31.8 hours: ask the Paris pendulum, and that's how long a day lasts

Part IA day that doesn't close

Chapter 1Thirty-one point eight hours

In March 1851, anyone in Paris could walk in and watch the Earth turn. Under the dome of the Panthéon, Léon Foucault hung a wire 67 metres long. At its end was a hollow brass sphere 17 cm across, filled with lead to a mass of 28 kg. A stylus under the sphere cut marks in ridges of sand at the end of each swing.¹ Swing after swing, the marks crept round. Seen from above, the line of the swing turned slowly clockwise. In a 2017 history of the experiment, the physicist Joël Sommeria writes that the demonstrations had "an immense popular impact".

The swing was not turning. The floor was. The Earth turns once relative to the stars in one stellar day of 23 h 56 min 4.1 s (23.9345 hours), which is 15.04° an hour, not 15°.² At the North Pole the floor would turn under the pendulum at that full rate. In Paris it turns more slowly, by the sine of the latitude (Chapter 3 explains why): at the Panthéon, at 48.846° N, the swing turns 15.04° × 0.753 = 11.33° an hour. So a full circle takes

23.9345 h ÷ sin 48.846° = 23.9345 h ÷ 0.75295 = 31.79 hThis page's arithmetic. At the Paris Observatory, about a kilometre to the south, the same arithmetic gives 31.79 hours.

That is the headline: thirty-one point eight hours for the swing to turn all the way round in Paris, not one day. The digits themselves carry a fact. With a careless 24-hour day in place of the stellar day, the answer would be 31.87 hours, which rounds to 31.9. The "31.8" already knows that the Earth takes about four minutes less to turn once against the stars than once against the Sun.

There are two ways to say how far short of a full circle the swing falls in "a day", and this page keeps them apart. After 24 hours, the Paris swing has turned 271.8°, leaving 88.2° to go. After one turn of the Earth, one stellar day, it has turned 360° × sin φ = 271.06°, leaving 88.94°. The 0.74° between them is what the swing turns in the extra four minutes (this page's arithmetic). The second number is the one with a meaning: Chapter 4 shows that 88.94° is the area of the polar cap north of Paris.

One more convention. A swing line has no arrowhead, so it already lies along its starting line after half a turn, 15.9 hours. "A full circle" on this page means 360°, the turn of a direction (an arrow) carried round, which is what the geometry of Chapter 4 counts.

The Panthéon pendulum took about 16.4 seconds for each back-and-forth swing, so a full circle took about 6,970 swings (this page's arithmetic). At each swing the plane moved on by about 0.05°; Sommeria gives the sideways step at the end of the swing as 2.3 mm, which is what a swing reaching about 2.5 m from the centre would give (this page's arithmetic). Sand was a good choice: a fresh cut a couple of millimetres along at every swing is hard to argue with.

It was not the first evidence that the Earth rotates. Chapter 8 lists what came before: a pendulum that swung more slowly near the equator in 1672, the flattening of the Earth measured in Lapland in 1736–37, and balls dropped down towers and mine shafts. What Foucault gave was the first unambiguous, public, easily repeated demonstration. Sommeria calls it "the first direct experimental proof of Earth rotation". It was repeated at once: at Rennes, in the cathedral of Reims, and in at least 25 cities in the United States during the summer of 1851.¹ Figure 1, in Chapter 3, lets you move the pendulum anywhere on the globe.

Chapter 2A wire in a cellar

The Panthéon was the third pendulum, not the first. Foucault described the first one in a note to the Académie des sciences. In the vault of a cellar, in January 1851, he fixed a heavy cast-iron piece; from a small block of tempered steel hung a steel wire 2 metres long and 0.6 to 1.1 mm thick. At its end was a ground and polished brass sphere of 5 kg, with a pointed tip below it.^3,1

The release was the delicate part. Foucault first took the twist out of the wire and stopped the sphere twisting to and fro. He then drew the sphere aside with a loop of thread tied to the wall, damped the leftover motion with an obstacle withdrawn little by little, and burned the thread. With nothing pushing it sideways, the pendulum set off on its own, swinging through an arc of 15 to 20 degrees.³ After half an hour, he wrote, the shift "leaps to the eyes". With a fixed pointer that the pendulum's tip just grazed at the end of its swing, it was visible in less than a minute: the tip kept arriving a little to the observer's left.

Thanks to François Arago, the director of the Paris Observatory, and the instrument maker Gustave Froment, Foucault then repeated the experiment with an 11-metre wire in the Observatory's meridian hall. He presented the results to the Académie on 3 February 1851, and his note appeared in the Comptes rendus issue of the same date.^3,1

The note states the physics at the pole, where the experiment must succeed "in all its purity", and then admits the hard part. At lower latitudes, Foucault wrote, the effect slows down; it vanishes at the equator and changes sense in the southern hemisphere. He wanted to draw "the attention of the géomètres", the mathematicians, to this complication, and although the law in between needed an analysis the note had no room for, he stated the result: the plane turns by the Earth's own turn in the same time, multiplied by the sine of the latitude.³ The mathematician Binet proved the sine law in two notes in the same volume of the Comptes rendus.⁴

The surprise is that nobody had done it before. Foucault himself pointed out that his result agreed with a memoir that Poisson had read to the Académie in 1837 on projectiles, which deflect to the right in the northern hemisphere: the bob, he reasoned, is a projectile deflected one way going out and the other way coming back, and the pendulum's advantage is that it accumulates the effect, swing after swing.³ Yet Poisson had looked at the pendulum and concluded that the effect was too weak to matter.¹ Earlier still, in the 1660s, members of Florence's Accademia del Cimento had noted that a pendulum's swing veered. They did not connect it with the Earth's rotation; they treated it as a nuisance and hung their pendulums from two cords to stop it.^5,1

Foucault was an outsider. He had trained in medicine, earned his living reporting the Académie's sessions for a Paris newspaper, and taught himself physics; in 1850, a few weeks ahead of his former collaborator Hippolyte Fizeau, he had shown that light travels more slowly in water than in air.¹

Why does a free swing keep its plane? The bob feels two forces: gravity, straight down, and the tension of the wire, along the wire. Both lie in the vertical plane in which the bob is moving. Neither has any component out of that plane, so, seen from a frame that does not rotate, nothing can turn it. The wire's twist does not matter either: a point-like bob on a thin wire has nothing for a twist to grip, and Foucault checked that turning the support had a negligible effect. He had been inspired by a thin flexible rod clamped in a lathe, which kept vibrating in a fixed plane while the lathe turned it.¹

At the North Pole the picture is complete. The plane stays fixed relative to the stars, and the Earth turns underneath it, so the swing seems to turn clockwise, seen from above, once every 23.93 hours. Everywhere else, something subtler happens.

