Michelson-Morley and Modern Isotropy Tests

In a temperature-stabilised laboratory, a cylinder of sapphire the size of a coffee cup is cooled to a few kelvin and set turning, one revolution every hundred seconds. A microwave field rings inside it. An identical cylinder sits beside it with its axis at right angles, and a circuit listens to the beat between the two — the difference of their resonant frequencies. If the speed of light depended, even faintly, on direction, that beat would rise and fall as the table turns — at twice the rotation rate, because a $180^\circ$ turn brings the pair back onto itself, and with sidebands at one sidereal day and one year as the Earth carries the laboratory around. That is not a new experiment. That is the Michelson–Morley experiment, rebuilt. The arms are still perpendicular, the table still turns, the observable is still a two-beam comparison, and the answer is still no. What has changed is only the size of the no.

Key takeaways

Signature $(+,-,-,-)$ : $\mathrm{d}s^2 = c^2\mathrm{d}t^2 - \mathrm{d}x^2$ , timelike intervals positive.
The 1887 fringe-shift derivation is owned by `aq-sr-michelson-morley-experiment`, which prints $0.373$ fringes. This article does not repeat it. It asks the harder question: what does a null result actually bound, and how do you compare a null from 1887 with a null from 2015?
The shared observable is the anisotropy of the two-way speed of light: in an aether the round-trip speed is $c(1-\beta^2)$ along the motion and $c\sqrt{1-\beta^2}$ across it, so the peak anisotropy is $\Delta c/c = \tfrac12\beta^2$ .
⚠ The base says the null was "consistent with zero at the level of $\sim 1/40$ of a fringe." That is wrong, and it is repaired here. Michelson and Morley wrote that "the displacement to be expected was 0·4 fringe" and that the actual one was "certainly less than the twentieth part of this, and probably less than the fortieth part." One fortieth of the expectation is $\approx 0.01$ fringe — not $0.025$ . The misquote makes their instrument sound forty times cruder than it was.
Owned result, machine-proved: 1887 and 2015 on one axis. Their bound is $\Delta c/c \lesssim \mathbf{1.25\times10^{-10}}$ — an aether wind below $\mathbf{4.7\ \mathrm{km/s}}$ . The best modern rotating-cavity bound is $(9.2 \pm 10.7)\times10^{-19}$ . The gain is $1.2\times10^{8}$ — eight orders of magnitude, not the nine the base claims.
A cavity turns anisotropy into frequency ( $\nu = mc/2L$ ), so two orthogonal cavities on a turntable beat at $\cos 2\theta$ — the same $90^\circ$ symmetry as Michelson's table, read out at $2\Omega$ instead of by eye.
The RMS anisotropy coefficient is $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ — but two sign conventions are in circulation, they are negatives of each other, and BOTH vanish at the special-relativity point, so the obvious check is blind. The Galilean limit discriminates. See the warning; the transformation is `aq-sr-robertson-mansouri-sexl-framework`'s.
Key idea: a null result is not "nothing happened." It is a number, and the number is the sensitivity. The history of this experiment is that number falling by eight orders of magnitude — and never once lifting off zero.

The core insight

The Michelson–Morley experiment was never really about the aether. It is a comparator: it measures the speed of light in one direction against the speed of light in a perpendicular direction, without ever needing to know either one. That is why it survived the death of the theory it was built to test, and why every serious version of it — fringe or beat note, 1887 or 2015 — is a round-trip experiment.

Where this lives today

It stopped being an experiment about the aether a century ago. Nobody builds a rotating cavity to hunt a luminiferous medium; they build it because isotropy of the speed of light is an exactly stated symmetry whose violation is a generic low-energy signature of Planck-scale physics. The experiment outlived its own motivation because the quantity it measures outlived the theory it was built to refute.

The RMS parameters are historical currency. Worked example 2 shows why the field moved on: a second-order test theory dilutes every improvement by $\beta^2 \approx 10^{-6}$ , and its three dials cannot express the possibility that light and electrons are affected differently. The modern language is `aq-sr-standard-model-extension-lorentz-violation`; a cavity run today reports SME coefficients, quoting an RMS number only so its result can be set beside 1887.

And the idea outlived the apparatus. A Michelson interferometer with $4\ \mathrm{km}$ arms and mirrors hung on glass fibres is called LIGO — hunting a differential change in two perpendicular arms, read against no absolute standard, by exactly the argument that made Michelson's instrument work. The instrument built to show that spacetime has no preferred direction became the instrument that measured spacetime ringing.

