The clock that measures a height

Two identical clocks, one on the floor and one on the bench, do not agree. The higher one runs faster, by $gh/c^2$, which near the ground is 1.09 parts in $10^{16}$ for every metre. The clock that runs slow lower down derives it: a photon climbing loses energy, energy is frequency, and a clock is a frequency.

For sixty years that number was a thing to be detected. It is now a thing to be corrected for, and then a thing to be used.

The instrument catching up with the effect #

The measurement history runs in an unusual direction. Most effects are first seen crudely over a large baseline and then measured precisely over a larger one. This one has been measured more precisely over shorter and shorter distances.

Pound and Rebka in 1960 needed the twenty-two-metre shaft of a university tower and the newly discovered Mössbauer effect — recoil-free gamma emission, whose linewidth is narrow enough to resolve a shift of two parts in $10^{15}$ — and confirmed the prediction to about ten per cent. Gravity Probe A in 1976 flew a hydrogen maser to ten thousand kilometres and got four digits.

In 2010 a pair of aluminium ion clocks measured the shift across thirty-three centimetres of height. That is a desk. The same paper measured time dilation for a clock moving at a few metres per second, which is walking pace, and both results are what a first-year course describes as unobservably small.

How an optical clock gets there #

The jump in accuracy that made all this possible is worth a paragraph, because it comes from one substitution.

A clock is an oscillator counted against a reference transition, and its accuracy is roughly the linewidth of that transition divided by its frequency, improved by however long the signal can be averaged. Caesium’s microwave transition is at 9.19 GHz. An optical transition in strontium or ytterbium or an aluminium ion is at a few hundred terahertz — five orders of magnitude higher — so the same fractional linewidth is five orders of magnitude finer in relative terms, and the same averaging buys far more.

Two further pieces were needed. The atoms have to be held still, or their motion shifts the line, which is what an optical lattice does: thousands of atoms trapped in a standing wave of light chosen at a wavelength where the trap shifts the two levels of the clock transition equally and therefore does not disturb the frequency being measured. And the optical frequency has to be counted, which was impossible until the frequency comb — a laser producing a ruler of evenly spaced optical frequencies whose spacing is a microwave that a counter can handle.

None of those three is about gravity, and together they are the reason a relativistic effect that needed a tower now fits on a bench. An instrument built for one purpose becoming the natural instrument for another is a common enough sequence to expect.

What the clock is actually measuring #

The useful part is not the confirmation. It is that a clock’s rate depends on the gravitational potential — and the potential is precisely the quantity that surveying is trying to determine and cannot measure directly.

Height above sea level is not a distance. It is defined as a difference of gravitational potential divided by gravity — because what “downhill” means is which way water flows, and water flows down a potential gradient rather than along a geometric line. So the surface that a levelled instrument is trying to refer everything to is a surface of constant potential, and the quantity being surveyed is a potential difference.

A theodolite and a staff cannot measure that directly. They measure it in pieces, one instrument set-up at a time, and add them up along a line — an integration of the gradient over a continent to recover the quantity being differentiated. A clock reads the potential at one point, with no path at all.

That is a change of kind rather than of precision. The same distinction runs through the parallelogram that will not close: a quantity that must be integrated along a path is exposed to everything that happens on the path, and one that is read locally is not.

Where the clock overtakes the theodolite #

The crossover is around a hundred kilometres, and the practical situation is worse for levelling than the figure suggests, because systematic errors — refraction, staff calibration, the slow tilting of the ground itself — do not obey a square root. National height systems disagree with each other by tens of centimetres. Each was levelled outward from its own tide gauge, each tide gauge sits at a slightly different potential because the sea surface is not level, and the disagreements have no way of being resolved from inside either system.

A clock network resolves them by measuring the potential at each reference point against a common standard. Two European laboratories linked by optical fibre have already compared their potentials this way, and the result agreed with the geodetic value to a few centimetres — which is a demonstration rather than a service, but the service is what the technique is for.

