The disc that cannot be spun

Length contraction is a statement about a measurement: a rod moving along its own length is measured shorter than the same rod at rest, and a rod moving across its length is not. Both halves of that are needed for what follows.

Take a disc and set it turning. Every element of the rim moves along the circumference, which is its own direction of motion. Every element of a radius moves across the radius.

So a measurement of the circumference made with rulers carried on the rim takes more rulers than a stationary observer counts, and a measurement of the radius takes the same number. The ratio is $\gamma$, and at a rim speed of nine-tenths of light it is $2.294$.

What is and is not being claimed #

Two things have to be separated at once, because the difficulty of this problem for a good many years was that they were not.

The spacetime is exactly flat. Nothing is curved, there is no gravitational field, and every event can be labelled with the coordinates of an inertial frame in which special relativity holds in its simplest form. The disc is a material object in flat spacetime.

The geometry a rotating observer measures is not Euclidean. That observer, making local measurements with rulers at rest with respect to the disc, builds up a three-dimensional geometry in which the circumference of a circle is $\gamma$ times $2\pi r$. The excess is real, in the sense that a rotating surveyor would find it.

The two are compatible because “the geometry a rotating observer measures” is not a slice through spacetime. It is what is got by adding up local measurements, and the local measurements cannot be assembled into a surface of simultaneity.

Why the slices do not close #

The obstruction is worth seeing, because it is the same one a ring interferometer detects.

Two beams sent opposite ways round a rotating ring return at different times, by an amount proportional to the enclosed area and the rotation rate. That difference is why the slices do not close: synchronising clocks neighbour to neighbour once round the rim arrives back at the start holding a time that disagrees with the one it set out from. There is no consistent “now” around a rotating loop, and the failure is measurable rather than philosophical — it is what a ring-laser gyroscope reads.

Synchronise a clock on the rim with its neighbour, and that one with the next, all the way round. Each synchronisation is local and each is done by the standard method, so each is unimpeachable. Come back to the starting clock and it disagrees with itself, by an amount proportional to the rotation rate and the enclosed area.

There is therefore no such thing as “the disc at one instant”. The rotating frame has no global time, and the three-dimensional geometry described above is what is obtained by giving up on one and using local measurements only.

That failure is not a defect of the procedure. It is the same quantity the Sagnac effect measures, and a fibre-optic gyroscope is a device for reading it.

The argument that took thirty years #

Ehrenfest stated the difficulty in 1909 in three sentences, and it was argued about until well after the second world war.

His formulation was a contradiction: the circumference must contract because it moves along itself, and it must not, because the radius is unchanged and the disc is a rigid body. Something had to give and it was not clear what.

The positions taken over the following decades included that the circumference does not contract; that the radius does contract; that the disc’s material rearranges itself; that the whole question is meaningless; and — from Einstein — that the geometry of the rotating frame is simply not Euclidean and that this is a fact about accelerated frames rather than a paradox.

Two things eventually settled it. The first was Herglotz and Noether’s theorem, in 1910, which showed that no Born-rigid motion can change a body’s rotation rate — so the premise “the disc is rigid” was not available and the contradiction had never been between two facts.

The second took longer and is about definitions. “The circumference of the rotating disc” is not a well-defined quantity until it is said who measures it, with what, and at what time — and in a rotating frame the last of those has no answer. Once the question is stated precisely, each version of it has a definite and uncontroversial answer, and the versions disagree with each other because they are different questions.

That is the shape of a great many relativistic paradoxes. What looks like a conflict between two results is usually a single phrase that has been used as though it named one thing.

Nothing can be spun up #

The most concrete consequence is about the material, and it forbids something rather than merely complicating it.

A disc at rest has a circumference of $2\pi r$ measured by rulers at rest with it. A disc rotating has a circumference of $2\pi r\gamma$ measured by rulers at rest with it. So between the two states every circle of material has had to change length in its own terms — by $8.3$ per cent at a rim reaching four-tenths of light speed, by $56.4$ per cent at nine-tenths — while staying the same distance from the axis.

That means the material is strained, necessarily, and by an amount that depends on radius. A body that keeps every internal distance fixed as measured by its own parts is called Born-rigid, and the theorem — Herglotz and Noether’s, from 1910 — is that a Born-rigid body has only three degrees of freedom rather than six and cannot change its rotation rate at all.

So a rigid disc is not a body that would be difficult to spin. It is a body that cannot exist.

That is a stronger statement than the one about the string between two accelerating rockets, and it has the same shape: a requirement of the geometry that no amount of engineering can satisfy, mistaken at first for a paradox about which of two answers is right.

There is a compact way of stating the whole result that avoids every ambiguity, and it is the one a modern treatment uses. The rotating frame’s spacetime metric is written down — it is flat spacetime in rotating coordinates, one line of algebra — and the spatial geometry a local observer measures is read off it as the part orthogonal to the observer’s worldline. That construction gives $2\pi\gamma$ for the circumference, gives the clock rates, and gives the failure of the clocks to close, all from the same expression. What the century of argument was about is what that construction means, and the answer is that it means exactly what it computes: the results of local measurements, which do not assemble into a slice.

