The various relationships that exist between time intervals, coordinates, and velocities in the special theory of relativity are conveniently illustrated via a system of coordinates in which on the axes we lay off either distance and time multiplied by the speed of light or time and distance divided by the speed of light. Curves that represent motion in such systems are known as world lines. Various world lines are shown in the figure in the $\frac{x}{c}$ vs. $t$ Coordinates. What does each line represent? Is there a line that contradicts the main principles of relativity theory?

             

Two systems are moving with respect to each other with a certain velocity. The motion of one system in terms of the coordinates $\frac{x}{c}$ and $t$ of the other system is represented by a world line directed at an angle $\theta$ to the $\frac{x}{c}$ axis. After a time interval $T_0$ reckoned from the origin of coordinates has elapsed, one system sends a signal to the other. After what time will the second system receive the signal?

Three systems, $A, B$ and $C$, are moving with respect to each other in such a manner that with respect to system $B$ the velocities of $A$ and $C$ coincide in magnitude and are directed towards $B$ (Figure (a)). When system $A$ comes alongside system $B$ (Figure (b)), the clocks in the two systems are synchronized. At this moment system $A$ begins emitting signals directed at $B$ and separated by equal time intervals $T_0$. This continues until $A$ comes alongside $C$ (Figure (c)), with $N$ signals being set over the entire interval between the encounters. At this moment the clock in $C$ is synchronized with the clock in $A$ and system $C$ starts to send signals directed at $B$ that are separated by the same time intervals $T_0$. Find the difference in readings of the clock in $B$ and $C$ when these two systems come alongside (Figure (d)).