Einstein's Two Postulates

Open Einstein's 1905 paper and the first thing you meet is not an interferometer. It is a magnet and a coil of wire. Faraday's Law of Induction records the coincidence that provoked him: move the magnet toward a stationary loop and the lab-frame story is an induced electric field pushing charges that stand still; move the loop toward a stationary magnet and the story is the magnetic force $q\mathbf{v}\times\mathbf{B}$ of the Lorentz Force Law, acting on charges precisely because they move — with no induced field anywhere. Different mechanism, different fields, identical current. The asymmetry, Einstein wrote, "does not appear to be inherent in the phenomena". Only the relative motion is real, yet the theory insists on knowing which body is really moving. That, and not the ether wind, is where the paper begins. His repair was not a new equation but a demolition. Take two statements — that the laws of physics look the same in every inertial frame, and that light in vacuum travels at $c$ regardless of how the source or observer moves — and treat them as axioms. Neither is exotic alone: The Principle of Relativity had been Galileo's since 1632, and the constancy of $c$ is what Maxwell's equations have said since 1865. What nobody had done was hold both at once and refuse to flinch — because holding both breaks a third assumption, so silent that in two centuries it was never written down: that all inertial observers share one universal time, $t' = t$ . This article proves the three cannot coexist. Not "time becomes strange" but a flat algebraic contradiction — two incompatible values forced on one and the same number. Absolute time is not revised by the postulates. It is refuted by them.

Key takeaways

Postulate 1 (relativity). The laws of physics take the same form in every inertial frame — old news for mechanics, but Einstein extended it to all of physics, electrodynamics included.
Postulate 2 (constancy). Light in vacuum travels at the same $c$ in every inertial frame, independently of the motion of the source and of the observer. The radical clause — and it does not say $c$ is the fastest speed, which is a consequence, not an axiom.
The owned result. Add absolute time ( $t' = t$ ) and you get a contradiction, not a theory. Linearity plus $t' = t$ forces $x' = a(x - vt)$ ; postulate 2 applied to a forward light ray demands $a = c/(c-v)$ , applied to a backward one it demands $a = c/(c+v)$ . One unknown, two answers, agreeing only at $v = 0$ . At $\beta = 0.6$ the same $a$ must be both $2.5$ and $0.625$ — a factor of $4$ apart, and the gap opens at first order in $\beta$ : the trouble is logical, not experimental.
Key idea: the two postulates do not make time relative — they make absolute time impossible. Granting $t' = t$ , postulate 2 hands you two different values for one scale factor. The only survivable move is to stop assuming that clocks in relative motion agree, and that single act of surrender is where the whole of relativity comes from.

The core insight

Two sentences, no apparatus: the laws are the same in every inertial frame, and light goes at $c$ in every inertial frame. They are consistent with each other — but not with the unspoken premise that everyone shares one clock. Assume $t' = t$ and postulate 2 forces the same scale factor to be both $c/(c-v)$ and $c/(c+v)$ . The contradiction is exact, algebraic, and unfixable except by deleting $t' = t$ — and everything the theory later says about clocks and rulers is the price of that one deletion.

Where this lives today

Postulate 2 is now the most brutally tested statement in physics, and it is tested by trying to break it: rotating cryogenic optical resonators — Michelson's stone slab rebuilt as perpendicular sapphire or silicon cavities, cooled to a few kelvin and spun on an air bearing — bound any directional dependence of $c$ at a fractional level near $10^{-18}$ . The most domestic application, though, is in your pocket. A GPS receiver fixes its position by comparing signals from satellites moving at several kilometres per second, and the solution assumes the signal travelled at exactly $c$ in the receiver's frame regardless of the satellite's motion — postulate 2, used as an engineering specification. If light picked up even a part in $10^{8}$ of the transmitter's velocity, position fixes would drift by metres within minutes. They do not. Two sentences from a patent office are load-bearing under a billion navigation devices.

