Why GPS would drift kilometres a day without relativity

Satellite in orbit

Your phone knows where you are because GPS satellites send radio signals.

Those signals carry the time from atomic clocks orbiting Earth.

But time runs at different rates depending on speed and gravity.

Without Einstein's equations, your phone would be lost in kilometres.

A GPS satellite orbits Earth 20,200 kilometres up.

It travels at 3.9 kilometres per second.

An atomic clock on board ticks out precise time.

That clock is the only link between the satellite and your position on the ground.

Time measurement

The GPS system measures distance by timing.

The satellite sends a signal and stamps it with the time.

Your phone receives it and compares the satellite's time to your own clock.

The delay tells you how far away the satellite is.

With signals from four satellites, your phone calculates your exact position.

Without vs. with relativity

Without relativity, you might think atomic clocks tick at the same rate everywhere.

If satellite clocks and ground clocks stayed perfectly in sync, the math would work.

But Einstein proved that clocks run slower when they move fast and near massive objects.

A satellite clock actually runs 38 microseconds faster per day than a clock on the ground.

38 microseconds × light speed gives the error: 11 kilometres per day of drift.

The 11-kilometre error

The satellite clock gains 38 microseconds per day.

Light travels 0.3 million kilometres per second.

Multiply them together: 38 microseconds times 300,000 kilometres per second.

The result is 11.4 kilometres of position error every single day.

Einstein's two effects

Einstein's special relativity says moving clocks run slow.

The satellite moves at 3.9 kilometres per second, so its clock loses 7 microseconds per day.

But general relativity says clocks run faster far from gravity.

The satellite is 20,200 kilometres up, where gravity is weaker, so its clock gains 45 microseconds per day.

The net effect: the satellite clock gains 38 microseconds per day.

Why GPS needs relativity

Without accounting for Einstein's relativity, GPS satellites would send signals with the wrong time.

In just one day, your phone would lose track by 11 kilometres.

In a week, the error would be 80 kilometres.

Every GPS receiver on Earth depends on relativity to work.