Lorentz–FitzGerald Contraction Hypothesis

In 1889 the American journal Science printed a letter of a few hundred words from an Irish physicist, George Francis FitzGerald. Michelson and Morley had just failed, spectacularly, to detect the Earth's motion through the ether, and FitzGerald offered "almost the only hypothesis" that could reconcile their null result with everything else physics believed: suppose the length of a body changes according to whether it moves through the ether or across it, by an amount depending on the square of the ratio of its speed to that of light. Shrink the interferometer's arm along the wind, and the wind hides itself. Three years later, knowing nothing of the letter, Hendrik Lorentz proposed the same thing in Leiden. What makes the idea more than a fudge is that the required shrinkage is not adjustable. Demand only that the two arms give equal round-trip times, and the algebra hands you exactly one answer: the arm along the motion must be shorter by the factor $\sqrt{1-\beta^2}$ , $\beta = v/c$ . Any other rule leaves a residual fringe shift that Michelson and Morley would have seen. The factor was right; the story around it was wrong. In the ether picture the contraction is an absolute, one-way deformation of matter streaming through a medium — one that must afflict sandstone and brass and glass identically, for no reason anyone could give, and that can never be measured, because every ruler you might use has shrunk by the same amount. In 1905 Einstein derived the identical $\sqrt{1-\beta^2}$ from two postulates, with no ether at all: same number, utterly different physics.

Key takeaways

The Lorentz–FitzGerald contraction hypothesis (FitzGerald 1889, Lorentz 1892) says a body moving at $v$ through the ether physically shortens along its line of motion by $\sqrt{1-\beta^2}$ , $\beta = v/c$ , while its transverse dimensions are untouched.
It was invented for one purpose: to explain why the Michelson–Morley experiment saw no fringe shift. Shrink the parallel arm and the two round-trip times become equal.
The factor is unique. Setting $t_\parallel(L_\parallel) = t_\perp$ gives a linear equation in $L_\parallel$ , so it has exactly one root — $L_\parallel = L_0\sqrt{1-\beta^2}$ . A factor $1 - k\beta^2$ leaves a residual $\Delta N = (2L_0/\lambda)\beta^2(1-2k)$ , which vanishes if and only if $k = \tfrac12$ .
At the Earth's orbital speed the factor is $1 - 5\times10^{-9}$ : Michelson's $11\ \mathrm{m}$ arm loses just $55\ \mathrm{nm}$ — invisible directly, yet exactly enough to erase the predicted $0.373$ -fringe shift. At $\beta = 0.6$ a $1\ \mathrm{m}$ rod becomes $0.80\ \mathrm{m}$ ; at $\beta = 0.866$ , $0.50\ \mathrm{m}$ .
Key idea: $\sqrt{1-\beta^2}$ is the one and only contraction that nulls the ether-wind experiment — and it is the same factor special relativity later obtains from its postulates. In one story it is an unexplained conspiracy of matter; in the other it is a theorem about spacetime. The arithmetic never changed; the ontology did.

The core insight

Ask what shrinkage of the parallel interferometer arm would make its light round trip take exactly as long as the transverse arm's. The equation is linear in the arm length, so it has one root and one only: $L_\parallel = L_0\sqrt{1-\beta^2}$ . That is the whole content of the hypothesis — a factor forced by the null result, not fitted to it. Its weakness was never the number but the story: an absolute deformation of all matter, relative to an ether nobody could detect, unmeasurable by any ruler that travels with it.

Where this lives today

The factor is now everyday engineering. At the Large Hadron Collider a proton carries $6.8\ \mathrm{TeV}$ , giving $\gamma = E/mc^2 \approx 7200$ and a contraction factor $1/\gamma \approx 1.4\times10^{-4}$ : in the laboratory frame the colliding bunches are flattened discs, and heavy-ion runs collide lead nuclei that are pancakes, whose overlap geometry must be modelled correctly or the initial conditions of the quark–gluon plasma come out wrong. What has not survived is the ontology, and the experiments that killed it are still running. If the contraction were an absolute deformation relative to a privileged frame, that frame's direction would eventually show itself; rotating cryogenic optical resonators now bound any anisotropy of the speed of light at $\Delta c/c \lesssim 10^{-18}$ , and Kennedy–Thorndike-type experiments bound any dependence on the laboratory's velocity too. Every one returns FitzGerald's null and Einstein's interpretation — a permanent lesson in the difference between fitting the data and understanding it.

