The clock that runs slow lower down

Every satellite navigation system in service carries atomic clocks whose rate has been deliberately altered before launch, because a clock in orbit does not keep the same time as a clock on the ground. The offset is about thirty-eight microseconds a day. Light travels three hundred metres in a microsecond, so an uncorrected system would be wrong by roughly ten kilometres after twenty-four hours, and by a hundred metres after fifteen minutes.

The remarkable feature of this is not that relativity turns out to matter in an engineering system. It is how little is needed to derive it. The whole of the height effect follows from a photon, a conservation law, and the sealed box of the previous rung.

The argument, in five lines #

Suppose a clock at the foot of a tower emits light of frequency $\nu$ upward, and a receiver at height $h$ measures what arrives.

Give the photon an effective inertia. Its energy is $E = h\nu$, and energy has inertia at the rate $E/c^2$, so climbing against gravity costs it

$$\Delta E = \frac{E}{c^2} g h.$$

Divide through by $E$: the fractional energy loss is $gh/c^2$, with the photon’s own properties cancelled out. Since the photon’s energy is proportional to its frequency and its speed cannot change, the loss shows up as a lower frequency:

$$\frac{\Delta \nu}{\nu} = -\frac{gh}{c^2}.$$

Now the step that turns a statement about light into a statement about time. The receiver counts wave crests arriving. If it counts fewer per second than the emitter emitted per second, and the crests are neither piling up nor being destroyed in between, then the two are disagreeing about what a second is. A frequency is a clock, and the only consistent reading is that the lower clock runs slow.

Two parts in a thousand million million, measured in a lift shaft #

That number looks unmeasurable and was, until the Mössbauer effect made it routine.

An atomic nucleus emitting a gamma ray normally recoils, and the recoil takes a share of the energy that varies from event to event, smearing the line far too broadly for a part in $10^{15}$ to be visible. Mössbauer’s discovery in 1958 was that a nucleus bound in a crystal lattice can emit with the whole crystal taking up the recoil — a recoil energy smaller by the ratio of the two masses, effectively zero — leaving a spectral line of extraordinary sharpness. For the 14.4 keV line of iron-57 the natural width is about three parts in $10^{13}$, which is only a factor of a hundred from what has to be resolved.

Pound and Rebka closed the gap by moving the source. A source drifting slowly toward the receiver adds an ordinary Doppler shift, and the speed needed to cancel a shift of $2.5\times10^{-15}$ is $c$ times that, or about 0.75 micrometres per second — a speed achieved with a loudspeaker cone and averaged over a cycle. The experiment found the predicted shift to within about ten per cent in 1960, and within one per cent in a refined version in 1964.

The design is worth admiring for what it avoids. It never measures a frequency. It finds the speed at which two effects cancel, which is a null measurement, and null measurements do not require an accurate instrument — only a stable one.

There is a second piece of cunning in it. The experiment was run in both directions — source at the top and source at the bottom — and the difference of the two results was taken. Anything that shifts the line for a reason unconnected with height, and there are several, including a temperature difference between the ends of a 22.5-metre shaft, appears with the same sign in both runs and cancels in the difference, while the gravitational shift reverses sign and doubles. The temperature effect is not hypothetical: the second-order Doppler shift from thermal motion of the nuclei is comparable in size with the whole quantity being measured, and it was the largest correction in the 1964 version.

What the argument does not use #

It is worth listing what the derivation above never touches, because the list is what makes the result so hard to escape.

It does not use a field equation. Nothing about how mass produces gravity appears anywhere: only that a photon climbing a height $h$ in a field of strength $g$ pays $gh$ per unit of effective mass, which is a statement about the field a laboratory can measure locally.

It does not use a specific theory of gravitation. Any theory in which energy has weight and energy is conserved predicts the same first-order shift, which is why the Pound–Rebka result is standardly described as a test of the equivalence principle rather than of general relativity. A theory could reproduce this result exactly and still be wrong about everything else.