Chapter 3Only the vertical share

Foucault's note already contains the right picture. At the pole, the local vertical turns on itself. Moving towards the equator, the horizon tilts against the Earth's axis, and the vertical, in his words, "describes a cone more and more open" as the Earth turns.³ A floor whose vertical sweeps a cone is not simply spinning.

Angular velocities add like arrows, so split the Earth's spin arrow Ω, which points along the axis, into two parts at latitude φ:

a vertical share, Ω sin φ, along the local up. It turns the floor about the vertical, like a turntable under the pendulum;
a horizontal share, Ω cos φ, pointing north along the ground. It tips the floor about a north–south line, like a seesaw. That carries the vertical round, and the swing with it, but it never turns the swing about the vertical.

So the floor turns under the swing at Ω sin φ, that is 15.04° × sin φ an hour. At the poles, |sin φ| = 1 and a full circle takes 23.93 hours. In Paris, sin φ = 0.753. On the equator, sin φ = 0: the Earth's spin there is all seesaw and no turntable, and the swing never turns. South of the equator the vertical share points down, and the turn runs anticlockwise.

Place	Latitude	Turn per hour	A full circle takes	Seen from above
North Pole	90° N	15.04°	23.93 h	clockwise
Reykjavik	64.15° N	13.54°	26.6 h	clockwise
London	51.51° N	11.77°	30.6 h	clockwise
Paris (Panthéon)	48.85° N	11.33°	31.8 h	clockwise
New York	40.71° N	9.81°	36.7 h	clockwise
Singapore	1.35° N	0.35°	42 days	clockwise
On the equator	0°	0	never	no turn
Quito	0.22° S	0.06°	260 days	anticlockwise
Sydney	33.87° S	8.38°	42.9 h	anticlockwise
South Pole	90° S	15.04°	23.93 h	anticlockwise

This page's arithmetic: 15.0411° × |sin φ| per hour, and 23.9345 h ÷ |sin φ| for a full circle, for an ideal pendulum.

The pole row has been tried. In the southern winter of 2001, Mike Town, John Bird and Allan Baker hung a 25 kg bob on a 33-metre wire in the stairwell of the South Pole station then being built, and timed a full turn of 24 hours ± 50 minutes: a demonstration rather than a precision measurement, but it lands on the right row of the table.⁶

Figure 1 puts one pendulum on a turning globe, with the floor under it drawn from above. Drag the latitude from Paris to the North Pole and then to the equator, press Play to watch two days go by, and switch on the split to see the two shares of the spin.

Figure 1🌍 Try it

One pendulum, any latitude

Drag the latitude

Latitude

Hours watched

Swing turns, per hour

A full circle takes

Seen from above

Turned so far

The ideal Foucault pendulum at the chosen latitude φ, at Paris's longitude. Globe: the Earth turns at Ω = 7.292 × 10⁻⁵ rad/s (15.04° an hour, one turn per stellar day of 23.93 h); the small disc is the floor under the pendulum, exaggerated in size, and the arrows (Show the split) are the local vertical share Ω sin φ and horizontal share Ω cos φ of the Earth's spin. Floor dial: the swing direction relative to the ground, which turns at Ω sin φ, clockwise seen from above in the north and anticlockwise in the south; one line per hour, with the brass bob at the end it was released from. A full circle (360°) takes 23.93 h / |sin φ|; a swing line already matches its starting line after half that. Ideal pendulum: no air, a symmetric support, small swings (Figure 4 shows what happens otherwise). Rates are this page's arithmetic.Drag the latitude, or tap a place. The little disc on the globe is the floor under the pendulum, and the dial shows that floor from above: each line is where the swing pointed, one per hour, and the brass ball marks the end it was let go from. In Paris it gets only about three-quarters of the way round in a day. Try the North Pole and the equator, and switch on the split to see which share of the Earth's spin does the turning.

The pole takes 23.93 hours, not 24. On the equator the dial's hourly lines stay on top of one another, and the split shows the whole spin arrow lying flat. Cross the equator and the dial runs the other way. And in Paris, at 24 hours, a brass slice of 88.2° is left uncovered: the day that doesn't close.

One popular sentence needs repair. "The swing stays fixed relative to the stars while the Earth turns under it" is exactly true only at the poles. On the equator the swing direction turns with the Earth, relative to the stars, once a day; in Paris it partly does. The statement that holds everywhere is narrower: the swing direction is never turned about the local vertical. Chapter 4 shows what that implies.

In equations: take x east and y north on the floor, small swings, and let the wire's tension absorb the vertical forces (the horizontal share of Ω only produces vertical ones; the centrifugal term is folded into g). The Coriolis force then gives

x″ − 2W y′ + ω₀² x = 0, y″ + 2W x′ + ω₀² y = 0,
W = Ω sin φ, ω₀ = √(g/L)With u = x + iy the pair becomes u″ + 2iW u′ + ω₀²u = 0.

u(t) = e^−iWt (A e^iω′t + B e^−iω′t), ω′ = √(ω₀² + W²)The bracket is an ordinary pendulum swing; the factor e^−iWt turns it clockwise at the rate W, whatever the length, the mass or the size of the swing.

A numerical integration of these equations for the Panthéon pendulum, released from rest 3 metres out, gives a turn of 11.3251° an hour, matching Ω sin φ to every printed digit. Released from rest, the bob traces a rosette whose petals miss the centre by about aW/ω₀: 0.43 mm for a 3-metre swing. Pushed off from the centre instead, it traces a star through the centre. Both turn at the same rate in these equations (all this page's arithmetic). They assume small swings, a perfectly symmetric support and still air. The largest effect they leave out shows in the fourth digit: the rosette's thin ellipse picks up Airy's precession (Chapter 7), which slows the turn by the fraction (3/8)(a/L)², so a bob let go from rest 3 metres out turns at about 11.317° an hour; the star, which has no ellipse, escapes it (this page's arithmetic). Chapter 7 is about what happens when the support and the release are not perfect.

