The Two Postulates of Special Relativity

On 30 June 1905 the Annalen der Physik received a thirty-page paper from a patent clerk in Bern. It contained no new experiment, no new force, and — for the first nine pages — almost no mathematics. It contained two sentences: the laws of physics take the same form in every inertial frame, and light in vacuum travels at the same speed c for every inertial observer, whatever the motion of its source. Raise those to axioms, said Einstein, and the rest follows. The historical drama belongs to Einstein's Two Postulates and The Constancy of the Speed of Light. This article is about the logical structure of the axioms — and one surprise: how little of postulate 2 is actually needed. Relativity, homogeneity, isotropy, and the demand that two boosts compose into a boost — four assumptions, none about light — already force a transformation containing a universal invariant speed K. They cannot tell you what $K$ is. Postulate 2's entire job is to answer that one question, empirically: $K$ is finite, and it equals $c$ . Set $K = \infty$ and Galileo comes back; set $K = c$ and you have relativity. Light is not special — it merely travels at $K$ , being massless. That theorem, Ignatowski's of 1910, is what this article owns, along with the reductio showing that the two postulates plus absolute time is a flat contradiction.

Key takeaways

Postulate 1 (relativity): the laws of physics — all of them, mechanics and electromagnetism alike — take the same form in every inertial frame. No experiment singles out a state of rest.
Postulate 2 (invariance of $c$ ): light in vacuum travels at $c$ in every inertial frame, whatever the motion of source or observer — the same number for everybody, not merely a definite one.
The two are consistent only if absolute time goes: assume $t' = t$ alongside them and the contradiction is immediate. That reductio is derivation A.
Relativity + homogeneity + isotropy + closure alone force $x' = \Gamma(x - vt)$ , $t' = \Gamma(t - vx/K^2)$ with a universal invariant speed $K$ ; postulate 2 only fixes $K = c$ . Everything downstream — simultaneity, dilation, contraction, the transformation itself — is then a theorem.
Key idea: the second postulate is not a statement about light. It is the empirical claim that the invariant speed $K$ — which the relativity principle alone already guarantees exists — is finite rather than infinite, and that its value is $c$ . Light is the messenger, not the message.

The core insight

The laws are the same in all inertial frames; $c$ is the same in all inertial frames. The two axioms contradict each other the instant you also assume a universal clock — because a universal clock forces the Galilean map, and that map hands the moving observer light at $c - v$ one way and $c + v$ the other. Only $t' = t$ can give. What replaces it is fixed almost entirely by symmetry, which already demands one universal invariant speed $K$ built into the transformation. Postulate 2 is the single experimental fact that sets $K = c$ rather than $K = \infty$ .

Where this lives today

Postulate 2 is tested along two axes. Isotropy — does $c$ depend on direction? Cryogenic sapphire resonators on turntables, compared as the Earth turns, bound any directional variation at $\Delta c/c \lesssim 10^{-18}$ : eight orders of magnitude beyond the 1887 interferometer. Source-independence — does $c$ depend on how fast the emitter moves? In 1964 Alväger and collaborators produced photons from neutral pions travelling at $0.99975c$ and timed them over a 30-metre flight path. Had the source's velocity added to the light's, they should have arrived at nearly $2c$ ; they arrived at $(2.9977 \pm 0.0004)\times10^{8}\ \mathrm{m/s}$ — the ordinary value, to four figures. The postulates earn their keep in engineering too. Accelerators are machines built on equation (3): each stage composes a boost with the beam's existing velocity, and the beam saturates rather than running away — at $7\ \mathrm{TeV}$ an LHC proton lags light by $2.7\ \mathrm{m/s}$ . And the metre has been defined since 1983 by fixing $c$ exactly, which a frame-dependent quantity could never be.

The physical picture

Sprint alongside a light beam at half the speed of light. Galileo says the beam now recedes at $c/2$ . Postulate 2 says it recedes at $c$ — full speed, exactly as when you stood still. Sprint at $0.999c$ and it still recedes at $c$ : no chasing it down, no closing the gap, no frame in which light is slow. That is a fact about the shape of spacetime, not about light. The two axioms are a vice — the only freedom left in the coordinate map is how the time coordinate depends on position, and the moment it does, simultaneity stops being absolute.

