Mar 19 2024

## Spherical Groups

With spherical groups of opposite charge signs, like emitted by the proton and electron in hydrogen, why would they not annihilate each other at different spots? For one thing they are segmented and pulsed, and spread out, and come out of a charged mass in pairs. These are convex regions, and “A region X open or closed, will be called convex if any two points in X are joined by at least one path which does not leave X.” ([1], pg. 7)

Additionally, “Let X be any open region and its closure.” ([1], pg. 7) It is like the groups have AI, and know how to avoid each other and know how to come back together. In footnote “*” on page 7, attributed to K. Menger, “We may think of X as filled with substance which conducts light along paths, all the space except X being opaque.”

At least in atoms or molecules, at close range, the groups are strong enough to avoid each other, before the polarization factor takes over. As far as hitting a target, the Coulomb field travels at the speed of light squared, and more groups come through fast.

[1] Whitehead, J. H. C., “CONVEX REGIONS IN THE GEOMETRY OF PATHS”, Princeton Press, (Received 15 August 1931)

Jan 20 2024

## Conical Angle

On page 9 of the April 2007 booklet, it is proposed that the magnetic field of the electron guides the rectification of incoming gravitons “within some inclusive conical angle that is greater than zero”. As the electron grows in an atomic or molecular orbital, the magnetic field of the electron becomes stronger.

The Cauchy integral formula is used in many applications. Here we are applying it to gravity.

Since z is the direction straight out of the earth, let us call f’’(z) the gravitational force. “The Cauchy integral formula in the theorem in Sec. 50 can be extended so as to provide an integral representation of derivatives of f at z0” ([1], pg. 165). Then the formula becomes:

f(z) = (1/2πi) ∫c [(f(s) / (s – z)] ds       ([1], pg. 166, formula (1))

Let us say z is a point at the center of the 3-dimensional electron, and (s – z) is the spin radius that starts off at 6.6676 x 10-16 m immediately after a spin flip at the end of τi. Let f(s) be the function that grows (s – z) as the orbital electron absorbs gravitons. Keep in mind that the conical angle grows also.

f’(z) is the rate at which the orbital electron absorbs gravitons, since an arc τ is often not directed at the center of the earth:

f’(z) = (1/2πi) ∫c [(f(s) / (s – z)2] ds       ([1], pg. 166, formula (2))

f’’(z) is Newton’s second law of motion, F = ma:

f’’(z) = (1/πi) ∫c [(f(s) / (s – z)3] ds       ([1], pg. 167, formula (4))

Since the electron is perfectly round, (s – z) still starts off at the radius of 6.6676 x 10-16 m at the beginning of an arc. As the electron grows in size in an orbital, at some point it is able to produce Pontrjagin classes, or higher k-planes, as long as f(s) is strong enough.

[1] Brown, James Ward and Churchill, Ruel V., “Complex Variables and Applications”, McGraw-Hill Higher Education, c. 2009

Dec 22 2023

## k-planes

As presented before, a magnetic field can bend a charge path, but not speed it up or slow it down. k-planes are produced by magnetic fields, whereby they manipulate the direction of electric fields. Due to a charged mass, k-planes are produced, and this means that electric fields of gravitons are bent into the same plane as the vector potential comes through. A vector potential can be a charged mass or a group traveling through the gamma ray field before acting on another charge.

When two graviton electric fields are combined it is called a “k-th Chern class ck(E)” ([1], pg. 309) and a “2k form γk“. When a “4k-form βk” ([1], pg. 309), it is called a “Pontrjagin class”. We may think of a Pontrjagin class as a flat picture of a mountain range with 4 mountains in the picture. It is not a sine or cosine curve, but has 4 lobes. The curve that defines the tops of the mountains can be thought of as a string. Statistically, lobes may be combined at times. There are “two elements (A, p) and (B, q)” ([2], pg. 216), and q is the distance the p moves laterally to help form a hypersurface k-plane. For example, there is a “2k-form on P” ([1], pg. 293) for a Chern class, and in any k class there is “oriented p-planes in Rp+q” ([1], pg. 271).

k-planes are created to do heavy work, and are parts of larger Lie groups which determine the chirality. As the group approaches another charge, it determines whether the charge is positive or negative. If it is of the same charge sign, the Lie groups instruct the k-planes to slap the target charge on the face. If the target is the opposite charge sign, the k-planes split, spin around, and slap the target charge on the back. After 18 years, we really need the String Theorists working on the mathematics of this.

The k-planes take up new gravitons quickly and leave others behind. It is its own entity within a Lie group, and “(g, h, σ) is effective” ([1], pg. 249), σ being the effect of the magnetic field.

