The Harmonic Neutron Hypothesis: The ratio of the frequency equivalents of the Bohr radius and the
neutron is a common coupling constant of the radii of hydrogen, the Schwarzschild radius of the
minimum black hole, and the velocity of the muon neutrino
Donald William Chakeres
The Ohio State University, Columbus, Ohio 43210
Email: [email protected]
Abstract:
The harmonic neutron hypothesis states that all of the fundamental constants are related to the ratios of
frequency equivalents of the electron, ve; Bohr radius, va0; and the Rydberg constant, vR; divided by the
neutron, vn. The only ratio associated with a distance or velocity is a0 and logically should be related to
multiple fundamental dimensions. va0/vn equals 2.4936654 x 10-5. The inverse of this ratio times the
Compton radius of the neutron is a0. The velocity of the muon neutrino is reported as 1+2.48 x 10-5 c. The
minimum mass of a black hole is approximately 1.4 to 3 solar masses 1.9892 x1030 kg. The associated
Schwarzschild radii range equals 4.1361806 x 103 m to 8.8632442 x 103 m. This dimension equals a0
times va02/vn. The ratio of rs of the minimum black hole divided by ao equals va02/vn. The ratio range of
these radii divided by the distance light travels in one second is 1.3797 x10-5 to 2.9564 x 10-5. Therefore
va0/vn to the known accuracies of these three subatomic, relativist, and cosmic entities is a common
coupling constant. These observations support the validity of the harmonic neutron hypothesis.
Key words: harmonic neutron hypothesis, neutron, muon neutrino, speed of light, Bohr radius,
Schwarzschild radius
Introduction:
The harmonic neutron hypothesis states that all of the fundamental constants are related to the ratios of
frequency equivalents of the electron, ve; Bohr radius, va0; and the Rydberg constant, vR; divided by the
neutron, vn 1-3. The only ratio of these three associated with a distance or velocity is a0. This ratio should
logically be related to multiple fundamental dimensions/ velocities if the hypothesis is correct. va0/vn
equals 2.49366539 x 10-5. It will be shown that this ratio is a common coupling constant that defines the
dimension of hydrogen and the atoms, the reported increase in the velocity of the speed of light of the
muon neutrino 4-10, and the Schwarzschild radius of the minimum mass of a black hole.
Methods and Results:
Most of the data for the fundamental constants are from website: http://physics.nist.gov/cuu/Constants/.
All of the calculations are done in normalized frequency equivalent units of Hz. All of the calculations are
done as dimensionless ratios. This includes the ratio of the Schwarzschild radius and the distance light
travels in one second. The increase in the velocity over the speed of light of the neutrino is dimensionless
ratio as well. va0 and vn equal 5.66525642 x1018 Hz, and 2.2718591 x1023 Hz.
β is the ratio of the velocity and c in a relativistic system, Equation (1). c is the unit velocity. Δ is used to
describe the increase over the unit velocity c, Equation (2). The subscript, t, indicates the tachyonic
domain. The coupling constant of the tachyonic velocity, Vt, with c equals 1+ Δ. Equation (3) is a classic
coupling constant format. The Δ reported for the muon neutrino is 2.48 x 10-5 1. The relative error is ±
0.28 (stat.) ± 0.30 (sys.). va0/vn is within the error of this measurement.
V
(1)
c
Vt
(1
Vt
c
)c (2)
(3)
c
The minimum mass of a black hole is approximated to be between 1.4 and 3 solar masses. One solar mass
is 1.9892x1030 kg. The range of masses for a minimum black hole is 2.78488 x 1030 kg to 5.9676 x 1030
kg. The Schwarzschild radius range is 4.1361 x103 m to 8.8632 x103 m. The range of ratios with the
distance light travels in one second is 1.3797 x 10-5 to 2.9564 x 10-5. This range encompasses the va0/ vn
coupling constant.
The predicted black hole Schwarzschild radius, rs, from the harmonic neutron hypothesis is the product of
c and va0/ vn. This equals 7.4758208 x 103 m, Equation (4). This predicted black hole rs can also be
derived without c, Equation (4). Equation (5) demonstrates how this relationship is related to unity.