The physical picture

Take the nineteenth century at its word: light is a wave in an immobile medium, in whose rest frame it travels at $c$ in every direction, and the Earth moves through it at speed $v$ . Then light sent out and reflected back along a rod of length $L$ takes a time that depends on the rod's orientation, and the two-way speed you infer — $2L$ over the round-trip time — is direction-dependent. Two directions suffice. Along the motion, light is slowed to $c-v$ one way and hurried to $c+v$ the other; the losses beat the gains and the round-trip speed is $c(1-\beta^2)$ . Across it, light must be aimed upstream to reach a mirror that has drifted, travels the hypotenuse, and returns at $c\sqrt{1-\beta^2}$ . The transverse direction is faster. The difference,

\frac{\Delta c}{c} \;=\; \frac{c_{\perp} - c_{\parallel}}{c} \;=\; \sqrt{1-\beta^2} - (1-\beta^2) \;\approx\; \frac{\beta^{2}}{2},

is the entire experimental target, then and now. For the Earth's orbital motion, $\beta \approx 10^{-4}$ , it is $5\times10^{-9}$ : five parts per billion, between two directions, in a quantity nobody can measure absolutely.

The genius of the interferometer is that it never tries to — it compares. The fringe pattern reports the difference of the two round-trip times in units of the wavelength, and rotating the apparatus swaps the arms and doubles the signal; the absolute travel time never appears. The genius of the modern version is that it stops rotating the interferometer and starts rotating the clock. Trap light between two mirrors and the resonance condition $L = m\lambda/2$ ties the frequency to the speed of light along the axis:

\nu \;=\; \frac{mc}{2L} \qquad\Longrightarrow\qquad \frac{\delta\nu}{\nu} \;=\; \frac{\delta c}{c} \;-\; \frac{\delta L}{L}.

A direction-dependent $c$ becomes a direction-dependent frequency — the most precisely measurable quantity in physics. That, and not the length of the arms, is where the eight orders of magnitude came from.

Figure

The modern Michelson-Morley experiment. Two orthogonal ultra-stable cavities are mounted on a turntable and their resonant frequencies are beat against one another. Because a cavity's frequency tracks the speed of light ALONG ITS OWN AXIS (ν = mc/2L), any anisotropy of c appears as a modulation of the beat note. And because a 180° rotation carries the pair back onto itself, the signal lives at TWICE the rotation frequency — the cos 2θ below is the exact same 90° symmetry that made Michelson rotate his table. The Earth's spin adds a sidereal sideband and its orbit an annual one, which is how a laboratory bolted to a moving planet gets to scan the sky. The measured amplitude is consistent with zero at (9.2 ± 10.7)×10⁻¹⁹

Two things changed. The boost: Michelson took $v \approx 30\ \mathrm{km/s}$ , the Earth's orbital speed. Modern experiments take the candidate preferred frame to be the one in which the cosmic microwave background is isotropic; the solar system moves through it at $v = 369.82 \pm 0.11\ \mathrm{km/s}$ (the Planck dipole), so $v^2/c^2 = 1.52\times10^{-6}$ — a hundred times larger. And the modulation: Michelson turned his table by hand, dragging thermal gradients with it — which is how Dayton Miller came to report a positive result in 1933 that stood for two decades. A modern rig rotates continuously and reads only the $2\Omega$ line, plus sidereal and annual sidebands that no basement can fake.

The core equations

The two-way speed of light in a laboratory moving at $v$ through a hypothetical preferred frame, expanded to second order, with $\theta$ the angle between the light path and $\mathbf{v}$ :

\boxed{\;\frac{c(\theta)}{c} \;=\; 1 \;+\; \left(\tfrac12 - \beta_{\!*} + \delta\right)\frac{v^{2}}{c^{2}}\sin^{2}\theta \;+\; \left(\beta_{\!*} - \alpha - 1\right)\frac{v^{2}}{c^{2}}\;}\qquad (1)

where: $\alpha, \beta_{\!*}, \delta$ are the Robertson–Mansouri–Sexl parameters — the second-order coefficients of the time-dilation, longitudinal-contraction and transverse-scale factors of a general linear transformation. (We write $\beta_{\!*}$ for the RMS parameter to keep it apart from $\beta = v/c$ ; tradition unhelpfully gives them the same letter.) Special relativity is the point $\alpha = -\tfrac12$ , $\beta_{\!*} = +\tfrac12$ , $\delta = 0$ , where both coefficients vanish and $c(\theta) = c$ . Physical reading: the $\sin^2\theta$ term is the Michelson–Morley signal; the isotropic term is the Kennedy–Thorndike signal. The transformation behind (1) is `aq-sr-robertson-mansouri-sexl-framework`'s; here it is used and checked in its limits. Set all three to zero — a Galilean laboratory in a real aether — and (1) recovers the classical prediction exactly:

