The clock that is wrong in two directions

A navigation satellite works by broadcasting the time. A receiver on the ground compares the times it hears from four of them and solves for its own position and its own clock error — four unknowns, four measurements, and no clock of its own worth trusting. The whole system is a comparison of clocks, so it is only as good as the assumption that the clocks agree — and they do not agree, for two separate reasons that push in opposite directions.

Two terms with opposite signs #

A clock’s rate relative to one far away, to first order in the small quantities, is

$$\frac{d\tau}{dt} \approx 1 + \frac{\Phi}{c^2} - \frac{v^2}{2c^2},$$

with $\Phi$ the gravitational potential (negative, and going to zero far away) and $v$ the speed in the chosen frame. The two corrections have opposite signs and quite different origins.

The second is special-relativistic time dilation: a moving clock runs slow, by $\gamma^{-1} \approx 1 - v^2/2c^2$, and this is the term everybody meets first.

The first is gravitational: a clock deeper in a potential well runs slow, by $\Phi/c^2$, which is why a clock higher up runs fast relative to one on the ground.

The velocity term at its simplest is a light pulse crossing a box. Set the box moving and the pulse travels a longer diagonal at the same speed, so the tick takes longer — the whole of special relativistic time dilation in one triangle. For a satellite this term makes the orbiting clock run slow, and it is the one of the two that everybody expects.

Putting the orbit in #

For a circular orbit at radius $r$ round a mass $M$, the speed is fixed by the orbit: $v^2 = GM/r$. Substituting, and measuring against a clock on the ground at radius $R_\oplus$ rather than against one at infinity, gives

$$\frac{\Delta f}{f} = \frac{GM}{c^2}\left(\frac{1}{R_\oplus} - \frac{1}{r}\right) - \frac{GM}{2rc^2} = \frac{GM}{c^2}\left(\frac{1}{R_\oplus} - \frac{3}{2r}\right).$$

The factor $3/2$ is where the two terms have merged. It carries the whole story: the potential contributes $1/r$ and the velocity contributes half of $1/r$ with the opposite sign, so the combination is $3/2$ of $1/r$ against the ground’s $1/R_\oplus$.

That the velocity term is exactly half the potential term, for a circular orbit, is not a coincidence and is worth naming. It is the virial theorem: for an inverse-square force the average kinetic energy is minus half the average potential energy, exactly, and for a circular orbit the averages are the instantaneous values. So the ratio of the two relativistic corrections for any circular orbit round any mass is fixed at two to one before either is evaluated — and it is that fixed ratio, rather than the sizes of the two terms, which puts the crossing at 1.5 planetary radii and nowhere else.

Setting it to zero:

$$r = \frac{3}{2}R_\oplus \quad\Longrightarrow\quad \text{altitude } = \frac{R_\oplus}{2} = 3{,}186\ \text{km}.$$

Everything has cancelled. Not $G$, not $M$, not $c$ — the radius at which a satellite’s clock keeps ground time is one and a half Earth radii, and it would be one and a half planetary radii for any planet whatever.

What 38 microseconds is worth #

A navigation receiver turns a time difference into a distance by multiplying by $c$. The arithmetic is brutal:

$$38.6\ \mu\text{s} \times 3\times10^8\ \text{m/s} = 11.6\ \text{km}.$$

Per day, and growing linearly. A system specified to a few metres would be out by the width of a city within twenty-four hours, and out by a kilometre within two hours.

The correction is applied in the simplest possible way: the satellite oscillators are built to run at 10.22999999543 MHz instead of 10.23 MHz, so that once they are in orbit and running fast they emit at the nominal frequency as seen from the ground. The offset is a factor of $4.4647 \times 10^{-10}$, and it is machined into the hardware before launch.

There is a further correction the offset cannot absorb, because it is not constant: real orbits are slightly elliptical, so both $r$ and $v$ vary round the orbit and the rate wobbles. For a typical eccentricity of 0.02 the wobble is about 46 ns in amplitude, which is 14 m of range — comfortably above the system’s specification — so receivers apply a periodic correction computed from the satellite’s own broadcast orbital elements.

It is worth noticing what that means about the division into two terms. On an elliptical orbit the satellite is fastest when it is lowest, so the two corrections vary in phase with one another rather than independently: the velocity term and the potential term reach their extremes at the same points, and the combined wobble is the difference of two things that are both changing. Adding the two separately computed corrections and then differentiating is the same as differentiating the exact expression, at this order — and it stops being the same two orders further in.

Why the receiver needs no good clock #

There is an asymmetry in the system that makes the whole relativistic correction necessary at one end and irrelevant at the other, and it is worth setting out because it is not obvious.

A receiver measures, for each satellite, the difference between the time stamped on the signal and the time on its own clock when the signal arrived. That difference multiplied by $c$ is not a range: it is a range plus whatever the receiver’s clock is wrong by, and the same offset contaminates every satellite equally because there is only one receiver clock. Such a quantity is called a pseudorange, and it has four unknowns behind it — three coordinates and the clock offset.

