1 centimetre: lift a clock that much and its time runs measurably faster

Part ITime runs faster upstairs

Chapter 1One centimetre

General relativity says that clocks lower in a gravitational field run slower than clocks higher up. Near the Earth's surface, two clocks a height Δh apart differ in rate by

Δf / f = gΔh / c² ≈ 1.09 × 10⁻¹⁶ per metre = 1.09 × 10⁻¹⁸ per centimetreStrictly the rate depends on the gravity potential W (gravitational plus centrifugal): Δf/f = −ΔW/c², with W larger deeper down.

This is not a quirk of clocks. It is a property of time itself, a consequence of the equivalence principle: hearts, chemical reactions and every other process run slower lower down. Clocks are simply the only things precise enough to notice. Some examples (this page's arithmetic):

Your head ages faster than your feet: 1.7 m apart, they differ by 1.86 × 10⁻¹⁶, about 6 nanoseconds a year, or half a microsecond over 80 years, if you spent them all standing up.
Living on the 10th floor (30 m up) instead of the ground gains you about 100 ns a year, and 8 µs in a lifetime.
Living in Denver (1,609 m) instead of at sea level gains you about 15 ns a day, and 0.44 milliseconds in a lifetime.

The effect was first convincingly measured in the laboratory in 1960 by Robert Pound and Glen Rebka, who sent gamma rays up and down a 22.5-metre tower at Harvard and saw their frequency shift by about 2.5 parts in 10¹⁵, as predicted, to about 10%.¹ In 2010, a team at the US National Institute of Standards and Technology (NIST) raised one of two aluminium-ion clocks by just 33 cm and saw it speed up; over 79 years, that adds up to about 90 billionths of a second.² In 2022, a team at JILA in Boulder measured the gravitational shift across a single cloud of strontium atoms about a millimetre tall.³ That was a special experiment: two parts of the same cloud were read out with the same laser, so the laser's own noise cancelled.

In September 2026, a team at the Centre for Quantum Technologies in Singapore published two lutetium-ion clocks whose error budgets are 1.2 and 1.3 × 10⁻¹⁹, the equivalent of about one millimetre of height.⁴ Put the popular way, such a clock would gain or lose less than a second in about 260 billion years (this page's arithmetic). The two ions sat at heights 3.97 mm apart, so the team had to correct their comparison by −4.3 × 10⁻¹⁹, more than three times one clock's uncertainty. The most accurate clocks ever built now need their level checked like a snooker table's.

Figure 1🏢 Try it

Which floor are you on?

Drag the clock up the tower

How much higher is the clock?

Fog: too small for the best clock of

Runs faster by

Gains per year

In an 80-year life

Can a clock tell?

A tower whose floors are powers of ten: each floor is ten times higher than the one below. The upper clock runs faster than the one on the ground by Δf/f = (GM/c²)(1/R − 1/(R + h)), which is gh/c² = 1.09 × 10⁻¹⁶ per metre near the ground (gravity only; the Earth's rotation and the clock's speed are ignored, so the GPS height shows the +45.7 µs/day gravity part alone). Gains are this page's arithmetic. The fog marks heights below the systematic uncertainty of the best clock of the chosen year (Figure 2), and "Can a clock tell?" names the first year, among the records of Figure 2 (from 2010), that a clock's budget reached the shift; a real height measurement needs two clocks and a link, and is usually less precise than either clock.Drag the clock up the tower, or tap a floor. Each floor is ten times higher than the one below, and the higher the clock, the faster it ticks. The fog shows what's too small for the best clocks of each year to notice. Press play to see how far ahead the upper clock gets in a lifetime.

One caution matters throughout this page. A clock does not measure its own height. Only the comparison of two clocks, through a link of optical fibre, radio or a transported clock, measures a difference in gravity potential, and the comparison is usually less precise than either clock. And what the clocks measure is height above the geoid, the level surface that the oceans would follow if they were still, not height above the smooth ellipsoid that GPS uses.⁵

Chapter 2Seventy times better in sixteen years

Since 1967 the second has been defined by caesium: 9,192,631,770 oscillations of the radiation from a transition in the caesium-133 atom. The best caesium "fountain" clocks reach about 1–2 × 10⁻¹⁶.⁶ Optical clocks tick tens of thousands to over a hundred thousand times faster, at the frequencies of visible, ultraviolet or near-infrared light, which splits each second far more finely. There are two families: single ions held in electromagnetic traps (aluminium, ytterbium, calcium, strontium, indium, lutetium) and thousands of neutral atoms held in a lattice of laser light (strontium, ytterbium, mercury).

Published	Clock	Systematic uncertainty	Height equivalent
2014	NIST caesium fountain F2	1.1 × 10⁻¹⁶	1 m
2010	NIST Al⁺⁷	8.6 × 10⁻¹⁸	8 cm
2019	NIST Al⁺, first below 10⁻¹⁸⁸	9.4 × 10⁻¹⁹	9 mm
2024	JILA Sr lattice⁹	8.1 × 10⁻¹⁹	7 mm
2025	NIST Al⁺¹⁰	5.5 × 10⁻¹⁹	5 mm
Feb 2026	Wuhan Ca⁺, cooled to about 80 K¹¹	4.4 × 10⁻¹⁹	4 mm
Sep 2026	Singapore Lu⁺⁴	1.2 × 10⁻¹⁹	1.1 mm

Height equivalent is this page's arithmetic (uncertainty ÷ 1.09 × 10⁻¹⁶ per metre). Not every intermediate record is listed.

Two different numbers describe a clock, and they are easy to confuse.

Systematic uncertainty, often called accuracy, is how well the team knows every shift that could pull the clock's frequency: the heat glow of its surroundings, the motion of the ion, stray electric and magnetic fields, and gravity itself. It does not average away. The records above are this number, and each is that team's own budget. The Wuhan calcium clock won by cooling its whole trap with liquid nitrogen, shrinking the dominant "blackbody" shift; the lutetium transition is unusually insensitive to it.
Instability is the random scatter, which averages down slowly, as one over the square root of the averaging time. The 2025 NIST clock, for example, would need about 28 hours of averaging to resolve a 1-cm step on its own (a comparison of two such clocks takes about twice as long; this page's arithmetic). The two lutetium clocks ran for about 200 hours and agreed to (−0.1 ± 5.7) × 10⁻¹⁹, still limited by statistics, the equivalent of about 5 mm.

