Length contraction and the muon lifetime experiment

Turning Points in Physics | AQA A-Level Physics

Key Definitions

Length contraction: An object moving relative to an observer appears shorter in the direction of motion than its proper length.

Proper length ($L_0$): The length of an object measured in the frame where the object is at rest. It is always the longest measured length.

The equation

$$L = \frac{L_0}{\gamma} = L_0 \sqrt{1 - \frac{v^2}{c^2}}$$

$L$ is the contracted length measured by the observer who sees the object moving.
$L_0$ is the proper length measured in the object's rest frame.
Since $\gamma \geq 1$, the contracted length $L$ is always less than or equal to the proper length $L_0$.
Contraction only occurs in the direction of motion. Dimensions perpendicular to the motion are unchanged.

The muon experiment from the muon's perspective

In the Earth's frame, muon survival is explained by time dilation (the muon's clock runs slowly, so it lives long enough to reach the ground).
In the muon's rest frame, the muon's lifetime is unchanged at 1.6 microseconds. Instead, the distance from the top of the atmosphere to the ground is contracted.
With $\gamma \approx 5$, the 10 km height contracts to about 2 km in the muon's frame. At $0.98c$, the muon can cover 2 km in its short lifetime.
Crucially, both explanations (time dilation from Earth's frame, length contraction from muon's frame) give the same physical prediction: muons reach the ground. This is a requirement of the first postulate, that the laws of physics are the same in both frames.

Derivation from the Lorentz transformation

Length contraction follows from time dilation using $\text{speed} = \text{distance}/\text{time}$.
In the rest frame: $v = L_0 / \Delta t$. In the moving frame: $v = L / \Delta t_0$.
Since $v$ is the same in both frames and $\Delta t = \gamma \Delta t_0$, we get $L_0 / (\gamma \Delta t_0) = L / \Delta t_0$, giving $L = L_0 / \gamma$.

Common Mistake

Students sometimes multiply by $\gamma$ for length contraction (as for time dilation). Remember: time dilation multiplies by $\gamma$ ($\Delta t = \gamma \Delta t_0$) but length contraction divides by $\gamma$ ($L = L_0 / \gamma$). Moving clocks run slow (times get longer); moving rulers get short (lengths get shorter).