The physical picture

Two spacecraft cross paths. Each pilot flicks on a laser and clocks the pulse. Postulate 1 says neither can discover which of them is "really" moving; postulate 2 says both read the same number for the pulse's speed. The pulse is running away from one pilot and chasing the other, and yet the gap closes at the same rate for both. The instinct is that something must be wrong with the light. It is not. What is wrong is a premise nobody stated, because it never occurred to anyone that it was a premise — that the two pilots' clocks can be assumed to agree on what time it is.

Figure

The two postulates — and everything they do NOT assume. Two sentences with no apparatus in them, and not even a speed limit: that is a consequence.

Now make the collision precise, because "something must give" is not physics. Grant the postulates, and grant, provisionally, absolute time. With $t' = t$ , linearity leaves exactly one free number: an overall stretch $a$ of the $x$ -axis. Send a light pulse to the right and postulate 2 tells you what $a$ must be. Send one to the left and it tells you again — a different answer.

Figure

PUNCHLINE — the contradiction machine. With t′ = t, one scale factor a must satisfy the forward light ray (a = 1/(1 − β)) and the backward one (a = 1/(1 + β)) at once. The branches touch only at β = 0; at β = 0.6 the same number must be both 2.5 and 0.625.

Look at what the figure is not saying. Not that absolute time is a poor approximation, or fails at high speed. It says the equations have no solution.

The core equations

The first postulate is a statement about the form of laws: write the correct equations in one inertial frame, rewrite them in the coordinates of a second frame gliding past at constant velocity, and you must get back the same equations, symbol for symbol, with no leftover $v$ . The second speaks about one quantity and is far more dangerous — it fixes a number, $c$ , and asserts that every inertial observer measures that same number for the speed of light in vacuum, however the lamp moves when it emits and however the observer moves when she catches it.

\textbf{P1:}\ \text{the laws take the same form in every inertial frame;} \qquad \textbf{P2:}\ |\mathbf{u}_{\text{light}}| = c \ \text{in every inertial frame} \qquad (1)

where: an inertial frame is one in which a free body moves at constant velocity (the operational test is Inertial Frames of Reference's); $c = 299\,792\,458\ \mathrm{m/s}$ exactly since the 1983 SI redefinition of the metre, so the number is a definition while P2 is the physical claim that it is the same for everyone. Physical reading: $c$ is not really "the speed of light" but a structural constant of spacetime, which light travels at because photons are massless. P2's two clauses are logically independent: source-independence is the subject of Emission Theories and Their Refutation, observer-independence was the target of the ether hunt. Now the collision. Adopt the standard configuration — $S'$ slides along $x$ at speed $v$ , origins coincident at $t = t' = 0$ — and add, provisionally, the premise the century before Einstein never questioned: one universal time, $t' = t$ . Homogeneity forces the map to be linear, since a nonlinear map would make a uniformly moving free particle accelerate in the other frame and so betray a preferred origin. With $t' = t$ fixed, the only remaining freedom is an overall stretch, and requiring the origin of $S'$ (the point $x' = 0$ ) to sit at $x = vt$ fixes its form completely.

x' = a\,(x - vt), \qquad t' = t \qquad (2)

where: $a$ — a single unknown number, possibly depending on $v$ but not on $x$ or $t$ . Physical reading: the whole Galilean world with one bolt loosened — setting $a = 1$ recovers Galilean Transformations exactly. We have granted ourselves the freedom to rescale the moving frame's ruler by any factor we like, and are about to find that even that is not enough. Fire a pulse to the right from the common origin, so $x = ct$ in $S$ . Then $x' = a(c-v)t$ while $t' = t$ , so its speed in $S'$ is $a(c-v)$ , and postulate 2 does not permit that to be anything but $c$ . Now fire one to the left, $x = -ct$ : in $S'$ it sits at $x' = -a(c+v)t$ , so its speed is $a(c+v)$ , and postulate 2 demands $c$ again. Two equations, one unknown.

a_{+} = \frac{c}{c-v} = \frac{1}{1-\beta}, \qquad a_{-} = \frac{c}{c+v} = \frac{1}{1+\beta}, \qquad \beta \equiv \frac{v}{c} \qquad (3)

where: $a_+$ — the value forced by the forward ray; $a_-$ — the value forced by the backward ray. Physical reading: these are not two approximations to be reconciled, nor two limits of one expression. They are two flat assertions about the same number, and $a_+ = a_-$ only when $v = 0$ . Each ray gets a vote, and they vote differently — so the theory as assembled is inconsistent, and the loosest bolt in it is $t' = t$ .

