Length Contraction

This then is a general feature of special relative. The length of objects
appear contracted when viewed from a moving frame. The
apparent length (*L'*) is proportional to the actual length, *L*, as measured in the objects frame of reference. The constant of proportionality is the inverse of the time dilation factor. In terms of equations:

Clearly the apparent length is always less than the actual length, so this effect is called length contraction. As with time dilation, the situation must be symmetrical. If objects, such as the atmosphere, that are at rest relative to the Earth appear contracted to the muon, then objects moving with the muon must appear shortened by the same factor when viewed from the Earth. Consider for example the size of the muon itself. Muons are very tiny particles and have virtually no extent, but let us for the sake of illustration assume that as measured in the muon's frame of reference, it has a width of one centimeter. Length contraction tells us that the size of a muon moving at 0.99c will appear contracted to one seventh of its true size (or just over a millimeter) when viewed by an Earthbound observer. Thus, as expected, lengths of objects attached to the muon appear contracted as seen from Earth, whereas lengths of objects moving with the Earth appear contracted as measured in the muon's frame.

One last thing to emphasize is that the entire discussion so far refers only to one dimensional motion. Length contraction applies only to lengths along the direction of motion. It is of course possible to apply the postulates of Special Relativity to three dimensional motion, but the analysis gets considerably more complicated, and no essential new features emerge.

1999-09-29