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 observer Y’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

$t=\gamma t_0\; \; \; \; \; \; \; \; 249$

where $\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ is called the Lorentz factor, which is useful in deriving the Thomas half.

Eq249 states that the time interval measured by an observer who is stationary relative to the moving clock for events occurring on the moving clock will always be longer than the time interval measured by someone at rest with respect to the clock. This longer time interval means that, from the perspective of the stationary observer, the moving clock is slowing down relative to the observer’s own clock. In other words, the stationary observer will see the moving clock as ticking more slowly, and the time measured by the moving clock will seem dilated or stretched from the viewpoint of the stationary observer.

How about the perspective of observer Y? If two observers move at constant velocity relative to each other, each sees the other’s clock running slower.

- X sees Y moving → Y’s clock is slow
- Y sees X moving → X’s clock is slow

This symmetry is called the reciprocity of time dilation. It may seem as if they are contradicting each other. However, as long as X and Y continue moving at a constant speed in a straight line, they can never stand next to each other to compare clocks. Their disagreement exists only while they are in different frames of reference. To compare clocks face-to-face, Y would have to return to X, which is best illustrated by the twin paradox.

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