The muon: relativity's smoking gun

Muons are unstable subatomic cousins of the electron, produced when cosmic rays slam into the upper atmosphere several kilometers above the ground, and they arrive at high-altitude detectors — and at sea level — in large numbers. Each muon, left alone in its own rest frame, decays with a mean lifetime of about τ₀ ≈ 2.2 μs. A muon born high in the atmosphere and racing downward at, say, v = 0.98c would — by plain Newtonian bookkeeping, distance = speed × time — travel only vτ₀ ≈ (0.98)(3.00×10⁸ m/s)(2.2×10⁻⁶ s) ≈ 6.5×10² m, about six hundred fifty meters, before decaying. That's nowhere near enough atmosphere to explain the muon flux routinely measured at sea level.

Special relativity closes the gap, and — this is the point of the exercise — it closes it two different ways depending on whose frame you do the bookkeeping in, and both ways agree on the observable outcome. From the ground observer's frame, nothing is wrong with distance = speed × time; what's wrong is using the muon's rest-frame lifetime. The muon's internal decay 'clock' is moving at 0.98c relative to the ground, so the ground observer must use the dilated lifetime γτ₀ instead of τ₀. At 0.98c, γ ≈ 5.03 (you'll compute this exactly in the problem set), stretching the survival distance by the same factor to roughly 3 km — enough for a large fraction of muons created a few kilometers up to survive all the way down.

From the muon's own point of view, its internal clock isn't dilated at all — it ticks at the ordinary rate τ₀, since the muon is at rest in its own frame. What it sees instead is the atmosphere rushing up to meet it at 0.98c, and that atmosphere's thickness is length-contracted by the very same factor γ. A slab of air that's several kilometers thick on the ground shrinks, in the muon's frame, to a fraction of that — short enough to cross in an ordinary, undilated lifetime. Time dilation and length contraction are two descriptions of one physical fact, each correct in its own frame, and the muon flux measured at sea level — first explained this way by Rossi and Hall in 1941 — remains one of the most direct everyday confirmations special relativity has.