The hard part is agreement between laboratories. Clocks of the same kind in the same building have been compared below 10⁻¹⁸. Between institutes, ytterbium-ion clocks at the UK's National Physical Laboratory (NPL) and Germany's PTB agreed within 7.7 × 10⁻¹⁸ in a 2023 campaign published in 2026, which the authors call the first international verification of two independently built clocks below 10⁻¹⁷.¹² A coordinated campaign published in 2025 compared ten optical clocks in six countries at once, by fibre and satellite, and reported 38 frequency ratios, four of them measured directly for the first time.¹² Chapter 8 explains why that gap matters so much.

Figure 2📈 Try it

Sixteen years, seventy times better

Tap a clock, drag through time

a single ion in a trapatoms in a lattice of lightcaesium (today's second)two labs compared

Best clock by then

Its error budget

Better than the 2010 record by

Published systematic uncertainties of leading optical clocks, by publication date (each team's own budget; the stepped line traces the record; not every intermediate result is shown), with the caesium fountain NIST-F2 (2014) for comparison and, as an open diamond, the agreement between two independently built clocks in different countries (NPL and PTB, measured in 2023 and published in 2026). Right axis: the equivalent height difference (uncertainty ÷ 1.09 × 10⁻¹⁶ per metre). Bands: the redefinition criterion for agreement between laboratories (5 × 10⁻¹⁸) and the peak-to-peak wobble of a ground clock caused by the solid-Earth tide (about 6–7 × 10⁻¹⁷, this page's model). Sources in Chapter 2, plus: JILA Sr 2014 (B. J. Bloom et al., Nature 506, 71) and 2015 (T. L. Nicholson et al., Nat. Commun. 6, 6896); PTB Yb⁺ 2016 (N. Huntemann et al., Phys. Rev. Lett. 116, 063001); NIST Yb 2018 (W. F. McGrew et al., Nature 564, 87); VTT MIKES Sr⁺ 2025 (T. Lindvall et al., Phys. Rev. Applied 24, 044082).Each dot is one of the world's most accurate clocks of its day; the line traces the record. Lower is better. Press play to watch the record shrink about 70-fold since 2010. The pale blue band is how much the Moon's tide can wobble a clock on the ground at the biggest tides, and the open diamond shows how well clocks in two different countries agree: not nearly as well as the best single clocks.

Chapter 3Clocks as height meters

If a clock's rate depends on height, comparing two clocks measures a height difference. Geodesists call it chronometric levelling, and the rule of thumb is simple: 10⁻¹⁸ in frequency is about 1 cm in height.⁵

2016, Tokyo: strontium clocks 15 km apart, at RIKEN and the University of Tokyo, linked by 30 km of fibre, measured their 15-metre height difference to about 5 cm, in agreement with conventional surveying.¹³ The same year, PTB and the French time laboratory SYRTE compared clocks 700 km apart through 1,415 km of fibre; they agreed within their uncertainty of 5 × 10⁻¹⁷, and the link reached 3 × 10⁻¹⁷ in just 1,000 seconds, ten times better than any other way of transferring frequency at the time.¹⁴
2018, the Alps: PTB drove a transportable strontium clock to the Modane underground laboratory in the Fréjus road tunnel, about 1,000 m above its partner clock in Turin; it measured the difference to the equivalent of about 18 m, a proof of principle.¹⁵
2020, Tokyo Skytree: two transportable strontium clocks at the base and on a floor 452.6 m up the tower. The top clock ran faster by 4.9 × 10⁻¹⁴, about 4.3 nanoseconds a day (this page's arithmetic), and the result tested general relativity to about one part in 10,000, comparable in precision to tests made in space.¹⁶
2024, across the Channel: in a campaign in March and April 2023, two transportable clocks compared NPL in Teddington with PTB in Braunschweig and gave their gravity-potential difference to better than the equivalent of 4 cm, on a par with the best classical method combining GPS with a model of the geoid. It is still a preprint.¹⁷

Why bother, when surveyors can level to millimetres? Because classical levelling drifts over long distances: by 0.2–0.25 m over about 1,000 km in the UK and France, and national height systems in Europe disagree by more than half a metre.⁵ The best geoid models are good to about 2 cm. A clock network does not drift with distance, which is why clocks may one day help define height systems, rather than being corrected by them. Today, the limit runs the other way: the Singapore team noted that comparisons with distant laboratories would be limited by how well the gravity potential is known, to a few centimetres.

A space test is under way. The Atomic Clock Ensemble in Space (ACES), launched to the International Space Station in April 2025, aims to compare ground clocks on different continents at about 10⁻¹⁷ and to test the redshift to a few parts per million. As of October 2026 we could find no published results.¹⁸

Part IIThe ground breathes

Chapter 4The solid Earth has tides

The Moon pulls a little harder on the side of the Earth facing it than on the Earth's centre, and harder on the centre than on the far side. Take away the average pull (the whole Earth is in free fall towards the Moon, as the pair circle their common centre) and what remains is the tidal potential, which stretches the Earth along the line to the Moon. Its main part is

V₂ = (GM r² / d³) × (3 cos²ψ − 1)/2M and d: mass and distance of the Moon (or Sun); r: the Earth's radius; ψ: the angle of the Moon from the zenith. It has two bulges, under the Moon and opposite it.

An ocean that could keep pace with the Moon instantly (the "equilibrium tide") would bulge by V₂/g: the main twice-daily lunar tide alone has an amplitude of about 24 cm at the equator. But the rock bulges too. The solid Earth is not rigid; it behaves like a very stiff jelly, and three "Love numbers" describe how it yields:¹⁹

h₂ ≈ 0.61: the ground rises by h₂V₂/g;
k₂ ≈ 0.30: the deformed Earth adds a potential of its own, k₂V₂;
l₂ ≈ 0.085: the ground also moves sideways.

