The Sagnac Effect and the Circulation It Counts
Michelson’s other experiment — why rotation is not null, and what the fringe shift is actually measuring
Michelson and Morley tested whether uniform motion through the medium betrays a preferred frame. It doesn’t — emergent Lorentz invariance sends the translational result to zero. But there is a second thing an interferometer can do to a medium that fills space: it can rotate. And rotation is not null. A ring interferometer spinning at angular velocity \Omega shows a fringe shift proportional to its enclosed area — the Sagnac effect (Sagnac 1913). If Michelson–Morley is the objection every skeptic raises first, the Sagnac effect is the one they raise second: if there is no medium, what is the rotating interferometer beating against?
The framework’s answer is the cleanest possible one. The Sagnac effect is the rotational companion to the Michelson–Morley null, and the two together partition exactly along the line that defines a superfluid: an interferometer responds to the circulation of the substrate flow enclosed by its loop — zero for translation, 2\vec\Omega\cdot\vec A for rotation. For matter waves the fringe count is not merely proportional to that circulation; it is the number of substrate circulation quanta threading the loop. The same experimenter who found nothing for translation found exactly this for rotation: Michelson, Gale & Pearson (1925) turned a 0.4\,\text{km}^2 evacuated pipe loop into an interferometer and read the Earth’s spin off the fringes, to \sim2\%.
One rule, two experiments
An interferometer splits a wave, sends the two halves around a closed loop in opposite senses, and recombines them. The phase difference on recombination is set by the difference in optical path the two senses experience. Write the substrate’s flow velocity as seen in the apparatus frame as \vec v(\vec x). A co-propagating and a counter-propagating wave are advanced and retarded by that flow, and to leading order the nonreciprocal phase is the line integral of \vec v once around the loop — the circulation enclosed,
\Gamma \;=\; \oint_{\partial A} \vec v\cdot d\vec\ell \;=\; \int_A (\nabla\times\vec v)\cdot d\vec A .
This single quantity decides both canonical tests, because the two experiments differ only in what \vec v is:
Uniform translation (Michelson–Morley). An apparatus gliding through the substrate at constant \vec V sees an apparent uniform flow \vec v = -\vec V. A uniform field is curl-free, so \Gamma = \oint(-\vec V)\cdot d\vec\ell = 0 around any closed loop. There is no enclosed circulation, hence no nonreciprocal phase — null, for a Sagnac-type loop exactly as for the classic crossed-arm geometry. (This is why a linearly moving, non-rotating gyroscope reads zero: Sagnac interferometers are blind to translation.)
Rotation (Sagnac). An apparatus turning at \vec\Omega sees the substrate as an apparent solid-body flow \vec v = -\vec\Omega\times\vec x, whose vorticity is uniform, \nabla\times\vec v = -2\vec\Omega. Now the circulation does not cancel: \Gamma \;=\; \int_A (-2\vec\Omega)\cdot d\vec A \;=\; -2\,\vec\Omega\cdot\vec A . The enclosed circulation is twice the rotation rate times the enclosed area — the Sagnac quantity, geometric and exact.
So Michelson–Morley and Sagnac are not two unrelated facts about light; they are the curl-free and the rotational readings of one line integral. That the null case is translation and the non-null case is rotation is precisely the irrotational-except-for-circulation signature of a superfluid: uniform flow leaves the condensate phase untouched, while rotation imprints circulation on it. The substrate does not merely survive the Sagnac effect — the Sagnac effect is the substrate’s own defining property read on a laboratory bench.
The optical fringe: a metric cross-term, the same one that drags frames
For light the two senses accumulate a time difference \Delta t \;=\; \frac{4\,\vec\Omega\cdot\vec A}{c^{2}},\qquad \Delta\varphi_\text{opt} \;=\; \omega\,\Delta t \;=\; \frac{8\pi\,\vec\Omega\cdot\vec A}{\lambda\,c}, famously independent of any refractive medium in the beam path — fiber-optic gyroscopes see the full geometric value with light that never leaves the glass. In the framework this independence is automatic and instructive: the effect is not a property of what light propagates through, it is a property of the effective metric of the rotating frame. Transforming the substrate’s acoustic metric into a frame turning at \Omega generates an off-diagonal term g_{t\phi} \;=\; -\,\Omega\, r^{2}\sin^{2}\theta , and \Delta t = -\tfrac{2}{c^2}\oint g_{t\phi}\,d\phi is the line integral of that term around the loop. This is the identical structure — the same gravitomagnetic g_{t\phi} cross-term — that carries frame-dragging in the gravity chapter, there sourced by a spinning mass’s entrained azimuthal flow v_\phi rather than by the apparatus’s own rotation. The Sagnac effect and the Lense–Thirring effect are one term of the substrate’s acoustic metric, read for two different sources of circulation.