Where the contradiction comes from

Derivation AAbsolute time is incompatible with the two postulates

The usual textbook line is that the postulates force time to be relative — which sounds like a discovery about clocks, something a fast enough clock might check. The claim proved here is stronger: assume the two postulates and absolute time, and the resulting equations have no solution at all for $v \neq 0$ . Not an approximate solution, not one that degrades at high speed — none. Absolute time is not falsified by the postulates; it is refuted by them, the way a contradiction refutes a premise. Nothing from S1 is borrowed here — no Lorentz factor, no Lorentz transformation, no light clock — and which transformation does satisfy both postulates belongs to Derivation of the Lorentz Transformation.

Step 1. Name the three premises, so we know which one dies. (a) Linearity: the map $(x,t) \to (x',t')$ is linear. (b) Absolute time: $t' = t$ . (c) Postulate 2: light has speed $c$ in $S$ and in $S'$ . Linearity permits $x' = Ax + Bt$ ; requiring the origin of $S'$ ( $x' = 0$ ) to sit at $x = vt$ gives $B = -Av$ , so with $a \equiv A$ the map collapses to $x' = a(x-vt)$ , $t' = t$ . Why: every scrap of freedom (a) and (b) allow is now packed into one number $a$ — so if the postulates over-determine $a$ , they over-determine the whole theory.

Step 2. Fire the forward ray, $x = ct$ . Substituting, $x' = a(c-v)t$ , and since $t' = t$ its speed in $S'$ is $a(c-v)$ . Postulate 2 says this equals $c$ , so $a_+ = c/(c-v) = 1/(1-\beta)$ . Why: $S'$ is not permitted to see the pulse creeping away at $c - v$ , which is exactly what Galilean addition predicts — so its ruler must be <strong>stretched</strong> to compensate.

Step 3. Fire the backward ray, $x = -ct$ . Then $x' = -a(c+v)t$ , so the pulse's speed in $S'$ has magnitude $a(c+v)$ , and postulate 2 demands $c$ once more: $a_- = c/(c+v) = 1/(1+\beta)$ . Why: postulate 2 is about every ray in every direction — and the backward ray is where the same ruler would have to be <strong>squeezed</strong>. One ruler cannot be stretched and squeezed at once.

Step 4. Collide them. The same $a$ must satisfy both, so $c/(c+v) = c/(c-v)$ , hence $2cv = 0$ , hence $v = 0$ . Equivalently $a_+ - a_- = 2cv/(c^2-v^2)$ , zero only at $v = 0$ . Why: this is the payload. The three premises are jointly satisfiable only for frames not moving relative to each other — only, that is, when there was nothing to transform.

Step 5. Read the size of the gap, then name the survivor. Expanding, $a_+ - a_- = 2\beta + 2\beta^3 + \cdots$ — the disagreement is first order in $\beta$ , so even at Earth's orbital $\beta = 10^{-4}$ the two demands differ by $2\times10^{-4}$ , ten thousand times the $O(\beta^2)$ ether signal The Michelson–Morley Experiment was built to hunt. Premise (a) is forced by homogeneity and (c) is an axiom, so premise (b) is what goes: $t' \neq t$ . Why: the flaw never hid behind experimental difficulty — it sat in the algebra, needing no apparatus, only someone willing to notice that $t' = t$ was an assumption. Granting the moving frame its own time coordinate gives the map a second unknown, and a solution exists; finding it is the business of Derivation of the Lorentz Transformation. Verified symbolically in verify_aq-sr-einsteins-two-postulates.py: sympy derives $a_\pm$ from the two light rays, and `solve(Eq(c/(c-v), c/(c+v)), v)` returns exactly `[0]`; the gap $a_+ - a_- = 2cv/(c^2-v^2)$ has series $2\beta + 2\beta^3 + \cdots$ ; the ratio is $(c+v)/(c-v)$ ; and the $c \to \infty$ limit sends both branches to $1$ .