The physical picture

Michelson's interferometer is a race between two beams of light. Down the arm lying along the ether wind, light fights upstream then rides back down; across the wind it must be angled upstream just to be carried straight over. Both arms are $L_0$ long, and yet the parallel beam loses — by $\Delta t \approx L_0\beta^2/c$ , which for the 1887 apparatus is $110\ \mathrm{nm}$ of extra optical path and a predicted shift of $0.373$ fringes on a $90^\circ$ rotation — twice the $110\ \mathrm{nm}/\lambda = 0.187$ of a single orientation, because rotating swaps the two arms and reverses the sign of the path difference, which is why equation (3) carries $4L_{0}/\lambda$ and not $2L_{0}/\lambda$ . That shift never appeared. FitzGerald's move is disarmingly simple. The parallel beam loses because its arm is too long — so shorten the arm, by exactly enough that the two round trips finish together. The remarkable thing is that the required shortening is not something you tune: it drops out of the algebra as a single unique number, and it is tiny. The $11\ \mathrm{m}$ arm need lose only $55\ \mathrm{nm}$ — about a tenth of a wavelength of sodium light — and its light path shortens by $110\ \mathrm{nm}$ , precisely the $110\ \mathrm{nm}$ the ether wind was adding. The books balance to the last digit because they are the same book.

Figure

Left: with equal arms the parallel beam loses 110 nm of light path and the fringes must shift by ΔN = 0.373. Right: shorten that arm by √(1−β²) — 55 nm out of 11 m — and its path drops by exactly 110 nm, so t∥ = t⟂ identically and ΔN = 0 at every angle and every β. Shortening drawn hugely exaggerated

So the hypothesis works — but look at what must be true for it to work, and the discomfort begins. The factor is not "roughly right": it is the only factor that nulls the experiment, and the 1887 data pin the coefficient of $\beta^2$ in it to better than three percent.

Figure

PUNCHLINE — the contraction factor √(1−β²) against β, with the numbers ON the curve: β = 0.1 → 0.995 (invisible), β = 0.6 → 0.800, β = 0.866 → 0.500. Michelson–Morley lived at β = 10⁻⁴, in the flat far-left corner where the factor is 1 − 5×10⁻⁹ — and only this curve nulls the experiment

The far-left corner of that curve is where all of nineteenth-century physics actually lived. At $\beta = 10^{-4}$ the factor differs from $1$ by five parts in a billion — no ruler, no micrometer, no comparison of standards could have caught it. The hypothesis therefore predicted a real deformation of matter that was, in 1889, undetectable by any means except the very experiment it was invented to explain.

The core equations

The hypothesis is a single statement about lengths, and it is anisotropic: only the dimension along the motion changes, while the transverse dimensions are left exactly as they were.

L_{\parallel} = L_0\sqrt{1-\beta^{2}}, \qquad L_{\perp} = L_{0\perp}, \qquad \beta \equiv \frac{v}{c} \qquad (1)

where: $L_0$ — proper (rest) length along the motion; $L_\parallel$ — its length while moving at $v$ through the ether; $L_{0\perp}, L_\perp$ — transverse dimensions, unchanged. Physical reading: the factor is at most $1$ and falls to $0$ as $\beta \to 1$ ; at everyday speeds it is $1$ to within a part in $10^{16}$ , which is why nobody had noticed. Everything rests on the two interferometer arm times, computed from Galilean velocity addition. The parallel arm's time is linear in its length; the transverse arm's is untouched by the hypothesis, since the contraction acts only along the motion.

t_{\parallel}(L_\parallel) = \frac{2L_{\parallel}}{c}\cdot\frac{1}{1-\beta^{2}}, \qquad t_{\perp} = \frac{2L_{0}}{c}\cdot\frac{1}{\sqrt{1-\beta^{2}}} \qquad (2)

where: $t_\parallel, t_\perp$ — round-trip light times along and across the ether wind. Physical reading: $t_\parallel$ grows in strict proportion to $L_\parallel$ , with the positive slope $2/[c(1-\beta^2)]$ , so it crosses the fixed value $t_\perp$ at exactly one length — and that is why the contraction factor is unique rather than merely convenient. Suppose the true factor were some other function $f(\beta)$ , so $L_\parallel = f L_0$ . The arms then fail to balance and a residual fringe shift survives the $90^\circ$ rotation.