Equation (6) demonstrates that the ratio of the dimension of a minimum black hole rs and a0 equals the
ratio of va02 divided by vn. This ratio equals 1.41272539 x 1014.
cva 0 s
vn
rs
1
va 0
vn
2
a0
1 cv a 0 vn
a0 v n v a 0 2
2
va 0
(4)
vn
rs vn
(5)
a0 va 0 2
rs
(6)
a0
Discussion:
There are only three ratio values utilized in the whole harmonic neutron hypothesis. The only ratio that
represents a ratio of two velocities or related to a length is va0/vn, Δ, 2.4936654 x 10-5. The product of a
length and a frequency is a velocity. The ratio of two velocities if they both have the same length is the
ratio of the frequencies alone. Therefore, there is no other logical possible coupling constant factor of the
harmonic neutron hypothesis, 1+ Δ, for the neutrino tachyonic velocity. This is an accurate prediction.
The Schwarzschild radius of a minimum black hole divided by the distance light travels in one second is
not a typical unit, but fully valid unit value. In this setting of coupling constants it is the natural method of
analysis. It is actually the method used to describe the increase above the speed of light of the muon
neutrino so it is valid 1. The predicted Schwarzschild radius from the harmonic neutron hypothesis is
7.4758208 x 103 m. This is within the known range. The ratio of the minimum black hole rs and a0 equals
the ratio of va02 divided by vn. a0 times va02 divided by vn equals the rs of a minimum black hole. This is
another example supporting the hypothesis since vn is fundamentally associated with the rs of the
minumum black hole.
The harmonic neutron hypothesis has accurately predicted the masses of many of the elementary particles
and bosons, the total energy lost in the neutron beta decay process, the mass of the proton, Planck time,
and the Newtonian gravitational constant from only these same three ratios 1-3. Therefore it is not
unexpected that other fundamental entities should be related to the identical coupling constants.
A unification model should utilize its foundation constants in multiple different setting if the model is
valid. The fewer variables necessary the more unified the model. In this case the coupling constant related
to dimension and therefore velocity simultaneously defines the dimensions of the atoms, the increase in
the muon neutrino velocity above c, and the dimension of the event horizon in space created by the
minimum black hole. Since all of the known range of coupling constants encompass the predicted ones
this supports the validity of the hypothesis. It is possible to experimentally confirm these predictions. It is
not likely that this is just a chance event since the scales of these dimensions and their setting are all very
different. This observation unifies sub atomic, relativistic, and cosmic entities all to the identical coupling
constant. The frequency of the neutron is the essential link to all of these different factors as proposed in
the hypothesis. These results support the validity of the harmonic neutron hypothesis.
Acknowledgements:
I would like to thank Mark Ricard Ph.D, Amir Abduljalil Ph.D, and Jane L. Weiss BSN for their help.
References:
1. D. W. Chakeres, Harmonic quantum integer relationships of the fundamental particles and bosons,
Particle Physics Insights, Particle Physics Insights. 2,1 (2009).
2. D. W. Chakeres, The Neutron Hypothesis: Derivation of the Mass of the Proton From the Frequency
Equivalents of a Neutron, Electron, Bohr Radius, and Ionization Energy of Hydrogen, Particle Physics
Insights 4, 19 (2011).
3. D. W. Chakeres, The Harmonic Neutron Hypothesis: Derivation of Planck Time and the Newtonian
Constant of Gravity from the Subatomic Properties of a Neutron and Hydrogen, Particle Physics Insights.
(2011). In press
4. T. Adama, et al, Measurement of the neutrino velocity with the OPERA detector in the CNGS beam,
arXiv:1109.4897, (2011).
5. O.M. Bilaniuk, E. C. Sudarshan, Particles beyond the Light Barrier, Physics Today 22, 43 (1969).
6. A. Chodos, Null experiments for neutrino masses, Physics Letters B 150, 431 (1985).
7. G. Simone, On velocities beyond the speed of light c, CERN, Geneva, arXiv:hep-ph/9712265v3 22 Jan
(1998).
8. A. Chodos et al., Mod. Phys. Lett. A7, 467 (1992).
9. B. Kayser, Neutrino mass, mixing, and flavor change, Particle Data Group. Retrieved 2007-11-25.
(2005).
10. C.L Cowan, F. Reines, F.B. Harrison, H.W. Kruse, A.D McGuire, Detection of the Free Neutrino: a
Confirmation, Science 124: 103, (1956).