B5 Muons 2: A Second Look
Let’s again consider the muons produced at an altitude of 15 km by cosmic radiation. Such a muon is at rest in its own inertial frame and thus has the usual half-life of 1.52 µs. This small half-life should only be enough for the earth to approach the muon by around 456 m. Why then do so many muons experience the arrival of the earth's surface?
Time dilation is no help here. But the 15 km thickness of the atmosphere extends in the direction of the earth's motion towards the muon and thus appears to the muon 'Lorentz contracted'. Using the formula of B3 we calculate for v = 0.99942·c a root factor of 0.03405. For our muon the 15 km diminishes to 511 m! The earth traverses these 511 m in somewhat more time than the half-life of a muon, i.e. nearly half of all muons experience the arrival of the earth's surface.
Seen by the muon the earth keeps its old cross sectional area. The earth’s diameter and also the thickness of its atmosphere shrink however in the direction of motion down to 3.4% of the rest value. The earth thus takes on the shape of a flat disk. [11-14f] offers no derivation of the length contraction, but it indicates in the text and in one of the many border illustrations (which make the book so attractive) that this contraction offers an explanation in the muon’s inertial frame. The border illustration on [11-15], shown below, contains however 2 errors, a harmless one in the picture and a worse one in the text. Can you find the two errors?
|Seen from the point of view of the muons, the earth which is approaching them at nearly the speed of light, appears greatly flattened. All distances in the direction of motion are shortened. The muons are thus able to travel the distance to the earths surface within their life time. (translation by Samuel Edelstein)|
In both representations (earth at rest with moving muons and muon at rest with approaching earth) one arrives at the same conclusion about the portion of muons which collide with the earth's surface. The reasoning is however completely different. The ‘history’ each tells differs strongly from that of the other one. 'History' is truly 'his story'… (wordplay personally communicated to me by Floyd Westermann). This however results in no contradictions or conflicts concerning the physics.