9. Lorentz Effect in the Gravitational Field

There is the equation $ds^2 = c^2dt^2-dx^2-dy^2-dz^2$ in special relativity(Special relativity). Special relativity theory assumes that gravity is constant. So, we can set $dr^2= dx^2+dy^2+dz^2$. If the $\gamma$ effect occurs along the arc of a circle with a constant diameter, it also affects the diameter $r$. In other words,

 

$2\pi{dr}=2\pi\gamma{dr'}=>dr=\gamma{dr'}$

 

Lorentz effect in the gravitational field

 

$r$ direction is the direction in which the acceleration of an object occurs as well as the gravity direction. Therefore, it is reasonable that applying that to gravitational field equation $dv/dt=v^2/r$.

 

In the special relativity, there is a relationship with a slight change $dx=\sin\theta\cos\phi{dr}, dy=\sin\theta\sin\phi{dr}, dz=\cos\theta{dr}$

(Dependent Variable, Independent Variable ).

That is,

 

(1)  $dr_{xy}=dr'=\sin\theta{dr}=1/\gamma{dr}, dr_z=\cos\theta{dr}=\beta{dr}$

 

This is the properties of light. The motion of $dr'$ is shown as frequency(or wave length) as if projected onto the $xy$ plane of our sight or measurement(Absolute Time, Relative Time).

 

In order for this relationship to be established, the following relationship must exist in a three-dimensional spherical coordinate system, .

 

$dr^2_{xy}=dx^2+dy^2=(d(r\sin\theta\cos\phi))^2 + (d(r\sin\theta\sin\phi))^2$

 

$=\sin^2\theta{dr^2}+r^2(\cos^2\theta{d\theta^2}+\sin^2\theta{d\phi^2}) + 2df(r, \theta, \phi)$

 

$dz^2=(d(r\cos\theta))^2=(\cos\theta{dr}-r\sin\theta{d\theta})^2$

 

$=\cos^2\theta{dr^2}+r^2\sin^2\theta{d\theta^2}-2r\cos\theta\sin\theta{d\theta}{dr} $

 

Assuming the length is a result of dot product of unit vectors, 

 

(2)  $2df(r, \theta, \phi) - 2r\cos\theta\sin\theta{d\theta}{dr} =0$

 

And the minimum condition in which the equation (1) is valid is that 

 

From $dx^2+dy^2+dz^2=\sin^2\theta{dr^2}+\cos^2\theta{dr^2} $,

 

the sum of the other terms,

$${d\theta^2}+\sin^2\theta{d\phi}^2=0, \phi=\pm i\ln\frac{\sin\theta}{1+\cos\theta}$$

 

A complex function between $\phi$ and $\theta$ is that one is perpendicular to the other($i$), and appears to physically represent an electromagnetic field. According to these results, we can conclude that $\phi$ is the dependent variable of $\theta$. Also, natural forces result from coordinate transformation.

 

In equation (2), all terms are eliminated.

The equation (2) is solved as follows.

 

$\cos\theta-\sin\theta=0, \sin\phi\cos\phi=0, r|\sin\theta|=r_N$. $r_N$ is constant and seems to be related with Planck's constant, strong force.

 

If $\theta=f(N)|\sin\theta|=GM_N/r_Nc^2, f(N)=\pi((N-1)!+1)/2N$ and $N$ is a prime number,

 

$r_N=GM_N/f(N)c^2$.

 

If $r_N/r_h=N$,

 

$GM_N/c^2=r_Nf(N)=Nf(N)r_h=\pi((N-1)!+1)r_h/2$.

 

Therefore, we can guess that $h=2GM_2/c^2 \times mc=2\pi mcr_h$ if $N=2$. When $N$ is larger than $2$, the result depends on what the substance is made of. That is, if the mass $M_N$ is the algebraic sum of $N$ photons, it will be $N\hbar/2$, otherwise it should be written as $((N-1)!+1)\hbar/2$. This may be related to whether that particles creates new material or are scattered.¹

 

Here, $r_h, m$ is Planck's length, mass of photon. The mass of photon can be calculated as $m \approx m_p/\beta$. Considering $\beta \approx \cos\theta$, $\theta \approx f(N)$. Otherwise, $1/\beta$ goes to infinity. This is thought to be related to the minimum size occupied by Planck's particles.

Lorentz_Effect_in_the_Gravitational_Field.pdf

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[Complex Vector]

 

Complex numbers or variables can be used for the matrix calculation, but there are some differences. Typically, it can be valid for angular momentums. In an electromagnetic field, angular velocity can be regarded as momentum around a centripetal point, and can be related to electric and magnetic potential. Operations between these can be replaced with addition and subtraction operations of complex numbers. Multiplication between complex vectors can be used to express the rotation of a system (local symmetry, elemetary particle or subatomic particle) dependent on a rotation system (global symmetry, electromagnetic field). For example, the spin-charged particles moving around the electomagnetic field is that. Complex can be divided, but vectors can't. But we can use like this.

 

$\vec{A} \times \vec{B}=\vec{C} => \vec{A}=\vec{C}/\vec{B}=\vec{C} \times \vec{B^*}/(\vec{B} \times \vec{B^*})$

($\vec{B^*}$: conjugate vector of $\vec{B}$)

 

When replacing a vector with a $(m+1) \times (n+1)$ matrix, the commutation law for $\vec{A}$ and $\vec{B}$ does not hold($m, n$: natural number).

 

Last Modified in 2023.12.03

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^{1. In subatomic field, if the radius $r$ is a prime multiple of $r_h$, then $\beta=\ln(r/r_0)=\ln aN=\cos(\theta-\delta)$. If $\theta=N\pi/2$, then $1/\beta=1/\sin\delta=1/\ln aN$. If $1/\beta=N$, then $a=e^{1/N}/N$ and $r_0=Ne^{-1/N}r_h \approx Nr_h$.
($a=r_h/r_0, \theta=GM/rc^2$).The reason mass or charge is not shown in this article is because the part where the conversion of internal or external product takes place is not covered. This part is explained through field theory and the Riemann zeta function.
Refer to Quantized Time and Distance, General Relativistic Quantum Mechanics, About the roots of Riemann zeta function.}