Everything so far has been theory — beautiful, but abstract. This is where nature signs off on it. Tiny particles called muons are created high in the atmosphere and should almost all decay long before reaching the ground. Yet detectors at the surface catch huge numbers of them. The only way to explain it is that time dilation (or, just as validly, length contraction) is genuinely real. The muons are living, decaying proof of special relativity.
📘 What you need to know
Muons are unstable particles (~200× the electron’s mass) made when cosmic rays strike the upper atmosphere, about 9–10 km up
They travel at about 0.98c with a half-life of about 1.6 µs
In one half-life at that speed they cover only ~470 m — so over 9–10 km, Newtonian physics predicts almost all decay before reaching the ground
Yet a large fraction is detected at the surface — direct evidence for special relativity
Earth’s frame: time dilation stretches the muon’s half-life (γ ≈ 5, so ~8 µs), so far fewer half-lives pass
Muon’s frame: length contraction squashes the atmosphere (~9 km → ~1.8 km), so there’s far less distance to cover
Both frames give the same surviving fraction = (½)n, where n = travel time ÷ half-life
The puzzle
Muons are born roughly 9–10 km up when cosmic rays smash into the top of the atmosphere. They rain straight down at about 0.98c. The trouble is they’re desperately short-lived: a half-life of just 1.6 µs means that at 0.98c they travel only about 470 m before half of them are gone.
A muon is born ~9 km up and dives toward the ground at 0.98c. Since it covers only ~470 m per half-life, Newtonian physics says it should almost never survive the trip — yet detectors catch plenty.
WE 1
Muons are created 9.0 km up, travelling at 0.98c, with a half-life of 1.6 µs. Ignoring relativity, how many half-lives pass on the way down, and what fraction reach the ground?
Step 1 — travel time (Newtonian): t = distance ÷ speedt = 9000 ÷ (0.98 × 3×10⁸) = 3.06×10⁻⁵ s = 30.6 µsStep 2 — number of half-livesn = t ÷ t½ = 30.6 ÷ 1.6 ≈ 19 half-livesStep 3 — fraction surviving(½)ⁿ = (½)¹⁹ ≈ 1.7×10⁻⁶≈ 0.0002% — essentially noneNewton’s prediction: the ground should be almost muon-free. Experiment says otherwise.
Two explanations, one answer
Relativity resolves the puzzle — and it does so in two different-looking but completely equivalent ways, depending on whose frame you sit in.
Left: Earth sees the full 9 km, but the muon’s half-life is dilated to ~8 µs. Right: the muon sees a normal 1.6 µs half-life, but the atmosphere is contracted to ~1.8 km. Both give the same ~3.8 half-lives.
WE 2
Now use special relativity in Earth’s frame (time dilation). Take γ = 5.0 for 0.98c. How many half-lives pass, and what fraction survive?
Step 1 — the muon’s half-life is dilatedt½ = γ × 1.6 µs = 5.0 × 1.6 = 8.0 µsStep 2 — the trip still takes 30.6 µs in Earth’s framen = 30.6 ÷ 8.0 = 3.8 half-livesStep 3 — fraction surviving(½)³‧⁸ ≈ 0.070≈ 7% reach the groundFrom 0.0002% to 7% — time dilation makes all the difference.
WE 3
Repeat in the muon’s frame (length contraction), and check you get the same answer. Use γ = 5.0.
Step 1 — the atmosphere is contractedL = L₀ ÷ γ = 9000 ÷ 5.0 = 1800 mStep 2 — trip time in the muon’s frame (half-life is the normal 1.6 µs here)t = 1800 ÷ (0.98 × 3×10⁸) = 6.1 µsn = 6.1 ÷ 1.6 = 3.8 half-livesStep 3 — fraction surviving(½)³‧⁸ ≈ 7% — identical to WE 2Two frames, two effects, one answer. Dilate the time or contract the distance — never both.
Keep the bookkeeping straight and you’ll never slip: Earth’s frame gets the longer time (dilated half-life, full distance); the muon’s frame gets the shorter distance (contracted atmosphere, normal half-life). Pick one frame and stick to it. A quick sense check: the relativistic answer must always let more muons survive than Newton — a longer time or a shorter distance for the muon to cross.
Counting the survivors
The survival fraction follows the usual decay law, fraction = (½)n, where n is the number of half-lives. Because the decay is exponential, the difference between n ≈ 19 (Newton) and n ≈ 3.8 (relativity) is the difference between “basically none” and “plenty”.
The exponential decay curve. Newton’s ~19 half-lives land on the dead flat tail (≈ 0 survive); relativity’s ~3.8 half-lives sit high up the curve (~7% survive). Detectors confirm the relativistic value.
This isn’t a thought experiment — it’s been measured. Classic muon-flux experiments (Rossi and Hall in 1941, and Frisch and Smith in 1963) counted muons at mountaintop and sea level and found exactly the relativistic survival rate, not the Newtonian one. Special relativity passed with flying colours.
Newton n ≈ 19 → ≈0%
add relativity
dilate time or contract length
same result
n ≈ 3.8 → ~7% arrive
🛠️ Solving a muon problem
Newtonian check.t = distance ÷ v; n = t ÷ half-life; fraction = (½)n (usually tiny).
Choose a frame. Earth or muon — then commit to it.
Earth frame. Dilate the half-life (× γ), keep the full distance: n = t ÷ (γ t½).
Muon frame. Contract the distance (÷ γ), keep the proper half-life: n = (L0/γ ÷ v) ÷ t½.
Same answer. Both give the same n and the same fraction = (½)n.
Quick recap: Muons made ~9–10 km up at 0.98c should nearly all decay (Newton), yet many reach the ground. Earth’s frame explains it with time dilation (half-life ~8 µs); the muon’s frame with length contraction (atmosphere ~1.8 km). Both give the same survival fraction — real, measured proof of special relativity.
💡 Top tips
~470 m per half-life. A muon covers only ~470 m in a rest-frame half-life — a handy anchor.
γ ≈ 5 at 0.98c. A convenient round value that pops up constantly here.
Two frames, same answer. Earth → longer time; muon → shorter distance. Never both in one sum.
Sense-check. The relativistic result must let more muons survive than Newton — longer time or shorter distance.
Fraction = (½)n.n doesn’t have to be a whole number.
⚠ Common mistakes
Mixing frames — dilating the time and contracting the distance in the same calculation (double-counting)
Pairing a dilated half-life with a contracted distance — match both to one frame
Forgetting n can be fractional in (½)n
Thinking the muon “feels” a longer life — in its own frame the half-life is the normal 1.6 µs; it’s the distance that shrinks
Expecting the Newtonian and relativistic answers to be close — they differ by orders of magnitude
And that’s the whole relativity story, start to finish: two innocent-looking postulates, the Lorentz transformations, the one invariant everyone agrees on, clocks that slow, rulers that shrink, a “now” that isn’t shared, the diagrams that draw it all — and finally, muons pattering onto detectors as living confirmation that it’s true. You’ve now got the complete HL relativity toolkit. Nicely done — from here, Topic 1 carries on into its next set of ideas, and every one of them will feel easier for having wrestled with this.
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