Fruechte Gravity Theory Blog

When I read on the internet how I should have studied harder in school, I find it funny.  Ask any of my high school or university friends how hard I studied.

“If you think education is expensive, try ignorance!”
–      Derek Bok, past president of Harvard
Keep ignoring the quark coincidence.

With the advent of gravitons serving as conjugate wave functions that hold subatomic particles together using gravitational pressure, we can go a step further.  Could it be that all material objects in our locale are made of gravitons?  If so, the graviton is the fundamental quantum unit, or otherwise we may call it quantum 1.
Kansas the band actually got closer to the truth than Oppenheimer did when it comes to particle theory relating to electrons when they sang “Everything is dust in the wind.”  They didn’t go quite small enough though if everything is gravitons.  Every material object in our solar system may indeed be made of gravitons, however we cannot go so far as to surmise that this may be true throughout the universe.
Newton’s law of gravitation still holds true.  The magnitude of the gravitational force exerted by one mass made up of atoms on another mass made up of atoms is:
F₁₂ = Gm₁m₂/r₁₂²
If the gravitational field were more uniform as one travels out from the earth we could call it the Higgs field, but because the force exerted is inversely proportional to the square of the distance between the masses we cannot.  Additionally, since gravitons are so small and only one half wavelength long we could call them strings, – if only they made loops once in a while.

The problem of what free charges, including nuclei, do with gravitons that arrive in a direct path has not been addressed to a great extent, though a physics PhD friend of mine has at least thought about it as relating to a free electron.  The necessity of dealing with the subject comes not only from when “energy is irretrievably radiated away by the acceleration fields” ([1], pg. 466), but also directly through electrostatics.
Let us first consider the acceleration fields.  As one example of power being radiated by an accelerated point charge, we have synchrotron radiation, the power formula for a free electron traveling near the speed of light being:
P = (2ke²γ⁴c) / (3r²)                      [2].
In Gaussian units, with k = 1, this becomes:
P = (2e²cβ⁴γ⁴) / (3ρ²)                   ([3], (14.31), pg 667)
In the second equation the Greek letter ρ (Rho) is used for the radius of the synchrotron accelerator, instead of r, and a β⁴ factor shows up which GSU left out because β = v/c is close to 1 in a synchrotron.
It could be reasoned that synchrotron radiation energy is replenished to the electron through electric and magnetic fields, by way of electric charges and currents produced by steam powered generators through the burning of hydrocarbon fuels, or nuclear fission.
Nevertheless, I do think that free charges absorb gravitons as part of the process of energy replacement when radiation is given off.  What keeps them stable is the main challenge here.  Why would they not keep absorbing gravitational energy until we end up with giant particles?  Is it gravitational pressure, in part, that keeps them at a certain charge and mass?
Fundamental charges and other ions most certainly allow gravitons to pass through, otherwise there would be scattering going on all the time and utter chaos.  Indeed, the gravitons passing through may provide conjugate wave functions that help keep the particles together.  It could be that gravity not only allows large masses to attract, but also provides for the continued existence of particle mass.
Moving on to electrostatics, the energy of a point charge is given as:
W = (ε₀/2) ∫_{all space} E² dτ                     ([1], (2.45), pg 94),
which turns out to be infinite as Griffiths shows ([1], Sect 2.4.4, pg 95).
Here the same process of gravitons providing conjugate wave functions would be taking place, and also we may have the electric field transmitting through the gravitational field, shouldering through the gravitons in a highly relativistic sense.  The infinite energy would no longer be a mystery, though it is not really infinite because the known universe is gradually losing gravitational energy as some gravitons escape into deep space.  The result is nothing new to physicists; our galaxy is expanding at an accelerating rate.
As far as gravitational conjugate wave functions relating to conservation of charge, it has been postulated that “the average number of gravitons in an electron, in an atomic orbital at the face of the earth, could possibly be 137/2 = 68.5” [4].  Not only for electrons in atomic orbitals then, but also for free electrons, the gravitational field providing conjugate wave functions may increase the number of gravitons involved in making a fundamental charge to 137.
Considering all, in the sudden absence of a gravitational field we are presented with an alternative to sequential atomic collapse in a slowly decreasing gravitational field, which is fundamental particles simply flying apart in a high energy gamma ray burst.  Gravitons like to follow gently arcing uniform magnetic field lines, so a rotating intense magnetic field deep in space, from a neutron star for example, could produce such a phenomenon if its path intersected the path of a more conventional mass.  These are not to be confused however with lower energy gamma ray bursts, such as those that come from the earth in the milliseconds before a lightning strike, due to the Compton scattering of gravitons.
[1] Griffiths, David J., Introduction to Electrodynamics, Third Edition, c. 1999, Prentice-Hall, Inc.
[2] GSU Hyperphysics: http://hyperphysics.phy-astr.gsu.edu/hbase/particles/synchrotron.html
[3] Jackson, J. D., Classical Electrodynamics, Third Edition, c. 1999 John David Jackson, John Wiley & Sons, Inc.
[4] https://www.fruechtetheory.com/blog/2008/04/01/wave-function-transfer-2-2/
 

