HyperGeometrical Universe Theory

Quantum Gravity



I am finally free of extra chores and will start developing the Quantum Gravity argument next.


Cheers,


MP

Infinite Dimensionality as a Bridge to Quantum Mechanics

In Physics, more specifically in Quantum Chromodynamics there are neat renormalization tricks, meant to ABRACADABRA some infinities (self-energy) etc…

One of these tricks is called Dimensional Renormalization. Normally it involves redeploying a theory in an N dimensional space and then making N go to 4 or 3. When that happens the infinities disappears or become meaningful within that new framework.

Recently, Dudley R. Herschbach, A.K.A. dude…:) (just kidding, Dudley) used a variation of Dimensional Renormalization in which instead of recovering regularized equations when N goes to 4, he obtained “Surprisingly Accurate Semiclassical Results” resembling Pure Quantum Mechanical Results by making N go to infinity…:)

Also surprisingly is that that same trick worked fine in the calculation of energy levels of atoms (molecules with a single positive center..:)

That is fine and dandy…:) but what is missing is the reasoning behind that coincidence…:) Nobody, including Dudley, is saying that the Universe is infinitely dimensional or the reason for Quantum Mechanics is the infinite dimensionality of the Universe. I should comment that he used the word Dimensional Scaling as opposed to Dimensional Renormalization, but that is a minor detail, just a name…

The unaddressed Hocus Pocus is the WHY? Why would an infinite dimensional Universe lead to Simple and Surprisingly Accurate Approach to the Chemical Bond Obtained from Dimensional Scaling …:)

To understand that, one should delve deep into the details of the derivation by Professor Herschbach. He did not found that any kind of infinitely dimensional topology yields Accurate Results in the case of Diatomic Molecules (e.g. H2) …:)

Only a special topology in which rho is infinitely dimensional, Phy is the dihedral angle between the planes containing the electrons and the internuclear axis of a H2 molecule. This special topology makes Rho perpendicular to the interatomic axis of H2.

If you remember that Quantum Lagrangian Principle (QLP), the basis for my Grand Unification Equations derivation, you would realize that that choice of topology is a requirement and the WHY for the Surprising Success of Infinite Dimensionality Scaling.

If you read equation (6) and (7) and understand Figure 2 on his paper, it should become clear that the difference between the incorrect equation (7) and the correct one (6) is the loss of dependence on Phy. This can be seen as if the two Hydrogen nuclei were generating cylindrical dilaton waves and that the electron wavefunction had a k_r component with a de Broglie wavelength r₁ and r₂. This is exactly what one would expect from the Quantum Lagrangian Principle.

IF electron 1 is at distance r₁ from nucleus 1, then it is an integer number of wavelengths from that nucleus or conversely, the de Broglie wavelength (5D dilaton projection onto the Hyperbolic 4D spacetime) is the distance to the nucleus divided by an integer n₁. This reasoning leads to the proper calculation of excited states.

Remember that the de Broglie wavelegth associated with the nuclei is much broader due to their 2000 times larger inertia. As you remember, the fundamental dilator has different inertia for the different phases of the coherence.

The QLP states that a dilator will always dilate in phase with the surroundings dilators. This works both in the 5D spacetime as well as in the 4D spacetime projection. The 5D dilatons project onto a 4D Hyperbolic Hypersurface as de Broglie Waves. This means that the Hydrogen atoms will be sitting on the crest of the electron de Broglie wave (at a given Rho-directed momentum).

I wrote this some time ago. I don’t have time to review it, but it is likely to be correct. Please review the reasoning and comment on it. You might have to read Dudley’s paper.

Cheers,

MP

 

Nothing like not being able to think to become vain..:)

I’ve the flu so I can barely think.. but writing about things I like and know should be a natural choice.

First, let’s celebrate a few milestones.  The first one is on the upper right panel containing the Blogged Editor’s review of my blog.  I got a Very Good rating and am thankful for that. For me this is a tremendous compliment, since the Editor reviewed the blog not for content although I hope he executed standard crackpot testing..:), but for style, readability and other literary measures. 