It does not use the light’s frequency, its polarisation, or what emitted it. The shift is a fraction, and fractions do not carry units. Gamma rays at 14.4 keV and optical light seven orders of magnitude lower in energy shift by the same fraction, which is a strong statement and a testable one.

And it does not use a preferred direction of travel. Send the photon down instead of up and the sign reverses: the receiver at the bottom sees a blueshift of the same size. The word “redshift” attached to this effect names the common case — light climbing out — and not a property of the effect, which is one of the standing sources of confusion about it.

The other half, which has the opposite sign #

A satellite is not only high. It is also moving, at about 3.9 km/s for a navigation satellite, and motion slows a clock by the Lorentz factor.

Motion alone costs time for a reason with no gravity in it. A light clock carried sideways has its pulse travel a longer, slanted path between the same two mirrors, and since the speed of light is the same for everybody the tick must take longer. That is the other half of the satellite correction, and it has the opposite sign to the gravitational one — height makes a clock run fast, speed makes it run slow, and a navigation satellite has both.

For an orbit at radius $r$ the speed is fixed by the orbit itself, $v^2 = GM/r$, so the two effects can be written against the same variable. The height term makes the satellite clock gain by $(GM/c^2)(1/R - 1/r)$ against a ground clock; the speed term makes it lose by $v^2/2c^2 = GM/2rc^2$. Adding them:

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

At the speeds in question the factor is barely a factor at all. A navigation satellite’s 3.9 km/s is $1.3\times10^{-5}$ of the speed of light, and the resulting $\gamma$ differs from one in the eleventh decimal place. Both corrections are of that size, they have opposite signs, and neither cancels the other — the gravitational term wins, and the net is 38 microseconds a day.

The bracket vanishes at $r = 3R/2$, an altitude of half an Earth radius: 3,186 km. Below it the speed term wins and an orbiting clock loses; above it the height term wins and the clock gains. At the navigation constellation’s 20,200 km the total is +38.5 microseconds a day, of which +45 is height and −7 is speed. At the space station’s 400 km it is about −25 microseconds a day, so those clocks run slow.

The crossing altitude is a genuinely surprising object. Nothing about the design of a satellite is involved in it; it is a property of the gravitational field alone, and any circular orbit at that height keeps the ground’s time regardless of what is flying there.

The engineering response to all this is blunter than the physics deserves. A navigation satellite’s caesium standard is manufactured to run at 10.229999995453 MHz rather than 10.23 MHz, so that once the relativistic gain is applied it emits at the nominal figure as seen from the ground. The correction is built into the hardware before launch and is not applied in software afterwards, which is the clearest statement available of how routine the effect has become: it is a manufacturing tolerance.

What the ground segment does still correct for, continuously, is the part the constant offset cannot cover. Real orbits are slightly elliptical, so both the height and the speed vary around each revolution, and the residual is a periodic term of a few tens of nanoseconds that depends on the eccentricity and on where the satellite is in its orbit. The general shape of the arithmetic is exactly what is drawn above; only the constant has been absorbed into the crystal.

The exact factor, and what $gh/c^2$ is an approximation to #

The formula derived above is a first term. The exact result for a static clock outside a spherical mass is

$$\frac{\mathrm{d}\tau}{\mathrm{d}t} = \sqrt{1 - \frac{r_s}{r}}, \qquad r_s = \frac{2GM}{c^2},$$

where $\tau$ is what the local clock reads and $t$ is what a clock infinitely far away reads.

The exact factor is worth seeing once, because it shows how far the everyday case sits from anything dramatic. At ten Schwarzschild radii a clock runs at 0.949 of the distant rate; at one and a half, 0.577. The Earth’s surface is at $7\times10^{8}$ of them, where the same curve is flat to eleven decimal places — so $gh/c^2$ is not an approximation that might fail, it is the first term of an expansion whose second term is unmeasurable.

Expanding for $r_s \ll r$ gives $1 - GM/rc^2$, and the difference between two heights in a weak field is $gh/c^2$. So the tower formula is the first term of a series whose full form only matters where $r$ is within a few multiples of $r_s$. That is the regime of a horizon, where the factor reaches zero and the argument on this page turns into something else entirely.

