Length Contraction

Length Contraction

In the previous section, Muon's ability to reach the Earth's surface could be explained in one of two ways, either by their lifetime being extended due to Time Dilation or by the distance they have to travel being shortened due to Length Contraction. 

Length Contraction is the second alternative effect of Special Relativity. Length Contraction again occurs in order to keep the speed of light a constant in all inertial frames of reference. 

The diagram above shows two different inertial frames of reference :

1. Passenger on the Train. Stationary relative to the train. 

2. Observer at the side of the track. Stationary in his frame of reference, observing the train travelling past him (right diagram). 

If the passenger uses a laser to measure the length of the train carriage using a mirror attached to the wall, the beam will follow the path as shown in the diagram. By using the time taken for the beam to return and knowing the speed of light, they can find the length of the carriage, shown as "L" in the diagram. 

The observer, who is standing at the side of the track, also sees the laser beam pass down the length of the train carriage, but due to the relative motion, they observe the beam travelling a shorter distance. As the distance travelled is used to measure the length of the carriage, the carriage appears contracted. 

Note - To find the length of an object, we must simultaneously know where each end is, then calculate the distance between them. For a stationary object, this is straightforward. 

For an object with relative motion, the position of both ends cannot be known simultaneously, which results in a 'contracted length' measurement. 

Length Contraction Calculations

The formula below is used to calculate the effects of Time Dilation:-

Where :-

l'  =  Length measured by an observer moving relative to the event (m) 

l  =   Length measured by an observer stationary relative to the Event (m) 

v  =  Speed of the moving object (ms-1) 

c  =  Speed of light (ms-1) 

The Lorentz Factor 

The scaling factor in the above equation is called the Lorentz Factor. The Lorentz Factor is used both in Time Dilation and Length Contraction to take into account the effects of relative speed. The Lorentz Factor is given the symbol Ɣ and is given by :-

This means that the Length Contraction formula above can be written in a simpler form :-

l'  = I / Ɣ 

Note - Due to the ratio of  v2/c2 within the Lorentz Factor, for low speeds the value of is approximately 1 and no Length Contraction is seen and classical mechanics can apply. However, at speeds greater than 0.1C, the value rises rapidly and Length Contraction can be observed. 

Example 1 - 

A rocket has a length of 10 m when measured on Earth. A stationary observer watches the rocket pass by with a relative velocity of 1.5x108 ms-1.  What is the apparent length of the rocket, as measured by the observer? 

l  =  10 m 

v  =  1.5x108 ms-1

c  =  3x108 ms-1