[Book Info] General Relativistic Quantum Theory

• This Book is for sale in Amazon.com
• Title: General Relativistic Quantum Theory
• Author: Sang-Young Joo
•  
• ASIN ‏ : ‎ B0DKG2WH29
• Publisher ‏ : ‎ Independently published (October 21, 2024)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 124 pages
• ISBN-13 ‏ : ‎ 979-8343491241
• Item Weight ‏ : ‎ 10.6 ounces
• Dimensions ‏ : ‎ 7 x 0.28 x 10 inches

 

 

This book is composed of general relativistic quantum mechanics(ISBN-13: 979-8864033456) and articles added afterward. In addition to removing unnecessary parts and correcting errors compared to before, I tried to create a more accurate theory. This content is not only about physics, but also about mathematical problems such as the Riemann hypothesis. The overall content is about solving general relativity and applying it to quantum mechanics. In the process, contents such as the Riemann zeta function and the Riemann hypothesis also appear. Since it is written by a non-specialist, it will not be difficult to read if you have basic knowledge of special relativity and quantum theory. This article is a thesis-like article, and as far as the author knows, most of the things have not yet been confirmed experimentally.

 

 

I would like to thank everyone who directly or indirectly helped make this book possible.

 

 

[Typos and Corrections]

■  In equation 3.28, 3.29,

$$r \frac{\partial \hat{e_r}}{\partial t}= r \frac{\partial \theta}{\partial t_\theta}\hat{e_\theta} $$

$$r \frac{\partial \hat{e_\theta}}{\partial t}=-r \frac{\partial \theta}{\partial t_r}\hat{e_r} $$

 

■  In page 60, $h, M_c$ which are expressed as $c2$ must be corrected to $c^2$.

 

■   Since all terms in equation 5.3 are eliminated and become 0, equation 5.5 is an incorrectly calculated result.

 

■  On page 44, the equation $2\Phi((1+\frac{\sin 2\Phi}{2\Phi})+ig(r))$ is corrected to $\Phi((1+\frac{\sin 2\Phi}{2\Phi})+ig(r))/2$.

 

■   In Section 3.8, the four types of potentials represent the boundary points between various forces, and it seems more accurate to call them spinor fields. When $\beta^2=1/2$, the values ​​1, 1/2, and 2 appear, because they are similar to the spin values ​​of bosons, fermions, and photons that are known to exist.

 

■ In Equation 3.21, Correct the line below that, $2\pi P_w=h\sin\theta$ => $2\pi rP_w=h\sin\theta$.

...

 

■ On page 107 of 'Artificial Intelligence', the number 9 of types of facts, 

'A says B and ...Z' must be corrected to  'A is B and ...Z',

'A says B is divided into C' must be corrected to 'A means that B is divided into C'.

---

 

[Relativistic Potential]

In page 33, 

$\sin2\theta, (\theta=GM/rc^2)$ is include in $\rho, \nu, p$. Here, $M$ is the mass acting on the surroundings at the time the element is created. These are terms created due to the relativistic effects of light. When we consider that the real and imaginary parts of the velocity of light $v=c(\cos\theta+i\sin\theta), (\theta=ct'/r_c-GM/rc^2)$ represent the gravitational and electromagnetic terms, $\theta$ used in the potential must also include the time term, $ct'/r_c$. This means that gravity, like the electromagnetic potential, is also expressed in the form of a wave function. The relationship with the relativistic wave function can be seen as $\psi=v/c=\cos\theta+i\sin\theta$. 

 

The Schrödinger equation is a non-relativistic wave equation, which is as follows:

$$i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi$$

The equation for conservation of energy is different for subatomic and non-subatomic molecules, and it is also different for charge and non-charge molecules. If $V$ represents the relativistic potential instead of the classical potential $GM/r, e^2/4\pi\epsilon_0 r$, it is thought that the meaning of the wave equation can be reinterpreted in a relativistic way and the non-relativistic Schrödinger equation can be converted into the relativistic Schrödinger equation. Then, a three-dimensional solution can be obtained from the relativistic Schrödinger equation.

Refer to equation 3.26, 3.27.

 

[Dimension extention of wave function]

In Equation 3.23, there is no distinction between gravity and electromagnetic force, and the phase difference is ignored when converting between Cartesian and curvilinear coordinate systems. Extending three-dimension, the electromagnetic potential is separated into trigonometric functions. Likewise, $\psi$ can be thought of as $\psi=\cos\theta\hat{e_r}+\sin\theta\cos\phi\hat{e_\theta}+\sin\theta\sin\phi\hat{e_\phi}, (\theta=GM/rc^2)$.