Figure

The axiom tree. Two postulates, one casualty (absolute time), four consequences — each derived in the article that owns it. Nothing below the dashed line is an extra assumption: every box is a theorem.

Now the punchline, and it is not the one most courses give. Ask only that boosts compose — a boost by $v_1$ followed by one by $v_2$ must itself be a boost — and add homogeneity, isotropy and relativity. Nothing about light. That already forces a saturating composition law with a fixed point: a speed that combines with anything and comes back unchanged. Galileo's rule, $v_3 = v_1 + v_2$ , is the member of the family whose fixed point has run away to infinity.

Figure

PUNCHLINE — postulate 2 does not say "light is special". It says a finite invariant speed EXISTS, and light travels at it. Composing two equal boosts β under the invariant-speed law gives 2β/(1 + β²) (blue, saturating) instead of Galileo's 2β (grey, straight, no ceiling). At β = 0.6: relativity gives 0.6 ⊕ 0.6 = 0.8824, Galileo gives 1.2. The whole content of postulate 2 is one number — K = c, finite, not K = ∞.

The core equations

Standard configuration: $S'$ slides along the $x$ -axis of $S$ at constant $v$ , axes parallel, origins coinciding at $t = t' = 0$ . Throughout $\beta \equiv v/c$ , $\gamma \equiv 1/\sqrt{1-\beta^2}$ , $c = 299\,792\,458\ \mathrm{m/s}$ (exact, by the SI definition of the metre since 1983). The postulates are sentences, not formulae; what can be written down is what they forbid and what they force. Galilean Transformations proves the first result imported here — push the light sphere through the classical map and it comes back centred in the wrong place.

x^2+y^2+z^2 = c^2t^2 \ \ \xrightarrow{\ t'=t\ }\ \ (x'+vt')^2 + y'^2 + z'^2 = c^2t'^2 \qquad (1)

where: the arrow is the Galilean map, $x = x'+vt'$ , $t = t'$ . Physical reading: still a sphere of radius $ct'$ , but centred at $x' = -vt'$ — the moving observer has run out from under her own wavefront, so light reaches her at $c - v$ forwards and $c + v$ backwards. Flat contradiction with postulate 2. What symmetry alone permits is far more constrained than one would guess. The most general map compatible with relativity, homogeneity, isotropy and closure is this:

x' = \Gamma\,(x - vt), \qquad t' = \Gamma\!\left(t - \frac{v x}{K^{2}}\right), \qquad \Gamma = \frac{1}{\sqrt{1 - v^{2}/K^{2}}} \qquad (2)

where: $K$ — a universal constant with the dimensions of a speed, the same for every pair of frames. Physical reading: Ignatowski's theorem, derivation B. Notice what is absent: light. No wave, no photon, no electromagnetism — only symmetry and the demand that boosts compose. The theory is fixed up to one number, still undetermined.

v_{3} = \frac{v_{1}+v_{2}}{1 + v_{1}v_{2}/K^{2}}, \qquad\quad K \oplus u = \frac{K+u}{1 + uK/K^{2}} = K \ \ \text{for every } u \qquad (3)

where: $v_3$ — the composite boost's velocity; $\oplus$ — composition under (3). Physical reading: every frame agrees on $K$ as a matter of algebra, before any experiment. An invariant speed is not a paradox; it is the fixed point of a group law. (Velocity composition as a tool belongs to Relativistic Velocity Addition Formula, the group law to Derivation of the Lorentz Transformation.) Now, and only now, does postulate 2 arrive — and all it does is set the value of $K$ .

K = \infty \ \Rightarrow\ \text{Galileo}\ (t'=t) \qquad\qquad \boxed{\,K = c\,} \ \Rightarrow\ \Gamma = \gamma,\ \ \text{relativity} \qquad (4)

where: $\gamma = 1/\sqrt{1-\beta^2}$ — the Lorentz factor, developed in The Lorentz Factor Gamma. Physical reading: the two great kinematics of physics are the two ends of one family, parametrised by $1/K^2$ ; classical mechanics is the $1/K^2 = 0$ member, and postulate 2's content is the measured statement $1/K^2 = 1/c^2 \ne 0$ .

