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

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

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

Jan 25 2021

## Final Mediator

Going back to the April 2007 paper once again: “… and the same Coulomb force being the final mediator of the gravitational force as proposed, the mass energy of the proton appears to be integral to gravitational field action.”, we can go further back in history to examine the path of how we arrived at this place.

In the scientific community, the cause of gravity is by many still considered as unknown.  As Hughes-Hallett et al. puts it in relation to the gravitational force: “How does acceleration come about?  How does the velocity change?  Through the action of forces.  Newton placed a new emphasis on the importance of forces.  Newton’s laws of motion do not say what a force is, they say how it acts.” 1

Newton himself says in The Principia: “…considering in this treatise not the species of forces and their physical qualities but their quantities and mathematical proportions, as I have explained in the definitions.” 2  In Definition 8 Newton says: “This concept is purely mathematical, for I am not now considering the physical causes and sites of forces.” 3

Roger Cotes, Plumian Professor of Astronomy and Experimental Philosophy at Cambridge University wrote in his Editor’s Preface to the Second Edition of The Principia: “But will gravity be called an occult cause and be cast out of natural philosophy on the grounds that the cause of gravity itself is occult and not yet found?” 4  Electromagnetic waves were not yet discovered in 1713 when Cotes wrote this, so it is difficult to imagine how anyone could have discovered the cause of gravity back then.

1  Hughes-Hallet, Gleason, McCallum et al., Calculus, John Wiley & Sons, Inc., 2005, p.304

2  Cohen, Whitman, Isaac Newton, The Principia, Mathematical Principles of Natural Philosophy, University of California Press, 1999, p. 588

3  Ibid., p. 407

4  Ibid,. p. 392

Apr 17 2020

## Russia’s DA-ASAT

You can read the news on this recently tested system for yourself. One concern is the extra amount of space junk it could produce. This web site tells how to pull down space junk, missiles, and satellites.

I really do not think that the USA would want to let Russia or China use www.fruechtetheory.com for weapons first, so the USA had better get on it!

Feb 09 2017

## E = mc^2

It occurred to me in January or February 2008, during my first foray into Quantum Mechanics, that the reason there is no 1/2 factor in front of mc^2 in Einstein’s formula E=mc^2, – like there is in the Newtonian formula for kinetic energy K. E. = (1/2)mv^2, is that there are gravitons inside a fundamental particle that are bouncing back and forth against gravitational pressure on the outside, which doubles the energy.

Jun 24 2014

## Macroscopic vs. Atomic G

Dan Fordice sent me two articles about newer experiments that were set up to measure the gravitational constant.  One of the articles referenced a paper by Tino et al. where the constant is determined using one mass type of hundreds of kilograms of tungsten, and the other being laser cooled rubidium atoms.  The apparatus involving the tungsten masses looks like it may be the same apparatus as was used for the Schwarz et al. experiment from 1998 [1].  The Tino et al. value for G is given as 6.667 x 10-11 m3kg-1s-2 [2] with statistical uncertainty and systematic uncertainty given in the paper.

When we have a macroscopic mass where atoms are chemically bonded, and masses are held together by various means, a gravitational field acting on one atom can have a component of force on another atom that is chemically bonded to it.  A single atom free of bonding to other atoms, on the other hand, has fewer instantaneous electron orbital path vectors than a macroscopic mass of several kilograms when we consider it as a whole.  Therefore one would expect that the value of G when measured on individual atoms would be lower than a conventional value of 6.672 x 10-11 m3kg-1s-2 [3].

The gravitational constant based on one third the mass of the proton is 6.6807 x 10-11 m3kg-1s-2, but does it ever get this high in reality?  Planets in orbit around the sun would get close to this value.

G = 6.672 x 10-11 m3kg-1s-2 is probably still a good value to use when considering macroscopic masses on the surface of the earth, or in the atmosphere, or in orbit around the earth.