The map above shows the distributions of readers, showing that little by little this theory is becoming part of the discussion which is why I wrote this blog.  The number of unique URL readers is actually around 43 thousand readers, a very large number for such an obscure and challenging subject.

The third reason to be thankful for is the presentation of this theory at the  7th Symposium in Honour of Jean-Pierre Vigier 

The  proceedings of the symposium were finally published by the American Institute of Physics. The abstract is here.

 This paper presents a simple and purely geometrical Grand Unification Theory. Quantum Gravity, Electrostatic and Magnetic interactions are shown in a unified framework. Newton’s, Gauss’ and Biot-Savart’s Laws are derived from first principles. Unification symmetry is defined for all the existing forces. This alternative model does not require Strong and Electroweak forces. A 4D Shock -Wave Hyperspherical topology is proposed for the Universe which together with a Quantum Lagrangian Principle and a Dilator based model for matter result in a quantized stepwise expansion for the whole Universe along a radial direction within a 4D spatial manifold. The Hypergeometrical Standard Model for matter, Universe Topology and a new Law of Gravitation are presented. ©2010 American Institute of Physics

This is the goal I had in mind when I started this blog. I wanted to provide an outlet for this idea.  If is right or wrong, that is to be decided on a debate among ideas not by censorship. 

As I mentioned in the prior blog, there is still a factor of two to be corrected in this paper.  I will review it before to long.  It is interesting since it will show that the Hypergeometrical Standard Model predicts a relationship between charge and spin..:) Well, one  can just review the Delta Plus Plus already discussed in prior blogs to have a glimpse of the future…:)

Cheers,

MP

Anonymous said…

are all your factors constants or functional approximations?

where is your evidence to confirm their definition for each factor, time, time change according to the history of time, what is space?

Is it constant or variable, or constant/variable, or other…

Thanks for the question. Every so often someone asks a kind of general question that it permits me to review the whole theory and many times to find a better way to explain my ideas..:)

The theory does not have factors other than the Planck’s constant, which relates 3D Mass (inertial mass) to a dilaton projection wavelength.

The fundamental components of this theory are :

1. Particles are modeled as dilators

   

   Dimensional notes associated with Electron and Positron. Vertical letters indicates perpendicularity with respect to our 3D Universe

2. Dilators are coherences between stationary deformation states of a local metric.

   

   Lower (higher) state is associated with Proton(Electron). The initial state (in phase with the 3D Universe) defines the particle nature (proton, antiproton,electron or positron).

    

3. The Hypergeometrical Universe topology (3D Universe is a lightspeed expanding hyperspherical shockwave)
4. Dilators are thin along the radial direction and spin at half the speed of tunneling, that is, when the wavefunction describing the dilator moves from one shape to another (180 degrees), the dilator profile rotates by 90 degrees.
5. Pseudo-time quantization results from item 4). Interaction only occurs when the dilator has a footprint on the 3D Universe.
6. Dilators motion along the 4D spatial manifold obeys the Quantum Lagrangian Principle which states that dilators always dilate in phase with surrounding dilaton

   field.

   

   At any de Broglie step of the Hypergeometrical Universe expansion (one step is equal to the Compton wavelength of a Hydrogen atom), the dilator has to move itself by distance x to satisfy the Quantum Lagrangian Principle, that is, to dilate in phase with the surrounding field. In this picture, I show two angles. Alpha1 is the arctan(x/Lambda1), where Lambda1 is the Compton Wavelength of a Hydrogen atom (around 10^-15meters). If you are a dilator, your k-vector is now along that new direction. If you are a neutral particle (e.g. a hydrogen atom or a neutron), you will also move yourself by distance x, but your k-vector will only point along angle alpha0 or thereabouts. Angle Alpha0 is arctan(x/(R0+lambda1), where R0 is the radius of the Universe (equal to 21.72 billion light years). The Universe is actually 21.72 Billion Years old.

