Consider two observers, A and B, moving with relative speed v. A has a torch and a mirror. A sends a flash of light towards the mirror at the instant when A and B are very close to each other.

From A’s point of view the situation is

Let t_o be the time taken for the light to go from the torch to the mirror as measured by A.

The distance between torch and mirror is therefore given by d = c×t_o

From B’s point of view the situation is as shown in the next diagram.

Let t be the time taken for the light to reach the mirror as measured by B.

The distance moved by the light as measured by B is ab and so

ab = c×t

During the time taken for the light to reach the mirror, A has moved a distance x in B’s frame of reference.

so, we can write ..........x = v×t

From the diagram it is clear that ........ab² = d² + x²

Substituting into this last equation it is easy to show that the relation between t_o (A’s time for the light to reach the mirror) and t (B’s time for the light to reach the mirror) is

1. The time measured by an observer at rest relative to the apparatus (t_o) is called the proper time*. (In this example A gives the "proper time".)

2. Other inertial observers give improper times.

3. Improper time > Proper time, hence the term time dilation.

4. If two observers are in relative motion each will think that the clock of the other observer is running slow when compared to his/her own clock.

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