Length Contraction - Basis in Relativity

Basis in Relativity

First it is necessary to carefully consider the methods for measuring the lengths of resting and moving objects, where "object" simply means a distance with endpoints that are always mutually at rest, i.e., that are at rest in the same inertial frame of reference. If the relative velocity between an observer (or his measuring instruments) and the observed object is zero, then the proper length of the object can simply be determined by directly superposing a measuring rod. However, if the relative velocity > 0, then one can proceed as follows: The observer installs a row of clocks that either are synchronized a) by exchanging light signals according to the Poincaré-Einstein synchronization, or b) by "slow clock transport", that is, one clock is transported along the row of clocks in the limit of vanishing transport velocity. Now, when the synchronization process is finished, the object is moved along the clock row and every clock stores the exact time when the left or the right end of the object passes by. After that, the observer only has to look after the position of a clock A that stored the time when the left end of the object was passing by, and a clock B at which the right end of the object was passing by at the same time. It's clear that distance AB is equal to length of the moving object.

Thus the definition of simultaneity is crucial for measuring the length of moving objects. In Newtonian mechanics, simultaneity is absolute and therefore and are always equal. Yet in relativity theory the constancy of light velocity in all inertial frames in connection with the relativity of simultaneity destroys this equality. So if an observer in one frame claims to have measured the object's endpoints simultaneously, the observers in all other inertial frames will argue that the object's endpoints were not measured simultaneously. The deviation between the measurements in all inertial frames is given by the Lorentz transformation. As the result of this transformation (see Derivation), the proper length remains unchanged and always denotes the greatest length of an object, yet the length of the same object as measured in another inertial frame is shorter than the proper length. This contraction only occurs in the line of motion, and can be represented by the following relation (where is the relative velocity and the speed of light)

For example, a train at rest in S' and a station at rest in S with relative velocity of are given. In S' a rod with proper length is located, so its contracted length in S is given by:

Then the rod will be thrown out of the train in S' and will come to rest at the station in S. Its length has to be measured again according to the methods given above, and now the proper length will be measured in S (the rod has become larger in that system), while in S' the rod is in motion and therefore its length is contracted (the rod has become smaller in that system):

Thus, as it is required by the principle of relativity (according to which the laws of nature must assume the same form in all inertial reference frames), length contraction is symmetrical: If the rod is at rest in the train, it has its proper length in S' and its length is contracted in S. However, if the rod comes to rest relative to the station, it has its proper length in S and its length is contracted in S'.