Einstein Clock Synchronization

Here is a fact that unsettles people the first time they meet it: nobody has ever measured the one-way speed of light, and nobody ever will. Not for want of trying, and not for want of better lasers. To time a pulse from $A$ to $B$ you need a clock at $A$ and a clock at $B$ reading the same time — and to set those two clocks you need to know how long light takes to get from $A$ to $B$ . The snake eats its tail: every "measurement" of the one-way speed smuggles in the answer through the clocks it uses. What experiment can do is bounce the pulse back. Emit at $t_1$ , reflect off a mirror $300\ \mathrm{m}$ away, receive at $t_3$ : the round trip takes $2.0014\ \mu\mathrm{s}$ , uses one clock, and needs no synchrony at all. That two-way speed is so solidly measurable that since 1983 it is not measured but defined — $c = 299\,792\,458\ \mathrm{m/s}$ exactly, with the metre built from it. So Einstein's rule — declare the reflection simultaneous with $t_2 = \tfrac12(t_1 + t_3)$ — is a convention, not a discovery. The article Einstein Synchronization of Clocks tells that story; this one owns the quantitative structure. It proves that Reichenbach's whole family $t_2 = t_1 + \varepsilon(t_3 - t_1)$ leaves the two-way speed at exactly $c$ for every $\varepsilon$ , so no round-trip experiment can ever pick one out. Then it proves the theorem that rescues $\varepsilon = \tfrac12$ from arbitrariness: a clock carried from $A$ to $B$ at speed $u$ lags by $(L/u)(1 - 1/\gamma_u) \approx Lu/2c^2$ , which goes to zero as $u \to 0$ . Two utterly different procedures, one synchrony. Finally it names the price: the $-\gamma vx/c^2$ term of the Lorentz transformation is not physics but gauge.

Key takeaways

Only the two-way speed of light is measurable — it uses one clock. The one-way speed cannot be measured without two synchronized clocks, and synchronizing them presupposes a one-way speed. The circle is structural, not technological.
Reichenbach's family $t_2 = t_1 + \varepsilon(t_3 - t_1)$ , $0 < \varepsilon < 1$ , gives one-way speeds $c_\rightarrow = c/2\varepsilon$ and $c_\leftarrow = c/(2 - 2\varepsilon)$ — and their harmonic mean is exactly $c$ for every $\varepsilon$ , so no round-trip experiment can distinguish them.
Slow clock transport is an independent route to synchrony: a clock carried at $u$ over $L$ lags by $(L/u)(1 - 1/\gamma_u) \approx Lu/2c^2 \to 0$ . Infinitely slow transport reproduces $\varepsilon = \tfrac12$ exactly.
The $-\gamma vx/c^2$ term of the Lorentz transformation is the synchronization gauge: it is what a row of clocks synchronized in one frame looks like from another — a linear staircase.
Key idea: $\varepsilon = \tfrac12$ is a convention, not an ignorance. No experiment can fix it — yet two independent procedures (light signals and slow transport) both land on it, which is why the convention is the only sane one to adopt and why the Lorentz transformation looks the way it does.

The core insight

Distant simultaneity is defined, not observed. The two-way speed of light is measurable with a single clock; the one-way speed is not measurable at all, because measuring it requires the very synchrony it would establish. Einstein's $\varepsilon = \tfrac12$ stipulates the one-way speed to be $c$ both ways. The whole $\varepsilon$ -family is experimentally equivalent — and yet slow clock transport, which uses no light signal, independently converges on the same $\varepsilon = \tfrac12$ .

Where this lives today

Every timekeeping network on Earth is a working answer to this article. International Atomic Time is assembled from hundreds of clocks that are not co-located and not at rest with respect to one another, and the two techniques used to compare them are exactly those of Derivation A and Derivation B: two-way satellite time and frequency transfer, which bounces signals off a geostationary satellite so the path is traversed both ways and the one-way speed cancels out, and portable-clock transport — slow-clock transport done with a caesium standard in a padded flight case. The two agree: the theorem, checked. GPS shows the same structure with the gauge visible. The system defines coordinate time in one non-rotating Earth-centred frame and applies Einstein synchrony there; the ground segment is not at rest in that frame, so the Sagnac correction — the $-vx'/c^2$ stagger of the punchline figure, integrated around the rotating Earth — must be subtracted, and it reaches hundreds of nanoseconds. Meanwhile the anisotropy experiments often mistaken for one-way measurements — rotating cryogenic optical resonators, Michelson–Morley's modern descendants — bound the two-way speed's directional dependence at parts in $10^{18}$ , and by Derivation A that is all they can ever bound.