Where the results come from

Derivation AThe postulates plus absolute time is a contradiction — and only absolute time can be blamed

The goal is a genuine reductio, not a plausibility argument. Assume all three of P1, P2 and the universal clock $t' = t$ . The temptation is to say "the Galilean map fails, but perhaps some other map with $t' = t$ survives" — and that is the step usually skipped. We show first that $t' = t$ with P1 alone forces the map to be exactly Galilean; then that the Galilean map contradicts P2. The three cannot coexist, and P1 and P2 are the axioms under test, so the casualty is $t' = t$ .

Step 1. Homogeneity makes the map linear — a free particle's straight worldline must stay straight, and only a linear map does that for every particle. With $t' = t$ imposed, the general linear map in standard configuration is $t' = t$ , $x' = A(v)(x - vt)$ ; the factor $x - vt$ is forced because the origin of $S'$ must sit at $x = vt$ . Why one unknown is all that survives: absolute time has consumed the whole time row of the matrix, and tracking the moving origin the space row but for its scale.

Step 2. Kill $A$ with P1. Isotropy says the coefficient cannot care about the sign of $v$ , so $A(-v) = A(v)$ . Relativity says $S$ sees $S'$ move at $+v$ exactly as $S'$ sees $S$ move at $-v$ , so the inverse map is the same map with $v \to -v$ : $A(v)A(-v) = 1$ . Together $A^2 = 1$ , and continuity with $A(0) = 1$ gives $A = 1$ — so $x' = x - vt$ , $t' = t$ , the Galilean transformation exactly, nothing adjustable left. Why this is load-bearing: it upgrades "the Galilean map fails" into "every map with a universal clock fails".

Step 3. Apply P2 and watch it break. By equation (1) the light sphere comes back centred at $x' = -vt'$ , so $S'$ finds light running at $c - v$ along her motion and $c + v$ against it, where P2 demands $c$ both ways. Why there is no escape hatch: the residual $2vt'x' + v^2t'^2$ vanishes only at $v = 0$ — a structural failure, first order in $v$ , not a small error to be absorbed.

Step 4. Convict the right assumption. The contradiction used P1 (twice, in step 2), P2 (in step 3) and $t' = t$ (in step 1). We will not surrender P1 or P2 — they are the axioms; and homogeneity is not optional, since without it inertial frames do not exist. One assumption is left in the dock. Absolute time must go: $t'$ has to depend on position as well as time. Why all of relativity is already implicit here: a time coordinate depending on $x$ means clocks synchronized in one frame are not synchronized in another — the relativity of simultaneity. *Verified in verify_aq-sr-two-postulates-special-relativity.py: sympy imposes $t' = t$ , reciprocity $M(v)^{-1} = M(-v)$ and the isotropy parities on the general linear map, and returns $A = 1$ — the Galilean map, uniquely. It then substitutes the map into $x^2+y^2+z^2-c^2t^2$ , extracts the residual $2vt'x'+v^2t'^2$ (zero only at $v=0$ ), locates the centre at $x' = -vt'$ and the axis speeds $c \mp v$ , and confirms that at $\beta = 0.5$ the forward light speed is $0.5c$ — half of what P2 demands.*

Go deeper · Derivation B · 3 minIgnatowski's theorem — relativity alone forces an invariant speed K; the light postulate only names it

Derivation BIgnatowski's theorem — relativity alone forces an invariant speed K; the light postulate only names it

This is the result the article owns, and it inverts the usual story. Take four assumptions, none of which mentions light: (i) relativity; (ii) homogeneity of space and time, so the map is linear; (iii) isotropy, so no direction is preferred; (iv) closure — a boost by $v_1$ followed by one by $v_2$ is itself a boost. These force equation (2), carrying a universal constant $K$ with the dimensions of a speed, identical for every pair of frames and the fixed point of the composition law. What they cannot give is the value of $K$ — and that one number is the whole empirical content of the second postulate.

Step 1. Write the general linear map on $(t, x)$ : $x' = \Gamma(v)(x - vt)$ , $t' = D(v)\,t + C(v)\,x$ — three unknown functions. Isotropy (send $x \to -x$ and $v \to -v$ ; the map must be unchanged) makes $\Gamma$ and $D$ even and $C$ odd. Why $C$ 's parity is the crux: it is the only coefficient allowed to be non-zero and odd, and it is precisely the one absolute time sets to zero. Everything hinges on whether we let it live.