If we express this as a differential equation, it has a similar form to the Dirac equation, but it is a little different because there is no charge. If this wave function is related to the electromagnetic potential $\phi$ as in Equation 5.4, it can be viewed as $\phi=\ln\frac{\sin\theta}{1+\cos\theta}$. If we transform this back into Cartesian coordinates, $ \hat{e_r}, \hat{e_\theta}, \hat{e_\phi}$ will have the relationship as $ \hat{e_r}, \hat{e_\theta}$ are expressed as trigonometric functions that include time. A way to solve this is to utilize equation 2.2, which also seems to be related to Maxwell's equations.

※ If light is an electromagnetic wave, it should react to electromagnetic fields, but it is said that light is only affected by gravity. So the book distinguishes between the Lorenz effect caused by light and the Lorenz effect caused by electromagnetic fields.

 

[Difference between first and second derivatives from coordinate transformation]

In Equation 2.1
$$\frac{d^2r'}{dt'^2}=\frac{1}{r'}(\frac{dr'}{dt'})^2=\frac{v^2}{r'}$$
Because $\frac{dr'}{dt'}=v$, if we consider $\frac{d^2r'}{dt'^2}=\frac{dv}{dt'} $, the entire equation remains unchanged. The reason why we get such completely different results is because of how to calculate the second derivative when performing the $\gamma$ transformation for $dr', dt'$. We can check which is correct by using the limit and $\Delta$ according to the definition of differentiation, but it seems that the difference occurs fundamentally because $r', t'$ are discontinuous variables.

 

One more thing to consider is that, although the total differential form is used in the case of two variables, if it is possible to express it in the partial differential form between discontinuous variables, the above equation can be written as follows.
$$\frac{\partial^2r}{\partial t^2}=\frac{1}{r}(\frac{\partial r}{\partial t})^2=\frac{v^2}{r}$$

The solution to this equation can be expressed as $r=r_ce^{i(ct/r_c+g(r))}$, and comparing it to the speed of light in Section 2.7, it can be written as $\theta=ct/r_c+GM/rc^2, g(r)=2GM/rc^2$. The reason why $g(r)$ is expressed this way is because many of the assumptions in Section 3.1 can be rationalized and the wavefunction can be prevented from being expressed only in time when it is transformed to  Cartesian coordinate. This can be expressed in the form of a gravitational field equation as follows.
$$ \frac{\partial^2r}{\partial t^2}=\frac{v^2}{r}\times\frac{r+2r_g}{r+r_g}$$

In case $r_g=2GMi/c^2$ is a very small constant value, it can be ignored. Here, the complex number $i$ is in the direction perpendicular to $r$. If this interpretation is correct, then the difference between total differentiation and partial differentiation can be seen as depending on the presence or absence of $r_g$. In the absence of $r_g$, the wave equation is expressed only in time in the Cartesian coordinate system. Furthermore, the fact that the wave equation describes motion on a sphere or concentric circles can be understood as light or gravity transferring energy between neighboring particles.

※ Compared to the contents in Section 3.1, this equation is considered to be related with the relativistic quark energy equation. The complex trigonometric function in Section 3.1 seems to be identical to the rotational effect in Equation 3.17.

 

[Various interpretations of the gravitational field equation]
Equation 2.3 is as follows.
$$\int\frac{cdt}{r}=-\frac{1}{\cos\theta}+\ln\frac{\sin\theta}{1+\cos\theta}, \beta=\cos\theta=v/c$$

If $1/\cos\theta$ represents gravity, it can be transformed as follows.
$$\int\frac{cdt}{r}=-\frac{1}{\cos\theta}=>\frac{dv}{dt}=\frac{v^2}{r}$$

This equation is identical to the equation derived in Equation 1.14, but it differs in that the Lorenz effect is applied once more. Therefore, $r, t$ in the two equations are not strictly the same. This can be thought of as the reason why a non-complex value of constant $b$ appears on page 56. The phrase "not considering the Lorenz effect" on this page means that the trigonometric expression is omitted. The exact difference between the two formulas is explained in Section 3.1, but there are many assumptions and there seems to be quantum theoretical effects unknown to the author.

