The speed that cannot be measured one way

Simultaneity is a choice of slicing, and different observers slice differently. This essay is about a choice that is available to a single observer, and about what it does to the most quoted number in physics.

The procedure, and the gap in it #

To measure a one-way speed, note the time a pulse leaves A and the time it arrives at B, and divide the distance by the difference. That requires the clock at B to be set correctly relative to the one at A.

How is it set? The standard method is radar: send a pulse from A at $t_1$, let it bounce off B, receive it back at $t_3$, and set B’s clock — for the moment of the bounce — to

$$t_2 = t_1 + \varepsilon(t_3 - t_1)$$

Einstein’s convention is $\varepsilon = \tfrac{1}{2}$: assume the pulse took as long going as coming back. That assumption is exactly the statement that the one-way speed is the same in both directions, which is what the experiment was supposed to determine.

The circularity is not a flaw in the procedure; it is a property of the situation. Any $\varepsilon$ strictly between 0 and 1 gives a self-consistent description in which the speed of light differs between the two directions, and no experiment distinguishes them, because everything measurable involves light going out and coming back.

Why every workaround fails in the same way #

The natural response is to find some other way of synchronising, and there are three standard attempts. All three fail, and it is worth seeing that they fail differently in appearance and identically in substance.

Slow clock transport. Synchronise two clocks side by side, then carry one slowly to the far position. Time dilation vanishes in the limit of zero speed, so the transported clock should still agree.

The objection is that the transported clock’s rate is described using coordinates that already contain the convention. If the one-way speed is anisotropic then so is the time dilation of a slowly moving clock, in exactly the way that reproduces the original convention. The procedure is self-consistent with any $\varepsilon$ and therefore selects none.

What the slow-transport argument needs is that a moving clock’s slowing depends only on its speed and not on the direction it travels in. That is true in the isotropic convention and false in every other one, by construction — the anisotropic conventions put the direction into the transformation explicitly. So the argument does not fail because transport is imperfect; it fails because it assumes the answer it is being used to establish.

A rigid rod. Lay a rod between the clocks and use its length. But the length of a moving rod, and the simultaneity of its two ends, are the quantities in question.

Astronomical events. Jupiter’s moons, pulsar timing, the fly-past of a spacecraft: all of them are one-way measurements of something, and all of them require a clock at the far end whose reading was set by one of the above.

There is a fourth response, which is to declare the question meaningless. That is Reichenbach’s position and it is defensible: a quantity no experiment can determine is not a physical quantity, and the isotropy of light is a definition rather than a fact. The opposing view is that the choice is so overwhelmingly natural that treating it as arbitrary is perverse. Both camps agree on every prediction.

What survives every convention is the cone. Which events lie inside an event’s future light cone, on it, and outside it is not conventional at all: it is decided by whether a signal at or below $c$ can get from one to the other, and every convention agrees. What is conventional is only how the interior is ruled into surfaces of constant time — which lines are drawn across the cone, not where the cone is.

The half that is not free #

It is worth marking out precisely which part of the description is conventional, because the claim is often overstated in both directions.

What is free is the labelling of events with a time coordinate — specifically, which set of events at different places is assigned the same label. What is not free is anything else: the ordering of causally connected events, the interval between any pair, the proper time along any worldline, and the outcome of any experiment.

A useful test is whether a quantity survives being computed at one place. Every convention-independent statement in relativity can be written as a comparison made at a single event: two clocks brought together and read; a pulse leaving and returning to one detector; a particle decaying after a proper time it carries with it. Everything that requires two places and one time is where the convention enters.

Every convention-free statement in relativity has one shape: a clock read twice at the same place. That is exactly what makes the round-trip speed measurable and the one-way speed not — the round trip begins and ends at one clock, and there is nothing to synchronise. A pulse leaving a detector and coming back to it, two clocks brought together and compared, a particle decaying after a proper time it carried with it: all one-place comparisons, all conventionally free.

That test is worth carrying beyond relativity. A quantity that can only be defined by comparing readings at two places, using a rule that had to be chosen, is a candidate for being a convention — and in physics the honest response is to find the version of the question that a single instrument can answer.

What is actually measured #

It is worth being precise about which experimental facts are untouched, because the list is long and the conventional part is small.

Round-trip constancy. Michelson and Morley, and every optical cavity experiment since, compares round-trip times along different directions. Modern versions bound the anisotropy of the round-trip speed at a part in 10¹⁸. That is a real measurement and it is not a statement about the one-way speed.

Time dilation between reunited clocks. The twin who comes back younger is a comparison of two clocks at the same place, twice. No synchronisation is involved and the result is convention-independent.

The interval. The quantity nobody argues about is invariant under any relabelling of coordinates whatever, which includes the ε-family.