Foucault was not fully satisfied with his pendulum, Sommeria writes, because the public found the dependence on the sine of the latitude difficult to understand. On a suggestion of the mathematician Louis Poinsot he turned to a purer device, a fast-spinning rotor in gimbals, and named it the gyroscope, from the Greek for "rotation" and "to observe". The gimballed rotor itself was older: Johann Bohnenberger had built one in 1817 to illustrate the Earth's motion. The name, and the use of it to show the Earth turning, were Foucault's. Froment's rotor was spun up with a hand crank and four stages of gears to as much as 200 turns a second, kept spinning for 10 minutes, and its slow drift against the room was read with a microscope. The experiment was done in May 1852 and reported in three notes in the Comptes rendus.^7,1 A spinning rotor holds its axis fixed in space, not just against turning about the vertical: at Paris a horizontal axis drifts in azimuth by about 1.9° in 10 minutes against the ground (this page's arithmetic).

Part IIThe shortfall is an area

Chapter 4Eighty-nine degrees of area

The sine law gives the rate. It does not say why one turn of the Earth leaves the Paris swing 88.94° short. The answer is geometry, and the physicist Michael Berry uses the Foucault pendulum as his opening example of what he calls "global change without local change".⁸

Carry an arrow along a path on a curved surface, keeping it tangent to the surface and never letting it turn about the surface's normal. Mathematicians call this parallel transport. On a flat sheet, an arrow carried round any closed loop comes back pointing exactly as it started. On a sphere it does not. It comes back turned, and the turn equals the area the loop encloses, measured on a sphere of radius 1: the solid angle, in steradians, read as an angle in radians. (This is the Gauss–Bonnet theorem for a sphere, whose curvature is 1 everywhere.) The failure of the arrow to return is called the anholonomy; Berry notes that, according to the Oxford English Dictionary, the word was first used by Heinrich Hertz in 1894, and that geometers usually call it "holonomy", a usage he considers "a barbarism".⁸

The pendulum is such an arrow. Once every stellar day, the Earth carries it round its circle of latitude, and its swing direction is never turned about the local vertical (Chapters 2 and 3). As Berry puts it, the direction of swing is slaved to the vertical by parallel transport, and the angle between the initial and final swing directions "is equal to the solid angle subtended at the Earth's center" by the circle of latitude.⁸ For the cap north of latitude φ:

α = 2π(1 − sin φ) = 1.5523 sr = 88.94° at ParisThis page's arithmetic, with φ = 48.846°. The research behind this page also carried a vector numerically round the circle in small steps and recovered 88.940°.

On the real, slightly flattened Earth the rule survives with one refinement. The turn is the cap's total curvature: the solid angle swept by the plumb line, which at the Panthéon leans 41.15° from the Earth's axis, so the φ in the formula is the geodetic latitude, the one a plumb line feels, used throughout this page. Seen from the Earth's centre, Paris's circle sits at a geocentric latitude of 48.66° and would give 89.73°, and the cap's true share of the flattened surface, 12.43%, would give 89.47° (this page's arithmetic); on a sphere all three are the same number.

That is the same position as the ground's count of Chapter 1. Relative to the ground, the swing has turned clockwise by 360° sin φ = 271.06°. A clockwise turn of 271.06° and an anticlockwise turn of 88.94° leave a line in the same place, because together they make 360°. The two descriptions are one fact.

The cap north of Paris covers 12.35% of the sphere: on a sphere the size of the Earth, about 63 million square kilometres (this page's arithmetic). The limits check out. On the equator the circle of latitude is a great circle, which encloses half the sphere, 2π steradians or 360°, so the arrow comes back unchanged: the swing never turns. At the pole the circle shrinks to a point, nothing is enclosed, and the arrow keeps its direction against the stars while the floor turns a full 360° under it.

A walk makes the rule concrete. Stand at the North Pole holding a spear that points south along a meridian. Walk down that meridian to the equator without turning the spear. Turn left and walk a quarter of the way round the equator, still never turning the spear; it now points sideways to your path. Then walk back up to the pole. Your path had three corners of 90°: 270° in all, where a flat triangle would have 180°. Back at the pole, the spear points 90° away from where it started. The 90° excess is the area you walked round, one-eighth of the sphere, π/2 steradians (this page's arithmetic). On a flat sheet the angles of every triangle add to exactly 180°; whether space itself is flat, measured for the whole universe, is the subject of Nº 6, Chapter 2.

There is also a way to see the 88.94° without any calculus. Put a paper cone on the globe so that it touches the sphere all along Paris's circle of latitude, like a hat. Its apex lies on the Earth's axis, at a distance R/sin φ from the centre, and its slanting side is R cot φ long: for the Earth, about 5,570 km at Paris (this page's arithmetic). Along the circle the cone and the sphere share their tangent planes, so carrying the arrow round the circle on the sphere is the same as carrying it round on the cone. But a cone is flat: cut it along one side and it unrolls onto a table without stretching. Laid flat, it is a fan of radius R cot φ whose arc has the length of the circle, 2πR cos φ. So its opening angle is

2πR cos φ ÷ R cot φ = 2π sin φ = 271.06°, missing slice 360°(1 − sin φ) = 88.94°This page's arithmetic. At 30° the fan is a half disc, missing 180°, and the cap above 30° is a quarter of the sphere.

On the flat fan, parallel transport is ordinary parallelism: the arrow simply keeps pointing the same way on the table. Glue the two cut edges back together and the arrow, arriving at the far edge, meets its starting point turned by the missing slice. This tangent-cone picture is a classical one; it was used for the Foucault pendulum by Hart, Miller and Mills in 1987, among others, and extended by Krivoruchenko to the Thomas precession of spinning particles.^9,10 Oprea, and von Bergmann and von Bergmann, give other geometric treatments of the pendulum.^11,12

Figure 2 carries an arrow once round a circle of latitude, beside the same cone cut open and laid flat. Press Play, drag the latitude, and then switch to the triangle walk and guess the spear's turn before you press Play again.

Figure 2🍰 Try it

The missing slice is an area

Carry the arrow round

Latitude of the circle

How far round

Area enclosed

The arrow comes back turned by

Foucault's count in one turn of the Earth

Parallel transport on a unit sphere. Circle mode: a tangent vector carried once round the circle of latitude φ, never rotated about the local vertical, returns rotated by the enclosed solid angle 2π(1 − sin φ), which at Paris is 1.552 sr, or 88.94°; a Foucault pendulum's swing is carried the same way once per stellar day, so its turn relative to the ground, 360° sin φ = 271.06° clockwise, is the same position (mod 360°). Fan: the cone tangent to the sphere along the circle (slant cot φ) unrolled into a plane fan of opening 360° sin φ; on the fan, parallel transport is ordinary parallelism, and the missing slice, 360°(1 − sin φ), is the turn. Triangle mode: the geodesic triangle pole–equator–90° E–pole has three right angles; the vector turns by the excess, 90° = π/2 sr. This page's arithmetic; see Berry (1990) for the anholonomy, and Hart, Miller and Mills (1987) and Krivoruchenko (2009) for the tangent cone.Press play to carry the arrow once round the circle through Paris. It is never twisted, yet it comes back turned by 89°, and 89° is exactly the area of the cap above the circle. The fan is a paper cone that sits on the globe along that circle, cut open and laid flat: the missing slice is the same 89°. Try the triangle walk too.

