Cosmic Muons: Time Dilation Made Visible
1 · Predict
Muons created by cosmic rays high in the atmosphere have a proper lifetime of only about 2.2 microseconds — not nearly enough time, at nearly light speed, to reach the ground unaided. Yet muons are detected at sea level in large numbers. Does time dilation explain this?
- Yes — a fast-moving muon's decay clock runs slow from Earth's frame, extending its observed lifetime enough for it to reach the ground.
- No — muons must be created closer to the ground than expected.
- This has nothing to do with relativity.
2 · Set Up
- Open the muon-time-dilation preset and press Reset. The muon's proper lifetime is fixed at 2.2 µs, speed β = 0.995.
- Enable the Lorentz factor and dilated-lifetime readouts.
- Set the muon's speed for each trial and record its dilated (Earth-frame) lifetime.
3 · Collect Data
| Speed fraction β = v/c | Lorentz factor γ | Dilated lifetime Δt (µs) |
|---|---|---|
| 0.9 | ||
| 0.995 | ||
| 0.999 |
Plot dilated lifetime Δt (y-axis) against γ (x-axis) for your three trials. Is the line straight through the origin?
4 · Analyze
- For one trial, compute γ = 1/√(1 − β²), then Δt = γ·(2.2 µs). Compare to the table. Confirm β = 0.995 gives roughly a tenfold lifetime extension.
- Without time dilation, even at nearly light speed a 2.2 µs proper lifetime lets a muon travel only about 660 m — far short of the ~15 km atmospheric height where most are created. Explain how the dilated lifetime you computed changes this picture.
5 · Extend
- From the muon's own reference frame, its lifetime is still just 2.2 µs — but the muon 'sees' the atmosphere length-contracted, so it doesn't need to travel as far. Explain why both explanations (Earth-frame time dilation, or muon-frame length contraction) predict the exact same experimental outcome: the muon reaches the ground.
- The 1941 Rossi-Hall experiment comparing muon counts at a mountaintop versus at sea level was one of the first direct experimental confirmations of special relativity's time dilation. Why might comparing counts at two different altitudes be a clever way to test this prediction?
The Physics Behind This Experiment
Muon Time Dilation
A muon moving at β = 0.995 (γ ≈ 10) has its 2.2 µs proper lifetime stretched to about 22 µs as observed from Earth — long enough, combined with its near-light speed, to traverse the atmosphere and reach detectors at sea level.