6.7 Length Contraction

Suppose that a meter stick is at rest in the reference frame x ?y ?z ?t ? moving through the laboratory at high speed. Figure 6.21 shows the worldlines of the front end and of the rear end of this meter stick. The meter stick occupies the region of spacetime that is shown hatched in Fig. 6.21. How long is this meter stick as seen in the laboratory frame? At a given instant ( t = 0), the rear end of the meter stick is at A, and the front end is at B. Since the point B is nearer the origin than the 1-meter calibration mark, its length is less than 1 meter as measured in the laboratory frame. By the construction of the calibration on the axes, the length contraction factor is 1/ ?:

Figure 6.21: Worldlines of the front end and of the rear end of a meter stick at rest in the x ?y ?z ?t ? frame.

For a meter stick at rest in the laboratory, the situation is as shown in Fig. 6.22. The measured length in the x ?y ?z ? t ? frame is AB, and this length is again shorter than 1 meter:

Figure 6.22: Worldlines of the front end and of the rear end of a meter stick at rest in the laboratory.

The general conclusion is, then, that...