The arrow is never turned, yet it arrives turned, and the readout's two numbers, the arrow's turn and the cap's area, agree at every latitude. The fan's missing slice is the same number again. On the triangle, the 90° excess over 180° is the turn.

This is the hallmark of a geometric phase. The turn depends only on the area the path encloses. It does not depend on how fast the Earth turns, how long the wire is, or how heavy the bob: a pendulum carried round Paris's circle in a year instead of a day would come back turned by the same 88.94°. Speed and length only enter through the imperfections of Chapter 7.

Chapter 5The same slice, in light

Light is a wave whose electric field points across its direction of travel. For linearly polarised light, that field direction plays the part of the pendulum's swing line. In a glass fibre that bends gently, the polarisation is carried along without being turned about the fibre's direction: the theory goes back to Sergei Rytov in 1938 and Vassily Vladimirskii in 1941, as Berry recounts.^13,8

This is parallel transport again, on a different sphere. Draw the fibre's direction at each point as a point on a sphere of radius 1, the sphere of directions. The polarisation is always perpendicular to that direction, so it is an arrow tangent to the sphere. As light travels along a coiled fibre, its direction traces a loop on the sphere, and the polarisation comes out turned by the solid angle inside the loop. For a helix whose tangent makes an angle θ with the helix's axis, the direction traces a circle at θ from the pole, enclosing 2π(1 − cos θ) per turn of the coil: 48.2° per turn at θ = 30°, 105.4° at 45° and 180° at 60° (this page's arithmetic).

Akira Tomita and Raymond Chiao measured exactly this in 1986, with a fibre wound into a helix, and titled their paper "Observation of Berry's topological phase by use of an optical fiber"; Chiao and Yong-Shi Wu gave the theory in the same issue of Physical Review Letters.^14,15 The rotation is the solid angle through which the fibre's tangent has turned.⁸ So any change to the coil that keeps that solid angle, such as a wider or narrower coil at the same lean, leaves the rotation alone.

The history deserves to be told straight. Berry notes that J. Neil Ross, at the Central Electricity Generating Board's laboratories in Leatherhead, England, had demonstrated the rotation earlier, in a 1984 experiment, implicitly assuming the parallel-transport law; and that Chiao's group themselves pointed out that "this appears to be a phenomenon of classical optics".^8,13 The 1986 experiment framed and tested the effect as Berry's phase; the effect itself had been anticipated decades earlier.

Now the exact match. A fibre whose tangent leans 41.15° from its coil's axis turns light, per turn of the coil, exactly as a Paris pendulum turns in one turn of the Earth: 88.94°. The reason is that 41.15° = 90° − 48.85°, and cos 41.15° = sin 48.85°. Over a day, Paris's local vertical sweeps round the Earth's axis at 41.15° from it: Foucault's "cone more and more open". The fibre's direction sweeps a cone of the same opening, and the same cap on the sphere of directions (this page's arithmetic).

Figure 3 shows a coil and, beside it, the sphere of directions. Lean the coil, choose the number of turns and send the light through. Start at "Like Paris", then try 60° and the flat ring.

Figure 3🌀 Try it

The same slice, in light

Lean the coil

How far the fibre leans from the coil's axis

Turns of the coil

Area on the sphere of directions, per turn

Polarisation turns by

Matches a pendulum's shortfall per turn of the Earth at

A right-handed helical fibre whose tangent makes angle θ with the helix axis. The tangent direction traces a circle of polar angle θ on the sphere of directions (right), enclosing 2π(1 − cos θ) per turn, and the polarisation, carried along the fibre without rotation about it (Rytov 1938, Vladimirskii 1941), turns by that solid angle: Tomita and Chiao (1986). The rotation depends only on the solid angle, not on the coil's radius or length; at θ = 41.15° = 90° − 48.85° it equals a Paris pendulum's shortfall per stellar day. The bar on the sphere is the same polarisation, carried round the circle. Linear polarisation looks the same after 180°; a left-handed coil turns it the other way. This page's arithmetic; the coil is not to scale. Ideal fibre: no birefringence (in a real coil, bending adds some).Light wiggles in one direction, its polarisation (the bar). Send it through the coiled fibre and watch the bar come out turned. On the right, the dot shows which way the fibre points as the light goes round: it draws a circle, and the area inside sets the turn. Lean the coil to the Paris setting: same 89° as the pendulum.

At 41.15° the bar comes out turned by 88.94°, as the pendulum does. At 60° it turns 180° per turn, which for a line looks like no turn at all: a polarisation direction, like a swing line, has no arrowhead. A flat ring (θ = 90°) sweeps the fibre's direction round a whole great circle, half the sphere, and turns the polarisation 360°, visibly nothing; a straight fibre (θ = 0°) sweeps no area and turns nothing. They are the equator and the pole of Chapter 3, in glass.

Chapter 6Berry's phase, Hannay's angle

The pendulum and the fibre belong to a family, and one of its early members was found by a 22-year-old. In 1956 S. Pancharatnam, of Bangalore, a nephew of C. V. Raman, worked out how to compare the phases of two beams in different states of polarisation. He found that a beam taken round a cycle of polarisation states, each step "in phase" with the last, can return with a different phase: half the solid angle the cycle encloses on the Poincaré sphere of polarisation states.^16,8

In a paper published in 1984, Berry found the quantum version in general. A quantum system whose surroundings are changed slowly round a closed loop, for example a spin in a magnetic field whose direction is slowly taken round a circuit, returns to its starting state with an extra phase on top of the usual one. The extra phase depends only on the geometry of the loop. For a spin state with component s along the field, it is −s times the solid angle the field direction swept: half the solid angle, with a sign, for a spin-½ neutron.^17,8 Barry Simon pointed out in 1983, working from Berry's preprint, that the extra phase is a holonomy: the same kind of turn as the spear's, in the space of quantum states.¹⁷ In 1987 Bitter and Dubbers saw it in the spin rotation of neutrons.¹⁸

The classical member came next. In 1985 J. H. Hannay showed that a classical oscillating system carried slowly round a loop picks up a geometric shift in its angle variables, and Berry related this angle to the quantum phase.^19,20 The Foucault pendulum is the textbook example: its 88.94° is the classical cousin of Berry's phase. The geometry is the same in all of them; what changes is the object carried round, a swing line, a polarisation, a quantum phase.

