The Conventionality of Simultaneity

In January 1893 Henri Poincaré joined the Bureau des Longitudes, and for two decades a mathematician best known for celestial mechanics spent his committee hours on a concrete problem: how do you make a clock in Paris and a clock in Berlin read the same time? The telegraph was the only instrument, and a telegraph signal takes time to arrive. Correct for the delay and the clocks agree — but to correct for it you must already know how long the signal took, and to know that you would need two clocks that already agree. The engineers cut the circle by sending the signal both ways and splitting the difference. In 1898 Poincaré wrote La Mesure du Temps and pointed out that the split was not a discovery. It was a rule we had chosen. Bounce a light pulse off a distant mirror and catch the echo: one clock tells you the round trip took $2L/c$ , and nothing you measured tells you when the pulse touched the mirror. Reichenbach turned that gap into a dial. Let $\varepsilon$ be the fraction of the round trip assigned to the outbound leg; the one-way speeds become $c/(2\varepsilon)$ and $c/(2(1-\varepsilon))$ , and their harmonic mean is exactly $c$ for every $\varepsilon$ . The two-way speed — the only speed anyone has measured — never budges. Turn the dial to $\varepsilon = 0.9$ and light crawls to the Moon at $0.556\,c$ and returns at $5\,c$ , and the radar echo still comes back in $2.564$ seconds, to the last digit.

Key takeaways

One clock measures the round-trip speed of light, $2L/(t_2-t_1)$ . A one-way speed needs a clock at the far end that already agrees with yours — which is what a synchronization rule has to supply. The circle is structural, not technical.
Reichenbach's rule $t_B = t_1 + \varepsilon(t_2-t_1)$ — globally, $t_\varepsilon = t + \kappa x/c$ with $\kappa \equiv 2\varepsilon-1$ — parametrizes every consistent way of cutting it. Einstein picks $\varepsilon = \tfrac12$ ; causality forces $0 < \varepsilon < 1$ , and nothing else constrains $\varepsilon$ .
The one-way speeds $c_+ = c/(2\varepsilon)$ and $c_- = c/(2(1-\varepsilon))$ are wildly anisotropic away from $\tfrac12$ — yet $2/(1/c_+ + 1/c_-) = c$ identically in $\varepsilon$ . At $\varepsilon = 0.9$ : out at $0.556\,c$ , back at $5\,c$ , Earth–Moon round trip still $2.564\ \mathrm{s}$ .
The relabelling is provably harmless: it makes the metric off-diagonal, $g_{0x} = -\kappa c$ , while leaving $ds^2$ along every curve unchanged. It tilts the light cone; it does not open it.
Key idea: what one clock measures is a fact; what two clocks measure is partly a stipulation. The two-way speed of light is a measured constant of nature; the one-way speed is a coordinate choice wearing the costume of a number — and no experiment, ever, can tell you which choice is right.

The core insight

Set a mirror a distance $L$ away, fire a pulse at your clock reading $t_1$ , catch the echo at $t_2$ . That is all the data. Giving the reflection a time is not reading an instrument — it is writing a definition, and $t_B = t_1 + \varepsilon(t_2-t_1)$ is the general form of every definition that respects the causal order. Standard synchrony, $\varepsilon = \tfrac12$ , is the sane choice and this article will not talk you out of it. What it shows is that the choice is a choice: the one-way speed of light is not a quantity awaiting a better measurement — it is a coordinate.

Where this lives today

The convention is running right now in every timing network on the planet. Two-way satellite time and frequency transfer links national timing laboratories by sending a signal each way through a geostationary satellite and taking the half-sum — Einstein's rule in hardware, and one of the ingredients from which International Atomic Time is assembled. The Network Time Protocol does the same arithmetic on every computer you own, estimating your clock's offset from a server's as half the difference of the two legs, on the explicit assumption that the network delay is symmetric. When that assumption fails — an asymmetric fibre route, a one-way queue — the error is not a broken clock but a broken convention, and the residual is exactly this article's $\kappa$ . The engineering has improved by fourteen orders of magnitude since Paris and Berlin; the epistemology has not moved.