Four satellites give four pseudoranges, and four equations in four unknowns are solvable. So the receiver’s clock error is not corrected; it is solved for, afresh at every measurement, along with the position. Which is why a navigation receiver can be built round an ordinary quartz oscillator costing a few pence and drifting by microseconds an hour, while the satellites carry caesium and rubidium standards costing as much as a car.

That asymmetry explains where relativity has to be dealt with. The satellites’ clocks must genuinely agree with one another and with the ground, because their readings are the input to the solve; a satellite whose clock is wrong by a microsecond injects three hundred metres of error into every position computed from it, and there is nothing in the four equations that could absorb it. The receiver’s clock may be wrong by anything at all, because its error is one of the answers.

And it explains why four satellites rather than three. A system in which every clock could be trusted would need three, and the fourth is bought entirely to pay for not trusting the receiver’s. It is a good trade: one extra satellite in view, against an atomic clock in every device.

Which clock is right #

The phrase “the satellite’s clock runs fast” needs care, because it is not a statement about the clock. Every clock measures its own proper time correctly; nothing is wrong with any of them, and proper time is the invariant that every frame agrees about.

What the system needs is a common time, and it uses one: coordinate time in a non-rotating frame centred on the Earth. Each satellite’s proper time is converted into that coordinate time by the factor above. The comparison is between a clock’s own reading and a bookkeeping convention, and the convention is chosen because it makes the arithmetic of signal propagation simple.

A common time has to be chosen rather than found, which is worth saying plainly because it sounds like an evasion and is not. Two events simultaneous in one frame are not in another, so “the time now at the satellite” has no frame-independent meaning at all. What the system does is pick a frame — an Earth-centred, non-rotating one — and define coordinate time in it. The clocks are then adjusted to agree with that definition rather than with each other.

The rotation of the Earth adds a third effect for the same reason. A receiver on the equator moves at 465 m/s in the chosen frame, so its clock runs slow too, and a signal travelling to a receiver arrives at a place that has moved during the flight. The correction for the second of those — the Sagnac term — is up to about 130 ns, or 40 m, and depends on the direction the signal came from. It is a consequence of choosing a rotating frame rather than of anything relativistic in the usual sense, and it would be present for sound in a rotating room.

Everything above is a small-quantity expansion of one factor, and the expansion is very good indeed. At satellite speeds $\beta$ is $1.3\times10^{-5}$ and $\gamma - 1$ is $8\times10^{-11}$, so the exact curve is indistinguishable from a horizontal line across the entire range that matters. The corrections are large only because they accumulate: eight parts in $10^{11}$ per second is microseconds per day, and microseconds are metres.

The dispute, and how it was settled #

The correction was not accepted in advance. When the first navigation satellite to carry a caesium clock went up in 1977, that clock carried a frequency synthesiser able to apply the offset, and the offset was left switched off — deliberately, because a section of the programme held that the effect would not appear.

The disagreement was not about the algebra. It was about whether a clock’s rate could depend on where it was, which some engineers regarded as a claim about the clock rather than about time, and which the atomic-clock community had never had an instrument good enough to settle from orbit. So the satellite was flown as an experiment: the clock ran unadjusted for twenty days, its rate against ground clocks was measured, and it came out at 442.5 parts in $10^{12}$ against a prediction of 446.5. The synthesiser was switched on and has been on in every satellite since.

The episode is worth recording because it is a rare case of a relativistic prediction being tested by an engineering programme that had a reason not to believe it, with the instrument in place and the switch left off on purpose.

Where relativity sits among the other errors #

Quoting 11.6 kilometres a day makes the relativistic correction sound like the dominant term in the system, and in one sense it is and in another it is not. The comparison is worth making, because it shows what kind of error each is.

Every position from a navigation system carries several error sources at once. The signal’s passage through the ionosphere delays it by a metre or two on a quiet day and by tens of metres through a disturbed one at low elevation; the troposphere adds a couple of metres; the broadcast description of where the satellite actually is carries a metre or so; and a receiver near buildings picks up reflections that can add tens of metres. Set against those, the residual relativistic effects after the built-in correction — the eccentricity term at fourteen metres, the rotation term at forty — are of the same order as everything else.

The difference is in the shape rather than the size. Every one of those other errors is bounded: it varies, it can be modelled or measured, and on average it goes nowhere. The uncorrected relativistic rate offset is a drift, and a drift integrates. It is at eleven and a half kilometres after one day, eighty after a week, and four thousand after a year, and nothing about the system brings it back.

That distinction is the useful one to carry, and it is not special to navigation. A bias can be calibrated out and a random error can be averaged down; a rate error can be done neither to, and it is the only kind that grows without limit. Which is why the correction had to be built into the hardware before launch rather than handled in the receiver’s software, and why the question of whether it was real had to be settled before the system could exist at all.

