The clock they slowed down before launch

One expression

A clock's rate depends on where it is and how fast it is going, and to the accuracy anything here needs, those two enter as a sum. Put a clock in a circular orbit of radius r and compare it with a clock at sea level. Being high makes it run fast; being fast makes it run slow. For a circular orbit the speed is not free — it is set by the radius, v² = GM/r — so both terms collapse into one variable and the whole of this page is the line below.

Δ = W₀/c² − 3GM / 2rc²

W₀ = 62 636 856 m²/s² is the potential of the geoid — the surface the sea would settle on. It is one number for the whole planet, and the third section says why that is not a simplification. The 3 is the whole story in one digit: two parts of height, one part of speed, and the speed is half the size of the height term and points the other way.

Drag it down towards the Earth and the net turns over. There is nothing special about the crossing except that the two terms happen to be equal there, and that is the next section.

The altitude where the clocks agree

Set Δ = 0 and the only unknown left is the radius:

r₀ = 3GM / 2W₀

There is an orbit that keeps time with the ground. It is · above the sea, it is not a resonance and not an equilibrium, nothing sits there, and a clock flown at that height would need no relativistic correction at all — not because relativity has stopped, but because two of its terms have the same size. Below it, orbital clocks run slow: the International Space Station loses · a day, and an astronaut really does come home younger, by a marginally different mechanism from the one that makes GPS run fast.

A cheaper version of the same number: pretend the Earth does not spin and is a sphere, so W₀ = GM/R. Then r₀ = 3R/2 exactly — one and a half Earth radii, no constants at all. That gives · against the honest ·, a difference of ·. The whole of the Earth's spin, its equatorial bulge and its lumpiness are worth a quarter of one percent of the answer, which is the correct order for all three of them and is not obvious in advance.

The curve is Δ(r) in microseconds a day. It is not a straight line and it does not level off at anything you would guess — the ceiling as r → ∞ is · a day, which is just W₀/c², the rate a clock infinitely far away sees the sea running slow.

A ladder, from a desk to the Moon

Two formulas, marked. A clock that stays put at height h only picks up gh/c². A clock in orbit picks up the expression at the top. They are the same physics and they cannot be swapped.

The first rung is the one that makes the rest believable, because you can do it indoors. Lift an optical clock by the thickness of a hardback and it runs fast by ·. NIST did exactly that in 2010, over 33 cm of laboratory bench, and measured (4.1 ± 1.6) × 10⁻¹⁷. The same expression that puts a satellite 38 microseconds a day out is measurable across a desk.

The frequency they build

GPS runs on a fundamental frequency of exactly 10.23 MHz, and it wants that frequency as received on the ground. The clock is in orbit, so the factory has to hand it a clock that is wrong by the amount the orbit will be wrong by, in the other direction. Multiply and see:

f = 10.23 MHz × (1 − Δ)

The top line is computed on this page from GM, W₀, c and the orbital period. The bottom line is the number in the specification, the number the crystal is cut to, quoted from memory and not used to compute anything above it. They agree to · printed digits. This is the closest thing on the wall to a measurement: a decision made in a factory in the 1970s, recovered from three constants.

A constant printed in a specification

Orbits are not circles. An eccentric orbit's clock speeds up and slows down twice a revolution, and no factory offset can absorb something that changes sign. So the system gives the job to your receiver: the public interface document tells every GPS receiver on Earth to add

Δt = F · e · √a · sin E

and prints the value of F. It is not a fitted number and the document does not say where it comes from. It is this:

F = −2√GM / c²

Ten digits, all of them. And there is a fossil in it: the agreement is exact only against GM = 3.986005 × 10¹⁴, the value the GPS specification itself names. Feed it the modern IERS figure and the eighth digit moves. The constant in the document is the 1984 value of GM, divided by the square of the speed of light, and you can read which almanac was on the desk.

