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Exact mathematical Formula that connect 6 dimensionless physical constants
Stergios Pellis
http://physiclessons.blogspot.com/
Department of Physics of the University of Ioannina
Greece
18 October 2021
Abstract
In this paper are a new formula for the Planck length ℓpℓ and a new formula for the Avogadro number NA. Also 9 Mathematical formulas that connect dimensionless physical constants. The 6 dimensionless physical constants are the Proton to Electron Mass Ratio μ,the Fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro number NA,the Gravitational coupling constant αG for the electron and the gravitational coupling constant αG(p) of proton. After a new formula for Gravitational Constant G and new exact formula for the Avogadro number NA. Finally 8 exact Mathematical formulas that connect 6 dimensionless physical constants and a new exact formula for Gravitational Constant G.
1. Introduction
In physics,a dimensionless physical constant is a physical constant that is dimensionless,a pure number having no units attached and having a numerical value that is independent of whatever system of units may be used. The term fundamental physical constant is used to refer to some universal dimensionless constants. A long-sought goal of theoretical physics is to find first principles from which all of the fundamental dimensionless constants can be calculated and compared to the measured values.
In the 1.920s and 1.930s,Arthur Eddington embarked upon extensive mathematical investigation into the relations between the fundamental quantities in basic physical theories,later used as part of his effort to construct an overarching theory unifying quantum mechanics and cosmological physics. The mathematician Simon Plouffe has made an extensive search of computer databases of mathematical formulas,seeking formulas for the mass ratios of the fundamental particles. An empirical relation between the masses of the electron,muon and tau has been discovered by physicist Yoshio Koide,but this formula remains unexplained.
2. Measurement of the 6 dimensionless physical constants
2.1. Measurement of the Fine Structure constant
The 2.018 CODATA recommended value of α is:
α=qe2/4∙π∙ε0∙ħ∙c=0.0072973525693(11)
With standard Uncertainty 0,0000000011×10-3 and Relative Standard Uncertainty 1,5×1010. For reasons of convenience,historically the value of the reciprocal of the fine-structure constant is often specified. The 2.018 CODATA recommended value is given by:
α-1= 137,035999084(21)
With standard Uncertainty 0,000000021×10-3 and Relative Standard Uncertainty 1,5×1010. There is general agreement for the value of α,as measured by these different methods. The preferred methods in 2.019 are measurements of electron anomalous magnetic moments and of photon recoil in atom interferometry. The most precise value of α obtained experimentally (as of 2.012) is based on a measurement of g using a one-electron so-called "quantum cyclotron" apparatus,together with a calculation via the theory of QED that involved 12.672 tenth-order Feynman diagrams:
α-1=137,035999174(35)
This measurement of α has a relative standard uncertainty of 2,5×10-10. This value and uncertainty are about the same as the latest experimental results. Further refinement of this work were published by the end of 2.020,giving the value:
α-1=137,035999206(11)
with a relative accuracy of 81 parts per trillion.
2.2. Measurement of the Proton to Electron Mass Ratio
The 2.018 CODATA recommended value of the Proton to Electron Mass Ratio μ is:
μ=1.836,15267343
With standard Uncertainty 0,00000011 and Relative Standard Uncertainty 6,0×10-11. The value of μ is known at about 0,1 parts per billion. The value of μ is a solution of the equation:
3·μ4-5.508·μ3-841·μ2+10·μ-2.111=0
The 2.018 CODATA recommended value of μ-1 is:
μ-1=me/mp=0,000544617021487
With standard Uncertainty 0,000000000000033 and Relative Standard Uncertainty 6,0×10-11.
