Before a GPS satellite is launched, its atomic clock is set to the wrong frequency. On purpose.

The clock is supposed to tick at 10.23 million times a second. Engineers set it to 10.22999999543 million. That’s slower by about 4.5 parts in ten billion. Without that change, the clock would gain about 39 microseconds a day on a ground clock, enough for light to travel more than 11 kilometers. Here’s why.

Two effects, pulling opposite ways

Compared with a clock on the ground, a clock on a GPS satellite is pulled two ways by relativity, in opposite directions.

It moves fast, about 3.9 km/s, and a moving clock runs slow. On its own, that would cost it roughly 7 microseconds a day.

It also sits high, where Earth’s gravity is weaker. The last post left gravity out, so take this one on trust for now: time runs slightly faster where gravity is weaker. On its own, that would add roughly 46 microseconds a day.

Interstellar pushes this to the extreme. In the movie, a planet orbits so close to a black hole that an hour on its surface is about seven years for the crew waiting far above. The film turns the dial way up, but it’s the same effect.

Gravity wins, about six to one. Net, the satellite’s clock runs about 39 microseconds a day fast. (The paper gives the net, 4.4647 parts in ten billion. The split into two pieces is the standard textbook one.)

So before launch, engineers lower the clock’s frequency by exactly that amount. A clock that would run fast in orbit is built to run slow on the ground, and once it’s up there, it ticks at 10.23 MHz as seen from the ground.

Wait, shouldn’t it run slow?

Close to the Earth, it does go the other way. On the International Space Station, the speed effect wins, even though weaker gravity still speeds the clock up a little, so the clocks run slow and an astronaut there ages a hair less than a twin on the ground. The weaker-gravity effect only takes over a few thousand kilometers up, and GPS orbits well above that.

Neither clock is broken, and a perfect clock would do the same. This isn’t about how the clock is built. Speed and gravity change how much time passes, and the difference is predictable. In 2010, NIST saw it in a lab: take two atomic clocks sitting still, raise one by just 33 centimeters, where gravity is very slightly weaker, and with no speed involved it ticks measurably faster.

Why a clock error becomes a distance error

A GPS receiver doesn’t measure distance. It measures time: how long a signal took to arrive from each satellite. Then it multiplies by the speed of light, the same limit from the last post. One microsecond is about 300 meters. One nanosecond is about 30 centimeters. (That’s also why a receiver needs at least four satellites: to work out its position and correct for its own imperfect clock.)

So 39 microseconds a day adds up. Light covers about 11.6 kilometers in 39 microseconds. Try it:

For scale, a good Cesium clock left alone for a day stays correct to within about 4 nanoseconds. The relativity correction is about ten thousand times bigger than that. The paper puts it this way: “Relativistic effects are huge compared to this.”

What it means for software people

GPS is a converter between clocks and distance, so a timing error turns into a distance error. The speed limit from the last post is doing a day job here.

Software depends on this too. Google’s Spanner database uses GPS and atomic clocks (it calls the system TrueTime) to keep transactions in order across data centers (paper).

The 39 microseconds is real. Two clocks counted different amounts of time. But “the same time” on a satellite and on the ground is a different question. GPS has to define it: pick a convention for which readings count as simultaneous, then correct for speed, gravity and signal travel time. Why that works, and what it says about “now”, is where I’m going next.