The physical picture

You stand at $A$ with a clock. A mirror sits at $B$ , a distance $L$ away, laid out beforehand with rods. At $t_1 = 0$ you fire a pulse; at $t_2 = 2L/c$ the echo returns. Two numbers, one instrument, no assumptions. Now answer a simple question: at what time did the pulse strike the mirror? You cannot look. To see the reflection you must wait for light to bring the news — which is the round trip you already timed. You could post a friend at $B$ with her own clock, but her clock must be set, and setting it is the problem. So you stipulate: you declare the reflection to have happened at $t_B = \varepsilon\,t_2$ , with $\varepsilon$ strictly between $0$ and $1$ — strictly, because the reflection lies causally after the emission and before the reception, and no relabelling may reverse that.

Figure

The ε-dial. ONE experiment: pulse out at t₁ = 0, echo back at t₂ = 2L/c. Those two clock readings are the entire dataset and are identical in all three panels — only the stipulated turnaround changes, and with it the surface of “now”

The dashed line is where the trouble lives. It is the set of events the observer at $A$ calls simultaneous with the mirror flash, and it swings through a fan of orientations as the dial turns — while $t_1$ and $t_2$ , the readings that generated it, sit unmoved. A quantity that changes when nothing measurable changes is not a measurement. It is a label. Now turn the dial hard: push $\varepsilon$ toward $1$ and the outbound light crawls toward $c/2$ while the return light diverges. Every one of those descriptions predicts the same echo, at the same instant, on the same clock.

Figure

PUNCHLINE — one-way speeds against ε, log scale. c₊ diverges as ε → 0, c₋ as ε → 1, and at ε = 0.9 light goes out at 0.556 c and returns at 5 c. The two-way speed, in green, is dead flat at exactly c across the whole dial — the only thing anyone has measured

The green line is not an approximation. It is an algebraic identity, proved next.

The core equations

Start with the only thing you own: two readings on one clock. Fire at $t_1$ , catch the echo at $t_2$ , divide the there-and-back distance by the elapsed time. This is the two-way speed of light — the quantity The Michelson–Morley Experiment bounded, and whose modern isotropy limits belong to The Constancy of the Speed of Light.

c_{\text{two-way}} \equiv \frac{2L}{t_2 - t_1} = c \qquad (1)

where: $L$ — the $A$ -to- $B$ distance, laid out with rods in the frame where both are at rest; $t_1, t_2$ — emission and reception, both on the single clock at $A$ . Physical reading: no second clock appears in this equation, which is why it is a fact about the world rather than about our bookkeeping — and why the 1983 SI defines the metre from it. Now the step that is not a measurement. To give the reflection a time we must invent one, and the general invention respecting the causal order assigns a fraction $\varepsilon$ of the round trip to the outbound leg. Globally, the same rule is a relabelling of time growing linearly with position, $t_\varepsilon = t + \kappa x/c$ .

t_B = t_1 + \varepsilon\,(t_2 - t_1), \qquad \kappa \equiv 2\varepsilon - 1, \qquad 0 < \varepsilon < 1 \qquad (2)

where: $t_B$ — the coordinate time stipulated for the reflection; $\varepsilon$ — Reichenbach's synchrony parameter; $\kappa \in (-1,1)$ — the anisotropy, zero at standard synchrony. Physical reading: the endpoints are excluded by causality, not taste. And $x$ is untouched — no rod moves, no clock is re-rated; only the offsets of distant clocks are re-set, in proportion to their distance, which is all a synchrony convention ever does. Feed (2) into the definition of a speed and the directions come apart: outbound, light covers $L$ in $\varepsilon\cdot 2L/c$ ; inbound, in the leftover.

c_+ = \frac{c}{2\varepsilon} = \frac{c}{1+\kappa}, \qquad c_- = \frac{c}{2(1-\varepsilon)} = \frac{c}{1-\kappa} \qquad (3)

where: $c_\pm$ — one-way speeds along $\pm x$ . Physical reading: at $\varepsilon = \tfrac12$ both equal $c$ ; away from it they are wildly unequal — yet their reciprocals, which are what add up over a round trip, sum to $2/c$ regardless. The anisotropy has nowhere to show itself. The last equation settles the physics, using the flat line element purely as a coordinate-change exercise: substitute $t = t_\varepsilon - \kappa x/c$ into $ds^2 = c^2dt^2 - dx^2$ . (What makes $ds^2$ invariant between frames is a different question, belonging to The Invariant Spacetime Interval.)

ds^2 = c^2 dt_\varepsilon^2 - 2\kappa c\,dt_\varepsilon\,dx - (1-\kappa^2)\,dx^2, \qquad g_{0x} = -\kappa c \qquad (4)

where: $g_{0x}$ — the off-diagonal time–space metric component, nonzero exactly when $\varepsilon \neq \tfrac12$ . Physical reading: the form of the metric changed; the value of $ds^2$ along any curve did not. Everything built from $ds^2$ — every clock reading, decay length, fringe count — is therefore blind to $\varepsilon$ . A convention that leaves the interval alone can leave no fingerprint on an experiment.

