Why Do Muons Reach the Ground? Time Dilation, Measured

Cosmic-ray muons are one of the cleanest everyday demonstrations that moving clocks run slow. They are born high in the atmosphere, they should not live long enough to reach the ground, and they reach the ground anyway. Our sticker Muonaut is that particle. This post works through the numbers and the experiments, and builds on our explainer Why Is the Speed of Light the Limit?.

What is a muon, and how long does it live?

A muon is an elementary particle like a heavy electron: same charge, spin ½, about 207 times the mass. It is unstable and decays, mostly to an electron and two neutrinos. Its mean life at rest, from the Particle Data Group average, is

τ₀ = 2.1969811 ± 0.0000022 μs

(PDG 2024 muon listing^[1]). A population of muons at rest decays as N(t) = N₀ e^−t/τ₀, where N₀ is the starting number and t the elapsed time.

Where do cosmic-ray muons come from?

Cosmic rays — mostly protons and nuclei from space — hit air nuclei high in the atmosphere and make showers of secondary particles. Charged pions in those showers decay into muons and neutrinos (π⁺ → μ⁺ + ν_μ, π⁻ → μ⁻ + ν̄_μ). The Particle Data Group’s cosmic-ray review notes that most muons with energies above a few GeV travel through the atmosphere without decaying and reach the ground (PDG 2024, Cosmic Rays^[2]). Feynman’s lecture on special relativity uses the same example, with muons created “some 10 kilometers up” (Feynman Lectures I-15^[3]).

Why shouldn’t they make it?

Ignore relativity for a moment. Even at nearly the speed of light, c = 299 792 458 m/s exactly (NIST CODATA^[4]), a muon living 2.2 μs travels about cτ₀ ≈ 659 m on average. Feynman makes the same point: in its short lifetime a muon “cannot travel, even at the speed of light, much more than 600 meters” (Feynman I-15^[3]). Ten kilometres is about fifteen of those average lifetimes. With N/N₀ = e^−t/τ, the surviving fraction over 10 km at 0.995c would be about e⁻¹⁵ — around two in ten million. Muon counters at sea level would be almost silent. They are not.

The time-dilation answer

Special relativity says a clock moving at speed v relative to you ticks slower, as you measure it, by the Lorentz factor

γ = 1 / √(1 − v²/c²)

so a muon’s mean life in the Earth frame is τ = γτ₀. At v = 0.995c, γ ≈ 10. The muon’s mean life as measured on the ground becomes about 22 μs, and its mean decay length about 6.6 km. Over the same 10 km, the surviving fraction becomes e^−1.5 ≈ 0.22 instead of 2 × 10⁻⁷. Feynman’s summary: from their own point of view they live only about 2 μs, from ours considerably longer, “enough longer that they may reach the earth” (Feynman I-15^[3]).

These are idealised numbers for a single speed. Real cosmic-ray muons arrive with a spread of energies and lose energy as they cross the air, so the experiments below fit full energy distributions rather than one γ.

What does the muon see?

In the muon’s own rest frame its clock is normal and it lives 2.2 μs. What changes is the atmosphere: the distance from the top of the shower to the ground is length-contracted by the same factor, L = L₀/γ, where L₀ is the length measured by someone at rest on Earth. At γ ≈ 10, ten kilometres of air becomes about one kilometre. Both frames agree on the one thing that can be checked: how many muons reach the detector (summary of the frame analysis^[5]).

Rossi and Hall, 1941: the first quantitative test

Bruno Rossi and David B. Hall compared slow and fast cosmic-ray “mesotrons” (the muon’s old name) by sorting them by how much lead they could penetrate. The softer group decayed about three times faster than the more penetrating group, “in agreement with the theoretical predictions based on the relativity change in rate of a moving clock”, and they derived a proper lifetime of (2.4 ± 0.3) × 10⁻⁶ s (Rossi & Hall 1941, Phys. Rev.^[6]). That proper lifetime is consistent, within their uncertainty, with the modern 2.197 μs.