The result is a tide in the ground itself. In this page's model, the ground under Geneva rises and falls by about 30 cm on a typical day, and by up to about 50 cm peak to peak at the most extreme spring tides; near the equator the extremes reach about 56 cm. Gravity changes by up to about 300 microgals peak to peak, three parts in ten million of g. Nobody feels it, because everything for thousands of kilometres around rises and falls together. At mid-latitudes the two daily highs are unequal, because the Moon usually sits north or south of the equator, which tilts the two bulges; and spring tides, when the Sun and Moon line up, are two to three times as large as neap tides. The Sun's effect is a little under half the Moon's.

Chapter 5When the ground rises, the clock slows

A clock standing on the ground feels the tide in three ways:

the tidal potential of the Moon and Sun itself, +V₂;
the extra potential of the deformed Earth, +k₂V₂;
the clock riding up with the ground, which takes it to where the Earth's own potential is weaker by h₂V₂.

Adding them up gives the change in potential at the clock, and so its change of rate:

ΔW = (1 + k₂ − h₂) V₂ ≈ 0.69 V₂, Δf/f = −0.69 V₂ / c²1 + k₂ − h₂ is the "diminishing factor" γ₂, the same factor that reduces the tide seen by a tide gauge.

Here is the surprise. When the Moon is overhead, V₂ is positive and the ground rises, so you might expect the clock, lifted, to speed up. It slows down. The level surfaces of the gravity field rise more than the rock does (by 1 + k₂ ≈ 1.30 times V₂/g, against 0.61 for the ground), so relative to the level surface, the clock has effectively sunk, by about 1.14 times the distance the ground rose. The explanation is this page's, from standard tidal theory; the sizes agree with published calculations for clock laboratories.²⁰

How big is it? Up to about 4.2 m²/s² of potential at a clock, or 4.7 × 10⁻¹⁷,²⁰ and in this page's model about 6–7 × 10⁻¹⁷ from peak to peak at the largest tides: the same as moving the clock up and down by about 60 cm. That is roughly 500 times the uncertainty of the best lutetium clock. Every clock on Earth is wobbled twice a day; for the best, by hundreds of times their own error budget.

Figure 3🌊 Try it

Two days of tide under a clock

Ground up, clock slow

Starting day

Ground rises and falls

Clock rate swings

Compared with the best clock

This page's model of the solid-Earth tide: the degree-2 (Moon and Sun) and degree-3 (Moon) tide-generating potential from low-precision ephemerides, with IERS Conventions Love numbers (h₂ = 0.608, k₂ = 0.30, l₂ = 0.085; degree 3: 0.292, 0.093, 0.015), the permanent tide removed, no ocean loading. Globe: the ground (solid outline) and the level surface of gravity (dashed), both displaced by the degree-2 tide and exaggerated four million times (one million times for the fluid Earth, which deforms about four times more); the Moon and Sun are not to scale. The gauge follows the chosen place: the ground rises by u = hV/g, the level surface by (1 + k)V/g, and the clock's fractional frequency shift is −(1 + k − h)V/c², plotted in units of 10⁻¹⁸ (about 1 cm of equivalent height). For pairs of laboratories the lower panel shows the difference between their clocks. "Rigid Earth" sets h = k = 0; "fluid Earth" uses the homogeneous-fluid values h₂ = 5/2, k₂ = 3/2, for which the ground follows the level surface and the clock feels nothing. Times in UTC from the chosen day in October–November 2026; the Moon symbols mark its highest point in the sky, and the phase is from the same ephemeris. "Spring tide" and "Neap tide" jump to the nearest two days centred on a new or full Moon, or on a half Moon; the Moon's distance and declination also change the range, so at mid-latitudes a neap tide can rival a spring tide (as in early October 2026). For pairs, the last tile averages the difference over 10 days from the starting day; the fortnightly and monthly tides survive the averaging.The Moon (and, a little less, the Sun) stretches the whole Earth into a slight rugby-ball shape, hugely exaggerated here. As the Earth turns, your city rides up and down through the bulges. The solid line is the ground; the dashed line is gravity's "level". The level rises more than the ground, so the clock ends up below it, and runs slow. Press play, try other cities, a pair of labs far apart (even averaged over ten days, the Moon doesn't cancel out), or an Earth that is perfectly rigid or completely liquid.

Why haven't the record clocks noticed? Because two clocks in the same laboratory ride the same tide, and the effect cancels: for the two NIST and JILA laboratories in Boulder, 1.5 km apart, the difference is only about 2 × 10⁻²⁰ (this page's arithmetic). The difference grows with distance: about 1.3 × 10⁻¹⁷ between PTB and SYRTE, and about 10⁻¹⁶ between continents, where the tides peak hours apart. Averaging helps less than you might hope, because the fortnightly and monthly tides change slowly: over a year, ten-day averages of an intercontinental comparison still wander by about ±1–4 × 10⁻¹⁸, most for labs far apart in latitude, such as Singapore and Braunschweig, close to the target for redefining the second (this page's arithmetic). So the long-distance comparisons have to model the tides.²⁰ The 2017 test of relativity with the SYRTE–PTB–NPL fibre network did exactly that.²¹

As far as we can find, no one has yet published a clean detection of the Moon in a clock comparison. Takano and colleagues called it easily observable with distant clock pairs, and a review suggests it as a natural test of a clock network.^13,5 It would also be useful: clocks are sensitive to 1 + k₂ − h₂, GPS to h₂ and gravimeters to a third combination, so together they could measure how the Earth yields directly.

Chapter 6The Moon in the particle beam

Long before optical clocks felt the tide, a particle accelerator did. CERN's Large Electron–Positron collider (LEP) ran from 1989 to 2000 in a 26.7-km tunnel under the French–Swiss border, colliding electrons and positrons at the energy needed to make Z bosons, about 91 GeV. Measuring the Z's mass precisely meant knowing the beam energy to about 20 parts per million.

The energy was measured with a clock of a kind: the spin of the electrons. In a storage ring, electrons slowly line up their spins, and the spins then precess in the ring's magnetic field, turning a number of times per lap set by the beam energy:

E = ν × 440.65 MeVν, the "spin tune", is the number of spin precessions per turn; 440.65 MeV is the electron's mass energy divided by its anomalous magnetic moment. At 45.6 GeV, ν ≈ 103.5.

