Clocks at different heights

The previous lesson described geometry in terms of distances, clock readings and free paths. We can now measure one part of that picture directly, by comparing clocks held at different heights above Earth.

Take two accurate clocks and compare them while they remain at different heights. In 2010, researchers at NIST measured a difference with one clock just 33 centimetres above the other. The higher clock ran slightly faster.

A difference in elapsed time

This effect is called gravitational time dilation. It is a difference in the elapsed time recorded by the clocks. Two ideal clocks that start together and follow the same journey agree, however they keep time. Separating their paths can produce different readings.

Bring the separated clocks back together and their readings can differ. To calculate the difference precisely, we also account for the journeys taken to separate and reunite them.

Near Earth’s surface, each metre of height changes the relative rate by about one part in ten quadrillion. The change across a shelf would amount to roughly 90 billionths of a second over a human lifetime.

Why navigation needs this correction

A GPS satellite’s higher altitude makes its clock gain roughly 45 microseconds per day relative to a clock on the ground. Its orbital motion produces an opposing effect, reducing the net gain to about 38 microseconds per day. Satellite navigation accounts for both.

Connecting clock rates with down

For this comparison, imagine Earth as a sphere with no rotation, and keep each clock at a fixed position above it. Moving a clock lower makes it run slower relative to one kept higher up. Down is also the direction in which a released object begins to fall.

The useful measurement is how the rate changes from one height to the next. This is the clock-rate gradient. The next lesson connects that gradient with falling, without treating a slow clock as something that pulls on a ball.

Einstein’s equations predict both effects from the geometry around Earth.