[1] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

Oct 27 2023

## Pauli Exclusion Principle

With the Pauli exclusion principle, “Only two electrons (with opposite spins) can occupy a given quantum state.” ([1], pg. 798) The magnetic moments being in opposite directions help keep the separation, though it is electric current loops in each electron doing the work.

The reason orbitals in an atom or molecule are limited to two electrons has to do with the fact that “σ, τ are any (local) bisections of G”. ([2], pg. 28) In mechanical engineering, σ is the symbol for stress. Likewise, σ is the symbol here for local negative charge stress in the gamma ray field, and we have “The map φ: σ → γ” ([3], pg. 226). Alternatively, there is “the sheaf of germs of local bisections of G.” ([2], pg. 133) Here we have another name for the graviton in “germ”, and a “sheaf” is a member of the h field, when two or more gravitons get combined to do the work.

With the spin of the electron, “the mapping Sσ → aσn Sσ” ([4], pg. 162) provides a matrix theory to the Coulomb field, with “cyclic groups (σi) of orders ni” ([4], pg. 130). This provides additional proof that Coulomb groups are pulsed.

Two other authors call the Pauli exclusion principle a “set of pairs (τ, J)” ([5], pg. 68), with “permutations σ” ([6], pg. 293). J is a charged mass:

https://www.fruechtetheory.com/blog/2023/06/29/vector-bosons-and-other-fleeting-field-particles/,

and in this case we are referring to an electron.

Physicists already knew most of this blog entry before it was entered. What many people do not know is the presence of a gamma ray field, though it is reasonable to know because of the gamma ray telescopes.

[1] Tipler, Paul A., “Physics”, Worth Publishers, Inc., 1976

[2] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

[3] Bishop, Richard L. and Crittenden, Richard J., “Geometry of Manifolds”, AMS CHELSEA PUBLISHING, Copyright 1964 held by the American Mathematical Society. Reprinted with corrections by the American Mathematical Society, 2001

[4] Zassenhaus, Hans J., “THE THEORY OF GROUPS”, Dover Publications Inc., 1999 (Originally published by Chelsea Publishing Co., 1958)

[5] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

[6] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

Oct 16 2023

## Atomic and Molecular Electron Arcs

Further to uniting Riemannian geometry, Lie groups, and symmetric spaces with gravity, τ is an atomic or molecular arc, and “τ is a segment” ([1], pg. 168). Also, “τ is minimizing” ([1], pg. 166).

Sometimes τ is called a complete orbital, and we “divide τ into a finite number of arcs, say, τ1, τ2, … , τk” ([1], pg. 191).

In an orbital arc the “endomorphisms A1, … , Ak are linearly independent” ([2], pg. 353), and k – 1 in this instance is the number of gravitons absorbed in an arc. “A” is the vector potential, and each time an electron absorbs a graviton in an orbital, its vector potential increases. We know that A1, … , Ak is not pulsed Lie groups in the gamma ray field, because there is no “…” after the Ak. In the same paragraph it talks about a “mapping ξ → Aξ“, therefore in a particle mass, and in groups or manifolds in the open gamma ray field, the gamma rays are blended and surjective.

If a function can be called “the growth of an orbital electron in size and charge”:

then “γ and f point in opposite directions” ([3], pg.165).

The Φ field is within atomic and molecular orbitals, including the boundary, and Ψ is outside of the orbitals. In an emitting antenna, it is the Ψ field as well, since the electrons are free. “Φ0 is isomorphic to Ψ0 in a natural manner” ([1], pg. 193), because the gamma ray field is normally constant in the area within and around the molecule.

Often in a molecule, or any type of p orbital, the Gaussian curvature, when ¾ through the arc compared to ¼ through the arc, is negative.

In the open gamma ray field “m = dim M and n = dim Δ” ([4], pg. 155), and m – n is the number of singularities in a locality. Stoker terms it “singularity in the coordinate system” ([5], pg. 84). A singularity is when the electric and magnetic fields of a gamma ray cross over the t axis, though when near the axis it could be called a singularity also.

If the polarization factor is greater than 2, as at the surface of the sun or Jupiter, then specific nuclei likely have more mass than on the face of the earth, and electrons in atomic or molecular arcs grow larger. It could be because of these factors the value of Newton’s gravitational constant G = 6.672 x 10-11 (N-m2)/ kg2 stays the same.

[1] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[3] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

[4] Boothby, William M., “An Introduction to Differentiable Manifolds and Riemannian Geometry”, Academic Press, 2003

[5] Stoker, James J., “Differential Geometry”, John Wiley & Sons, Inc., c. 1969

Sep 30 2023

## Michelson-Morley experiment

It has been said that “The electric field travels faster the denser a gravitational field is, though the speed difference may not be discernable.” Also: “What we have is an infinitesimal zigzag pattern, though when we back out to the classical level, it does not matter for any application.”

