Fruechte Gravity Theory Blog

It happened again; a Greek letter did not come through properly.  In “Subatomic Particle Structure” it was the letter nu.  I try to be very careful when copying referenced formulas into the blog, and the character was not a mistake, just a software incompatibility.  Sorry.

Research on the internal structure of subatomic particles has been ongoing since the early days of particle accelerators and cosmic ray experiments, and there are several points of view that merit study.  The view promoted here has not much new, and should sound familiar.  The very stable proton, as CERN would agree, is a worthy starting point.

First of all, “there is no hard core to the proton. … It could be like jelly, or it could be like a strawberry, with seeds scattered throughout, but no accumulation of them at the centre.”  When scattering e^– + p is elastic, the resulting Bjorken scaling “is interpreted as indicating that the scattering takes place off point like constituents of the proton, called partons.”, where “the structure factors are functions of ω only.” ([1], pgs 16-18).

Since “q is the photon 4-momentum”, for “an individual parton of mass m” we can use the relativistic mass of the graviton, m_g = 5.575 x 10^-28 kg [2], in which case the dimensionless ω = (2Mν)/q², with q² = 2mν ([1], pg 17), gives ω = m_p/m_g = 3.

One may think of the internal structure of the proton as a crystal lattice, with nodes, or partons, that are basically zero points of eigenvectors, where internal and pass through wave functions add or subtract momentum.  If you are a string theorist, added head to tail you may choose to look at the vector structure as vibrating strings.  If you have a Mechanical or Civil engineering degree you may think of the structure as finite elements, – not necessarily tetrahedral however.  The nodes are the seeds of Ryder’s strawberry.

The defined boundary of a subatomic particle results from gravitational pressure at its barrier domain.  Let us view then the particle as a wave packet “when the wave packet is subject to the influence of a parabolic potential.  Physically, this result arises from the fact that the tendency of the wave packet to spread is compensated by the potential, whose effect is to push the wave packet towards the origin from regions where V(x) is large.” ([3], Compl. G_v, 3.c., pg 572)  Due to gravitational pressure, V(x) would be large at the barrier domain of the particle.  With consistent frequency, ω_g =  4.75 x 10²³ rad. sec^-1 [2], gravitons step through the barrier easily.

Baryons and fermions would be similar in structure, though differing greatly in density.  Scattering can happen through barrier elasticity or partial penetration with node to node scattering.  Deeper node penetration on a large nucleus can result in alpha decay.

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

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

[3] Cohen-Tannoudji, Dui, Laloë, Quantum Mechanics, Hermann, 1977, Paris, France

In converting the graviton energy to mass, we can apply 5.011 x 10^-11 J = m_gc², or m_g = 5.575 x 10^-28 kg, though due to relativistic effects the graviton is never converted completely to mass in the natural world.  The muon mass, m_μ = 207 m_e = 1.86 x 10^-28 kg ([1], Compl. A_v, 4.c., pg 527), is m_g/3 and m_p/9, where m_p is the mass of the proton.

To quote the Frenchmen: “A muon μ^– which has been slowed down in matter can be attracted by the Coulomb field of an atomic nucleus and can form a bound state with the nucleus.” ([1], Compl. A_v, 4., pg 525).  Before going any further then, and at the risk of sounding trite, like the quark and positron, the muon’s real identity may be that of a graviton.

Taking the concept of mass further, a conjugate wave graviton would slow down much more when passing through a proton or another nucleus than when it passes through an electron.  This increased compaction helps explain why the proton mass is 1836.5 times the electron mass.

Again relating to the muon, “the spread of the ground state, if the well were perfectly parabolic, would be on the order of: √(ћ/(2m_μω)) ≈ 4.7 x 10^-13 cm” ([1], Compl. A_v, 4.c., pg 528).  This of course is very close to the wavelength of a free graviton: 3.965 x 10^-13 cm.