The physical picture

Two people stand $300\ \mathrm{m}$ apart, each holding a stopwatch. There is no way to shout the time across instantly. The best they can do is send the fastest signal there is — light — and agree on a rule for interpreting the delay. Einstein's rule is the fairest one available: assume the signal took the same time each way, so the distant event happened at the midpoint of send and receive. Notice what has happened: the assumption of equal times is not tested by the experiment, it is put in by the rule. What comes back out — that light travels at $c$ in both directions — was installed, not discovered.

Figure

The radar protocol on a 300 m baseline. A emits at t₁ = 0; the pulse reflects at B and returns at t₃ = 2.0014 μs — a round trip read off ONE clock, needing no prior synchrony. Einstein's rule (green) declares the reflection simultaneous with A's clock reading t₂ = ½(t₁ + t₃) = 1.0007 μs. Reichenbach's ε = 0.6 (amber) declares it simultaneous with t₂ = 1.2008 μs instead — a tilted line, a slower outbound leg and a faster inbound one. No round-trip measurement can tell the two apart.

Now push the consequence. Every frame adopts the rule, each with its own light signals and its own rest clocks. The rule is the same; the frames are not. So a row of clocks that one frame calls synchronized, another finds staggered in a straight line — and that stagger is the convention showing its hand.

Figure

PUNCHLINE — synchrony is a gauge. Four clocks at rest in S′, spaced 300 m apart and Einstein-synchronized THERE, read at one instant of the ground frame S with β = 0.6. Their faces are staggered linearly in x′ by −vx′/c²: 0, −600.4, −1200.8, −1801.2 ns — exactly 600.4 ns per 300 m, and leading clocks lag. That staircase IS the −γvx/c² term of the Lorentz transformation (the γ cancels against the contracted spacing): the fingerprint of the ε = ½ convention, not a fact about clocks.

The core equations

Frame $S$ has a clock at every place; $\beta \equiv v/c$ , $\gamma \equiv 1/\sqrt{1-\beta^2}$ ; $c = 299\,792\,458\ \mathrm{m/s}$ exactly. Clock $A$ sits at the origin, mirror $B$ at rest a distance $L$ away.

t_2 = \tfrac{1}{2}\,(t_1 + t_3) \qquad (1)

where: $t_1$ — the emission time on $A$ 's clock; $t_3$ — the reception time of the echo, on the same clock; $t_2$ — the time assigned to the reflection at $B$ . Physical reading: Einstein's definition, from §1 of the 1905 paper. It is a stipulation, and the word is exact: $t_2$ is not read off any instrument, it is decreed.

c \equiv \frac{2L}{t_3 - t_1} \qquad (2)

where: $L$ — the rest distance from $A$ to $B$ , laid off with rods; $t_3 - t_1$ — the round-trip time on the single clock $A$ . Physical reading: the two-way speed — the only speed of light any experiment has ever measured. One clock, so no synchrony is presupposed and no circularity arises. Since 1983 the relation runs the other way: $c$ is fixed by definition and (2) defines the metre.

t_2 = t_1 + \varepsilon\,(t_3 - t_1), \qquad 0 < \varepsilon < 1 \qquad (3)

where: $\varepsilon$ — Reichenbach's synchronization parameter, from the 1924 Axiomatik; $\varepsilon = \tfrac12$ recovers (1). Physical reading: the general convention. Nothing forbids $\varepsilon \ne \tfrac12$ : the bounds $0 < \varepsilon < 1$ only require the reflection to be assigned a time between emission and reception — what causality demands, and no more.

c_{\rightarrow} = \frac{c}{2\varepsilon}, \qquad c_{\leftarrow} = \frac{c}{2 - 2\varepsilon}, \qquad \frac{2}{1/c_{\rightarrow} + 1/c_{\leftarrow}} = c \ \ \text{for every } \varepsilon \qquad (4)

where: $c_\rightarrow$ — the one-way speed $A \to B$ implied by $\varepsilon$ ; $c_\leftarrow$ — the one-way speed $B \to A$ . Physical reading: the one-way speeds are free — as $\varepsilon \to 0$ the outbound trip becomes instantaneous — but their harmonic mean is pinned at $c$ , identically in $\varepsilon$ . That identity is the whole reason no experiment can fix $\varepsilon$ . Only $\varepsilon = \tfrac12$ makes light isotropic.