2.3. Measurement of the Gravitational coupling constant αG for the electron
In physics,the gravitational coupling constant αG is a constant that characterizes the gravitational pull between a given pair of elementary particles. For the electron pair this constant is denoted by αG. The choice of units of measurement,but only with the choice of particles. The gravitational coupling constant αG is a scaling ratio that can be used to compare similar unit values from different scaling systems (Planck scale,atomic scale,and cosmological scale). The gravitational coupling constant can be used for comparison of length,range and force values. The gravitational coupling constant αG is defined as:
αG=G∙me2/ħ∙c
There is so far no known way to measure αG directly. The value of the constant gravitational coupling αG is only known in four significant digits. The approximate value of the constant gravitational coupling αG is:
αG≃1,751809945728109×10-45
2.4. Measurement of the Gravitational coupling constant αG(p) for the electron
The gravitational coupling constant for the proton is produced similar to the electron,but replaces the mass of electrons with the mass of the protons. The gravitational coupling constant of the proton αG(p) is defined as:
αG(p)=G∙mp2/ħ∙c
The approximate value of the constant gravitational coupling of the proton αG(p) is:
αG(p)≃5,906151273795571x10-39
2.5. Measurement of the ratio Ν1 of electric force to gravitational force between electron and proton
The enormous value of the ratio of electric force to gravitational force was first pointed out by Bergen Davis in 1.904. But Weyl and Eddington suggested that the number was about 1040 and was related to cosmological quantities. The electric force Fc between electron and proton is defined as:
Fc=qe²/4∙π∙ε0∙r²
The gravitational force Fg between electron and proton is defined as:
Fg=G∙me∙mp/r²
So:
Ν1=Fc/Fg
Ν1=qe2/4∙π∙ε0∙G∙me∙mp
Ν1=ke·qe2/G∙mp·me
Ν1=α·ħ∙c/G∙mp·me
The approximate value of the ratio Ν1 of electric force to gravitational force between electron and proton is:
Ν1≃2,26866072471432x1039
2.6. Measurement of Avogadro number NA
The most accurate definition of the Avogadro number NA value involves the change in molecular quantities and,in particular,the change in the value of an elementary charge. The exact value of the Avogadro number NA is:
NΑ=6,02214076×1023
The value of the Avogadro number NA can also be written in numerical expressions:
NΑ≃279=6,044629098×1023
NΑ=84.446.8853=6,02214076×1023
3. New Formulas
3.1. New Formula for the Planck length ℓpℓ
The Bohr radius α0 is defined as:
α0=ħ/α∙me∙c
The reduced Planck constant ħ is:
ħ=α∙me∙α0∙c
So:
ħ2=α2∙me2∙α02∙c2
So:
(ħ∙G/c3)=α2∙me2∙α02∙(G∙me2/ħ∙c)
So:
(ħ∙G/c2)=α2∙α02∙(G∙me2/ħ∙c)
So:
ℓpℓ2=α2∙α02∙αG
So the new formula for the Planck length ℓpℓ is:
ℓpℓ=α∙α0∙αG1/2 (1)
3.2. New formula for the Avogadro number NA
Jeff Yee in his paper «The Relationship of the Mole and Charge» [10],the mole and charge are related by deriving Avogadro’s number from three constants,the Bohr radius,the Planck length and Euler’s number. The Avogadro number NA can be calculated from the Planck length ℓpℓ,the Bohr radius α0 and Euler's number e:
NΑ=α0/2·e·ℓpℓ
So:
NΑ=α0/2·e·α∙α0∙αG1/2
So the new Formula for the Avogadro number NA is:
NΑ=(2·e·α∙αG1/2)-1 (2)
So:
2∙e ∙NA∙α∙αG1/2=1 (3)
4. Mathematical Formulas that connects dimensionless physical constants
