Somewhere with no gravity, a box accelerates upward at . Send light from the ceiling toward the floor.
The light takes time to reach the floor, and the box keeps accelerating meanwhile. The floor receives the light while moving upward faster than when the light set out.
Because the floor receives the light while moving toward its source, the Doppler effect raises the frequency there.
With the ceiling a height above the floor, the light takes about to arrive. Meanwhile the floor's speed increases by .
The frequency rises by this speed divided by , a fraction .
The frequency of light, set by how many times it oscillates per second, also serves as the ticks of a clock. A higher frequency received at the floor means that, for someone at the floor, the ceiling clock runs fast.
By the equivalence principle, a box accelerating upward cannot be told apart from a box held in gravity with acceleration . What holds in the accelerating box holds in gravity too.
In gravity, clocks higher up run faster and clocks lower down run slower. This is called gravitational time dilation.
In a change in an atomic clock's rate was measured when it was raised by just cm.
With the time ticked by the higher clock and by the lower one,
For a height difference of m this is , so even after a day ( seconds) the difference is under nanoseconds.
No speed of the clocks appears in this formula. Clocks at rest still run at different rates, which makes this an effect separate from the time dilation of special relativity.
GPS satellites orbit at an altitude of about km. Gravity is weaker there, so the satellite clocks run about microseconds a day faster than clocks on the ground.
The satellites move at about km/s, so special relativistic time dilation makes them lose about microseconds a day. The net result is a gain of about microseconds a day.
Light travels about km in microseconds, so without a correction the position calculation would drift by more than km a day. The satellite clocks are therefore set before launch to tick at MHz on the ground instead of MHz.