The fundamental commutator relation [x, p] = iћ, between the operators of coordinate and momentum, provides a way to show how a graviton can add linear momentum to an electron.
An energy relation for a synchronous encounter by a graviton with an oncoming electron in an atomic orbital can start with:
[x, p_g]² = i²ћ² = -ћ²,
the added kinetic energy being ћ²k_i²/2m_i, and the added momentum -√(ћ²k_i²).
The added momentum, as shown by the minus sign, is in the opposite direction of that in which the graviton was traveling at the speed of light in a vacuum before it was absorbed by the electron.
Internal to the electron we can use the {N} representation to form the basis of a set of wavefunctions forming orthonormal vectors: │0 >, │1 >, …, │n >, …, with eigenvalues of N: 0, 1, …, n, … [Messiah, XII.16, pg 436].  The graviton in the process of being absorbed by an electron in a quantum atomic orbital can then be seen as a raising operator, where
a^†φ_n = φ_n+1 and a^†φ_n+1 = φ_n+2 ,
and the release of a graviton a lowering operator, with
aφ_n = φ_n-1 and aφ_n-1 = φ_n-2 .
The Hamiltonian for such a system is represented as:
H φ_n = ћω₀(a^†a + ½) φ_n        [Liboff, Section 7.2],
with energy eigenvalues
E_n = ћω₀(n + ½)        n = (0, 1, 2, …, 68, …n_max).
Here 68 represents the average n value at the face of the earth, and n_max depends on the orbital.
If the mass of the electron diminishes as the gravitational field diminishes, the characteristic wavenumber β of the electron also diminishes.  For each graviton internal to the electron β_i² = m_iω₀/ћ, and for the mass of the electron at the face of the earth we have Σm_i = m_e = 9.1095 x 10^-31 kg.

In October 1976 the musical group Kansas came out with its fourth album, called Leftoverture.  At social gatherings and on the radio that autumn of my freshman year at the University of Wisconsin it was not uncommon to hear the song “Carry On Wayward Son”.  Around the time I was finishing up three semesters of calculus and two of physics as a sophomore, still living in the lakeshore dorms, the group had given us “Point Of Know Return” and “Dust In The Wind” from a fifth album.
Two years ago near the end of this month, I took a circuit drive with copies of my physics paper.  Kansas State University was one of the places I stopped.  It was a fruitful visit because on Ocotober 3, 2006 I received an analysis of the paper from KSU Physics, – only the second one offered independent of the journals.  Something in the KSU analysis prompted me to look for a way to clean up the Rotational Energy section of my paper, which I then did.  That section however, does not have anything to do with the calculation of the frequency of a graviton, but rather is there only to show that there is enough energy in the electron to do the job.
If you see Kerry Livgren, Steve Walsh, Robby Steinhardt or the others of Kansas the band please tell them the ‘point of know return’ in physics has come.  “How long?” does not have to be asked anymore, – it took 28 years.

Many physics experiments on earth are set up horizontal to the earth’s surface, so the preferred z axis in quantum mechanics is the same axis as the direction of a flood of gravitons coming out of the earth.  It is certainly not the only direction gravitons are traveling, just the dominant direction and the direction of a vector summation.
Most measurements of the de Broglie wavelength, I presume, are also measured horizontally, as matter with velocity weaves through the gravitons.  This brings us to the question of whether or not the de Broglie wavelength is dependent upon the strength of the gravitational field, and if so, what in the formula for this wavelength, λ = h/mv, is allowed to vary?  If it is the mass, then transporting a particle to a weaker gravitational field will result in it measuring less in mass, and longer in wavelength for a given velocity, neglecting the relativistic effect.
In an area of space where the gravitational field is much weaker than the levels that we know of in our solar system, the concept of mass may have less relevance.  A neutron star in such an area may not have mass as we know it and the de Broglie wavelength may not have as much meaning.
Values that would not change anywhere in the universe are Planck’s constant and the speed of light in a vacuum.  The energy of a photon, which has no mass, would then of course remain as Planck’s constant times its frequency, anywhere in the universe.