Step 2. Impose relativity: the inverse of the boost by $v$ is the boost by $-v$ , so $M(v)^{-1} = M(-v)$ . Multiplying out with the parities of step 1 gives two consequences at once — $\det M = 1$ and $D(v) = \Gamma(v)$ . The map is down to two functions, $x' = \Gamma(x - vt)$ and $t' = \Gamma(t - \kappa(v)\,x)$ with $\kappa \equiv -C/\Gamma$ odd, and $\det M = 1$ reads $\Gamma^2(1 - \kappa v) = 1$ , pinning $\Gamma = 1/\sqrt{1 - \kappa(v)\,v}$ . Why the light postulate is nowhere in sight: reciprocity is a statement about the frames' equal standing, not about signals — and it has already done most of the work.

Step 3. Impose closure, and $K$ drops out. Compose $M(v_2)M(v_1)$ and demand the product have the same form — which, by step 2, means its $(t,t)$ entry must equal its $(x,x)$ entry. Those entries are $1 + \kappa(v_2)\,v_1$ and $1 + \kappa(v_1)\,v_2$ , so closure demands $\kappa(v_2)\,v_1 = \kappa(v_1)\,v_2$ : the ratio $\kappa(v)/v$ takes the same value for every $v$ . Call it $1/K^2$ . Then $\kappa(v) = v/K^2$ and $\Gamma = 1/\sqrt{1 - v^2/K^2}$ — equation (2), derived. Why this is the theorem in one line: a quantity forced to be the same for all velocities and all frames is a universal constant of nature by construction — and this one has the dimensions of a speed. Relativity manufactures an invariant speed whether you asked for one or not.

Step 4. Read off what $K$ does. Composition gives $v_3 = (v_1+v_2)/(1 + v_1v_2/K^2)$ , and setting $v_1 = K$ returns $v_3 = K$ for every $v_2$ : $K$ is the fixed point, the one speed all frames agree on. Three branches survive. $1/K^2 = 0$ : Galilean, $\Gamma \equiv 1$ , absolute time restored. $1/K^2 > 0$ : Lorentzian, a finite invariant speed, $\Gamma > 1$ . $1/K^2 < 0$ : Euclidean rotations of spacetime, in which a large enough boost reverses the order of causally connected events. Why the third branch is discarded honestly: it is excluded not by symmetry but by demanding that cause precede effect in every frame — a fifth assumption, worth stating out loud rather than smuggling in.

Step 5. Now let the light postulate speak. It says there exists a signal whose speed is the same in all frames, and that speed is $c$ . By step 4 such a signal's speed is the fixed point — so $K = c$ and $\Gamma = \gamma$ . That is postulate 2's entire contribution: one measurement, selecting the finite branch and setting its scale. It does not say light is special; it says $K$ is finite, and hands you the number. Why the historical reading changes: Ignatowski (1910) obtained the transformation with no light postulate at all, but could not extract the value of $K$ from symmetry — nor can anyone, because it is not in there. *Verified in verify_aq-sr-two-postulates-special-relativity.py: sympy builds the general map with unknown $\Gamma(v), D(v), C(v)$ ; imposes $M(v)^{-1} = M(-v)$ with the isotropy parities and recovers $\det M = 1$ , $D = \Gamma$ ; composes $M(v_2)M(v_1)$ and shows the same-form condition reduces to $\kappa(v_1)/v_1 = \kappa(v_2)/v_2$ , hence $\kappa = v/K^2$ and $\Gamma = (1-v^2/K^2)^{-1/2}$ ; confirms $K \oplus u = K$ identically, that $K \to \infty$ returns the Galilean map and $K = c$ the Lorentz coefficients; and checks $0.6 \oplus 0.6 = 0.8824$ , $0.75 \oplus 0.75 = 0.96$ , $0.99 \oplus 0.99 = 0.999949$ , $1 \oplus 1 = 1$ .*