   This difference in change of direction rate (the change in direction happens at each de Broglie step of the expansion) explains the difference in strength between electromagnetic and gravitational forces. Discrepancies are associated with physical quantities (elasticity of space in the case of gravitation and elasticity anisotropy in the case of electromagnetism). From the elasticity of space derived from the Gravitational constant, one derives the natural frequency of Gravitational Waves. From the anisotropy of elasticity one derives the phase matching conditions required for coherent nuclear fusion, antimatter creation etc…J

   If you look at the plot below for a neutron dilation levels you can find the meaning of electron neutrinos. They correspond to the subcoherence between protons and electrons. These subcoherences (half-electron neutrino each) correspond themselves to rotations within the 3D space (as opposed to tumbling withing the 4D spatial manifold). The dephasing associated with this rotation defines the mass of all isotopes, that is, one parameter is used to calculate the mass of everything that is composed of protons and neutrons.

   

   Hyperons requires the knowledge of another dephasing parameter from the pions composition shown below:

   

   Pion Minus diagram.

   Here the red lines correspond to the already know half-electron-neutrino. The blue lines correspond to a new dephasing process.

   

   This dephasing process happens by rotations on the upper states and correspond to rotations of the electron state within the 3D Universe. With this dephasing parameter all other particles in the Univers can have their masses calculated. If you know how to split a neutrino, you can make antimatter..:)

7. Items 6, 5 and the embedding of the 3D lightspeed expanding hyperspherical shockwave Universe within a 4D spatial manifold allows for the uncertainty that creates Quantum Mechanical behavior. This means that the Hypergoemetrical Universe provides the reasons for the observed Quantum Mechanics.
8. The two cross-sections shown at the top of this and almost any other posting here, shows that the Hypergeometrical Universe contains an Absolute Time, a preferential direction but the absolute time cannot be used to time anything other than the Universe expansion. The preferential direction cannot be seen within the 3D expanding Hypersphere where we live.

    

Now let’s talk about constants. The theory derives formulas for Cosmological Constants like Newton’s Gravitational constant G given by:

= space elasticity

P=3

K=elasticity anisotropy

Derivation is on the pdfs.

The natural laws for Gravitation and Electromagnetism are derived from first principles as:

or

Where C1 and C2 are the charges traveling at V1 and V2 and c is the speed of light.

For Gravitational one has exactly the same equation (Gyrogravity). This equation is consistent with Gravitational Lensing, the Precession of Mercury Perihelion and explains the Galactic Puzzle (the constant angular speed of mass distributions within spiral galaxies). All these demonstrations are posted within this blog.

This means that one equation describes Electrostatics, Magnetism and Gravitation and this equation was derived from first principles using the fundamental hypotheses presented above.

The Hypergeometrical Universe theory explains the particle physics aspects of the Universe including the puzzling aspects of Cosmology. Currently there are two inconsistent observations. Hubble measurements using short distance calibration indicates that the Universe is around 14 billion years old. Standard Stellar Candles (Supernovae type 1A) which are supposed to have constant luminosity through all ages of the Universe life, indicates that the furthest observed Stellar Candle to be around 50-60 Billion light years away. This conclusion is arrived by observing an exceedingly weak supernovae type 1A explosion. The weaker the supernovae luminosity the larger the distance (apparent luminosity decays with the inverse of the distance).

This inconsistency is easily explained by the Hypergeometrical Universe theory. Just look at the dependence of the Gravitational “constant” G. It decays with the inverse of the Radius of the Universe. This means that at early epochs, when the radius of the Universe was half (quarter) of today’s radius (21.72 billion light years as derived from the Pioneer Anomaly results), G would be twice (four) times larger. A larger value of G means that the supernovae 1A will contain a smaller mass in the early Universe. This correction indicates that the perception of 60 Billion Years away supernova, just means that that supernova is at 2.68 billion years old Universe, no paradox there… NO NEED FOR INFLATION….:)

To prove that my theory provides the reason for Quantum Mechanics is trivial and will be presented later in this blog.

