The ring where the two beams disagree

Simultaneity is a choice of slicing, and different observers slice differently. The rung this ladder starts from makes that case for two observers in uniform motion, and the second rung asks which of two events came first. Both are arguments about frames that are inertial, and both end with a slicing that exists — every inertial observer has a perfectly good notion of “now” covering the whole of space, and the disagreement is between one such notion and another.

Put the observer on a turntable — a frame where the forces that are not there appear — and the notion itself stops existing. The demonstration is an interference experiment with a number in it.

The derivation, done in a frame that is not rotating #

The temptation is to do the calculation on the turntable, and it is unnecessary. Work in the inertial laboratory, where light goes at c in both directions and the only thing moving is the apparatus.

Light leaves a beam splitter on the rim of a ring of radius R. While the co-rotating beam travels round, the splitter moves forward, so the beam has to cover $2\pi R + R\Omega t_+$ before it catches up:

$$ct_+ = 2\pi R + R\Omega t_+ \quad\Longrightarrow\quad t_+ = \frac{2\pi R}{c - R\Omega}.$$

The counter-rotating beam meets a splitter coming to meet it, so $t_- = 2\pi R/(c + R\Omega)$. Subtracting,

$$\Delta t = \frac{4\pi R^2\Omega}{c^2 - R^2\Omega^2} \approx \frac{4A\Omega}{c^2}.$$

Three things about that are worth pausing on.

It is first order in Ω. Nearly every relativistic effect an experimenter can reach is second order in a velocity — time dilation, length contraction, the transverse Doppler shift all carry $v^2/c^2$ — and are correspondingly hard to see, length contraction among them. This one carries $v/c$ once, which is why an apparatus with an area of 866 square centimetres turning twice a second produced a visible fringe shift in 1913 with a mercury lamp.

The second-order effects are worth putting beside it for scale. A light clock’s tick lengthens by $\gamma$, which at 0.6c is 1.25 and at ordinary speeds is one part in $10^{14}$ — the reason time dilation waited for atomic clocks and cosmic rays. The Sagnac shift at the same rim speed is enormous by comparison, and the reason is structural: a rotating loop compares two paths against each other, and a comparison of two things that differ at first order never needs the second.

The area appears and the shape does not. Redo the calculation for a square, a triangle or a coil of fibre wound to any outline and the same expression comes out with A the enclosed area. What matters is the flux of the rotation vector through the loop, which is why a fibre gyroscope multiplies its sensitivity by winding the same fibre round many times: a thousand turns of a ten-centimetre coil enclose fifty square metres.

And the refractive index cancels. Light in glass goes at $c/n$, which lengthens both transit times, and it is dragged along by the moving medium, which shortens the difference between them — and to first order the two effects cancel exactly. So a fibre gyroscope and a vacuum ring of the same enclosed area give the same reading, which is a result about how a moving medium carries light rather than about what a medium does to it at rest and was not obvious before it was checked.

The first-order-ness has a second consequence. Because the effect is linear in Ω, its sign reverses with the direction of rotation, so an instrument reading it distinguishes clockwise from anticlockwise. A second-order effect could not: γ does not know which way a thing is going. That is what makes the Sagnac effect an instrument rather than a curiosity — it returns a signed rate rather than a magnitude, which is the whole requirement for navigating.

The same number, said as a failure of synchronisation #

The interference result is a fact about two beams. The general statement is about clocks, and it is stronger.

That is what “no global simultaneity” means, and it is a much more concrete statement than the usual one. It is not that the rotating observer chooses a different slicing from the inertial one; it is that on a rotating platform there is no slicing at all, because the procedure that defines one is path-dependent. Carry the definition of “now” round the rim and it comes back changed.

