How introduction of the natural geometric unit helps to better understand the reality?
The point of origin for the further reasoning are intuitive hypotheses
that:
- all of the current SI units are interdependent and
thus finding relations between them is possible, and that
- the underlying relations are of geometric nature, so all the SI units can be brought to one common denominator, being a natural geometric unit.
The proposed natural geometric unit u (fast derivation is shown
further down at the end of the text) is a dimensionless unit. Reduction of
the dimensions of units is achieved by setting the four main physical constants
(speed of light in vacuum c, Newtonian gravitational constant G, Coulomb constant k and Planck constant h) to 1, jointly numerically and dimensionally - not only
numerically, as it is being done in case of the Planck units system. Such a joint - numerical and dimensional - normalization of the fundamental constants to 1 is not a new approach in scientific analysis, it is being used frequently (with variations in details)
in works on the general relativity, e.g. in the article by Kim & Kim Geometrical interpretation of
electromagnetism in a 5-dimensional manifold (section 8. Unit system, p. 14.)
The only remaining fundamental physical constant used in description of electromagnetic and gravitational interactions, the fine structure constant α, is in the SI already dimensionless, so we do not need to normalize it. The π number is a geometric dimensionless constant by definition (btw. Euler's number e is defined by π: eiπ+1=0).
When we set c = G = k = h = 1, where k = μ0c2/4π ( μ0 denoting the permeability of vacuum or magnetic constant understood according to the pre-2019 SI definition μ0 = 4π/107 [H/m = kg/(C2·m)] ) and define the unit u as equal to √h what translates to conversion factors to m, s, C, Wb and kg
u = √|hG/c3| m, u = √|hG/c5| s,
u = √|107h/c| C, u = √|hc/107| Wb,
u = √|107hG2/c| kg,
then √(α/2π) u equals to the elementary charge
e = √(α/2π) u = 1.602 176 634·10-19 C
( the exact value defined by CODATA 2018 ),
where
α ≈ 0.007 297 352 569 28
( CODATA 2018: α ≈ 0.007 297 352 569 3(11) )
and 27α u = 28π · α/2π u equals to Bohr magneton
( CODATA 2018: μB ≈ 9.274 010 075 5·10-24 ... 9.274 010 081 1·10-24 [J/T = C·m2/s]
the unit u here is u = √|107h/c| C · |hG/c3| m2 / (√|hG/c5| s)).
We can also note, that surprisingly but precisely,
electron magnetic moment equals to
≈ 9.284 764 698 1·10-24 J/T
(CODATA-2018: -μe ≈ 9.284 764 704 3(28) ·10-24 J/T, but for this value to be comparable with the SG value above, so to preserve the validity of the Ampère’s force law in the translation from the SI to the SG, we need to divide the CODATA value by the factor 1.000 000 000 55, which is a relation of the 2019 SI magnetic constant μ0 value to its pre-2019 SI value. We get then the corrected CODATA-2018 value of -μe ≈ 9.284 764 699 2(28) ·10-24 J/T.)
The factor α/2π is apparently the base wavelength divided by 2π (expressed in radians of a circle of a length of a base unit of length u) of an electromagnetic wave in vacuum that is characteristic to electrons at rest. A - kind of - standing wave in the spacetime (maybe a soliton type solution to nonlinear Schrödinger equation? Further insight is required to propose relevant mathematics here).
Then, using the same core α/2π, if we take CODATA-2018 bracket middle value for
electron rest mass (qe = -1 e): me = √1-1 e / μB · 1 / 4π u ≈ 9.109 383 70·10−31 kg ≈ 0.511 MeV/c2,
we get the Newtonian constant of gravitation
G ≈ 6.673 655 205 · 10-11 m3/(kg · s2),
and we can guess
quark u rest mass(qu = 2⁄3 e): mu = √(2⁄3)1 e / μB · √2 / π u ≈ 2.360 MeV/c2 and
quark d rest mass (qd = -1⁄3 e): md = √(1⁄3)-1 e / μB · √2 / π u ≈ 5.007 MeV/c2.
We can see then also that the electron Zitterbewegung period te = μB/e u ≈ 6.440 443 341 0·10-22s.
Let us call the values predicted using the system of the natural geometric unit (SG system) the SG values. How they fit then into current state of experimental knowledge?
The big G
It is common knowledge that some unknown factors influence measurements of big G resulting in discrepancies between results of individual experiments far greater than the predicted error margins for the experiments. Even the authors of the most precise experiments till date admit that. Citing after conclusion of J. Wu et al. 2018 paper* that presented the most precise measurement of the Newtonian constant of gravitation to this day: “(...) unfortunately, G still remains the least precisely known among all fundamental constants in the latest CODATA-2014 adjustment. To make matters worse, the discrepancy among all 14 G values adopted in CODATA-2014 adjustment reaches 0.05%. This measurement status has confused experimental physicists over the years.”
