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,

t=t_0\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{2L_0}{c}\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\; \; \; \; \; \; \; \; 253

Equating eq251 with eq252,

L=\frac{L_0}{\gamma}\; \; \; \; \; \; \; \; 254

where \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}.


Next article: Thomas precession
Previous article: Time dilation
Content page of quantum mechanics
Content page of advanced chemistry
Main content page

Leave a Reply

Your email address will not be published. Required fields are marked *