Muon decay & time dilation
Cosmic-ray muons counted at any altitude you like: far more arrive than a muon’s lifetime allows, and only a slow-running clock accounts for it.
Nuclear & quantum — Radioactive decay, the absorption of radiation, photons, and energy levels in the atom.
Experiments already set up on it
Counting muons on the way down
The headline: far more reach sea level than they possibly should.
- A muon lives 2.2 µs, so at 0.98c it should manage about 650 m
- They arrive from 15 km anyway, in numbers nothing classical can explain
- Their clocks run slow by γ, so the trip costs them γ times less of their life
Faster muons last longer
Not because they decay more slowly, but because their clocks run slow.
- γ climbs very steeply as v approaches c
- The surviving fraction depends on it far more than on the extra speed
- A muon at rest and a muon at 0.999c decay at exactly the same rate — in their own time
Counting for longer, up a mountain
Thin counts high up: the √N lesson applies here as much as anywhere.
- A count of N carries an uncertainty of √N, whatever is being counted
- So a thin rate needs a long count
- Which is why cosmic-ray experiments run for weeks
What you can measure
Each of these can be recorded over time, aggregated over a run, or read at the moment something happens — through an instrument that carries its own uncertainty.
- Timet (s)
- Counts so farN
- Count rateR (1/s)
- Fraction that gets heref
- Fraction if clocks did not slowf_c
- Lorentz factorγ
- Trip time, by our clockst_u (s)
- Trip time, by the muon’s clockt_m (s)
- How far it would get without dilationd_c (m)
- Altitudez (m)
What you can change
Any of these can be the thing you vary, and every combination of the ones you choose is run as its own trial.
- Made at a height ofh₀ (km)
- Speedv/c
- Muons setting out, per secondΦ (1/s)
- Detector altitudez (m)
- Background rateR_b (1/s)
- Count fort_g (s)
The equations behind it
What muon decay & time dilation is built on. Every one of these is in the lab's formula reference too, where it copies straight into a write-up.
Special relativity
- Lorentz factor
- Time dilation — Δt₀ is measured in the frame where the two events happen in the same place.
- Length contraction — L₀ is the length in the frame where the object is at rest.
- Lorentz transformation of position
- Lorentz transformation of time
- Adding velocities
- Spacetime interval — The same in every frame.
- Total energy
- Relativistic kinetic energy
- Energy, momentum and mass
Also in Nuclear & quantum
Radioactive decay
A source counted with a Geiger tube, decay by decay, with the √N scatter of real counting statistics and a background that does not decay.
Absorption of radiation
Alpha, beta and gamma through absorbers of your choosing: only one of the three is absorbed exponentially, and the counter still does not subtract the background.
Photoelectric effect
Light on a metal plate and a retarding voltage turned up until the current stops: the stopping voltage is found rather than calculated, and brightness changes nothing about it.
Electron diffraction
Electrons accelerated through a voltage and diffracted by graphite: the rings move as 1/√V, which is de Broglie’s λ = h/p made visible.
Atomic spectra
Hydrogen through a grating spectrometer: each line is an electron falling between two levels, and a graph of them gives the Rydberg constant.
Rutherford scattering
Alpha particles onto a thin foil, counted at an angle: the 1/sin⁴(θ/2) law is what a point nucleus gives, and a head-on alpha says how small it is.