8]


Mr. Gehrcke wants to make us believe that the Perihelion shift of mercury can be explained without the theory of relativity. So there are two possibilities. Either you invent special interplanetary masses. [...] Or you rely on a work by Gerber, who already gave the right formula for the Perihelion shift of mercury before me. The experts are not only in agreement that Gerber's derivation is wrong through and through, but the formula cannot be obtained as a consequence of the main assumption made by Gerber. Mr. Gerber's work is therefore completely useless, an unsuccessful and erroneous theoretical attempt. I maintain that the theory of general relativity has provided the first real explanation of the perihelion motion of Mercury. I have not mentioned the work by Gerber originally, because I did not know it when I wrote my work on the perihelion motion of Mercury; even if I had been aware of it, I would not have had any reason to mention it.[C 1]

Thus the Hypergeometrical Universe Theory also "predicts" the Precession of Mercury's Perihelion, the Gravitational Lensing angle, the White Orifice phenomena and thus equates and surpasses General Relativity in its achievement. Needless to say, it provides the reason for having the speed of light as the limit, provides an alternative model for the Gravity induced metric (spacetime) deformation.

Gerber model is a limit of my theory where one considers that body 1 is not rotating. It yields the correct value for the precession constant (42.3 arc seconds per century).




Gerber's Results:

Paul Gerber reached an identical Gravitational Potential and from there derived the precession of Mercury's Perihelion

According to Gerber, the relation of the speed of gravity (c) and the Perihelion shift (Ψ) is:

ε=Eccentricity, a=Semi-major axis, τ=Orbital period.

It was noted by the Einstein- and relativity critic Ernst Gehrcke in 1916,[A 3] that this formula is mathematically identical to Albert Einstein's formula (1915) for general relativity.[A 4]

, where e=Eccentricity, a=Semi-major axis, T=Orbital period.

Data on Mercury

Mercury is the closest planet to the Sun and the eighth largest. Mercury is slightly smaller in diameter than the moons Ganymede and Titan but more than twice as massive.

orbit: 57,910,000 km (0.38 AU) from Sun

diameter: 4,880 km

mass: 3.30e23 kg

Mercury's orbit is highly eccentric; at perihelion it is only 46 million km from the Sun but at aphelion it is 70 million. The position of the perihelion precesses around the Sun at a very slow rate. 19th century astronomers made very careful observations of Mercury's orbital parameters but could not adequately explain them using Newtonian mechanics. The tiny differences between the observed and predicted values were a minor but nagging problem for many decades. It was thought that another planet (sometimes called Vulcan) slightly closer to the Sun than Mercury might account for the discrepancy. But despite much effort, no such planet was found. The real answer turned out to be much more dramatic: Einstein's General Theory of Relativity! Its correct prediction of the motions of Mercury was an important factor in the early acceptance of the theory.

Data on Sun

Our Sun is a normal main-sequence G2 star, one of more than 100 billion stars in our galaxy.

diameter: 1,390,000 km.

mass: 1.989e30kg

temperature: 5800 K (surface)

15,600,000 K (core)

The Sun is by far the largest object in the solar system. It contains more than 99.8% of the total mass of the Solar System (Jupiter contains most of the rest).

Orbital period

Under standard assumptions the orbital period () of a body traveling along an elliptic orbit can be computed as:


  • is standard gravitational parameter,
  • is length of semi-major axis.


  • The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis (),
  • The orbital period does not depend on the eccentricity (See also: Kepler's third law).