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.

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

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

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

Mar 29 2022

## Transmission of the Coulomb Field

The gamma ray field we live in is extremely rich and dense.  For the forms we find in nuclei and assorted particles, there is all the energy needed to drive all physical processes.

A Calabi-Yau shape within a nucleus or particle needs an external energy supply to maintain it. Gravity provides the energy. Here we are talking about force and pressure within a nucleus or particle, with only indirect connection to the outside, or connection at a point, curve, or surface.  There may also be tears joining and reforming.

Occasionally we refer to neutrons, protons, electrons, and nuclei.  A proton can be a hydrogen nucleus, though we list it separately when we talk about free protons, such as in the solar wind, particle colliders, or elsewhere.  Let’s take an Oxygen nucleus for example with the makings of 8 protons and 8 neutrons. Inside the nucleus, at the top, parachutes with baskets attached through ropes, or strings, instead of a parachutist, may cause some gravitons to loop around the insides of the parachutes, or branes, and into the baskets with enough force to hold the parachutes against the highest flux density of gravitons. Then the gravitons would find ways to tunnel through the baskets, pushed from behind.  In the motions of O2 in air, the parachutes may slide around to stay opposite the maximum flux.

This may also help explain weak interaction parity violation, because as an electron forming within a nucleus tries to escape, out the bottom is easier, due to escape out the top involving going through the gaps in the parachutes.  More than 50% would come out downward.

The manifold of the sun’s gamma ray field, the manifold of the earth’s gamma ray field, and likewise with other celestial bodies, provides a combination of symmetric spaces. During the day, at noon let’s say, the vectors of the sun’s manifold are in the opposite direction as the vectors of the earth’s terrestrial manifold. The Coulomb field uses all vectors of all manifolds to propagate, because all vectors, within a distance of 10 meters at least, are frozen in time for phonon transmission.

Let’s say M1 is the earth’s manifold, and M2 is the combination of the earth’s and sun’s manifolds. “…a diffeomorphism F: M1 → M2 of manifolds oriented by Ω1, Ω2, is orientation-preserving if F*Ω2 = λΩ1, where λ > 0 is a C function on M.” ([1] pg. 209) In our example here, λ > 1, and we have neglected the earth’s moon for simplification.

We may call a negative charge a left coset space, and a positive charge a right coset space. Each creates its own homomorphism in the dense gamma ray field, by a diffeomorphism on the electric fields of the gamma rays.  For one thing, there is circular polarization. For another, perpendicular to the greatest flux density of gamma rays the electric fields of the gamma rays may have skewed sine wave lobes, somewhere between a normal sine wave and a sawtooth. The Coulomb field acts tangent to the R vector sphere, and “(∇XY)p depends not on the vector field X but only on its value Xp at p.” ([1] pg. 309]  The way that the Coulomb field transmits radially is by centrifugal force through the gamma ray field.

The inside of an atom may be called a geodesic.  An electron path in an atomic orbital may also be called a geodesic, and “a long geodesic may not be minimal.” ([2] pg. 62)  This is due to the Lorentz force:

https://www.fruechtetheory.com/blog/2010/12/23/electron-orbitals-and-the-lorentz-force/

Gravity is an integral manifold.  Each orbital arc is a line integral absorbing gravitons.  The Coulomb field, on the other hand, is a charge induced diffeomorphism in the gamma ray field. Substantially outside of neutral atoms there is a propensity for positive and negative charges to cancel, though in the near field we have van der Waals forces.

Phonons for the Coulomb interaction are generated inside a charge.  The field created, that acts on another charge, may act on the outside of another charge, possibly only 5% of the diameter deep.  The fields may also act in the interspace, producing backflush to the charges that generate the fields.  Phonons of opposite chirality attract, and of the same chirality repel.

As points meet for the Coulomb force, the acceleration would be periodic, and relates to the vector potential.  A Fourier Series can be applied to the vector potential, with the direction of force being the side of the ‘x’ axis where the sine or cosine function has larger lobes.  Often a geodesic is called piecewise smooth, due to gravitons being separate, though on a classical scale the motion is smooth.

Two electrons can occupy the same atomic orbital if they have opposite half-integer spin projections.  This is the Pauli exclusion principle.  In terms of tensor math, “the subspaces are mutually orthogonal and each is a nontrivial irreducible subspace.” ([1] pg. 242)

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

[2] J. Milnor, based on lecture notes by M. Spivak and R. Wells, Morse Theory, Princeton University Press, 1969

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

Feb 09 2022

## Core of an Electron or Proton

We can see from the calculation of the diameter of a free electron that as the density of the gravitational field goes down, the diameter increases.  This would be because of less gravitational pressure on the outside of the electron.

As gravitons enter a proton, electron, neutron, or nucleus, as conjugate waves or to take residence, the buildup takes on a fuzzy look that makes them look larger.  If we take a core diameter of 1.3335 x 10-15 m, the part that produces the fundamental charge, and add one graviton wavelength, we arrive at 5.30 x 10-15 m diameter, which is close to the classical diameter of the electron, 5.64 x 10-15 m *.  One graviton wavelength is used because one-half wavelength is on one side of the electron and one-half wavelength of a different graviton is on the other side.

We may call these outer layer gravitons tentacles or strings.  When nuclear fission occurs, the de Broglie wavelength of a neutron can come in at an angle where the strings on each entity hook and help pull the neutron into the nucleus. The cross section for this process is larger for slow neutrons vs fast neutrons in part because of the longer de Broglie wavelength.

* Jackson, J. D., Classical Electrodynamics, Third Edition, c. 1999 John David Jackson, John Wiley & Sons, Inc., p. 695

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