Length contraction

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 $t_1$ (indicated by the top purple arrow) is

$ct_1=L+ut_1\; \; \; \; \; \; \; \; 250$

where $c$ is the speed of light, $L$ is length of the clock according to X and $u$ is the speed of the train.

When the photon returns from B to A in $t_2$ , A would have travelled a distance of $ut_2$ . Thus, the distance travelled by the returning photon from B to A (indicated by the bottom purple arrow) is

$ct_2=L-ut_2\; \; \; \; \; \; \; \; 251$

The total time taken is $t=t_1+t_2$ . Substitute eq250, where $t_1=\frac{L}{c-u}$ , and eq251, where $t_2=\frac{L}{c+u}$ , in $t$ ,

$t=\frac{2L}{c}\frac{1}{1-\frac{u^{2}}{c^2}}\; \; \; \; \; \; \; \; 252$

For observer Y, who is on the train, the total time $t_0$ measured by him is $t_0=\frac{2L_0}{c}$ , where $L_0$ 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,