## 2 Light Clock
### Time Measurement from Universal Frame of Reference  
Let us consider the situation as shown in Figure 3.  
We have several identical light clocks. One of these clocks is placed in the inertial system \(U^{\prime}\), while the other clocks are placed in the universal system \(U\). The clocks in the universal system \(U\) are synchronized with the light, which in the universal system \(U\) has a speed of known one-way value \(c\) (assumption II). The clock from \(U^{\prime}\) system passes by the clocks from the \(U\) system; thus, their indications can be compared. It will be shown below that although all clocks are identical, at the time when the clock from \(U^{\prime}\) system follows one cycle, the clocks from the \(U\) system will follow the other number of cycles. This is the phenomenon of time dilation. For this purpose, we will analyze the distance covered by the light of the clock in the \(U^{\prime}\) system from the observer's perspective in the \(U\) system (Figure 4). We consider the case when the arm of the clock is parallel to the vector of velocity \(v\) with which the clock moves. Dimensions parallel to velocity \(v\) will be marked with index \(x\), so in this case, the length of the clock arm is \(D_{x}(0)=D^{\prime}{}_{x}\) in \(U^{\prime}\) and \(D_{x}(v)\) in \(U\). Figure 3: Comparison of time elapse: (**a**) clock moving in relation to ether; (**b**) motionless clocks in relation to ether. Figure 4 in part \(A\) shows the moment when a light impulse is emitted from a source (S). Part \(B\) shows the moment when the light impulse is reflected by the mirror (M). Part \(C\) shows the moment when the light impulse returns to the source (S). From the perspective of the \(U^{\prime}\) system, the light travels a distance to the mirror of length \(D_{x}(0)=D^{\prime}{}_{x}\) at time \(t^{\prime}{}_{1}\), at a speed \(c_{x}{}^{+}\). Returning to the source, the light follows a path of the same length \(D_{x}(0)=D^{\prime}{}_{x}\), at time \(t^{\prime}{}_{2}\), at a speed \(c_{x}{}^{-}\). If the clock follows one cycle, then according to the observer in the same inertial system \(U^{\prime}\), the time elapse \(t^{\prime}=t^{\prime}{}_{1}+t^{\prime}{}_{2}\) with value (5). As the dimensions of bodies moving in relation to the universal frame of reference can change, hence from the perspective of the \(U\) system, the distance between the light source and the mirror will be marked by \(D_{x}(v)\). From the perspective of the \(U\) system, light travels a distance to a mirror of \(L_{1}\) length, at time \(t_{1}\), at a speed \(c\). Returning to the source, the light travels a distance of \(L_{2}\), at time \(t_{2}\), at the same speed \(c\). From Figure 4 (part \(b\)), we obtain  
\[L_{1}=D_{x}(v)+v\cdot t_{1},\qquad L_{2}=D_{x}(v)-v\cdot t_{2} \tag{6}\]  
\[t_{1}=\frac{L_{1}}{c}=\frac{D_{x}(v)+v\cdot t_{1}}{c},\qquad t_{2}=\frac{L_{2 }}{c}=\frac{D_{x}(v)-v\cdot t_{2}}{c} \tag{7}\]  
Relations (7) must be resolved in respect of \(t_{1}\) and \(t_{2}\). Then we obtain the time and the path of light flow in the system \(U\) in the form of  
\[t_{1}=\frac{D_{x}(v)}{c-v},\qquad t_{2}=\frac{D_{x}(v)}{c+v} \tag{8}\]  
\[L_{1}=c\cdot t_{1}=D_{x}(v)\frac{c}{c-v},\qquad L_{2}=c\cdot t_{2}=D_{x}(v) \frac{c}{c+v} \tag{9}\]  
Figure 4: Light clock: (**a**) the path of light impulse seen from \(U^{\prime}\) system in which the clock is located; (**b**) the path of light impulse seen from \(U\) system in which the light propagates. When the light pulse follows one cycle, then the light signal for the observer from the \(U\) system travels a distance of \(L_{1}+L_{2}\). Therefore, for the observer from the \(U\) system, the time \(t\) elapses, which the light needs in this system to cover a distance of \(L_{1}+L_{2}\).