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We find a simple precise formula for the gravitational constant G relating it to the electron charge, electron mass, the vacuum dielectric constant and the speed of light (or magnetic permeability of the vacuum) in power of the fine structure constant i.e. relating the gravitational constant to the Planck constant through others which also well exist without the quantum mechanics therefore relating two fundamental constants as not independent through the parameters of the electron and the electromagnetic properties of the vacuum.

The ratio of the proton to the electron mass was once noticed [

μ = m p m e = 6 π 5 = 1836.12 (1)

The 1951 ratio can be further adjusted to the most accurate current value by the simple tuning multiplicative factor 2 36821 i.e. 36821-st root of 2

μ = 6 π 5 2 36821 = 1836.152673 (2)

We find here a simple precise analytical formula for the gravitational constant G relating it to the electron charge, electron mass, the vacuum dielectric constant and the speed of light (or magnetic permeability of the vacuum) in power of the fine structure constant as it was only a constant of the Quantum Electrodynamics (QED) [

G s 0 = ℏ c m e 2 = 3.80994 × 10 34 m 3 kg ⋅ s 2 (3)

2) It is proportional as the QED perturbation expansion dimensionless coupling constant i.e. in power of the fine structure constant α so it formally looks like the n-th order perturbation correction to the electron-electron Coulomb interaction through the Yukawa exchange mechanism [

The preliminary formula yields

G = 4 3 α 20 ( e m e ) 2 1 4 π ε 0 = 4 3 α 21 ℏ c m e 2 = 6.79769 × 10 − 11 m 3 kg ⋅ s 2 (4)

i.e. gravitational constant is proportional in (4/3) of the 20-th power of the fine structure constant α = e 2 / ( 4 π ε 0 ℏ c ) ≈ 1 / 137 , to the square of the ratio between the electron charge e and its mass m e divided by 4 π times the vacuum dielectric constant ε 0 or in 21-th power of the fine structure constant to the strong gravitational constant defined here as

G s = 4 3 ℏ c m e 2 = 5.07992 × 10 34 m 3 kg ⋅ s 2 , (5)

which shows immediately the electron-electron gravitational interaction as an effective ultra-small electromagnetic (electrostatic) interaction of some order in α and that the gravitational energy corrections to the electron-electron electromagnetic interactions are in the 20-th power of the fine structure constant. The value is that obtained by Newton [

G = 1 2 n 2 ℏ c m n 2 (6)

i.e. electron for

n ≈ ( 1 α ) 2 1 / 2 ( 3 8 ) 1 / 2 (7)

n = 16740933742287336243200

as a Big Bang ultra-high random “Rydberg toss” excitation which is approximately assuming that the elementary particle is a discharged Black Hole which self binds itself orbiting itself on a circular orbit with the Schwarzschild radius without the relativistic mass gain with the speed of light on its own mass with the fractional closed multiple-spiral near-circular orbit “hydrino” [

1 N λ N c = 1 N h m N c = 2 π r s = 2 π × 2 G m N c 2 (8)

or

G = 1 2 N ℏ c m n 2 (9)

or

N = n 2 . (10)

It also agrees up to the small multiplicative factor 2 2 with self-consistently solving the exact nonrelativistic Schrödinger equation for the Black Hole to obtain its possible quantum mass spectrum [

The leading order of the Lamb shift [

δ E g = − 2 × 6.8 × 4 3 α 20 eV = − 3.3 × 10 − 42 eV (11)

and so 2 × 6 × π 5 of that for the Hydrogen and twice of that for the Deuterium ground state respectively. Assuming the somehow perfectly Mössbauer-rigid nucleus in both cases not to change the Kepler reduced mass and eliminate the native isotope effect it leads mathematically to the Trojan or anti-Trojan Wave Packet generation [^{68} meters radius (10^{42} times the size of the Universe)i.e. with

n = R ∞ δ R ∞ , (12)

n = 10 39 . It also takes about 10^{27} seconds (10^{9} ages of the Universe) (and so only 10^{−27} of it within one second) to cause one interference fringe shift assuming two counter rotating Trojan Wave Packets around the principal quantum number n = 10 which belong to two Hydrogen atom of two different isotopes can be interfered between each other with the method of delta pulse ionization described in [

The simple correcting prefactor 1 / 2 38 (inverse of the 38-th root of 2) is further adjusting the value to the best known experimental [

G = 1 2 38 4 3 α 21 ℏ c m e 2 = 6.674330065689801 × 10 − 11 m 3 / ( kg ⋅ s 2 ) ≈ ( 130923 / 100000 ) α 21 ℏ c m e 2 . (13)

It may be interpreted as a ratio factor between the gravitational self energy radius r g and the electromagnetic electron radius r e when the gravitational self energy deficiency in modulus is assumed to be the ( 4 / 3 ) α 20 quantum fraction of the electrostatic one i.e.