And it is a purely local failure adding up. Each pair of neighbouring clocks is an ordinary special-relativistic problem — two observers moving with a small relative velocity, close enough that the platform is indistinguishable from an inertial frame — and each pair is handled correctly. The discrepancy is not in any step. It is in the sum, and a quantity whose sum round a closed loop is nonzero while every increment is fine is exactly what a physicist calls a holonomy.

And the two statements are the same statement. Synchronising a chain of clocks by exchanging pulses is sending light both ways round the loop, one step at a time. The interference experiment measures the closure failure directly, in one go, by letting the two beams do the whole circuit and comparing them at the end; the clock chain measures it in pieces and adds them up. There is no separate Sagnac effect and separate synchronisation problem — there is one geometric quantity with two names.

That is the reason the expression is so indifferent to what the apparatus is made of. A holonomy is a property of the loop and of the geometry, and the material carrying the signal round it can be glass, vacuum, water, a chain of clocks, or an acoustic wave in a rotating tube — the answer is 4AΩ/c² for all of them, and for a sound wave it is 4AΩ/c_s² with the sound speed in place of light’s, which is enormously larger and has been measured.

What it is used for #

A rotating platform is an accelerating frame, and an accelerating frame is indistinguishable from a gravitational field. A rotation rate is the one quantity a vehicle cannot measure by looking outside itself if it is sealed, and an instrument that returns it with no moving parts is worth a great deal.

There is another way to read the Earth’s rotation from inside a building, and comparing the two is instructive about what each costs. A Foucault pendulum’s plane turns at $\Omega\sin\lambda$ — a full turn in 23.9 hours at the pole, in 31.8 hours at 48.85°, and never at the equator — so it measures the same component of the same rotation as Michelson and Gale’s rectangle did, by an entirely mechanical route. What it needs that a ring does not is a suspension good enough to run for days, and what it cannot do is give a reading in a second.

There is a design consequence worth drawing out of the expression. Sensitivity is proportional to enclosed area, and a coil of N turns of fibre of total length L on radius R encloses NπR² = LR/2 — so it is proportional to the product of the fibre length and the coil radius, and a given length of fibre is better used on a large coil than a small one. That is why a navigation-grade fibre gyroscope is a few kilometres of fibre on a coil the size of a saucer rather than a great deal more fibre on something smaller.

The sensitivity follows straight from the figures. A kilometre of fibre on a ten-centimetre coil encloses fifty square metres, and at the Earth’s own rotation rate that gives a time difference of 1.6 × 10⁻¹⁹ seconds — a phase of two hundred microradians at 1.55 µm. Reading a phase to a microradian is routine, so such an instrument measures a hundredth of the Earth’s rotation rate, which is more than enough to navigate with.

The Earth’s rotation is a nuisance as well as a signal. Any timing system that transfers time between points on a rotating Earth has to correct for this, because a signal path from a satellite to a receiver sweeps out an area in the rotating frame and picks up 2AΩ/c². For a signal crossing a substantial fraction of the Earth the correction reaches a hundred nanoseconds, which is thirty metres of position — so a navigation system that ignored it would be wrong by the width of a street.

One of the other corrections the same system carries gives the scale. A clock in orbit runs fast by the gravitational shift and slow by its speed, and the net is about 38 microseconds a day — five orders of magnitude larger than the Sagnac term. It is nevertheless in a different category, because a rate offset can be trimmed once, at manufacture, and never thought about again. The Sagnac term depends on where the signal is going, so it has to be computed for every signal.

Lock-in, and the instrument that has to be shaken #

The ring laser is the best form of the instrument and it has a failure mode that no amount of care with the optics removes, and the failure is at exactly the rotation rates a navigator most wants.

Two counter-propagating modes in one cavity are not independent. Any backscatter — from a mirror, from a speck of dust, from the gas in the cavity — couples a little of each into the other, and coupled oscillators pull each other into step. Below some rotation rate the frequency difference the Sagnac effect is trying to impose is smaller than the coupling can bridge, the two modes lock to a common frequency, and the beat note disappears entirely.