Tickling the beam with a weak oscillating magnetic field at exactly the spin's frequency destroys the polarisation, so finding the frequency at which it vanishes gives the energy to about 0.2 MeV. This "resonant depolarisation" became available in 1991, and the measured energies wandered by about one part in 10,000, in step with the local pull of gravity. The tide had already been suggested as a cause by Gerry Fischer of SLAC and Albert Hofmann of CERN.^22,23

On 11 November 1992, the day after a full Moon, the LEP team followed the energy for 24 hours. It rose and fell as a tide model predicted: the energy changed by 0.869 parts per million for every microgal change in local gravity (falling as gravity fell, when the Moon was high), against 0.882 predicted.²² The mechanism is subtle. The tide stretches and squeezes the ground sideways, by about 2 parts in 100 million, so LEP's circumference swung by up to about a millimetre between its shortest and its longest.²⁴ But the beam's path length is fixed by the radio-frequency system that drives it. So when the ring of magnets grows, the beam is pushed off-centre in the focusing magnets, which changes its energy:

ΔE/E = −(1/α_c) × ΔC/Cα_c, the "momentum compaction factor", was about 1.85 × 10⁻⁴ for LEP, so the ring amplified a strain by about 5,400.

A strain of 2.2 × 10⁻⁸ became an energy change of about 120 parts per million: about 5 MeV at 45.6 GeV. In this page's model, a change of just 0.11 mm in LEP's circumference shifts the beam energy by 1 MeV. The model, the same one used for the clocks in Chapter 5, reproduces the size of the published LEP numbers: a circumference swing of up to about a millimetre from shortest to longest, and energy changes of up to about 120 parts per million. The swing is lopsided, as Figure 4 shows: the energy drops furthest, by up to about 6 MeV (130 parts per million), when the Moon is high, and rises only about half as far when it is low in the sky. Uncorrected, the tide would have shifted the collision energy of individual measurements by 10 MeV or more, several times the final error on the Z mass. Corrected, it vanished from the error budget: the final LEP value, 91.1875 ± 0.0021 GeV, carries an uncertainty of 1.7 MeV from the beam energy, almost none of it from the Moon.²³

Figure 4🌕 Try it

The ring that breathed with the Moon

CERN, November 1992

Time

Moon's height in the sky

Ring circumference

Beam energy

Collision energyBoth beams together

This page's tide model (Figure 3) at CERN for 10–12 November 1992, starting at 00:00 UTC on the day of full Moon. The shaded day, 11 November 1992, is when the LEP team followed the energy for 24 hours; the paper does not give the exact start time. The ring's fractional circumference change is half the horizontal areal strain, ΔC/C = ΣV_n(2h_n − n(n+1)l_n)/(2gR), applied to LEP's 26.659 km; the beam energy follows ΔE/E = −(ΔC/C)/α_c with α_c = 1.85 × 10⁻⁴ (the 1992 optics), shown in MeV at the Z-pole beam energy of 45.6 GeV, with the matching circumference change on the right axis. Changes are relative to the long-term mean. The map is schematic (lake, river and Jura foothills approximate); the ring's change is exaggerated six million times, and the beam's offset in the magnified magnet is schematic. The sky dial shows the Moon's direction and height seen from CERN. The 1992 measurement (Arnaudon et al.) found 0.869 ppm of beam energy per microgal of tidal gravity change (energy falling with gravity), against 0.882 predicted. The last tile is the change in collision energy, twice the beam's, because both beams circle the same ring and meet head-on; it is compared with the final ±2.1 MeV error on the Z mass.This is CERN's 27-km ring under the Geneva countryside as the Moon went by in November 1992 (the ring's breathing is hugely exaggerated). The shaded day is when the team measured it. When the Moon is high, the ground stretches, the ring grows by a fraction of a millimetre, the particles end up slightly off-centre in the magnets, and their energy drops. Press play, or drag along the graph. The last box adds both beams together and compares the shift with how precisely the Z boson's mass was finally measured.

The ring felt other things too. The circumference changed by up to 2 mm with the seasons, with rainfall, groundwater and the level of Lake Geneva. And in 1995 the beam energy was found to creep up through each fill by about 5 MeV, but not at night: the cause was about one ampere of stray current from the railway line between Geneva and Bellegarde, leaking into the ground and along LEP's vacuum chamber.^24,23

The Large Hadron Collider in the same tunnel feels the same tides, about a millimetre, but a feedback system adjusts its radio frequency automatically. In November 2016, during a long fill, the tide showed up in the LHC beam's momentum, and the large earthquake in New Zealand on 13 November showed up alongside it.²⁵ The proposed 91-km electron–positron Future Circular Collider (FCC-ee), chosen by CERN Council in May 2026 as its preferred next flagship but not yet approved, plans to measure the Z mass to about 100 keV; in the 2019 design study, the tide would swing the beam energy by about 120 MeV from peak to peak, and the ring's mean radius would need to be known to half a micrometre.²⁶

Part IIIEveryday relativity, and the second

Chapter 7GPS: relativity in the bookkeeping

The best-known application of all this is GPS. Its satellites orbit at about 20,200 km altitude (26,560 km from the Earth's centre), moving at 3.87 km/s. Relativity changes their clocks in two opposite ways:²⁷

higher up, at a higher gravitational potential, they run faster: +45.7 microseconds a day;
moving fast, they run slower: −7.1 µs a day, once the ground clock's own motion with the rotating Earth is allowed for.

+45.7 − 7.1 = +38.6 µs per day (4.4647 × 10⁻¹⁰)The two effects would cancel for an orbit about 9,545 km from the Earth's centre. Below it, satellite clocks run slow (as on the International Space Station); above it, fast.

So the satellite clocks are offset: the 10.23 MHz reference is meant to run at 10.22999999543 MHz as measured beside it in orbit, so that, seen from the ground, it ticks at 10.23 MHz. Caesium clocks were set this way before launch; rubidium clocks, which now run almost the whole constellation, jump in frequency during launch, so their rate is adjusted on board once they are in orbit, and the broadcast clock corrections take care only of the small remainder.^27,28 The first caesium clock in orbit, on the NTS-2 satellite in 1977, confirmed the predicted offset to about 1%. Here relativity is bookkeeping, a correction engineers allow for; in a car battery it is the chemistry itself, about 1.7 of each cell's 2.1 volts (Nº 11).

