1. Introduction

2. Method

2.1. Premise of Analysis: The Maximum Anisotropy of the Speed of Light is Not Strictly Equal to Zero

Our analysis is grounded on the premise that the maximum anisotropy of the speed of light, measured as $\epsilon <{10}^{-17}$ in the Michelson-Morley experiment[9], is indeed small, but not strictly equal to zero. We hypothesize that the true factor may be concealed under the assumption that this small but non-zero value of the maximum anisotropy of the speed of light reflects the underlying physical reality.

Based on the Pythagorean Theorem, the relationship between t and $t_{\theta }^m$ can be expressed by the following equation:

$${\left(v·t+c·t_{\theta }^m·s_v·cos\theta \right)}^2+{\left(c·t_{\theta }^m·s_p·sin\theta \right)}^2={\left(c·t\right)}^2$$

(1)

According to Equation (1), for $t_{\theta }^m\ge 0$, the following relational equation is satisfied:

$$t_{\theta }^m={\displaystyle \frac{\sqrt{\frac{v^2}{c^2}·{cos}^2\theta ·{s_v}^2+\left({cos}^2\theta ·{s_v}^2+{sin}^2\theta ·{s_p}^2\right)·\left(1-\frac{v^2}{c^2}\right)}-\frac{v}{c}·cos\theta ·s_v}{{cos}^2\theta ·{s_v}^2+{sin}^2\theta ·{s_p}^2}}t$$

(2)

Figure 1. The red dotted lines depict the dimensions of the object as observed in the moving frame M, where h represents the dimension perpendicular to the direction of motion and w represents the dimension along the direction of motion. The blue solid lines illustrate the shape of the object as observed in the rest frame R under the scaling effects of special relativity. Here, $s_v$ is the dimensional scaling factor in the direction of motion, and $s_p$ is the scaling factor perpendicular to the direction of motion, with $s_v=1$ and $s_p=\sqrt{1-v^2/c^2}$ under the Lorentz factor. The angle $\theta$ is between the light ray observed by observer M and the velocity of motion, ranging from $[0,\pi ]$, and it is not affected by velocity-induced scaling, as the object’s size remains constant to the observer in the moving frame. Points $D^m$ and D represent the same location on the moving object. Point $D^m$ is where the light ray reaches as observed by observer M in the moving frame along the $\theta$, and point D is where the light ray reaches as observed by observer R in the rest frame under the scaling effect. The variable t denotes the time for light to travel from O to D as recorded by the clock R in the rest frame, and $t_{\theta }^m$ is the time for light to travel from $O^m$ to $D^m$ as recorded by the clock M in the moving frame.

Figure 1. The red dotted lines depict the dimensions of the object as observed in the moving frame M, where h represents the dimension perpendicular to the direction of motion and w represents the dimension along the direction of motion. The blue solid lines illustrate the shape of the object as observed in the rest frame R under the scaling effects of special relativity. Here, $s_v$ is the dimensional scaling factor in the direction of motion, and $s_p$ is the scaling factor perpendicular to the direction of motion, with $s_v=1$ and $s_p=\sqrt{1-v^2/c^2}$ under the Lorentz factor. The angle $\theta$ is between the light ray observed by observer M and the velocity of motion, ranging from $[0,\pi ]$, and it is not affected by velocity-induced scaling, as the object’s size remains constant to the observer in the moving frame. Points $D^m$ and D represent the same location on the moving object. Point $D^m$ is where the light ray reaches as observed by observer M in the moving frame along the $\theta$, and point D is where the light ray reaches as observed by observer R in the rest frame under the scaling effect. The variable t denotes the time for light to travel from O to D as recorded by the clock R in the rest frame, and $t_{\theta }^m$ is the time for light to travel from $O^m$ to $D^m$ as recorded by the clock M in the moving frame.

where the angle $\theta$ is defined as the angle between the direction of light propagation as observed by the observer M within the moving frame and the velocity of the object. This angle $\theta$ ranges from 0 to $\pi$ and is not affected by the scaling effect due to changes in the object’s speed, as the object’s size remains constant to the observer M in the moving frame. When $\theta =0$, the direction of light propagation is the same as the direction of the object’s motion. Conversely, when $\theta =\pi$, the direction of light propagation is entirely opposite to the direction of the object’s motion. It is important to note that the angles $\theta$ and $2\pi -\theta$ represent the same physical situation, both corresponding to the angle $\theta$.

We define the unidirectional time scaling factor between $t_{\theta }^m$ and t as ${\delta }_{\theta }$, where:

$${\delta }_{\theta }={\displaystyle \frac{t_{\theta }^m}{t}}$$

(3)

The unidirectional time scaling factor $\left({\delta }_{\theta }\right)$ characterizes the relative passage of time for different directions within a moving object. Notably, ${\delta }_{\theta }=0$ signifies complete stasis of time in the direction characterized by angle $\theta$. Values of $0<{\delta }_{\theta }<1$ correspond to a slowdown in the passage of time, while ${\delta }_{\theta }=1$ indicates no relative change. Conversely, ${\delta }_{\theta }>1$ suggests an acceleration of time in that specific direction.

