Planck constant h and the gravitational constant G

The relationship between Planck's constant h and the gravitational constant G .               


Planck's constant h defines the limits of applicability of classical and quantum physics.  It shows how applicable classical mechanics is to a given physical system. Currently, since 2019, the value of the Planck constant is considered fixed and exactly equal to the value      

         h = 6.62607015 x10^-34 kg x m^2 x s^-1 (J•s).

The reduced Planck constant, equal to Planck's constant divided by 2 pi and denoted as "h with a dash", is also widely used.
The gravitational constant G is a fundamental physical constant, a constant of gravitational interaction. 
Numerically, it is equal to the modulus of the gravitational force acting on a point body of a unit mass from another similar body located at a unit distance from it.

The accuracy of measurements of the gravitational constant is several orders of magnitude lower than the accuracy of measurements of other physical quantities.
In units of the International System of Units (SI), the value of the gravitational constant recommended by the Data Committee for Science and Technology (CODATA) for 2020:

G = 6.67430(15)x 10^-11 m3 x s^-2 x kg^-1, or H x m^2 x kg^-2.

In August 2018, physicists from China and Russia published the results of new measurements of the gravitational constant with improved accuracy (error 0.0012%) in the journal "Nature". Two independent methods were used — measurement of the swing time of the torsion bar and measurement of angular acceleration, G values were obtained, respectively:

G = 6.674184(78) x 10^-11 m3 x s^-2 x kg^-1;
G = 6.674484(78) x 10^-11 m3 x s^-2 x kg^-1.

Both results are within two standard deviations of the recommended CODATA value, although they differ from each other by ~ 2.5 standard deviations.
If we compare the value of the gravitational constant currently accepted and the value of Planck's constant, dividing G by h, ignoring the dimension of the values of the constants, we get the following value :   

G / h = 6.6743015 x 10^-11/ 6.62607015 x 10^-34 = 1.00728 x 10^23

In the resulting value of 1, 00728 x 10^23, the value of 1.00728 can be represented as ( 1 + 0.00728 ).  Comparing the value of 0.00728 with the value of the fine structure constant equal to L = 0.00729735, it can be seen that they practically coincide.  Such a coincidence cannot be an accident.  If we now take and substitute the value of the fine structure constant into equality

h x(1+L) x 10^23 = G ,

then the following value will be obtained :               

6,62607015 x 10^-34 x(1+0,00729735)x 10^23 = 6,6744229 x 10^-11

From the result obtained, it can be seen that the value of 6.6744229 fits into the value of the gravitational constant obtained in 2018 in the experiment with a change in angular acceleration. This value is the exact value of the gravitational constant G, obtained by calculation.   
  At first glance, the discrepancy between the dimensionality of physical quantities in the formula can be explained by the fact that, as in Coulomb's law, the coefficient K has a value of 10^-7 Gn /m contributing its dimension to this formula, so in the above formula 10^23 contributes its dimension.    The dimension 10^23 should be equal to m x s^-1 x kg^-2.

The magnitude and dimension of h x ( 1 + 0.00729735 ) x 10^23 coincide with the dimension of G if ( 1 + 0.0072973525 ) x 10^23 is represented as:

 ( 1 + 0.00729735 ) x 10^23 = c/( 2 x n x Mp^2 ), where 

c - is the speed of light in a vacuum,

n - is the number of pi ( 3.14159 )

Mr - is the Planck mass, currently considered equal to 2.176434 x 10^-8 kg.After recalculation, a value equal to Mp = 2.1764143 x 10 ^-8 kg is obtained.


It can be said that Planck's constant h and the gravitational constant G are not determined by the characteristics of particles and are not determined by their properties, but are determined by the properties of the four-dimensional space in which these particles exist, and the four-dimensional space determines the energy properties of particles.