\Delta N(f) = \frac{4L_{0}}{\lambda}\left(\frac{f}{1-\beta^{2}} - \frac{1}{\sqrt{1-\beta^{2}}}\right) \qquad (3)

where: $\Delta N$ — fringe shift on a $90^\circ$ rotation; $\lambda$ — wavelength of the light. Physical reading: $\Delta N$ is a strictly increasing linear function of $f$ , crossing zero at $f = \sqrt{1-\beta^2}$ and nowhere else. Put $f = 1$ and it returns the uncontracted prediction, $2L_0\beta^2/\lambda = 0.373$ fringes. Finally, the size of the effect. Expanding for small $\beta$ shows the contraction is second order, exactly like the signal it cancels.

\frac{\Delta L}{L_0} = \sqrt{1-\beta^{2}} - 1 \approx -\frac{\beta^{2}}{2} \qquad (4)

where: $\Delta L = L_\parallel - L_0 < 0$ — the shortening. Physical reading: at the Earth's orbital speed $\Delta L/L_0 = -5\times10^{-9}$ — a one-metre rod loses five nanometres, and the $11\ \mathrm{m}$ interferometer arm loses $55\ \mathrm{nm}$ .

Derivation ASolving for the contraction factor — and finding it is unique

This is the intellectual heart of the hypothesis. The textbook move is to assert the factor $\sqrt{1-\beta^2}$ and then verify that it happens to null the experiment — which is backwards, and makes the idea look like a lucky guess. Instead we leave the contracted arm length as an unknown, impose the one condition the null result demands, and solve. The equation turns out to be linear, so it has exactly one root, and that root is $L_0\sqrt{1-\beta^2}$ . No freedom, no family of nearly-working factors, no adjustable parameter anywhere.

Step 1. Set up the unknown. Let the arm along the ether wind have some length $L_\parallel$ , to be determined, and let the transverse arm keep its rest length $L_0$ — whatever the hypothesis turns out to be, it acts only along the motion. Why: refusing to assume the answer is the whole point; substitute $\sqrt{1-\beta^2}$ now and we can only ever check it, never derive it.

Step 2. Impose the null. The fringes did not move at any orientation, which requires $t_\parallel = t_\perp$ . From equation (2), $\dfrac{2L_\parallel}{c}\cdot\dfrac{1}{1-\beta^2} = \dfrac{2L_0}{c}\cdot\dfrac{1}{\sqrt{1-\beta^2}}$ . Why: this is the only condition the experiment supplies — the fringes are blind to the common part of the two times and sensitive only to their difference.

Step 3. Notice the structure before solving. The left side is linear in $L_\parallel$ with slope $2/[c(1-\beta^2)] > 0$ ; the right side is a constant. A strictly increasing straight line crosses a horizontal line exactly once. Why: this settles uniqueness before any algebra — whatever the root is, there is precisely one, so no rival contraction rule can also null the experiment — given the hypothesis's own assumption that transverse lengths are unchanged, since the null result by itself fixes only the ratio $L_\parallel/L_\perp$ .

Step 4. Solve. Cancelling $2/c$ , $\dfrac{L_\parallel}{1-\beta^2} = \dfrac{L_0}{\sqrt{1-\beta^2}}$ , hence $L_\parallel = L_0\dfrac{1-\beta^2}{\sqrt{1-\beta^2}} = \boxed{L_0\sqrt{1-\beta^{2}}}$ . Why: the $(1-\beta^2)$ of the parallel arm and the $\sqrt{1-\beta^2}$ of the transverse arm differ by exactly one square root — and the contraction is that square root.