Frisch and Smith, 1963: the famous mountain experiment

David H. Frisch and James H. Smith counted muons of a selected energy band on Mount Washington in New Hampshire and again near sea level in Cambridge, Massachusetts, and made a teaching film of it. Their paper is “Measurement of the Relativistic Time Dilation Using μ-Mesons”, Am. J. Phys. 31, 342 (1963; Crossref DOI 10.1119/1.1969508). As summarised from that paper, they counted about 563 muons per hour at the summit, near 1,900 m. Without time dilation, fewer than 30 per hour should have survived the 6.4 μs descent; about 400 per hour arrived. That implies a time-dilation factor near 9, consistent with the γ expected from the muons’ measured energies once energy loss in the air is included (Wikipedia summary of Frisch–Smith^[5]).

CERN, 1977: muons in a storage ring

The cleanest test does not use the sky at all. At CERN’s muon storage ring, J. Bailey and colleagues circulated positive and negative muons at γ = 29.33 and measured lifetimes of τ⁺ = 64.419(58) μs and τ⁻ = 64.368(29) μs. Divide by γ and you get back the at-rest lifetime: the Einstein time-dilation factor agreed with experiment to a fractional error of 2 × 10⁻³ at 95% confidence (Bailey et al. 1977, Nature^[7]). Those muons were moving in a circle under enormous acceleration, and the dilation was still set by speed alone — a direct check of what relativists call the clock hypothesis (cross-check^[5]).

Same physics, different clock: GPS

Muons are natural clocks. Satellites carry artificial ones. GPS satellites move fast enough that their clocks lose time from special-relativistic dilation, and sit high enough that general relativity makes them gain more, so the system must correct for both. That is the subject of our companion post How Does GPS Correct for Relativity?, and of Orbitick, Muonaut’s neighbour on the On Cosmic Time sticker sheet. The rest of the set is in Relativity & Cosmos.

FAQ

Does the muon feel its time slow down?

No. In its own frame it lives the normal 2.2 μs. Time dilation is a comparison between frames, and in the muon’s frame the atmosphere is contracted instead.

Is muon time dilation just a theory?

No. It has been measured with cosmic rays since 1941 and in accelerator storage rings to parts per thousand. The 1977 CERN result matched special relativity to 2 × 10⁻³.

Could you build a muon clock?

In a sense every muon detector is one: the decay rate of a sample is a clock. That is why muons are a standard undergraduate relativity lab.

References

1. Particle Data Group (2024), Review of Particle Physics, muon listing (mean life 2.1969811 ± 0.0000022 μs). https://pdg.lbl.gov/2024/listings/rpp2024-list-muon.pdf
2. Particle Data Group (2024), Review of Particle Physics, Ch. 30 “Cosmic Rays”. https://pdg.lbl.gov/2024/reviews/rpp2024-rev-cosmic-rays.pdf
3. R. P. Feynman, R. B. Leighton & M. Sands, The Feynman Lectures on Physics, Vol. I, Ch. 15: The Special Theory of Relativity (Caltech online edition). https://www.feynmanlectures.caltech.edu/I_15.html
4. NIST CODATA 2022, speed of light in vacuum (exact). https://physics.nist.gov/cgi-bin/cuu/Value?c
5. Wikipedia, “Experimental testing of time dilation” (summary of Rossi–Hall 1941 and Frisch–Smith 1963; cross-check only). https://en.wikipedia.org/wiki/Experimental_testing_of_time_dilation
6. B. Rossi & D. B. Hall (1941), “Variation of the Rate of Decay of Mesotrons with Momentum,” Phys. Rev. 59, 223. https://journals.aps.org/pr/abstract/10.1103/PhysRev.59.223
7. J. Bailey, K. Borer, F. Combley, H. Drumm, F. Krienen, F. Lange, E. Picasso, W. von Ruden et al. (1977), “Measurements of relativistic time dilatation for positive and negative muons in a circular orbit,” Nature 268, 301. https://www.nature.com/articles/268301a0

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.