Fruechte Gravity Theory Blog

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.

Ref: https://www.fruechtetheory.com/blog/2022/02/14/spin-drive/

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 S_σ → a_σⁿ S_σ” ([4], pg. 162) provides a matrix theory to the Coulomb field, with “cyclic groups (σ_i) of orders n_i” ([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

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, … , τ_k” ([1], pg. 191).

Ref: https://www.fruechtetheory.com/blog/2008/06/28/gravity-and-the-uncertainty-principle-2-2/

In an orbital arc the “endomorphisms A¹, … , A^k 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, … , A^k is not pulsed Lie groups in the gamma ray field, because there is no “…” after the A^k. In the same paragraph it talks about a “mapping ξ → A_ξ“, 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”:

https://www.fruechtetheory.com/blog/2022/08/27/concentrated-group-action/ ,

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

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.”

Ref: https://www.fruechtetheory.com/blog/2023/01/17/action-of-the-electric-field/

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

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 T_s^r “ ([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 + … +s_k” ([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

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

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 “M_W = 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 “M_Z = 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).

Reference: https://www.fruechtetheory.com/blog/2023/03/17/nuclear-ideals/

The “renormalized charge e_r” ([1], pg. 336) and “renormalized mass m_r” 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

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

Let us say there is an imaginary horizontal disk centered on a vertical, unshielded emitting antenna. Cosine waves of various frequencies and amplitudes go out in all directions centered on the disk. As a cosine wave travels away from the disk, it imbues EM waves of the same frequency perpendicular outward to the electric part of the cosine wave in a push. As the cosine wave comes back toward the disk, there is no push, preventing double signals. Each torus grows continually until it runs out of momentum, and in a complex set of signals there are many interspersed tori.

The amplitude of each cosine signal, as it multiplies, may not be constant throughout the torus, though frequency is. For a given location of a receiving antenna, the amplitude ratios of all the signals are the same.

By the 80:20 rule, 80% of a cell phone, radio, or television EM wave travels through the h field, and 20% travels by using the g field. According to Morse Theory, 100% travels through the h field:

“ h_– + n(I) = dim H ≤ a(I) = i(I) + n(I), so h_– ≤ i(I) “      ([1], pg. 233)

This is from the proof of Theorem 6, and in the next section it is written: “the Morse index theorem says that the inequality of theorem 6 is an equation.” ([1], pg. 233) With the torus action: “t passes from 0 to b” ([1], pg. 234) in the positive and negative directions. What is meant by augmented index, a(I), is that the cosine waves, as they are emitted, go out horizontally in all directions from the antenna.

In outer space, the way signals can travel long distances, the Morse index theorem comes very close to reality. In earth’s atmosphere the tori run out of momentum faster.

[1] 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

An electric field of an electromagnetic wave does the work to extend the magnetic field of the same wave. What makes the electric field turn around must have something to do with running out of energy to extend the magnetic field further. Griffiths says: “Magnetic forces do no work” ([1], pg. 207), and that is why it is said that transmission of the Coulomb field is “a diffeomorphism on the electric fields of the gamma rays”:

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

Magnetic fields can act as guides however, and can help hold together a groupoid in the gamma ray field so it can act transitively. There is “energy stored in the magnetic field” ([1], pg. 317] and “Magnetic forces may alter the direction in which a particle moves, but they cannot speed it up or slow it down.” ([1], pg. 207) It is the same in Coulomb groups, spherical or concentrated, that carry the Coulomb field, – there are electric currents that are altered in direction by magnetic fields. Another example of this is gravitational lensing.

An involution may be a charged particle, or nucleus, with mass, as it absorbs gravitons for the energy to send out Coulomb groups, or it may be a Coulomb group itself in an open field. As a spherical group travels, for example, it takes on new gamma rays and leaves some behind, and the new gamma rays may be called an involution as they become part of the Coulomb group.

When it is said that with Coulomb phonon transmission, the gamma rays are “frozen in time” up to “10 meters at least”:

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

it is in relation to travel, though they may travel a miniscule amount. It is torsion that transmits the Coulomb field, and the angular velocity, ω, is higher the stronger the field.

In a Cartan decomposition, “g₁ = t₁ + p₁ and g₂ = t₂ + p₂“ ([2], pg.517), p is the peak point of the electric field of a graviton. In a Riemannian globally symmetric space of type I, p follows the peak of a sine wave, and it also follows the peak in a Riemannian globally symmetric space of type II.

[1] Griffiths, David J., “Introduction to Electrodynamics”, Prentice Hall, 1999

[2] Helgason, Sigurdur, “Differential Geometry, Lie Groups, and Symmetric Spaces”, American Mathematical Society, 2012

When a molecule is formed, each nucleus senses the one(s) closest by its spherical pulses. Then each nucleus starts sending out alternating concentrated groupoids toward the nearest nuclei in the molecule.

In a Coulomb attraction, the groupoid decides how to bisect by the spin of a target. The two brackets then compress against other gamma rays and subsequentially spring back and squeegee along the backside of the target in what is called a pullback. Past the target, the brackets “re-emerge as action morphisms of Lie algebroids” ([1], pg. 152), and join a spherical group.

The scalar potential has units of J/s, which is energy per time. The electric field has units of N/C, and Force = mass x acceleration per Newton’s second law. The acceleration is less for a larger mass of charge, and there are neutrons in most nuclei which makes the effect greater. The electric field travels faster the denser a gravitational field is, though the speed difference may not be discernable.

We can have “a π-saturated open set” ([1], pg. 97) with “saturated local flow”, though the gravitons will be at various phases on sine waves when an electric field comes through. Thus, in terms of analytic coordinates, “such coordinates do not usually exist for Lie groupoids.” ([1], pg.  pg. 142) 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 said earlier, Coulomb repulsion acts on the frontside of another charge. The electric field travels much faster than the charged mass it pushes, in part due to inertia, so likewise, after the push, the brackets join another spherical group behind the target. A nuclear concentrated groupoid may join a spherical groupoid once it passes a target.

In both cases, Coulomb attraction or repulsion, the spherical group from which the brackets came mends itself.

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