Two things about that formula are worth more than the digits. The first is that e and a are in it but the satellite's identity is not: the correction is a property of the ellipse. The second is that the mean of it over one revolution is zero — eccentricity contributes nothing to the average rate. A clock on an ellipse of semi-major axis a keeps, on average, exactly the time of a clock on a circle of radius a, however squashed the ellipse is. That is not obvious, it is not an approximation, and it is why one factory offset serves a whole constellation whose orbits are all slightly different shapes.

The wave is drawn against time, not against angle, so it leans: the satellite loiters at apogee. It is a sine in the eccentric anomaly and a lopsided thing in the clock on your wall, and the area above the axis still equals the area below it.

What the famous number is worth

Everyone has heard that GPS without relativity would be wrong by 11 kilometres a day. Multiply · microseconds a day by the speed of light and you get ·, so the number is the right size. It is also, as a claim about your position, close to wrong, and the reason is worth more than the number.

A receiver does not know what time it is. It solves for four unknowns — three of position and one of its own clock — from four satellites. An error that every satellite shares goes entirely into the fourth unknown and moves your position not at all. If every clock overhead drifted at exactly 38.6 µs a day, your receiver would absorb the whole of it and never notice. The 11 km is a real drift of GPS time, and it would wreck the timing service — the banks and the power grid and the telephone network that use GPS as a clock — long before it moved a car on a map.

What actually reaches your position is the part that differs between satellites, and there are two of those. The constellation's semi-major axes are not identical; a spread of 20 km in a is a rate difference of ·, which is · of range a day and does not cancel. And the eccentricity term above is per-satellite by construction: at e = 0.02 it reaches ·, which is · of range, instantaneously, on one satellite and not the others.

Which is exactly the shape of the engineering: the common part is dealt with in the factory, once, because it is common; the differing part is dealt with in every receiver, continuously, because it is not. The two mechanisms are not redundant and the design tells you which part of the folklore is true.

The ground under the clock

One loose end. The expression at the top uses W₀ and never asks where the ground clock is. A clock at the equator is 21 km further from the centre than one at the pole, which should make it faster, and it is being carried east at 465 m/s, which should make it slower. The claim built into W₀ is that those cancel exactly — that is what a geoid is: the surface on which they do. Sea level is not defined by the sea, it is defined by clocks agreeing.

That is checkable, and it is the one thing on this page that does not come out clean. Build the potential from the Earth's zonal harmonics and compare the equator with the pole:

J₂ alone gets it to 1.5 parts per million — the flattening really is doing almost the whole job of making a spinning planet's clocks agree. Adding J₄ more than halves what is left. Adding J₆ makes it worse, and that is not a bug in the arithmetic: I am mixing the real Earth's harmonic coefficients with a surface defined by an idealised ellipsoid, and past the second term those two objects are no longer the same shape. The residual is a statement about my truncation and not about the physics, and the honest thing is to print all three rows rather than the one that flatters.

The leftover is small in a way that has stopped being academic. A part per million of W₀/c² is · a day, which no clock on a moving platform will ever see — but a laboratory optical clock now resolves a centimetre of height, and at that resolution the geoid is no longer a surface you assume. It is a thing you measure with clocks, which is the same equation read from the other end.

The turn of the Earth, while the signal is in flight

One more term, of a different kind, and it belongs here because it is the one people leave out after remembering the other two. A signal takes time to arrive, and the ground moves while it does. In a frame turning with the Earth that shows up as a path-dependent offset proportional to the area swept by the signal's path — not its length. Send a pulse all the way round the equator eastward and back, and it disagrees with a pulse sent westward by

Δt = 2ΩA / c² = ·

which is · of range: a hundred times the accuracy your phone claims. It has been measured directly, by flying caesium clocks around the world in both directions and by fibre loops on a bench, and every GPS receiver applies it. Same constant c, same rotating Earth, third distinct way of noticing.

What could have contradicted this

Every left-hand number below is computed on this page from constants that know nothing about clocks in orbit. Every right-hand number was published by somebody else. The tolerances are stated rather than chosen after the fact.