[1]  Schwarz, Robertson, Niebauer, Faller, “A Free-Fall Determination of the Newtonian Constant of Gravity”, Science, 282, 2230-2234; 1998: http://www.ngs.noaa.gov/PUBS_LIB/BigG/bigg.html

[2]  G. Lamporesi, A. Bertoldi, L Cacciapuoti, M. Prevedelli, G.M. Tino, “Determination of the Newtonian Gravitational Constant Using Atom Interferometry”, http://arxiv.org/abs/0801.1580, 2013

[3]  Tipler, Paul A., Physics, Worth Publishers, Inc., 1976, inside back cover

Sep 01 2011

## Fermilab Magnets

With the Fermilab Tevatron shutting down this month, I wonder if its magnets could be used for a space debris vacuum.  The problem pops up in the news periodically, and did again today:

http://www.usatoday.com/tech/science/space/story/2011-08-31/Solutions-sought-for-growing-space-junk-problem/50207662/1

Jan 12 2011

## Gradient of a Scalar Potential

In Thomas’ Calculus we find the definition: “If F is a vector field defined on D and F = del f for some scalar function f on D, then f is called a potential function for F.” ([1], pg 921)

Kaplan states that the “gravitational field is the gradient of the scalar f = kMm/r” ([2], Prob. 4, pg 180), which for General Relativity is indeed the case.  With gamma ray energy exchange however, the scalar is f = k1M/r, in form much like the scalar potential k2q/r with electrodynamics.  This tells us that both gravitational and Coulomb forces are of independent action, and the corresponding fields are gradients of scalar potentials, which can be considered as more evidence for unification.

The vector fields we are looking at have units of N/kg for gravity, and of course E = N/C for an electric field.  A gravitational field calculation is most applicable when the object in the field is small compared to, or far away from, the source of the field, such as Jupiter compared to the sun.

Aside from the King James Version of the Bible, Wilfred Kaplan’s Advanced Calculus is still my favorite book.  By the way, we used Thomas’ Calculus book at the University of Wisconsin in 1976 and 1977.

[1] Thomas, George B., as revised by Weir, Maurice D. and Hass, Joel, Thomas’ Calculus, Twelfth Edition, Addison-Wesley of Pearson Education, Inc., 2010, 2005, 2001

[2] Kaplan, Wilfred, Advanced Calculus, Fifth Edition, Addison-Wesley of Pearson Education, Inc., 2003

Dec 14 2010

## Independent Action

In one of the FGT YouTube videos of February 2008 it was mentioned that unlike a tug of war where the tension in the rope is the same throughout when the rope is stationary, gravity is one mass acting independently on another.  In the case of two masses of spherical shape the force is known to be F12 = Gm1m2/r122, which we can equate to m1a or m2a, depending on whether we want to calculate the acceleration of m1 or m2, and where r12 is the distance between the centers of the two masses.  With planetary masses that are close to spherical, and the objects they pull, this formula has been used most reliably, while it has been known for a long time by learned mathematical and scientific people that shape matters, and using center of mass with the standard physics book formula can cause increased error in some instances.

As an analysis that has already been done, let us use Kline Chapter 16, Sections 5, 6, and 7.  Starting with Section 5, “Gravitational Attraction of Rods”, the example given is that of a “rod 6 feet long and of mass 18 pounds which is uniformly distributed” and “so thin that we think of it as extending in one dimension only.  Three feet from one end of the rod and along the line of the rod is a small object of mass 2 pounds which we shall regard as located at one point.”   Kline first calculates the force that the rod exerts on the 2-pound mass as though the entire mass of the rod were concentrated at its center, according to the standard F12 = Gm1m2/r122 formula, which comes out as equaling G poundals.  Then he does the calculation properly using an integration over the rod, showing that the force that the rod exerts on the 2-pound mass is actually (4/3) G poundals.  In Section 6, “Gravitational Attraction of Disks” [1], again a difference from the standard formula is shown which for comparison includes Exercise 1 from that section.  Finally, in Section 7 it is shown that the standard physics book formula can be used with spheres.

Since the title of the referenced book includes “An Intuitive and Physical Approach”, intuition may tell us that, since the rod was integrated along its length to obtain a correct answer, every piece of a nearly one dimensional rod in the limit of a Riemann sum exerts a gravitational force independently on a separate point mass.

With General Relativity, unproven to date, a mass produces surface curvature in space, which may contact a series of points on other surfaces, due to another mass, through a tensor product [2] that reduces to a force vector.  This may be interpreted as space-time surface interaction, and not necessarily complete independent action.  At least two multi-million dollar projects are built and running ([3], [4]) and at least one is being developed [5] in attempts to prove General Relativity.

Special Relativity, on the other hand, E = mc2, has long been proven correct, and the gravitational theory presented on this web site would not work without it.  Some physicists in their writings seem to make a transition from Special to General Relativity as though the two are somehow linked, and in reality they bear no relationship to each other.

[1] Kline, Morris, Calculus, An Intuitive and Physical Approach, John Wiley and Sons, Inc., 1967, 1977; Dover (1998) unabridged republication.

[2] Rainich, George Yuri, Mathematics of Relativity, John Wiley and Sons, Inc., 1950, Chapter 4

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