Everything about ordinary rotation survives unchanged, and it is worth saying what does not: the assumption that the body can be treated as a fixed set of distances between its parts. A rigid body in the Newtonian sense has a shape that does not depend on its motion, and no such object exists — so a disc cannot be spun up from rest while remaining rigid, and the question of what happens to its circumference has no answer until the material’s own dynamics are supplied.

The impossibility is also worth stating in a way that removes the geometry from it. A rigid body is one in which a push at one point moves every other point at once, which is a signal at infinite speed — so rigidity is forbidden in relativity for reasons that have nothing to do with rotation. What the disc adds is that the forbidden thing is not merely a limiting case reached at high acceleration; it is unavailable at every rotation rate, because the requirement is on lengths rather than on rates of change.

What Einstein took from it #

The disc appears in Einstein’s own account of how he got to general relativity, and the step he made is worth stating exactly.

A rotating frame is an accelerated one, and no local experiment distinguishes acceleration from gravity. That is what Einstein took from the disc: if a rotating frame has a geometry that is not Euclidean — and it does, since the rim contracts and the radius does not — then a gravitational field has one too. The disc was the bridge from special relativity to a curved spacetime, and it was a thought experiment rather than a measurement.

An accelerated frame is locally indistinguishable from a gravitational field. A rotating frame is an accelerated one. And a rotating frame demonstrably has a non-Euclidean geometry — which is not a matter of interpretation but of what rulers on the disc measure.

If accelerated frames are gravitational fields, and accelerated frames have non-Euclidean geometry, then gravitational fields have non-Euclidean geometry. That is the inference, and Einstein described the disc as the argument that convinced him the theory would have to be written in curved geometry rather than in ordinary coordinates.

The inference is not a proof and he did not offer it as one. What it did was tell him what kind of mathematics was needed, which was the obstacle: the four years between that recognition and the field equations were spent learning and adapting Riemannian geometry.

It is worth noticing what the disc does not show. It does not produce curvature — the spacetime remains flat, and a rotating frame in flat spacetime is not a gravitational field in any global sense. What it produces is a non-Euclidean spatial geometry in an accelerated frame, which was enough to indicate the direction.

The clocks, and an instrument built on them #

The rim’s clock rate is the same factor inverted, and that half of the effect is measured routinely.

Every clock on the rim is a moving clock, running slow at a rate set by its distance from the axis. That is not a complication to be corrected away — it is the reason a rotating frame cannot have a single time coordinate agreeing with all its clocks, and it is measured routinely: a stored beam of muons circulating in a ring lives longer by exactly the factor its speed requires.

A clock at radius $r$ runs slow by $\gamma(\Omega r)$ relative to one on the axis. For a centrifuge that is a large effect: the Mössbauer experiments of the early 1960s put an emitter at the centre of a rotor and an absorber on the rim, and measured the frequency shift as the rotation rate was varied — the shift that survives at right angles, isolated by a geometry in which there is no first-order term to subtract.

The result matched the transverse Doppler prediction, and the arrangement is unusually clean: the emitter and absorber stay at a fixed distance, so there is no first-order shift at all, and what is measured is the second-order term alone.

The same effect is a correction that has to be applied to any clock on a rotating body — a laboratory on the Earth’s surface, for instance, whose rate depends on latitude through both this term and the gravitational one, in a combination that turns out to be constant over the geoid — which is why a clock’s rate on the Earth’s surface depends on its altitude above sea level and not on its latitude.

What a surveyor on the disc would actually find #

It is worth describing the measurement concretely, because “the geometry is non-Euclidean” is an abstract-sounding conclusion drawn from an entirely mundane procedure.

Give the surveyor a supply of short rigid rods, all identical, all at rest with respect to the disc. Lay them end to end round a circle at radius $r$ and count how many are needed: the answer is $2\pi r\gamma/\ell$. Lay them along a radius from the axis to that circle and count: the answer is $r/\ell$.

Dividing gives $2\pi\gamma$, which is not $2\pi$.

Nothing in that procedure requires any theory. The surveyor has done what a surveyor does, with rulers at rest in the laboratory, and has got a number that a Euclidean plane cannot produce. Doing the same at several radii gives a ratio that varies with radius, which is what a curved geometry looks like from inside — the same kind of measurement that would detect a tidal field in a falling laboratory.

The rate at which it departs from Euclid is the useful number. At small rim speeds $\gamma - 1$ is $\tfrac12\beta^2$, so the excess circumference is half the square of the rim speed in units of $c$ — a part in $10^{18}$ for a laboratory centrifuge, and utterly unmeasurable. The disc is a thought experiment because the effect is quadratic in a quantity that no material lets get large.

The number worth carrying away from that is how sharply the requirement bites at ordinary speeds. A steel flywheel at its bursting speed has a rim moving at about half a kilometre a second, which is a Lorentz factor differing from one by $1.4\times10^{-12}$: the hoop strain relativity requires is a millionth of a millionth, against a yield strain of about a thousandth. Relativity’s contribution to the stress in a real flywheel is nine orders of magnitude below the ordinary centrifugal one, which is why nothing in engineering has ever had to notice it and why the problem stayed a matter of principle.

