A moving clock loses time. Not what a camera sees: what the clock reads.

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.

/**
 * Clock loss per coordinate second: 1 - 1/gamma = beta^2 / (1 + sqrt(1 - beta^2)).
 * printedSecondOrder is (1/2) beta^2.
 */
export function dilationLossPerSecond(
  beta: number,
): KinematicResult<Readonly<{ exact: number; printedSecondOrder: number }>> {
  const bad = refuseBeta(beta);
  if (bad) return bad;
  if (isMode1904()) refuseModernSurface("gamma");
  const root = Math.sqrt(1 - beta * beta);
  return ok({
    exact: (beta * beta) / (1 + root),
    printedSecondOrder: 0.5 * beta * beta,
  });
}

/** Each inertial frame reports the same dilation factor for the other clock's rate -- the
 * symmetric statement this bead requires, without constructing a desynchronized separated-
 * reading comparison (that needs events.ts's simultaneity apparatus, out of this bead's Scope
 * to build). */
export function reciprocalRates(beta: number): KinematicResult<ReciprocalRatesSummary> {
  const g = gamma(beta);
  if (g.status !== "value") return g;
  return ok({ dilationFactor: g.value });
}

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.