Where the freedom comes from, and why it costs nothing

Derivation AThe epsilon family, the one-way speeds, and why the two-way speed cannot notice

The claim is sharper than "the one-way speed is hard to measure". It is that the one-way speed is not a measurable quantity at all: the number you would quote is fixed by a stipulation made before the experiment began, and every stipulation reproduces the identical round-trip datum. Establishing that means showing the two-way speed is not merely approximately unaffected but identically $c$ as a function of $\varepsilon$ across the whole open interval — an algebraic identity, with no expansion in it anywhere.

Step 1. Set the data. A pulse leaves $A$ at $t_1 = 0$ , reflects at $B$ a rod-measured distance $L$ away, and returns at $t_2 = 2L/c$ on the single clock at $A$ . That is the complete record: one number. Why: nothing here presupposes a synchrony, because nothing here involves a second clock — which is why the round trip, and only the round trip, is empirical bedrock.

Step 2. Stipulate the turnaround: $t_B = \varepsilon t_2 = 2\varepsilon L/c$ , with $0 < \varepsilon < 1$ forced by the chain emission $\prec$ reflection $\prec$ reception. Why: that is the only constraint the world imposes, and within it the assignment is free — "free" being a technical claim we are about to cash.

Step 3. Read off the one-way speeds. Outbound, $c_+ = L/t_B = c/(2\varepsilon)$ ; the return leg occupies $t_2 - t_B = (1-\varepsilon)\,2L/c$ , so $c_- = c/(2(1-\varepsilon))$ . At $\varepsilon = 0.9$ these are $0.556\,c$ and $5\,c$ — a factor of nine apart. Why: a one-way speed is nothing but $L$ divided by a number we wrote down ourselves; its "anisotropy" is the anisotropy of our own bookkeeping.

Step 4. Now compute the two-way speed back out of the one-way speeds — the only honest test of whether the convention has teeth. The round trip is $L/c_+ + L/c_- = 2\varepsilon L/c + 2(1-\varepsilon)L/c = 2L/c$ : the $\varepsilon$ cancels exactly, so $c_{\text{two-way}} = c$ and the harmonic mean $2/(1/c_+ + 1/c_-) = c$ identically. Why: the round trip weights the legs by precisely the times we assigned them, and those times were built to sum to the measured total. The cancellation is the definition biting its own tail.

Step 5. Draw the moral. An experiment distinguishing $\varepsilon = 0.9$ from $\varepsilon = 0.5$ would have to time a one-way transit, needing two already-synchronized clocks, needing an $\varepsilon$ . Slow clock transport looks like an escape and is not: Einstein Synchronization of Clocks shows a transported clock agrees with standard synchrony as its speed goes to zero — but that agreement is computed in the standard time coordinate. Why: every operational road to the one-way speed runs through a synchrony convention. There is no back door. *Verified symbolically in verify_aq-sr-conventionality-of-simultaneity.py: $c_\pm$ from the $\varepsilon$ -assignment; $L/c_+ + L/c_- - 2L/c$ simplifies to $0$ for symbolic $\varepsilon$ ; the harmonic mean simplifies to $c$ identically, and numerically at 25 values of $\varepsilon$ ; and $0.556\,c$ , $5\,c$ at $\varepsilon = 0.9$ .*

Go deeper · Derivation B · 2 minWhy the relabelling is provably harmless: an off-diagonal metric with an unchanged interval

Derivation BWhy the relabelling is provably harmless: an off-diagonal metric with an unchanged interval

Showing that the round-trip datum cannot detect $\varepsilon$ is necessary but nowhere near sufficient — a physicist is entitled to ask whether some other experiment could. The strongest answer available is structural: show the entire $\varepsilon$ -family is a change of coordinates on one and the same flat spacetime, so the invariant from which every outcome is built is untouched by construction. Then nothing is left for any experiment to grip.