As it turns out, the speed difference may have been indirectly discerned by the Michelson-Morley experiment in 1887, and the “infinitesimal zigzag pattern” is less for a denser gravitational field.

A picture and description of the Michelson-Morley experiment tells it was mounted on a large block of sandstone, for stability, and floated on an annular trough of mercury for rotation.

The block that the experiment was on, and the sensors and brackets in the forward direction of travel of the whole apparatus, would have helped decompress the gamma rays toward the center of the apparatus, near the forward brackets. The brackets behind would have compressed the gamma rays. What was probably happening was a slow-fast travel of the electromagnetic waves in one direction, and a fast-slow travel in the opposite direction.

ηi is when the center t of a gamma ray moves one way in an alternative direction. ξi is a smooth electric field, and ηi and ξi work together to smear electric fields of the gamma rays into electromagnetic waves of larger dimensions, as a laser, emitted cell phone wave, etc. As a tornado takes up air molecules and expels others, these waves of lower frequency than a graviton take up gravitons they reach. “Assume that ξ is affine” ([1], pg. 377), and “ξ is a (column) vector in Rn” ([2], pg. 269). The gravitons that are expelled can take off in almost any direction. “ξi and ηi are orthogonal” ([3], pg. 315], because they are independent. “t” is not an electric field, it is a singularity.

ξi is a member of the H field, though it is specific to an emitted wave, subject to “the compatibility conditions which ξ and H are obliged to satisfy” ([1], pg. 362), and “ξ is a pure translation” ([4], pg. 193).

In the Michelson-Morley experiment, in the denser gamma ray field the ξ field is more efficient and moves faster, and the wavelength is slightly shorter than average. In the less dense gamma ray field, the wavelength is slightly longer than average.

[1] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[3] Stoker, James J., “Differential Geometry”, John Wiley & Sons, Inc., c. 1969

[4] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

Aug 03 2023

## Ricci tensor field S

When a free electron accelerates, it may be able to increase in mass, charge, and diameter for the increase in work it must do. Again, we think of an emitting antenna.

We know that “s” can stand for spin, and that electrons have spin. The electrons in the antenna may impart spin into a “tensor space Tsr “ ([1], pg. 209], where “r” is the vector away from the antenna, and “s” is the spin. What we can liken this to is a corkscrew in a gravitational field. Each corkscrew “s is a direct sum of simple ideals: s1 + … +sk” ([2], Appendix 5, pg. 279)

To send these corkscrews out in all directions from an antenna is a phenomenal amount of work. It is not absolutely necessary that accelerated free electrons expand for this to occur, though they would at least absorb gravitons at a greater rate than a free electron at rest or traveling at constant velocity in a straight line.

It is not known what percentage of these corkscrews would be left-handed. When two electrons are near each other, “(βi) is invariant by the left translation” ([2], pg. 207), and they repel each other.

Furthermore to the Ricci tensor field containing spin, there are the following two corollaries:

“Corollary 5.5   If M is a compact Riemannian manifold with vanishing Ricci tensor field, then every infinitesimal isometry of M is a parallel vector field.” ([2], pg. 251)

“Corollary 5.6   If a connected compact homogeneous Riemannian manifold M has zero Ricci tensor, then M is a Euclidean torus.” ([2], pg. 251)

As the output from an emitting antenna turns into tori, there is a “concatenation of paths” ([3], pg. 229].

Of course, as these tori wear out, they disintegrate, because of “a theorem of Weyl that any representation of a semisimple Lie algebra is completely reducible” ([2], Appendix 5, pg. 279]. We now know that these Lie algebras are reducible to gravitons.

[1] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

[3] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

Jul 06 2023

## Polarization Factor

It has been said that the Coulomb field transmits by torsion and centrifugal force:

https://www.fruechtetheory.com/blog/2023/01/29/magnetic-fields-as-effecting-coulomb-groups/

https://www.fruechtetheory.com/blog/2022/03/29/transmission-of-the-coulomb-field/

In the near field of molecules there are van der Waals attractive forces “proportional to 1/r7.“ ([1], pg. 390) This is a fast reduction, in part due to the polarization factor, which is very strong, though not as strong as the Coulomb force of a concentrated charge of mass when another concentrated charge of mass is nearby.

Kobayashi and Nomizu put it this way:

“Theorem 2.2.  Every Riemannian manifold admits a unique metric connection with vanishing torsion.” ([2], pg. 158)

Additionally, there is an “invariant Riemannian metric which is naturally reductive”. ([3], pg. 377)

It has been said before that gauge invariance is local at a given time. When it is referred to as a “unique metric connection”, the instantaneous density of the gravitational field at an area of the surface of any planet is unique.

One reason the potential energy of a point charge is not infinite was given here:

https://www.fruechtetheory.com/blog/2009/03/15/electric-charges/

In real time, the polarization factor is the reason the potential energy of a point charge is not infinite.