4.1. Mathematical Formulas that connects 3 dimensionless physical constants
The exact mathematical formula that connects the mass ratio of proton to electron,the fine-structure constant α and the proton-proton gravitational coupling constant αG(pp) is:
α7=μ7·[αG(pp)·log2(2·π)]
The exact mathematical formula that connects the mass ratio of proton to electron,the fine-structure constant α and the proton-electron gravitational coupling constant αG(pe) is:
α7=μ8·[αG(pp)·log2(2·π)]
The exact mathematical formula that connects the mass ratio of proton to electron,the fine-structure constant α and the gravitational coupling constant of electrons-electrons αG(ee) is:
α7=μ9·[αG(pp)·log2(2·π)]
The exact mathematical formula that connects the Proton to Electron Mass Ratio μ,the fine-structure constant α,the Gravitational coupling constant αG for the electron and the gravitational coupling constant of proton αG(p) is:
αG(p)=μ2·αG
The mathematical formula that connects the Proton to Electron Mass Ratio μ,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro number NA,the Gravitational coupling constant αG for the electron and the gravitational coupling constant of proton αG(p) is:
2∙e ∙NA∙α∙αG1/2=1 (4)
4.2. Mathematical Formulas that connects 4 dimensionless physical constants
The exact mathematical formula that connects the Proton to Electron Mass Ratio μ,the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton and the Gravitational coupling constant αG for the electron is:
α=μ⋅Ν1⋅αG (5)
The exact mathematical formula that connects the Proton to Electron Mass Ratio μ,the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton and the gravitational coupling constant of proton αG(p) is:
α⋅μ=Ν1⋅αG(p) (6)
The exact mathematical formula that connects the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton,the Gravitational coupling constant αG for the electron and the gravitational coupling constant of proton αG(p) is:
α2=N12∙αG ∙αG(p) (7)
The exact mathematical formula that connects the Proton to Electron Mass Ratio μ,the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton and the Avogadro number ΝA is:
μ∙Ν1=4⋅e2⋅α3⋅ΝA2 (8)
4.3. Mathematical Formula that connects 5 dimensionless physical constants
The exact mathematical formula that connects the Proton to Electron Mass Ratio μ,the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro number NA and the Gravitational coupling constant αG for the electron is:
4⋅e2⋅α⋅μ⋅αG2⋅ΝA2⋅Ν1=1 (9)
The exact mathematical formula that connects the Proton to Electron Mass Ratio μ,the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro number NA and the gravitational coupling constant of proton αG(p) is:
μ3=4⋅e2⋅α⋅αG(p)2⋅ΝA2⋅Ν1 (10)
The exact mathematical formula that connects the Proton to Electron Mass Ratio μ,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro number NA,the Gravitational coupling constant αG for the electron and the gravitational coupling constant of proton αG(p) is:
μ=2⋅e⋅αG1/2⋅αG(p)⋅ΝA⋅Ν1 (11)
4.4. Mathematical formula that connects 6 dimensionless physical constants
The mathematical Formula that connects the Proton to Electron Mass Ratio μ,the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro number NA,the Gravitational coupling constant αG for the electron and the gravitational coupling constant of proton αG(p) is:
μ=4⋅e2⋅α⋅αG⋅αG(p)⋅ΝA2⋅Ν1 (12)
5. New formula
