The clocks the sea keeps together

Near the end of the first section of the 1905 paper on special relativity, among the deductions, Einstein observed that a clock at the equator should run slower than an identical clock at one of the poles. The reasoning was a direct application of the clock that has to slow. A clock moving in a circle at speed $v$ with respect to one at the centre falls behind by $v^2/2c^2$, and the Earth’s rotation carries an equatorial clock around at 465 m/s. So the difference is about $1.2 \times 10^{-12}$: about 104 nanoseconds a day, or a millisecond in 26 years. It was far too small to be measured in 1905, and it sat in the paper as a modest prediction for many years.

Modern atomic clocks can resolve that in a fraction of a second. When they were compared around the world, clocks at sea level in different latitudes were found to agree. Nobody has seen the difference. Nor could they, because it is not there. Einstein’s claim was correct about the effect, and he had not thought of the shape of the Earth, or, at that date, of gravity. It would be nearly a decade before he could incorporate that. The surprise in this case is what happens when he did: two effects that come from different parts of physics cancel, to about one part in $10^{4}$ of their size, all the way down to the tiny variations in actual sea level, and the cancellation has a reason.

Two effects from one number #

Take a clock on the surface of a rotating Earth, compared with a hypothetical clock very far away and at rest. Its rate differs by two terms. One is the term of special relativity, the one in the 1905 paper. The other is the one from the clock that runs slow lower down: a clock deep in a gravitational well ticks more slowly, by a fractional $\Phi/c^2$ with $\Phi$ the gravitational potential. So to the order of interest,

$$\frac{d\tau}{dt} = 1 - \frac{GM}{rc^2}\Big(1 + \cdots\Big) - \frac{v^2}{2c^2}.$$

Now take $v = \omega r\cos\phi$, the equatorial rotation at latitude $\phi$. The expression is minus one over $c^2$, multiplying

$$W = \frac{GM}{r}(1 + \cdots) + \frac{1}{2}\omega^2 r^2\cos^2\phi,$$

and that quantity has another name. It is the gravity potential of geodesy, including the centrifugal term. It is the one that any surveyor uses to define what “level” means. A surface of constant $W$ is the surface on which a marble will not roll in a rotating frame, and on which water would come to rest.

The ocean is such a surface, to a good approximation. Water runs downhill, and it keeps running until there is no more downhill to run. Over geological time so does the Earth itself, a fluid on long enough timescales. So the sea surface is an equipotential of $W$, and $W$ is exactly the combination that sets a clock’s rate.

The calculation shows the cancellation with real numbers. It uses the World Geodetic System’s ellipsoid and the leading term $J_2$ in the Earth’s gravity, with no knowledge of clocks. The two curves are nearly mirror images: the speed effect goes to nothing at the poles, and the gravity effect goes from zero at the pole to the largest difference at the equator. They add to a flat line, and what is left over at a part in $10^{14}$ belongs to the higher terms in the Earth’s field that a two-term model omits. A standard value for the combination, the rate of a clock on the geoid, is the constant $L_G = 6.969290134 \times 10^{-10}$, and it comes out correct here to within a tenth of a per cent. International atomic time is defined by that surface. Since 1977 the clocks of the world have been corrected for their height above the geoid, and not otherwise for latitude.

A rigid world would be different #

The clearest way to see what the Earth is doing is to imagine a world that cannot do it.

So on a truly rigid and spherical Earth with no deformation at all, Einstein’s 1905 prediction would have been right. A marble placed anywhere but on the poles on such a world would roll towards the equator, as the combined gravity and centrifugal force would tilt the apparent vertical. All the sea would flow there, in a band several kilometres deep, with bare land near the poles. With the ocean in that band, in which the sea surface is an equipotential, the clocks in that band would of course agree. But those on a tall peak near the pole would not agree with those at the equator, in just the way that one would expect from the height of the peak above the equipotential.

Now let the rock flow too, as it has over the Earth’s life. That is the actual bulge.