Step 5. Confirm exactness, and measure the tolerance. Substituting back gives $t_\parallel = (2L_0/c)(1-\beta^2)^{-1/2} = t_\perp$ identically — not to order $\beta^2$ , but at every $\beta$ and every orientation. Were the factor instead $f = 1 - k\beta^2$ , equation (3) would leave $\Delta N = (2L_0/\lambda)\beta^2(1-2k)$ , vanishing only at $k = \tfrac12$ ; the 1887 bound $\Delta N < 0.01$ against a $0.373$ prediction pins $k = 0.500 \pm 0.013$ . Why: uniqueness is what raises the hypothesis from an excuse to a quantitative claim — and what makes its later re-derivation from relativity so striking. *The nulling condition is degree 1 in $L_\parallel$ , confirming a single root $L_0\sqrt{1-\beta^2}$ ; back-substitution gives $t_\parallel - t_\perp = 0$ identically; the residual formula and the tolerance $k = 0.500 \pm 0.013$ follow directly.*

Go deeper · Derivation B · 2 minFrom an ad hoc rescue to a theorem about spacetime

Derivation BFrom an ad hoc rescue to a theorem about spacetime

Getting the right number is not the same as having the right theory, and this is the classic case. What follows computes no new formula; it audits the one we have, asking what the hypothesis commits us to, why physicists found those commitments intolerable, and how within sixteen years the same $\sqrt{1-\beta^2}$ reappeared as an unavoidable consequence rather than an assumption. The point is not that FitzGerald and Lorentz were foolish — it is that a formula and its interpretation are separable, and the second is where the physics lives.

Step 1. Audit the universality. The interferometer is sandstone, brass, glass and silver, and the light path is set by the mirror positions those materials hold. From Step 5 above, the shrinkage coefficient must equal $\tfrac12$ to within three percent — for every one of them, since a mismatch would displace the mirrors and reappear as a residual shift. Why: the hypothesis is not "matter contracts a bit", it is "all matter contracts by precisely the same amount, regardless of what holds it together" — and nothing in the physics of 1889 required that.

Step 2. Audit the observability. Take a rod of rest length $L_0$ and a ruler of rest length $\ell_0$ , both moving at $v$ . The rod contracts to $L_0\sqrt{1-\beta^2}$ — and so does the ruler, to $\ell_0\sqrt{1-\beta^2}$ . The rod therefore spans $\dfrac{L_0\sqrt{1-\beta^2}}{\ell_0\sqrt{1-\beta^2}} = \dfrac{L_0}{\ell_0}$ ruler-lengths, independent of $\beta$ . Why: the contraction is invisible to every co-moving measurement. The only instrument that responds compares light travel times in two directions — the Michelson–Morley experiment itself. A hypothesis testable only by the experiment it was invented to explain is unfalsifiable by construction.

Step 3. Watch Lorentz turn the guess into a consequence. If cohesive forces are electromagnetic and the field of a moving charge flattens along its motion, the equilibrium spacing of matter should flatten by the same factor. His 1895 Versuch introduced the "local time" $t' = t - vx/c^2$ and a theorem of corresponding states, correct to first order in $\beta$ ; by 1904 he had a full transformation, valid to all orders, in which the contraction is a corollary rather than a postulate. Why: real progress — the factor is now derived from a dynamical theory. But the ether survives, the contraction stays absolute, and "local time" is still a mathematical convenience rather than time.

Step 4. Watch Einstein remove the scaffolding. In 1905 he obtained the same $\sqrt{1-\beta^2}$ from two postulates — the relativity principle and the constancy of $c$ — with no ether, no privileged frame, and no assumption about the forces inside matter. The contraction becomes reciprocal: each observer finds the other's rulers short, with no contradiction, because they disagree about which end-readings are simultaneous. Why: identical arithmetic, transformed physics. In Lorentz's world matter is deformed by its passage through a medium; in Einstein's, nothing is deformed at all, and the factor tells you how measurements of length translate between frames. See Length Contraction and The Lorentz Transformation. *The co-moving reading simplifies to $L_0/\ell_0$ with no residual $\beta$ dependence.*

Limiting cases

Case	Result	Meaning
Galilean limit $c \to \infty$	$\beta \to 0$ , $\sqrt{1-\beta^2} \to 1$	infinitely fast light has nothing to hide and needs no contraction; the effect exists only because $c$ is finite
Low speed ( $\beta \ll 1$ )	$\sqrt{1-\beta^2} \approx 1 - \tfrac12\beta^2$	second order, like the signal it cancels; at $\beta = 10^{-4}$ that is $5\times10^{-9}$
Ultra-relativistic ( $\beta \to 1$ )	$\sqrt{1-\beta^2} \to 0$	flattens without limit: a $6.8\ \mathrm{TeV}$ proton has $1/\gamma \approx 1.4\times10^{-4}$
Any factor $f \neq \sqrt{1-\beta^2}$	$\Delta N \neq 0$	a residual fringe shift survives — with transverse lengths held fixed, the null result admits exactly one contraction rule

