Length Contraction - Derivation

Derivation

Length contraction can simply be derived from the Lorentz transformation as it was shown, among many others, by Max Born:

In an inertial reference frame S', and shall denote the endpoints for an object of length at rest in this system. The coordinates in S' are connected to those in S by the Lorentz transformations as follows:

and

As this object is moving in S, its length has to be measured according to the above convention by determining the simultaneous positions of its endpoints, so we have to put . Because and, we obtain

(1)

Thus the length as measured in S is given by

(2)

According to the relativity principle, objects that are at rest in S have to be contracted in S' as well. For this case the Lorentz transformation is as follows:

and

By the requirement of simultaneity and by putting and, we actually obtain:

(3)

Thus its length as measured in S' is given by:

(4)

So (1), (3) give the proper length when the contracted length is known, and (2), (4) give the contracted length when the proper length is known.