The measurement that was celebrated and wrong #

The shift’s most famous early confirmation was not one, and the way it failed is worth recording because both halves of it went wrong in the same direction.

Eddington argued in 1924 that Sirius’s faint companion had to be extraordinarily dense — a star of about a solar mass in a body the size of a planet — and pointed out that its gravitational redshift would be a test of the claim. From the radius he had, he expected a shift equivalent to about twenty kilometres a second.

Adams measured it at Mount Wilson the following year and reported nineteen. The agreement was celebrated, reported as a confirmation of general relativity and of the white-dwarf interpretation together, and stood for nearly fifty years.

Both numbers were wrong. Eddington’s radius was too large by a factor of about two and a half, so the correct prediction is around eighty kilometres a second rather than twenty. And Adams’s spectrum was contaminated: Sirius B sits a few arcseconds from a star ten thousand times brighter, and scattered light from the primary dominated the plate. The two errors were of comparable size and pulled the same way, and the agreement was a coincidence between a bad prediction and a bad measurement.

It was sorted out in 1971, when Greenstein and colleagues re-measured it with the contamination controlled and obtained a much larger shift, and settled definitively in 2005 with a space-based spectrum: eighty and two-thirds kilometres a second, agreeing with the modern mass and radius.

The moral is not about carelessness — Adams was a first-rate observer working at the limit of his instrument. It is that agreement between a measurement and a prediction is evidence only when the two were arrived at independently, and here neither was secure. A test that passes for the wrong reason is harder to detect than one that fails, because nobody goes looking.

The Sun is a more accessible target and was scarcely easier. Its predicted shift is two parts in a million, equivalent to 636 metres a second, and it is buried under an effect of comparable size that has nothing to do with gravity: the photosphere is convecting, the rising granules are hotter and brighter than the sinking lanes between them, and the resulting weighted average of Doppler shifts blueshifts every line by a few hundred metres a second — by an amount that depends on which line is used and on where on the disc it is measured. Separating a gravitational shift from a convective one required modelling the convection well enough to subtract it, and a clean measurement on integrated sunlight was only published in 2020. It gives 638 metres a second.

The second, and where it is defined #

The effect stopped being a correction some time ago and became part of the definition of the unit.

The SI second is a count of caesium oscillations, and the definition specifies the atom at rest, at zero field and at absolute zero — but says nothing about where. Since a clock’s rate depends on its gravitational potential, an unqualified definition would leave every laboratory realising a different second, and by an amount that dwarfs their disagreements about anything else.

The resolution is a convention. International Atomic Time is defined as a coordinate time on the rotating geoid — the surface of constant effective potential that mean sea level approximates — so every contributing clock’s rate is corrected to what it would read there before its ticks are counted.

The size of that correction settles how seriously it has to be taken. A laboratory a thousand metres above sea level runs fast by about eleven parts in $10^{14}$, which is nine microseconds a year. The best optical clocks are stable to about a part in $10^{18}$, which is ten thousand times finer. So the altitude of a timing laboratory is not a detail to be estimated; it has to be known to a few centimetres for its clock to contribute at its own precision.

That requirement has produced an unusual inversion. The geoid itself — the reference surface everything is corrected to — is known globally to only a centimetre or two, and that uncertainty is now the limiting term when clocks in different countries are compared. Geodesy has become the accuracy limit on timekeeping.

The proposed way out is the one the previous section describes from the other end: use the clocks to determine the surface. A network of optical clocks connected by fibre measures potential differences directly, which is the quantity a geodesist wants and which no levelling survey delivers over continental distances. The redefinition of the second onto an optical transition, expected within the decade, will make that the working arrangement rather than a demonstration — and the relation on this page, derived from a photon climbing a tower, will be the thing that ties the world’s clocks and the world’s surveys into one system.