t' = \gamma\left(t - \frac{v x}{c^{2}}\right) \quad\Longrightarrow\quad \Delta t'\big|_{\text{one instant of }S} = -\,\frac{v\,x'}{c^{2}} \qquad (5)

where: $x'$ — the proper position of a clock at rest in $S'$ ; the offset is what $S$ finds that clock reading, relative to the clock at $x' = 0$ , at one instant of $S$ . Physical reading: the gauge. Each frame runs the $\varepsilon = \tfrac12$ rule with its own light signals, and the result is a linear stagger $-vx'/c^2$ , the leading clock behind; the $\gamma$ cancels against the $1/\gamma$ of the contracted spacing. That term is the convention's fingerprint, not a fact about clocks.

Where the results come from

Derivation ANo experiment can fix ε — the two-way speed is c for every choice

The goal is to make the conventionality of simultaneity a theorem rather than a slogan. The claim is strong: for every $\varepsilon$ in $(0,1)$ the round-trip speed of light comes out at exactly $c$ — not approximately, not to leading order, but identically. So a laboratory that measures light speeds by bouncing pulses off mirrors can never discriminate $\varepsilon = \tfrac12$ from $\varepsilon = 0.6$ . The choice is underdetermined not by present data but by all possible round-trip data.

Step 1. Fix what is measured. Emit at $t_1$ , receive the echo at $t_3$ , both on the one clock at $A$ ; call $T = t_3 - t_1$ . Lay off $L$ with rods. The two-way speed $c = 2L/T$ is now an experimental number, and $L = cT/2$ . Why this is the honest starting point: one clock has been used, so nothing about distant simultaneity has been assumed. This is the entire empirical content of "the speed of light".

Step 2. Adopt the convention. Declare the reflection to happen at $t_2 = t_1 + \varepsilon T$ : the outbound leg is assigned $\varepsilon T$ and the inbound leg $(1 - \varepsilon)T$ , which sum to $T$ for any $\varepsilon$ . Why the bounds matter: $0 < \varepsilon < 1$ says only that the reflection lies between emission and reception. Nothing narrower is forced.

Step 3. Read off the one-way speeds. Outbound: $c_\rightarrow = L/(\varepsilon T) = (cT/2)/(\varepsilon T) = c/2\varepsilon$ . Inbound: $c_\leftarrow = L/[(1-\varepsilon)T] = c/(2 - 2\varepsilon)$ . At $\varepsilon = 0.6$ these are $2.498\times10^{8}$ and $3.747\times10^{8}\ \mathrm{m/s}$ — one leg slower than $c$ , the other faster. Why "faster than $c$ " is not a scandal: $c_\leftarrow$ is a coordinate speed under a chosen labelling of distant events. No signal arrives before it was sent; the round trip is untouched.

Step 4. Now the theorem. Time the round trip using the one-way speeds: $L/c_\rightarrow + L/c_\leftarrow = (2\varepsilon L/c) + (2 - 2\varepsilon)L/c = 2L/c$ . The $\varepsilon$ cancels identically. Hence the two-way speed is $2L \div (2L/c) = c$ , whatever $\varepsilon$ was — equivalently, the harmonic mean of $c_\rightarrow$ and $c_\leftarrow$ is $c$ for every $\varepsilon$ . Why this closes the case: any experiment that starts and ends at the same clock measures only this harmonic mean, and the harmonic mean is $\varepsilon$ -blind. The circularity is structural, not technological.

Step 5. Say precisely what $\varepsilon = \tfrac12$ buys. Setting $c_\rightarrow = c_\leftarrow$ in (4) gives $2\varepsilon = 2 - 2\varepsilon$ , so $\varepsilon = \tfrac12$ uniquely, and then both one-way speeds equal $c$ . Isotropy of the one-way speed is therefore installed by the convention, not proven. Why this is worth being blunt about: a student who thinks Michelson–Morley measured a one-way speed has misread it. The canon article records what it did measure — "a moving ether would make the two-way speed of light depend on direction at order $\beta^2$ " — and a first-order effect "in a two-way light path is forbidden by symmetry, not merely suppressed." *Verified in verify_aq-sr-einstein-clock-synchronization.py: sympy derives $c_\rightarrow = c/2\varepsilon$ and $c_\leftarrow = c/(2-2\varepsilon)$ from $L = cT/2$ , simplifies the harmonic mean to $c$ with $\varepsilon$ absent from the free symbols, and solves $c_\rightarrow = c_\leftarrow$ for the unique root $\varepsilon = \tfrac12$ . Numerically the $\varepsilon = 0.6$ speeds $2.498$ and $3.747\times10^{8}\ \mathrm{m/s}$ average back to $299\,792\,458\ \mathrm{m/s}$ exactly.*

