Saved by Relativity

Coming at you from space, flying close to the speed of light, about 1 muon passes through your head every second. Probably not the best thing for your health, but also not enough to really harm you.

image from nsf.gov / J. Yang

Muons happen to be very unstable: their half-life is near 1 millionth of a second (microsecond or μs). 100 muons become 50 in 1μs and 25 in another. They decay so fast that if every particle in the universe were a muon, within 1 millisecond (1000μs) they'd all be gone!

These muons fly toward Earth at near the speed of light, at about 1000 ft/μs. So, if every second 1 muon goes through your head on Earth's surface, calculations show (due to their decay) 2 should go though your head at 1000 feet, 4 at two thousand feet, and 2^30 at 30000 feet -- that's over 1 Billion muons, enough to kill you instantly!

But planes fly at 30000 feet all the time. So where did we go wrong?

To figure that out, we need some relativity. The special theory of relativity postulates that the laws of physics are the same in all inertial (not accelerating) reference frames and that the speed of light is always constant. Its consequences include:

1. Moving clocks appear to go slower to stationary observers.
2. An observer will measure the length of a moving object as shorter in the direction of its relative motion.
3. E = mc^2 (which we don't need for this problem).

The first two of these consequences have a noticeable effect only at very high velocities and a strong effect at near the speed of light.

Remembering that muons travel fast enough to experience relativistic effects, it becomes clear what's going on. From our point of view they decay (what turns out to be) 10x slower. So instead of 30 halvings, we only see them go through 3. At 30000 feet only 2^3 = 6 muons per second go through your head, and you can survive that just fine!

This just leaves one puzzle: from the muons' reference fames it is our clocks that slow down, not their own. How do we explain them halving only thrice between airplane height and the Earth's surface from their point of view?

This post is inspired by a 2002 Princeton physics lecture by Peter Meyers. A similar story appears on the Stanford SLAC website.

Update (5/11/10): my following post answers the muon riddle.