If you want to understand the Cosmogenesis, please let me know..:) The explanation is simple and based upon these two plots..:)

Before I forget:

SPACE

What is space. Space and Absolute time are quasi-constants, that is, they exist always as they are now, but they exist after the dimensional transition that created the extra four spatial dimensions…>J

Space itself as far as I am concerned is just a Cartesian medium with presents no friction and has a restoring force (elasticity). I am agnostic with respect to any sub-structure since I don’t believe any experiment could probe any substructure. The reason is that colliding particles also follow the Quantum Lagrangian Principle, that is are always I quantized positions determined by the interference pattern of the dilaton field.

Huygens Principle

Huygens and Newton debated how to describe the dynamics of interacting bodies. Newton won.

Huygens was biased by his optical interference experiments and wanted to explain all motions (interaction) as being the sole product of interacting waves. If you think that the Quantum Lagrangian Principle is equivalent to the Huygens Principle you are wrong. The QLP describes what is happening on the RXYZ cross-section shown in the first figure in this posting. We don’t live there…J We live in the TauXYZ cross-section, where Tau is the proper time. I modified Newton’s Second Law to present it as a Spatial Stress-Strain relationship between these two cross-sections

Let’s make Area _4D =1 then Area3D is just the overlapping of the unitary dilator 4D volume with the FS.

Notice the hyperbolic tangent on the 3D Strain. This is a requirement for relativity.

Even though the dilator moves in the RXYZ following an interference pattern, the dynamics has to be described in the TauXYZ. In that cross-section, wavelengths vary with velocity and one cannot simply imagine motion through an interference pattern. Here one has to take into consideration Doppler shifts of the dilator wavelength while creating the 3D Lissajous figures where dilators will pop-up at each de Broglie step of the Universe expansion.

If Huygens had followed this simple reasoning, he would have discovered Quantum Mechanics.

Cheers,

MP

The Pioneer Anomaly

The solution to the Pioneer Anomaly can be derived directly from the Hypergeometrical Universe topology. It is our current understanding that only matter curves spacetime, this means that locally the curvature is quite well defined by the inexistence of localized matter and equal to zero, that is, without the obvious occurrence of matter we believe that we live in a locally flat space. The addition of another spatial dimension in the Hypergeometrical Universe theory changes things, since there are now many curvatures to talk about.

Let’s see what happens when one measures the speed of light within our standard paradigm. The measure time delay between shooting the laser pulse and measuring the reflection is equal to twice the distance divided by the speed of light.

For the reflected radiation bouncing back from the mirror at rest,

Now if we place the mirror in motion we arrive at the standard derivation of the Doppler Effect:

The Doppler shift is given by:

The derivation is different if one allows v to vary:

Resulting:

The derivation is different still if one allows both c and v to vary:

Resulting:

This means that both accelerations on the speed of light or the actual speed of the Pioneer spacecraft would contribute to the Doppler Shift. This also means that if you are not considering that the effective speed of light vary with distance, you will be missing the contribution.

Under the current paradigm, the time it takes for light to bounce back from a mirror is given by 2L/c and doesn’t change with the distance L. This considers that light and the mirror are traveling or existing in the same dimension (3D) which is not the Hypergeometrical Universe Paradigm.

There, light (a spatial modulation of the source of a dilaton field) travels within the 5D spacetime (4D spatial manifold). Anything else just flows within the 3D Hyperspherical Universe, which itself travels through the 4D spatial manifold.

If you consider our topology, it is clear that there is a curvature and that is equal to 1/R0 and it is independent of the existence of matter locally. Remember that a 5D Spacetime has more than one radius of curvature. The RX radius of curvature is R0, and so it is the ФX radius of curvature. The situation changes one consider the τX (τ is the local or proper time). The τX radius of curvature will depend upon which laws do you use to describe your Universe (Gauss Electrostatic Law, Newton’s Gravitational Law). If you were to use the Quantum Lagrangian Principle to describe dilator dynamics, then you could use a Cartesian reference system for the local proper time. The usage of Newton’s, Gauss’ or similar laws to describe dilator dynamics makes it necessary to make local proper spacetime metric to be Lorentzian.