To understand the time dilation effect, it is essential to define a clock as a timekeeping instrument based on periodic reciprocating motion. This principle underlies all clocks, whether mechanical, electronic, or atomic. In a mechanical clock, the periodic motion is provided by a pendulum or balance wheel, which drives the clock’s gears and hands. An electronic clock relies on the oscillation of a quartz crystal, which vibrates at a precise frequency when an electric current is applied, converting these vibrations into electrical impulses for timekeeping. In an atomic clock[10,11], vibrations of atoms like cesium or rubidium, when exposed to an electromagnetic field, oscillate at a highly stable frequency, making atomic clocks the most accurate.

Applying this principle to the analysis of time dilation, we introduce the Einstein Light Clock[2,4], which is a theoretical apparatus for measuring time intervals based on the behavior of light. It consists of two mirrors separated by a fixed distance h, which is the distance light travels from one mirror to the other in a light clock. A beam of light travels from one mirror to the other and back again, completing a round trip. Each round trip constitutes one "tick" of the clock. This setup allows time to be measured based on the number of ticks the light completes between the mirrors. To analyze the relationship between the time dilation effect and the $\theta$ angle, we consider placing the light clock at a $\theta$ angle to the direction of motion, as shown in Fig. Figure 2. Here, $\theta$ ranges from 0 to $\pi$. When light propagates from one mirror to the other at an angle $\theta$, it returns from the other mirror to the starting mirror at an angle $\pi -\theta$. This placement allows for examining how the effect of time dilation on the light clocks varies with $\theta$.

When the light clock in the moving frame is positioned at an angle $\theta$ to its direction of velocity, the time interval for each tick of the light clock in the moving frame can be measured in two different frames of reference. The time interval measured in the rest frame is denoted as $T_{\theta }^r$, and the time interval measured in the moving frame is denoted as $T_{\theta }^m$. These intervals can be expressed respectively as follows:

$$T_{\theta }^r=\left({\displaystyle \frac{1}{{\delta }_{\theta }}}+{\displaystyle \frac{1}{{\delta }_{\pi -\theta }}}\right)·{\displaystyle \frac{h}{c}}$$

(4)

$$T_{\theta }^m={\displaystyle \frac{2h}{c}}$$

(5)

The time scaling factor of the light clock $\zeta \left(\theta \right)$, which represents the ratio of the time intervals measured by the clocks in the moving frame and the rest frame, is defined as follows:

$$\begin{array}{cc}\hfill {\zeta }_{\theta }& ={\displaystyle \frac{T_{\theta }^m}{T_{\theta }^r}}\hfill \\ \hfill & ={\displaystyle \frac{1-\frac{v^2}{c^2}}{\sqrt{\left({cos}^2\theta ·{s_v}^2+{sin}^2\theta ·{s_p}^2\right)-\frac{v^2}{c^2}·{sin}^2\theta ·{s_p}^2}}}\hfill \end{array}$$

(6)

The time scaling factor of the light clock ${\zeta }_{\theta }$ is related to both the velocity v of the moving object and the angle $\theta$. The correlation with the velocity v reflects the time dilation effect on moving objects as described by special relativity. Specifically, the time dilation’s effect on a light clock depends on the angle $\theta$, demonstrating how the effect of time dilation on the light clock is distributed in three dimensions within a moving frame.

When the time scaling factor of the light clock ${\zeta }_{\theta }=0$, the time of the light clock in that $\theta$ direction is completely stationary. When $0<{\zeta }_{\theta }<1$, the time of the light clock in that $\theta$ direction slows down. When ${\zeta }_{\theta }=1$, the time of the light clock in that $\theta$ direction is no relative change in time compared to the rest frame.

A noteworthy aspect of the light clock’s operation arises when ${\delta }_{\theta }>0$ and ${\delta }_{\pi -\theta }=0$. In this scenario, light traveling along direction $\theta$ within the clock can propagate from one mirror to the other. However, the return path along $\pi -\theta$ becomes impossible. This effectively traps the light, preventing it from completing a full round trip (tick) within the clock. Consequently, the displayed time freezes, appearing to the rest frame observer as a completely static clock (${\zeta }_{\theta }=0$).

It’s crucial to emphasize that this time stasis on the light clock does not contradict the ability of photons to propagate along $\theta$ when ${\delta }_{\theta }\ne 0$ within the moving frame. The halted clock reading signifies a standstill in the timekeeping mechanism, not the cessation of light propagation in all directions. Notably, in directions corresponding to ${\delta }_{\theta }\ne 0$, light propagation can be remains feasible.

