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Clock Lab: the maths

Special relativity for the light clock; the weak-field approximation of general relativity for clocks in circular orbit. Computed live.

Equations

γ = 1 / √(1 − v²/c²)

The time-stretch factor. A clock moving at speed v ticks once for every γ ticks of a clock at rest beside you.

Δt_moving = Δt_rest / γ

Ticks counted on the moving clock during Δt_rest of yours.

Δf/f ≈ GM/c² · (1/R − 1/r) − GM/(2rc²)

An orbiting clock's rate against a clock on the ground: the first term is gravity (higher runs faster), the second is orbital speed v² = GM/r (faster runs slower). Multiplied by 86,400 × 10⁶ for microseconds per day.

Constants

c	299,792,458 m/s	exact, by definition
GM	3.986004 × 10¹⁴ m³/s²	Earth's gravitational parameter (IERS)
R	6,371 km	Earth's mean radius

Worked examples ✓ checked on every change

γ at 0.87 of light speed	2.028	the formula above
γ at 0.943 c (a third of the rate)	3.005	the formula above
A GPS clock's gain per day	+38.6 µs/day	Ashby (2003): +45.7 from gravity, −7.1 from speed

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• The light clock is ideal: mirrors that don't flex, light that doesn't scatter.
• Orbits are circular; Earth is a non-rotating sphere, so the small effect of the ground clock's own motion is left out.

Where it breaks

Near black holes or neutron stars the weak-field formula fails (use full general relativity — see the River). The light clock itself never breaks: γ applies to every clock.

Sources: A. Einstein (1905, 1915); N. Ashby, Living Rev. Relativ. 6, 1 (2003).