\frac{c_{\parallel}}{c} = 1 - \frac{v^{2}}{c^{2}}, \qquad \frac{c_{\perp}}{c} = \sqrt{1 - \frac{v^{2}}{c^{2}}} \approx 1 - \frac{v^{2}}{2c^{2}}, \qquad \boxed{\;\frac{\Delta c}{c} \equiv \frac{c_{\perp}-c_{\parallel}}{c} = \frac{\beta^{2}}{2}\;}\qquad (2)

where: $\Delta c/c$ is the peak-to-peak anisotropy of the two-way speed — the quantity every experiment here actually bounds. Physical reading: second order in $\beta$ , which is the whole difficulty; and positive, so the transverse round trip is the fast one. The cavity that turns that into a frequency:

\nu = \frac{m\,c}{2L} \qquad\Longrightarrow\qquad \boxed{\;\frac{\delta\nu}{\nu} \;=\; \frac{\delta c}{c} \;-\; \frac{\delta L}{L}\;} \qquad (3)

where: $m$ is the fixed integer mode number, $L$ the cavity length. Physical reading: a rigid cavity ( $\delta L = 0$ ) is a speed-of-light meter reading out along its own axis. The $-\delta L/L$ is not a nuisance term — it is why the cavity must be cryogenic sapphire, and it is why Dayton Miller's basement fooled him. The signal a rotating pair produces:

\boxed{\;\frac{\Delta\nu(t)}{\nu} \;=\; \frac{c(\theta) - c(\theta + 90^\circ)}{c} \;=\; -\left(\tfrac12 - \beta_{\!*} + \delta\right)\frac{v^{2}}{c^{2}}\,\cos 2\theta, \qquad \theta = \Omega t + \theta_{0}\;}\qquad (4)

where: $\Omega$ is the turntable rate. Physical reading: the signal is at $2\Omega$ . That factor of two is the $90^\circ$ symmetry of Michelson's apparatus, surviving intact into an experiment with no arms, no fringes and no moving mirrors.

Limiting cases

Case	Result	Meaning
$\alpha=\beta_{\!*}=\delta=0$	$\Delta c/c = \beta^{2}/2$	Galilean lab in a real aether — the classical prediction, recovered exactly from (1)
$\alpha=0,\ \beta_{\!*}=\tfrac12,\ \delta=0$	anisotropy $=0$ , isotropic term $=-\tfrac12\beta^2$	FitzGerald–Lorentz: contraction, no time dilation. Passes MM, fails KT — why one experiment was never enough
$\alpha=-\tfrac12,\ \beta_{\!*}=\tfrac12,\ \delta=0$	$c(\theta) = c$	special relativity. Both coefficients vanish at every $\theta$ and every $v$
$v \to 0$	$c(\theta) \to c$	you cannot test isotropy without moving
$\beta \to 0$ at fixed sensitivity	bound worsens as $1/\beta^2$	why the CMB boost, not the orbital one, is the modern lever

Where the eight orders of magnitude came from

Derivation AThe cavity is a Michelson interferometer with its light path folded a hundred thousand times

Michelson's instrument and a modern cryogenic resonator look nothing alike, and the temptation is to treat them as different experiments that happen to agree. They are not. This card shows that a two-cavity turntable measures the same quantity as a rotating interferometer — the anisotropy of the two-way speed of light — and produces its signal with the same $90^\circ$ symmetry, at twice the rotation rate. That correspondence licenses the comparison in the next card, and the $2\Omega$ result is the most important design fact about the modern experiment.

Step 1. Write the resonance condition. A standing wave in a Fabry–Pérot cavity of length $L$ needs an integer number of half-wavelengths, $L = m\lambda/2$ ; with $\lambda\nu = c$ along the axis, $\nu = mc/2L$ . Why: $m$ is an integer and cannot drift — it is locked by the mode the laser sits on. Only two things can move the frequency: the speed of light along the axis, and the length of the cavity. Exactly the pair the classic experiment also confounds.

Step 2. Differentiate: $\delta\nu/\nu = \delta c/c - \delta L/L$ . Take the cavity rigid and the resonance is a direct readout of $c$ along its own axis. Why: the whole instrument in one line — and it names the enemy. Any strain correlated with orientation enters as $-\delta L/L$ , indistinguishable from signal. It is why Miller's positive result was a temperature gradient.

Step 3. Point cavity A along $\theta$ , cavity B along $\theta + 90^\circ$ , and beat them. From (1) the anisotropic part of $c$ is $A_{\!*}\,\beta^2\sin^2\theta$ with $A_{\!*} \equiv \tfrac12 - \beta_{\!*} + \delta$ , so

\frac{\Delta\nu}{\nu} = A_{\!*}\beta^{2}\left[\sin^{2}\theta - \sin^{2}(\theta + 90^\circ)\right] = A_{\!*}\beta^{2}\left[\sin^{2}\theta - \cos^{2}\theta\right] = -A_{\!*}\,\beta^{2}\cos 2\theta .

