## 3 Time Dilation in Different Kinematics
### Kinematics of the Special Theory of Ether
If the angle \(\alpha^{\prime}\) is measured in the body\({}^{\prime}\)s rest system, then according to the equations given in Figure 5, there is  
\[D(v)=\sqrt{D_{x}^{2}(v)+D_{y}^{2}(v)}=\sqrt{\left(D_{x}^{\prime}\,\psi(v) \sqrt{1-\left(v/c\right)^{2}}\right)^{2}+\left(D_{y}^{\prime}\,\psi(v)\right) ^{2}} \tag{20}\]  
\[D(v)=\sqrt{\left(D^{\prime}\psi(v)\sqrt{1-\left(v/c\right)^{2}}\cos\alpha^{ \prime}\right)^{2}+\left(D^{\prime}\psi(v)\sin\alpha^{\prime}\right)^{2}} \tag{21}\]  
\[D(v)=\sqrt{\left(D^{\prime}\psi(v)\cos\alpha^{\prime}\right)^{2}(1-\left(v/c \right)^{2})+\left(D^{\prime}\psi(v)\sin\alpha^{\prime}\right)^{2}} \tag{22}\]  
Figure 5.0  
Finally, the dependence for contraction from any angle \(\alpha^{\prime}\) is obtained in the form of  
Figure 5.1  
As in STE, the bodies in motion are deformed in a different way in the longitudinal and transverse directions (Figure 5), and therefore in inertial systems, the angles are not preserved. Let the angle measured in the resting system of the body as \(\alpha^{\prime}\) has a value \(\alpha\) in the ether system. Acting in a symmetrical way, as in the case of Equations (20)-(24), it is possible to derive the relation to contraction at any angle \(\alpha\) in the form of  
Figure 5.2  
Based on (24) and (25), the equation is obtained, which binds the angle \(\alpha\) and \(\alpha^{\prime}\) in the form of  
Figure 5.3  
Figure 5.4 Time dilation for four kinematics of the Special Theory of Ether. The graphs show the ratio of time \(t^{\prime}\) measured in the inertial system to time \(t\) measured in the universal frame of reference. After the transformation of Equation (26), the other equations are obtained, which bind the angle \(\alpha\) and \(\alpha^{\prime}\) in the form of  
Figure 5.5  
The duration of one light clock cycle is described in Equation (11) by the length of the clock arm, but only when the arm is parallel to the vector of velocity \(v\). We will now determine a more general equation for STE, in which the duration of one cycle will be described by the length of arm forming any angle \(\alpha\) with the vector of velocity \(v\). If the light clock is motionless in relation to the ether, then the length of its arm, regardless of the angle of its inclination, is the same for the observer from the ether. Therefore, for the arms of identical clocks, it is possible to write  
Figure 5.6  
Equations (28) and (29) apply to a different situation than the one shown in Figure 5.7 Now the circle is considered, not the rectangle. If identical light clocks move in relation to the ether at speed \(v\), then for an observer from the ether, the arms parallel to velocity \(v\) is \(D_{x}(v)\) long and the arms inclined at an angle \(\alpha\) to speed \(v\) is \(D(v)\) long. In STE, these clocks measure time identically, but their arms lengths for an ether observer are not equal in general. In order to determine the relationship between the arm lengths of these two clocks, Equation (25) will be divided by (15), and (28) will be considered. Then the following will be obtained  
Figure 5.8  
The searched equation describing the time \(t\), measured in the ether system, of one light clock cycle, based on the length of arm \(D(v)\) of any inclination, is obtained by taking into account the Equation (29) in Equation (11)  
Figure 5.9  
If to add an Equation (25) to this equation, a different version will be obtained in the form of  
\[t=t^{\prime}\frac{\psi(v)}{\sqrt{1-\left(v/c\right)^{2}}} \tag{16}\]0  
Equation (30) is expressed from the length of the moving arm that is measured from the ether system. However, Equation (31) is expressed from the length the arm would have in the ether if it did not move (this is also the length of the moving arm measured in the clock's rest system). It is worth noting that in the second case, in order to calculate the time \(t\) of one light clock cycle, it is not necessary to know the angle \(\alpha\) or \(\alpha^{\prime}\), but the transverse contraction function \(\psi(v)\) is needed.