Step 1. Write the relabelling as the coordinate map $t_\varepsilon = t + \kappa x/c$ , with $x$ unchanged, and check it reproduces the rule: a pulse leaving $(0,0)$ arrives at $x = L$ at standard time $t = L/c$ , so $t_\varepsilon = (1+\kappa)L/c = 2\varepsilon L/c$ — Step 2 of the previous card. Why: the local rule and the global coordinate change are the same object, which licenses treating $\varepsilon$ as a choice of coordinates rather than a hypothesis about light.

Step 2. Invert and differentiate: $c\,dt = c\,dt_\varepsilon - \kappa\,dx$ . Substitute into $ds^2 = c^2dt^2 - dx^2$ : $ds^2 = (c\,dt_\varepsilon - \kappa\,dx)^2 - dx^2 = c^2dt_\varepsilon^2 - 2\kappa c\,dt_\varepsilon dx - (1-\kappa^2)dx^2$ , which is equation (4). Why: a cross term has appeared. The metric is genuinely non-diagonal, $g_{0x} = -\kappa c$ — the signature of a "now" that is not orthogonal to the observer's worldline.

Step 3. Notice what did not change. The substitution is an identity, so along any curve whatever the number $ds^2$ evaluates the same in both coordinate systems. The metric components changed; the metric did not. Why: this is the whole ballgame. Anything computable from $ds^2$ alone — what a carried clock reads, how far a muon gets before decaying, how many fringes a beam splitter counts — is identical for every $\varepsilon$ . The formal machinery is Proper Time (S1); we need only the invariance.

Step 4. Recover the light cone. Set $ds^2 = 0$ and solve $(1-\kappa^2)u^2 + 2\kappa c\,u - c^2 = 0$ for $u \equiv dx/dt_\varepsilon$ : the roots are $c/(1+\kappa) = c/(2\varepsilon)$ and $-c/(1-\kappa) = -c/(2-2\varepsilon)$ — exactly $c_+$ and $-c_-$ from (3). Why: the two derivations meet. The off-diagonal metric hands back the same anisotropic one-way speeds: the cone has been <strong>tilted</strong>, not opened, and its two edges still bound every causal process.

Step 5. State the boundary. As $\varepsilon \to 0$ or $1$ , $|\kappa| \to 1$ and $(1-\kappa^2) \to 0$ : the cone degenerates and one one-way speed diverges. Why: conventionality is bounded by causality. You may relabel "now" as freely as you like; you may not relabel "before" into "after", and the metric knows it. *Verified symbolically in verify_aq-sr-conventionality-of-simultaneity.py: the substitution reproduces (4) with $g_{0x} = -\kappa c$ ; $ds^2$ in $(t_\varepsilon, x)$ simplifies back to $c^2dt^2 - dx^2$ identically in $\kappa$ ; solving $ds^2 = 0$ returns $c/(2\varepsilon)$ and $-c/(2-2\varepsilon)$ ; and $(1-\kappa^2) \to 0$ at both endpoints.*

Limiting cases

Case	Result	Meaning
Standard synchrony, $\varepsilon = \tfrac12$	$\kappa = 0$ , $c_+ = c_- = c$ , $g_{0x} = 0$	the isotropic choice: metric diagonal, "now" orthogonal to the worldline
$\varepsilon \to 0^+$ or $1^-$	one speed $\to \infty$ , the other $\to c/2$	endpoints excluded: they collapse emission $\prec$ reflection $\prec$ reception
Two-way speed, any $\varepsilon$	$2/(1/c_+ + 1/c_-) = c$ exactly	the identity that makes the whole family empirically indistinguishable
Galilean limit, $c \to \infty$	$\kappa x/c \to 0$ , so $t_\varepsilon \to t$	instantaneous signals collapse the dial and absolute simultaneity returns — conventionality is a consequence of light being finitely fast (recall Absolute Space and Absolute Time)

Common trap“The one-way speed of light has never been measured” is true — and it does NOT mean light might really be anisotropic

Half the sentence is right. No experiment has measured a one-way light speed, and none can: every such measurement needs two synchronized clocks, and synchronizing them presupposes the answer itself. But the usual moral — so maybe light really does go faster one way than the other and we cannot tell — is wrong, and instructively so. There is no fact there to be ignorant of. The $\varepsilon \neq \tfrac12$ description is not a rival hypothesis we lack the technology to test; it is the same world, written in coordinates whose time axis is sheared, and the second derivation proves it in the strongest available form: $ds^2$ , from which every observable is built, is numerically unchanged. Compare: "the prime meridian might really be elsewhere". It is not that Greenwich is unverifiable — it is that "really" has no work to do.

