Proper Length

Proper Length

One thing all observers agree upon is relative speed. Even though clocks measure different elapsed times for the same process, they still agree that relative speed, which is distance divided by elapsed time, is the same. This implies that distance, too, depends on the observer’s relative motion. If two observers see different times, then they must also see different distances for relative speed to be the same to each of them.

The muon discussed in this example illustrates this concept. To an observer on the Earth, the muon travels at \(0.950c\) for \(7.05\phantom{\rule{0.25em}{0ex}}\mu s\) from the time it is produced until it decays. Thus it travels a distance

\({L}_{0}=v\Delta t=(0.950)(3.00×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s})(7.05×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{s})=2.01\phantom{\rule{0.25em}{0ex}}\text{km}\)

relative to the Earth. In the muon’s frame of reference, its lifetime is only \(2.20\phantom{\rule{0.25em}{0ex}}\mu s\). It has enough time to travel only

\(L=v\Delta {t}_{0}=(0\text{.}\text{950})(3\text{.}\text{00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s})(2\text{.}\text{20}×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{s})=0\text{.627 km}.\)

The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.

The Earth-bound observer measures the proper length \({L}_{0}\), because the points at which the muon is produced and decays are stationary relative to the Earth. To the muon, the Earth, air, and clouds are moving, and so the distance \(L\) it sees is not the proper length.

(a) The Earth-bound observer sees the muon travel 2.01 km between clouds. (b) The muon sees itself travel the same path, but only a distance of 0.627 km. The Earth, air, and clouds are moving relative to the muon in its frame, and all appear to have smaller lengths along the direction of travel.