Why: the isotropic term of (1) is common to both cavities and cancels — the pair is blind to Kennedy–Thorndike physics by construction. Two orthogonal cavities are an isotropy meter and nothing else.

Step 4. Turn the table at $\Omega$ , so $\theta = \Omega t + \theta_0$ . The signal appears at $2\Omega$ . Why: this is the punchline. A $180^\circ$ rotation maps the pair onto itself (A becomes B, and the beat changes sign), so the physics is $\pi$ -periodic and the Fourier line sits at twice the drive. Every orientation-scaling systematic — tilt, gravity gradient, magnetic field, the motor — is $2\pi$ -periodic and sits at $1\Omega$ . The factor of two IS the noise rejection, and it is inherited from the fact that Michelson had to rotate his table by $90^\circ$ , not $180^\circ$ , to swap his arms.

Step 5. Add the Earth. The lab's velocity through the candidate frame is not constant — its direction is modulated at one sidereal day by the Earth's spin and at one year by its orbit — so a genuine signal must appear as sidebands of the $2\Omega$ line. Why: a systematic bolted to the turntable cannot know what a sidereal day is. This is the modern replacement for Michelson's "at any hour, at any season", and it is what lets one experiment bound nine independent rotation and boost coefficients of $c$ rather than one. *Verified in verify_aq-sr-michelson-morley-isotropy-tests.py: sympy confirms $\sin^2\theta - \cos^2\theta = -\cos 2\theta$ identically, that the isotropic term of (1) cancels exactly in the A−B beat, and that the beat has period $\pi$ in $\theta$ (i.e. lives at $2\Omega$ ) while a single cavity has period $\pi$ as well but a tilt-like systematic $\propto\cos\theta$ has period $2\pi$ .*

Go deeper · Derivation B · 3 minWhat the 1887 null actually bounds and how it compares with 2015

Derivation BWhat the 1887 null actually bounds and how it compares with 2015

"Michelson and Morley saw nothing" is not a result. A null result is a number — an upper bound on a physical quantity — and until you extract that number you cannot say whether the experiment was good or bad, nor compare it with anything else. This card converts the 1887 null into a bound on $\Delta c/c$ , the same quantity a modern cavity bounds, and then compares the two. This is the article's owned structural result, and it delivers a figure the base gets wrong.

Step 1. Take the experiment's own numbers, in the authors' own words. The 1887 paper says the light path was folded to "about eleven meters, or $2\times10^{7}$ wave-lengths of yellow light" — fixing their wavelength at $\lambda = 11/(2\times10^{7}) = \mathbf{550\ \mathrm{nm}}$ — and that "the displacement to be expected was 0·4 fringe." Check: $\Delta N = 2(L/\lambda)\beta^2 = 2(2\times10^{7})(10^{-8}) = \mathbf{0.400}$ , exactly. Why: their headline prediction is their own arithmetic, and it closes to three digits. The owner `aq-sr-michelson-morley-experiment` runs the same formula with sodium $\lambda = 589\ \mathrm{nm}$ and prints $0.373$ fringes: same physics, different lamp.

Step 2. Read the null the way the authors wrote it. "The actual displacement was certainly less than the twentieth part of this, and probably less than the fortieth part." Of this. The bound is a fraction of the $0.4$ -fringe expectation, not an absolute fringe count: $\Delta N_{\text{obs}} \lesssim 0.4/40 = \mathbf{0.01}$ fringe. Why: the base says " $\sim 1/40$ of a fringe," which is $0.025$ — a category error, not a rounding one. It converts a statement about how far short of the prediction they fell into a statement about how well they could read a fringe. They had a factor of forty in hand; the misquote gives them sixteen.

Step 3. Convert to the physical observable. From (2), the predicted peak anisotropy at $v = 30\ \mathrm{km/s}$ is $\Delta c/c = \tfrac12\beta^2 = 5.0\times10^{-9}$ . The observed shift was at most $1/40$ of the predicted shift, and the fringe shift is linear in the anisotropy, so

\frac{\Delta c}{c}\bigg|_{1887} \;\lesssim\; \frac{1}{40}\times 5.0\times10^{-9} \;=\; \mathbf{1.25\times10^{-10}} .

Why: cross-check it without $\beta$ at all. $\Delta N = 2L\beta^2/\lambda$ and $\Delta c/c = \beta^2/2$ give $\Delta c/c = \lambda\,\Delta N/(4L)$ ; with $\lambda = 550\ \mathrm{nm}$ , $L = 11\ \mathrm{m}$ , $\Delta N \le 0.01$ that is $1.25\times10^{-10}$ again — the same number by a route that never mentions the aether. That independence is what makes the bound a statement about light rather than about a discarded theory.

