Archive for the 'Mathematics' Category

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

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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.

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 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

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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).

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

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”:

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

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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.”

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

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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

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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

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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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Mar 06 2023

Emitting Cell Phone, Radio, and Television EM Waves

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

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Nov 17 2022

The Coulomb Gauge    

There is another name for a free graviton, – it is “the identity isomorphism idEx, here denoted 1x, and the elements 1x, x ϵ M, act as unities for any multiplication in which they can take part” ([1], pg. 4). We see that unlike π, idEx has some degree of circular polarization and/or skewed sine waves. In some writing instances π is the same as idEx and I am not trying to dictate how they should be used.

In the “Coulomb, radiation, or transverse gauge. This is the gauge in which ∇ · A = 0” ([2], pg. 241), we have a classical description. In the tensor sense, we have the forms Χij. The direction we choose for Χ is always transverse to the radial electric field at a chosen point, and the coordinate frame Ui is picked centered on the same point, creating a k-plane. We have that “The forms Χij are the transition forms for the Lie algebroid atlas {Ui, ψi, Θi}” ([1], pg. 206), and Θi varies with the density of the gamma ray field:

https://www.fruechtetheory.com/blog/2022/10/05/the-vector-potential/

Considering the transition form TP/G [1], we may here call G the density of the gravitational field. It is seen that as the density goes up the transition angle Θi decreases for a given charge and distance from the charge.

In Jackson’s problem 6.19 (b), “the original and space-inverted vector potential differ by a gauge transformation” ([2], pg. 291). Though the earth catches some of the sun’s gravitons all the time, the sun’s gravitons during the day are greater at the face of the earth than at night, and inverted, changing the Coulomb gauge.

With the “Lorenz condition (1867), ∇ · A + (1/c2) ẟφ/dt = 0” ([2], pg. 240), it is mathematically shown that the system {Ui, ψi, Θi} acts fast compared to the gradient of A, and
           ιX (φ ˄ ψ) = ιX(φ) ˄ ψ + (-1)i φ ˄ ιX(ψ)              ([1], pg. 306)
Also, as small as gravitons are, we may as well call the k-planes “flat connections Θi“ ([1], pg. 206).

Since we have “t the fixed point set of θ” ([3], pg. 401), t is on the center line of a gamma ray, and “g0 = t0 + p0 is a Cartan decomposition of g0“ ([3], pg. 184). In certain situations the center can shift as well, in which case “c0 is the center of t0” ([3], pg. 452) as t0 moves back and forth.

With the polarization factor, it is interesting to call h the vector summation of two gamma ray electric fields. When a gravitational field is yet more compact, h is the summation of more than 2 electric fields, so that “f: MH be a smooth map” ([1], pg. 183), and “Let h be a proper subalgebra of g of maximum dimension” ([3], pg. 160).

Incidentally, the identity isomorphism reminds us of quantum 1:

https://www.fruechtetheory.com/blog/2009/09/16/the-fundamental-quantum-unit/

[1] Mackenzie, Kirill C. H., “General Theory of Lie Groupoids and Lie Algebroids”, c. 2005 Kirill C. H. Mackenzie, London Mathematical Society
[2] Jackson, J. D., “Classical Electrodynamics, Third Edition”, c. 1999 John David Jackson, John Wiley & Sons, Inc
[3] Helgason, Sigurdur, “Differential Geometry, Lie Groups, and Symmetric Spaces”, American Mathematical Society, 2012

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Oct 05 2022

The Vector Potential

In electrodynamics we find that “A quantum-mechanical description of photons necessitates quantization of only the vector potential” ([1], pg. 242), as in the summation of all the manifolds of gravitational fields at a given location. In a more densely packed summation of manifolds, the action of an electric charge will have a lesser rotational effect on the electric fields of the gamma rays than on a less dense field. The power of the rotation is the same in either field however, as long as we are referring to a gravitational field that is not too sparse for electric fields to propagate.

“The definition of B = ∇ x A specifies the curl of A, but it doesn’t say anything about the divergence – we are at liberty to pick that as we see fit, and zero is ordinarily the simplest choice.” ([2], pg. 235) The reason we may pick the divergence as zero is that the manifolds “are frozen in time for phonon transmission”:

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

As far as group action, Mackenzie [3] calls these “groupoids”, such as an ellipsoid, a spheroid, or another 3-dimensional shape. The definition of a spheroid I find is that it is like a sphere, but not a perfect sphere, and in the present case we have “oscillations and accordion motion in multiple axes”:

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

On a side note, though related to manifolds of gravitational fields, the Nobel Prize in Physics is being given this year for essentially this:

https://www.fruechtetheory.com/blog/2014/05/30/quantum-entanglement/

[1] Jackson, J. D., “Classical Electrodynamics, Third Edition”, c. 1999 John David Jackson, John Wiley & Sons, Inc
[2] Griffiths, David J., “Introduction to Electrodynamics, Third Edition”, c. 1999, Prentice-Hall, Inc.
[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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