Go deeper · Derivation B · 2 minThe magnet and the conductor — the asymmetry that opens the 1905 paper

Derivation BThe magnet and the conductor — the asymmetry that opens the 1905 paper

Einstein did not begin with Michelson and Morley. He began with a magnet and a coil, and the goal here is to show that the opening is not a rhetorical flourish but a precise statement of the disease. We compute one measurable EMF by two completely different mechanisms — one purely magnetic, one purely electric — and get the same answer to the last digit. Postulate 1 says a theory that tells two irreconcilable stories about one relative motion has a defect in it, and the defect is not in the experiment. Faraday's Law of Induction and Lorentz Force Law own the laws used here; both are recalled by name, neither re-derived.

Step 1. Set the stage. A conducting rod of length $\ell = 0.10\ \mathrm{m}$ closes a circuit in the uniform field $B = 0.50\ \mathrm{T}$ of a magnet, and rod and magnet approach at $u = 2.0\ \mathrm{m/s}$ . Only the relative velocity is specified. Why: nature is handed one situation. We are about to hand it two theories, and watch the theory become confused about which body is moving.

Step 2. Story A — magnet at rest, rod moving. The lab field is static, so $\partial\mathbf{B}/\partial t = \mathbf{0}$ and Faraday's law gives an induced electric field of exactly zero, everywhere. What pumps the charge is the magnetic half of the Lorentz force: each carrier rides at $u$ and feels $q\,\mathbf{u}\times\mathbf{B}$ , giving $\varepsilon = \oint(\mathbf{u}\times\mathbf{B})\cdot d\boldsymbol{\ell} = B\ell u = 0.10\ \mathrm{V}$ . Why: there is no electric field in this story at all — the agent is magnetism, acting on the charges precisely because they move.

Step 3. Story B — rod at rest, magnet moving. Now every carrier has $\mathbf{v} = \mathbf{0}$ , so the magnetic force $q\,\mathbf{v}\times\mathbf{B}$ is identically zero and magnetism does nothing whatever. But the field at the rod changes in time, so Faraday's law supplies a circulating, non-conservative electric field; the flux sweeps at $|d\Phi/dt| = B\ell u$ , giving $\varepsilon = 0.10\ \mathrm{V}$ . Why: the agent is now an electric field, acting on the charges precisely because they stand still. The mechanism has been swapped for its opposite — and the galvanometer has not moved by one microamp.

Step 4. State the scandal. The stories share no ingredient: A has $\mathbf{E} = 0$ and a nonzero magnetic force; B has a nonzero $\mathbf{E}$ and zero magnetic force. Yet both give $0.10\ \mathrm{V}$ , and $\varepsilon = B\ell u$ holds for every $B$ , $\ell$ and $u$ — an algebraic identity, not a coincidence at these values. Why: as Faraday's Law of Induction puts it, two unrelated mechanisms share one formula, and "nothing in the nonrelativistic framework requires their formulas to match." Einstein took the coincidence as data.

Step 5. Turn it into an argument for P1. The experiment sees only the relative motion; the theory insists on knowing which body is absolutely moving and hands you a different mechanism for each answer. If the observable is blind to the distinction then — postulate 1 — the distinction is not physical. Why: the actual repair is that $\mathbf{E}$ and $\mathbf{B}$ are two aspects of one object whose split depends on the observer, so boosting from the magnet's frame to the rod's converts part of $\mathbf{B}$ into exactly the $\mathbf{E}$ story B needs — bookkeeping that belongs to Magnetism as a Relativistic Effect. S0's job is to make the crisis undeniable, not to close it. *Verified numerically in verify_aq-sr-einsteins-two-postulates.py: $B\ell u = 0.50 \times 0.10 \times 2.0 = 0.10\ \mathrm{V}$ by both routes, with $\varepsilon_{\text{Lorentz}}(B,\ell,u) = \varepsilon_{\text{Faraday}}(B,\ell,u)$ checked as a symbolic identity, not merely at these numbers.*