Limiting cases

Case	Result	Meaning
$K \to \infty$ in equation (2)	$\Gamma \to 1$ , $x' = x-vt$ , $t' = t$	Galilean kinematics is the $1/K^2 = 0$ member of the family — not a different theory, just a different value of one constant
$K = c$ , $\beta \ll 1$	$\gamma = 1 + \tfrac12\beta^2 + O(\beta^4)$	at $\beta = 10^{-4}$ the correction is $5\times10^{-9}$ — why two centuries never noticed
$\beta \to 1$	$\gamma \to \infty$	no finite stack of boosts reaches $K$ : $0.99 \oplus 0.99 = 0.999949$ , never $1$
$u = c$ in equation (3)	$c \oplus v = c$ for every $v$	the group law's fixed point — postulate 2 made algebraic
A massless particle	speed $= K = c$ in every frame	light is not privileged, only massless; gravitational waves do the same

Common trap“The second postulate is about light — that's why light is special in relativity.”

It is not, and it is not. Read the axiom again with derivation B in hand: what it actually asserts is that a finite invariant speed exists, and its value is $c$ . The existence half was never postulate 2's to give — relativity, homogeneity, isotropy and closure hand you a universal invariant $K$ for free, with no signal of any kind mentioned. All postulate 2 supplies is the empirical fact that $K$ is finite rather than infinite, plus its value. Light appears for exactly one reason: the photon has zero rest mass, and a massless particle has no choice but to move at $K$ in every frame — gravitational waves do the same. The right slogan is not "nothing can go faster than light" but "there is one invariant speed, and light happens to travel at it" — which is why $c$ turns up in $E = mc^2$ , where no light is present at all.

Worked examples

1 · Stacking boosts: 0.75c and 0.75c

A rocket leaves Earth at $0.75c$ and launches a probe forward at $0.75c$ relative to itself. Galileo says the probe moves at $1.5c$ over the ground. What does the invariant-speed law say — and can the difference be measured?

Step 1. A composition of two boosts, so equation (3) applies with $K = c$ : $\beta_3 = (\beta_1+\beta_2)/(1 + \beta_1\beta_2)$ , $\beta_1 = \beta_2 = 0.75$ .

Step 2. Numerator $0.75 + 0.75 = 1.5$ ; denominator $1 + (0.75)(0.75) = 1.5625$ . So $\beta_3 = 1.5/1.5625 = \boxed{0.96}$ — comfortably below $c$ , where Galileo's $\boxed{1.5c}$ is comfortably above.

Step 3. Test the boundary: $1 \oplus 1 = (1+1)/(1+1) = \boxed{1}$ . Even light plus light returns light. The law does not merely happen to stay below $c$ ; it cannot leave it, because $c$ is the group's fixed point.

Step 4. Is the difference observable? Emphatically: every synchrotron composes boosts for a living. If velocities added Galileo's way, a few dozen accelerating stages would carry the beam past $c$ and the machine would not work at all. The discriminating experiment is the accelerator itself.

2 · How fast is fast enough? The Michelson–Morley arithmetic

The two kinematics disagree — but by how much at speeds a nineteenth-century laboratory could reach? Use the Earth's orbital motion, and ask what the disagreement looks like in an interferometer.

Step 1. Get $\beta$ . The Earth orbits at $v_\oplus \approx 29.8\ \mathrm{km/s}$ , so $\beta = v_\oplus/c = 9.94\times10^{-5} \approx \boxed{10^{-4}}$ — the fastest vehicle a Victorian physicist could ride moved at a ten-thousandth of the invariant speed.

Step 2. Find the order of the effect. Round-trip light times are even in $\beta$ — first-order errors cancel between outbound and return legs — so the leading disagreement is $O(\beta^2) = \boxed{10^{-8}}$ . A part in a hundred million: no clock of the era could see it, which is why an interferometer was needed.

Step 3. Convert to the observable. The Michelson–Morley Experiment derives the shift on rotating the apparatus through $90^\circ$ : $\Delta N = 2L\beta^2/\lambda$ . With its $L = 11\ \mathrm{m}$ folded path and sodium light at $\lambda = 589\ \mathrm{nm}$ , that is $\Delta N = \boxed{0.373} \approx 0.4$ fringes — the figure Michelson and Morley themselves quoted.