The ocean feels the same vertical share. Water set moving by a gust of wind and then left alone circles under the Coriolis force at the frequency f = 2Ω sin φ. Its period, the inertial period, is a stellar day divided by 2 sin φ, which the oceanographer Robert Stewart's textbook describes as one half the time a local plane on the Earth takes to rotate. At Paris's latitude that is 15.9 hours (this page's arithmetic), half a pendulum day; the circling runs clockwise in the northern hemisphere, the same sense as the pendulum. Stewart calls inertial currents the most common currents in the ocean.²¹

Part IIIReal pendulums, other proofs

Chapter 7Why Foucault needed 67 metres

If the rate does not depend on the length, why hang 67 metres of wire? Sommeria gives the reasons: a long pendulum reduces the influence of spurious perturbations and gives a larger step at each swing, and a heavy, compact bob suffers less from air drag.¹ The deeper reason is that real pendulums lie, and short ones lie more.

The first lie is built into any pendulum. In 1851 the astronomer G. B. Airy showed that a spherical pendulum whose bob traces a thin ellipse instead of a straight line turns that ellipse by itself, in the same sense as the bob goes round it, at a rate that grows with the ellipse's area.²²

Ω_A = (3/8) ω₀ ab / L²a and b: the semi-major and semi-minor axes of the ellipse; L: the length; ω₀ = √(g/L). Airy's result as restated by Salva and Salva (2025), who give the time for a full Airy turn as 8L²T/(3ab).

This turning has nothing to do with the Earth. Set it equal to the Earth's term W = Ω sin φ and you get the ellipse that would fake the whole Foucault effect: a semi-minor axis b* = 8WL²/(3ω₀a). For the Panthéon with a 2.5-metre swing, b* is 687 mm (573 mm with a 3-metre swing); for the 11-metre Observatory pendulum swinging 1 metre, 19 mm; for a 2-metre cellar pendulum swinging 20 cm, 1.32 mm (this page's arithmetic). The cellar pendulum's bob needs to stray only 1.3 mm to each side of its line to double or cancel the Earth's effect. At the same ratio of swing to length, the tolerance relative to the swing grows as √L, so a 67-metre pendulum forgives about √(67/2) ≈ 5.8 times more; with the swings assumed here, 0.27 of the swing against 0.0066, about forty times more (this page's arithmetic).

The Coriolis force itself makes only a tiny ellipse, about aW/ω₀, some 0.4 mm for the Panthéon (Chapter 3). Even that ellipse has Airy's precession, which slows the turn by the fraction (3/8)(a/L)²: 0.05% for the Panthéon swinging 2.5 metres, 0.4% for a 2-metre pendulum swinging 20 cm (this page's arithmetic). Figure 4 leaves it out. The danger is everything else: a sideways nudge at release (hence Foucault's burned thread), air currents, and the support.

The second lie is the support. The first thorough theory of the imperfect pendulum came in 1879, in the doctoral thesis of Heike Kamerlingh Onnes, who later liquefied helium and discovered superconductivity. He had noticed that a swing which starts as a straight line turns as expected but, after typically an hour, "deteriorates" into an ellipse. He explained it by a slight difference between the pendulum's natural frequencies along two horizontal directions, from a geometrical flaw or the elasticity of the support, which makes the swing beat between two modes; with his analysis he corrected his own pendulum.^1,23

How much difference is too much? The research behind this page integrated the equations of motion with two slightly different frequencies, for small swings. For those, the threshold is a fractional difference Δω/ω₀ = 2W/ω₀. Below it, the swing still turns round, but swells into an ellipse with b/a ≈ Δω/2W. Above it, the swing locks to the support's axes and only rocks to and fro (by ±23° at 1.39 times the threshold). For the 67-metre pendulum the threshold is 287 parts per million: the two periods must agree to 4.7 ms out of 16.42 s. For 11 metres it is 116 ppm (0.77 ms out of 6.65 s); for 2 metres, 50 ppm (0.14 ms out of 2.84 s) (this page's arithmetic). With swings as large as Foucault's, Airy's precession turns the ellipse that the support opens before it can lock, as Figure 4 shows.

Figure 4 combines both lies. It uses this page's slow-amplitude model: write the swing as u = A e^iω₀t + B e^−iω₀t, two circular motions in opposite senses whose amplitudes drift slowly,

dA/dt = −iW A + iΩ_AA + iκB̄, dB/dt = −iW B + iΩ_AB − iκĀ,
Ω_A = (3/8)(ω₀/L²)(|A|² − |B|²), κ = (ω₁ − ω₂)/2The ellipse has semi-axes |A| + |B| and ||A| − |B||, and its long axis points at half the sum of the phases of A and B. The model reproduces the full integration to 0.1°. It leaves out air drag, the slow loss of amplitude and the Coriolis-made ellipse. The swings (2.5 m, 1 m and 20 cm) are this page's assumptions. Foucault's own cellar pendulum swung wider, through an arc of 15 to 20 degrees (Chapter 2), at least 26 cm each way, so 20 cm flatters the short pendulum: at 30 cm the stray that doubles or cancels its turn shrinks from 1.32 mm to 0.88 mm (this page's arithmetic).

Give the Panthéon pendulum and a 2-metre cellar pendulum the same tiny oval, a bob straying 2 mm to each side, and the same support, and watch 32 hours. Then reverse the sense of the loop, and make the support a little lopsided.

Figure 4📏 Try it

Why Foucault needed 67 metres

Same flaw, two lengths

Panthéon: 67 m wire, 2.5 m swing

a 687 mm stray doubles or cancels its turn

Cellar: 2 m wire, 20 cm swing

a 1.3 mm stray doubles or cancels its turn

Oval: how far the swing strays to each side

The bob loops

Support stiffer one way (swing-time difference, ppm)

Hours watched

Panthéon, 67 m

Cellar, 2 m

The Earth alone

Top view of the swing's ellipse every 20 minutes for two Paris pendulums with the same flaws, from this page's slow-amplitude model (equations in the text), checked against a direct integration of the full pendulum. Oval: the semi-minor axis at release, with the bob looping in the chosen sense; Airy's precession, (3/8)ω₀ab/L², turns the ellipse in that sense, and a stray of 8WL²/(3ω₀a) (687 mm, 19 mm and 1.32 mm for the three pendulums) doubles or cancels the Earth's turn. Support: the two horizontal frequencies differ by the chosen fraction (east–west stiffer; released east–west). For small swings the line would lock to the support above 2W/ω₀ (287 ppm for 67 m, 116 ppm for 11 m, 50 ppm for 2 m; Kamerlingh Onnes, 1879); with these swings, Airy's precession turns the ellipse that the support opens before it can lock, so the line whirls round too fast instead, and only the Panthéon pendulum locks, above about 550 ppm (this page's model). Swing amplitudes are this page's assumptions (2.5 m, 1 m, 0.2 m); each panel is scaled to its own swing. Dashed: the Earth alone, 11.33° an hour. Rates are averages over the hours watched. No air drag; real pendulums also lose amplitude.Two pendulums in Paris, seen from above: Foucault's 67 m one and a 2 m one. Give both the same tiny flaw, a swing that strays 2 mm to the side, and press play. The dashed line is where the Earth alone would turn them. Then make the support a little lopsided and see which one goes wrong first.

