Muon Decay: Time Dilation vs. Length Contraction

Source

Collection: Wiki SR simulations — time dilation in cosmic-ray muons.
Concept: The lab frame and the muon's rest frame give one invariant proper time τ: the lab uses time dilation (τ = t_lab/γ), the muon uses length contraction (τ = (L/γ)/(βc)). Both descriptions agree on τ, on the survival fraction f = e^−τ/τ₀, and on the invariant interval s² = (cτ)².
Serves: Resolving the muon-decay "paradox": why cosmic-ray muons reach the ground even though their proper lifetime is far shorter than L/(βc) would classically allow.
Inquiry: As β → c, does the survival fraction f rise or fall? Do the two frames disagree about τ? Where does the invariant hyperbola meet the muon's worldline?
Misconception: That "faster ⇒ crosses more atmosphere ⇒ fewer muons survive" — the classical picture, which ignores that the muon's own clock runs slow (or, equivalently, that the atmosphere is contracted).

Pedagogy principles applied

ID	Principle	How it shows up here
H1	Misconception is structural, not a text card	The "faster ⇒ more atmosphere ⇒ lower f" error is a wrong prediction choice; its rebuttal appears only if the student picks it.
D6/D7	Guided inquiry reveals in stages	Five step-cards run predict → observe the lab frame → checkpoint predict → observe the muon frame (revealed here) → resolve; the muon panel appears exactly when the step introduces it.
J	Invariant panel scoped to an invariance lesson	The lesson is that τ is frame-invariant, so the readout shows τ_lab = τ_muon and the Minkowski overlay renders the invariant hyperbola (cτ) through the arrival event.
A2	Fewest objects	Opens with the lab frame alone; the muon frame and its readouts appear only once the inquiry reaches them.
C	Every number earns its place	β, γ, L, L/γ, t_lab, τ and f are each read or compared by a step; the two-column readout is the crux of the equivalence.

Speed

Lab frametime dilation · τ = t_lab/γ

Guided Inquiry

1 · Predict the survival fraction

The survival fraction f = e^−τ/τ₀ is the fraction of muons that live long enough to reach the ground; it depends on the proper time τ the muon experiences. As you push the muon's speed β closer to c, what happens to f?

2 · Watch the lab frame

In the lab frame the muon races down through the full atmosphere L = 15 km. Compare the two clocks: the lab clock (red) races ahead while the muon's proper clock (teal) ticks slowly by the factor γ. That slow proper clock is time dilation — watch the teal proper-time bar fill.

3 · Checkpoint: switch frames

Now sit in the muon's rest frame. Here the muon is still and the atmosphere is length-contracted to L/γ, rushing up past it. Will this frame give the same proper time τ that the lab frame just found?

4 · Watch the muon frame

The muon panel is now open. The muon sits at rest while the contracted atmosphere L/γ sweeps up past it — one clock, ticking normally. This is length contraction. Its proper-time bar fills to the same τ the lab frame reported.

5 · One invariant, two descriptions

Both frames land on τ_lab = τ_muon and the same survival fraction f. Time dilation and length contraction are two views of a single fact: the invariant interval s² = (cτ)². Turn on the Minkowski overlay to see the invariant hyperbola pass through the arrival event, or open ∑ Formal for the algebra. Hit Finish to explore freely.

Controls

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v = 0.9900c γ = 7.09

0.9 c0.9999 c

Sync frames

Readouts

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Atmosphere · lab15.00 km

Atmosphere · muon (L/γ)— km

Lorentz factor γ (invariant)—

Coord. time · lab— µs

Coord. time · muon— µs

proper time τ · lab— µs

proper time τ · muon— µs

survival f · lab—

survival f · muon—

Minkowski overlay

τ₀ = 2.2 µs · c = 0.29979 km/µs · L = 15 km (production altitude).

Lorentz factor & proper time

Dilation ⟺ contraction

Survival & invariant interval

Rapidity —