5. New formula for Gravitational Constant G
The 2.018 CODATA recommended value of Gravitational Constant G is:
G=6,67430×10-11 m3/kg∙s2
With standard Uncertainty 0,00015×10-11 m3/kg∙s2 and Relative Standard Uncertainty 2,2x10-5. The Gravitational Constant G is:
G=αG∙ħ∙c/me2
From (5):
G=(αG⋅αG(p)⋅Ν1/μ⋅α)⋅(ħ⋅c/me2)
From (13):
G=ħ⋅c/4⋅e2⋅α2⋅ΝA2⋅me2
So the new formula for Gravitational Constant G is:
G=(4⋅e2⋅α2⋅ΝA2)]-1⋅(ħ⋅c/me2) (13)
6. New exact formulas
6.1. Correction number
We need to find a correction number to have more accurate equations. We propose:
2⋅e≃2⋅e⋅(6⋅7⋅φ/52⋅e)13/2 (14)
6.2. New exact formula for the Avogadro number NA
From (2) and (14) the new exact formula for the Avogadro number NA is:
NΑ=[(6⋅7⋅φ/52⋅e)13/2·α∙αG1/2)]-1 (15)
6.3. Mathematical formulas that connects dimensionless physical constants
From (4) and (14):
2∙e ∙(6⋅7⋅φ/52⋅e)13/2⋅NA∙α∙αG1/2=1 (16)
From (8) and (14):
μ∙Ν1=4⋅e2∙(6⋅7⋅φ/52⋅e)13⋅α3⋅ΝA2 (17)
From (9) and (14):
4⋅e2⋅(6⋅7⋅φ/52⋅e)13⋅α⋅μ⋅αG2⋅ΝA2⋅Ν1=1 (18)
From (10) and (14):
μ3=4⋅e2⋅(6⋅7⋅φ/52⋅e)13⋅α⋅αG(p)2⋅ΝA2⋅Ν1 (19)
From (11) and (14):
μ=2⋅e∙(6⋅7⋅φ/52⋅e)13/2⋅αG1/2⋅αG(p)⋅ΝA⋅Ν1 (20)
6.4. Exact mathematical formula that connects 6 dimensionless physical constants
From (12) and (14) the exact mathematical formula that connects 6 dimensionless physical constants is:
μ=4⋅e2⋅(6⋅7⋅φ/52⋅e)13⋅α⋅αG⋅αG(p)⋅ΝA2⋅Ν1 (21)
So:
μ=215⋅313⋅5-26⋅713⋅φ13⋅e-11⋅α⋅αG⋅αG(p)⋅ΝA2⋅Ν1 (22)
So:
α⋅μ-1⋅αG⋅αG(p)⋅ΝA2⋅Ν1=2-15⋅3-13⋅526⋅7-13⋅φ-13⋅e11 (23)
6.5. New exact Formula for Gravitational Constant G
From (13) and (14) the new exact formula for Gravitational Constant G is:
G=[4⋅e2⋅(6⋅7⋅φ/52⋅e)13⋅α2⋅ΝA2]-1⋅(ħ⋅c/me2) (24 )
With exact value:
G=6,67430×10-11 m3/kg∙s2
7. Conclusions
In this paper are a total of 22 new formulas. Α new formula for the Planck length ℓpℓ:
ℓpℓ=α∙α0∙αG1/2
Α new formula for the Avogadro number NA:
NΑ=(2·e·α∙αG1/2)-1
9 Mathematical formulas that connect dimensionless physical constants:
α=μ⋅Ν1⋅αG
α⋅μ=Ν1⋅αG(p)
α2=N12∙αG ∙αG(p)
μ∙Ν1=4⋅e2⋅α3⋅ΝA2
2⋅e⋅α⋅ΝA⋅αG1/2=1
4⋅e2⋅α⋅μ⋅αG2⋅ΝA2⋅Ν1=1
μ3=4⋅e2⋅α⋅αG(p)2⋅ΝA2⋅Ν1
μ=2⋅e⋅αG1/2⋅αG(p)⋅ΝA⋅Ν1
μ=4⋅e2⋅α⋅αG⋅αG(p)⋅ΝA2⋅Ν1
Α new formula for Gravitational Constant G:
G=(4⋅e2⋅α2⋅ΝA2)]-1⋅(ħ⋅c/me2)
Α new exact formula for the Avogadro number NA:
NΑ=[(6⋅7⋅φ/52⋅e)13/2·α∙αG1/2)]-1
8 exact Mathematical formulas that connect 6 dimensionless physical constants:
2∙e ∙(6⋅7⋅φ/52⋅e)13/2⋅NA∙α∙αG1/2=1
μ∙Ν1=4⋅e2∙(6⋅7⋅φ/52⋅e)13⋅α3⋅ΝA2
2⋅e ∙(6⋅7⋅φ/52⋅e)13/2⋅α⋅μ⋅ΝA⋅αG1/2=1
4⋅e2⋅(6⋅7⋅φ/52⋅e)13⋅α⋅μ⋅αG2⋅ΝA2⋅Ν1=1
μ3=4⋅e2⋅(6⋅7⋅φ/52⋅e)13⋅α⋅αG(p)2⋅ΝA2⋅Ν1
μ=2⋅e∙(6⋅7⋅φ/52⋅e)13/2⋅αG1/2⋅αG(p)⋅ΝA⋅Ν1
μ=4⋅e2⋅(6⋅7⋅φ/52⋅e)13⋅α⋅αG⋅αG(p)⋅ΝA2
α⋅μ-1⋅αG⋅αG(p)⋅ΝA2⋅Ν1=2-15⋅3-13⋅526⋅7-13⋅φ-13⋅e11
A new exact formula for Gravitational Constant G:
G=[4⋅e2⋅(6⋅7⋅φ/52⋅e)13⋅α2⋅ΝA2]-1⋅(ħ⋅c/me2)
References
[3] www.math.stackexchange.com/
[4] www.mathworld.wolfram.com/
[6] http://physics.nist.gov/cuu/Constants/
[7] Stergios Pellis Unification Archimedes Constant π,Golden Ratio φ,Euler's Number e and Imaginary Number i
https://vixra.org/pdf/2110.0043v1.pdf, 2.021
[8] Stergios Pellis Exact Formula for the Fine Structure Constant α in Terms of the Golden Ratio φ
https://vixra.org/pdf/2110.0053v1.pdf, 2.021
[9] Stergios Pellis Exact Mathematical Expressions of the Proton to Electron Mass Ratio with Symmetrical Shape and Physical Meaning
https://vixra.org/pdf/2110.0071v1.pdf, 2.021
[10] Jeff Yee The Relationship of the Mole and Charge.