Go deeper · Derivation B · 2 minSlow clock transport — a second, independent route to the same synchrony

Derivation BSlow clock transport — a second, independent route to the same synchrony

If $\varepsilon$ is free, why does everyone choose $\tfrac12$ ? Not for tidiness. There is a second procedure that never sends a light signal at all: synchronize two clocks side by side, then carry one from $A$ to $B$ and set $B$ by what it reads on arrival. This route uses no convention about light propagation — only the clock's own proper time. The theorem is that in the limit of infinitely slow transport it reproduces Einstein synchrony exactly. Two operationally unrelated procedures, one answer: that is why $\varepsilon = \tfrac12$ , a convention in principle, is not arbitrary in practice.

Step 1. Set up. Clocks $A$ and $B$ are at rest in $S$ , a distance $L$ apart, with $B$ already Einstein-synchronized to $A$ — we are testing the two procedures against each other, so one must supply the yardstick. A third clock, matched to $A$ at $x = 0$ , is carried to $B$ at constant speed $u \ll c$ . In $S$ the journey takes coordinate time $t = L/u$ , which is what the stationary clock at $B$ shows on arrival. Why this first half needs no relativity: it is distance over speed in a single frame, using clocks that frame has already set.

Step 2. Count the traveller's own time. The moving clock ticks its proper time, dilated by $\gamma_u = 1/\sqrt{1 - u^2/c^2}$ , so on arrival it reads $\tau = t/\gamma_u$ . The discrepancy is $\Delta = t - \tau = \dfrac{L}{u}\left(1 - \dfrac{1}{\gamma_u}\right)$ : the transported clock lags. Why time dilation and nothing else enters: the clock hypothesis says a clock's rate depends on its instantaneous speed, not its acceleration, so the gentle push-off and stop contribute nothing.

Step 3. Expand for small $u$ . With $1/\gamma_u = \sqrt{1 - u^2/c^2} \approx 1 - u^2/2c^2$ , the bracket is $\approx u^2/2c^2$ , so $\Delta \approx \dfrac{L}{u}\cdot\dfrac{u^2}{2c^2} = \dfrac{L\,u}{2c^2}$ . The two factors of $u$ fight: the trip takes longer as $u$ falls, but the dilation per unit time falls faster, as $u^2$ . The product goes to zero. Why the linear-in- $u$ result is the theorem in miniature: the error is proportional to the transport speed, so you can make it as small as you please by walking more slowly.

Step 4. Take the limit. $\lim_{u \to 0}\ (L/u)(1 - 1/\gamma_u) = 0$ — verified symbolically, not just asymptotically. A clock transported infinitely slowly arrives reading exactly what the Einstein-synchronized clock at $B$ reads: slow transport and $\varepsilon = \tfrac12$ define the same synchrony. At $u = 1\ \mathrm{m/s}$ over $L = 300\ \mathrm{m}$ the offset is already $1.67\times10^{-15}\ \mathrm{s}$ , below any laboratory's noise floor. Why this does not un-do the conventionality: a Reichenbach partisan can say slow transport merely inherits the same convention. The honest statement is that two procedures with no shared assumption agree — a reason of coherence, not of measurement. *Verified in verify_aq-sr-einstein-clock-synchronization.py: sympy forms $\Delta = L/u - (L/u)/\gamma_u$ , confirms it equals $(L/u)(1 - 1/\gamma_u)$ , series-expands it to $Lu/2c^2 + O(u^3)$ , and evaluates `sp.limit(delta, u, 0, '+')` $= 0$ — the theorem. The numbers $1.67\times10^{-15}\ \mathrm{s}$ ( $u = 1\ \mathrm{m/s}$ , $L = 300\ \mathrm{m}$ ) and $1.39\ \mathrm{ns}$ ( $u = 250\ \mathrm{m/s}$ , $L = 1000\ \mathrm{km}$ ) are recomputed in extended precision.*

