Time dilation

Time dilation is the difference in time measured by an observer who is moving relative to another observer. Consider two frames of reference, each with an observer and a clock, which consists of a photon bouncing between two mirrors A and B. If the frame of observer X is at rest, while the frame of observer Y is moving away from it, observer Y sees the photon of his clock travelling vertically between the mirrors (diagram I) and measures $t_0=\frac{2L}{c}$ , where $t_0$ is known as the proper time, $L$ is the distance between the mirrors and $c$ is the speed of light. Observer X, however, sees the same photon moving in a triangular path (as the observer X’s clock moves away from him) and measures $t=\frac{2D}{c}$ (diagram II), where $t$ is the time recorded using his own clock.

Substituting $D=\sqrt{\left ( \frac{vt}{2}\right )^{2}+L^{2}}$ , where $v$ is the speed of the moving frame relative to the rest frame, and $L=\frac{ct_0}{2}$ in $t=\frac{2D}{c}$ , we have