Limiting cases

Case	Result	Meaning
Galilean limit $c \to \infty$	$a_+ \to 1$ , $a_- \to 1$	infinitely fast light removes the contradiction: absolute time and P2 coexist, and Galilean Transformations is right
Low speed, $\beta \to 0$	$a_+ - a_- \approx 2\beta \to 0$	the contradiction shrinks in size but never vanishes: Newtonian physics is an excellent approximation to an impossible theory
$v = 0$	$a_+ = a_- = 1$	the only exact solution — two frames not moving relative to each other
First order in $\beta$	coefficient $= 2$ , not $0$	the flaw is $O(\beta)$ : ten thousand times the $O(\beta^2)$ ether signal of The Michelson–Morley Experiment
Drop $t' = t$	a solution exists	see Derivation of the Lorentz Transformation

Worked examples

1 · The magnet and the conductor, both routes, one voltage

A rod of length $\ell = 0.10\ \mathrm{m}$ closes a circuit in the field $B = 0.50\ \mathrm{T}$ of a magnet. Rod and magnet approach at $u = 2.0\ \mathrm{m/s}$ . Compute the EMF twice: once with the magnet at rest, once with the rod at rest.

Step 1. Magnet at rest, rod moving. The lab's field is static, so $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t = \mathbf{0}$ : no induced electric field exists anywhere in this problem. The carriers move with the rod at $u$ , and the magnetic force per unit charge along the rod is $|\mathbf{u}\times\mathbf{B}| = uB$ . Integrating, $\varepsilon = uB\ell = (2.0)(0.50)(0.10) = \mathbf{0.10\ V}$ .

Step 2. Rod at rest, magnet moving. Every carrier has $\mathbf{v} = \mathbf{0}$ , so $q\,\mathbf{v}\times\mathbf{B}$ is identically zero and magnetism contributes nothing at all. The field at the rod is now time-dependent, and the flux changes at $|d\Phi/dt| = B\ell u$ , so the flux rule gives $\varepsilon = (0.50)(0.10)(2.0) = \mathbf{0.10\ V}$ . The agent here is an induced electric field; the magnetic force is exactly zero.

Step 3. Compare. Same voltage, same current, two mechanisms with nothing in common — and no experiment can tell you which one "really" happened, because "which body is moving" is not a question the world answers. That is the asymmetry the 1905 paper opens by refusing to tolerate.

2 · Chasing a light beam at 0.999 c — and why Maxwell forbids the view

Einstein said he first felt the problem at sixteen: what would a light wave look like if you ran alongside it? Take $\beta = 0.999$ , and reason first with Galileo, then with Maxwell.

Step 1. Galileo's answer. Galilean velocity addition subtracts your speed from the wave's, so the beam creeps away at $c - v = 0.001\,c = 2.998\times10^{5}\ \mathrm{m/s}$ — a mere $300\ \mathrm{km/s}$ . Push $\beta$ to $1$ and it stops dead: you would be looking at a frozen electromagnetic wave, a static field pattern oscillating in space but not in time, hanging beside you like a corrugated wall.

Step 2. Ask Maxwell whether that object exists. A pattern that does not change in time has $\partial_t\mathbf{E} = \partial_t\mathbf{B} = \mathbf{0}$ ; feed that to the four source-free vacuum equations and every right-hand side dies, leaving $\nabla\times\mathbf{E} = \mathbf{0}$ , $\nabla\times\mathbf{B} = \mathbf{0}$ , $\nabla\cdot\mathbf{E} = 0$ , $\nabla\cdot\mathbf{B} = 0$ . Test the candidate — the transverse sinusoid $\mathbf{E} = E_0\sin(kx)\,\hat{\mathbf{z}}$ . It is divergence-free, so it passes two of the four conditions; but its curl is $\nabla\times\mathbf{E} = -E_0k\cos(kx)\,\hat{\mathbf{y}}$ , which is not zero unless $E_0k = 0$ : unless the amplitude vanishes, or the wave has no wavelength at all.