 

[Quantization and Differential Equation Calculation Priority]

In Equation 3.10,
$$\frac{\vec{F}}{m}dr=\frac{c}{r}S(c\hat{e_i})rd\theta', dr=rd\theta', \theta'=(GM/rc^2)^2$$
When calculating $dr=rd\theta'$, if $r, \theta'$ are treated as independent variables, $r=Ae^{\theta'}$ will be obtained, but if $\theta'=(GM/rc^2)^2$ is entered into the differential equation first and calculated, $r$ will be calculated as a constant. It seems that treating each other as independent variables in a non-continuous differential approximation equation is more consistent with the actual phenomenon, and this is thought to be an important computational element in quantization. Also, since $rd\theta'$ represents the length of the arc, we can guess that this force is exerted on a spherical surface.

 

[Changes in the gravitational constant]

The right side of Equation 3.4 may actually be a value other than 0. We considered the case where $\Phi$ is greater than 1, but the same assumption cannot be made when it is less than 1. This is because the result is $\Phi^2$, which is different from the general form of gravity. In this case, we must make a different assumption, as follows.

$$\frac{1}{2}(f(r,t)+1/f(r,t))\sin2\Phi=G(\Phi)(1+\frac{1}{2\Phi}\sin2\Phi)$$

If $\Phi$ is small, it can be approximated as $\sin 2\Phi \approx 2\Phi$, which gives the following equation.

$$\frac{1}{\cos\theta} \approx \frac{G(\Phi)(1+\frac{\sin 2\Phi}{2\Phi})}{\frac{G^2(\Phi)}{\Phi^2}-\cos^2 2\Phi} \approx \frac{G(\Phi)(1+\frac{\sin 2\Phi}{2\Phi})}{\frac{G^2(\Phi)}{\Phi^2}-1}$$

$$=k\Phi(1+\frac{\sin 2\Phi}{2\Phi}), k=1/2, G(\Phi)=(1 +\sqrt{2})\Phi$$
We assume $k=1/2$, but it can be a different value or not be a constant. The reason for this difference may be that the contribution of the valueof $f(r,t),1/f(r,t)$ to gravity varies depending on the size of $r$.

 

Compared to 'Difference between first and second derivatives according to coordinate transformation's, we can see that $f(r,t)=1/\Phi=rc^2/GM, r_c=GM/c^2$. In addition, $r_c$ means Planck length and the boundary of a subatomic particle, and among $f(r,t)+1/f(r,t)$, $f(r,t)$ is the potential that acts in a short section, whereas $1/f(r,t)$ acts in the entire section outside the boundary. Considering that $(1+\sin 2\Phi/2\Phi)$ is the Lorenz effect, we can guess that the source of gravity is $1/f(r, t)$. In addition, in subatomic scale, $1/\beta$ is the mass generation and is generally treated as $\beta=\cos\theta$, but if $f(r,t)$ plays this role, it can be inferred as $1/\beta=f(r,t)$.

 

[Complex potential]

Complex potential is used when describing the behavior of quarks or in the generation of mass and charge. This is because $\Phi$ above has the form of a complex function, which can be explained by the extended Riemann zeta function and the gamma function. And as the $r$ value increases, the magnitude of the imaginary part decreases. In 'Various interpretations of the gravitational field equation', the mass appears in the form of $e^{f(\theta)}$, and the charge and electromagnetic potential are due to the Lorenz effect.

 

[Potential energy of a charged body]

In the case of gravity and electromagnetic potential, the energy expression is generally as follows.
$$E=\int Fdr= \int \frac{GMm}{r^2}dr+ \int \frac{kq_1q_2}{r^2}dr$$
In Section 3.4, the energy expression is assumed as
$$E=r\frac{\partial P}{\partial t}$$
, but when calculating with $\Phi=GMm/r+kq_1q_2/r$, the result is the same as $\int Fdr=Fr$, but this is not the case when the $\Phi$ expression is different. Comparing with Equations 3.26 and 3.27, the energy differential expression in the potential is not $Fdr$, but $d(Fr)=Fdr+rdF=>F_gdr+rdF_e$, and it can be inferred that $rdF_e=0$ in the case of no charge. According to this result, the matrix in Equation 3.11 is applicable not only to forces, but can also be converted to an energy-related matrix by multiplying both sides by $r$.

