The Sagnac effect

The Sagnac effect

Georges Sagnac, 1913

The Michelson–Gale–Pearson apparatus at Clearing, Illinois, after the accounts of 1925: 610 metres one way, 340 the other, and pumped empty of air to find out whether the Earth was turning. The reading depends on the area enclosed, which is why it had to be a rectangle and not a line.

Observed	A fringe shift between counter-propagating beams
Proportional to	The enclosed area × the rate of rotation
Formula	Δt = 4AΩ/c²
General form	Δt = (2/c²) ∮ v·dl
Predicted by relativity	Yes
Cited as disproving relativity	Frequently
What it is, in the theory	The relativity of simultaneity
Sold commercially as	A rotation sensor
Reports the Earth's rotation	Once a day, without fail
Reports the Earth's orbit	Never

The Sagnac effect is the difference in travel time between two beams of light sent in opposite directions around a closed path which is rotating.^[1] It was demonstrated by Georges Sagnac in 1913, it is the operating principle of the ring laser gyroscope, and it is measured daily in every aircraft that carries one.

It is also the most frequently offered experimental disproof of special relativity, on the ground that a fringe shift proportional to velocity ought not to appear if the speed of light is the same for everyone.^[2] Relativity predicts the effect, has predicted it since before it was measured, and the quantity it predicts is the one thing in the theory the objectors like least.

The effect

Light is split and sent both ways around a loop enclosing an area $A$, which rotates at angular rate $\Omega$. The beam travelling with the rotation must chase a detector that is moving away from it; the beam travelling against it meets a detector coming to meet it. The arrival times differ:

The effect does not depend on the medium, the shape of the loop, or the position of the centre of rotation. It is small: a metre-square loop turning at one radian per second gives a time difference of about $4\times10^{-17}$ seconds, which is why it took until 1913 to see and why it is now measured with lasers rather than fringes.

The more useful statement of it is a line integral around the path:

This form matters for what follows. It is not a formula about rotation. It is a formula about the motion of the path, and rotation is the case in which the integral is easy to evaluate.

The relativistic account

The objection assumes that relativity should predict nothing here, because the speed of light is constant for inertial observers. The apparatus is not an inertial observer. It is rotating, and a rotating frame is not inertial in relativity for the same reason it is not inertial in Newton: the rotation is detectable from inside, without reference to anything outside.^[3]

The passage usually cited against this is in §4 of the 1905 paper, where Einstein carries a clock around a closed curve at constant speed and finds it returns slow. That is read as licensing the rotating apparatus as an inertial frame. It does no such thing: the calculation is performed in the stationary frame, about a clock that goes round and comes back, and the quantity it yields is the clock's lost time. It is the seed of the clock paradox, not a promotion of the turntable. Einstein is also careful there in a way the citation is not, flagging the step from a polygonal path to a smoothly curved one as an assumption rather than a result.^[12]

Written in rotating coordinates, the flat-spacetime metric acquires a cross term between time and angle:

That cross term is the whole effect. Its presence means that the surfaces of constant coordinate time are not orthogonal to the observer's worldline, and the practical consequence is that clocks cannot be consistently synchronised all the way around a closed path in a rotating frame. Carry the synchronisation round the loop and you return to your starting clock with a discrepancy. The discrepancy is

which evaluates, to first order, to $4A\Omega/c^{2}$: the Sagnac time difference, arrived at without reference to light beams at all.^[4]

The Sagnac effect is therefore not an embarrassment to relativity. It is the relativity of simultaneity, in a form you can photograph. The failure of global simultaneity, which is the part of the theory most often rejected as a philosopher's trick, is the quantity the interferometer returns.

This is worth stating in one sentence, because it is the position the argument ends in: the effect held up as the strongest evidence against relativity is a direct measurement of the specific consequence of relativity most often denied.^[5]

Michelson, twice

The clearest demonstration is that the same man performed both experiments.

Albert Michelson, whose 1887 interferometer found no trace of the Earth's motion through the aether, built another one in 1925 with Henry Gale and Fred Pearson: a rectangular loop of pipe 610 by 340 metres, laid out on a tract of land at Clearing, Illinois. He had proposed the design in 1904 and did not act on it for twenty-one years, and then only at another man's urging.^[11] A trial at Mount Wilson in 1923 had established that the pipe would have to be emptied of air, so the whole half-kilometre of it was pumped out.

It was not a null result. The predicted shift from the Earth's rotation at that latitude was 0.236 ± 0.002 of a fringe. The observed shift was 0.230 ± 0.005.^[6]

The pair is the entire argument. An apparatus that detects rotation and not translation is not a broken aether detector. It is a working instrument, reporting the difference between two kinds of motion that relativity says are different and that Newton also said were different, for once in agreement.

The generalised effect

The strongest version of the objection rests on the work of Ruyong Wang, who showed in a series of experiments from 2003 that the effect is not confined to rotation.^[7]

Wang's apparatus was a fibre-optic conveyor: a length of fibre carried on a moving belt, with the light path partly on the moving section and partly off it. He found a time difference proportional to the velocity of the moving segment, and a second configuration – a parallelogram in which the source and detector stay still and only one straight portion moves – gave the same result. Linear motion, not rotation, and a shift proportional to $v$.

This is a real and well-executed result, and it is presented as the death of the inertial-frame defence: if a straight-moving segment produces the shift, the "rotating frames are not inertial" reply appears to fail.