There is a second reason it had to be settled first. The other errors were all discovered by operating the system and looking at the residuals — the usual way an engineering programme finds what it has missed. A drift of this size would not have shown up as a residual; it would have shown up as the system not working, with no indication of which of a hundred possible causes was responsible.

Where the model stops #

The expansion is first order. Both corrections are kept to lowest order in $v^2/c^2$ and $\Phi/c^2$. The next terms are of order $10^{-20}$ and would matter after $10^{10}$ years; they are not the reason anything is left out. What is left out at the relevant level is everything of order $10^{-13}$ and larger, and that list is longer than the two terms this essay is about — it includes the receiver’s own height above sea level, the solid-earth tide, and the difference between the geoid and an ellipsoid.

The Earth is spherical and static. Its oblateness gives the potential a quadrupole term, which changes the gravitational shift by a few parts in $10^{13}$ over the orbit — small, and not zero, and included in the highest-precision timing work. The tidal potentials of the Sun and Moon contribute at a similar level.

The ground clock is at rest in the chosen frame. It is not: the Earth rotates, so a clock on the equator moves at 465 m/s and one at a pole does not. Their rates would differ by 10⁻¹² if that were the whole story — and they do not differ at all, measurably, because the Earth’s surface is an equipotential of the effective potential including the centrifugal term, and the equator’s extra speed is exactly compensated by its extra distance from the centre. Sea level is a surface of constant clock rate, which is a fact about the shape the planet settled into rather than about relativity.

The frame is the Earth’s, not the Sun’s. Working in a solar-system frame instead adds a large common term for the Earth’s orbital motion and its position in the Sun’s potential, which cancels between the satellite and the receiver to the accuracy required. That cancellation is why the Earth-centred frame is usable at all, and it stops being adequate for interplanetary navigation.

In the regime where neither term is small they stop being corrections and become the whole description. A body falling towards a black hole is both deep in a potential and moving fast, and its proper time diverges from coordinate time without limit — so there is no expansion to truncate and no useful sense in which one clock is nearly right. The satellite case works because both terms are tiny; the strong-field case is not the same calculation done more carefully.

The Earth’s surface sits at about $7\times10^{8}$ Schwarzschild radii, far out where the exact factor is flat to eleven decimal places. That is the honest reason the two-term expansion is adequate for navigation: not that the effects are negligible — they are not — but that the curve is straight there, so the first two terms are the whole of it.

What the pictures cannot show #

The rate curves are drawn for circular orbits, which lets the speed be replaced by $\sqrt{GM/r}$ and reduces the whole problem to one variable. A real orbit has two, and the eccentric wobble described above cannot appear on any of these axes.

None of the figures shows accumulation. A rate is a slope; the quantity that ruins a navigation system is the integral of the rate over a day, and the reason it is a problem is that it does not average out. A plot of rate against altitude looks small and a plot of accumulated error against time does not.

And the crossing at 3,186 km is drawn as a point on an axis, which makes it look like an operating regime. Nothing is easier about a satellite there; the corrections at that altitude are individually 21 µs a day and merely happen to cancel, so an eccentric orbit through that radius would still have a wobble of the full size.

The instrument turned round #

Once the correction is trusted, the arithmetic can be run backwards and the clock becomes an altimeter.

The gravitational term is $\Delta f/f = g\,\Delta h/c^2$ for small height differences, which is $1.09 \times 10^{-16}$ per metre. An optical lattice clock reaching a fractional accuracy of $10^{-18}$ therefore resolves a centimetre of height — not by measuring anything mechanical, but by comparing its own tick with another clock’s. Two such clocks in different laboratories measure the difference in gravitational potential between them, which is what a geodesist actually wants and what levelling instruments have always approximated.

That is a genuine reversal of the relationship. For sixty years the gravitational shift was a small correction to be applied to clocks; it is now a signal, and the clock is the instrument. Chronometric levelling has already been demonstrated over hundreds of kilometres, and it measures a quantity — the geopotential difference — that no chain of spirit levels can obtain directly at all.

Where the ladder goes next #

The two rungs below established each effect separately: a moving clock runs slow, and a clock lower in a potential runs slow. This rung is what happens when both apply at once to the same object and the answer has to be a single number.

The rungs above are the ones where the separation stops working. Very near a compact object the two terms are not small and cannot be added; proper time and coordinate time diverge without limit, and the distinction between “moving” and “deep” stops being meaningful because the metric mixes them.

The habit worth carrying out of it: when two corrections of opposite sign have different dependence on a parameter, look for where they cancel. The point is rarely useful in itself, and the arithmetic of finding it usually reveals which combination of the underlying constants the problem actually depends on. Here it revealed that the answer contains no constants at all.