A popular claim says that without relativity, GPS positions would drift by 10 km a day. Light covers 11.6 km in 38.6 µs, so that is the raw size of the error, but positions would not actually drift that way. An error shared by all the satellites' clocks shifts every distance equally, and a receiver solves for its own clock offset anyway (that is why it needs four satellites, not three). And the ground control segment measures every satellite clock and broadcasts corrections. If nobody corrected the offset, what would break first is GPS as a time service: satellite time would run ahead of world time by 38.6 µs a day.²⁷ The service promises time to within 30 ns (95% of the time), a promise the uncorrected offset would break in about a minute (this page's arithmetic).²⁹ The relativity that matters for position is subtler: each satellite's slightly elliptical orbit makes its clock speed up and slow down, so that it runs up to about ±23 ns ahead or behind for an orbital eccentricity of 0.01 (about ±7 m of range), and the Earth's rotation during the signal's flight (the Sagnac effect) adds up to about 130 ns. Receivers correct both.²⁸ Relativity is built into the bookkeeping.

That orbit-shape term once became a test of Einstein. In August 2014, a fault in the rocket's upper stage left two satellites of Europe's Galileo system in elliptical orbits, so their hydrogen-maser clocks sped up and slowed down with every orbit, far more than a GPS clock does. Two teams followed them for about three years and confirmed the gravitational redshift to a few parts in 100,000, four to six times better than Gravity Probe A, a maser clock flown on a rocket in 1976.³⁰

The clocks are not what limits your phone. In 2024, the satellites' clock and orbit errors together were 0.93 m (95th percentile) for the standard signal, while a phone under open sky is typically good to about 5 m; the difference comes from the atmosphere, reflections and the buildings around you.²⁹

Figure 5🛰️ Try it

Switch off relativity

Switch a correction off

youa real receiver (works out its own clock error)a receiver that trusts the satellites' clocks

Days since the correction was switched off

Satellite clocks off by

Light travels that far in that time

A real receiver puts you off by

GPS time versus world time

The rate of a GPS satellite clock compared with a clock on the ground (Ashby 2003): +45.7 µs/day from the higher gravitational potential, −7.1 µs/day from the satellite's speed (net of the ground clock's rotation), +38.6 µs/day (4.4647 × 10⁻¹⁰) in all, removed by offsetting the clocks to 10.22999999543 MHz (on board: before launch for caesium clocks, once in orbit for rubidium clocks; the broadcast clock corrections handle only the small remainder). The globe shows a 24-satellite constellation (six planes at 55°, 26,560 km from the Earth's centre) in the Earth's rotating frame, with a receiver at 46° N; lines join it to the satellites more than 10° above its horizon. The map shows this page's least-squares position fixes from those satellites, with every other error source left out. "Clock-rate correction off": every satellite clock gains 38.6 µs a day and no one corrects it (in reality the control segment would). A receiver that solves for its own clock offset, as real ones do, absorbs the shared error, so only GPS time goes wrong; a receiver with a perfect clock that trusted the satellites would be thrown off by kilometres, mostly upwards. "Orbit-shape correction off": the per-satellite eccentricity term (±23 ns, about ±7 m of range, for eccentricity 0.01, given to every satellite here for illustration) is left uncorrected, and the fix wanders by metres, or by tens of metres when few of this model's 24 satellites are in view or they bunch together in the sky (the real constellation has about 31 satellites, so receivers usually see more). GPS time itself is unaffected, because receivers make this correction. The Sagnac correction, also applied by receivers, is not modelled.GPS satellites gain nearly 39 millionths of a second every day because of relativity. Switch the correction off and press play. Your map wouldn't wander off by kilometres, because your phone works out its own clock error and all the satellites are wrong together. What breaks is GPS time. The part of relativity that really moves your dot on the map is different for each satellite: try the orbit-shape switch.

Chapter 8Redefining the second

Until 1960 the second was a fraction of the mean solar day; then briefly a fraction of the year 1900; and since 1967, 9,192,631,770 oscillations of caesium, the transition that Louis Essen and Jack Parry first used as a frequency standard at NPL in 1955.³¹ Optical clocks now beat caesium fountains by a factor of about a hundred (the CGPM's cautious 2026 figure) to nearly a thousand (comparing the best teams' own error budgets). But no clock can realise the second better than its definition allows: today an optical clock's frequency in hertz can only be known as well as caesium can measure it, about 10⁻¹⁶. So metrologists want a new definition. Whatever they choose will be fixed so that the second keeps its present length, as closely as caesium can measure it.³²

The Consultative Committee for Time and Frequency, which advises the International Committee for Weights and Measures, has a roadmap, first drawn up in 2022 and updated in 2025, with mandatory criteria to be met first.^32,33 The hardest are about agreement. One asks for at least five frequency ratios between different optical clocks, each measured at least twice by different institutes, agreeing to better than 5 × 10⁻¹⁸. Ratios matter because a ratio of two frequencies is a pure number, which needs no definition of the second to measure. (Nº 1 makes the same point about lengths: only ratios are measurable.)

Progress is real but uneven. In 2026, NIST and JILA reported three ratios, between aluminium-ion, ytterbium and strontium clocks, with uncertainties of 2.2–3.2 × 10⁻¹⁸.³⁴ All three are below the roadmap's 5 × 10⁻¹⁸ threshold for ratios (PTB had reached 4.4 × 10⁻¹⁸ for an indium-ion/ytterbium-ion ratio in 2025,³⁵ and 4.3 × 10⁻¹⁸ for an ytterbium-ion/strontium ratio in an August 2026 preprint³⁶), but the criterion needs five such ratios, each confirmed by a second institute. And the same measurement moved the strontium ratios by about 1 × 10⁻¹⁶, 14 standard deviations away from the same group's 2021 values.³⁷ An independent PTB measurement (still a preprint) disagrees with the 2021 value by 8.6 standard deviations, but agrees with the 2026 value within 1.2 standard deviations.³⁸ So the revision now has independent support, though each of these ratios has still been measured only once at the required level. The consultative committee summed it up in 2025: comparisons below 5 × 10⁻¹⁸ between same-type clocks built by different laboratories have "not yet been achieved", and there is no consensus on whether to base the new second on one species or on an ensemble of several.³⁹

Gravity is now part of the definition problem. International Atomic Time is meant to tick like a clock on a conventional level surface close to mean sea level (the potential W₀ = 62,636,856.0 m²/s² behind the definition of Terrestrial Time), so every contributing clock is corrected for its height above it.¹⁹ The roadmap requires each laboratory's gravity potential to be known to about 10⁻¹⁷ (9 cm of height) for its clocks to contribute to International Atomic Time, and, for comparisons between the best clocks, well enough not to spoil the comparison: a few centimetres or better.³² To redefine the second, you have to know how high your laboratory is, and you have to model the Moon. On 15 October 2026, the CGPM set its sights on a proposal for the new definition at its next meeting, in 2030 (Chapter 9).