Now let’s get down to the Pioneer Anomaly and how do we measure how long the Universe is alive from the observed Pioneer Spacecraft deceleration.

The first thing to consider is what would be the perceived speed of light deceleration resulting from the geometry. To do this, we need to consider how much light has to travel back as a function of the farthest distance traveled by the spacecraft.

This figure shows what would be the path traversed by each pulse emitted by the Pioneer spacecraft. The angle of the light ray is always 45 degrees. The leftmost radial line represents Earth position. This means that when Pioneer is far from Earth by 45 degrees or PI/4*R0, light will never come back (the light ray will be parallel to the Earth radial line. Two parallel lines never meet.

Let’s say that at time zero Pioneer is at R0 from the 4D Center of the Universe and L from Earth and let’s derive the deceleration from the simple geometry.

Let R(t) be the Radius of the Universe at the time t:

Where time is being measured after a light pulse is emitted from the Pioneer while it is at distance from the 4D Center of the Universe equal to R0.

The Cosmological Angle associated with Pioneer at time t given by:

for calculating the deceleration of the speed of light

And

for calculating the deceleration of the speed of the Pioneer spacecraft

The angle alpha is called Cosmological Angle and it is measured from the actual Center of the Universe. The time that measures the expansion of the Universe is the Cosmological Time φ, but for quasi-relaxed fabric of space (low velocities), the Cosmological Time is a reasonable approximation to the proper time.

The equation for the photon trajectory leaving the Spacecraft and reaching Earth later is given by:    

Where y(t,t’) is the horizontal distance represented in the figure above of the reflected light beam as a function of time t’, which starts counting at the reflection moment. The time t refers to the t governing the expansion of the Universe from some given initial condition (R₀, L).

Equating this equation to zero and solving for t’, one obtains the time it takes for light to come from Pioneer to Earth at any given time or Pioneer position.

The perceived distance traversed is given by x=c.t’:

Taking the second derivative, expanding in Taylor Series and simplifying to obtain the acceleration as:

The observed acceleration is 8.75e-10 m/s2, the speed of light = 299792458 m / s. This yields a value for R₀= 2.05E+26 meters. Since the Pioneer velocity is much smaller than c and the second term can be neglected.

Since the 4D observed velocity is m this means that the Universe is years old or 1.5364E+10 years or

15.364 Billion Years Old…J

PerceiveAge=15.364 Billion Years Old…:) (using the Hubble Constant as a measure of the age of the Universe).

The important result is that the Universe is actually older that the age perceived by inspecting the Hubble Constant:

RealAge=15.36*sqrt(2)= 21.65 Billion Years Old:)

One salient point is that the Hypergeometrical Lightspeed Expanding Shockwave Universe expands at the observed speed of light, this means that the material world travels slower than the dilaton field, by a factor of cos(45) or .

This conclusion is very important because it breaks the symmetry between dilators and dilatons, in the sense that they do not travel at the same speed. On the other hand, this velocity relationship is strictly necessary for the Quantum Lagrangian Principle to make all dilators to surf the surrounding dilaton field.

Dilators interact with other dilators through retarded potentials or retarded dilaton fields.

The Beginning of Times

At time zero, the initial macroscopic metric fluctuation (the one that followed the dimensional transition) decayed into a huge number of dilators. Many recombined and released Gamma Rays. The initial radiation blast accelerated the extremely small Universe both radially and tangentially with the same strength. Dilators flowed with the retarded dilaton field in all directions of the 4D spatial manifold, creating the 3D Lightspeed Expanding Shockwave Universe. The 4D Big Bang also synchronized all dilators’ spinning. Only dilators that were flush (in perfect overlap) with the 3D Hypersphere at time zero, were accelerated into c.

Up to now, I considered that dilators and dilatons always traveled at the only allowed speed (c).

