Jun 03 2024

Isometry and Homotopy

In Chemistry, “This phenomenon of two or more compounds having the same molecular formula but different structures is called isomerism.” ([1], pg. 405) In a positive charge groupoid, traveling through an open gamma ray field, coming from a given nucleus, there is an isometry. Though the homotopy comes across “an m-dimensionable manifold M of class Cr” ([2], pg. 646), the gravitons have different phases as the groupoid travels on, in an outer tangent space, and “the homotopy problem is equivalent to an extension problem.” ([3], pg. 175)

“16.2. LEMMA.  If (E,S) is a cell and its boundary, then (E X 0)(S X I) is a retract of E X I.” ([3], pg. 84)

Thus, “π is an infinite cyclic group,” ([3], pg. 199). Steenrod calls S “A system S of coordinates” ([3], pg. 22), or “a bundle of coefficients.” ([3], pg. 190) In this case, with “E X I“, I is an isotopy, and the zero space is the tangent space. There is “no left distributive law” ([3], pg. 122), because orbital electrons absorb gravitons at various rates.

It may be that the tangent space is all that the charged particle puts out, and that Lie groups form in the open gamma ray field.

[1] Hein, Morris, “Foundations of College Chemistry, Fourth Edition”, DICKENSON PUBLISHING COMPANY, INC., c. 1977

[2] Whitney, Hassler, “Differential Manifolds”, The Annals of Mathematics, Second Series, Vol. 37. No. 3 (Jul., 1936) pp. 645-680

[3] Steenrod, Norman, “The Topolgy of Fibre Bundles”, Princeton University Press, c. 1951

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

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

Jan 29 2023

Magnetic Fields as Effecting Coulomb Groups

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”:

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

Jan 17 2023

Action of the Electric Field

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

Dec 15 2022

Fusion

At Lawrence Livermore Laboratory a fusion reaction was produced using 192 lasers. By a factor of 1.5, more energy was produced than the energy put in by the lasers.
Early in my education at the University of Wisconsin – Madison, we learned of conservation of energy. If a reaction can absorb enough gravitons during a process, then it would appear that conservation of energy is violated, though it was not really violated.

Aug 27 2022

Concentrated Group Action

There is another slightly different view in which the Coulomb field transmits when it comes to nuclei as parts of molecules, and that is by pulsating, concentrated Weyl groups or frames of a small conical angle, toward another nucleus. It is not wholly different from the last blog entry because spherical pulses could also be seen as groups, and for a given charge these concentrated groups are in addition to spherical groups.

When an O2 or N2 binuclear molecule forms, or let us say a benzene molecule, each nucleus senses the other nuclei closest. This is a strong repulsion, so the nuclei may start sending out groups concentrated toward the other nuclei for efficiency, while the electron cloud in between the nuclei offers attraction and keeps the molecule from flying apart. This also changes the Calabi-Yau structures within the nuclei.

For Coulomb attraction, a frame may wrap around another charge. For Coulomb repulsion, there may be partial contact and some backflush. As two close nuclei in a molecule sense each other, there may also be alternating, concentrated, group pulses between the two. This is likened to a synchronization between the nuclei, without the need for backflush. With phonon transmission this is a very fast process and transmits without intercepting electrons in orbitals. When we compare the size of nuclei and electrons to molecular size, there is a lot of empty space filled with gravitons, so this synchronization is reasonable.

In larger nuclei there are more compact spaces and more affine connections between them. For a nucleus we may call these irreducible representations, where the exception is fission, as a “reduced root system in V” ([1], pg. 461). The nuclear charge manufactures springboard groups repeatedly, with oscillations and accordion motion in multiple axes. A stable nucleus in a molecule is an isomorphism, though we must be careful here because as orientations change, there may be slight changes in structure. Particle colliders are excluded from this discussion.

A nucleus consists of involutive automorphisms, the summation adding to the entity’s spin, as it absorbs gravitons for the energy to send out groups or frames. Boothby calls these “inner automorphisms of G” ([2]. Pg. 237). A Weyl chamber is part of atomic mass, while a Weyl group transmits as a packet in the not so compact space of the gamma ray field.

We see that “π is a continuous and open mapping.” ([1], pg. 120]. In certain areas of deep space we may call this Riemannian globally symmetric space I, with perfect sine waves and no circular polarization. “Riemannian globally symmetric spaces of type II” ([1], pg. 516) are due to the bi-invariant structure of the Coulomb field. In both cases there is a “strong orthogonality” ([1], pg. 576) which produces a polarization factor of 2, as used in the blackbody radiation formula and in G = 4hf/3. π may be called a free graviton, since it is one half wavelength long.

As far as the Coulomb field produced by an electron in an atomic or a molecular orbital: “Let N0 be a bounded star-shaped open neighborhood of 0 ϵ g which exp maps diffeomorphically onto an open neighborhood Ne of e in G.” ([1], pg. 552) Let e be the electron, star-shaped be the lobes of orbitals, and exp be the growth of an orbital electron in size and charge. The increase in size of an electron in the orbital enables it to absorb more gravitons at a given time, thus increasing gravitational pull in the second half of the arc.

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

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

Feb 11 2022

Isotropic Property of the Coulomb Potential

In the vicinity of where our machines have been, we know that electric current will flow in any designated direction and is not particular to the direction of the highest flux density of gravitons.