 

[Center of Potential]

When calculating the potential acting between masses $m_1, m_2$, the potential is usually expressed as $\Phi=Gm_1m_2/r$ using the distance between the two objects. When considering kinetic energy, the two objects move around the center of potential. The center is considered to be the point where $m_1/r_1=m_2/r_2$ (where $r_1$, $r_2$ are the distances from the center of potential). If the potential center is not considered, when using the relativistic potential, the potentials calculated by the distance $r$ between each point $m_1, m_2$ are $m_2c^2e^{\theta_1}$, $m_1c^2e^{\theta_2}$, ($\theta_1=Gm_1/rc^2$, $\theta_2=Gm_2/rc^2$, the trigonometric terms are omitted), which are not equal. To solve this problem, the following assumptions are made. The reason why the entire potential is expressed as a sum is because the kinetic energy is expressed as a sum.
$$m_2\Phi_1+m_1\Phi_2=m_1m_2c^2e^{\theta_1+\theta_2}$$
Using the potential center, $\Phi_1=\Phi_2=\Phi$, therefore
$$\Phi=\frac{m_1m_2c^2}{m_1+m_2}e^{\theta_1+\theta_2}$$

 

[Waves and Potentials - August 16, 2025]

Equation 3.24, $cdr/r=dv$, is thought to be related to the weak force and electromagnetic waves. However, what is its connection to $cdt/r$, which appears not only in Equation 2.3 but also in many other places in the book when calculating potentials? In Equation 3.21, we assumed $2\pi r=\lambda\sin\theta$. If $\sin\theta$ is a transverse wave like electromagnetic waves, and there are also weakerlongitudinal waves, we can write it as a complex number, $2\pi r=\lambda(\cos\theta+i\sin\theta)$. In this case, $r$ is a complex number. Therefore, Equation 2.3 can be rewritten as follows:

$$\int \frac{c}{r}dt= \int \frac{2\pi c}{\lambda(\cos\theta+i\sin\theta)}dt=-\frac{1}{\beta}+\ln\frac{\sqrt{1-\beta^2}}{1+\beta}$$

Here, $1/\beta$ is related to mass generation, gravity, and other factors, depending on the scale. In this equation, the potential can be interpreted as the result of integrating wave energy over time. In order for the left-hand side integral result of this equation to be a real number rather than a complex number as in the right-hand side, it seems that $dt$ must be a complex number, as in Equation 3.30. The reason why the general relativistic effect is produced even though only the Lorenz effect $\gamma$ is applied to the gravitational field equation as in Equation 2.1 is because the gravitational field equation includes $1/\beta$. Therefore, although the equation is a scalar equation, it can be seen as including a vector form. Also, considering that light changes discontinuously, the integral on the left-hand side is unlikely to follow the general integration method.

Correlation_with_existing_quantum_theory.pdf

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[Entropy density function and wave equation, 2025.12.21]

The entropy (or energy) density function is as follows.

$$\int\frac{c}{r}dt=-\frac{1}{\beta}+\ln\frac{\sqrt{1-\beta^2}}{1+\beta}, \beta=v/c$$

Let's explore the connection between this equation and the Schrödinger wave equation. Differentiating both sides with respect to time $t$ yields the following:

$$\frac{c}{r}=(\frac{1}{\beta^2}-\frac{i}{1-\beta^2})\frac{\partial \beta}{\partial t}$$

Considering that the $1/\beta^2$ term is the particle energy term related to mass production or gravity, the remainder of imaginary part is the wave energy. If we isolate the wave energy component and multiply both sides by $\hbar$, and let $\lambda=2\pi r$, then:
Equation (1)
$$i\hbar\frac{\partial \beta}{\partial t}=-(1-\beta^2)\frac{2\pi\hbar c}{\lambda}=-(1-\beta^2)h\nu=-(1-\beta^2)E= -\frac{E}{\gamma^2}$$
$E$ represents energy. The Schrödinger wave equation is as follows:

$$i\hbar\frac{\partial}{\partial t}\psi=(-\frac{\hbar^2}{2m}\nabla^2+V(r))\psi$$

If $E=-\psi V(r)$ and $$\frac{\hbar^2}{2m}\nabla^2\psi=\beta^2E$$ are matched, the two equations become similar. And we can infer that $\psi=\beta=v/c$

Furthermore, equation (1) can be transformed as follows if $i\hbar\partial\beta/\partial t=-\hbar^2\partial^2\beta/(\partial t^2E)$.
$$\hbar^2\frac{\partial^2 \beta}{\partial t^2}=(1-\beta^2)E^2$$
$$E^2\beta^2=E^2-\hbar^2\frac{\partial^2 \beta}{\partial t^2}$$
Here, let $\beta=\psi$ and $E\beta=mc^2\psi$,
$$(m\psi)^2=\frac{E^2}{c^4}-\frac{\hbar^2}{c^4}\frac{\partial^2 \psi}{\partial t^2}$$, which is the basic form of the Dirac equation.