According to Equation (6), the time dilation’s effect varies when the light clock is placed at different angles $\theta$ along the direction of motion within the moving frame. To eliminate the variation in time dilation’s effect caused by different $\theta$ angles in three-dimensional space inside the moving frame, and to ensure a comprehensive consideration of the time scaling factor of the light clock ${\zeta }_{\theta }$, we introduce a consensus time scaling factor of the light clock. This factor is determined by placing a light clock in each direction indicated by the uniformly distributed arrows from the origin to the external three-dimensional space, as illustrated in Figure 3. The mean value of the time scaling factors from all these light clocks is then taken to establish the consensus time scaling factor of the light clock, denoted as $\zeta$. This consensus time scaling factor of the light clock satisfies the following expression:

$$\zeta ={\displaystyle \frac{\underset{0}{\overset{\pi }{\int }}{\zeta }_{\theta }·sin\theta d\theta }{\underset{0}{\overset{\pi }{\int }}sin\theta d\theta }}$$

(7)

By substituting formula (6) into formula (7), we obtain the expression for the consensus time scaling factor of the light clock $\zeta$ as follows:

$$\zeta ={\displaystyle \frac{\left(1-\frac{v^2}{c^2}\right)·ln\frac{{\left(s_v+\sqrt{{s_v}^2-{s_p}^2·\left(1-\frac{v^2}{c^2}\right)}\right)}^2}{{s_p}^2·\left(1-\frac{v^2}{c^2}\right)}}{2\sqrt{{s_v}^2-{s_p}^2·\left(1-\frac{v^2}{c^2}\right)}}}$$

(8)

where ${s_v}^2-{s_p}^2·\left(1-{\displaystyle \frac{v^2}{c^2}}\right)⩾0$

The consensus time scaling factor of the light clock aligns with the principles of atomic clocks [10,11], which depend on the isotropic absorption and emission of photons during atomic vibrations. These vibrations occur uniformly in three dimensions, ensuring consistent and reliable timekeeping. Thus, applying a similar comprehensive consideration to the time scaling factor of the light clock in the three-dimensional space of a moving frame ensures more accurate and uniform time measurement, as supported by the three-dimensional isotropic properties fundamental to atomic clocks.

3. Discussion

3.1. Visualization of the Scaling and Time Dilation Effect for Four Categories

In order to intuitively visualize the scaling and time dilation effect corresponding to different combinations of $\alpha$ and $\beta$ in the six categories, we select one combination of $\alpha$ and $\beta$ values from each category for plotting. Specifically, we choose the following combinations:

• Category I: $(\alpha =0,\beta =1)$
• Category II: $\alpha =0,\beta \in (1,1.7999999999]$, we select $(\alpha =0,\beta =1.5)$
• Category III: $\alpha \in (0,3.3333333\times {10}^{-10}],\beta ={\displaystyle \frac{1}{2\alpha +1}}$, we choose $(\alpha =2.0\times {10}^{-10},\beta ={\displaystyle \frac{1}{2\alpha +1}})$
• Category IV: $\alpha \in (0,3.3333333\times {10}^{-10}],\beta \in ({\displaystyle \frac{1}{2\alpha +1}},{\beta }_{\epsilon }^{\alpha }]$, we select $(\alpha =2.0\times {10}^{-10},\beta =1.5)$

Figure 5. Four representative $(\alpha ,\beta )$ combinations are examined at speeds of 0, 0.5, 0.9, 0.999, and 1.0 times the speed of light to visualize the time dilation effect, ${\delta }_{\theta }$, in three dimensions as light rays travel at an angle $\theta$ relative to the direction of motion in a moving frame. Subfigures (a) and (c) are similar, with the key difference being that in (c), as the speed reaches the speed of light, ${\delta }_{\theta }$ remains non-zero for $\theta \in (\pi /2,\pi ]$, whereas in (a), ${\delta }_{\theta }$ is non-zero only at $\theta =\pi$. Similarly, subfigures (b) and (d) both show non-zero ${\delta }_{\theta }$ values for $\theta \in (\pi /2,\pi ]$.

Figure 5. Four representative $(\alpha ,\beta )$ combinations are examined at speeds of 0, 0.5, 0.9, 0.999, and 1.0 times the speed of light to visualize the time dilation effect, ${\delta }_{\theta }$, in three dimensions as light rays travel at an angle $\theta$ relative to the direction of motion in a moving frame. Subfigures (a) and (c) are similar, with the key difference being that in (c), as the speed reaches the speed of light, ${\delta }_{\theta }$ remains non-zero for $\theta \in (\pi /2,\pi ]$, whereas in (a), ${\delta }_{\theta }$ is non-zero only at $\theta =\pi$. Similarly, subfigures (b) and (d) both show non-zero ${\delta }_{\theta }$ values for $\theta \in (\pi /2,\pi ]$.

(a)Category I: $\alpha =0,\beta =1$	(b)Category II: $\alpha =0,\beta =1.5$
(c)Category III: $\alpha =0.0000000002,\beta ={\displaystyle \frac{1}{2\alpha +1}}$	(d) Category IV: $\alpha =0.0000000002,\beta =1.5$

The consensus time scaling factors $\zeta (\alpha ,\beta )$ are visualized in Figure 6. The figure illustrates that the different $(\alpha ,\beta )$ combinations of scaling factors exhibit a similar pattern: they all decrease as the speed of the object increases and approach zero as the speed nears the speed of light. The minimal differences between these factors suggest that even if the Lorentz factor is not the exact correct factor in nature, it closely approximates the true factor with an extremely small margin of error, effectively capturing the correct scaling effect.

4. Conclusions

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