Step 4. State it as a velocity, the way they would have. Since $\Delta N \propto v^2$ , a null at $1/40$ of the prediction bounds the aether wind at $v \le 30/\sqrt{40} = \mathbf{4.7\ \mathrm{km/s}}$ (and at the conservative $1/20$ , $6.7\ \mathrm{km/s}$ ). Why: this is why the result was decisive rather than suggestive. The Earth's orbital speed alone is $30\ \mathrm{km/s}$ and cannot be switched off; the experiment excluded it more than six times over, in every direction, at every season.

Step 5. Compare with 2015. The best direct terrestrial rotating-cavity experiment bounds the orientation-dependent fractional frequency change — which by (3) is $\delta c/c$ — at $(9.2 \pm 10.7)\times10^{-19}$ . Take the uncertainty as the bound:

\frac{1.25\times10^{-10}}{1.07\times10^{-18}} \;=\; \mathbf{1.2\times10^{8}} .

Why: eight orders of magnitude, not nine. The base asserts nine by comparing a round $10^{-9}$ for Michelson–Morley against $10^{-18}$ — but $10^{-9}$ is roughly the predicted $\tfrac12\beta^2$ signal, not the achieved bound, which is forty times smaller. Comparing a prediction against a bound is exactly the kind of quiet order-of-magnitude inflation this article exists to prevent. The honest number is $1.2\times10^{8}$ — still one of the most remarkable improvements in the history of measurement. *Verified in verify_aq-sr-michelson-morley-isotropy-tests.py at mp.dps = 50: $\lambda = 550.0$ nm recovered from the paper's own " $11$ m $= 2\times10^{7}$ wavelengths"; $\Delta N = 0.400$ exactly; the owner's $0.373$ reproduced from $\lambda = 589$ nm; $\Delta c/c \le 1.25\times10^{-10}$ by two independent routes agreeing to $10^{-20}$ ; the $4.7$ km/s wind bound; and the factor $1.17\times10^{8}$ — every one artefact-guarded against perturbations of $L$ , $\lambda$ , $\beta^2$ and the null fraction.*

Figure

PUNCHLINE — one axis, one observable, one hundred and twenty-eight years. Every point is a bound on the SAME quantity: the fractional anisotropy of the TWO-WAY speed of light, Δc/c. The dashed line is the aether's prediction for the Earth's orbital motion, Δc/c = β²/2 = 5×10⁻⁹ — the thing that was supposed to be there. Michelson and Morley pushed the bound a factor of forty below it in 1887 and it has fallen by another eight orders of magnitude since, without ever once lifting off zero. Note what the improvement is NOT made of: the arms did not get longer. They got SHORTER. The gain is entirely in frequency stability

Common trapKill the "one fortieth of a fringe" misquote — it libels the instrument

The AllQFT base states that the 1887 null was "consistent with zero at the level of $\sim 1/40$ of a fringe." It was not. Michelson and Morley expressed their result as a fraction of the predicted displacement, and said so in the same sentence: "the displacement to be expected was 0·4 fringe … The actual displacement was certainly less than the twentieth part of this, and probably less than the fortieth part." One fortieth of $0.4$ is $\approx 0.01$ fringe. The misquote recasts a sensitivity ratio as an absolute resolution, making 1887 sound like a near-miss instead of the forty-fold rout it was. A corpus sweep finds it only in this article's base; the owner `aq-sr-michelson-morley-experiment` states it correctly. It dies here.

Common trapThe anisotropy coefficient has TWO sign conventions in the literature and they are negatives of each other

You will meet the Michelson–Morley RMS combination written both as $\left(\beta_{\!*} - \delta - \tfrac12\right)$ and as $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ . Both are correct — of different quantities. The first is the coefficient of $\sin^2\theta$ in $\,c/c(\theta)$ ; the second, its coefficient in $\,c(\theta)/c$ . The two are reciprocals to $O(v^2/c^2)$ , hence negatives. Both vanish at the special-relativity point, so the obvious check cannot tell them apart — which is exactly what lets the error propagate. The discriminating test is the Galilean limit, where the aether answer is known independently: only $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ reproduces $c_\parallel = c(1-\beta^2)$ , $c_\perp = c\sqrt{1-\beta^2}$ . Equation (1) uses it. Experimental papers quote the magnitude $\left|\beta_{\!*} - \delta - \tfrac12\right|$ , and this article follows them when quoting bounds.

Worked examples

1 · What did Michelson and Morley actually prove?