That identity is not academic. A large ring-laser gyroscope pinned to the Earth beats primarily against the Earth’s rotation (the ordinary Sagnac term), but at the level of parts in \sim10^{9} its Sagnac frequency also carries the geodetic and Lense–Thirring corrections — the substrate’s entrained flow around the spinning Earth mass. This is exactly what the GINGER / Gran Sasso and Wettzell “G” ring-laser programs (Schreiber; Di Virgilio et al.) are built to measure. The framework reads their target plainly: the general-relativistic shift a ring laser chases is the same entrained azimuthal substrate flow that already fixes Gravity Probe B’s frame-dragging to within its error bars — a mass-sourced circulation superposed on the apparatus-sourced one, the two indistinguishable in kind because they are the same g_{t\phi}.
The matter-wave fringe: counting substrate circulation quanta
The sharpest statement lives with matter-wave interferometers — neutrons and cold atoms sent around a Sagnac loop. There the phase is \boxed{\;\Delta\varphi_\text{matter} \;=\; \frac{2m}{\hbar}\,\vec\Omega\cdot\vec A\;} larger than the optical phase for the same loop by the enormous factor mc^{2}/\hbar\omega \sim 10^{10}, which is why atom-interferometer gyroscopes (Gustavson, Bouyer & Kasevich 1997) reach navigation-grade sensitivity in a tabletop. Now recall the framework’s foundational identity: \hbar is the substrate’s circulation quantum. A superfluid whose constituents have mass m carries circulation in units of the Onsager–Feynman quantum \kappa = h/m. The enclosed circulation the rotating loop presents to the wave is \Gamma = 2\vec\Omega\cdot\vec A, so the number of circulation quanta threading the loop is N \;=\; \frac{\Gamma}{\kappa} \;=\; \frac{2\,\vec\Omega\cdot\vec A}{h/m} \;=\; \frac{2m\,\vec\Omega\cdot\vec A}{h}, \qquad\Longrightarrow\qquad \Delta\varphi_\text{matter} \;=\; 2\pi N .
The matter-wave Sagnac phase is 2\pi times an integer count of substrate circulation quanta. A neutron interferometer rotating in the lab is, on this reading, a device that literally counts the quantized circulation of the medium enclosed by its arms — one fringe per quantum h/m_n. The mass in the Sagnac formula, long treated as an awkward reminder that inertial mass appears where one expected a purely kinematic effect, is exactly the mass that sets the circulation quantum \kappa = h/m: the two occurrences of m are the same m. The optical case is the same statement carried by the photon’s effective circulation quantum \kappa_\gamma = h/(\hbar\omega/c^{2}) = \lambda c; matter simply makes the count enormous and the interpretation unmissable.
This closes a small circle the framework has been drawing since Aharonov–Bohm: a closed-loop phase is always a count of something threading the loop. Aharonov–Bohm counts flux quanta of the substrate’s emergent gauge field; the Sagnac effect counts circulation quanta of the substrate flow itself. Same topology, two faces of the one medium.
Why this is a consistency triumph, not a vulnerability
A theory that puts a real medium back into empty space owes an account of every way an interferometer might catch that medium out. Michelson and Morley tried translation and found nothing; the superfluid’s emergent Lorentz invariance explains the nothing. Sagnac, Michelson–Gale, and every ring-laser and atom gyroscope since have tried rotation and found something — and that something is precisely 2\vec\Omega\cdot\vec A, the enclosed circulation of the substrate flow, with a matter-wave fringe count equal to the number of circulation quanta enclosed. The framework predicts both outcomes from the one line integral \oint\vec v\cdot d\vec\ell, with the split between them (curl-free vs. rotational) being the very property that makes the medium a superfluid rather than a classical aether. The effect a naive medium theory would fear most turns out to be the substrate’s cleanest self-portrait.
Summary
| Test | What rotates / moves | Enclosed circulation \Gamma=\oint\vec v\cdot d\vec\ell | Result | Substrate reading |
|---|---|---|---|---|
| Michelson–Morley | uniform translation \vec V | 0 (curl-free) | null | irrotational flow leaves the condensate phase untouched |
| Sagnac (optical / ring laser) | apparatus rotation \vec\Omega | 2\vec\Omega\cdot\vec A | \Delta\varphi=\dfrac{8\pi\,\vec\Omega\cdot\vec A}{\lambda c} | g_{t\phi} cross-term of the rotating-frame acoustic metric |
| Matter-wave Sagnac (neutron / atom) | apparatus rotation \vec\Omega | 2\vec\Omega\cdot\vec A | \Delta\varphi=\dfrac{2m}{\hbar}\vec\Omega\cdot\vec A=2\pi N | fringe count N=\Gamma/(h/m) = substrate circulation quanta enclosed |
| Ring-laser GR term (frame-dragging) | spinning Earth mass | entrained v_\phi | Lense–Thirring shift (\sim10^{-9}) | same g_{t\phi}, sourced by mass instead of apparatus |
Michelson–Morley showed the substrate keeps no preferred frame under translation. The Sagnac effect shows why: not because the medium is absent, but because it is a superfluid — irrotational under uniform flow, and imprinting exactly 2\vec\Omega\cdot\vec A of quantized circulation the moment you turn it. The same Michelson measured both. One is the substrate hiding; the other is the substrate counting.