Chapter 9Three votes about time

The General Conference on Weights and Measures (CGPM), where the member states of the Metre Convention vote on the definitions of units, met at Versailles from 13 to 15 October 2026. It adopted seven resolutions; the three about time were adopted on 15 October 2026.³³

Resolution 4, "On the future definition of the second". It does not redefine anything, choose a species, or choose between one species and an ensemble. It records that "a consensus has not been reached", that there are "noticeable inconsistencies" in some comparisons of optical clocks and in the measured frequency ratios, and that some mandatory criteria are not yet met, such as optical clocks contributing to the calibration of International Atomic Time, though they "could realistically be fulfilled by 2030". It asks the International Committee for Weights and Measures (CIPM) to work towards a proposal for the new definition, and a date for bringing it in, at the next conference, in 2030. In a statement after the vote, the consultative committee's president said that two options remain, the frequency of a single optical transition or a weighted geometric mean of several, and that two of the mandatory criteria are still far from being met.³³

Resolution 5, "On the technical actions needed to ensure the continuity of UTC", ends leap seconds. From 20 May 2027, World Metrology Day, Coordinated Universal Time (UTC) runs continuously: about eight years ahead of the 2035 deadline the conference set in 2022. The gap between atomic time and the Earth's rotation, UT1 − UTC, may now grow to 3,600 seconds (one hour), which the resolution says keeps UTC continuous "for several centuries". The 27 leap seconds since 1972 were all positive, added because the Earth's day has usually been slightly longer than 86,400 atomic seconds. (A ring laser in Bavaria can feel the day lengthen or shorten by a fraction of a millisecond within hours: Nº 16.) The long-term reason is the tide: tidal friction alone would lengthen the day by about 2.3 milliseconds per century (from the Moon's measured recession), and 1.8 ms per century is observed over 2,700 years.⁴⁰ At the average pace since 1972, about half a second a year, the gap would grow by under a minute per century; because the braking makes the day ever longer, it grows faster over time, and would reach an hour in roughly a thousand years (this page's estimate). But from 2020 to 2025 the Earth spun, on average, slightly fast, and the resolution cites an estimate, made at a March 2025 workshop, that the chance of the first-ever negative leap second would reach 30% by 2035; preparing for one, it notes, would cost industry about as much as preparing for the millennium bug. Melting polar ice had already postponed it by a few years.⁴¹ Then, in 2026, the Earth slowed again: UT1 − UTC, +0.09 s in October 2025, was −0.04 s on 15 October 2026.⁴² No leap second will be added at the end of 2026, and the IERS predicts the gap will stay far below the 0.9 s that would call for one before 20 May 2027. So the leap second added on 31 December 2016 is set to be the last.

Figure 6⏳ Try it

The Earth clock falls behind

Drag through the years

civil time (UTC)the Earth's turning (UT1)IERS prediction

Year

Leap seconds so far

Earth behind atomic time

Gap UT1 − UTC

Atomic time (TAI) against civil time (UTC, stepped) and the Earth's rotation (UT1, smooth), from IERS data: monthly UT1 − UTC from the EOP 20 C04 series (January 1972 to September 2026), the value measured on 15 October 2026 (−0.037 s, Bulletin A) and the IERS prediction (Bulletin A, 22 October 2026) to 20 May 2027 (dashed). Each step is one of the 27 leap seconds, all positive. The lower strip shows UT1 − UTC. "After 2027": UTC no longer steps (CGPM Resolution 5, 2026), so the gap grows; the band is this page's estimate, with today's day 0–1.7 ms longer than 86,400 s and lengthening by 1.8 ms (observed, Stephenson et al. 2016) to 2.3 ms (tidal friction alone) per century, ignoring decade-scale swings of a few milliseconds.Each step is a leap second: 27 since 1972, all added because the Earth's spin keeps falling behind atomic clocks. Drag to the year you were born to count the leap seconds you have lived through. Look at 2020–2025, when the Earth sped up a little, and at 2026, when it slowed again. Then switch to "After 2027": with no more leap seconds, the gap just grows, slowly at first, reaching an hour in roughly a thousand years.

Resolution 6, "On the definition of an international lunar reference time scale and its traceability to UTC", does not yet set a time for the Moon. It notes studies showing that Lunar Coordinate Time (TCL), defined by the International Astronomical Union in 2024, "without any scaling would meet the scientific requirements". It encourages space agencies to share their experience and, whenever they compare a lunar time with UTC, to describe the relativistic model, the ephemerides and the uncertainties they used, and it recommends that the CIPM follow the results of these pioneering projects and act as necessary. The aim, the consultative committee's president said, is one coherent approach rather than competing lunar time scales.³³ It is the same physics as the rest of this page: a clock on the Moon runs faster than one on Earth by about 56 µs a day.⁴³

So the Moon runs through all three: the tide that wobbles the clocks, the tidal braking behind the leap seconds, and the clocks that will one day keep time on the Moon itself.

Chapter 10The next clock: a nucleus

Every clock on this page ticks with electrons. One atomic nucleus, thorium-229, has an excited state so low in energy, 8.36 electronvolts, that an ultraviolet laser can reach it. A nucleus is tens of thousands of times smaller than an atom and shielded by its electrons, so it should be far less disturbed by stray fields, and it can even be embedded by the billions of billions in a transparent crystal. Its transition is also roughly a thousand times more sensitive than the most sensitive atomic clocks to any change in the fine-structure constant,⁴⁴ which makes it a candidate detector for some kinds of dark matter. (Carbon-14, a nucleus that decays over thousands of years, keeps a much slower kind of time: it dates tree rings and even your own cells, Nº 12.)