Limiting cases

Case	Result	Meaning
$\varepsilon = \tfrac12$	$c_\rightarrow = c_\leftarrow = c$	Einstein synchrony: the unique choice that makes the one-way speed isotropic
$\varepsilon \to 0^{+}$ (or $1^{-}$ )	$c_\rightarrow \to \infty$ , $c_\leftarrow \to c/2 = 1.499\times10^{8}$ (or the mirror image)	one leg becomes instantaneous — an "absolute" now with $B$ . Still legal, still untestable
any $\varepsilon \in (0,1)$	two-way speed $= c$ exactly	the harmonic mean is $\varepsilon$ -blind: no round-trip experiment can choose
$u \to 0$ (slow transport)	$\Delta = (L/u)(1 - 1/\gamma_u) \to 0$	an independent procedure lands on $\varepsilon = \tfrac12$
$v \to 0$	stagger $-vx'/c^2 \to 0$	two frames agree on synchrony — the Galilean limit, where "now" is universal

Common trap“We just haven't measured the one-way speed of light yet — give the experimentalists another decade.”

You cannot, and no decade will help. A one-way measurement needs a clock at $A$ to stamp the emission and a clock at $B$ to stamp the arrival, reading a common time — and every way of making them read a common time either sends a signal (whose one-way speed is what you were trying to measure) or transports a clock (whose reading must be corrected using a speed you have not yet established). Assume $\varepsilon = \tfrac12$ and you will "measure" $c$ both ways: you put it in. Assume $\varepsilon = 0.6$ and you will "measure" $2.498$ and $3.747 \times 10^{8}\ \mathrm{m/s}$ — and every round-trip check will still return exactly $c$ , as Derivation A proves. The obstruction is not metrology; it is that "the one-way speed" is not a well-defined quantity until a convention has been chosen.

Worked examples

1 · The 300-metre baseline, done twice

A mirror sits $L = 300\ \mathrm{m}$ from clock $A$ . A pulse is emitted at $t_1 = 0$ and the echo returns at $t_3$ . (a) Find $t_3$ and the Einstein time $t_2$ . (b) Redo the labelling with $\varepsilon = 0.6$ : what are the one-way speeds, and what does the round trip come to?

Step 1. The round trip. $t_3 = 2L/c = 600/(2.998\times10^{8}) = \boxed{2.0014\ \mu\mathrm{s}}$ — about $2.0\ \mu\mathrm{s}$ , read off the single clock at $A$ . Nothing has been synchronized yet, and nothing needs to be.

Step 2. Part (a), Einstein. Equation (1): $t_2 = \tfrac12(0 + 2.0014) = \boxed{1.0007\ \mu\mathrm{s}}$ . Clock $B$ is set to read that at the moment the pulse bounces off it, so each leg is assigned $1.0007\ \mu\mathrm{s}$ for $300\ \mathrm{m}$ and both one-way speeds come out at $c$ — by construction.

Step 3. Part (b), $\varepsilon = 0.6$ . The legs are re-labelled $\varepsilon T = \boxed{1.2008\ \mu\mathrm{s}}$ out and $(1-\varepsilon)T = \boxed{0.8006\ \mu\mathrm{s}}$ back. The implied one-way speeds are $c_\rightarrow = c/1.2 = \boxed{2.498\times10^{8}\ \mathrm{m/s}}$ and $c_\leftarrow = c/0.8 = \boxed{3.747\times10^{8}\ \mathrm{m/s}}$ — light "crawls" one way and "sprints" the other.

Step 4. Check the round trip — the only thing anyone can measure. Total time $= 300/(2.498\times10^{8}) + 300/(3.747\times10^{8}) = 1.2008 + 0.8006 = 2.0014\ \mu\mathrm{s}$ , the same measured number, so the two-way speed is $\boxed{299\,792\,458\ \mathrm{m/s}} = c$ exactly. Both conventions survive intact, and no third experiment separates them.

2 · How slowly is "slow"?

A clock is carried from $A$ to $B$ to set $B$ by transport. (a) Over $L = 300\ \mathrm{m}$ at a walking pace $u = 1\ \mathrm{m/s}$ , how far does the carried clock end up behind an Einstein-synchronized clock at $B$ ? (b) A caesium clock is flown $1000\ \mathrm{km}$ at $u = 250\ \mathrm{m/s}$ . Is that still "slow"?