What it costs, and where the model stops #

It needs a static field. Every statement here compares two clocks that stay put and exchange signals for ever, and the comparison is only meaningful because nothing about the geometry is changing. In a spacetime that is not static there is no such thing as the rate of one clock against another far away — there is only what a particular signal did on a particular path. The tidy potential $\Phi$ in $\Delta f/f = \Delta\Phi/c^2$ exists because the field is unchanging, and it does not survive into the general case.

There is no global time to be slow with respect to. The phrase “runs slow” is a comparison between a local reading and a bookkeeping coordinate. It is a real, measured, one-way disagreement — bring the clocks back together and they genuinely differ — but the coordinate itself is a convention, and stating a rate without saying against what is meaningless.

The photon derivation is a heuristic and gets away with it. Assigning a mass $E/c^2$ to a photon and dropping it into Newtonian gravity is not a legitimate step in either theory. It happens to give the right first-order answer because the redshift depends on the time part of the metric alone. The same trick applied to the deflection of light gets half the right answer, and the reason is exactly that the deflection depends on the space part too.

The effect is not confined to gravity’s weak field. In the strong-field regime the redshift saturates the entire structure: light emitted close enough to a horizon arrives with arbitrarily little energy, and the practical consequence is that the region is dark rather than merely dim.

Clocks that were flown around, and what they proved #

Between the lift shaft and the satellites there is an experiment worth its space in the history, because it made the abstraction concrete in the crudest possible way: in 1971 Hafele and Keating put four caesium clocks on scheduled airline flights and flew them round the world twice, once eastward and once westward, then compared them with the clocks that had stayed at the United States Naval Observatory.

The prediction is a sum of the two terms above with a third complication: the ground station is itself moving, because the Earth rotates, so the relevant speed for each clock is its speed in a non-rotating frame rather than its speed relative to the ground. An eastward flight adds to the rotation speed and a westward one subtracts, which makes the speed term differ in size between the two directions while the altitude term is much the same. The predicted differences were therefore asymmetric: about −40 nanoseconds eastward and +275 westward, and the measurements came out at −59 and +273 with uncertainties of a few tens of nanoseconds.

The uncertainties were large, the clocks were commercial, the aircraft were airliners on ordinary schedules, and the experiment was criticised at the time and since for exactly those reasons. What makes it worth recording is not its precision but its arrangement: the twin comparison drawn on a spacetime diagram as a thought experiment was performed with luggage, and the clocks came back reading different times, in nanoseconds anybody could read off a printout.

Where it stopped being a correction and became an instrument #

For fifty years this was a test of relativity. It is now a measurement tool, because optical lattice clocks reached a fractional stability around $10^{-18}$, and $gh/c^2$ at one part in $10^{18}$ corresponds to a height difference of about one centimetre.

That inverts the logic. A clock is no longer something to be corrected for its height; it is a device for measuring height, or rather for measuring gravitational potential, which is what a geodesist actually wants. Two clocks connected by an optical fibre now compare potentials between distant laboratories directly, without a survey, and the technique — chronometric levelling — has been demonstrated between mountain and valley laboratories.

The origin of all of it is one box that cannot tell acceleration from gravity, with light crossing it. Over a thirty-metre baseline the sag is four times what it is over fifteen, and the effect goes as the square of the crossing — which is what made Pound and Rebka’s tower experiment possible at all, and what makes the effect a correction on Earth and the dominant term near anything dense.

The same reversal happened to the Mössbauer effect and to nearly every precise null experiment in physics: what is built to test a theory becomes, once the theory is believed, the most sensitive instrument available for measuring something else.

The ladder from here #

Later rungs on this anchor: the redshift of light from a compact object, where the factor is not a correction; the Shapiro delay, which is the same metric component read as a time of flight rather than as a frequency; the twin paradox with gravity in it, in which the higher twin ages faster and the effect has been measured with aircraft-borne clocks; the redshift as a test of local position invariance, which is what a null result would actually be constraining; and clocks as gravimeters, where the whole relation is used backwards.

The neighbouring ladder is special relativity’s own time dilation, which supplies the term with the opposite sign in every orbiting clock — and the fall through a horizon, where the factor above goes to zero and the disagreement between two clocks stops being a correction and becomes the entire phenomenon.