Draft:Theoretical Formula for Calculating G Source: en.wikipedia.org/wiki/Draft:Theoretical_Formula_for_Calculating_G

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The gravitational constant, denoted by G, is an empirical physical constant involved in the calculation of gravitational effects, primarily in Newton's law of universal gravitation and Albert Einstein's general relativity. It is also known as the universal gravitational constant or the Newtonian constant of gravitation.

Until 2024, no theoretical equation to calculate G was known. In 2024, Dr. Policarpo Yoshin Ulianov developed a gravitation model ^[1] that deduced a formula to calculate G based on three fundamental constants of quantum mechanics:

$${\displaystyle G={\frac {P_{P}L_{P}^{4}}{m_{P}^{2}}},}$$

where: $${\displaystyle P_{P}={\text{Planck Pressure}}(4.633\times 10^{113}\,{\text{Pa}}).}$$ $${\displaystyle L_{P}={\text{Planck Length}}(1.616\times 10^{-35}\,{\text{m}}).}$$ $${\displaystyle m_{P}={\text{Planck Mass}}(2.176\times 10^{-8}\,{\text{kg}}).}$$

By substituting these Planck values into the equation, the calculated theoretical value of G is:

$${\displaystyle G=6.67418400\times 10^{-11}\,{\text{m}}^{3}\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}.}$$

In August 2018, a Chinese research group announced new measurements based on torsion balances^[2]:

• Time-of-Swing (TOS) method: 6.674184(78)×10⁻¹¹ m³⋅kg⁻¹⋅s⁻²
• Angular Acceleration Feedback (AAF) method: 6.674484(78)×10⁻¹¹ m³⋅kg⁻¹⋅s⁻².

These values are considered the most accurate measurements of G as of 2024. However, since the two values have the same uncertainty but differ significantly, one of them is likely affected by systematic errors. An average value (accurate only to three decimal places) need be considered:

$${\displaystyle G=6.674(33)\times 10^{-11}\,{\text{m}}^{3}\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}.}$$

Given that the theoretical G value matches the TOS measurement, the most accurate (accurate to six decimal places) value of G currently know is the one measured in the TOS method experiment:

$${\displaystyle G=6.674184(77)\times 10^{-11}\,{\text{m}}^{3}\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}.}$$

This suggests that the AAF measurement might involve systematic errors or overlooked uncertainties.

Considering that the constants in the G formula have the following errors: $${\displaystyle {\text{Er}}{L_{P}}={\text{Theoretical error in the value of Planck Length}}(0.0000000130\times 10^{-35},{\text{m}}).}$$ $${\displaystyle {\text{Er}}{m_{P}}={\text{Theoretical error in the value of Planck Mass}}(0.0000000130\times 10^{-8},{\text{kg}}).}$$ $${\displaystyle {\text{Er}}_{P_{P}}={\text{Theoretical error in the value of Planck Pressure}}(0.0000000130\times 10^{113},{\text{Pa}}).}$$ The error associated to the theoretical calculation of G can be calculated using:

$${\displaystyle {\text{Er}}_{G}={\sqrt {(4\cdot {\text{Er}}{L_{P}})^{2}+(2\cdot {\text{Er}}{m_{P}})^{2}+{\text{Er}}{P_{P}}^{2}}}.}$$

Substituting the values:

$${\displaystyle {\text{Er}}_{G}=2.298197854791\times 10^{-18}.}$$

This generates the result:

$${\displaystyle G=6.67418400\times 10^{-11}\pm 2.30\times 10^{-18}\,{\text{m}}^{3}\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2},}$$

which can be expressed as:

$${\displaystyle G=6.674184(23)\times 10^{-11}\,{\text{m}}^{3}\cdot {\text{kg}}^{-1}\cdot {\text{s}}^{-2}.}$$

Note that it is the same G value measured by the TOS method experiment but which a half of error and so the Ulianov equation that calculate G provide the most accurate value of G know until 2024.

As the equation:

$${\displaystyle G={\frac {P_{P}L_{P}^{4}}{m_{P}^{2}}},}$$

has no adjustment coefficients and generates a value more precise than the most accurate experiment conducted to date, this cannot be considered a mere coincidence.

This equation was developed within the context of the Le Sage's theory of gravitation, the concept of Einstein-Rosen bridges (wormholes), and the calculation of the density and pressure of the Higgs field. Therefore, it is based on solid historical foundations, but within a framework that is different and not yet recognized in modern physics.

Thus, it is better to consider the Ulianov equation for calculating G as an empirical formula that works precisely, even though it was derived through a path unfamiliar to modern physics.

In this way, the Ulianov theoretical equation for G maybe provide a key to unlocking not only the true meaning of gravity, but also as the first equation of a new Theory of everything, or a new Unified field theory that connects Newtonian mechanics, General relativity, and Quantum mechanics.