Step 1. The formula. From Derivation B, $\Delta = (L/u)(1 - 1/\gamma_u) \approx Lu/2c^2$ — first order in $u$ , and vanishing with it.

Step 2. Part (a). $\Delta \approx (300)(1)/[2(2.998\times10^{8})^2] = \boxed{1.67\times10^{-15}\ \mathrm{s}}$ — one and a half femtoseconds, far below anything a laboratory can resolve over $300\ \mathrm{m}$ . Walk the clock across the room and you have Einstein-synchronized $B$ to a precision nobody can challenge.

Step 3. Part (b). $\Delta \approx (10^{6})(250)/[2(2.998\times10^{8})^2] = \boxed{1.39\ \mathrm{ns}}$ — enormous for a caesium clock, which resolves picoseconds, and $42\ \mathrm{cm}$ of light travel, which matters for navigation.

Step 4. Read the moral. "Slow" is slow relative to the precision you demand: the error scales as $Lu/2c^2$ , so halving the speed halves it, all the way down. You never need the limit — only enough of it.

Physics of the portrait

The portrait is a procedure, not a person, and it is worth pausing on how strange that is. Physics usually names its portraits after discoveries. Clock synchronization is not a discovery; it is a decision, taken once, that made a coordinate time exist at all. Everything downstream — the Lorentz transformation's form, the very question "when did that distant thing happen" — depends on it.

Origins · 2 minSynchronization and how this was discovered

Synchronization

1898–1977 · Poincaré's convention · Einstein's definition · Reichenbach's ε · Malament's reply

The idea belongs to four people and remains argued over. Henri Poincaré (1854–1912) saw it first: in La mesure du temps (1898) he wrote that we have no direct intuition of the simultaneity of distant events, and that the rule we actually use — correcting a signal for its transit time — presupposes that light takes the same time each way. That, he insisted, is a postulate adopted because it makes the equations simple; in 1900 he gave Lorentz's "local time" a physical reading as what clocks show when observers moving with the Earth synchronize them by light signals on exactly that assumption. Albert Einstein (1905) put the definition on page one of Zur Elektrodynamik bewegter Körper and made it the foundation of the kinematics rather than a footnote. Hans Reichenbach (1891–1953) asked how much freedom was hiding in the choice, and in the Axiomatik der relativistischen Raum-Zeit-Lehre (1924) introduced the $\varepsilon$ -definition as Definition 2 of his axiom system; Adolf Grünbaum defended the thesis in Philosophical Problems of Space and Time (1963). The argument has not ended: in 1977 David Malament showed that $\varepsilon = \tfrac12$ is the only simultaneity relation definable from the causal structure of Minkowski spacetime alone.

History

1898

— Poincaré, "La mesure du temps", Revue de métaphysique et de morale 6, 1–13: the simultaneity of distant events must be fixed by a rule, and the rule presupposes light takes the same time each way.

1900

— Poincaré, "La théorie de Lorentz et le principe de réaction": Lorentz's "local time" read physically, as what clocks show when moving observers synchronize them by light signals.

1905

— Einstein, "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17, 891–921 (received 30 June, published 26 September): §1 defines simultaneity by the light-signal rule.

1913

— Sagnac's rotating interferometer (Comptes Rendus 157): light-signal synchronization around a closed rotating loop does not close on itself.

1924

— Reichenbach, Axiomatik der relativistischen Raum-Zeit-Lehre: the ε-definition of simultaneity appears in print, as Definition 2.

1963

— Grünbaum, Philosophical Problems of Space and Time: the conventionality thesis developed and defended.

1977

— Malament, Noûs 11, 293–300: ε = ½ is the only simultaneity relation definable from the causal structure of Minkowski spacetime.

1983

— the 17th CGPM fixes c = 299 792 458 m/s exactly and defines the metre from it: the two-way speed of light is no longer measured at all.

Going further

Texts. Einstein, Annalen der Physik 17, 891 (1905), §1 · Rindler, Introduction to Special Relativity, 2nd ed., §2 · Taylor and Wheeler, Spacetime Physics, 2nd ed., ch. 1 · Reichenbach, The Philosophy of Space and Time (Dover, 1958), §19–20.

