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Length Contraction
Our friends, A and B, of time dilation and relativity of simultaneity fame, have put their heads together again and dreamed up another very-difficult-but-theoretically-not-impossible-to-perform 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 to.

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 2to , 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 to is

This means that the relation between L (length of rod in A's frame of reference) and Lo (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, Lo . 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).
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