The frequency of a muon is also close to that of a graviton:

ω ≈ 1.3 x 10²² rad. sec^-1  ([1], Compl. A_v, 4.c., pg 527),

whereas equivalent units for a graviton come out as:

ω_g = (7.562 x 10²² Hz) (2π rad/cycle) = 4.75 x 10²³ rad. sec^-1

It cannot be overlooked that the rotational component of a conjugate wave graviton passing through a fundamental particle may be zero when considering gravity alone.  Similar to the metal pump tops sold in the ‘50’s and ‘60’s, of which I remember operating one as a boy, the pump action is linear and the top spin rotational.  Should there be a rotational action of a pass through graviton, it may be related only to Coulomb field production.  This Coulomb field action, produced by charge, is in total balance.  Griffiths states it this way: “… plus and minus charges occur in exactly equal amounts, to fantastic precision, in bulk matter, so that their effects are almost completely neutralized.  Were it not for this, we would be subjected to enormous forces: a potato would explode violently if the cancellation were imperfect by as little as one part in 10¹⁰.” ([2], pg xiv)

While it is difficult to find in books or on the web how fast the Coulomb force is transmitted, my guess is that it is nearly instantaneous, transmitting through the gravitational field in wave packets at group velocity, by phase shift and chirality, the combination of which determines positive and negative charge.

[1] Cohen-Tannoudji, Dui, Laloë, Quantum Mechanics, Hermann, 1977, Paris, France

[2] Griffiths, David J., Introduction to Electrodynamics, Third Edition, c. 1999, Prentice-Hall, Inc.

Google has a live Pac-Man game as their logo today to mark the 30th anniversary.  For kids, Pac-Man can be thought of as an electron gobbling up gravitons, except he does not spit them out at a turn as he would if really an electron in an atomic orbital.

The ‘ghosts’ in the game try to stomp on the theory.

In the complex number system (a – bi) is the conjugate of (a + bi).  For a wave function, e^-ikx is the conjugate of e^ikx, where:

e^ikx = cos kx + i sin kx, and

e^-ikx = cos kx – i sin kx

Gravitons passing through an electron, proton, or neutron create potential wells for the standing waves inside these particles, which internal waves are in bound states.  To borrow from Kronig-Penny Hamiltonian math, in the “well domain of the potential array” ([1], pg 295) we may have:

φ_I = Ae^ik₁x + Be^-ik₁x,

and in the “barrier domain”:

φ_II = Ce^ik₂x + De^-ik₂x,          ([1], (8.65), pg 307)

The speed of a conjugate wave graviton slows to less than the speed of light in a vacuum as it passes through an electron or a nucleus.  Its frequency stays the same while its amplitude increases.  Coming free out the other side, the graviton resumes the speed of light in a vacuum and returns to lower amplitude.  Gravitons arriving isotropically, with the exception of those absorbed and becoming mass, help cradle the particle mass and, in summation, gravitational pressure at the boundary between the well and barrier domains holds particle mass together and keeps it from flying apart.

The letter k is used for wave number, which is in units of radians per meter.  Since k₁ > k₂, the full wavelength is shorter in distance for φ_I compared to φ_II, the conjugate wave graviton then being compacted within a subatomic particle.  The conjugate graviton would also have a rotational component in phase with a corresponding rotational component constituent to the mass and earlier written [2], in the well domain.

In terms of a Fourier transform of dimension inside the particle mass we have:

d(x-x₀)  =  (1/(2πћ)) ∫_-¥^+¥ dp e^ip(x-x0)/ћ  =  (1/(2π)) ∫_-¥^+¥ dk e^ik(x-x0)           ([3], (34), pg 1473)

where x₀ is the position of the particle, k is k₁ from above, and p is the momentum of a graviton otherwise known as p_g [2].

As some gravitons escape into deep space, entropy in the universe is always increasing from a state of original creation, – no “big bang”.  The force we have all been aware of our entire lives may be associated with the single source of all energy and mass.

