Our friends, A and B, of time dilation
and relativity of simultaneity fame,
have put their heads together again and dreamed up another
verydifficultbuttheoreticallynotimpossibletoperform
experiment (a thought experiment). 

This time, the intention is to find out how the measured length of a
rigid rod depends on the state of its motion relative
to the person making the measurement. 





A carries a "light beam clock" (as seen in
time dilation). 

A and B have a relative velocity v directed at 90° to
the path of the light pulses. 

A is going to use the clock to measure the length of a
rigid rod carried by B. 

The time, measured by A, for the light pulse to go from
torch to mirror is t_{o}. 






A and B carefully arrange things such that
the light pulse leaves the torch (not shown in the diagrams below)
at the instant when it is just next to the end 1 of the rigid rod
and returns to the torch when it is just next to the other end. 





This means that in a time (measured by A)
equal to 2t_{o }, B moved a distance equal to the length of
the rod, L. 

Therefore, the length of the rod, as
measured by A is 





Now, how does this length compare with a
measurement made by B of the same rigid rod, using the same light beam clock? 

The diagrams above are
all drawn from A's point of view. As in the time dilation situation,
B considers that the light took a different path, as shown below. 



If the time taken for
the light to go to the mirror from B's point of view is t, then B
will conclude that the length of the rod is given by 



Dividing the first equation above by the
second gives 



and, as has been shown, the relation between
t and t_{o} is 



This means that the relation between L
(length of rod in A's frame of reference) and L_{o} (length
of rod in B's frame of reference) is 



usually written as 



and, using the abbreviation mentioned on the
time dilation page, this becomes 





Points to note: 

The length of a rod as measured by an
observer who is at rest relative to it is called the proper
length, L_{o} . In this example, B gives the "proper
length". 



Other inertial observers moving relative to
the rod will find lengths less than the proper length, hence the
term length contraction in the title. 

This effect is often called the
Fitzgerald contraction after the Irish mathematician who
predicted it using a different theory. 



This contraction only affects the dimension
of the rod which is parallel to the direction of the relative
motion. 





One way of explaining why two current
carrying conductors attract or repel each other is based on this
effect (see here). 
