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.

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 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.

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.

Earlier it was postulated that the number of turns in an atomic orbital, referenced to the nuclear spin axis of an atom, – or to the nucleus to nucleus axis of a diatomic molecule, is equal to 2n + 1, n being the principal quantum number.  In some instances this may be a little hard to believe, such as when looking at some of the Leighton sketches of Liboff Figure 10.17 [1].  The hydrogen 4F, m = 0, orbital, for example, is one that looks symmetrical with six lobes when sectioned in reference to a plane that includes the z axis of the atom.
Let us say that the electron starts out in the 4F orbital on the negative z axis, which we will also allow to be the negative z axis of the nuclear spin at the same point in time.  It then arcs past the nucleus on its way to the first turn of the orbital where it reverses intrinsic spin orientation as it gives off one or more gravitons.  As the electron spends more time in the areas that show a higher whiteness density in the Leighton sketch, the original -z axis of the atomic nucleus does not have to be aligned with the -z axis of the sketch and the electron spin axis again until the electron enters the 9^th turn.  The nuclear spin axis would be allowed to change azimuthal angle back and forth in quantum angular steps.  The electron can be at the original -z location more than once in the course of the orbital, but not at the same instant in time as the -z axis of the nucleus until the last turn.
The raising and lowering operators of the total angular momentum of the state | E₀, L, S, J, M>, are known as:
J_± | E₀, L, S, J, M> = ћ sqrt[J(J+1) – M(M±1)] | E₀, L, S, J, M±1>  [2] Chpt. 10, Complement D_x , 3.a.
The operators J_± may exist in part to balance the energy so that a steady-state, steady-flow process can occur during the absorption of gravitons by the nucleus and the subsequent release of energy expended by the nucleus in creating a magnetic field that keeps electrons in orbit.  When the electron reverses its intrinsic spin axis at the apex of a turn, it arcs in the opposite direction because its magnetic dipole is reversed, and the magnetic field vector and strength from the nucleus is the same in that locale as just before the turn took place.
From a measurement standpoint E_n looks stable, however the vector operator M ± 1, with 2l + 1 possible integer values of the magnetic quantum number, m_l ([1] Table 10.4, pg 451) , may help keep the energy the same once the uncertainty principle is taken into account.  The shift through integer values of m_l, with [J_z, J_±] = ±ћJ_± ([1] (9.20), pg 358) helps keep the energy of the atom stable.  For example, the energy of the total orbital angular momentum for the D term is 6ћ²/2I ([1] Fig. 9.8, pg 364) through all eigenvalues of L_z, however it cannot be certain at any given instant in time what the value of L_z is for any orbital with higher total orbital angular momentum, L, than the S orbital.
As far as quantum mechanics goes, there does not appear to be anything currently written that will need to change.  For example, take the commutation relations [J_x, J²] = [J_y, J²] = [J_z, J²] = 0  ([1] (9.18), pg 358).  Every soccer match between J² and J_z will continue to result in a tie score and no one will win, or better yet the game is forfeited before it starts.  That is not to say that nothing will get added.  The wave functions going on inside the electron, and how they interact with the Coulomb potential, can be an additional field of physics.
[1] Liboff, Richard L., Introductory Quantum Mechanics, Fourth Edition, Addison Wesley, 2003
[2] Cohen-Tannoudji, Dui, Laloë, Quantum Mechanics, Hermann, 1977, Paris, France

When a stationary permanent magnet holds a steel object, a paper clip for example, against the force of gravity, where does the energy come from to continually hold the clip?  Since work is force times distance, an undergraduate student may say there is no distance moved and therefore no work is done.  On a quantum level however, things are not motionless, and it does take energy to hold the clip.  The answer I think starts with the entry on this blog called “The Nucleus and Gravitons”.  The energy expended through the individual atomic dipole moments of the permanent magnet to hold a ferromagnetic material object against the pull of gravity can come from gravity itself.  The nuclei of the atomic dipoles of the permanent magnet may be absorbing gamma rays at and very near 7.562 x 10²² Hz.

There is another possibility, as opposed to total deflection, for what happens when gravitons encounter a nucleus.  Since the makings of an electron exist inside a neutron, and neutrons and protons have their own spin functions going on inside, it is not outside the realm of possibilities that atomic nuclei, each when part of its own functional atom, absorb gravitons.
What a nucleus in a gravitational field, the earth’s let’s say, would do with all this absorbed energy is not too hard to imagine.  The Coulomb field of an atom is full of electromagnetic wave activity involved with keeping electrons in orbit, and some of that energy may escape.  Therefore we would have gravitational energy replenishing Coulomb energy through both the electrons and the nucleus of an atom.
The synchronization involved as a graviton enters a nucleus would have to be just as smooth as when one enters an electron in a quantum atomic orbital in order for the nucleus to not have been pushed by the absorbed gravitons.  Also, inflow of energy must balance outflow in order for there to be a steady-state, steady-flow process.  In this case it is only those gravitons which pass near the edge of a nucleus that would have to be deflected by its local magnetic field.
If you have read it and remember, my April 2007 paper already has the Coulomb force as the final mediator of the gravitational force.