Worked examples

1 · The 55 nanometres that hid the ether

The effective arm was $L_0 = 11\ \mathrm{m}$ , the light sodium at $\lambda = 589\ \mathrm{nm}$ , and the Earth's orbital speed gives $\beta = 10^{-4}$ . By how much must the parallel arm contract, and does that really erase the predicted $0.373$ -fringe shift?

Step 1. Evaluate the factor: $\sqrt{1-\beta^2} = \sqrt{1 - 10^{-8}} = 1 - 5.0\times10^{-9}$ . The shortfall from unity is $\beta^2/2 = 5\times10^{-9}$ , exactly as equation (4) predicts.

Step 2. Convert to a length: $\Delta L = L_0(1 - \sqrt{1-\beta^2}) = 11\ \mathrm{m} \times 5.0\times10^{-9} = \mathbf{55\ nm}$ — a mere $0.09$ of a wavelength of sodium light, a shortening no mechanical measurement of 1887, or of today, could detect on an optical bench.

Step 3. Convert to light path, and compare with what must be cancelled. The parallel round trip is $c\,t_\parallel = 2L_\parallel/(1-\beta^2)$ , so losing $\Delta L$ of arm removes $2\Delta L/(1-\beta^2) = \mathbf{110\ nm}$ of optical path — while the ether wind's extra path was $c\,\Delta t = L_0\beta^2 = 110\ \mathrm{nm}$ . They agree, and not by luck: the identity $2L_0(1-\sqrt{1-\beta^2})/(1-\beta^2) = 2L_0[(1-\beta^2)^{-1} - (1-\beta^2)^{-1/2}]$ holds exactly at every order in $\beta$ . The $0.373$ -fringe prediction becomes exactly zero.

Step 4. Scale it up. Nothing about $\sqrt{1-\beta^2}$ is small in principle — it is small only because the Earth is slow. At $\beta = 0.6$ the factor is $\sqrt{0.64} = 0.80$ , so a $1\ \mathrm{m}$ rod becomes $\mathbf{0.80\ m}$ ; at $\beta = 0.866$ it is $\mathbf{0.50}$ — half length, and exactly one half at $\beta = \sqrt{3}/2$ .

2 · Why no other factor will do

Suppose FitzGerald had guessed differently. Take two plausible wrong rules — $f = 1 - \beta^2/4$ (half the required shrinkage) and $f = 1 - \beta^2$ (the naive "cancel the $1/(1-\beta^2)$ " guess). What fringe shift does each leave behind?

Step 1. Use equation (3), with $4L_0/\lambda = 44/(5.89\times10^{-7}) = 7.47\times10^{7}$ : $\Delta N(f) = \dfrac{4L_0}{\lambda}\left(\dfrac{f}{1-\beta^2} - \dfrac{1}{\sqrt{1-\beta^2}}\right)$ .

Step 2. Half the shrinkage, $f = 1 - \tfrac14\beta^2$ . Expanding, $f/(1-\beta^2) \approx 1 + \tfrac34\beta^2$ while $(1-\beta^2)^{-1/2} \approx 1 + \tfrac12\beta^2$ , leaving a bracket of $\tfrac14\beta^2 = 2.5\times10^{-9}$ . Hence $\Delta N = 7.47\times10^{7} \times 2.5\times10^{-9} = \mathbf{+0.187\ fringes}$ — nineteen times the observed bound of $0.01$ , impossible to miss.

Step 3. Over-contraction, $f = 1 - \beta^2$ . Now $f/(1-\beta^2) = 1$ exactly, so the bracket is $1 - (1-\beta^2)^{-1/2} = -5.0\times10^{-9}$ and $\Delta N = \mathbf{-0.373\ fringes}$ : a shift as large as the uncontracted prediction, in the opposite direction. Over-shrink the arm and the fringes swing the other way, just as visibly.

