6. Absolute Time, Relative Time

Relative Energy of Wave and Particle

Modifying Lorentz factor $\gamma=1/\sqrt{1-\beta^2}$ slightly, we can write it like this:
$$\beta^2+1/\gamma^2=(P_p/P)^2+(P_w/P)^2=1$$Here, $P_p$ is the momentum in the gravitational direction, $P_w$ is the momentum in the direction perpendicular to that, and $P$ is the total momentum in subatomic world. It also means mass (gravitational potential) and charge (electromagnetic potential).

 

Velocity of Light

This also applies to the velocity of light, which is as follows.

$$\vec{v}=c(\cos\theta\hat{e_r} - \sin\theta\hat{e_\theta}), (\theta=\frac{GM}{rc^2})$$

Here, unit vectors are those in tangent-normal coordinate system. The derivative of light velocity becomes the source of gravity and electomagnetic force like this.
$$\frac{\vec{F}}{m}=\frac{\partial\vec{v}}{\partial{t}}=-\frac{GM}{r^2}(\cos^2\theta\hat{e_r}-\sin^2\theta\hat{e_\theta})$$

 

Gravitational Field Equation

In the process of deriving the force, the following equation is derived, which expresses the gravitational field without electromagnetic effects.

$$\frac{dv}{dt}=\frac{v^2}{r}$$

 

General Relativistic Potential

If we apply the Lorentz factor due to electromagnetism again, it becomes as follows.
$$\int{\frac{c}{r}dt}=-\frac{1}{\beta}-\ln\gamma(1+\beta)+C=-\ln{\frac{\Phi}{c^2}}$$$1/\beta$ represents gravitational potential, $\ln\gamma(1+\beta)$ represents electromagnetic potential. If we set $\cos\theta=\beta$, $\theta$ is the curved angle in tensor field.

 

Momentum, Force and Energy

$Fdr=cdP$ in subatomic field including photon.
$Fdr=vdP+Pdv$ in non-subatomic field.

 

In the subatomic world, $P=mv$ and in the electromagnetic, gravitational field, it is $\gamma{mv}$. $Pdv$ is considered to be an effect of the amount of charge and becomes 0 if there is no charge. The relativistic force in the gravitational field can be defined as $F=\gamma^3mdv/dt=-\gamma^3mv^2/r$ and identical to centrifugal force. $r$ represents the direction in which acceleration occurs. Considering only natural forces, $r$ can be seen as representing a centripetal direction such as gravity or electromagnetic force, and when considering artificial forces, that represents a virtual centripetal point. In cases where the change in momentum and acceleration are constant(e.g. orbital motion) and the velocity is small with respect to$c$, it can be written as follows respectively.

$F=cP/r$

$F=vdP/dr=dP/dt=\gamma^2vP/r \approx vP/r$ (when $v$ is samll compared to $c$)

 

Relative, Absolute Time

If absolute time is the time where Lorentz effect is not applied, and only gravity exists, the time vector $t'$ or $\vec{t}$ can be defined as follows.

$t'=t(\cos\theta\hat{e_r}+\sin\theta\hat{e_\theta})=t(\beta\hat{e_r}+1/\gamma\hat{e_\theta})$

 

This means that objects, including light, have different time degeneracy rates with respect to the direction in which they travel. $\beta$ represents the direction of progress (or direction of gravity), and $1/\gamma$ represents the direction perpendicular to it. And the measured speed of light is assumed to be $\beta c$. Therefore, time differnece between absolute coordinate and relative coordinate is as follows.($t\gamma$ is the time resulting from special relativity and that is relative time)

$t_\beta=\beta{t}$

 

If $\beta$ is same, $\Delta{t_\beta}=\beta{\Delta{t}}$, relative time $t_\beta$cannot bigger than absolute time $t$. Time degeneracy in special relativity refers to the $\hat{e_\theta}$ component. In special relativity, the relative speed between two objects A, B are as follows.

$$v_{AB}=\frac{v_A-v_B}{1-v_Av_B/c^2}, \beta_{AB} =\frac{v_{AB}}{c}=\frac{\beta_A-\beta_B}{1-\beta_A\beta_B} $$If the speed of gravity is $c\cos\theta$, the speed of the object under that gravity is as follows.
$$v_g=\frac{v-c\cos\theta}{1-vc\cos\theta/c^2}, \beta_g=\frac{v_g}{c}=\frac{\beta-\cos\theta}{1-\beta\cos\theta}$$If two objects are moving at different speeds under different gravity, you will need to compensate for the gravitational effect. If the gravity-corrected relative speed is calculated, $1/\gamma_g=\sin\theta/\gamma(1-\beta\cos\theta )$ can be obtained, which affects the frequencies of light such as red and blue shifts. In cases such as time differences at different altitudes or artificial satellites, centrifugal force must be considered. The time it takes for light to travel in a gravitational field can be calculated as follows.
$$\int{\frac{c}{r}dt}=-\frac{1}{\beta}, \Delta{t}=-\int{\frac{r}{c}d(\frac{1}{\beta})}$$

 

Reference: General Relativistic Quantum Mechanics