Worked examples

1 · Turn the dial to 0.9 and fire a laser at the Moon

The Apollo and Lunokhod retroreflectors sit $L = 3.844\times10^{8}\ \mathrm{m}$ away. Find the outbound, return and round-trip times under standard synchrony and under $\varepsilon = 0.9$ . Take $c = 2.998\times10^{8}\ \mathrm{m/s}$ .

Step 1. Get the datum — the only thing the observatory records. The round trip is $2L/c = 2(3.844\times10^{8})/(2.998\times10^{8}) = \mathbf{2.564\ s}$ (with the exact SI $c$ : $2.5644\ \mathrm{s}$ , the same number). A photodiode registers this and nothing else.

Step 2. Standard synchrony, $\varepsilon = \tfrac12$ : each leg gets $1.282\ \mathrm{s}$ — out at $c$ , back at $c$ , the Moon " $1.28$ light-seconds away".

Step 3. Set $\varepsilon = 0.9$ . The outbound leg is stipulated to take $0.9 \times 2.5644 = \mathbf{2.308\ s}$ , the return the remaining $0.1 \times 2.5644 = \mathbf{0.256\ s}$ . The implied speeds are $c_+ = c/1.8 = \mathbf{0.556\,c}$ going and $c_- = c/0.2 = \mathbf{5\,c}$ returning — on this bookkeeping the echo outruns light by a factor of five.

Step 4. Add them: $2.308 + 0.256 = \mathbf{2.564\ s}$ , identical to Step 1, digit for digit. The photodiode cannot tell the stories apart, because there is only one story — the same photons, the same telescope, the same tick. Lunar ranging is good to millimetres, and none of that precision bears on $\varepsilon$ .

2 · The same rod, two lengths — recalling the epsilon-length result

Operational Definition of Distance derives, for a rod of proper length $L$ moving at $\beta = v/c$ and measured by marking both ends at equal $\varepsilon$ -time, the result $L_\varepsilon = L/(1 + \beta(2\varepsilon-1))$ . Evaluate it at $\beta = 0.6$ , and separate what is physical from what is not.

Step 1. Recall the formula by name — it is derived in Operational Definition of Distance and reused here. With $\kappa = 2\varepsilon-1$ , $L_\varepsilon = L/(1+\beta\kappa)$ . It is first order in $\beta$ : a simultaneity effect, not a $\sqrt{1-\beta^2}$ effect, and nothing to do with length contraction, which Length Contraction (S1) owns and which is second order. At $\varepsilon = \tfrac12$ , $\kappa = 0$ and $L_\varepsilon = L$ .

Step 2. At $\varepsilon = 0.9$ : $\kappa = 0.8$ , so $1 + (0.6)(0.8) = 1.48$ and $L_\varepsilon = L/1.48 = \mathbf{0.676\,L}$ . At $\varepsilon = 0.1$ : $\kappa = -0.8$ , so $1 + (0.6)(-0.8) = 0.52$ and $L_\varepsilon = L/0.52 = \mathbf{1.923\,L}$ . The same rod, in the same experiment, "measures" $0.676\,L$ or $1.923\,L$ — a factor of $2.8$ apart.

Step 3. Separate fact from label. Marking both ends of a moving rod "at the same time" needs a simultaneity, so the answer inherits the convention: the factor of $2.8$ is bookkeeping. Measuring the rod in its own rest frame, where its ends never move, needs none at all — you lay a ruler alongside at leisure. That number, the proper length, is independent of $\varepsilon$ for every $\varepsilon$ : set $\beta = 0$ and $L_\varepsilon = L$ identically. What one clock measures is a fact; what two clocks measure is partly a stipulation.

Physics of the portrait

Poincaré found the door and did not walk through it. Between 1898 and 1905 he stated in print that simultaneity at a distance is a convention; that clocks are synchronized by assuming light equally fast both ways; that Lorentz's "local time" is what such clocks actually read; and that the principle of relativity — his phrase — might be an exact law. He never discarded the ether. But every philosophical question in this article is his.