Step 4. Read the general rule. For $f = 1 - k\beta^2$ the residual is $\Delta N = 0.373\,(1-2k)$ fringes: $k = 0$ gives $+0.373$ , $k = \tfrac12$ gives $0$ , $k = 1$ gives $-0.373$ . The bound $|\Delta N| < 0.01$ forces $|1-2k| < 0.0268$ , so $k = 0.500 \pm 0.013$ — the null result does not merely permit a contraction, it prescribes one to within three percent in its only free coefficient.

Physics of the portrait

The portrait here is not a person but an idea. The contraction has FitzGerald's insight and Lorentz's name attached to it, but its real biography is the sixteen years in which one factor migrated from a rescue, through a dynamical theory, into the geometry of spacetime.

Origins · 2 minLorentz contraction and how this was discovered

Lorentz contraction

1889–1905 · FitzGerald · Lorentz · Larmor · Einstein

The idea began in a letter. George Francis FitzGerald (1851–1901), professor of natural philosophy at Trinity College Dublin, wrote to Science in 1889 proposing that bodies change length as they move through the ether by an amount depending on the square of $v/c$ — a few hundred words that would carry his name for a century. He was so unsure the letter had been printed that when Hendrik Antoon Lorentz, who reached the same idea independently in 1892, wrote in 1894 to ask, FitzGerald told him to take the credit; Lorentz declined, and named it after them both. Lorentz then did what FitzGerald had not: he sought a reason. If intermolecular forces are electromagnetic and the fields of moving charges flatten, matter should flatten with them — an argument that carried him from the "local time" of the 1895 Versuch to the complete transformation of 1904, with Joseph Larmor reaching much of the same structure in Aether and Matter (1900). Then in 1905 Albert Einstein took two postulates and no ether, and the same $\sqrt{1-\beta^2}$ fell out as a theorem. The factor survived; the ether did not. FitzGerald never saw it — he died in 1901, aged forty-nine.

History

1887

— Michelson and Morley find no ether wind: the predicted 0.4-fringe shift is absent, the observed shift under 0.01.

1889

— FitzGerald, in a brief note to Science titled "The Ether and the Earth's Atmosphere", suggests that moving bodies change length by an amount depending on the square of their speed divided by that of light.

1892

— Lorentz, unaware of FitzGerald's letter, proposes an almost identical contraction to reconcile the null result with a stationary ether.

1894

— Lorentz learns of the letter and credits FitzGerald, who had not been sure it was ever printed.

1895

— Lorentz's Versuch introduces "local time" and the theorem of corresponding states, correct to first order in v/c.

1900

— Larmor, in Aether and Matter, carries the transformations to second order and finds a time-dilation factor.

1904

— Lorentz publishes the full transformation, valid to all orders, with the contraction as a consequence rather than an assumption.

1905

— Einstein derives the same factor from the relativity principle and the constancy of the speed of light, with no ether and no privileged frame.

Going further

Texts. Rindler, Introduction to Special Relativity, 2nd ed., §§1–3 · Taylor & Wheeler, Spacetime Physics, 2nd ed., Ch. 3 · Griffiths, Introduction to Electrodynamics, 4th ed., §12.1 · G. F. FitzGerald, "The Ether and the Earth's Atmosphere", Science 13 (1889) 390 · H. A. Lorentz, "Electromagnetic phenomena in a system moving with any velocity smaller than that of light" (1904). All numbers here are computed from the anchors $c = 2.998\times10^8\ \mathrm{m/s}$ , $v_\oplus = 29.8\ \mathrm{km/s}$ ( $\beta = 9.94\times10^{-5} \approx 10^{-4}$ ), sodium $\lambda = 589\ \mathrm{nm}$ , $L_0 = 11\ \mathrm{m}$ . Key check-values: the unique root of $t_\parallel(L_\parallel) = t_\perp$ is $L_0\sqrt{1-\beta^2}$ ; $\sqrt{1-\beta^2} = 1 - 5\times10^{-9},\ 0.995,\ 0.800,\ 0.500$ at $\beta = 10^{-4}, 0.1, 0.6, 0.866$ ; the $55\ \mathrm{nm}$ arm loss and the cancelling $110\ \mathrm{nm}$ of light path; the residuals $+0.373$ , $+0.187$ , $-0.373$ fringes for $f = 1,\ 1-\beta^2/4,\ 1-\beta^2$ ; and $k = 0.500 \pm 0.013$ . Note on wavelength: some sources quote $\lambda \sim 500\ \mathrm{nm}$ for the Michelson–Morley light; the correct value is the sodium D line at $589\ \mathrm{nm}$ , which gives a predicted shift of $0.373$ fringes rather than $0.44$ .

