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. Amy Farrah Fowler - Stationary in her frame of reference (left diagram).
2. Sheldon Cooper - Stationary in his frame of reference, observing Amy travelling past him (right diagram).
If Amy 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, Amy can find the length of the carriage, shown as "L" in the diagram.
Sheldon, 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, Sheldon observes the beam travelling a shorter distance. As the distance travelled is used to measure the length of the carriage, then the arriage appears contracted.
Note - To find the length of an object, we must simultaneously know where each end is, then calculating 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 Visually
The diagram below shows the effect of Length Contraction on an object for varying values of velocity:-
Note - It is only the Length of the object in the direction of travel that experiences Length contraction. The vertical height of the object is unchanged.
Length Contraction Calculations
The formula below is used to calculate the effects of Time Dilation:-
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