Origins · 2 minHenri Poincaré and how this was discovered

Henri Poincaré

1854–1912 · Nancy · École Polytechnique · Sorbonne · Bureau des Longitudes

Born in Nancy in 1854, Poincaré was the last universalist: he reshaped celestial mechanics, founded topology, and found deterministic chaos in the three-body problem. He also sat on committees, and one mattered more for relativity than any of it. From January 1893 until his death he served on the Bureau des Longitudes, the body responsible for French timekeeping and for fixing the longitudes of a colonial empire — which meant, in practice, running telegraph signals between distant observatories and arguing about how to subtract the transmission delay. In 1898 he analysed the telegraphic longitude determination between Paris and Berlin, and in the same year published La Mesure du Temps</strong>, which contains the thesis of this article: we have no direct intuition of the simultaneity of distant events, and the rule by which we establish it — assume the signal takes equal times both ways — is a definition, adopted because it makes the equations simple, not because nature dictates it. In 1900, at the Lorentz jubilee, he read Lorentz's local time as precisely what light-synchronized clocks on a moving Earth would show; in 1902 he set the conventionalist programme out in La Science et l'Hypothèse</strong> — on number, space, geometry and mechanics, but not on time, which stayed in the essays; he named the principle of relativity in 1904. What he would not do was give up the ether — and so he assembled nearly every part of special relativity while declining to build it.

History
1893
— Poincaré joins the Bureau des Longitudes, and spends two decades on the practical problem of synchronizing distant clocks by telegraph.
1898
— Poincaré, "La Mesure du Temps", Revue de Métaphysique et de Morale 6, 1–13: simultaneity at a distance rests on a rule we adopt, not a fact we perceive.
1900
— Poincaré, at the Lorentz jubilee, reads Lorentz's local time as what light-synchronized clocks show, given that light is assumed equally fast both ways.
1902
— Poincaré, La Science et l'Hypothèse: conventionalism about number, space, geometry and mechanics, for a general readership. The 1898 essay on time is not in it; it is reprinted in La Valeur de la Science (1905).
1905
— Einstein, "Zur Elektrodynamik bewegter Körper", §1: the synchronization rule is introduced explicitly as a definition, not a result.
1928
— Reichenbach, Philosophie der Raum-Zeit-Lehre, introduces the ε-parametrization (English translation, The Philosophy of Space and Time, 1958).
1963
— Grünbaum, Philosophical Problems of Space and Time (2nd, enlarged edition 1973): the conventionality thesis in its strongest form.
1967
— Ellis and Bowman, Philosophy of Science 34, 116–136: slow clock transport, they argue, singles out standard synchrony.
1977
— Malament, Noûs 11, 293–300: under stated premises, ε = ½ is the unique simultaneity relation definable from causal structure plus an observer.
1999
— Sarkar and Stachel, Philosophy of Science 66, 208–220, challenge Malament's premises. The debate remains open.

Going further

Texts. Reichenbach, The Philosophy of Space and Time (1928; English 1958), §19 · Grünbaum, Philosophical Problems of Space and Time, 2nd ed. (1973) · Malament, Noûs 11 (1977) 293 · Janis, "Conventionality of Simultaneity", Stanford Encyclopedia of Philosophy · Rindler, Introduction to Special Relativity, §§2–5 · Galison, Einstein's Clocks, Poincaré's Maps (2003).

Provenance. The $\varepsilon$ -rule, the one-way speeds and the harmonic constraint are Reichenbach's (1928, §19) as developed by Grünbaum (1973); Malament's theorem is quoted from Noûs 11 (1977) 293–300, the objections from Sarkar & Stachel (1999) and Janis, the slow-transport argument from Ellis & Bowman (1967). Every number is recomputed in verify_aq-sr-conventionality-of-simultaneity.py, which also checks these bibliographic dates as a string table so a wrong year fails the build: Earth–Moon $3.844\times10^{8}\ \mathrm{m}$ at $c = 2.998\times10^{8}\ \mathrm{m/s}$ ⇒ round trip $2.564\ \mathrm{s} = 2.308 + 0.256\ \mathrm{s}$ at $\varepsilon = 0.9$ , with $c_+ = 0.556\,c$ , $c_- = 5\,c$ ; and $L_\varepsilon = L/(1+\beta(2\varepsilon-1))$ , recalled by name from Operational Definition of Distance, giving $0.676\,L$ and $1.923\,L$ at $\beta = 0.6$ . It confirms symbolically that $L/c_+ + L/c_- = 2L/c$ identically in $\varepsilon$ , that the coordinate change reproduces $ds^2$ , and that $ds^2 = 0$ returns $c/(2\varepsilon)$ and $-c/(2-2\varepsilon)$ .

