Rossi and Hall (1941) compared cosmic-ray muons at two altitudes in Colorado and found more reached the lower detector than the muon lifetime allowed. Frisch and Smith (1963) counted muons at the top of Mount Washington (about 1.9 km above sea level) and at sea level. They counted about 563 per hour on the mountain and about 412 per hour at sea level, where only 27 per hour were expected if there were no time dilation (using a 2.2 µs mean lifetime). Their measured time dilation factor of 8.8 ± 0.8 agreed with the prediction. The muons that reached sea level had lost some energy on the way, so their speed was a little lower than at the top.

The model. The muon has a half-life of 1.52 µs on its own clock (mean lifetime 2.197 µs). Muons are made about 10 to 20 km up and travel straight down at v. The fraction left after n half-lives is (1/2)ⁿ. The 1000 muons in the grid have decay times chosen to follow this law exactly, scattered at random across the grid. The real speed of cosmic-ray muons varies, and they lose some energy in the atmosphere, so the real numbers differ from this model.

Earth's explanation. On Earth clocks the trip takes t = h/v. A moving clock runs slow, so the muon's clock reads only t₀ = t/γ.

The muon's explanation. The muon is at rest and its clock is normal. The atmosphere moves past it, so its thickness is contracted to L = h/γ, and the ground reaches it after t₀ = L/v.

Why they agree. L/v = h/(γv) = t/γ. Both observers find the same t₀, the same number of half-lives and the same number of survivors. They disagree about why, because they disagree about which clock or which ruler is moving. They do not disagree about what happens to the muon.