Cosmic-ray muon time-dilation experiment

Atmospheric cosmic rays generate muons whose mean lifetime at rest is about $2.2\mu \mathrm{s}$. For a selected speed $v$ and production-to-detection distance $h$ in the terrestrial frame, special relativity predicts survival fraction $e^{-h/(\gamma v{\tau }_0)}$, where $\gamma$ is the Lorentz factor. Without time dilation, the exponent would instead be $-h/(v{\tau }_0)$. Comparing altitude-dependent counts at controlled energies, allowing for acceptance and energy loss, tests these alternatives. In the muon rest frame the terrestrial distance is shortened to $h/\gamma$ by length contraction, so the proper flight time and survival fraction agree with the terrestrial calculation. This frame comparison does not turn the same decay measurement into two independent tests.

An unstable particle of rest lifetime ${\tau }_0$ traversing distance $H$ at constant speed $v$ has survival probability $p=\exp [-H/(\gamma v{\tau }_0)]$. Requiring a flight time no greater than one dilated mean lifetime selects $p\ge e^{-1}$. In the ultrarelativistic approximation $v\simeq c$, this gives $\gamma \gtrsim H/(c{\tau }_0)$ and $E\gtrsim m{c}^{2}H/(c{\tau }_0)$. It is a specified survival-probability criterion, not a sharp cutoff for individual random decays. Continuous energy loss requires integrating the decay probability along the path.