Their apparatus had an effective arm $L = 11\ \mathrm{m}$ and, by their own reckoning, $2\times10^{7}$ wavelengths in that arm. They predicted a $0.4$ -fringe shift and saw less than one fortieth of it. Turn that into a bound on the isotropy of the speed of light, and into a bound on the Earth's velocity through any aether.

Step 1 — identify. The observable is the peak two-way anisotropy $\Delta c/c$ , equation (2); the transfer function $\Delta N = 2L\beta^{2}/\lambda$ is owned by `aq-sr-michelson-morley-experiment` and used here.

Step 2 — the wavelength, from their own words. $\lambda = 11/(2\times10^{7}) = \mathbf{550\ \mathrm{nm}}$ , so $\Delta N = 2\times(2\times10^{7})\times10^{-8} = \mathbf{0.400}$ fringes — the $0.4$ they printed, to three figures.

Step 3 — the bound in fringes. Less than the fortieth part of this: $\Delta N_{\text{obs}} \le 0.4/40 = \mathbf{0.01}$ fringe.

Step 4 — the boundary number. Eliminate $\beta$ between $\Delta N = 2L\beta^2/\lambda$ and $\Delta c/c = \beta^2/2$ :

\frac{\Delta c}{c} \;=\; \frac{\lambda\,\Delta N}{4L} \;=\; \frac{(550\times10^{-9})(0.01)}{4\times 11} \;=\; \mathbf{1.25\times10^{-10}} .

And as a velocity: $v \le 30\ \mathrm{km/s} \times \sqrt{1/40} = \mathbf{4.7\ \mathrm{km/s}}$ . In 1887, in a basement, with a sandstone slab floating on mercury, two people proved that the speed of light is the same in perpendicular directions to better than one part in $8\times10^{9}$ .

2 · What does a modern cavity need, and what does it buy?

A rotating pair of cryogenic sapphire resonators reaches a fractional frequency resolution of $10^{-18}$ on the $2\Omega$ line. What bound does that place on the RMS anisotropy parameter, given the CMB boost?

Step 1 — identify. From (4) the beat amplitude is $\left|A_{\!*}\right| \beta^{2}$ with $A_{\!*} = \tfrac12 - \beta_{\!*} + \delta$ . The parameter bound is the frequency bound divided by $\beta^2$ — the boost is a lever, and a weak one.

Step 2 — the lever. Through the CMB rest frame, $v = 369.82\ \mathrm{km/s}$ , so $\beta = 1.2336\times10^{-3}$ and $\beta^{2} = \mathbf{1.52\times10^{-6}}$ .

Step 3 — divide. $\left|A_{\!*}\right| \lesssim 10^{-18}/1.52\times10^{-6} = \mathbf{6.6\times10^{-13}}$ . Published analyses quote $\beta_{\!*} - \delta - \tfrac12 = (4 \pm 8)\times10^{-12}$ — the same order, once the sidereal decomposition and boost model are done properly.

Step 4 — the boundary number, and the lesson. A part in $10^{18}$ in frequency buys only a part in $10^{12}$ in the test-theory parameter, because the signal is suppressed by $\beta^2 \approx 10^{-6}$ . That dilution is inescapable in any second-order experiment, and it is why the field moved to the Standard-Model Extension, where coefficient bounds are not throttled by one universal boost. But hold on to the raw number: two crystals in a room, disagreeing about the speed of light by less than one part in a billion billion — and the disagreement is zero.

Physics of the portrait

The instrument, not the person, is the protagonist of this chapter — which is fitting, because the modern experiment's authorship is a list of institutions and the 1887 experiment's fame belongs to an apparatus.

Origins · 2 minOptical cavity and how this was discovered

Optical cavity

1899–2015 · Fabry and Pérot build the etalon · Michelson folds his beam eleven metres · sapphire and silicon carry it to a part in 10¹⁸

What it is. Two mirrors facing each other, and a light wave that survives between them only if a whole number of half-wavelengths fits. Charles Fabry and Alfred Pérot built the first practical one at the end of the 1890s. Its resonance condition is why the Michelson–Morley experiment could be rebuilt as a frequency measurement — and frequency is the quantity human beings measure better than any other. Why it displaced the interferometer. Michelson's fringes are a position readout — estimate where a dark line sits, to perhaps a hundredth of a fringe. A cavity is a frequency readout: lock a laser to it and count beats, and modern locks hold to a part in $10^{16}$ . Same physics, but one is limited by a human eye and the other by the thermal noise of a crystal lattice. Nothing about the light path got better. The clock did. What it costs. In $\delta\nu/\nu = \delta c/c - \delta L/L$ the second term is merciless: a fractional length change of $10^{-18}$ is a displacement of $10^{-19}\ \mathrm{m}$ on a $10\ \mathrm{cm}$ cavity — a fraction of a nuclear diameter. Hence sapphire and single-crystal silicon; hence cryogenic temperatures, where the thermal expansion coefficient passes through zero; hence the $2\Omega$ line, which an orientation-correlated strain cannot follow. The history of this experiment since 1887 is almost entirely the history of making $\delta L/L$ small enough to see $\delta c/c$ .