In 2024 the transition was excited by a laser for the first time⁴⁵ and its frequency was measured against a strontium clock to about 10⁻¹².⁴⁶ In June 2026 two groups, in Europe and in China, reported clocks with a laser locked to the nuclear transition (apparently the first working nuclear clocks); both papers were published in Nature on 7 October 2026. Their instabilities are 3 × 10⁻¹² and 5 × 10⁻¹³ over one second, and the European clock approached 10⁻¹⁵ over a day.⁴⁷ That is still thousands of times less stable than optical clocks, and the crystal's frequency shifts with temperature, so reaching 10⁻¹⁸ would need the crystal held steady to millionths of a degree. A nuclear clock beating the lutetium ions is a hope for the coming decade, not a result.

Meanwhile, the optical clocks are already looking for new physics: if a fundamental constant drifted, clocks of different kinds would drift apart, and comparisons now limit any drift of the fine-structure constant to a few parts in 10¹⁹ per year.⁴⁸ Over a far longer span, the rock of a natural nuclear reactor that ran in Gabon about two billion years ago shows the fine-structure constant matching today's to about 1 part in 10⁸, if nothing else changed (Nº 13 sets the rock against the clocks). To look for tiny wiggles like that between distant clocks, the Moon's daily wiggle has to be taken out first.

Chapter 11Glossary and sources

Fractional frequency, Δf/f: How much faster or slower a clock runs, as a fraction. 10⁻¹⁸ is one part in a billion billion.
Gravitational redshift: Clocks deeper in a gravitational field run slower: 1.09 × 10⁻¹⁸ per centimetre near the Earth's surface.
Gravity potential, geoid: The energy per kilogram it would take to lift something from that point out of the Earth's gravity (plus rotation), so it is larger deeper down. The geoid is the level surface the oceans would follow at rest.
Systematic uncertainty: How well a clock's team knows all the effects that shift its frequency. It does not average away.
Instability: A clock's random scatter, which averages down with time.
Frequency ratio: The ratio of two clocks' frequencies: a pure number, measurable without any definition of the second.
Chronometric levelling: Measuring height differences by comparing clocks.
Love numbers h, k, l: How much the solid Earth rises, adds potential and moves sideways under a tidal pull.
Momentum compaction: How much a storage ring's orbit length changes with beam energy; it turns a tiny stretch of the ring into a large energy shift.
Spin tune: The number of times an electron's spin turns per lap of a ring; it measures the beam energy.
UTC, UT1, leap second: UTC is the world's atomic civil time; UT1 follows the Earth's actual rotation. Leap seconds kept them within 0.9 s; from 20 May 2027 there will be no more, and the gap may grow to an hour.
CGPM: The General Conference on Weights and Measures, which decides the definitions of the SI units.