The main reason for this discrepancy is that the Schrödinger wave equation is non-relativistic and three-dimensional, and its potential term is arbitrary, changing depending on whether the system is subatomic, atomic, or molecular. In contrast, the entropy density function is a potential field expressed in terms of light and elementary particles. In particular, the book mentions that $\beta^2$ and $1/\gamma^2$ can represent forces and can also be used in energy equations. There are many places where the velocity of light can be derived, but a simple example is as follows:
$$\frac{GM}{r^2}dr+c\gamma dv=0$$
This equation can be rewritten as
$$\frac{GM}{r^2c^2}+\gamma \frac{d\beta}{dr}= \frac{\theta}{r}+\gamma\nabla\beta=0, \theta=\frac{GM}{rc^2}$$. Comparing this equation to the Dirac equation, $\psi$ can be seen as both representing $\beta$ and $\theta$. This phenomenon appears only when calculating the velocity of light, and appears elsewhere in the form $\psi=v/c$.

 

Although not derived from the gravitational field equations, another entropy density function can be constructed from $L_3$ transformation.

From $a_{21}$ and $a_{22}$ of $L_3$, we can write it as follows:
Equation (2)
$$\int \frac{cdt}{r^*}=-\gamma+\ln\frac{\gamma+1}{\beta\gamma}$$
The symbol * in the integral sign distinguishes it from the conventional entropy density function. Using complex number notation, we can rewrite it as follows:

$$\frac{c}{r^*}=\frac{\partial }{\partial t}(-\gamma+i\ln\frac{\gamma+1}{\beta\gamma})$$

To make this equation into a form alike to a wave equation, we take only the imaginary part and transform it into $\gamma$ instead of $\beta$, and rearrange it as follows.

$$\frac{c}{r^*}=(\frac{1}{\gamma+1}-\frac{\gamma}{\gamma^2-1})\frac{\partial \gamma}{\partial t}$$

$$\hbar\frac{\partial\gamma}{\partial t}=-(\gamma^2-1)E$$

Instead of $\gamma$, we can use $\psi^*$ and write it as follows.

Equation (3)

$$i\hbar\frac{\partial\psi^*}{\partial t}=(-\frac{\hbar^2}{m}\nabla^2+V(r))\psi^*$$

If there is a relationship $(\psi^*)^2=1/(1-\nabla^2\psi)$, or $\nabla^2\psi^*=1/(1-\nabla^2\psi)$, then $\psi$ can be obtained. To transform equation (2) like a gravitational field equation, differentiate both sides with respect to $t$ and rearrange them to obtain the following:

$$\frac{dv}{dt}=-\frac{c^2}{r}\frac{\beta}{\gamma^3}$$ 

Just as we did the $\gamma$ transformation in the gravitational field equation, we can write this function as follows by using $1/\beta$ instead of $\gamma$.

$$\frac{dv}{dt}=-\frac{2c^2}{r}\frac{\beta}{\gamma^3}=-\frac{2c^2}{r}\sin^3\theta\cos\theta$$

In this process, two types can be applied, such as $dr'=\beta dr$, $dr'=dr/\beta$, and I used the one where one side does not become 0. In the gravitational field equation, only the sign of the logarithm term changes no matter which of the two is applied, but here the results are completely different. If we differentiate $\sin^3\theta\cos\theta$ with respect to $\theta$ and find the point where it becomes 0, we get $\sin\theta=\pm \sqrt{3}/2$, which means that $\theta$ has a maximum or minimum at $\pi/3$, which could mean three quarks. If, for some reason, the phase changes so that $v/c=\sin\theta$, the equation becomes:
$$\frac{dv}{dt}=\frac{c^2}{r}\frac{\beta^2}{\gamma^2}$$
This is because the entropy density function used here is derived from $\vec{v}=c(\sin\theta+i\cos\theta)$, not $\vec{v}=c(\cos\theta-i\sin\theta)$.