Thought Toys · Waves & rhythm · Exhibit 66
Two identical clocks, each a photon bouncing between two mirrors. One sits still; the other glides sideways. Nothing inside either clock changes — but the moving one's photon has to travel a longer, slanted path, and light refuses to go any faster to make up for it. So it ticks less often. That's the whole argument.
Two light clocks, watched from the same seat the moving clock your clock, at rest the light's path
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A light clock is the simplest honest clock anyone can build: two mirrors facing each other, and one photon bouncing between them. One round trip is one tick. If the mirrors are a distance L apart, a tick takes 2L⁄c — however long light needs to go up and back.
Now slide the whole clock sideways. While the photon climbs from the bottom mirror to the top one, the mirrors have moved, so the photon doesn't go straight up — it goes up and along, on a diagonal. That diagonal is longer than the straight climb. In ordinary life this wouldn't matter: the light would just be carried along and arrive on time, the way a ball thrown straight up inside a train lands back in your hand. But light doesn't work that way. Its speed is the same c for everyone, whatever the source is doing. A longer path at the same speed takes longer. So the tick stretches.
How much longer is pure Pythagoras — the diagonal, the vertical climb and the sideways drift make a right triangle — and it turns into the factor γ below. Push the slider to 0.866 and γ hits exactly 2: the moving clock ticks once for every two of yours. Push it toward 1 and γ has no ceiling at all. Pull it back to walking pace and the effect is still there, just quadratically tiny — which is exactly why nobody noticed for two and a half centuries.
And nothing has been done to the moving clock. Its mirrors are the same distance apart, its photon is the same photon. From its own seat it ticks along perfectly normally — it's your clock that looks slow to it, by exactly the same γ. Press Ride with the other clock and you'll see it: the roles swap, the picture is the mirror image, and the number on the verdict doesn't move at all. There is no experiment either of you can do to settle which one is "really" moving, because the question has no answer.
What both of you do agree on is the spacetime interval between the two events — the photon leaving the bottom mirror and coming back to it. You each measure a different elapsed time Δt and a different distance Δx, but the combination c²Δt² − Δx² comes out identical. That, not anyone's clock, is the thing that's actually absolute — and it's why this is a fact about the shape of spacetime rather than a defect in a clock.
improve/verify/66-time-dilation.js): solving the triangle
numerically for T — by bisection, without ever substituting the γ formula — reproduces γ·T₀ at twelve speeds
to within 1e-11, so the formula is derived here rather than assumed; an independent step-by-step
photon flight, given the velocity components that keep its total speed exactly c, lands on the same
tick time to 1e-9; the landmark factors are exact (β=0.6→1.25, β=0.8→5⁄3, β=√3⁄2→2); and the spacetime
interval c²Δt²−Δx² comes out identical in both frames despite completely
different Δt and Δx. Negative controls (three): the classical picture, where light is simply
carried along, predicts no dilation whatsoever at any speed — and requires the photon to exceed
c, which is the assumption experiment rules out; a γ with the square root dropped is rejected
outright by the very same triangle at every tested speed; and the effect is confirmed to be quadratic in β,
not linear, so a "slows a bit per unit speed" model is refuted by the same numbers. A 250 m/s airliner flown
for a year falls about 11 microseconds behind — the same formula, thirteen orders of magnitude down.
Also in Waves & rhythm: The angle past which light refuses to leave →
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 66.