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Dr Karl: Super-accurate clocks can measure height

11 August 2023
Part of innovation powering future industries
We’ve had clocks for thousands of years. But clever scientists have worked out how to use them to measure height!

Einstein told us that time is affected by both speed and gravity. As an everyday example, GPS satellites would be uselessly inaccurate, if they did not use Einstein's theories. Compared to us down on the ground, they are moving faster, and are in a weaker gravitational field.

The Global Positioning System (GPS) is a network of a few dozen satellites orbiting the Earth at around 14,000 kilometres per hour, and about 20,000 kilometres above the surface. They continuously broadcast their exact position and their exact time, which they get from a super-accurate atomic clock on-board.

Once the GPS unit in your car, or smart phone, receives the signal from four or more satellites, it uses simple geometry (triangulation) to work out your location to an accuracy of around 5 metres.

First, Einstein’s 1905 Special Theory of Relativity tells us that the faster you travel, the more your clock will slow down. GPS satellites orbit at a speed of around 14,000 kilometres per hour, relative to the ground.

Einstein's maths tells us that the on-board GPS atomic clocks will run slower than clocks sitting stationary on the ground. They will lose about 7 millionths of a second (7 microseconds) each day – thanks to their faster speed.

Second, Einstein's 1915 General Theory of Relativity says that the stronger your gravitational field, the more time will slow down. The Earth’s gravitational field weakens with distance, so the satellites GPS satellites (20,000 kilometres above us) are in a weaker gravitational field – relative to us.

Compared to ground-based clocks, the clocks in GPS satellites gain about 45 microseconds each day – thanks to weaker gravity.

So when you factor in the 7 microseconds lost from speed, and the 45 microseconds gained from weaker gravity, overall, the clocks in GPS satellites gain about 38 microseconds each day – relative to our atomic clocks down here on the ground.

So we deliberately 'fudge' the atomic clocks in the GPS satellites, so that we (down here on the ground) can know our position accurate to 5 metres, or better.

If we did not fudge the clocks in the GPS satellites, after just one day, the position they give us would be wrong by about 10 kilometres.

Atomic clocks have since become more accurate. In 2010, scientists were able to detect clocks speeding up when they were lifted by a tiny elevation of just 30 centimetres.

In 2022, physicists made atomic clocks even more accurate – by cooling 100,000 atoms of strontium 87 to about one hundred billionths of a degree above absolute zero, and then separating them into two layers – like pancakes. The layers of atoms were about a millimetre apart. The atoms in the upper pancake layer oscillated faster, because they were in a weaker gravitational field - yes, by one tiny millimetre!

These super-super-accurate clocks are so precise that you would have to wait 300 billion years for them to be off by even one second.

Our current GPS is accurate to 5 metres. But if the clocks were a thousand times more precise, GPS could measure to 5 millimetres.

The industrial implications are limited only by our imagination –parking your car, measuring the side of a volcano bulging, and much more.