With the default 2 mm oval, looping anticlockwise, the Panthéon pendulum turns at 11.29° an hour, 0.3% off the Earth's 11.33°. The cellar pendulum turns the wrong way, at 5.80° an hour anticlockwise; looping clockwise instead, it turns at 28.45° an hour, two and a half times the Earth's rate. The 11-metre Observatory pendulum lands in between, 11% off. With no oval, a support lopsided by 100 ppm makes both pendulums lie more: the ellipse it opens picks up Airy's precession, so the Panthéon pendulum turns at 16.3° an hour, 44% too fast, and the cellar pendulum whirls round at about 96° an hour, more than eight times the Earth's rate. With these swings only the Panthéon pendulum locks, above about 550 ppm (this page's model, checked against a direct integration of the full pendulum). Length buys honesty, not a different rate.

Modern museum pendulums cheat back. They are kept swinging by a magnet on the pendulum that receives impulses from a coil driven by a feedback circuit, and a mechanical device damps the growth of the ellipse; the turn then matches the Earth's rotation "with a precision of the order of 1%", Sommeria writes, citing a 2010 design by Salva and colleagues.^1,24 In 2025 Salva and Salva measured a 4.8-metre test pendulum and modelled the Airy and Coriolis precessions together with the support's two periods: for certain swing amplitudes the precession can stop entirely while the pendulum keeps swinging.²²

Chapter 8Before and after Foucault

By the end of the seventeenth century, Sommeria writes, scientists had no real doubt that the Earth turns. The evidence was indirect. In 1672 the astronomer Jean Richer found that a pendulum swings more slowly at Cayenne, near the equator, than in France, which Huygens explained by the centrifugal force; Newton predicted that the spinning Earth must be flattened at the poles, and Maupertuis's expedition to Lapland measured the flattening in 1736–37 and published it in 1738.¹ What was missing was a direct proof. Laplace wrote in 1805 that, although the rotation was established, a direct proof should interest mathematicians and astronomers. His treatise on celestial mechanics already contained the mathematical form of the force that Gaspard-Gustave Coriolis would describe in 1835; Coriolis died in 1843, and nobody in 1851 connected his work with the pendulum.^1,25,26

The search for a direct proof had focused on falling bodies. A ball dropped from a height starts out moving east a little faster than the ground below it, because it is farther from the Earth's axis, so it should land slightly east of the point straight below. Giovanni Borelli argued this in the 1660s, qualitatively. Guglielmini measured deviations east and south for lead balls dropped inside Bologna's Asinelli Tower at the end of the eighteenth century, and Benzenberg repeated such drops in 1802 and 1804.^27,1 The best known, Sommeria writes, are Ferdinand Reich's, made in 1831 in a 158-metre mine shaft at Freiberg in Saxony and published in 1832. Over 106 drops, the mean eastward deviation was 2.8 cm, "consistent with theory", but the scatter around it was larger than the mean.^28,1 The theory, (1/3) Ω g cos φ t³ without air drag, gives 27.5 mm (this page's arithmetic). Right on average and unconvincing drop by drop: that is why a pendulum you could watch, accumulating its effect swing by swing, mattered.

In 1913 Arthur Holly Compton, then at the University of Wooster and later a Nobel laureate, published "A laboratory method of demonstrating the Earth's rotation". Foucault's pendulum, he wrote, is "inapplicable in many laboratories" because there is no "convenient place to hang a sufficiently long and heavy pendulum". He bent glass tubing with a 1.3 cm bore into a ring 99.3 cm in radius, about 2 metres across, filled it with water seeded with drops of oil of the same density, and held it horizontal until the water was still. Then he flipped the ring over through 180° about an east–west axis. The water keeps its share of the Earth's motion, so it drifts round the tube; Compton followed an oil globule through a micrometer microscope. For a horizontal ring his formula is V = Ωr sin φ, the same sine as the pendulum's. He measured 0.0513 mm a second against the 0.0484 mm a second he computed, "a difference of 5 per cent".²⁹ (A frictionless fluid would reach twice the value of his formula, 2Ωr sin φ, by this page's arithmetic, so his close agreement shows the sense and the size of the effect rather than a precise test.)

Neither the pendulum nor Compton's ring is the tabletop proof it is sometimes called. A Foucault pendulum needs height (Foucault's shortest was 2 metres), and Compton's ring, about 2 metres across, was a laboratory alternative to it.

Light came next. Two beams sent opposite ways round a loop that rotates arrive at slightly different times: the Sagnac effect. In 1925 Albert Michelson and Henry Gale laid out a rectangle of 12-inch pipe 2,010 by 1,113 feet (612.6 by 339.2 metres) on open ground at Clearing, Illinois, and sent light both ways round it. They observed a shift of 0.230 ± 0.005 of an interference fringe, against 0.236 ± 0.002 computed for the Earth's rotation.^30,31 The difference in arrival times was about 0.45 femtoseconds (this page's arithmetic, Δt = 4AΩ sin φ/c², the same sine again). It is often called the Michelson–Gale–Pearson experiment; the paper's authors are Michelson and Gale.

