The Light Clock · The Cavern

lab frame — two light-clocks, one rest & one moving

A light-clock is two mirrors a distance L apart with a photon bouncing between them; one round trip is one tick. Set it moving and the photon must travel a slanted path — longer, so each tick takes longer. By Pythagoras the stretch is exactly γ. Light is the one speed everyone agrees on, so time itself bends to keep it constant.

Velocity of the moving clock

β = v/c 0.600

Lorentz factor γ1.250

rest tick t₀ = 2L/c2.000 L/c

moving tick t = γ·t₀2.500 L/c

moving clock runs1.25× slower

ticks elapsed (rest / moving)0 / 0

Length contraction — L′ = L/γ

rest length L1.000 L

moving length L′0.800 L

The invariant interval — s² = (ct)² − x²

lab framemoving frame ct1.400— x0.500—

s² = 1.710 = 1.710 invariant ✓

The boost tilts the t′ and x′ axes toward the 45° light-line like scissors, sliding the event along its invariant hyperbola — new coordinates, same s². Drag β toward c and watch the moving clock fall behind. The chip top-right proves all of this to machine precision.