For various reasons, we cannot have protons and electrons continuously flipping, – the Stern-Gerlach experiment proves that they do not. There must be internal processes of the proton and electron which produce isotropic electric fields. Some of this was previously addressed in two blog entries:

https://www.fruechtetheory.com/blog/2010/12/09/field-line-curvature/

https://www.fruechtetheory.com/blog/2010/06/15/muonic-states/

It is possible that not all conjugate wave gravitons pass straight through a proton or electron, or that even with a free proton or electron that the gravitons leaving have just entered. Some may make a horseshoe pattern and come out near the same point entered. They may be able to come back out at any angle. As compressed as the gravitons become inside a particle, almost any shape can occur. Gravitational pressure dictates a consistent size of a free proton or electron.

With the flux density coming out of the face of the earth, we seem to have a conundrum with the idea of gravitational pressure, one side having much greater pressure than the other. Why do gravitons not burst out the top, resulting in particle collapse? It also begs the question as to why electrons are perfectly round, and not teardrop shaped:

https://www.fruechtetheory.com/blog/2011/05/28/free-electrons-perfectly-round-3/

Possibly, branes form at the top of an electron and reform in a spin flip.  These branes would be linked inside the particle so that they do not bust out, and may deflect some exiting gravitons at various angles. These branes may also help keep the electron round. Here we are designating “top” as away from the highest flux density of gravitons.

As far as isotropic fields, at this point we must say that it is designed internal to the proton or electron and is of consistent pattern.  The open field starts just outside the particle, so it is maintained that electric and magnetic fields transmit openly by “phase shift and chirality” or “phase shift and parity”.  The Coulomb force is considered instantaneous at reasonable distances:

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

It appears as though this is necessary, because then the speed that free gravitons travel at, the speed of light in a vacuum, does not effect the electric and magnetic fields generated.

Feb 10 2022

Unification

Inside a proton or electron, events may approach the Planck length.  The frequency of the waves would not change from that of a free space graviton, though wavelength and amplitude do change.

Waves inside a particle may make loops in certain circumstances, not necessarily around the perimeter, though internally, and required because of all the traffic.

Certainly, the ways these vibrations set up in a proton or electron determines whether we have a positive or negative charge.  If we did not have any loops and curves, the versatility needed would be hard to set up.  It is somewhat like a Hilbert space with wrapped up dimensions.

Put another way: “A string vibrating in one particular pattern might have the properties of an electron, while a string vibrating in a different pattern might have the properties of an up-quark, a down-quark, or any of the other particle species in Table 12.1.  It is not that an “electron string” makes up an electron …Instead the single species of string can account for a great variety of particles …” *

If you peruse this website, you will find other areas of unification.

* Greene, Brian, The Fabric of the Cosmos, c. 2004 Vintage Books, a division of Random House, Inc., p. 346-347

Aug 14 2011

Topology of the Vacuum

Whether we are looking at nuclear fission or the results of scattering experiments, the way spin-parity assignments are often kept in order in nature would be similar to the cause of a de Broglie wavelength.  Rotational states ratchet through the gravitational flux, with potential wells rising and falling in one of the most fundamental of quantum phenomenons that exist.

During and shortly after high flux, high velocity hadron collisions at Fermilab or the CERN LHC nevertheless, some of the scattering resonances seen may be due to a blitz through the gravitational field, not organized very well in a manner, for example, such as an electric field.  The static we typically see in Goldhaber plots generated from hadron colliding experiments may in part be due to a cascade of momentum generated through the gravitational field.

Another evidence of the gravitational field is the Bohm-Aharonov effect.  As Ryder puts it, “the Bohm-Aharonov effect owes its existence to the non-trivial topology of the vacuum, and the fact that electrodynamics is a gauge theory.  In fact, it has recently been realized that the vacuum, in gauge theories, has a rich mathematical structure, with associated physical consequences,” ([1], pg 101).

Astronomically, and for the sake of history, it is somewhat reminiscent of the luminiferous aether.

Another concept related here is that “the configuration space of the vacuum is not simply connected.” ([1], pg 102)  When we speak of ‘one loop’ consequences, we can liken it to the Cauchy integral, which Greiner calls “The surprising statement of the integral formula (4.16), namely, that it is sufficient to know a function along a closed path to determine any function value in the interior,” ([2], pg 109).  For those more willing to trust the mathematicians for pure math, the Cauchy integral formula is presented in Brown and Churchill:

f(z) = (1/2πi) ∫ (1/(s-z)) f(s) ds                     ([3], pgs 166 and 429)

With “the gauge invariance of electrodynamics” ([1], pg 97), the perfect balance of charge that exists in the near universe, – possibly the entire universe, and the quantum steps of the Coulomb force by phonon transmission, the Bohm-Aharonov effect does indeed show us that there are physical consequences to the vacuum that are non-trivial, relating to the gravitational field in which the Bohm-Aharonov test and other tests are set up and run.

As a final thought, it is probable that planar electromagnetic waves would not turn into spherical electromagnetic waves were it not for traveling through a gravitational field.

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

[2] Greiner, Walter, Classical Electrodynamics, First German edition, Klassische Elektrodynamik, 1991 Verlag Harri Deutsch. 1998 Springer-Verlag New York, Inc.

[3] Brown, James Ward and Churchill, Ruel V., Complex Variables and Applications, Eighth Edition, McGraw-Hill Higher Education, 2009

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