Why the potential is the honest quantity #

There is a conceptual tidiness here that is easy to miss and worth stating, because it explains why the clock’s quantity is the right one rather than merely a convenient one.

Gravity has no absolute meaning at a point. A freely falling observer measures none at all, and the term free fall cannot remove shows that what survives is the tidal difference rather than the field itself. So an instrument that reads $g$ is reading something frame-dependent, and its answer at a point says as much about the instrument’s support as about the Earth.

The potential difference between two points is not like that. It is what a clock comparison returns, it is what water responds to, and it is what a height is defined as. A gravimeter measuring $g$ has to be integrated to become useful; a clock measuring a potential difference is already the useful quantity.

The rule of thumb is that the quantity an instrument reports directly is usually the one the theory considers fundamental, and where it is not, the instrument is probably measuring something about its own mounting. That is worth applying in the other direction too: the fact that clock comparisons are so awkward to do over long distances is a statement about signal transfer and not about the quantity.

What such a clock cannot help seeing #

An instrument sensitive to a centimetre of height is sensitive to everything that moves the potential by that much, and the list is longer than it looks.

The solid Earth tide is the big one. The ground rises and falls by tens of centimetres twice a day as the Moon and the Sun deform the planet, and the potential moves with it, so a clock keeping time has to be corrected for the Moon — which is an odd sentence and an entirely routine correction.

The rest of the list turns the nuisance into an instrument. Groundwater, snowpack, ocean loading and the slow rebound of land that was under ice all move mass, mass moves the potential, and a clock reports the change at a point. A gravimeter also sees these, but it reads the gradient, which is dominated by whatever is nearest; a clock reads the potential, which weights distant mass more heavily and is therefore a different and complementary measurement.

The tower experiment that reads as surveying #

The clearest demonstration to date is a pair of strontium lattice clocks placed 450 metres apart in height, one at the base of a broadcasting tower in Tokyo and one on an observation deck near its top, linked by fibre.

The two disagreed in rate by about five parts in $10^{14}$, as they must. Turning that round, the clocks report the height difference as about 452.6 metres, against 453.0 metres from conventional levelling. The two agree, and the point is not the agreement but the direction of the sentence: the clocks were used to measure the tower.

The same experiment tested the redshift itself to about one part in $10^5$, which is a competitive result, and did so outside a laboratory with a clock that had been transported and reassembled. Both readings come from one measurement, and which one is the result depends on what is already known: with the height known, it tests the physics, and with the physics assumed, it measures the height. That is the ordinary condition of a good instrument.

Why this is not a strong test of general relativity #

It is worth being clear about what the measurement establishes, because the answer is less than it is usually credited with.

The shift follows from the equivalence principle almost alone. A photon climbing in a uniform field can be analysed in a freely falling frame, where nothing happens to it, and transformed back — which is the floor that cannot be told from gravity used as a calculating device. Any theory in which gravity acts on energy and in which local physics is special-relativistic gives the same first-order answer, and there are several such theories.

So a redshift measurement tests the equivalence principle and does not choose between metric theories. The measurements that do — light deflection, the perihelion advance, the Shapiro delay — depend on the second-order structure of the metric, which the redshift at this order does not touch.

That does not make the measurement uninteresting; it makes it a different kind of thing. A quantity that follows from a principle rather than from a detailed theory is a good quantity to build an instrument on, because the instrument’s calibration does not depend on which detailed theory turns out to be right.

What it does to the definition of a second #

There is a consequence for timekeeping that arrives whether or not anyone wants a geodetic instrument.

International Atomic Time is a weighted average of clocks in laboratories around the world, and every one of them runs at a rate set by its own potential. To combine them at all, each contribution is corrected to what it would be on the geoid — the surface of constant potential that defines mean sea level — and that correction needs the laboratory’s potential to be known.

At the caesium level of a part in $10^{16}$, knowing it to a metre is enough, and a metre is easy. At a part in $10^{18}$, a centimetre is needed, and a centimetre is not: the geoid itself is known to a few centimetres over much of the world, so the definition of the second would become limited by knowledge of the shape of the Earth.