Of course, I noticed that dilators have inertia with respect to lateral motion but I thought that the radial motion was the natural motion of a metric deformation, thus requiring no initial push to be achieved.

If I accept that the dilaton speed or light speed is actually larger than the observed speed of light then the picture associated with the initial Big Bang changes from a simple decaying process into a tangential explosion which set the 3D Hypersphere to travel at the observed speed of light.

It is clear that a dilaton field cannot accelerate dilators to its speed (sqr(2)*c) because interaction has to occur through RETARDED POTENTIALS. Retarded Potentials always propagate at c (Lightspeed) due to the initial geometry.

The interaction of dilators with the retarded dilaton field from other dilators produces a distribution of tangentially dilators traveling at c which is also a distribution of radially traveling dilators at that same speed. In other words, a lightspeed traveling Hyperspherical Universe and an sqr(2)*c dilaton-dilator interaction form a self-consistent solution to the problem.

This salient point can be summarized as:

The speed of light is SQR(2) times C or light is faster than the speed of light..J

This is due to motion be part of a coherence that was placed in motion at the beginning of times through the interaction of near field dilatons. The interference pattern of near field radially propagating homogeneous dilatons travels along 45 degrees. This is another very salient point.

Cheers,

MP

The Galactic Puzzle

 I found this posting lying around. I suspect I reached a conclusion here and forgot to post it.  It explains the Galactic Puzzle where the edges of Galaxies rotate in the same angular speed as the centers of the Galaxies, in contradiction to Newton’s Gravitational Law..:) .

Spiral galaxies present a puzzling characteristic. Their most external stars are rotating at speeds that are too high to be still attached to the galaxy if one considers Newton’s law.

Of course, if we consider the Hypergeometrical Gyrogravitational Law there is no anomaly. The Gyrogravitational Potential is given by:

Equating the Gyrogravitational Force to the Centrifugal Force one derives the Equilibrium Radius:

As you can see, this equation is quite more complex than the standard one from Newton’s Law of Gravitation.

Let’s consider that all velocities are tangential, thus perpendicular to R. If one considers that for the Galaxy to keep its equilibrium shape while rotating and

Under those conditions the equilibrium R is given by:

Life is Stranger than Fiction…:)

is the ratio between the Newtonian Gravitational Force and the Centrifugal Force.

The Galactic Puzzle is how to explain situations where this ratio seems to be smaller than one, that is, it seems that there is equilibrium but the Centrifugal Force is too strong. Under those conditions

Or

Or there is a lag of more than 90 degrees between the rotation of the slowly rotating SuperMassive Black Hole and the star on the edge of the Spiral Galaxy.

This is the condition for the creation of the Spiral.

This is a simplified derivation. A proper analysis would take into consideration the Galactic Mass Distribution; integrate over it and possibly the effect of retarded potentials.

Despite of being a simplified derivation it already shows how in the vicinity of the SuperMassive Black Hole the beginning of the Spiral Galaxy takes shape.

Cheers,

MP

Hubble Expansion and the Hypergeometrical Universe

Given that the Hypergeometrical Universe proposes a dynamic (lightspeed expanding hyperspherical Universe) topology for our Universe, it shouldn’t be surprising that we have to revisit the interpretation of the Hubble constant and how one measures distance across the vast expanses.

Figure 1 displays the geometry for peering into the past. From the past blog, we established that the actual speed of light is sqr(2) times the observed speed of light.

I struggled a lot with this problem because it requires a Gedanken Experiment which I wasn’t 100% sure about the outcome. The question is:

1. Can one make waves on a wave and see them propagating along the wave?
2. Can two pilots traveling in an open cockpit at the speed of sound hear each other speaking?

These seemingly simple questions (for a fluid dynamicist) bothered me a lot. If the two pilots could not hear each other that would imply that the speed of light is actually faster (sqr(2)* c) than the observed speed of light.