Builds on. The Michelson–Morley Experiment · The Luminiferous Ether · Ether Drift and the Absolute Rest Frame · Galilean Transformations. Leads to. Einstein's Two Postulates · The Constancy of the Speed of Light · Length Contraction · The Lorentz Transformation.

If everything contracts by the same factor, including my ruler, how could anyone ever detect it?

In the ether picture, you could not — and that is the hypothesis's deepest weakness, not a technicality. A rod of rest length $L_0$ and a ruler of rest length $\ell_0$ both shrink by $\sqrt{1-\beta^2}$ , so the rod still spans $L_0/\ell_0$ ruler-lengths exactly as before. Every co-moving measurement returns the uncontracted answer. The only instrument that responds is one comparing light travel times along two directions, because light does not contract with the apparatus — which is to say, a Michelson interferometer.

Was FitzGerald's original proposal really the exact factor √(1−β²)?

Not in the printed letter. FitzGerald's 1889 note to Science says the lengths of material bodies change "by an amount depending on the square of the ratio of their velocities to that of light" — the correct order, but no explicit formula. The precise factor emerged as the idea was worked out, in his own later lectures and in Lorentz's papers from 1892 onward. This changes the character of the achievement rather than diminishing it: FitzGerald saw that a second-order length change of the right size would do the job, and the uniqueness argument above shows that once you demand a genuine null rather than a mere reduction, the exact factor stops being a matter of guesswork altogether.

Does a contracted rod feel squeezed? Is there a stress inside it?

No. The contraction is not a compression, and no internal stress accompanies it. This is easy to get wrong because Lorentz's motivation was dynamical — electromagnetic cohesive forces flatten, so matter flattens with them — which sounds mechanical. But even in his account the rod sits in equilibrium at its new length: nothing pushes on it, nothing heats up, and no material property such as stiffness or density enters. In relativity the point is sharper still — the rod is entirely unchanged in its own frame, and the contraction describes only how a measurement made in another frame comes out. Nothing whatever is done to the rod.

Why does only the parallel dimension contract? Why not all three?

Because only the parallel dimension is implicated in the null result. The Michelson–Morley discrepancy comes from the difference between $(1-\beta^2)^{-1}$ along the motion and $(1-\beta^2)^{-1/2}$ across it, and it is cancelled by removing exactly one square root's worth from the arm along the motion; contract the transverse arm too and you reintroduce a mismatch. There is also a symmetry argument Lorentz himself gave: the motion picks out one axis and one only, so an isotropic transverse change has no preferred direction to point in, and nothing in the theory supplies a separate reason for one. A moving sphere therefore becomes an oblate spheroid, flattened along its velocity and untouched across it.

If the factor is exactly right, why is the hypothesis considered wrong?

Because "the formula is right" and "the theory is right" are different claims, and physics cares about both. The hypothesis asserts a real, absolute deformation measured against an ether frame; relativity asserts no deformation at all, and reads the same factor as a relation between two observers' measurements. The two disagree about reciprocity — relativity says each observer finds the other's rulers short, which is flatly impossible if there is a fact about who is really contracted — and about whether a privileged frame exists.

How much does the null result actually pin down the factor, quantitatively?

Very precisely, which is the surprise. Write the factor as $1 - k\beta^2$ and let $k$ float. The residual fringe shift on a $90^\circ$ rotation is $\Delta N = (2L_0/\lambda)\beta^2(1 - 2k) = 0.373\,(1-2k)$ fringes for the 1887 apparatus. Michelson and Morley's bound $\Delta N < 0.01$ then requires $|1 - 2k| < 0.0268$ , so $k = 0.500 \pm 0.013$ — the coefficient is fixed to within about three percent. A single experiment that measured nothing determined the leading coefficient of the contraction law to two significant figures.