Sources

R. V. Pound and G. A. Rebka Jr., "Apparent weight of photons," Phys. Rev. Lett. 4, 337 (1960).
C. W. Chou, D. B. Hume, T. Rosenband and D. J. Wineland, "Optical clocks and relativity," Science 329, 1630 (2010).
T. Bothwell et al., "Resolving the gravitational redshift across a millimetre-scale atomic sample," Nature 602, 420 (2022), arXiv:2109.12238.
K. J. Arnold et al., "Lu⁺ optical frequency references with accuracy verified at the 19th digit," Nature 658, 60 (2026), doi:10.1038/s41586-026-11072-8, arXiv:2512.07346.
T. E. Mehlstäubler, G. Grosche, C. Lisdat, P. O. Schmidt and H. Denker, "Atomic clocks for geodesy," Rep. Prog. Phys. 81, 064401 (2018), arXiv:1803.01585.
T. P. Heavner et al., "First accuracy evaluation of NIST-F2," Metrologia 51, 174 (2014); V. Gerginov et al., on the caesium fountain NIST-F4, Metrologia 62, 035002 (2025).
C. W. Chou et al., "Frequency comparison of two high-accuracy Al⁺ optical clocks," Phys. Rev. Lett. 104, 070802 (2010).
S. M. Brewer et al., "²⁷Al⁺ quantum-logic clock with a systematic uncertainty below 10⁻¹⁸," Phys. Rev. Lett. 123, 033201 (2019).
A. Aeppli, K. Kim, W. Warfield, M. S. Safronova and J. Ye, "Clock with 8 × 10⁻¹⁹ systematic uncertainty," Phys. Rev. Lett. 133, 023401 (2024).
M. C. Marshall et al., "High-stability single-ion clock with 5.5 × 10⁻¹⁹ systematic uncertainty," Phys. Rev. Lett. 135, 033201 (2025), arXiv:2504.13071.
B.-L. Zhang et al., "Liquid-nitrogen-cooled ⁴⁰Ca⁺ ion optical clock with a systematic uncertainty of 4.4 × 10⁻¹⁹," Phys. Rev. Lett. 136, 053202 (2026), doi:10.1103/vngc-c1xv.
M. Pizzocaro et al., "International optical clock comparison using the European optical fiber network," Phys. Rev. Research 8, 033250 (2026), doi:10.1103/l4bh-ryxs, arXiv:2604.27963; T. Lindvall et al., "Coordinated international comparisons between optical clocks connected via fiber and satellite links," Optica 12, 843 (2025), doi:10.1364/OPTICA.561754, arXiv:2505.06763.
T. Takano et al., "Geopotential measurements with synchronously linked optical lattice clocks," Nat. Photonics 10, 662 (2016).
C. Lisdat et al., "A clock network for geodesy and fundamental science," Nat. Commun. 7, 12443 (2016).
J. Grotti et al., "Geodesy and metrology with a transportable optical clock," Nat. Phys. 14, 437 (2018).
M. Takamoto et al., "Test of general relativity by a pair of transportable optical lattice clocks," Nat. Photonics 14, 411 (2020).
ICON Collaboration (A. Amy-Klein et al.), "International comparison of optical frequencies with transportable optical lattice clocks," arXiv:2410.22973 (2024, preprint).
European Space Agency, Atomic Clock Ensemble in Space (ACES) mission pages (2025–2026); CNES, PHARAO page (April 2026).
G. Petit and B. Luzum (eds.), IERS Conventions (2010), IERS Technical Note 36 (2010), chapters 6–8 and 10.
C. Voigt, H. Denker and L. Timmen, "Time-variable gravity potential components for optical clock comparisons and the definition of international time scales," Metrologia 53, 1365 (2016).
P. Delva et al., "Test of special relativity using a fiber network of optical clocks," Phys. Rev. Lett. 118, 221102 (2017).
L. Arnaudon et al., "Effects of terrestrial tides on the LEP beam energy," Nucl. Instrum. Methods A 357, 249 (1995); L. Arnaudon et al., "Effects of tidal forces on the beam energy in LEP," Proc. 1993 Particle Accelerator Conference, p. 44 (1993).
ALEPH, DELPHI, L3, OPAL and SLD Collaborations et al., "Precision electroweak measurements on the Z resonance," Phys. Rep. 427, 257 (2006).
J. Wenninger, "Observation of radial ring deformations using closed orbits at LEP," Proc. 1999 Particle Accelerator Conference, THP28 (1999).
J. Wenninger, "Linear imperfections," CERN Accelerator School (2018), arXiv:2004.14001 (Fig. 9: tides in the LHC beam momentum, November 2016); LHC Beam Operation Committee records (CERN login required; the earthquake of 13 November 2016 on the mean LHC orbit).
A. Blondel et al., "Polarization and centre-of-mass energy calibration at FCC-ee," arXiv:1909.12245 (2019); CERN Council statement on the Future Circular Collider (22 May 2026).
N. Ashby, "Relativity in the Global Positioning System," Living Rev. Relativ. 6, 1 (2003), doi:10.12942/lrr-2003-1.
IS-GPS-200 Revision N, Navstar GPS Space Segment/Navigation User Interfaces (US Space Force).
GPS.gov, "GPS accuracy"; US Department of Defense, Global Positioning System Standard Positioning Service Performance Standard, 5th ed. (April 2020); Applied Research Laboratories, University of Texas at Austin, GPS standard positioning service performance report for 2024 (December 2025).
P. Delva et al., "Gravitational redshift test using eccentric Galileo satellites," Phys. Rev. Lett. 121, 231101 (2018), arXiv:1812.03711; S. Herrmann et al., "Test of the gravitational redshift with Galileo satellites in an eccentric orbit," Phys. Rev. Lett. 121, 231102 (2018), arXiv:1812.09161.
L. Essen and J. V. L. Parry, "An atomic standard of frequency and time interval: a caesium resonator," Nature 176, 280 (1955).
N. Dimarcq et al., "Roadmap towards the redefinition of the second," Metrologia 61, 012001 (2024).
BIPM, resolutions adopted at the 28th meeting of the CGPM (15 October 2026): Resolution 4, "On the future definition of the second," doi:10.59161/CGPM2026RES4E; Resolution 5, "On the technical actions needed to ensure the continuity of UTC," doi:10.59161/CGPM2026RES5E; Resolution 6, "On the definition of an international lunar reference time scale and its traceability to UTC," doi:10.59161/CGPM2026RES6E; N. Dimarcq, statement by the CCTF President (15 October 2026).
A. Aeppli et al., "Atomic clock frequency ratios with fractional uncertainty ≤ 3.2 × 10⁻¹⁸," Phys. Rev. Lett. 137, 033201 (2026), arXiv:2512.21428.
H. N. Hausser et al., "¹¹⁵In⁺–¹⁷²Yb⁺ Coulomb crystal clock with 2.5 × 10⁻¹⁸ systematic uncertainty," Phys. Rev. Lett. 134, 023201 (2025), arXiv:2402.16807.
C. Vishwakarma et al., "Interspecies clock comparison below 5 × 10⁻¹⁸ uncertainty with a transportable clock," arXiv:2608.01916 (2026, preprint).
Boulder Atomic Clock Optical Network (BACON) Collaboration, "Frequency ratio measurements at 18-digit accuracy using an optical clock network," Nature 591, 564 (2021).
F. Dawel et al., "An Al⁺ clock with 1.6 × 10⁻¹⁸ systematic uncertainty and its frequency ratios," arXiv:2606.23151 (2026, preprint).
Consultative Committee for Time and Frequency, Recommendation CCTF 24-3 (2025); BIPM, "FAQ on the redefinition of the second," version 2.1 (8 October 2026).
F. R. Stephenson, L. V. Morrison and C. Y. Hohenkerk, "Measurement of the Earth's rotation: 720 BC to AD 2015," Proc. R. Soc. A 472, 20160404 (2016).
D. C. Agnew, "A global timekeeping problem postponed by global warming," Nature 628, 333 (2024).
IERS, Bulletin C 72 (6 July 2026) and Bulletin A, vol. XXXIX nos. 042 (15 October 2026) and 043 (22 October 2026); IERS EOP 20 C04 Earth-orientation series, iers.org.
N. Ashby and B. R. Patla, "A relativistic framework to estimate clock rates on the Moon," Astron. J. 168, 112 (2024).
K. Beeks et al., "Fine-structure constant sensitivity of the Th-229 nuclear clock transition," Nat. Commun. 16, 9147 (2025).
J. Tiedau et al., "Laser excitation of the Th-229 nucleus," Phys. Rev. Lett. 132, 182501 (2024).
C. Zhang et al., "Frequency ratio of the ²²⁹ᵐTh nuclear isomeric transition and the ⁸⁷Sr atomic clock," Nature 633, 63 (2024).
L. Toscani De Col et al., "A thorium-229 optical nuclear clock with feedback loop," Nature 658, 897 (2026), doi:10.1038/s41586-026-11084-4, arXiv:2606.04997; B. Huang et al., "A nuclear clock synchronized to ²²⁹Th," Nature 658, 903 (2026), doi:10.1038/s41586-026-11122-1, arXiv:2606.08870.
M. Filzinger et al., "Improved limits on the coupling of ultralight bosonic dark matter to photons from optical atomic clock comparisons," Phys. Rev. Lett. 130, 253001 (2023).

1 cm higher, and time runs faster

One centimetre

The race to 19 decimal places

Clocks that measure mountains

The ground breathes

Up is slow

The Moon in the particle beam

Relativity in your pocket

A new second

Three votes about time

The next clock