History

1881

— Michelson runs the first version alone in Potsdam; the sensitivity is marginal, and Lorentz later shows his theory of the apparatus was wrong by a factor of two.

1887

— Michelson and Morley, in Cleveland, report the definitive null: expected displacement 0.4 fringe, observed "certainly less than the twentieth part of this, and probably less than the fortieth part."

1889

— FitzGerald proposes that bodies moving through the aether contract along the motion by exactly the factor that would null the fringe shift; Lorentz reaches the same hypothesis independently in 1892.

1930

— Georg Joos repeats the experiment in Jena with a quartz-framed interferometer and photographic readout, tightening the bound by about an order of magnitude.

1933

— Dayton Miller publishes a claimed positive ether-drift signal from Mount Wilson (Rev. Mod. Phys. 5, 203) — the most serious challenge the null result ever faced.

1955

— Shankland, McCuskey, Leone and Kuerti re-analyse Miller's raw data (Rev. Mod. Phys. 27, 167) and trace the signal to statistical scatter and temperature gradients in his shed. The null stands.

1979

— Brillet and Hall lock a laser to a rotating Fabry-Perot cavity (Phys. Rev. Lett. 42, 549), improving on the best previous laser test by a factor of about 4000. The cavity era begins.

2009

— Herrmann and co-workers run a rotating optical cavity at the 10 to the minus 17 level (Phys. Rev. D 80, 105011), bounding the RMS combination beta minus delta minus one half at parts in 10 to the 12.

2015

— Nagel and co-workers reach (9.2 plus or minus 10.7) times 10 to the minus 19 with rotating cryogenic sapphire oscillators (Nat. Commun. 6, 8174), bounding nine rotation and boost anisotropies of the speed of light at once.

Going further

Primary sources. Michelson & Morley, Am. J. Sci. 34, 333 (1887) — short, readable, and the source of every 1887 number above · Shankland, McCuskey, Leone & Kuerti, Rev. Mod. Phys. 27, 167 (1955) · Brillet & Hall, Phys. Rev. Lett. 42, 549 (1979) · Herrmann et al., Phys. Rev. D 80, 105011 (2009) · Nagel et al., Nat. Commun. 6, 8174 (2015) · Planck Collaboration, A&A 641, A1 (2020) for the CMB dipole, $v = 369.82 \pm 0.11\ \mathrm{km/s}$ . Texts: Rindler ch. 1 · Will, Theory and Experiment in Gravitational Physics §2.4 (RMS).

Provenance. Signature $(+,-,-,-)$ . Every number is recomputed at 50 digits in verify_aq-sr-michelson-morley-isotropy-tests.py, which reads this file, regexes the numbers out of the prose and asserts them. Base defects repaired: (i) " $\sim 1/40$ of a fringe" is a misquote — the bound is one fortieth of the $0.4$ -fringe expectation, $\approx 0.01$ fringe; (ii) the base states the SR values as $\beta_{\!*} = 1/2,\ \delta = -1/2$ , which do not null the base's own anisotropy formula ( $\tfrac12 - \tfrac12 - \tfrac12 = -\tfrac12 \ne 0$ ); the correct value is $\delta = 0$ ; (iii) the base's nine-order improvement compares a prediction against a bound — the correct figure is $1.2\times10^{8}$ ; (iv) the base evaluates the fringe shift with $\lambda \approx 500\ \mathrm{nm}$ , giving $0.44$ , while asserting $0.4$ elsewhere.

Flagged, not fixed (live canon). `aq-sr-michelson-morley-experiment` prints $\Delta N = 0.373$ fringes; with $L = 11\ \mathrm{m}$ , $\lambda = 589\ \mathrm{nm}$ , $\beta = 10^{-4}$ it is $0.3735$ , which rounds to $0.374$ . The owner truncated. This article quotes $0.373$ verbatim by coordinator ruling; the repair belongs to a dedicated pass on the owner.

Scope. The 1887 fringe-shift derivation is owned by `aq-sr-michelson-morley-experiment` and used, not repeated. Equation (1)'s transformation is `aq-sr-robertson-mansouri-sexl-framework`'s; the trio's other two legs are `aq-sr-kennedy-thorndike-ives-stilwell`'s; the successor is `aq-sr-standard-model-extension-lorentz-violation`.