There are two ways out and both are being pursued. One is to improve the geoid, which is what satellite gravity missions do — the same measurement made from above, where what is read is the potential’s effect on an orbit rather than on a clock, as in the two clocks that flew in opposite directions. The other is to stop referring clocks to the geoid at all and let a network of them define its own reference surface — which is chronometric levelling used the other way round, and would make the timekeeping and the surveying one problem instead of two.

The transfer, which is the hard part #

Comparing two clocks that are not in the same room is a separate problem from building either, and for most of this history it was the limiting one.

Satellite links — the method used to combine the world’s timing laboratories — reach a few parts in $10^{16}$ per day, which is a metre of height and is not enough to use an optical clock at all. Optical fibre does far better, because the whole link can be made to correct itself: send the light to the far end, reflect part of it back, compare the returned phase with the outgoing one, and drive a correction that cancels whatever the fibre did. What the fibre did is common to both directions, so it cancels; what it did to the light in one direction only does not, and that residue sets the limit.

Links of a thousand kilometres and more have been run this way at parts in $10^{19}$, which is better than the clocks at either end. That is the right place for a link to be, and it means the accuracy of a chronometric height difference is set by the clocks rather than by the distance — which is exactly the claim the crossover figure makes and the reason it is a flat line.

The awkward case is a link across an ocean, where there is fibre but not the kind that can be interrogated this way, and where the current answer is to fly a transportable clock or to wait for a satellite link good enough. Neither is satisfactory, and it is the reason a global chronometric height system does not exist yet while a continental one nearly does.

Where the model stops #

The formula used here is the first-order one. The shift between two points is really the difference of a metric potential, and $gh/c^2$ is its leading term with $g$ taken as constant over the height. That is excellent over a building and inadequate over a satellite orbit, where the potential’s variation with radius has to be integrated and the clock that is wrong in two directions is where the two competing terms are separated.

Motion has been left out. A clock that is moving also runs slow, and for anything on a rotating Earth the two effects appear together: what a comparison actually measures is the difference in the combined potential including the centrifugal term. That combination is what defines sea level, so the geodesy works out, but the two effects cannot be separated by a clock alone.

The comparison link is idealised. Transferring a frequency along a thousand kilometres of fibre without corrupting it at the eighteenth digit is the hard part of the whole enterprise, and it needs the fibre’s own length fluctuations — from temperature, from traffic vibration — actively cancelled by sending a reference back the other way.

And the clocks are assumed to agree when co-located. Two clocks of different construction have systematic differences of their own, and separating a real potential difference from a difference in the clocks requires bringing them together, or comparing three, or running the experiment with the two exchanged.

What the pictures cannot show #

The scale figure draws a clock’s accuracy as a single number, and an accuracy is a statement about a systematic budget with a dozen entries in it — the black-body shift from the room’s thermal radiation, the collisions between the atoms themselves, the lattice light that holds them. Each is corrected for, each correction has an uncertainty, and the height a clock can resolve is set by whichever of those is worst rather than by anything about gravity.

The tide figure draws the amplitude of each signal and not its period, and the periods are what makes them separable: the Earth tide is twice a day, the atmosphere is weather-band, the groundwater is annual, the rebound is a trend. A clock record long enough to resolve those periods can take them apart; a single measurement cannot.

Where the ladder goes next #

The gravitational-redshift ladder began with the clock that runs slow lower down, which derives the shift from a photon climbing, and continued to the parallelogram that will not close, where the failure of a closed circuit of comparisons to return to itself is the sign that no global time exists. This rung asks what the effect is good for once it is large compared with the instrument, and the answer is that it measures the shape of the Earth.

The rung after it is the clock in orbit rather than on the ground, where the shift and the motion contribute with opposite signs and comparable sizes, and where a clock is used to look for changes in the potential from above. The habit worth carrying is the one this rung is built on: an effect stops being a test and becomes an instrument at the moment the measurement of it is better than the knowledge of what it depends on — and after that, the interesting quantity is the thing it depends on.