There are two meta analyses proposing to solve the conundrum by including into the error elimination procedures:
1. some unknown factors (probably related to movement of liquid masses below the Earth’s crust) that result in otherwise measurable effect of the length of day changes with period of 5.9 years (proposal by a group related to Jet Propulsion Laboratory**)
conjectured G value after the error elimination
6.673 899 ± 0.000 069 · 10 −11 m3/(kg·s 2)
SG value
6.673 655 205 · 10 −11 m3/(kg·s 2)
diff -25…-47 ppm
2. Sun’s (and possibly Jupiter’s) dragging effects with a period of one year (proposal by prof. J.L. Parra from Florida International University***)
conjectured G value after the error elimination
6.673 52 ... 6.673 54 · 10 −11 m3/(kg·s 2)
SG value
6.673 655 205 · 10 −11 m3/(kg·s 2)
diff +19 ppm
There is plausible thesis then, to be verified, that both of the meta analyses found real effects and they both may be affecting the measurement results in combination, partially offsetting each other influence. The value of G predicted by the natural geometric unit method findings fits very well into the joint meta analyses conjecture, close to their simple average.
The rest masses of quarks u and d
The recent 2020 Particle Data Group values are the following:
mu = between 1.9 and 2.65 MeV/c2 (most probable value 2.16), SG value 2.360 MeV/c2
md = between 4.5 and 5.15 MeV/c2 (most probable value 4.67), SG value 5.007 MeV/c2
In addition to the value brackets PDG publishes experimentally verified relations between quarks u, d and s rest masses. The relations narrow the experimentally allowed values for quarks u and d rest masses:
mu/md = 0.47 + (−0.07…+ 0.06), SG value 0.47
(mu+md )/2 = 3.45 + (−0.15…+ 0.55) MeV/c2 and ms / ((mu + md )/2) = 27.3 + (−1.3…+ 0.7), where ms = 93 + (− 5…+ 11) MeV/c2 what together translates to
(mu+md )/2 = 3.45 + (−0.15…+ 0.40) MeV/c2, SG value 3.68 MeV/c2
The values of mu and md predicted by the natural geometric unit method findings fit very well into the narrowed brackets, close to their middle values.
If we add two mu and multiply by md, we get 2mumd = √2 / (π · μB/e)2 u2 = 26√2 me2 what gives a hint, that also the full proton rest mass can be later derived with use of similar type of SG based formula, maybe containing non-linear, superluminal components. SG based formulas for values of rest masses of other charged fermions (the unstable fermions) should contain factors related to their lifetimes (decay rates). When they are found they would then represent a proposal of unification of electromagnetic and strong interactions.
And there is more evidence to it...
The Schwinger limit instead of the Planck critical values
In the SG Planck length lp = √(2π) u is equal to other Planck units: Planck charge, Planck mass, Planck energy and Planck time. In a natural way the unit u represents all the Planck units critical values divided by √(2π).
The Planck charge qp/√(2π) = u becomes a natural representation of elementary charge (the quantum of electric charge) e = √(α/2π) u and double magnetic flux quantum 2Φ0 = √(2π/α) u, being their geometric average √(e·2Φ0) = u.
The Planck mass mp/√(2π) = u together with Planck length lp/√(2π) = u become natural representations of electron rest mass me = 1/(29π√(2πα)) u and electron Compton wavelength λCe = 29π√(2πα) u, being their geometric average √(meλCe)= u.
The Planck time tp/√(2π) = u together with Planck energy Ep/√(2π) = u become natural representations of electron Zitterbewegung period te = 27√(2πα) u and electron rest energy mec2 = 1/(27√(2πα) u, being their geometric average √(temec2) = u.
The Schwinger limit calculated in the SI as electromagnetic field flux density has value that is not consistent with the real world phenomena: ultra high frequency radiation was observed in 2019 that is stronger than the Schwinger limit predicted. But when we calculate the Schwinger limit in SG as a minimum wavelength limit and then translate back to SI, we get a value of order of the Planck length, the indisputably real limit of wavelength.
In the natural geometric unit system the Schwinger limit of linearity of electromagnetic waves becomes equal to 1/(29πα)2 · √(α/2π) 1/u = e / (4π·μB)2, understood as the electromagnetic wavenumber limit in m-1 (translation of the Schwinger limit to natural geometric unit in shown on p. 3 of the article [b], see the articles list below). So the limit for the spacetime observability scale is close to Planck constant but in reality it is the Schwinger limit, Planck constant being only a scaling factor in conversion to the SI from the natural geometric unit system.
The 28 factor and 8 'dimensions'
Professors A. Dragan (of University of Warsaw and National University of Singapore) and A. Ekert (of Oxford University and National University of Singapore) conjecture based on their general relativity extension into superluminal realm**** (based on acknowledging of the existence of superluminal solutions to the Lorentz transformation) that the subluminal spacetime in our light cone (where we can easily recognize what is past and what is future, although the present moment may differ for different observers) and which has three spatial and one temporal dimension (3S+1T) needs to be extended by adding superluminal spacetime (or timespace 😊) outside of our light cone with three time-like and one space-like dimension, altogether totaling to 8 (4 spatial and 4 temporal) dimensions, with subluminal realm (3S+1T) being causal, and superluminal realm (3T+1S) being non-causal because superluminal observers can not differentiate the past from the future. The boundary between the realms being the light (the electromagnetic waves in vacuum).
