QED-Like Simple High Order Perturbative Relation between the Gravitational Constant G and the Planck Constant h

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.


Introduction
The ratio of the proton to the electron mass was once noticed [1] as magical 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) [2], i.e. relating the gravitational constant to the Planck con-How to cite this paper: Kalinski stant 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. We asked for the following conditions: 1) The gravitational constant is proportional to the natural strong gravity [3], gravitational constant for the electron which already can be constructed from the dimensional analysis of units, simply 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 [4] of the logitudinal photons but is insensitive to the sign of charge.
3) The front rest coefficient is around 1. 4) In similarity to the proton electron mass ratio, it can be expressed as a simple combination of the integer and the irrational numbers and 5) The analytical formula is perfectly matching the best known experimental CODATA value.

The Preliminary Newtonian Value
The preliminary formula yields 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 [5] by the rough estimate of the Earth density as the 6 times that of water. This is in agreement with [6] predicting the approximate (when the complex i, i.e. electron for Journal of High Energy Physics, Gravitation and Cosmology 16740933742287336243200 n = 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" [7] Bohr quantization i.e. is a fractional Planck particle when only the integer fraction of its Compton wavelength is its Schwarzschild circumference ( 2π times the Schwarzschield radius) of its mass i.e.
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 [8] treated as the self-bound Hydrogen atom while equalizing its Rydberg energy to the rest relativistic energy of the minus half of its effective mass but considering also the "hydrino" fractional quantum numbers i.e.  . 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 10 n = 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 [12].

The Refined Value
The simple correcting prefactor 38  It may be interpreted as a ratio factor between the gravitational self energy radius g r and the electromagnetic electron radius e r when the gravitational self energy deficiency in modulus is assumed to be the ( ) 20 4 3 α quantum fraction of the electrostatic one i.e.
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 16894315429949215866880 n = (Table 1). We so find (Figure 1 where G is the gravitational constant,    value only if the right side quantities are also the CODATA values. This expresses explicitly that the gravitational interaction between the electrons may be considered as an exotic ultra-weak Coulomb attractive electromagnetic (electrostatic) interaction of the 20-th order in the fine structure constant α and therefore is normally neglected in QED as much below the highest 7-th order in α normally considered there. It also defines electron strong gravity constant as The further ultra-fine tuning prefactor placing the value perfectly at the center of the CODATA error field can be found as 160000 1 2 (hundred sixty thousandth root of 2) i.e.
with the method of power 2 approximants of the number which is close to 1 as