8. Quantized time and distance

The following equation is the relation between force and energy in subatomic particles.

 

$dv/dt=cv/r$

 

Applying Lorentz effect, this becomes a wave equation. Therefore, above equation is showing the motion in subatomic field. Modifying the equation, that is similar to prime-counting function.

 

$$ct=\int{\frac{dr}{\ln{(r/r_0)}}}= \int{\frac{dr}{\beta}} \approx \pi(r). (\beta=v/c=\ln{(r/r_0)})$$

 

If radius $r$ is quantized, it is possible that we assume $\beta, \gamma$ are quantized and $ct$ is proportional to prime-counting function and $r$ is related with prime number. This phenomenon also appears in the distribution function mixed with particle and wave energy.

$$\Pi(\theta)=\int_{\theta}^{\frac{\pi}{2}}{\frac{\sin\theta}{1+\cos\theta}e^{-1/\cos\theta}}d\theta=-\int_{0}^{x}{\frac{dt}{\ln{t}}}+e \int_{0}^{\frac{x}{e}}{\frac{dt}{\ln{t}}} $$

Meanwhile, in the gravitational field, the following equation holds true, and light(or photon) is also satisfied here as well as in the subatoms above.

 

$dv/dt=v^2/r$

 

And the solution to this differential equation is $r=r_0e^{at}$, and in the case of light, $a$ must be in the form of a pure imaginary number. It is not yet clear what this means if $a$ is not a purely imaginary number, especially if it is faster than the speed of light. In the book, there is a comparison that the velocity of light derived from a gravitational field and the velocity of light derived in a subatomic field are the same. The reason is because light has the same velocity both in a subatomic and in a gravitational field.

 

Reference:

Dependent Variable, Independent Variable

General Relativistic Quantum Mechanics

General Relativistic Quantum Mechanics