3. Relativstic Force

Relativistic Centrifugal Force

It is presumed that the impetus for conceiving force in the past may have originated from universal gravitation – gravity. Therefore, in the theory of relativity, it is necessary to redefine force. The following is derived to express that weightlessness due to centrifugal force and weightlessness in free fall are the same.

 

if kinetic energy $E_k=mc^2(\gamma-1)/k$ and centrifugal force $F_c/m=f(\beta, \gamma)v_\theta^2/r$ then, $mc^2(\phi-1)=E_k=mc^2(\gamma-1)/k$. Differentiating the equation,

$$-c^2\frac{d\phi}{dr}=-\frac{c^2}{k}\frac{d\gamma}{dr}= -\frac{\gamma^3}{k}\frac{vdv}{dr}= \frac{\gamma^3}{k}\frac{v^2}{r}=\frac{F_c}{m}$$

$$(\frac{dv}{dt}=v\frac{dv}{dr}=-\frac{v^2}{r})$$

 

if $v^2=v_\theta^2$ and $k=1$, then $f(\beta, \gamma)=\gamma^3$

$$\frac{\vec{F}}{m}=\gamma^3\frac{d\vec{v}}{dt}=-\gamma^3\frac{v^2}{r}\hat{e_r}=-c^2\frac{d\phi}{dr}\hat{e_r} $$

Considering that the acceleration $d\vec{v}/dt$ is in the r direction, there is fundamentally no difference between centrifugal force and gravity. Except in cases where it must be described in terms of force, it is much more concise to express it in terms of energy.

 

Quantum Gravity

From the theory of 'general relativistic quantum mechanics', gravitational potential is as follows.

 

$\ln{\Phi}=-\frac{1}{\beta}-\ln{\gamma(1+\beta)}+C$

 

$1/\beta$ represents gravitational potential, $\ln\gamma(1+\beta)$ represents electromagnatic potential. If $\Phi=\phi\phi_e$($\phi$: gravitational potential, $\phi_e$: electromagnetic potential),

 

$\phi = c^2(e^\rho-1)$

 

$(\rho=\theta(1+\frac{1}{4\theta}\sin4\theta), \theta=\frac{GM}{2rc^2})$

 

Gravitational force $F_g=d\phi/dr=-\frac{GM}{r^2}e^\rho\cos^22\theta \approx -\frac{GM}{r^2}$ when r is very large. If electrons are moving in the surface where $2\theta=n\pi$ in atomic world, it can be regarded as quantum gravity.