The simple formula $v^2/2g$ gives 11 km. The observed bulge is 21.4 km. This difference used to be the subject of a dispute between Newton and the Cassinis, and in fact Newton got close to this number by a beautiful calculation of a self-gravitating fluid in the Principia. The factor of about two arises because a body that bulges has more mass at its equator, and that mass has gravity of its own. In a fluid that attracts itself, displacing mass raises the potential of the equator further, and the shape needs to rise more. A homogeneous Earth would give a factor of 5/2, and the real figure of about 2 reflects the Earth’s denser core. All of which is irrelevant to the clocks. A clock does not care why the equator is 21 km further out, or what it is made of. It cares only about the potential, and that is constant by the very mechanism that made the bulge, since rock and water were moving until it was.

So there is a neat logical inversion here. A clock comparison across the world would have to find no difference, regardless of the Earth’s internal structure and regardless of how fast it spins, as long as the sea has had time to settle. It is not a test of the two terms being equal. It is a test that time follows the same potential that governs which way water runs. Hence it can be said to confirm the equivalence principle in a rather elegant way, on the scale of the planet.

The surface is now a thing to measure #

Once the cancellation is understood as a statement about an equipotential, what is left is what stops a real clock from sitting on that equipotential, and that has become an instrument.

The real geoid departs from a smooth ellipsoid by up to around 100 m, because mass within the Earth is not uniformly distributed. There is a low of 106 m in the Indian Ocean south of India, and sea level above a seamount rises several metres over it, and the shape of the surface is a sensitive measure of what lies below, as the sea that stands lower because water gives would suggest for another reason. Traditionally geodesists measure these by surveying and gravity. A clock could now do it. In 2020 Tokyo Skytree was used as a tower for this: a pair of portable optical clocks at its base and at the 450 m observation deck read the difference at about 5 × 10⁻¹⁴ to a precision equivalent to a few centimetres.

Two clocks on the Earth differ, from now on, by more than their own errors, because of the tides. The solid Earth rises and falls by about 30 cm twice a day under the Moon, and ocean tides load the crust. A modern optical clock on a good day is sensitive to those, through a daily variation of a few parts in $10^{17}$. So at precision of $10^{-18}$ a clock is a geophysical sensor, and comparing clocks in different countries runs up against what the geoid actually is at the site, known to only a few centimetres. It is a curious development that the definition of the second may be limited by how well the shape of the ocean is known. A planned redefinition of the second using optical clocks has to deal with this very point.

The same terms in orbit #

Above the surface there is no sea to enforce the equality, and the terms need not cancel.

A satellite has the orbital speed its height demands, $v^2 = GM/r$. So the speed term and the gravity term are related in a fixed ratio of one to two, and there is a single height at which they combine to match the ground rate. It comes out at about 3,170 km above the surface, and an orbit up there would have a clock in step with sea level. A low orbit, like the International Space Station’s, has a clock running slow, as speed dominates. GPS, at 20,200 km, gains 447 parts in $10^{12}$, or 38.6 microseconds a day. That has been built into the system since the first satellites, with a factory offset to the frequency and a further small correction for the orbit’s eccentricity. The satellites launched into the wrong orbit turned an accident of two wrong orbits into a test of the height term, and the two clocks that flew in opposite directions used the same arithmetic in an airliner, including the speed term that is the same as the Earth’s rotation. Those clocks did see a difference, since aeroplanes fly at speeds of their own and at heights well away from the equipotential.

It is a pleasing small fact that a satellite at about half an Earth radius above the surface would not need any correction at all. It happens that hardly any satellites orbit at that height, because it sits in the inner radiation belt.

What every clock lab already corrects for #

The cancellation has a practical side that shows up in the way national clocks are compared. A clock in Boulder, Colorado, stands about 1,650 m above the geoid, and it runs fast against a clock at sea level by about 1.8 parts in $10^{13}$. It is corrected for that, every day, before its reading goes into international atomic time. A clock in Bogotá needs a bigger correction than one in Paris, for its altitude. Neither needs any correction for its latitude. Being at a different speed around the Earth’s axis has already been absorbed into the surface that the height is measured from. So the arithmetic that took Einstein until 1907 to begin and until 1915 to finish is now one line in a spreadsheet, and the only input is a surveyor’s height.