Step 3. Close the door in general. With $\nabla\times\mathbf{E} = \mathbf{0}$ and $\nabla\cdot\mathbf{E} = 0$ , the identity $\nabla(\nabla\cdot\mathbf{E}) - \nabla\times(\nabla\times\mathbf{E}) = \nabla^2\mathbf{E}$ makes every Cartesian component of $\mathbf{E}$ harmonic — and a harmonic function bounded everywhere and decaying at infinity is identically zero. A frozen light wave is not a thing Maxwell's equations have. Galilean kinematics predicts a view electromagnetism says cannot exist, so the two theories contradict each other about an observation — and it is the Galilean Transformations that must go.

Physics of the portrait

The Einstein of the photographs — the hair, the sockless sandals, the sage — is a man in his fifties. The one who wrote this article's physics was twenty-six, a third-class examiner at the Swiss Patent Office in Bern, rejected from every academic post he had applied for. It shows: the 1905 paper reads like the work of someone paid all day to ask whether a machine really does what its inventor claims. No new force, no new field — an audit.

Origins · 2 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · Bern · Zürich · Berlin · Princeton

Born in Ulm and schooled in Munich, Einstein failed to secure a teaching post after graduating from the Zurich Polytechnic in 1900 and took a job assessing patent applications in Bern. From that desk, in the miracle year of 1905, came four papers in Annalen der Physik: the light-quantum hypothesis (which later brought him the Nobel Prize), a determination of molecular dimensions, the theory of Brownian motion, and — received on 30 June, published on 26 September — "Zur Elektrodynamik bewegter Körper", the paper whose postulates this article states. It cites no experiment and no colleague; its bibliography is empty. Three months later, on 27 September, he sent in a three-page afterthought and answered its title question with $E = mc^2$ . Asked late in life what had started it all, he pointed back to a puzzle from his teens — what a light wave would look like if you could run beside it — and to the magnet and the conductor. Not to any interferometer.

History

1632

— Galileo's Dialogo: no mechanical experiment below decks reveals a ship's uniform motion. The principle of relativity is born.

1687

— Newton's Principia, Corollary V, states the same invariance for the laws of motion — while Book I quietly installs absolute time in the background.

1865

— Maxwell's A Dynamical Theory of the Electromagnetic Field fixes the speed of light from bench constants alone, with no frame attached to it.

1887

— Michelson and Morley find no ether wind: predicted fringe shift 0.373, observed under 0.01.

1904

— Lorentz publishes the transformation that bears his name; Poincaré, at the St Louis Congress, names "the principle of relativity".

1905

— Einstein, "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17, 891 — received 30 June, published 26 September. It opens with the magnet and the conductor, cites nothing, and takes the two postulates as axioms.

1905

— a three-page sequel, received 27 September (Ann. Phys. 18, 639), extracts

E = mc^2

from the same two postulates.

1908

— Minkowski's "Raum und Zeit" recasts the whole structure as a geometry, and the postulates acquire a spacetime.

Going further

Texts. Taylor & Wheeler, Spacetime Physics, 2nd ed., Ch. 3 · Rindler, Introduction to Special Relativity, 2nd ed., §§1–7 · Griffiths, Introduction to Electrodynamics, 4th ed., §12.1 · Einstein, "Zur Elektrodynamik bewegter Körper", Ann. Phys. 17 (1905) 891.

Provenance. The postulates follow Einstein (1905) §§1–2 and Rindler §1; $c = 299\,792\,458\ \mathrm{m/s}$ is the SI-defined constant, fixed in 1983. Everything numeric is recomputed in verify_aq-sr-einsteins-two-postulates.py: the branches $a_\pm = 1/(1 \mp \beta)$ and their collision at $v = 0$ (sympy `solve` returns exactly `[0]`); $2.5$ and $0.625$ at $\beta = 0.6$ ; the series $a_+ - a_- = 2\beta + 2\beta^3 + \cdots$ ; the magnet–conductor EMF $B\ell u = 0.10\ \mathrm{V}$ by both routes; and the frozen-wave obstruction. Two base errors, corrected here. The base's three derivation cards derived the Lorentz factor, the Lorentz transformation and the relativity of simultaneity — all belonging to S1 articles, none of them this article's to prove; they are dropped for the postulates' own owned result. And its claim that Michelson–Morley is "the classic experimental motivation" for the postulates overstates the record.

