* Length contraction* is the shortening of the length of a moving object, as compared to its proper length, which is measured by an observer at rest with respect to the object being measured. Consider a moving train with a clock, which consists of a photon bouncing horizontally between two mirrors A and B (see diagram below).

For observer X, who is at rest on a platform and seeing the passing train, the total distance travelled by the photon from A to B in (indicated by the top purple arrow) is

where is the speed of light, is length of the clock according to X and is the speed of the train.

When the photon returns from B to A in , A would have travelled a distance of . Thus, the distance travelled by the returning photon from B to A (indicated by the bottom purple arrow) is

The total time taken is . Substitute eq250, where , and eq251, where , in ,

For observer Y, who is on the train, the total time measured by him is , where is called the * proper length*, which is the length measured by an observer who is at rest with respect to the object being measured. From eq249,

Equating eq251 with eq252,

where .