The invariant interval is the geometric statement of the same thing. Drawn as hyperbolae, it is what every observer agrees on — and changing the synchronisation convention changes which straight lines are called “constant time” while moving nothing on those curves at all. That is as sharp a division between the conventional and the physical as the subject offers: one alters the ruling, the other would alter the geometry, and only the first is available.

And every causal relation. Whether one event can influence another is a question about the light cone, and the light cone is untouched. Which of two events came first is conventional only for pairs that could not have influenced each other, which is the same statement relativity was making already.

Writing the theory the other way #

The clearest way to see that nothing physical is at stake is to write special relativity in an anisotropic convention and see what happens. It has been done, thoroughly, and the answer is that the equations get uglier and the predictions do not change.

In such a formulation the metric acquires cross terms between space and time; the coordinate speed of light depends on direction; and the transformation between frames is no longer the familiar Lorentz form. Every measurable quantity — the round-trip time, the Doppler shift, the aberration angle, the outcome of any experiment — comes out identical.

That is the technical content of the claim that the convention is a convention: not that the question is unimportant, but that it is a choice of coordinates, in the same sense that using polar coordinates rather than Cartesian is a choice. Coordinates are not measurements.

The composition law is where the cost of the other conventions shows. In the isotropic convention velocities combine by the standard formula; in an anisotropic one they combine by something more elaborate, with the direction of travel appearing explicitly. Both give the same answer for every closed circuit — which is what a measurement is — and the standard form is preferred because it is tidier, not because it is truer.

The one place it might not be a convention #

There is a caveat worth stating, because it is where the debate is live rather than settled.

The argument above concerns conventions — relabellings with no physical content. It does not cover the possibility of a genuine, physically real anisotropy of space, in which the round trip would also be affected at some order and the effect would be detectable.

Those are searched for constantly. Modern tests using pairs of optical cavities, or comparing atomic clocks as the Earth rotates and orbits, bound Lorentz-violating anisotropies at extraordinary levels — parts in 10¹⁸ and better in some coefficients. A real anisotropy is a measurable thing and has not been found.

The distinction is the one to keep. A convention cannot be measured because there is nothing there; an anisotropy has not been measured because it appears not to exist. The two conclusions look alike from a distance and are entirely different in kind.

Telling a change of frame from a change of convention is the whole content of this essay, and the two are easy to confuse because both amount to relabelling. A change of frame is real in its consequences: lattice spacings measured from two frames genuinely differ, and the difference is what an experiment reports. A change of convention alters no measurement anywhere. The test is the one above — does the quantity survive being read at a single place.

Why the natural choice is natural #

Nothing above says the isotropic convention is a bad one. It is overwhelmingly the right choice, and saying why is not the same as saying it is forced.

It makes the equations simplest: the metric is diagonal, the Lorentz transformation has its familiar form, and the composition of velocities is the standard one. It is the only choice for which the description is invariant under spatial rotations, so it is the only one that does not build a preferred direction into a theory that has none. And it is the choice under which the transformation between two inertial frames has the same form as its own inverse.

Those are strong reasons of the kind that decide every choice of coordinates in physics, and they are aesthetic and practical rather than empirical. The situation is the same as with any other convention that is not arbitrary in practice: measuring angles in radians is a convention, and using anything else in a Taylor series is a mistake.

Where the same distinction matters elsewhere #

In GPS. The system defines a global coordinate time and synchronises to it, and the definition is a convention — chosen for convenience, adjusted for the Sagnac effect around a rotating Earth, and not a measurement of anything. The positioning works because the convention is applied consistently, not because it is true.

In the rotating frame, where the convention becomes visibly awkward: going all the way round a turntable and synchronising neighbour to neighbour by radar returns a discontinuity, which is the ring where the two beams disagree. No consistent global simultaneity exists there at all, which makes the arbitrariness of the flat-space choice easier to see.

And in cosmology, where “the age of the universe” is a proper time along a particular family of worldlines with a particular time slicing, and quoting it without saying which is a convention silently made.

The experiment that keeps being proposed #

Every few years a scheme appears that claims to measure the one-way speed, and the schemes are worth examining because their failures are all the same failure wearing different clothes.

Two clocks synchronised at one place and separated. This is slow transport, and the objection above applies.

A single clock and a distant mirror, with the mirror moved. Every timing is still a round trip; moving the mirror changes the round-trip distance and nothing else.

Two-photon entanglement. Correlations between entangled photons cannot signal, so they cannot be used to synchronise clocks any more than they can carry a message. This one recurs particularly often.

Comparing arrival times of light and a particle of known speed. The particle’s speed is itself known only from a round-trip calibration, or from a theory that has the convention in it.