Builds on. Inertial Frames of Reference · Galilean Relativity · The Principle of Relativity · Electromagnetism and the Failure of Galilean Invariance. Leads to. The Constancy of the Speed of Light · Derivation of the Lorentz Transformation · The Lorentz Factor Gamma.

Did the Michelson–Morley experiment lead Einstein to the postulates?

Not in the way the textbooks imply, and the 1905 paper settles it by omission: it never cites the experiment. It opens with the magnet and the conductor, and refers only in passing to "the unsuccessful attempts to detect a motion of the earth relative to the light medium". Einstein's later statements were equivocal — as The Michelson–Morley Experiment records, he told Robert Shankland in the 1950s that he had learned of the result through Lorentz's writings, was unsure when he first heard of it, and had assumed it true without consciously building on it; the experiments he named as decisive were stellar aberration and Fizeau's measurement of light in moving water. Michelson–Morley made the crisis undeniable. It was not the clue.

If the postulates contradict absolute time, how did nobody notice for two centuries?

Because nobody wrote $t' = t$ down. It was not a hypothesis anyone defended but the water everyone swam in — so thoroughly assumed that Galilean Transformations states it as a clause of a coordinate change rather than a claim about the world. And the second ingredient was missing: you need a law that fixes a speed without reference to a frame, and no such law existed before Maxwell wrote one in 1865. What was hard was not the mathematics but believing that $t' = t$ was the thing to give up — rather than Maxwell's equations, or the ether's state of motion, or the rigidity of matter, all tried first.

Isn't "the laws are the same in every frame" empty? Any equation can be rewritten in new coordinates.

Rewritten, yes — but the postulate demands more: that the rewritten equations be the same equations, symbol for symbol, with no leftover $v$ anywhere. That is a sharp constraint, and it fails all the time. Electromagnetism and the Failure of Galilean Invariance shows exactly how: push the vacuum wave equation through a Galilean boost and a cross term appears that was not there before, so the boosted equation is not the one you started with, and it propagates light at $c \mp v$ . Form-invariance is a demand a theory can flunk, and Galilean electrodynamics flunks it.

Postulate 2 sounds like it needs a light source. What if there were no light?

Then $c$ would still be there. The invariant speed is a property of spacetime's structure, not of photons: light travels at $c$ because photons happen to be massless, and if they had a tiny mass — which experiment bounds but does not exclude — light would travel slightly below the invariant speed and every conclusion of relativity would survive untouched. Modern treatments therefore state the postulate as "there exists a finite invariant speed $c$ ", then note as a separate empirical fact that light travels at it. Gravitational waves travel at $c$ too, measured to a part in $10^{15}$ from the 2017 neutron-star merger.

Why is it legitimate to assume the transformation is linear?

Because space and time are homogeneous: no point in empty space and no instant in empty time is distinguished from any other. Suppose the map were nonlinear. Then a free particle moving in a straight line at constant speed in $S$ — as the definition of an inertial frame requires — would come out accelerating in $S'$ , with an acceleration depending on where and when it happened to be. That would let you locate a preferred origin by watching free particles, contradicting the homogeneity you started from. Linearity is what homogeneity means, written as an equation.

The contradiction is first order in β. Could a nineteenth-century experiment have caught it?

The contradiction is first order — but a contradiction is not an observable. What is first order is the disagreement between two things the theory demands; to convert that into a measurement you would have to time a light pulse one way, over a known distance, with clocks at both ends synchronised by something other than light itself. Nobody could do that in 1887, which is exactly why Michelson built a two-way instrument and settled for a $\beta^2$ effect. The lesson runs the other way: the fatal flaw in classical physics was ten thousand times larger than the effect Michelson spent a decade hunting, and it sat in plain sight.