Gravitational Time Dilation

General relativity says gravity slows time itself: a clock deep in a gravitational well ticks slower than one higher up.^[1] In 1959 the Pound–Rebka experiment caught it in a Harvard tower just 22 metres tall — light climbing against gravity lost energy (shifted redder) by the minuscule predicted amount.^[2] Precise clocks now show the effect over a change of just centimetres in height. It stacks on top of special relativity's motion-based time dilation, and GPS satellites correct for both — their onboard clocks run fast from weaker gravity and slow from orbital speed, and getting either wrong would ruin the fix.^[3]

Deep dive · Why a GPS clock ticks 38 microseconds a day fast

A GPS satellite carries an atomic clock orbiting at radius $r \approx 26{,}600\,\text{km}$ (altitude ~20,200 km) at speed $v \approx 3.87\,\text{km/s}$ . Two relativistic effects push its clock in opposite directions relative to one on the ground at Earth's radius $R \approx 6{,}370\,\text{km}$ .

Gravity (general relativity) — runs fast. Higher in the well, time speeds up. In the weak field, the fractional rate difference is the difference in gravitational potential $\Phi = -GM/r$ divided by $c^2$ :

\left(\frac{\Delta t}{t}\right)_{\text{grav}} = \frac{\Phi_{\text{sat}} - \Phi_{\text{ground}}}{c^2} = \frac{GM}{c^2}\left(\frac{1}{R} - \frac{1}{r}\right) \approx +5.3 \times 10^{-10}.

Motion (special relativity) — runs slow. The orbiting clock moves, so it ticks slower by $v^2/2c^2$ :

\left(\frac{\Delta t}{t}\right)_{\text{vel}} = -\frac{v^2}{2c^2} \approx -0.8 \times 10^{-10}.

Net. Adding them, the satellite clock gains

\frac{\Delta t}{t} \approx +4.5 \times 10^{-10} \quad\Longrightarrow\quad (4.5 \times 10^{-10}) \times 86{,}400\,\text{s/day} \approx +38\ \mu\text{s/day}.

That sounds tiny, but light travels ~30 cm per nanosecond, so 38 µs of clock error is about 11 km of position error piling up every day. GPS would be useless in minutes. Engineers correct it at the source: the satellite oscillators are deliberately set slightly slow — to 10.229999995 43 MHz instead of exactly 10.23 MHz — so that once in orbit they tick at the right rate for us on the ground.^[4]