The rim a rotor slows

In 1960, in the same weeks that Robert Pound and Glen Rebka were reporting the redshift of gamma rays climbing a tower at Harvard, a group at Harwell in England — Hay, Schiffer, Cranshaw and Egelstaff — did something that looks quite different and is the same experiment. They put a source of iron-57 gamma rays near the axis of a rotor, an absorber of iron-57 near its rim, and spun it. As the speed rose, the absorber absorbed less, because the gamma rays arriving at it were no longer at quite the frequency it was tuned to absorb. The size of the mismatch was a few parts in ten trillion, and it followed the square of the rotation speed.

That experiment can be described in two ways, and the interest of it lies in the fact that both descriptions are correct. In the laboratory, the absorber is moving, and moving clocks run slow. On the rotor, the absorber is at rest, but it sits in a centrifugal field, lower in a potential than the source at the axis, and clocks lower in a potential run slow. The floor that cannot be told from gravity said that an accelerated frame and a gravitational field cannot be told apart locally. The rotor experiment is the equivalence principle demonstrated with a frame whose acceleration is centripetal.

A line sharp enough to see it #

The effect is tiny. A point on a rotor’s rim at radius $r$, turning at angular speed $\omega$, moves at $v = \omega r$, and its clock runs slow by $v^2/2c^2$. For a rim six or seven centimetres from the axis turning five hundred times a second, the speed is about two hundred metres a second and the fractional slowing is about $2.4\times 10^{-13}$.

Measuring a frequency shift of that size needs a reference sharper still, and before 1958 there was none. The gamma ray its own kind cannot absorb told how Rudolf Mössbauer found that nuclei bound in a crystal can emit and absorb gamma rays without recoil, so that the emitted line is as narrow as the nuclear state’s lifetime allows. For iron-57’s 14.4 keV line the width is about $3\times 10^{-13}$ of the frequency. A rotor shift comparable to that width moves the source’s line off the absorber’s, and the absorption falls.

The measurement is a count. A detector behind the absorber counts the gamma rays that get through. At rest, many are absorbed resonantly, and few get through; as the rotor spins faster the absorber’s resonance slides off the source’s line and more get through. Hay’s group, and Walter Kündig in 1963 with a more refined arrangement, found the change in transmission following the square of the rotation rate with the size the formula predicts, to within the few-per-cent precision their apparatus allowed.

No first-order Doppler shift #

There is a puzzle in that result. A moving absorber should see a Doppler shift, and the ordinary first-order Doppler shift is a part in a million for two hundred metres a second, a million times larger than the effect being measured. Why does it not swamp everything?

Because the absorber’s velocity is perpendicular to the line along which the gamma rays arrive. A source on the axis sends its gamma rays outward along a radius; the absorber on the rim moves tangentially, at right angles to that radius. The first-order Doppler shift depends on the component of velocity along the line of sight, which is zero. What is left is the second-order, transverse Doppler shift, which is pure time dilation. The rotor geometry is a way of switching off the large effect to reveal the small one. Any misalignment that gives the absorber a velocity component along the line of sight brings back a first-order shift, a million times larger, so the experimenters had to place the source precisely on the axis. They could check that they had: a first-order shift reverses sign when the rotor’s direction is reversed, while time dilation, depending on the square of the speed, does not.

The general formula, for a source at radius $r_s$ and an absorber at $r_a$, is

$$\frac{\Delta\nu}{\nu} = \frac{\omega^2(r_a^2 - r_s^2)}{2c^2}.$$

It depends only on the two radii. D. C. Champeney and P. B. Moon tested the most striking consequence in 1961: they put source and absorber at opposite ends of a diameter, both on the rim, moving in opposite directions at several hundred metres a second, and found no shift at all. Two clocks at the same radius on a rotor tick together, whatever their velocities relative to each other. In the laboratory frame that is because both are moving at the same speed and so both are dilated by the same factor. In the rotor’s frame it is because both sit at the same centrifugal potential.

Two descriptions of one number #

The two descriptions deserve to be set side by side, because the equivalence principle is in the comparison.

In the laboratory, an inertial frame, the rotor’s rim moves at $v = \omega r$ and its proper time runs at $1/\gamma$ of the laboratory’s, with $\gamma = 1/\sqrt{1 - v^2/c^2}$. The source at the axis is not moving. So a clock on the rim ticks slow relative to one at the axis by $\gamma - 1 \approx \omega^2 r^2/2c^2$.