Matter waves followed light. In 1979 Werner, Staudenmann and Colella saw the effect of the Earth's rotation on the quantum-mechanical phase of neutrons, and in 1997 Schwab, Bruckner and Packard detected it using the phase coherence of superfluid helium.^32,33 In 2024 a team in Vienna did it with maximally path-entangled photons in a fibre coil enclosing an effective 715 square metres, in what they describe as a table-top experiment. Their sensitivity, 5 microradians per second, is about 7% of the Earth's rate (this page's arithmetic); the authors call it "the highest rotation resolution ever reached with optical quantum interferometers".³⁴ Cold atoms have gone much further. In 2022 a team at the Paris Observatory, the institution in whose meridian hall Foucault's 11-metre pendulum had swung, sent caesium atoms through an interferometer enclosing 11 square centimetres and found the phase shift caused by the Earth's rotation in agreement with theory to 25 parts per million.³⁵

And the bathtub? The Coriolis acceleration on water moving at 1 cm a second in Paris is about 1.1 × 10⁻⁶ m/s², a ten-millionth of g. For a 30 cm basin draining at that speed, the ratio of the water's own swirl to the Earth's (the Rossby number) is about 300 (this page's arithmetic): the shape of the basin and the way it was filled win easily. Convincing evidence of a Coriolis-driven drain vortex came only in 1908, from Ottokar Tumlirz, and it needs very quiet initial conditions in a metre-sized tank, not a home bath.^36,1 Ascher Shapiro demonstrated it in 1962, in what Sommeria calls a classic educational film, and Trefethen and colleagues repeated it in the Southern Hemisphere in 1965.^37,38

Part IVWatching the Earth spin today

Chapter 9A ring of light under a Bavarian field

A ring laser turns the Sagnac effect into a musical note. Mirrors at the corners of a closed loop make a laser cavity in which two beams run in opposite directions. When the loop rotates, the two beams need slightly different frequencies to fit the cavity, and they beat against each other at

Δf = 4AΩ_n / (λP)A: the enclosed area; P: the perimeter; λ: the laser wavelength; Ω_n: the component of the rotation along the loop's normal. For a horizontal ring that component is Ω sin φ, exactly the pendulum's share.

The ring in this story is the "G" ring at the Geodetic Observatory Wettzell in Bavaria: a horizontal square, 4 metres on a side, in an underground laboratory.^39,40 Its two beams beat at about 348.5 hertz, close to the F above middle C, and a change of 4 ppb in the Earth's spin moves that note by about 1.4 microhertz (this page's arithmetic, for a 16 m² square of red helium–neon light at 49.1° N). It measures rotation against local inertial space, with no stars or quasars in sight. In 2023 Ulrich Schreiber of the Technical University of Munich and colleagues reported that it followed variations in the Earth's rotation rate of about five parts per billion over three months of continuous running, each data point averaging more than three hours, and overlaid them on the data of the International Earth Rotation and Reference Systems Service (IERS).^41,42

In September 2025 the same team reported 250 days of operation, with the laser inside a pressure-stabilising vessel. Because the ring measures rotation relative to local inertial space, the slow precession and the nutation of the Earth's axis are contained in the signal. Optical interferometry, they write, "yields an accuracy limit for rotation sensing of 48 parts per billion", and over about three hours the measurements are steady to about 4 parts per billion (an Allan deviation of 4 × 10⁻⁹).⁴⁰ The dominant error is light scattered at the mirrors from one beam into the other.⁴⁰ The press office put the time resolution at "less than an hour instead of a day", without a network of radio telescopes on different continents.³⁹ Note the words: 48 ppb is an accuracy limit over months, not a short-term precision. In a May 2026 conference abstract, not yet a paper, the team adds that G now runs stably for more than a year.⁴⁰

What does a part per billion of spin mean? A day that is longer than 86,400 seconds by ΔLOD corresponds to a spin slower by ΔΩ/Ω = −ΔLOD/86,400 s. So a day 1 ms longer is the Earth spinning 11.6 parts per billion slower; the ring's 4 ppb in three hours is 0.35 ms of day length, its 2023 figure of 5 ppb is 0.43 ms, and the 48 ppb limit is 4.1 ms (all this page's arithmetic). The Moon's fortnightly tide changes the length of the day by about 0.36 ms either way, which is 4.2 ppb, and the day's average lengthening over the past 2,700 years, about 1.8 ms a century (tidal braking alone would give 2.3 ms), is 21 ppb a century.^2,43 Leap seconds made up for the slow part of such changes, the day's excess over 86,400 SI seconds added up day after day; they are what the General Conference on Weights and Measures voted on 15 October 2026 to end from 20 May 2027; none is due before then, so the leap second of 31 December 2016 is set to be the last (see Nº 9, Chapter 9).⁴⁴

The honest summary: a ring laser can see the day lengthen or shorten by a fraction of a millisecond within hours. It cannot keep the Earth's clock by itself. A ring laser measures the rate of rotation; UT1, the time scale set by the Earth's actual rotation angle, is an angle, and it is kept by very-long-baseline interferometry (VLBI), radio telescopes on several continents timing distant quasars. The two have been compared side by side: during a 2017 VLBI campaign, the G ring's rotation data were set against VLBI's.⁴⁵

Figure 5 places the main experiments of Chapter 8 and this chapter on one scale: how finely each pinned down the spin, as a share of the Earth's rate. Tap an experiment to see what it measured, then slide the day longer and watch where the line falls against the ring laser.

Figure 5🔦 Try it

From the right sign to parts per billion

Tap an experiment

Make the day longer by

The Earth spins slower by

Does the 2025 ring laser see it within 3 hours?

Adds up to one second in

Each point is the experiment's own figure of merit as a fraction of the Earth's rotation rate: for Reich, the scatter of single drops (his 106-drop average was better); for Compton, the 5 per cent he claimed, with a bar up to the 47 per cent by which he fell short of a frictionless fluid (this page's arithmetic); for Michelson and Gale, the stated uncertainty and agreement with prediction; for modern instruments, the stated resolution (entangled photons, 5 µrad/s), agreement with prediction (cold atoms, 25 ppm) or stability and accuracy (Wettzell G: about 5 ppb over three months in 2023; in 2025, an Allan deviation of 4 × 10⁻⁹ at 3 hours and an accuracy limit of 48 ppb over 250 days). These are not the same quantity, so the chart shows orders of magnitude; the 2023 and 2025 ring-laser marks are nudged apart sideways so both can be seen. Lines: the spin change that matches a day longer by the chosen amount (1 ms = 11.6 ppb); third tile: the days for that excess over 86,400 s to add up to one second (this page's arithmetic); frame dragging at the surface, about 0.46 ppb of the Earth term a horizontal ring sees, and the de Sitter precession of the Earth's orbit, 0.04 ppb (this page's estimates). Foucault's pendulum (1851) and gyroscope (1852) showed the sense and rate; no figure of merit is plotted. Sources in Chapters 8 and 9.Each dot is a way people felt the Earth spin, placed by how finely it pinned the spin down: lower is better. Tap a dot to see what it was. Slide the day longer: a day 1 millisecond longer is the Earth spinning about 12 parts in a billion slower, and the ring laser can see that within hours.