Builds on. Inertial Frames of Reference · The Constancy of the Speed of Light · Einstein Synchronization of Clocks · Operational Definition of Distance. Leads to. The Relativity of Simultaneity · The Lorentz Transformation Equations · Minkowski Spacetime and the Light Cone.

Didn't Malament prove in 1977 that epsilon = 1/2 is the only possible choice?
He proved a real theorem, under premises worth stating exactly. Give yourself only causal connectibility — $p$ and $q$ are related when one lies on or inside the other's light cone — and the worldline $O$ of one inertial observer. Require a simultaneity relation $S$ to be (i) definable from those two, i.e. invariant under every causal automorphism of Minkowski spacetime mapping $O$ to itself; (ii) an equivalence relation; (iii) non-trivial. Malament showed that under (i)–(iii), $S$ must be standard synchrony. The critics — Grünbaum, Janis, Sarkar and Stachel — do not dispute the mathematics; they dispute premise (i), arguing that invariance under the full automorphism group (temporal reflection included) is stronger than definability from bare causal order, and smuggles in the isotropy of time the convention was free to violate. This article does not adjudicate. The debate is live.

Doesn't slow clock transport measure the one-way speed and settle this?
It is the best argument against conventionality and it is not decisive. Walk a clock slowly from $A$ to $B$ and its lag relative to the light-synchronized value vanishes as the transport speed does — Einstein Synchronization of Clocks derives this, and Ellis and Bowman (1967) pressed it hard. The conventionalist reply: the calculation showing the lag vanishes is itself carried out in the standard time coordinate, so it agrees with light synchrony because it assumed isotropic coordinates. Redone in $\varepsilon$ -coordinates it reproduces $\varepsilon$ -synchrony. Whether that is deep or merely circular is what the literature argues about.

Why is epsilon restricted to the open interval, and never 0 or 1?
Because relabelling times may not reorder causes and effects. The reflection at $B$ happens after the emission at $A$ and before the reception at $A$ — not a convention but a fact about which events can influence which. Setting $\varepsilon = 0$ gives the reflection the same coordinate time as the emission, and $\varepsilon = 1$ the same time as the reception; either way a strict causal chain collapses into a simultaneity. Geometrically, $|\kappa| \to 1$ drives $(1-\kappa^2)$ to zero and the light cone degenerates. Conventionality is bounded by causality.

Does any of this touch the twin paradox, or muon lifetimes, or GPS?
Not by a hair, and the second derivation says why in a line: $ds^2$ is unchanged along every curve. The twins' differential ageing compares two clock readings at the same place, when they meet — a local coincidence, and coincidences are $\varepsilon$ -blind. A muon's decay length is set by what its own clock reads. GPS is subtler, because its architecture adopts Einstein synchrony in the Earth-centred inertial frame, so $\varepsilon$ sits in its coordinate conventions — but the corrections it applies are proper-time effects, and those are invariant. The convention moves the labels, never the readings.

Is this the same thing as the relativity of simultaneity?
No, and conflating them is the commonest error in the area. The relativity of simultaneity (The Relativity of Simultaneity) is a claim about two frames: fix $\varepsilon = \tfrac12$ everywhere and observers in relative motion still disagree about which distant events are simultaneous — a disagreement forced by the postulates and visible in real experiments. Conventionality is a claim about one frame: even standing still, which distant events you call simultaneous is not fixed by the data. The first is physics you cannot escape; the second is bookkeeping you cannot pin down.

Poincaré said all this in 1898. Why is Einstein credited with relativity?
Because saying that simultaneity is conventional is not the same as building a theory on it. Poincaré had the convention, the operational reading of local time, the phrase "principle of relativity", and by 1905 the Lorentz group — and he kept the ether, treating the moving observer's time as apparent and the ether frame's as true. Einstein threw the ether out, promoted the two postulates to axioms, and let the transformation follow. The philosophy of this article is Poincaré's; the rest of the track needed someone willing to say the moving observer's time is not apparent but real.