A light clock is two mirrors a fixed distance L apart with a photon bouncing between them; one round trip is one "tick." In the clock's own rest frame the photon just travels straight up and down, taking:
τ = 2L / c (proper time — the ship/muon's own clock)
Now watch the same clock fly past you sideways at speed v = βc. Because light always travels at c in every frame, but the mirrors have also moved sideways during the bounce, the photon must trace a longer diagonal (zig-zag) path in your frame. Pythagoras on that right triangle gives:
(c·t/2)² = (v·t/2)² + L²
→ t = 2L / (c·√(1 − v²/c²)) = γ · τ
γ = 1 / √(1 − β²) (Lorentz factor, β = v/c)
- β slider — sets the clock's speed relative to you (the lab frame); the right-hand "lab view" always shows the zig-zag, longer path.
- Presets — real β values: GPS satellites (~14,000 km/h), cosmic-ray muons produced in the upper atmosphere (their time dilation is why enough of them survive to reach the ground before decaying), and two illustrative fast cases.
- τ vs t — τ is what the moving clock itself reads; t is what you read on your own clock while watching it. t is always ≥ τ: moving clocks run slow, never fast, and the effect is symmetric only when you swap which observer is "moving."
This is the standard textbook derivation of special-relativistic time dilation (Einstein, 1905) — no gravity involved, purely a consequence of the constancy of the speed of light for every inertial observer.