A.5.15 (HL)—Muon decay evidence

Start from the observation

Muons created high in Earth’s atmosphere have a short proper lifetime, yet many are detected at the ground while travelling at speeds close to $c$. Without relativistic effects, the flight time through the atmosphere would exceed the muon lifetime and far fewer would survive.

Explain it in the Earth frame

In the Earth frame the moving muon’s lifetime is dilated: $\Delta t=\gamma\Delta t_0$. The increased lifetime allows more muons to travel the atmospheric distance before decaying. This is experimental evidence for time dilation.

Explain it in the muon frame

In the muon’s frame, the atmosphere is moving and its thickness is length-contracted: $L=L_0/\gamma$. The shorter distance can be crossed within the muon’s proper lifetime. Both frames predict the same detection rate; together they support time dilation and length contraction.

Evidence calculation from local Question Bank row 30447

Muons are produced $2.0\,\mathrm{km}$ above ground, move at $0.98c$, have proper lifetime $2.2\,\mu\mathrm{s}$ and $\gamma=5.0$.

$$t_{flight}=\frac{2000}{0.98(3.0\times10^8)}=6.8\,\mu\mathrm{s}$$
$$t_{Earth}=\gamma t_0=(5.0)(2.2)=11\,\mu\mathrm{s}$$

The dilated Earth-frame lifetime exceeds the flight time, explaining why many more muons reach the ground than the non-relativistic model predicts.

Common trap

Do not claim that the muon’s own clock runs slow in its rest frame. The proper lifetime is measured by the muon; the Earth observer measures the dilated lifetime, while the muon observer measures a contracted atmosphere.