The light clock and moving clocks

A clock that travels away and comes back reads less time than an identical clock that stayed put. At 60 percent of the speed of light, a trip that takes 10 seconds on the clock at home takes only 8 seconds on the traveller.

Section 4 takes a clock at rest at the origin of the moving system k and asks how fast it runs, judged from the resting system K. Its position is x = vt, and the transformation gives its reading as τ = t√(1 − v²/c²), so each second it falls behind by 1 − √(1 − v²/c²) seconds, or ½(v/c)² to within terms of fourth order. Einstein draws the consequence: a clock carried from A to B and set beside one that stayed there lags by ½t(v/c)², and so does a clock taken round a closed path back to its start. The instrument computes that reunion for an out-and-back trip or a circle at constant speed. At 0.6c over 10 s of the resting clock, the traveller reads 8 s: the exact loss is 0.2 s per second, 2 s in all, where the printed approximation gives 0.18 s per second, 1.8 s. A light clock with an arm of 1 light-second ticks every 2 s in its own frame and every 2.5 s as judged from K, the same factor 1.25. The comparison is of clock readings at a shared event; it is not what a camera would see.

Take the resting system K and a clock moving along x at speed v = 0.6c, in seconds and light-seconds, so c = 1. The clock's position is x = vt. Section 3's transformation gives the moving system's time as τ = (t − vx/c²)/√(1 − v²/c²). Put x = vt into the top: t − v × vt/c² = t(1 − v²/c²), and dividing by √(1 − v²/c²) leaves τ = t√(1 − v²/c²). At 0.6c, v²/c² = 0.36, 1 − 0.36 = 0.64 and √0.64 = 0.8: the moving clock reads 0.8 s for every second of K, and loses 1 − 0.8 = 0.2 s each second. Einstein's approximation uses √(1 − ε) ≈ 1 − ε/2 for small ε, so the loss is about ½ × 0.36 = 0.18 s each second: very good at everyday speeds, 10 percent low at this one. For the trip, out for 5 s and back for 5 s of K's time at 0.6c both ways, the traveller reads 0.8 × 10 = 8 s against the home clock's 10 s, a lag of 2 s; the approximation says 1.8 s. The turn does not change this, because the rate depends only on the speed, so any path at 0.6c for 10 s gives 8 s. A light clock shows the same factor from geometry. In its own frame light crosses an arm of 1 light-second and returns, a tick of 2 s. Seen from K the arm moves at 0.6c while the light crosses, so each leg is the long side of a right triangle: if a leg takes a time T, then (cT)² = 1² + (0.6cT)², so 0.64T² = 1 and T = 1.25 s, a tick of 2.5 s, and 2.5/2 = 1.25 = 1/0.8. At the equator the ground moves at 465 m/s, so (v/c)²/2 = 1.20 × 10⁻¹², about 104 ns a day in special relativity alone. To lose a whole second a day a clock would need ½β² = 1/86 400, so β = 0.0048, about 1440 km/s.

Einstein called the result a peculiar consequence. He proved it for a path made of straight pieces and assumed it for a curve, with a clock whose rate depends only on its speed, an assumption later named the clock hypothesis. His closing example, a balance-wheel clock at the equator running slower than one at a pole, holds in special relativity alone; on the real Earth the difference in gravitational potential cancels it, and clocks at sea level keep the same rate everywhere, which general relativity explains. The light clock is not in the paper: Lewis and Tolman used it in 1909. Hafele and Keating flew atomic clocks round the world in 1971 and found the lags predicted when both effects are included.

Model: an ideal clock whose rate depends only on its instantaneous speed. Not modeled: gravitational time dilation, real clock mechanisms under acceleration, rotating-frame synchronization, the geoid's actual shape, atomic-clock physics, and clock noise.