There is an instructive contrast with the clock that is wrong in two directions, where each observer sees the other’s clock slow. That symmetry needs two inertial frames. Clocks on a rotating Earth are not in such frames, and there is no symmetry between them and a clock on the axis, which is why the comparison gives a definite answer and can be accumulated. The rotating ring in the ring where the two beams disagree makes the same point differently: a clock carried once around a ring, in the direction of the Earth’s spin, comes back behind one carried against it. A network of clocks synchronised around the equator by light signals fails to close, by about 207 nanoseconds per circuit. That is a true property of a rotating frame and is corrected for in every global time transfer.

Nor is the Earth special. The same relation holds on any rotating planet with a liquid surface, and it can be rewritten for a planet with no ocean at all, in that the equipotential surface still exists whether or not any water marks it. Jupiter’s cloud tops are flattened by about 6.5 per cent, far more than the Earth, and the speed term at its equator is around a hundred times larger. Those large numbers go in and the same equality comes out. The Moon, which spins slowly, has nearly no term of this kind and a clock on it has a different problem, a rate offset from the Earth’s by about 58 microseconds a day, on which the agencies planning lunar navigation have recently had to agree a convention. There is no sea on the Moon to enforce agreement between places on it. The rocks there have frozen in a shape that is not an equipotential, and so clocks at different places on the Moon really would drift apart by more than their elevation above any smooth figure suggests.

What this does not establish #

The sum assumes a slowly rotating weak field, in which the two terms are small and simply add. For a neutron star or a black hole, the same principle remains but the separate terms mix, and the idea of an additive “speed” and “height” stops being accurate. So the cancellation is a weak-field result, and the actual relation is about Killing vectors, as a formal treatment of a stationary spacetime would have it. An equipotential of the formula above is then a surface on which all static clocks agree, for all stationary spacetimes that a fluid can come to rest in.

The figures use a two-term field, and a flat result good to about $10^{-14}$ corresponds to around 80 metres of height, which is simply what the leading terms of the field cannot resolve. It is not a measurement of how well actual clocks at the poles and at the equator agree. Very few such comparisons have been made directly, and they are limited by how well the height of each clock above the true geoid is known.

The ocean is also not actually an equipotential. Currents hold it out of balance by up to a metre or so, and it is also out of balance in the short term due to tides, wind and pressure. The dynamic topography of the sea, which oceanographers measure by comparing satellite altimetry against the geoid, is exactly this departure. So “the sea keeps the clocks together” holds down to about a metre, or a part in $10^{16}$. Below that level it would be more accurate to state that a clock on the geoid does so, and the geoid is a mathematical surface that the sea only approximates. The clocks are now better than the sea.

Finally, the argument says nothing about clock mechanisms. It assumes that all ideal clocks follow proper time, regardless of how they work, and this is itself something to test. The clocks that must all slow together was on that subject, and a difference between clock types at different potentials would be a violation of the equivalence principle, currently limited by experiment to a few parts in $10^{7}$ or better of the redshift itself.

Still open: whether a network of clocks can replace a geoid #

An operational definition of height using clocks, called chronometric levelling, has been a theoretical possibility for decades, and it is now on the edge of being practical. Surveying heights by traditional means has accumulated errors over continents, of many decimetres between national height systems, since they are linked only across oceans by indirect means. A fibre-linked network of optical clocks, of which Europe already has parts, could compare potentials directly, without intervening measurements. The clocks would be read as a kind of spirit level spanning continents. Whether such a network could really beat satellite gravity measurements, once real instruments and their systematic errors are included, is an active question. And if the second is redefined in terms of optical clocks by the end of this decade, as is being discussed, its realisation will depend on a surface whose position depends on the state of the ocean and the planet’s slow redistribution of ice and water. Time would then depend directly on geophysics, and a metre of sea-level change would appear in the 16th decimal place of the definition.

Speed and height both enter a clock’s rate through one potential, and a settled sea is a surface of that potential, so the 1.20 parts in 10¹² lost to the equator’s 465 m/s is paid back by its greater height, and clocks agree along the sea to about the limit of the model here, while a rigid ball would have shown 104 ns a day. Einstein’s 1905 prediction was right about clocks and wrong about the Earth. The habit worth carrying is to check whether a surprising agreement is a coincidence or a consequence, because a consequence goes on working when the numbers change, and here the numbers could be anything, and water would still have run downhill.