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.
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