Builds on. The Michelson–Morley Experiment · The Luminiferous Ether · Lorentz–FitzGerald Contraction. Leads to. Kennedy–Thorndike and Ives–Stilwell · The Robertson–Mansouri–Sexl Test Theory · The Standard-Model Extension.

If the modern experiment is so much better, why are its arms so much shorter?

Because arm length was never the bottleneck. Michelson's fringe shift grows as $2L\beta^2/\lambda$ , so a longer arm does buy signal — hence the beam folded to eleven metres on a one-metre table. But a cavity reads a frequency, not a fringe position, and a frequency readout is limited by how stably the cavity holds its length, not by how long that length is. A longer resonator is harder to hold rigid. So the apparatus shrank — eleven metres in 1887, a few centimetres today — while the sensitivity improved a hundred-million-fold. The gain is entirely in $\delta L/L$ and the frequency reference.

Does a null Michelson-Morley result prove special relativity?

No — and this is the most important thing to understand about it. The FitzGerald–Lorentz hypothesis (a real aether, real length contraction, no time dilation) predicts a null Michelson–Morley result exactly, at every velocity and orientation: read it off the limiting-cases table, where at $\alpha = 0,\ \beta_{\!*} = \tfrac12,\ \delta = 0$ the anisotropy coefficient vanishes identically. What it cannot survive is Kennedy–Thorndike, whose signal is the isotropic term of (1) and equals $-\tfrac12\beta^2$ there — nonzero. Isotropy of $c$ is one leg of a tripod. `aq-sr-kennedy-thorndike-ives-stilwell` supplies the other two.

Why does the experiment care about the cosmic microwave background?

Because a test of isotropy is worthless if you happen to be at rest in the preferred frame. Equation (1) scales as $v^2/c^2$ , so sensitivity to the test-theory parameters is the frequency resolution divided by the square of the lab's velocity through whatever frame the anisotropy is defined against. Michelson assumed the Sun ( $v \approx 30\ \mathrm{km/s}$ ). The CMB rest frame — where the microwave sky has no dipole — is the only observationally-defined candidate, and the solar system moves through it at $369.82\ \mathrm{km/s}$ , giving $v^2/c^2 = 1.5\times10^{-6}$ : a hundred times the orbital value. Adopting it improves every RMS bound by two orders of magnitude for free, and makes the sidereal and annual modulations calculable rather than assumed.

What was Dayton Miller's positive result, and why does it matter?

Miller ran interferometers at Mount Wilson through the 1920s and reported, in 1933, a persistent ether-drift signal of about $10\ \mathrm{km/s}$ with a sidereal period. It was neither fraud nor incompetence: Miller took hundreds of thousands of readings and was taken seriously enough that Einstein wrote about it. In 1955 Shankland and co-workers obtained his raw data sheets and showed the signal tracked the temperature gradient across his observation hut. It matters because it is precisely the failure mode equation (3) predicts — an orientation-correlated $\delta L/L$ is indistinguishable from an orientation-correlated $\delta c/c$ in a single measurement. Cryogenics, continuous rotation, the $2\Omega$ line, the sidereal sidebands: every design feature of the modern experiment exists to defeat Miller's ghost.

Which sign is right for the Michelson-Morley RMS coefficient?

Both — and the trap is that they belong to different quantities. $\left(\tfrac12 - \beta_{\!*} + \delta\right)$ is the coefficient of $\sin^2\theta$ in $c(\theta)/c$ ; $\left(\beta_{\!*} - \delta - \tfrac12\right)$ is its coefficient in the reciprocal, $c/c(\theta)$ . To second order they are negatives, and both vanish at the special-relativity point, so "it goes to zero for SR" discriminates nothing. The Galilean limit does: with $\alpha = \beta_{\!*} = \delta = 0$ you must recover $c_\parallel = c(1-\beta^2)$ and $c_\perp = c\sqrt{1-\beta^2}$ , so $c(\theta)$ must increase with $\sin^2\theta$ . Only the first form does. Papers quote the magnitude $\left|\beta_{\!*} - \delta - \tfrac12\right|$ and are unharmed either way.

Is the two-way speed of light the only thing you can measure?

Operationally, yes — and this is deeper than it looks. To measure a one-way speed you must know the far clock agrees with the near one, and to synchronise distant clocks you must send a signal and assume how long it took. The assumption you make is the one-way speed you then "measure." A round trip escapes the circle: one clock, one place, no convention. It is why the RMS framework carries a free synchronisation parameter $\varepsilon$ that cancels identically from every two-way quantity. Isotropy of the one-way speed of light is a convention; isotropy of the two-way speed is a fact — and it is the fact this article measures.