Provenance. The rule $t_2 = \tfrac12(t_1+t_3)$ is Einstein 1905 §1; the $\varepsilon$ -family is Reichenbach's Axiomatik (1924), Definition 2; the slow-transport equivalence is standard in the conventionality literature. `verify_aq-sr-einstein-clock-synchronization.py` re-derives everything in sympy: the one-way speeds $c/2\varepsilon$ and $c/(2-2\varepsilon)$ , the two-way harmonic mean $= c$ with $\varepsilon$ absent from the free symbols, the unique root $\varepsilon = \tfrac12$ of $c_\rightarrow = c_\leftarrow$ , the transport offset $(L/u)(1 - 1/\gamma_u) \to 0$ as $u \to 0^+$ , and the LT reading $t/\gamma - vx'/c^2$ at proper position $x'$ . Every printed number is asserted, roundings included. Note: $c/(2-2\varepsilon)$ at $\varepsilon = 0.6$ is $3.7474\times10^{8}$ , i.e. $3.747\times10^{8}\ \mathrm{m/s}$ to four figures, not $3.748$ .

Builds on. The Two Postulates of Special Relativity · The Light Clock and Time Dilation · Einstein Synchronization of Clocks. Leads to. Relativity of Simultaneity · Derivation of the Lorentz Transformation · The Lorentz Transformation Equations · The Invariant Spacetime Interval.

If ε is genuinely free, is the one-way speed of light "really" c or not?

The question has no frame-free answer, and that is the point. "The one-way speed" is not a bare fact waiting to be uncovered; it is a number that exists only once you have said how distant clocks are to be set. Choose $\varepsilon = \tfrac12$ and it is $c$ in every direction; choose $\varepsilon = 0.6$ and it is $2.498\times10^{8}$ one way and $3.747\times10^{8}$ the other. Both predict identical outcomes for every experiment performable. Asking which is true is like asking whether the prime meridian really runs through Greenwich.

Doesn't a one-way speed greater than c break causality?

No, because nothing is transported faster than light. The label attached to one leg makes its coordinate speed exceed $c$ , but the pulse still leaves before it arrives, the round trip still takes $2L/c$ , and the causal order of every timelike pair is untouched — that is exactly what the bounds $0 < \varepsilon < 1$ guarantee. What you may not do is use a non-standard $\varepsilon$ and then quote the Lorentz transformation in its familiar form: in an $\varepsilon \ne \tfrac12$ chart the transformation looks different (asymmetric) while predicting the same measurable outcomes.

Doesn't slow clock transport prove ε = ½ after all, and destroy the conventionality thesis?

It is the strongest argument for $\varepsilon = \tfrac12$ , but it does not settle the philosophy. A determined conventionalist replies that analysing the transported clock's proper time already assumes a metric structure encoding the standard synchrony, so slow transport is not fully independent of the light convention. What is not in dispute is the physics: two procedures with no shared apparatus converge on the same synchrony in the limit, and the offset $Lu/2c^2$ says exactly how slowly you must walk to get there. Malament's 1977 result pushes the same way from the causal side.

Why 300 m and 2.0014 μs rather than 2.0000 μs? Where does the extra come from?

Because $c$ is $299\,792\,458\ \mathrm{m/s}$ , not $3\times10^{8}$ . The round trip covers $600\ \mathrm{m}$ , so $t_3 = 600/299\,792\,458 = 2.0014\ \mu\mathrm{s}$ . Since 1983 $c$ is exact by decree, so this arithmetic carries no experimental uncertainty at all — only a unit conversion.

Michelson and Morley compared light going different directions. Why doesn't that fix ε?

Because every path in the interferometer is a round trip: the light goes out along an arm, reflects, and returns to the beam splitter. As the canon article on the experiment puts it, "a moving ether would make the two-way speed of light depend on direction at order $\beta^2$ " — the two-way speed, in both arms. Derivation A shows that speed is $\varepsilon$ -blind, so the experiment could not have discriminated between conventions even in principle.

What actually goes wrong on a rotating platform, and why do people bring up the Sagnac effect here?

Einstein synchrony is defined pairwise, and on a rotating platform the pairwise rule does not close. Synchronize neighbouring clocks around a circle, return to your starting clock, and it is out of step with itself by an amount proportional to the enclosed area and the rotation rate. Nothing is broken: it means no globally consistent Einstein-synchronized time exists on a rotating frame, so timekeeping must be referred to an underlying inertial frame — which is what GPS does, and why the Sagnac correction is in its specification. Sagnac saw the effect in 1913.