How clock rates connect with falling

Clocks held lower down run slower, while a released ball falls towards the ground. To connect these observations, return to the accelerating rocket from What a scale measures. It lets us compare clock signals and a released ball in one setting.

Clocks in the rocket

The cabin accelerates towards its ceiling. A clock on the floor sends a regular series of light pulses upwards, one for each tick.

The pulses take time to reach the ceiling. While they travel, the rocket gains speed. The ceiling receives the light at a lower frequency, and the floor clock’s ticks arrive farther apart than ticks from an identical clock beside the ceiling.

For observers staying at fixed places in the cabin, the floor clock runs slower than the ceiling clock. Release a ball in the same cabin and it travels freely while the floor accelerates towards it. The ball appears to fall towards the end where the clocks run slower.

The equivalence principle lets us apply this comparison to a small laboratory held above Earth. Both the clock-rate difference and the falling ball fit the same description. No attraction from the clock is needed.

How the difference gives an acceleration

Near Earth, we can calculate the acceleration of a ball released from rest using the change in clock rate with height:

acceleration ≈ c² × fractional increase in clock rate per metre

Here, c is the speed of light. The measured rate changes by about 1.09 × 10⁻¹⁶ per metre. Multiplying by c² gives about 9.8 metres per second squared, the familiar acceleration near the ground.

This relationship comes from Einstein’s equations. We are using their approximation for a weak gravitational field that is not changing with time, and for a ball moving slowly compared with light.

The clocks measure a feature of the region the ball moves through. Their readings connect the ball’s motion with the support needed to remain at a fixed height.

The next lesson looks at what else is needed to describe light.