Ok so let's make some further statements: Using the relationships we obtained before, the square of the charge flux is:

[math]\phi^2 = \frac{n\hbar^2}{4 e^4} = \frac{1}{4c^2}\frac{G^2m^4}{e^4} = \frac{1}{c^2} = \epsilon_0 \mu_0[/math]

and I am being dragged away, will update the post instead of continuing in a new one later.

**Cont.**

Since the index of refraction is

[math]n = \sqrt{\frac{\epsilon_G \mu_G}{\epsilon_0 \mu_0}}[/math]

(from previous investigations into the gravitational aether), the ratio is easily seen to be dimensionless and related to the index of refraction [theoretically] as:

[math]\frac{1}{n^2} = \frac{\phi^2}{\epsilon_G \mu_G} = \frac{1}{\epsilon_G \mu_G}\frac{n\hbar^2}{4 e^4} = \frac{1}{4c^2}\frac{1}{\epsilon_G \mu_G} \frac{G^2m^4}{e^4} = \frac{\epsilon_0 \mu_0}{\epsilon_G \mu_G}[/math]

What I found interesting was that a variation in the flux will led to different energy levels, very similar to the instructional equations I built for the transition of a black hole under discrete quantum processes:

[math]\frac{\Delta \phi}{\epsilon_G \mu_G} = \frac{\phi^2_1 - \phi^2_2}{\epsilon_G \mu_G} = \frac{1}{n^2_1} - \frac{1}{n^2_2} [/math]

This type of aether theory had the unusual properties discovered by various asian scientists that the speed of light is not technically a constant and that the permittivity and permeability, both for the electric and gravitational interpretations equally can vary: [math]\frac{\epsilon_0 \mu_0}{\epsilon_G \mu_G}[/math] - this allowed the new aether theory to allow light to escape black holes, similar to how it takes about 40,000 years for a photon to be radiated from the sun from a core, the radiation of a black hole insists the speed of light can only approach zero but never reach it. Using similar arguments from the transition formula for the black hole, the density of nodes (or fluctuations within a given volume) is calculated in the following way:

[math]\frac{n \mathbf{J} c}{\epsilon_G \mu_G}(\phi^2_1 - \phi^2_2) = \alpha\ \int\ (\frac{n\hbar c}{n^2_1} - \frac{n\hbar c}{n^2_2})\ k\ dk^3[/math]

with [math]\mathbf{J}[/math] as the spin density and [math]k[/math] being the wave number, or alternatively can be written with the difference of the flux charge density [math]\Phi[/math], and from gauge invariance, [math]n\hbar c = Gm^2[/math],

[math]\frac{n \hbar c}{\epsilon_G \mu_G}(\Phi^2_1 - \Phi^2_2) = \beta\ \int\ (\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2})\ k\ dk^3[/math]

Moreover, we created the transition formula for a black hole under discrete quantum processes from a set of laws which gave rise to the use of the Reynolds number - we know now from previous exploration of the drag formula that the Bejan number is related to the Reynolds number as:

[math]R_e = \frac{\rho \ell v}{\mu}[/math]

[math]\mu[/math] - dynamic viscocity with the drag related to the Bejan and Reynolds number as:

[math]f = \frac{2F}{\rho v^2 A} = \frac{A_b}{A_f}\frac{B}{R^2_e}[/math]

with [math]B[/math] being the Bejan number and [math]R^2_e[/math] being the square of the Reynolds number. The transition formula for the black hole used the same quantum principles predicted from Reynolds own transition formula:

[math]\frac{1}{\Delta \lambda} = R_e(\frac{1}{n^2_1} - \frac{1}{n^2_2}) [/math]

Let's just cover those arguments for the transition formula for the black hole, as to derive the quantization of the black hole, in terms of discrete quantum mechanical processes, I defined the Rydberg constant in terms of the gravitational coupling constant we get:

[math]R_e = \frac{\alpha_G^2}{4 \pi \lambda_0} = \frac{1}{\hbar c}\frac{Gm^2}{4 \pi \lambda_0} = \frac{1}{\hbar c} \frac{Gm^3c}{4 \pi \hbar} = \frac{Gm^2}{\hbar c}\frac{p}{4 \pi \hbar}[/math]