From Reich's balls to the Wettzell ring, the points drop by about eight powers of ten. The line for a day 1 ms longer sits above the ring's three-hour floor, so the ring sees such a change within hours; the fortnightly tide sits right at that floor. The points are not the same kind of number (scatter, stated uncertainty, resolution, stability, accuracy), so the chart compares orders of magnitude, not fine detail.

The work goes on. On 27 March 2026 the German Research Foundation announced a new research unit, RING, led from LMU Munich with the geophysicist Heiner Igel as spokesperson, with about four million euros for its first four years (the Foundation's research units can run for up to eight), built on the ROMY ring laser at Fürstenfeldbruck and the Wettzell ring.⁴⁶ That is funding for future measurements, not a result.

Chapter 10Holding still, relative to what?

At the pole, this page has said, the swing holds still relative to the stars. Strictly, it holds still relative to the local inertial frame, the frame in which free bodies move in straight lines nearby. General relativity says that frame is not quite fixed to the distant universe. A spinning mass drags it round a little (frame dragging, or the Lense–Thirring effect), and a gyroscope carried round a mass on a curved path is turned by the curvature of spacetime (the geodetic precession).

At the Earth's surface, Sommeria writes, a gyroscope is entrained by the Earth's rotation at 0.2 arcseconds a year.¹ This page's estimate is 0.22 arcseconds a year at the pole; for a horizontal ring laser at any latitude it is about 0.46 parts per billion of the Earth term the ring sees. The geodetic precession of the Earth's own orbit round the Sun is smaller still, about 0.003 picoradians a second (19 milliarcseconds a year), 0.04 ppb (this page's estimates). The Wettzell team's 2025 abstract puts their accuracy "less than an order of magnitude away from the regime in which relativistic effects can be measured", and their press office says a tenfold improvement would let them measure the Earth's frame dragging.^40,39 By this page's estimate the gap for that steady term is wider: 0.46 ppb is about a hundredth of the 48 ppb accuracy limit. In the paper itself the team's yardstick is the noise floor, which they put about ten times above their estimate of the de Sitter term, 0.003 picoradians a second. They also note that matter-wave and entanglement experiments are not within six orders of magnitude of what such a measurement needs.⁴⁰

In orbit, it has been done. Gravity Probe B, launched on 20 April 2004, carried four cryogenic gyroscopes in Earth orbit and collected data until 14 August 2005. The final analysis, published in 2011, gave a geodetic drift of −6,601.8 ± 18.3 milliarcseconds a year against the predicted −6,606.1, and a frame-dragging drift of −37.2 ± 7.2 against −39.2: uncertainties of about 0.3% and 19% of the measured values (this page's arithmetic).⁴⁷ Laser-ranged satellites have seen the Earth drag their orbits too: the two LAGEOS satellites in 2004, and LAGEOS with LARES in 2019, with a claimed agreement of a few per cent, though how far their error budgets can be trusted is still argued over.⁴⁷ The Wettzell team compares the two instruments directly: after 10 hours of averaging, Gravity Probe B's gyroscopes resolved 1.35 × 10⁻¹³ radians a second and the G ring resolves 3 × 10⁻¹³, about a factor of 2 apart, with the ring on the ground.⁴⁰ The geodetic precession is parallel transport in curved spacetime: the same idea as the pendulum's cap, with the curvature of spacetime in place of the curvature of the sphere. It is also called the de Sitter precession, after the Dutch astronomer Willem de Sitter, whose name is also on the ever-expanding universe that Nº 3 follows to its end.

Back under the dome of the Panthéon, nothing turns the swing. The floor turns, and the curve of the Earth keeps the count: a slice of 88.94°, every time the Earth turns once.

Chapter 11Glossary and sources

Stellar day: One turn of the Earth relative to the distant stars: 86,164.0989 s, about 23 h 56 min 4.1 s. The pendulum keeps this day.
Sidereal day: One turn relative to the equinox, 8.4 ms shorter than the stellar day because the equinox drifts. Both are "about 23.93 hours".
Latitude, φ: The angle of a place north (positive) or south (negative) of the equator. The swing turns at 15.04° × sin φ an hour.
Swing plane, swing line: The vertical plane in which the pendulum swings, and the line it traces on the floor.
Parallel transport: Carrying an arrow along a path on a surface, keeping it tangent and never turning it about the surface's normal.
Anholonomy (holonomy): The turn an arrow picks up when it is parallel-transported round a closed loop. On a sphere of radius 1 it equals the enclosed area.
Solid angle, steradian: An area on a sphere of radius 1, measured in steradians (sr). The whole sphere is 4π sr; the cap north of Paris is 1.552 sr.
Geometric phase: A shift that depends only on the geometry of the loop a system is carried round, not on how fast.
Berry phase: The geometric phase of a quantum state carried slowly round a loop of its surroundings (Berry, 1984).
Hannay angle: The classical counterpart: a geometric shift in the angle of an oscillating system carried slowly round a loop (Hannay, 1985).
Coriolis force: The apparent sideways force on a body moving in a rotating frame, −2mΩ × v.
Airy precession: The turning of an elliptical pendulum swing by itself, in the sense of the bob's loop, at (3/8)ω₀ab/L².
Sagnac effect: The time or frequency difference between two light beams going opposite ways round a rotating loop.
Ring laser: A laser whose cavity is a closed loop; rotation makes its two counter-running beams beat at a frequency proportional to the rotation rate.
VLBI: Very-long-baseline interferometry: radio telescopes on several continents timing distant quasars. It is the only technique that measures UT1 directly, and the standard one for the nutation, which the Wettzell ring now also senses (Chapter 9); satellite methods supply most of the polar-motion data.
UT1: The time scale set by the Earth's actual rotation angle.
Length of day (LOD): The excess of the day over 86,400 SI seconds; a day 1 ms longer means a spin 11.6 ppb slower.
Parts per billion (ppb): Here, of the Earth's rotation rate: 1 ppb is 0.073 picoradians a second.
Frame dragging: The turning of local inertial frames by a spinning mass (the Lense–Thirring effect).
Geodetic (de Sitter) precession: The turning of a gyroscope carried round a curved path in curved spacetime.

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31.8 hours

31.8 hours

Only the up-and-down share counts

The missing slice is an area

Light does it too

Why the wire was 67 metres

Other ways to feel the spin

A ring of light that feels the day stretch

Holding still, compared with what?

Thirty-one point eight hours