Step 4. Read the verdict. The observed shift was under $\boxed{0.01}$ fringes — a roughly forty-fold null. Galilean kinematics predicted four-tenths of a fringe; nature delivered nothing. Postulate 2 predicts exactly zero, at any speed and any orientation. Already at $\beta = 10^{-4}$ it is the only kinematics that fits.

Physics of the portrait

The portrait is not a person but a pair of sentences, the second of which has been quietly misread for a century. Einstein's 1905 wording is scrupulous — light is propagated in empty space "with a definite velocity $c$ which is independent of the state of motion of the emitting body" — and read as physics rather than as a slogan, that is a claim about the structure of spacetime.

Origins · 2 minPostulates of special relativity and how this was discovered

Postulates of special relativity

1632–1910 · Galileo's principle · Einstein's two sentences · Ignatowski's theorem

The first postulate is nearly three centuries older than the second. Galileo (1632) argued it as physics in the Dialogo: below decks on a smoothly sailing ship, no experiment reveals the ship's motion. Poincaré, at the St Louis Congress in 1904, named it the principle of relativity and raised it to a general guide for all of physics. The second postulate is the new one, and the one Einstein (1905) had to be brave about: not that light has a definite speed — an ether would have granted that — but that every inertial observer measures the same one. He took both sentences literally, following them into the wreckage of absolute time rather than patching the machinery to save it. Then in 1910 Vladimir Ignatowski showed something nobody expected: relativity, homogeneity, isotropy and the group property of boosts already force a transformation carrying a universal invariant speed, with light nowhere in the argument. What symmetry could not give him was the value of that speed — for that he had to appeal to moving electrostatic fields. That is the honest modern reading of postulate 2: not an axiom about light at all, but the one empirical number symmetry leaves open.

History

1632

— Galileo's Dialogo: the sealed ship's cabin. Uniform motion is undetectable from within.

1865

— Maxwell, "A Dynamical Theory of the Electromagnetic Field": the field equations carry a wave speed set by the electric and magnetic constants, matching the measured speed of light.

1887

— Michelson and Morley find no fringe shift: about 0.4 fringes predicted, under 0.01 observed.

1904

— Poincaré, at the St Louis Congress of Arts and Science, names the principle of relativity and elevates it to a general principle of physics.

1905

— Einstein, "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17, 891–921; received 30 June, published 26 September. The two postulates stated as axioms, and the ether declared superfluous.

1908

— Minkowski's Cologne lecture Raum und Zeit recasts the postulates as the geometry of a four-dimensional spacetime.

1910

— Ignatowski, "Einige allgemeine Bemerkungen über das Relativitätsprinzip", Physikalische Zeitschrift 11: the transformation derived from relativity and group closure, with no light postulate — the invariant speed left undetermined.

1911

— Max von Laue publishes Das Relativitätsprinzip, the first textbook of the theory.

1964

— Alväger and collaborators time photons from pions moving at 0.99975c: the measured speed is (2.9977 ± 0.0004) × 10⁸ m/s. The source's motion adds nothing.

Going further

Texts. Einstein, "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17 (1905) 891 · Taylor and Wheeler, Spacetime Physics, 2nd ed., ch. 1–3 · Rindler, Introduction to Special Relativity, 2nd ed., §§1–2 · J.-M. Lévy-Leblond, "One more derivation of the Lorentz transformation", Am. J. Phys. 44 (1976) 271 — the modern form of Ignatowski's argument.

Provenance. Postulate statements follow Einstein (1905) §§1–2. Every relation is re-derived in `verify_aq-sr-two-postulates-special-relativity.py`: the reductio ( $t' = t$ plus reciprocity and isotropy forces $A = 1$ ; the light sphere then acquires the residual $2vt'x'+v^2t'^2$ , a centre at $x' = -vt'$ , axis speeds $c \mp v$ ) and Ignatowski's theorem ( $M(v)^{-1} = M(-v) \Rightarrow \det M = 1$ , $D = \Gamma$ ; closure $\Rightarrow \kappa(v)/v = 1/K^2$ ; $\Gamma = (1-v^2/K^2)^{-1/2}$ ; $K \oplus u = K$ ; the limits $K \to \infty$ , $K = c$ ). Numbers: $0.75 \oplus 0.75 = 0.96$ , $0.6 \oplus 0.6 = 0.8824$ , $0.99 \oplus 0.99 = 0.999949$ , $1 \oplus 1 = 1$ ; $\beta_\oplus = 9.94\times10^{-5}$ , $\Delta N = 0.373$ against an observed bound of $0.01$ ; LHC $7\ \mathrm{TeV}$ proton, $2.7\ \mathrm{m/s}$ slower than light; $c = 299\,792\,458\ \mathrm{m/s}$ (exact). The Michelson–Morley fringe arithmetic and the Galilean light-sphere result are quoted from the articles that own them.