The 8-dimensional (4D+4D) approach is consistent with the factor of 24c which is an expression in the CGS-Gaussian system of a difference of one order of the factor √(2πα) in the natural geometric unit system.
From the analysis of the magnetic moment anomaly above we can understand that the component α/2π, less concisely but more informatively presented as (√(2πα)/α)-2, is a component oscillating with a speed of light. Then it would mean that the electrostatic charge being of order (√(2πα)/α)-1 is an observable phenomenon reflecting non oscillating real feature of the base object (kind of soliton, maybe) and the (√(2πα)/α) being the double magnetic flux quantum also represents non-oscillating feature.
Then, the rest mass phenomenon would be a feature of the base object that oscillates with squared speed of light α(2-8√(2πα)-3) also meaning that it oscillates (rotates, vibrates, …) in all 8 dimensions, hence the 28 factor.
The
factor 28 appeared already in 1956 in a formula for - then so
called - electron self-energy 2⁄3 e2/(28λ)
proposed by professor R. L. Ingraham*****, as a consequence of his conformal relativity
theory published during his tenure at Institute for Advanced Study, Princeton,
a few years earlier. The conformal relativity theory is also a generalization
of general relativity theory (as is the Dragan-Ekert theory and 5D extensions
of Einstein’s equations to incorporate electromagnetism) assuming conservation
of angles only instead of conservation of angles and 4D lengths as in the
original Einstein’s approach.
The duality of electromagnetic and gravitational actions solving ZPE paradox
Let us note, that elementary charge is of order 1/√(2πα) while electron rest mass is of order 1/√(2πα)3, what then leads to imminent difference in interpretation of electromagnetic action and gravitational action, thus solving the zero-point energy paradox apparent in the quantum electrodynamics. We can see, that electromagnetic action quantum (product of double magnetic flux quantum and elementary charge) 2Φ0e = 1 u2 and electron gravitational rest action (product of electron Compton wavelength and electron rest mass) λCeme = 1 u2 are equal numerically, but their elements are of different real nature, since electromagnetic elements are of order √(2πα) – these do not curve the spacetime - and gravitational elements are of order √(2πα)3 – these do curve the spacetime. The order difference between 1/√(2πα) and 1/√(2πα)3 translates to the difference of the same order of two between the speed of light in vacuum factors in the CGS-Gaussian system counterparts of the formulas for the two types of action: 2Φ0e/c = h and λCemec = h. The SI system lost this difference in translation of the unit of electric charge statC (statcoulomb or Fr - franklin) used in CGS-Gaussian system to the unit of electric charge C (coulomb) used in the SI.
The fast derivation of u:
Let us equalize Newton's
and Coulomb's forces for an equally distanced, respectively: a pair of equal
masses and a pair of positive and negative electric charges with equal moduli,
so
Gm2/r2 = FN = FC
= kq2/r2,
set c = G = k = 1 and
require q = m,
meaning q equals m in a
sense of exerting equal forces at equal distances.
From this set of conditions we get G = 1 = kg/C √|k/G|,
where || denotes dimensionless - numerical only – value, what translates to
m3/(kg · s2) =
kg/C · √|k/G3|,
equivalent to
cm3/(g · s2) = (statC/g)2 from definition of the CGS-Gaussian system of units.
Then we can hypothesize, that electron rest mass relates
to elementary charge as follows
me = G/(m3/(kg · s2))
· kg/C · e/(29πα),
and (using for me, e and α their CODATA-2018
brackets middle values) get
G ≈ 6.673 655 205 · 10-11 m3/(kg
· s2),
After
setting additionally h = 1 we can define u as √h and get conversion factors to
kg, C and m:
u = √|107hG2/c| kg,
u = √|107h/c| C,
u = √|hG/c3| m.
A more detailed introduction to the above findings and an attempt to interpret them is presented in the four articles:
c. Electromagnetic interpretation of spacetime.
d. Natural geometric unit system and eight dimensions.
Janusz "Jani" Kowalski
* WU J., LI Q., LIU J., XUE C., YANG S., SHAO C., TU L., HU Z., LUO J., Progress in Precise Measurements of the Gravitational Constant, Annalen der Physik, 531, (2019) 1900013
** ANDERSON J., SCHUBERT G., TRIMBLE V., FELDMAN M., Measurements of Newton's gravitational constant and the length of day, Europhysics Letters, 110 (2015) 10002
*** PARRA J.L., The Implications of the Sun’s Dragging Effect on Gravitational Experiments, International Journal of Astronomy and Astrophysics, 7 (2017) 174-184
**** DRAGAN A., EKERT A., Quantum principle of relativity, New Journal of Physics, 22 (2020) 033038 (implications of the superluminal solutions of the Lorentz transformation)
***** INGRAHAM R., Classical Maxwell Theory
with Finite Particle Sources, Physical Review 101 (1956) p. 1411.
the post was last updated on Mar 26, 2021 ( on Mar 15, 2021 introduced μB = 27α u )
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