So the instrument has a dead band. It reads correctly above a threshold rate and reads exactly zero below it, which is the worst possible behaviour for a device meant to integrate rotation into an attitude: a slow turn accumulates unrecorded.

The standard fix is to make sure the instrument is never slow. The whole laser block is mounted on a flexure and dithered — oscillated back and forth through a small angle at a couple of hundred hertz — so that it spends almost none of its time near zero rate, and the dither is subtracted from the output afterwards. That is why a ring laser gyroscope buzzes audibly, and why a navigation unit containing three of them has a characteristic hum.

The dither leaves its own residue, because the instrument does pass through zero twice per dither cycle and locks briefly each time, accumulating a small random error. Randomising the dither amplitude turns that from a systematic drift into a random walk, which is a much better thing to have. Every one of those steps is a way of arranging for a defect to be noise rather than bias, and the sequence is a fair picture of how a physical effect becomes an instrument.

The experiments, and what each of them settled #

Sagnac, 1913. An interferometer on a turntable, 866 square centimetres enclosed, spinning at two revolutions a second, with a photographic plate recording the fringes. Reversing the sense of rotation doubles the shift, and the difference he measured was about 0.07 fringe against the 0.067 the expression gives. Sagnac believed he had demonstrated an ether; what he had demonstrated is a result that special relativity predicts unchanged, and the interpretation and the measurement have had quite separate afterlives.

Michelson and Gale, 1925. The harder version: not a turntable but the Earth, whose rotation cannot be reversed. A rectangle of evacuated pipe 613 metres by 339 was laid out at Clearing, Illinois, enclosing 207,800 square metres, and a smaller rectangle inside it served as the reference. Only the component of the Earth’s rotation about the local vertical contributes, which at that latitude is 4.86 × 10⁻⁵ radians a second, and the prediction is 0.236 fringes. They reported 0.230 ± 0.005.

That is the measurement the first figure marks, and it is worth being clear about what it establishes: the Earth rotates with respect to the local inertial frame, measured inside a sealed building with no reference to anything outside it. A gyroscope does the same thing mechanically; this does it with no moving parts and with an answer computable from the dimensions of the pipe.

The same loop, with atoms #

Nothing in the derivation used the fact that the two things going round the loop were light. It used only that they travel in opposite senses round an enclosed area, and the same argument applies to any wave — including the wave of a massive particle.

Run the calculation for matter and the phase difference comes out proportional to $mA\Omega/\hbar$ rather than to $A\Omega\omega/c^2$. The ratio between the two, for the same enclosed area and the same rotation, is the particle’s rest energy divided by the photon’s energy — which for a caesium atom against a near-infrared photon is about eleven orders of magnitude.

That is an enormous factor and it is not eleven orders of magnitude of improvement, because the areas differ too. An atom interferometer’s enclosed area is set by how far apart the two paths can be pushed and for how long the atoms are in free fall, and it is square centimetres at best against the fifty square metres a coil of fibre reaches. Multiplying the two ratios leaves the best atom gyroscopes comparable with the best optical ones rather than vastly beyond them.

What the atom instrument has instead is a different set of systematics. Its scale factor is set by $m/\hbar$ and the geometry — constants and a length — rather than by a wavelength and an optical path that drifts with temperature, so it is a candidate for an absolute measurement rather than one needing calibration. And the same apparatus measures acceleration as well as rotation, from a different combination of the same phases.

The comparison is worth making because it isolates what the effect actually depends on. Two instruments using entirely different waves, with rest masses differing by every order of magnitude there is, measure the same rotation through the same geometric quantity — and the only thing the particle contributes is the constant relating its phase to the loop.

The gyroscope that is not this at all #

For completeness, the rotation sensor in a telephone is not a Sagnac device and works by a mechanism this essay has nothing to do with, which is worth knowing before the two are confused.