E g = − G m e 2 r g = − 4 3 α 20 E e (14)

E e = m e c 2 = 1 4 π ε 0 e 2 r e (15)

i.e. r e = 2 38 r g . This is in inequality r e > r g ( r e = r g 2 8 ) agreement with the classical model of the electron with the spin ℏ / 2 assuming slightly different rotating Gaussian mass and charge distributions inside the electron

ρ e ( r ) = e N e − r 2 / r e 2 (16)

ρ m ( r ) = m e N e − r 2 / r g 2 (17)

predicting the anomalous electrong-factor 2 if only

( r e r m ) 8 = 2 , (18)

as it was built from the solid electrified matter but with the locally variable density and non-proportional variable level of electrification. The refined “Rydberg” quantum number of the electron as the excitation itself including the correction is now n = 16894315429949215866880 (

We so find (

G = 4 3 2 38 ℏ c m e 2 α 21 = 1.30923 × ℏ c m e 2 α 21 = 6.67433 × 10 − 11 m 3 kg ⋅ s 2 , (19)

where G is the gravitational constant,

ℏ = 1.05457181710 − 34 J ⋅ s (20)

Level | G_{s} formula | G_{s} value (m^{3}·kg^{−1}·s^{−2}) | G = G s α 21 value (m^{3}·kg^{−1}·s^{−2}) |
---|---|---|---|

(1) | ℏ c m e 2 | 3.80994 × 10^{34} | 5.09789 × 10^{−1}^{1} |

(2) | 4 3 ℏ c m e 2 | 5.07992 × 10^{34} | 6.79769 × 10^{−11} |

(3) | 4 3 2 38 ℏ c m e 2 | 4.98811 × 10^{34} | 6.67433 × 10^{−11} |

(4) | 4 3 2 38 2 160000 ℏ c m e 2 | 4.98809 × 10^{34} | 6.67430 × 10^{−11} |

c = 299792458 m s

m e = 9.1093837015 × 10 − 31 kg

α = 1 / 137.035999206 = 7.2973525693 × 10 − 3 ,

ℏ is the reduced Planck constant, c is the speed of light, m e is the electron mass and α is the fine structure constant i.e. with the perfect agreement with the most recent CODATA [

G s = 4 3 2 38 ℏ c m e 2 = 4.98811 × 10 34 m 3 kg ⋅ s 2 . (21)

The further ultra-fine tuning prefactor placing the value perfectly at the center of the CODATA error field can be found as 1 / 2 160000 (hundred sixty thousandth root of 2) i.e.

G = 4 3 2 38 2 160000 ℏ c m e 2 α 21 = 1.30923 × ℏ c m e 2 α 21 = 6.67430 × 10 − 11 m 3 kg ⋅ s 2 , (22)

with the method of power 2 approximants of the number which is close to 1 as for example for π / 3

π = 3.141592653589793 (23)

π ≈ 3 2 15 / 2 7509 ⋯ = 3.1415923582882304

etc. for which the ultra-refined

G s = 4 3 2 38 2 160000 ℏ c m e 2 α 21 = 4.98809 × 10 34 m 3 kg ⋅ s 2 . (24)

We find here the “Theory of Everything” formula relating the gravitational constant to the Planck constant through the electron mass, speed of light and the fine structure constant (electric charge and the dielectric permittivity of the vacuum) suggesting that the electron quantum mechanics is a result of the gravitational and dielectric properties of the vacuum or alternatively the gravity of electrons is an ultra-weak QED effect of longitudinal virtual quantum photons interacting with the 1-st scalar component of the electron 4-current and the Yukawa-like interaction between charges, which is insensitive to the sign of the charge. The 4/3 and 1/2^{1/38} (1 over 38-th root of 2) coefficients adjusting the 21-st α power order of the so defined strong gravity constant G s 0 = ℏ c / m e 2 to the first Newtonian 15-teen century value and than the perfect CODATA value are easy to remember as having the 4/3 factor of the sphere volume and 1/38 in the 2 exponent as the calculator LCD display altered value of the approximate fine structure constant 1/137. Yet another interpretation is that the parameters of the electron as the elementary particle like the mass m and charge e are fully determined by the properties of the vacuum and the quantum mechanics i.e. the value of ℏ by some Planck particle like mechanism of de Broglie standing wave Black Hole self no-escape for example with fractional angular momentum. Perfect CODATA values of all involved quantities must be used for the G CODATA value agreement.

Author would like to thank W. Tarkowski, the author of [

The author declares no conflicts of interest regarding the publication of this paper.

Kalinski, M. (2021) QED-Like Simple High Order Perturbative Relation between the Gravitational Constant G and the Planck Constant h. Journal of High Energy Physics, Gravitation and Cosmology, 7, 595-601. https://doi.org/10.4236/jhepgc.2021.72034