On the rotor, a rotating frame, everything is at rest. A body at rest at radius $r$ feels a centrifugal force $m\omega^2 r$ outward, which can be described by a potential $\Phi = -\tfrac12\omega^2 r^2$. The forces that are not there introduced the centrifugal force as a feature of a rotating frame; the equivalence principle says that a potential of that kind affects clocks exactly as a gravitational potential does, with a clock lower in the potential running slow by $\Delta\Phi/c^2$. A clock at the rim is lower than one at the axis by $\tfrac12\omega^2 r^2$, so it runs slow by $\omega^2 r^2/2c^2$.

The two calculations give the same answer. They part only at speeds approaching light, and there it is the potential formula that needs repair: the rotating-frame metric gives a clock rate of exactly $1/\gamma$, which corresponds to a potential $-c^2\ln\gamma$, of which $-\tfrac12\omega^2 r^2$ is the first term. No rotor will ever reach that regime — a steel rotor bursts at rim speeds below a kilometre a second, a few millionths of light — so for every rotor that can be built the two descriptions are one number, read either as motion or as a field.

This is the same equivalence that the clocks the sea keeps together used on a planetary scale: a clock on the equator moves faster than one at the pole, and is also further from the centre, and on the surface of a settled sea the two effects cancel exactly because the sea’s surface is itself an equipotential of the combined gravitational and centrifugal field. The rotor is that calculation without the gravity.

A twin paradox on a turntable #

A rotor is also the cleanest possible version of the twin paradox. Put one clock at the axis and another on the rim, spin the rotor for a day, stop it and compare them. The rim clock has run slow throughout, by $2.4\times 10^{-13}$ of every second, and after a day it is about 21 nanoseconds behind. Nothing about the comparison is ambiguous: the two clocks start together and end together, side by side on one machine, and the rim clock has aged less.

The usual twin paradox is argued over because the travelling twin turns round, and the turn seems to be where something happens. The longest way round is the shortest clock showed that the turn is a red herring: the travelling clock’s deficit is set by its whole path through spacetime, and the path that departs from straight, inertial motion is the one with less proper time. The rim clock departs from inertial motion all the time, gently and uniformly, and accumulates its deficit steadily rather than at a turning point. There is nothing to argue over. Each second on the rim is shorter than each second at the axis, by the same fraction, all the way round.

The rotor makes plain what the symmetry objection gets wrong. To an observer riding on the rim, the axis is going round them, so why should the axis clock not be the slow one? Because the rim observer is not inertial: they feel a centrifugal force pressing them outward, and they can measure it with a spring balance. The two clocks are not in symmetric situations, and the rotating observer’s account — that they sit lower in a potential — gives the same answer as the inertial account. Every frame agrees on which clock is behind.

Why it needed a nucleus #

The rotor shift had been calculable since 1905; it waited fifty-five years to be measured because no clock was good enough. An atom’s spectral line in a gas is broadened by the atoms’ thermal motion to about a part in a million of its frequency, so a shift of a part in ten trillion is ten million times smaller than the line’s width. A nucleus bound in a crystal, emitting without recoil, gives a line ten million times narrower, at a part in ten trillion. The Mössbauer effect turned a nucleus into a clock with the necessary precision almost overnight, and the rotor experiments, along with Pound and Rebka’s tower, were among the first things it was used for.

That the clock was a nucleus rather than an atom or a pendulum is itself part of the result. The clocks that must all slow together argued that every kind of clock, whatever its mechanism, must be dilated by the same factor in motion, or the principle of relativity would fail: a moving observer could tell their speed by comparing two kinds of clock. The rotor experiments used the strong-force-bound states of iron nuclei, a mechanism entirely unlike the electromagnetic transitions of atomic clocks or the gravity of a pendulum, and found the same dilation. That universality is what makes “the clock runs slow” a statement about time rather than about clockwork.

The precision has since moved far past the rotor’s few per cent. Atomic clocks flown in aircraft in 1971 measured time dilation and gravitational redshift together to about ten per cent; Gravity Probe A in 1976 measured the redshift to about a part in ten thousand; optical clocks today see the redshift between benches a few centimetres apart in height. But the rotor remains the case in which the kinematic and gravitational descriptions are most cleanly the same physics, because it has no gravity in it at all.