Even though the Rydberg constant was first applied to hydrogen atoms, it could be derived from fundamental concepts (according to Bohr). In which case we may hypothesize energy levels:

[math]\frac{1}{\Delta \lambda} = R_e(\frac{1}{n^2_1} - \frac{1}{n^2_2})[/math]

Plugging in the last expressions we get an energy equation:

[math]\Delta E_G = \frac{n\hbar c}{\Delta \lambda} = \frac{1}{4 \pi }\frac{m_0v^2}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{4 \pi \lambda_0}(\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2}) = \frac{p}{4 \pi \hbar}(\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2})[/math]

The relativistic gamma appears from the definition of the deBroglie wavelength. The ultimate goal of creating this equation, was to explore an idea I can trace back to Lloyd Motz suggesting a stable black hole. Black holes, even the microscopic kind, cannot be mathematically ruled out from physics (but later I will show in a logical sense why the would be unstable) - and if the work of Hawking is to be taken seriously, including analysis provided by Motz, then a black hole system really is deduced from a discrete set of quantum processes - those discrete processes always lead to an increase in the entropy of a system like a black hole.

I noticed that the equation we derived:

[math]\frac{n \mathbf{J} c}{\epsilon_G \mu_G}(\phi^2_1 - \phi^2_2) = \alpha\ \int\ R^2_e(\frac{n\hbar c}{n^2_1} - \frac{n\hbar c}{n^2_2})\ k\ dk = \alpha\ \int\ B(\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2})\ k\ dk [/math]

is expressible by using the square of the Reynolds number, since it has units of inverse length and so by consequence is related to the Bejan number: From here, we can speculate that the wavenumbers carry the same information as the inverse of the front area:

[math]\frac{n \mathbf{J} c}{\epsilon_G \mu_G}(\phi^2_1 - \phi^2_2) = \alpha\ \int\ R^2_e(\frac{n\hbar c}{n^2_1} - \frac{n\hbar c}{n^2_2})\ \frac{1}{A_f} = \beta\ \int\ B(\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2})\ \frac{1}{A_f} [/math]

Dividing through by the back area term we obtain

[math]\frac{1}{\epsilon_G \mu_G}\frac{nJ c}{R}(\phi^2_1 - \phi^2_2) = \alpha\ \int\ R^2_e(\frac{n\hbar c}{n^2_1} - \frac{n\hbar c}{n^2_2})\ \frac{A_b}{A_f} = \beta\ \int\ B(\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2})\ \frac{A_b}{A_f} [/math]

We can see how we might obtain a drag term from the last two equalities by dividing through by the Reynolds number squared,

[math] \alpha\ \int\ (\frac{n\hbar c}{n^2_1} - \frac{n\hbar c}{n^2_2})\ \frac{A_b}{A_f} = \beta\ \int\ \frac{A_b}{A_f}\frac{B}{R^2_e}(\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2})\ [/math]

Since the Reynolds number is inverse, it has units of length squared as we obtain a simplification:

[math]\alpha\ \int\ (\frac{n\hbar c}{n^2_1} - \frac{n\hbar c}{n^2_2})\ \frac{A_b}{A_f} = \beta\ \int\ \frac{A_b}{A_f}\frac{B}{R^2_e}(\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2})\ [/math]

By using the formula

[math]f = \frac{2F}{\rho v^2 A} = \frac{A_b}{A_f}\frac{B}{R^2_e}[/math]

It can be written in the following way:

[math]\alpha\ \int\ (\frac{n\hbar c}{n^2_1} - \frac{n\hbar c}{n^2_2})\ \frac{A_b}{A_f} = \beta\ \int\ \frac{2F}{\rho v^2 A}(\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2}) = \beta\ \int\ f\ (\frac{Gm^2}{n^2_1} - \frac{Gm^2}{n^2_2}) [/math]

**Edited by Dubbelosix, 07 August 2019 - 07:53 AM.**