A vibrating mass in a rotating frame experiences a Coriolis force perpendicular to its motion, so a structure driven to oscillate along one axis begins to oscillate along a perpendicular one at a rate proportional to the rotation. Sensing that second oscillation gives the rate. The whole device can be etched into silicon a millimetre across and costs pence.

It is not a comparison of two counter-propagating waves, it has no enclosed area in its scale factor, and its performance is worse by several orders of magnitude — its bias drifts with temperature, with age and with mechanical stress, in a way that an optical loop’s does not because an optical loop’s scale factor is a geometry. What it has is that it is small, cheap and needs no light source.

The division of labour follows from those properties rather than from any preference. Anything that has to know its attitude for hours without an external reference — an aircraft, a submarine, a missile — carries an optical gyroscope. Anything that needs to know which way up it is for a second or two carries the silicon one. Both are called gyroscopes and neither contains a spinning wheel.

What the argument does not settle #

It does not measure a one-way speed of light. Both beams traverse the full loop — each of them a wave whose crests carry no signal on their own — and return to the same point, so the experiment compares two closed paths and needs no clock at the far side. That is exactly why it works, and it is why the result says nothing about whether light goes at c in each direction separately — a question that cannot be answered without synchronising two separated clocks, which requires assuming an answer to it.

It is not a rotation relative to anything nearby. The Ω in the expression is the rotation with respect to a local inertial frame, and Michelson and Gale’s result establishes that the Earth turns with respect to that — not with respect to the walls of the building, which turn with it. What defines the local inertial frame is a separate and much deeper question.

What does not fail to close is proper time. Every clock reads its own, no synchronisation is involved, and carrying a clock round the rim and back gives a definite number that nobody disputes. What is path-dependent is the comparison between clocks at different places, not the reading of any one of them — and the Sagnac closure failure is a statement about the comparison. A quantity read at one place is safe; a quantity assembled from readings at many places is not.

It does not measure an absolute rotation either. What the expression contains is the rotation of the apparatus with respect to the local inertial frame — the frame in which free bodies move in straight lines — and that frame is defined by the physics in the neighbourhood rather than by anything at infinity. An instrument reading zero is one that is not turning with respect to free fall, which is a local statement, and whether that frame is itself determined by the distant matter of the universe is a question the measurement does not touch.

And nothing here needs general relativity. The whole argument is kinematics in flat spacetime with an apparatus that happens to be turning. Rotation is an acceleration, so a rotating frame is not inertial and the machinery of special relativity does not apply globally to it — which is what the closure failure is a symptom of — but no curvature and no field equation appears anywhere.

The nearest relative of the effect is a close one. Two boosts in different directions compose to a boost and a rotation, so a sequence of velocity changes that returns a body to rest leaves it turned. That is another quantity whose increments are each correct and whose sum round a loop is not zero — and the two are the same phenomenon wearing different clothes: a geometry in which going round and coming back is not the identity.

A ring laser makes the same measurement differently, and better. Instead of interfering two beams that have gone round once, build the loop into a laser cavity and let both directions oscillate: the two counter-propagating modes — only some of which the cavity allows — then have slightly different frequencies, because they see slightly different cavity lengths, and their beat note is directly proportional to the rotation rate. A metre-square ring laser at the Earth’s rotation gives a beat of a few tens of hertz — a frequency, counted rather than a fringe estimated, which is why the largest such instruments resolve one part in 10⁸ of the Earth’s rate and can watch the length of a day change.

Where this ladder goes next #

Three rungs have now been about how simultaneity is defined and where the definition fails. The next asks what survives it. Every observer, rotating or not, has a local notion of now — a small patch of spacetime in which the tilting is negligible and the ordinary rules apply — and the size of that patch is computable: it is set by how fast the tilting accumulates, which for a turntable is the rim speed over c per radian. Everything special relativity says is a statement about such a patch, and the useful question is not whether a global slicing exists but how big the patch is.