Length Contraction & Time Dilation

Length Contraction: Two dramatic results of the special theory of relativity are that a meter stick is shorter when moving than at rest and the moving clock runs slow. These results are quite real.

           Consider a stick at rest in the system lying along the axis with its ends at and . The length of the stick is, 
is called the rest or proper length of the system.

Now let us determine the length of the stick in system, where the observer is at rest. According to the observer in the S system, the stick moves with velocity towards right. The length of the stick is distance between its ends at the same instant of time. According to Lorentz transformation

from (2) and (3),  

is shorter than   ; as

This shortening is known as length contraction or Lorentz contraction and it occurs only along the direction of motion. If the stick lay along the -axis we would use = and =

Time Dilation : Let us investigate the effect of motion on time. Consider a clock at rest in the system and two events A and B both occurs at the same point .
the time interval is the time interval between two events in system or rest system. It is called the proper time. The corresponding time interval in laboratory system can be calculated as follows:

The last two equations are written based on the argument that for an observer at system, it would appear that system moves with velocity -.


Therefore, the time interval in the laboratory system is greater than that of rest system (). The moving clock runs slower and this effect is known as time dilation.