Acceleration does not enter #

The rotor’s rim is accelerating, enormously. A rim moving at two hundred metres a second at a radius of 6.6 centimetres is accelerating towards the axis at sixty thousand times the acceleration of gravity.

The shift depends on the speed and not on the acceleration. Two rotors with the same rim speed and very different radii give the same shift, though their rims accelerate at very different rates. The clocks in question are iron nuclei, whose internal time-keeping is set by nuclear forces vastly stronger than any acceleration a rotor can supply, and they keep time as their speed alone dictates. That is the clock hypothesis, which the clock that does not feel the turn tested with muons held in a storage ring at 10¹⁸ times the acceleration of gravity; the rotor experiments were an earlier, cruder confirmation of it at a gentler 10⁵.

In the rotor’s frame, the same statement reads differently: the clock’s rate depends on the potential, not on the field strength. A clock at a point where the centrifugal field is huge but the potential difference from the axis is small runs nearly at the axis rate. The gravitational analogue is familiar from the clock that runs slow lower down, where what mattered was height, not the strength of gravity at the clock.

Rotating frames are not inertial #

The rotor teaches one more thing, about the geometry of rotating frames. The disc that cannot be spun described Ehrenfest’s paradox: a disc spun up from rest has its rim contracted in the laboratory frame while its radius is not, so a rigid disc cannot be spun up at all. Clocks on a rotating disc add a further oddity. Clocks at different radii run at different rates, and clocks at the same radius cannot be consistently synchronised around the rim: carry a time signal around the rim and it arrives back out of step with itself, the Sagnac effect that the ring where the two beams disagree followed in light. A rotating frame has no global time, only local rates and a twist.

The equivalence principle makes the rate part of that look like gravity. The twist has no gravitational counterpart near a non-rotating mass; it is the signature of rotation, the part of a rotating frame that is not equivalent to a static gravitational field. Near a rotating mass, general relativity does predict such a twist, frame-dragging, which Gravity Probe B measured around the Earth. The rotor’s clocks show the static part of the equivalence; its rim’s synchronisation problem shows the part that is not static.

What the picture cannot show #

The formula is the second-order result for rigid rotation, with source and absorber treated as point clocks and the rotor perfectly rigid. Real rotors stretch under their own centrifugal stress, which changes radii slightly with speed, and vibrate, which adds first-order Doppler broadening to the line; the early experiments were limited more by vibration than by anything else. The absorption curve assumes ideal, unbroadened lines of the natural width; real Mössbauer sources and absorbers have lines several times broader from crystal imperfections and thickness, so the measured absorption falls more slowly with speed than the figure shows.

The comparison between descriptions assumes that a clock’s rate in a potential is given exactly by the potential, which is the equivalence principle’s content and what the rotor tests. Some authors have argued from later rotor experiments that an extra contribution appears beyond $\omega^2 r^2/2c^2$, and their claims have been disputed and attributed to systematic effects; the figures show the standard prediction, which the original experiments confirmed within their precision.

Still open: rotating clocks at modern precision #

The rotor experiments reached a few per cent. Optical clocks now compare rates to parts in 10¹⁸, and a precision rotor carrying such a clock, or a trapped ion orbiting in a ring, could test the rotating-frame prediction a hundred million times more precisely than the iron-57 experiments. The obstacle is that an optical clock’s frequency is sensitive to the very accelerations and vibrations a rotor imposes, through the shifts they cause in the atoms’ energy levels and the lasers that interrogate them. Ion storage rings carrying fast ions have tested the time dilation of moving clocks to parts in a billion, but a rotating-frame test with a clock at rest on a spinning structure, comparing the two descriptions directly at high precision, has not been made.

A rotor slows a clock on its rim relative to one at its axis by $\omega^2(r_a^2 - r_s^2)/2c^2$ — 2.4 × 10⁻¹³ at 6.6 cm and 500 turns a second, measured with iron-57’s recoil-free gamma rays in 1960 — which is time dilation in the laboratory and a centrifugal potential on the rotor, one number in two descriptions, independent of the rim’s acceleration and zero between two clocks at the same radius. The habit worth carrying is to compute an effect in the frame where it looks simplest, and then in another, because when two descriptions agree, the agreement is itself a law.