The axis that turns without being turned #

There is a second thing that happens to anything carried round a circle, and it is not about lengths at all. It is about directions, and it is the reason an atomic energy level comes out right.

Carry a gyroscope round a circular path at constant speed, with no torque on it anywhere. It does not keep pointing the same way. Its axis precesses, backwards relative to the orbital motion, by $2\pi(\gamma-1)$ per circuit — a purely kinematic effect with no force behind it.

The reason is the same one that makes the rotating frame’s clocks fail to close. The gyroscope’s rest frame at each moment is reached from the laboratory by a boost, and the direction of that boost is turning as the body goes round. Two boosts in different directions do not compose to a boost: they compose to a boost and a rotation. Going round the circle is an unending sequence of such compositions, and the leftover rotations accumulate.

The effect is Thomas’s, from 1926, and it arrived as the resolution of an embarrassment. The spin–orbit splitting of atomic fine structure had been computed by transforming to the electron’s rest frame, and the answer came out exactly twice the measured value — a discrepancy nobody could remove and everybody could see. The electron orbiting a nucleus is a gyroscope carried round a circle, so its spin axis precesses by this amount as well, and the correction is a factor of one half. The number that had been twice too large became right.

It is worth noticing what kind of explanation that is. Nothing was added to the physics: no new interaction, no new constant. What had been missed was that the electron’s sequence of instantaneous rest frames does not fit together the way the calculation assumed, which is exactly the failure this essay is about, appearing in a rotation rather than in a length.

Where the clocks not closing is an operational nuisance #

The Earth is a rotating frame, and the failure of its clocks to close is not a thought experiment: it is a line in the procedure by which international time is kept.

Carry a clock slowly eastward round the equator, keeping it synchronised with each station it passes, and bring it home. It disagrees with the clock it started with. Carry it westward instead and it disagrees the other way. The size of the discrepancy is $2\Omega A/c^2$, with $A$ the area enclosed by the path projected on the equatorial plane — the same expression the Sagnac effect gives, with a transported clock in place of a light beam.

For a circuit of the equator that is 207 nanoseconds, so the eastward and westward trips differ by 414. A nanosecond is thirty centimetres of light travel and a satellite navigation fix needs several of them, so this is not a subtlety to be filed away. Every time transfer between distant laboratories — by portable clock, by satellite link, by two-way exchange — carries an explicit Sagnac correction computed from the path’s enclosed area, and getting its sign wrong is a classic way to be out by twice the right amount.

The rotating disc’s central negative result is therefore a working constraint rather than a curiosity. There is no such thing as “now, everywhere on the rotating Earth”, so the world’s timekeeping is not built on one: it is built on a defined non-rotating frame, with each rotating clock’s reading converted into it, and the conversion is where the enclosed area appears.

Where the model runs out #

“The circumference” needs defining before it can be measured. The number $\gamma\cdot2\pi r$ is what a chain of rulers at rest on the rim reports. A stationary observer photographing the disc measures $2\pi r$, because the rim’s rulers are contracted in that frame and the circle they lie on is not moving anywhere. Both are correct answers to different questions, and stating which is meant is most of the work.

Everything measured on a rotating disc is a proper quantity, taken along the material’s own worldline, and comparing two of them requires a convention about which events count as simultaneous. That is where the model runs out: the question “what is the circumference of the rotating disc” is not answerable until one says whose circumference, measured when — and different reasonable answers give different numbers.

The disc is assumed to be in a steady state of rotation. Everything above is about a disc that has always been turning. The spin-up itself is a genuinely dynamical problem, involving elastic waves travelling round the material, and there is no general solution.

Real materials fail long before any of this matters. A rim at a tenth of the speed of light requires a hoop strain of half a per cent, which exceeds the yield strain of every material there is by an order of magnitude. The disc is a thought experiment and there is no prospect of any other kind.

And the excess circumference is not a curvature of anything. The three-dimensional geometry a rotating observer builds does have a non-zero curvature, and it is a curvature of a construction rather than of spacetime, which is flat. Confusing the two is the commonest error in accounts of this problem, and it leads to the wrong conclusion that rotation curves spacetime — which it does not, at any speed.

The ladder from here #

Later rungs on this anchor: Born rigidity and the Herglotz–Noether theorem, which is where the impossibility is proved rather than argued; the Sagnac effect as a measurement of the failure of clocks to close, and the fibre gyroscopes built on it; frame dragging, which is a genuinely curved-spacetime effect of rotation and is a different thing entirely; and the rotating-frame metric, where the whole of this essay is one line of algebra once the geometry is written down.

The neighbouring ladders are the length that depends on when, which is the contraction this applies twice, the string that breaks between two rockets, which is the same impossibility in a straight line, and the ring where the two beams disagree, which measures the failure of the rotating frame’s clocks to close.