It does not, for a reason visible in the formula given above. The general expression was always a line integral of $\mathbf{v} \cdot d\mathbf{l}$ along the path, and it has never contained the word rotation. Wang did not overturn the Sagnac formula; he measured the general case of it, and the general case was the formula. What his apparatus requires is that part of the light path move with respect to another part, and in every configuration he built, it did.^[8]

Which yields a test, and the test has an answer. If the device registers motion with respect to space, it should register the motion the Earth is known to have. Set it on a bench and leave it there, and it reports the belt. It has never reported thirty kilometres a second.

Ring lasers, and GPS

The effect is not a curiosity. It is an industry.

Every ring laser gyroscope and fibre-optic gyroscope in commercial aviation, shipping and guidance is a Sagnac interferometer, sold and certified as a rotation sensor. The large research instruments do better: a ring laser fixed to bedrock measures the Earth's rotation continuously, resolves the variation in the length of the day, and tracks the wobble of the pole.^[9]

It matters also to satellite navigation. Signals from a satellite reach a receiver that has moved during the transit, because the Earth turned underneath it, and a Sagnac correction must be applied to the range equation for the position to come out right; at the equator it reaches a couple of hundred nanoseconds, which is tens of metres.^[10] The correction is standard, documented, and applied by every receiver on the planet.

The use made of it

Two claims are made, and they should be separated.

The first is that the effect shows light does not travel at $c$ in all frames. It shows that light does not travel at $c$ in rotating frames, which relativity states, and which is why the rotating case is treated with the rotating metric.

The second is that the apparatus measures motion with respect to absolute space. It measures motion of one part of the apparatus with respect to another. This is not a subtle distinction and it can be settled without argument: the instrument is manufactured in quantity, mounted in vehicles, and read continuously, and the reading it gives when nothing on the vehicle is turning is zero. An absolute-space detector bolted to the Earth would show 30 km s⁻¹ from the orbit, 370 km s⁻¹ from the microwave dipole, and the rotation besides. It shows the rotation.

See also

• Michelson–Morley experiment – the null on translation, by the same experimenter
• Newton's bucket – the older argument that rotation is absolute
• Albert Einstein – whose least popular consequence this measures
• Hendrik Lorentz – the transformations the objection declines to apply
• Luminiferous aether – what a positive result was once expected to reveal
• The CMB dipole – an effect that genuinely does measure motion with respect to something
• GPS – where the correction is applied several billion times a day
• Special relativity – the theory whose least popular consequence this measures
• Foucault's pendulum – the mechanical counterpart, and the older one
• General relativity – where the rotating frame is handled properly, and still turns
• The speed of light – including the half of it that has never been measured at all
• Interferometry – the parent technique, and why a common path cancels everything but rotation

References

1. ^ G. Sagnac, 1913. Michelson had set out the ring configuration as a means of detecting the Earth's rotation twenty-one years before he built one, in the Philosophical Magazine for 1904, and says so in the first paragraph of the 1925 paper.
2. ^ The claim is generally made in the first-order form: the shift is proportional to v rather than to v², and first-order effects were what the aether theories predicted and the Michelson–Morley apparatus failed to find. The two experiments are not looking for the same thing.
3. ^ This is the point of Newton's bucket, and it is one of the few propositions Newton, Mach and Einstein can be got to agree on, though for three different reasons.
4. ^ The rotating-coordinate form is the Langevin metric. The derivation is standard and appears in the literature in at least two independent versions, one geometric and one by direct velocity composition, which agree.
5. ^ The community from which the objection usually comes also circulates the argument that the relativity of simultaneity is false. Both positions are held at once, and the instrument does not appear to have been consulted about the contradiction.
6. ^ Michelson, Gale and Pearson, 1925. The uncertainties overlap. The experiment is, incidentally, one of the earliest confirmations of special relativity in the low-velocity limit, which was not its purpose.
7. ^ R. Wang and others, from 2003. The papers are careful and the apparatus is sound; nothing in this section disputes the measurement, which has been reproduced.
8. ^ A relativistic treatment of the generalised case was published in 2014 and is not addressed in the arguments that cite Wang, which tend to stop at the measurement.
9. ^ The large ring lasers are geodetic instruments. They are used to measure the Earth's rotation because that is what they are good at, and their published output is a rotation rate with error bars.
10. ^ The correction arises because the receiver's frame is rotating with the Earth, which is the same reason as everything else on this page.
11. ^ Ludwik Silberstein pressed him to it. The 1925 paper records the fact in its abstract, in the tone of a man noting that the idea had been available the entire time.
12. ^ The section is about a travelling clock and gives its lag as half tv²/c². Nothing in it says that a rotating observer measures light at c in his own coordinates, which is the proposition the objection requires and which the theory denies.

External links

• Michelson, "The Effect of the Earth's Rotation on the Velocity of Light, I" – Astrophysical Journal 61 (1925), 137
• Michelson, Gale & Pearson, "The Effect of the Earth's Rotation on the Velocity of Light" – Astrophysical Journal 61 (1925), 140
• Michelson–Gale–Pearson experiment – general reference
• Ashby, "Relativity in the Global Positioning System" – Living Reviews in Relativity (2003)
• Rizzi & Ruggiero, "The relativistic Sagnac effect: two derivations" (2003)