[1] Tipler, Paul A. and Llewellyn, Ralph A., “Modern Physics Sixth Edition”, W.H. Freeman and Company, New York, c. 2012

[2] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume I”, John Wiley & Sons, Inc., c. 1963

[3] Kobayashi, Shoshichi and Nomizu, Katsumi, “Foundations of Differential Geometry Volume II”, John Wiley & Sons, Inc., c. 1969

Jun 29 2023

## Vector Bosons and Other Fleeting Field Particles

In the case of “ ‘intermediate vector bosons’: the charged bosons W+ and W“ ([1], pg. 363), it is possible that they may be Coulomb field inductors. Since it takes many gravitons to make up an inductor, it is reasonable to measure “MW = 80.40 GeV” at a test setup sensor.

When two positive inductors meet head on in the open gamma ray field, they annihilate each other. The same is true for two negative inductors. If two inductors of the same charge annihilate at a sensor, the mass may be higher, as in “MZ = 91.19 MeV”, which may be the “neutral Z0 boson”. Some extra gravitons may get into the act here, and since the vector summation of the gamma ray field is in the Z direction, this boson is appropriately named.

It is not being said that all field particles are due to the Coulomb field, just some of them. The time these particles exist is “ħ /mc2“ ([2], pg. 654). Mandl and Shaw allude that “some quantity in the vacuum is non-vanishing” ([1], pg. 404). It has been known since 2005 that the vacuum is not really a vacuum.

As far as J representing charged mass, Ryder mentions “The source J(t) plays a role analogous to that of an electromagnetic current, which acts as a source of the electromagnetic field.” ([3], pg. 175).

The “renormalized charge er” ([1], pg. 336) and “renormalized mass mr” can be thought of as the charge and mass of the electron immediately after a spin flip at the end of an atomic orbital arc. Somewhere in the middle of an electron orbital arc these are called “ ‘running mass’ and ‘running charge’ “.

Zee says that “free quarks have not been observed” ([4], pg. 377), though with the quark coincidence, something with the same energy has been observed with the gamma ray telescopes.

[1] Madl, Franz and Shaw, Graham, “Quantum Field Theory”, John Wiley and Sons, Ltd., 2011

[2] Krane, Kenneth S., “Introductory Nuclear Physics”, c. 1988, John Wiley & Sons, Inc.

[3] Ryder, Lewis H., “Quantum Field Theory, Second Edition”, Cambridge University Press, 1996

[4] Zee, Anthony, “Quantum Field Theory in a Nutshell”, Princeton University Press, 2010

Mar 17 2023

## Nuclear Ideals

On April 10, 2008, it was put forth that “we would have gravitational energy replenishing Coulomb energy through both the electrons and the nucleus of an atom”, through absorption of gamma rays:

https://www.fruechtetheory.com/blog/2008/04/10/the-nucleus-and-gravitons-2/

It wasn’t until 2022 that mathematics was applied to the process, and of course free nuclei and electrons would be absorbing gravitons as well to produce Coulomb fields.

In Krane’s work, the section on Spin-Orbit Potential starts on page 123. There, in the language of atomic physicists applied to nuclear physics: “total angular momentum j = l + s” ([1], pg. 124). In the same way, in the language of differential geometry, Lie groups, and manifolds, J can be called a nucleus, or “an isometry J of N” ([2], pg. 207). There is a “closed embedded submanifold N ⊂ M” ([3], pg. 165), and “N is closed in M” ([2], pg. 236). In this case N is the space atoms and molecules take up, to the outer reaches of the electron orbitals.

In “its ideal system J = (J, R(f), θ) is uniform” ([3], pg. 170). This infers that each different nuclei has a unique structure, and R(f) is the radius of the nuclear fibration. Since nuclei are normally parts of systems of atoms and molecules, θ refers to the angle of an electron arc, and each nucleus knows when to send spin flip signals for each electron arc for which it is responsible. The structure of nuclei is so complex that “covariant derivatives of J vanish” ([2], pg. 142).

As far as holding nuclear material and groups together, magnetic fields are mostly responsible, and serve as the function of chirality in Coulomb fields. As an example, “inductors correspond to, and characterize, pullback groupoids” ([3], pg. 83), as put out from a charged mass. Also, “(φ,f) is an inductor if φ!! is a diffeomorphism” ([3], pg. 176).

[1] Krane, Kenneth S., “Introductory Nuclear Physics”, c. 1988, John Wiley & Sons, Inc.

[2] Bishop, Richard L. and Crittenden, Richard J., “Geometry of Manifolds”, AMS CHELSEA PUBLISHING, Copyright 1964 held by the American Mathematical Society. Reprinted with corrections by the American Mathematical Society, 2001

[3] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society

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