When I first derived the “solution” to this problem I simplified the problem just by considering a static hyperspherical reference frame. That introduced another constant into the problem (pi/3). The reason was the at any given Cosmological Angle, light travels in straight lines through the 4D spatial manifold, while we traveled along an arc. By the time light traveled all the distance to the 4D Center of the Universe, we would have traveled around the arc on an equilateral triangle inscribed inside a circle. I have to confess that this seems to be an artifact of how I tried to solve that problem. It gives a slightly older Universe than what we obtain when using the proper geometry:

The full treatment of the Pioneer Anomaly problem doesn’t seem to require that constant.

Figure 1. Looking deep into the past.

This picture is quite telling. It shows the L0 radius of curvature of the time when the first stars lightened. At that time and at that radius of curvature, There was light… It is important to notice that light always travels at 45 degrees with the Fabric of Space – 45 degree with the tangential line at point A.

Under those conditions one can write the following equations for the relating L0, R0, Beta, Lambda_Obseved, C_Observed – yes even the speed of light depends upon what are you looking at…

Let’s start with the speed of light. Normally we are observing things close, in our neighborhood… solar system, galaxy… Under those conditions, beta is 45 degrees and point A is almost right radially aligned with B. The two hyperspheres look like two hyperplanes and can be represented by the figure below:

Figure 2. Pioneer Radar Gun Experiment. Looking into a few minutes in the past.

This picture shows the Pioneer experiment. Light, microwaves leave Earth on point A and bounce back from the Pioneer spacecraft on point B. This all happens as the Hyperspherical Ligthspeed Expanding Universe expands at velocity C_observed. The observed value for the speed of light is c, what we are used to think at the velocity of light. It just happens that the observed velocity depends upon the angle of observation. If you are observing the beginning of times, them the angle Beta will be very small, otherwise it will be around 45 degrees.

where beta ranges from 45 degrees to zero degrees

is the actual speed of light which is faster than the speed of light we are used to… by a sqrt(2) factor.

Similarly, the wavelength of light depends upon what is the angle beta of observation.

This is expected, since the actual frequency of light shouldn’t change. This also means that Hubble constant should depend upon how you calibrate it. Calibration is done using parallax and red shift or stellar candles and red shifts. These two calibrations are inconsistent since they look at different angles. One should use the near field calibration and then extrapolate it using the hypergeometrical argument.

Hubble equation relates distance with the receding velocity:

The Shifted Frequency is related to Hubble constant by the following equation:

Doppler effect equation.

The value of H is obtained when L is Ro. Under those considerations v becomes Cactual:

In summary, here we presented a geometric model for the Doppler Shift, that is, the change in wavelength has only to do with the 4D angle of observation. This angle cannot be drawn within the 3D Hypersphere since when looking into the past, one is looking outside the hyperspherical hypersurface.

Numerically, the modeled Hubble coefficient is a match to the experimental Hubble coefficient.

H=2.0637E-18 s^-1

Cheers,

MP

PS – The most recent observational determination of the proportionality constant obtained in 2009 by using the Hubble Space Telescope (HST) yielded a value of H₀ = 74.2 ± 3.6 (km/s)/Mpc.^[4] The results agree closely with an earlier measurement of H₀ = 72 ± 8 km/s/Mpc obtained in 2001 also by the HST.^[5] In August 2006, a less-precise figure was obtained independently using data from NASA’s Chandra X-ray Observatory: H₀ = 77 (km/s)/Mpc or about 2.5×10⁻¹⁸ s⁻¹ with an uncertainty of ± 15%.^[6] NASA summarizes existing data to indicate a constant of 70.8 ± 1.6 (km/s)/Mpc if space is assumed to be flat, or 70.8 ± 4.0 (km/s)/Mpc otherwise.[7]

1 megaParsec = 3.08568025 × 1022 meters

1megaparsec/Km=3.08568025 × 10¹⁹

Thus

H=2.0637E-18 s^-1=2.0637E-18 s^-1* 3.08568025 E19 km/s/mpc=63.63 km/s/mpc

Where discrepancies are likely due to NASA averaging values from different epochs (beta angles).