Work and World

Impossible Muons

Every second, thousands of cosmic rays – mostly
hydrogen and helium nuclei – strike every square meter of the earth’s upper atmosphere
. We don’t really know where they come from, but we do know that when cosmic rays crash
into air molecules in the atmosphere, they create a shower of other fundamental particles:
pions, kaons, positrons, electrons, neutrons, neutrinos, gamma and X rays, and muons. We know this because we have particle detectors
in labs down on the surface that detect the directions and energies of the particles in
these showers, and use them to study the original cosmic rays. But there’s something fascinating about
the fact that we detect a lot of the muons from cosmic rays down on the surface of the
earth. Because muons, if you make them in a laboratory,
only have a 1.5 microsecond half life before they spontaneously decay into an electron
or positron and some neutrinos. Oh yeah, the greek symbol, mu is both used
for “muon” AND for “microsecond”, which can certainly be a little confusing;
but the lifetime of muons is really close to a microsecond, so it’s also kind of beautifully
appropriate/fitting. Anyway, the point is that if you have a bunch
of muons, More specifically, if you have a bunch of muons, you’ll only be left with
about 50% after 1.5 microseconds, and 25% after 3 microseconds, and after 10 microseconds
there will only be 0.1% of the muons left. Muons don’t live very long -2.2 microseconds
on average! To put that into perspective, light, which
travels fast enough that it could go around the earth 7 times in a second, only travels
660 meters, or less than half a mile, in 2.2 microseconds. So even muons traveling at essentially the
speed of lighta , wouldn’t make it more than a kilometer or two before the vast majority
of them decayed . Which is far less than the 10 or 20 or 30 kilometers that muons DO regularly
travel from the upper atmosphere to the ground. So how do muons travel dozens of kilometers
through the atmosphere without spontaneously decaying, when in fact they should only be
able to travel less than one kilometer? Time dilation. Yes – because the muons are traveling close
to the speed of light, their time literally passes more slowly – at a speed of 99.5% the
speed of light, 2.2 microseconds for them would be ~22 microseconds for us , enough
time for the average muon to travel at least 6km (instead of half of a kilometer) before
decaying. And even higher-energy muons going even faster
would even more easily reach our detectors on the earth’s surface before they decayed
– at 99.995% the speed of light, the average muon would live for 220 microseconds and travel
at least 66 kilometers before decaying. So from our perspective, the fact that so
many cosmic ray muons reach our detectors on the earth’s surface is direct evidence
for special relativity and time dilation! But what about from the muons’ perspectives,
where they DO only live on average 2.2 microseconds? Well, for them the answer to the apparent
paradox is also relativistic – relativistic length contraction. From the muon’s perspective, it’s the
earth and the atmosphere which are moving – at 99.995% the speed of light – towards
the muon. And the lengths of moving objects are literally
contracted by a factor dependent on their speed – in this case, 50km of our atmosphere
is, to the muon, literally only half a kilometer – aka 500 meters – thick. Which is thin enough for even a muon with
a lifetime of 2.2 microseconds to traverse – well, actually from this perspective the
atmosphere moves past the muon – but at a speed of 300 meters per microsecond and at
a distance of only 500 meters, the ground has no problem reaching the muon before the
muon decays. This, in my mind, is one of the most awesome
experimental verifications of special relativity: the unequivocal time dilation (or length contraction,
depending on your perspective) for objects moving close to the speed of light. The
specific time dilation and length contraction factors I talked about can be calculated using
the time dilation and length contraction formulas – once you know how to use them, you can plug
in any speed you want and see how much distances and time intervals will be distorted. And, this video’s sponsor,
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