A P orbital is closer on average to the nucleus than an S orbital for a given atom, and the electron has additional curvature as it arcs in an attractive Coulomb field.  A hydrogen electron in an S orbital therefore absorbs more gravitons in a single arc than the same electron when in a P orbital, both because it is on an arc of less curvature and because the arc is longer in distance than in the P orbital.
As an electron in an atomic orbital passes its closest to the nucleus it continues to exist in a graviton absorption mode and, if traveling toward a gravitational source, continues to increase in mass, magnetic dipole moment, and charge until it comes time to make a turn.  This is likely the reason why Willis Lamb found in 1951 a greater energy in the 2S_1/2 orbital compared to the 2P_1/2 orbital.*  The electron in the 2S_1/2 orbital would be slightly larger in diameter on average and carry a slightly more negative charge on average compared to the electron in a 2P_1/2 orbital.
It may be that the Coulomb force within an atom is not photon mediated as originally thought, but rather due only to magnetic field vectors which could not be produced continually without the nucleus and orbiting electrons absorbing gravitons.
* http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/Lamb.html

When the frequency of a transverse magnetic field is twice the Larmor precessional frequency of an electron in a constant ‘z-axis’ magnetic field, the electron undergoes a resonant spin-flip behavior.  If, on the other hand, the transverse magnetic field occurs only long enough to produce one flip, the electron will reverse intrinsic spin orientation just once in a localized process.
In Hydrogen, a quantum angular step of the single proton nucleus may be the first action in the process of making an electron turn and start off on a new trajectory.  In larger atoms, Oxygen for example, it may be each alpha unit in the nucleus that is controlling the initiation of turns of two electrons.  It is important to note however that it is not likely the same alpha unit/electron combination in the initiation of turns over time, nor would it be required that the same two protons and two neutrons stay together over time as representing a definitive alpha unit in the nucleus.
The activity and equations of the wave functions inside the alpha unit, – and without as the whole nucleus must stay together, would produce not only the central Coulomb potential, but also transverse magnetic field pulses necessary to initiate electron turns.  This synchronized action would give further reason for the energy of a graviton to be linked to an integer ratio of the mass energy of the proton.  As it is, the energy of a graviton is precisely one third the mass equivalent energy of the proton.
Using the Schrödinger picture, wavefunctions are varying and operators are constant.  The graviton can be viewed as a momentum operator.

For the electron in a central Coulomb potential, we have the Schrödinger equation in polar coordinates written as:
[p_r²/2m + l²/2mr² + V(r)] ψ(r, θ, φ) = Eψ(r, θ, φ)                 ([1], (IX.13), pg 348)
The term containing the square of radial linear momentum gains its prominence due to the fact that with Schrödinger wave mechanics there is no such thing as a circular electron orbit.  The trajectory followed by the electron in an attractive Coulomb field is shown in Messiah Fig. VI.2 (b), page 229.  If we can imagine the curve as being shown concave upward, instead of concave downward as in the figure, we could think of the electron as on a ski jump and rolling forward at the end of the path.  By kicking off one or more gravitons in this process, and having its magnetic dipole moment reversed in direction, both actions help place the electron on another arcing path back toward the nucleus, but in a different plane relative to the first arc.  A quantum angular step in the orientation of the nucleus, which aids in the production of a changing magnetic field, would be a third process that helps the electron start out and complete its next trajectory.
The eigenfunctions for the simplest case, of hydrogen, are:
Φ_nlm(r, θ, Ф) = R_nl(r) Y_l^m(θ, Ф)    ([2], (10.130), pg 452),
Something that can again be pondered about the stability of the electron trajectory in a quantum atomic orbit, is gravitons approaching in transverse and rear directions.  As for how some gravitons may be able to step through in these cases, there may be a spherical spread of the electron in the regions between turns, with the electron contracting as it enters a turn and expanding as it comes out of the turn in its opposite spin orientation.  This is proposed as an accordion type behavior of the sphere that contains the electron’s internal wave functions.
A spread of the particle wave function would also then be taking place for free fermions in the earth’s gravitational field, in order to allow gravitons to pass through.  An exception is when electrons are held in high voltage, such as with the GLAST instrument designed to measure gamma rays greater in energy than 100 MeV, or in the ground in the milliseconds before a lightning strike.  In these cases the electron would be concentrated enough in volume to produce Compton scattering.
[1]  Messiah, Albert, Quantum Mechanics, Two Volumes Bound as One, Dover Publications, Inc., paperback, 1999
[2] Liboff, Richard L., Introductory Quantum Mechanics, Fourth Edition, Addison Wesley, 2003

Aug 14, 2008, Rev. A: Added “radial” to second paragraph; added eigenfunction equation.