[1] Liboff, Richard L., Introductory Quantum Mechanics, Fourth Edition, Addison Wesley, 2003

[2] https://www.fruechtetheory.com/blog/2008/04/01/wave-function-transfer-2-2/

[3] Cohen-Tannoudji, Dui, Laloë, Quantum Mechanics, Hermann, 1977, Paris, France, Appendix II

PS: The Dirac delta symbol and plus and minus infinity limits of integration in the Cohen-Tannoudji, Dui, Laloë equation do not transfer into this blog properly.  Go to reference [3] if you want more clarity.

According to what one reads, the CERN LHC is set to start colliding 3.5 TeV proton beams tomorrow, if they can get the beams, running for days now, lined up by then.  The LHC will obtain a variety of downscattered energies, however photons of the same energy and phase can add together in the same measurement.  In terms of discrete photons, it should be a continuous spectrum with a peak near the highest energy that the instrument can effectively measure.

With particle masses on the other hand, nobody knows for sure.  It could be an extension upon the “zoo of elementary particles that the experimentalists were discovering in their particle accelerators” * prior to the development of the Standard Model of particle physics.

In some ways it would appear easier to work with one elementary particle instead of sixteen or more.

* Smolin, Lee, The Trouble with Physics, Houghton Mifflin Company, c. 2006, p. 54

(The Standard Model of particle physics was not the main focus of Smolin’s book.)

Using Tipler’s notation starting on page 917, we begin with a particle of charge q and mass M, in a circular Bohr orbit of radius r.  Never mind that there is no such thing as a circular electron atomic orbit; the results are valid.

Particle velocity, v, and orbital period, T, are related by:

vT = 2πr

The charge traveling in a circle produces a current of:

I = q/T = qv/2πr

Multiplying this current by the enclosed area, the magnetic moment is obtained:

m = (qv/2πr) πr² = (qvr)/2                    [1], 39-16, pg 918

Angular momentum for the mass is:

L = Mvr = M (2m/q)

Rearranging and writing magnetic moment and angular momentum vectors in bold type:

m = (q/2M) L                                        [1], 39-17, pg 918

Having used a circular orbit of a particle with charge and mass to obtain this formula, it can also be used when spin angular momentum is inserted for L.  For the electron, the spin angular momentum is known to be ћ/2.  Also, still following Tipler, we can now insert e for the charge of the electron in place of q, and m_e for the mass of the electron in place of the general mass M.  There is just one problem when doing this however.  “For electron spin, the magnetic moment is twice that predicted by this equation.  This extra factor of 2 is a quantum-mechanical result which has no analog in classical mechanics.” ([1], pg 918)

The extra factor of 2 is the electron spin g factor, and is more precisely found to be:

g_e = -2.002 319 304 3622        [2]

The magnetic moment of an electron can then be written as:

m = g_e (e/2m_e) (ћ/2),

or equivalently:

μ_S = g_eμ_B(S/ћ)         [3]

This extra factor of 2.002 319 304 3622 shows that the gravitons which pass through electrons, and are not absorbed, serve as conjugate wave functions that double the expected magnetic moment of the electron.

[1] Tipler, Paul A., Physics, Worth Publishers, Inc., 1976

[2] http://physics.nist.gov/cgi-bin/cuu/Value?gem|search_for=g+factor

[3] https://www.fruechtetheory.com/blog/2008/01/05/electron-g-factor/

In the Stern-Gerlach experiment, originally performed in 1922, vaporized silver atoms in a high temperature oven are allowed to leave through a small opening, some then passing through a collimating slit so as to travel in a straight line along a horizontal axis.  Next in the path of the silver atoms is created a magnetic field, symmetrical with respect to the yOz plane ([1], Figure 1, pg 388) and having a significant gradient in the z direction which is perpendicular to the surface of the earth.  Beyond the magnetic field in the path of the silver atoms is placed a condensing surface which in the original experiment was a cold metallic plate.  The whole apparatus is placed inside a high vacuum.