Builds on. Inertial Frames of Reference · The Principle of Relativity · Galilean Transformations · The Michelson–Morley Experiment · Einstein's Two Postulates. Leads to. The Light Clock and Time Dilation · Relativity of Simultaneity · Derivation of the Lorentz Transformation · The Invariant Spacetime Interval.

If the second postulate is really just "K is finite and equals c", why does every textbook state it in terms of light?

History, then habit. Einstein wrote in 1905, when the crisis was an electromagnetic one: Maxwell's equations carried a definite speed, the ether was supposed to supply the frame in which it was measured, and Michelson and Morley had failed to find that frame. Stating the axiom in terms of light put it exactly where the wound was, and Ignatowski's theorem came five years later, by which time the formulation was set. The two versions are equivalent in content — but only one tells you which part is symmetry and which part is measurement.

Doesn't the Ignatowski argument make the second postulate redundant? Could we just drop it?

No — and it is worth being precise, because this is easy to overclaim. Symmetry gives you the family of transformations, parametrised by $1/K^2$ ; it cannot tell you which member the world uses. Choosing between the Galilean branch ( $1/K^2 = 0$ ) and the Lorentzian one ( $1/K^2 > 0$ ), and fixing $K$ in the second, is a measurement — no amount of group theory will do it for you. The postulate survives, but shrunken: from "a deep fact about light" to "the one number symmetry leaves open".

The first postulate says the laws are the same in all inertial frames. But c is a number, not a law — why must it be the same too?

Because $c$ lives inside the laws. Maxwell's equations do not merely permit waves; they force them to travel at a speed set by the electric and magnetic constants of the vacuum. If those equations hold unchanged in every inertial frame — exactly what postulate 1 demands, once electromagnetism counts as physics — then the speed they force is the same in every frame. So postulate 2 is very nearly a corollary of postulate 1 plus the claim that Maxwell's equations are laws. The escape hatch: one could insist they hold only in the ether frame, demoting them to frame-dependent rules. Einstein slammed it shut.

If each observer says the other's clock runs slow, isn't that an outright contradiction?

It looks like one; it is not — though the resolution is not this article's to give. The two observers are not performing the same comparison. To time a single moving clock against your own you need two of your clocks, at two places, synchronized — and the other observer denies that they are synchronized. That denial is the relativity of simultaneity, and it accounts for the discrepancy exactly; it is derived in Relativity of Simultaneity, with the reciprocity bookkeeping in Time Dilation. The postulates do not collapse: consistency is restored by giving up a habit, not an axiom.

Could there be a particle faster than c without breaking the postulates?

The postulates alone do not forbid one; no axiom says "no signal exceeds $c$ ". What they forbid is accelerating something from below $c$ to above it: equation (3) has $c$ as a fixed point, so no finite stack of boosts arrives there. A particle born above $c$ — a tachyon — is not excluded by the kinematics. What excludes it is causality: a boost can reverse the order of events separated faster than $c$ , so a superluminal signal would let some frame see the reply before the message. Every search has come up empty, and the price of admitting one is cause and effect itself.

Postulate 2 says "the speed of light in vacuum". Light slows down in glass — does relativity break there?

Not in the least, because the $c$ of relativity is not really "the speed of light": it is the invariant speed $K$ of the spacetime geometry. Light in vacuum travels at $K$ because the photon is massless. Light in glass travels more slowly because it is no longer free propagation — the wave is continually absorbed and re-radiated by the medium's charges, and the collective excitation moves at $c/n$ , while the photons between scatterings still do $c$ . This is why a fast electron in water can emit Cherenkov radiation with no postulate violated: it is beating $c/n$ , not $c$ .