bei48482_FM

16 Chapter One

As found by an observer

moving with the muon, the

ground is L below it, which is

a shorter distance than L 0.

As found by observer

on the ground, the

muon altitude is L 0.

L 0

L

Figure 1.9Muon decay as seen by different observers. The muon size is greatly exaggerated here; in fact,

the muon seems likely to be a point particle with no extension in space.

Although its lifetime is only t 0 2.2 s in its own frame of reference, a muon can

reach the ground from altitudes of as much as 10.4 km because in the frame in which

these altitudes are measured, the muon lifetime is t34.8 s.

What if somebody were to accompany a muon in its descent at 0.998c, so that

to him or her the muon is at rest? The observer and the muon are now in the same

frame of reference, and in this frame the muon’s lifetime is only 2.2 s. To the observer,

the muon can travel only 0.66 km before decaying. The only way to account for the

arrival of the muon at ground level is if the distance it travels, from the point of view

of an observer in the moving frame, is shortened by virtue of its motion (Fig. 1.9). The

principle of relativity tells us the extent of the shortening—it must be by the same

factor of  1 ^2 c^2 that the muon lifetime is extended from the point of view of a

stationary observer.

We therefore conclude that an altitude we on the ground find to be h 0 must appear

in the muon’s frame of reference as the lower altitude

hh 0  1 ^2 c^2

In our frame of reference the muon can travel h 0 10.4 km because of time dilation.

In the muon’s frame of reference, where there is no time dilation, this distance is

abbreviated to

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