The electron configuration of silver is [Kr] 5s¹ 4d¹⁰, with the outer 5s electron giving silver a quantized magnetic moment of one Bohr magneton, μ_b.  Straight from the periodic table, silver atoms are of neutral charge, so there is no Lorentz force on the atoms passing through the magnetic field.  The only force involved is

F_z = μ_z ∂В_z/∂z = – ( μ_b ∂В_z/∂z ) cos θ,

where θ is the angle between the z axis and the magnetic dipole moment of each atom, the group of which, in a time interval, would be expected, according to Quantum Mechanics texts, to race through the magnetic field of the apparatus oriented isotropically, and a continuous vertical pattern to condense on the plate.  Instead, what happens is that half of the atoms condense at the upper bound, and half at the lower bound, with none condensing in between.  Instead of a statistical pattern centered on the y axis, there are two patterns, centered at each of the expected upper and lower bounds, N₁ and N₂ ([1], Figure 3, pg 391).  The width of each of the two separate spots “corresponds to the effect of the dispersion of the velocities and of the width of the slit F” ([1], A.1.c., pg 391).  Evidently, there are only two discrete forces acting on the atoms as they traverse the magnetic field:

F_z = ± μ_b ∂В_z/∂z                                       [2], pg 525

S_z “is a quantized physical quantity whose discrete spectrum includes only two eigenvalues. … these eigenvalues are + ћ/2 and – ћ/2” ([1], A.1.c., pg 392).

One thing that puzzles me about what is found in text books is the classically expected pattern of the Stern-Gerlach experiment.  Since silver atoms are paramagnetic, I don’t understand why it would not have its center skewed off the y axis in the z direction.  Nevertheless, it is the quantum result, not the classically expected (which doesn’t happen anyway), that I would like to make an attempt at explaining.

What appears to be happening here is that the nuclei of the free silver atoms make effort to line up with the gravitational field in one of two spin orientations, in order to minimize energy, and to aid in the pass through of the gravitons that are not absorbed, which help pump nuclear spin.  Additionally, the orbiting electrons make effort to get out of the way of the highest flux density of gravitons, making orbital angular momentum also aligned as much as possible with the z axis, and monitored in two patterns, mirror images of each other, each relating to one orientation of the nucleus.

[1] Cohen-Tannoudji, Dui, Laloë, Quantum Mechanics, Hermann, 1977, Paris, France, Chapter IV

[2] Liboff, Richard L., Introductory Quantum Mechanics, Fourth Edition, Addison Wesley, 2003

It has been said that the Coulomb force is the final mediator of the gravitational force.  What this means is that in the final analysis we find that the force that produces gravity is the Coulomb force.  This is because “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.” *  Thus we can see that it is not until the electron passes its closest to the nucleus, in graviton absorption mode, that it activates the gravitational force in the direction that the nucleus is pulled the most.

Emission of gravitons during a turn process, and at the end of a spin-flip, serves to put the brakes on the electron in the direction of travel through conservation of momentum.  The Coulomb force then begins to bring the electron on another path back toward the nucleus.  Momentum balance is taken care of by the graviton, and the positron is not needed.

To sum it up, the Coulomb force produces the gravitational force, and the gravitational field produces the force that holds subatomic particles together.

With all this going on, it was a much more manageable problem to try to figure out the cause of gravity using the principles used to calculate blackbody radiation in the days of Rayleigh, Jeans, and Planck, than to try it, let’s say, by using the Dirac equation internal to subatomic particles, and the Schrödinger equation external.

* https://www.fruechtetheory.com/blog/2008/08/07/the-lamb-shift/

Thirty five minutes before or after a quantum measurement, part of Schrödinger’s cat could be part of the planet Jupiter, because gravitons travel at the speed of light in a vacuum.  One point of view that agrees with gravity is that the cat is not the same cat, no matter the time differential, as long as the time differential is not zero.

Erwin Schrödinger came up with his thought experiment in 1935, and by now part of the original cat may be headed back.  Since the moon and Jupiter are in fairly close alignment right now it may end up part of the moon instead of part of the cat again.