Interference is what most optical schemes reduce to, and it is the clearest case of the same failure. An interference measurement compares two paths that begin together and end together, so it is a round-trip comparison however elaborately the paths are folded. No arrangement of interfering beams determines the split of a round trip into halves, because the quantity it reports is a phase difference at one place.

The pattern is that every proposal either contains a round trip, or contains a second clock whose setting was chosen, or assumes some other quantity whose measurement contained one of those. That is not a coincidence: it is a theorem about what can be measured with signals bounded by a light cone, and finding the hidden assumption in a new scheme is a reliable exercise.

Rømer’s measurement, and the assumption in it #

The astronomical methods were dismissed above in a sentence, and the oldest of them deserves better, because it looks like a genuine one-way measurement and the reason it is not is instructive.

Rømer noticed in 1676 that the eclipses of Jupiter’s moon Io do not arrive on schedule. Io is eclipsed every 42.5 hours, and the observed intervals are shorter while the Earth is approaching Jupiter and longer while it is receding, accumulating to a discrepancy of about a quarter of an hour over half a year. He attributed it correctly to the changing distance the light has to cross, and Huygens turned it into a speed within a year.

Every timing in that measurement is made at the observer, with one clock. Nothing is synchronised, no signal is sent out and brought back, and no second clock exists. It looks exactly like the thing this essay says cannot be done.

The assumption is hiding in the phrase every 42.5 hours. What the observation gives directly is that the interval between successive eclipse-observations changes as the Earth moves. Turning that into a speed requires the premise that the eclipses themselves happen at equal intervals — that Io’s orbit is uniform — and that is a statement about the rate of a clock four astronomical units away.

In the isotropic convention, a uniform orbit at Jupiter means eclipses at equal coordinate times, the observed variation is entirely light travel time, and the speed comes out isotropic. In an anisotropic convention, “equal intervals at Jupiter” means something different, because coordinate time at Jupiter has been defined differently — and the same observations, analysed with the same premise, return the anisotropic speed the convention was built with.

So Rømer’s method does not contain a round trip and does not escape the problem. It substitutes an assumption about a distant clock’s rate for an assumption about a distant clock’s setting, and the two are the same kind of thing. That is the general shape of every astronomical proposal: somewhere in the analysis, something distant is assumed to be regular, and regularity at a distance is exactly what a synchronisation convention decides.

What the measurement genuinely established is not in doubt and is enormous: that light takes time to travel at all, against the prevailing view that its propagation was instantaneous, and that the delay is of the order of a quarter of an hour across the Earth’s orbit. Those are convention-independent facts about delays and distances. The step from them to a directed speed is the step that needs the stipulation.

The strongest argument on the other side #

The essay has presented the debate as two defensible positions that agree about every prediction, which is fair as far as it goes and leaves out the sharpest technical result in the argument.

Malament proved in 1977 that standard simultaneity is not merely one choice among many but the only one definable from the causal structure alone. Take Minkowski spacetime, keep nothing but the relation “this event can causally influence that one”, add a single inertial worldline, and ask which equivalence relations on events can be built from those ingredients while respecting every symmetry that preserves them. The answer is: the trivial ones, and Einstein’s.

That is a genuinely strong result, and it changes what the conventionalist can claim. It is no longer tenable to say that $\varepsilon = 1/2$ is picked out only by convenience and habit; it is picked out by the causal structure of spacetime, which is exactly the part everybody agrees is not conventional. The other values of $\varepsilon$ remain consistent, and they are no longer on an equal footing.

The objections are technical and are worth naming, because they are not evasions. The theorem requires the simultaneity relation to be an equivalence relation — reflexive, symmetric, transitive — and to be invariant under the full group of causal automorphisms fixing the worldline, which includes spatial reflections. Conventionalists have argued that demanding invariance under reflections is close to assuming isotropy, so that the conclusion is partly built into the premises; and that “uniquely definable from the causal structure” is a claim about definability rather than about physics, since a relation can be definable without being the one nature uses.

The position that survives all of this is narrower than the one usually stated and is worth having precisely. No experiment distinguishes the conventions, which is the claim this essay defends and which nothing in Malament’s theorem touches. The choice is nonetheless not arbitrary, because one of the options is singled out by the invariant structure and the others are not. Those two statements are compatible, and holding both is the honest position.

What the pictures cannot show #

The ε-family is not the most general convention. ε may also vary with direction and with position, and the full space of admissible synchronisations is larger than one parameter. The one-parameter family is enough to make the point and not enough to survey the possibilities.

Nothing here is about the value of c. The numerical value is fixed by definition — the metre has been defined from the second and c since 1983 — which is a third, separate kind of convention and is often confused with this one.

And the experiments quoted bound anisotropies of the round trip. No experiment is cited above as bounding the one-way anisotropy, because none can, and papers that claim to have measured it have without exception assumed a synchronisation somewhere in the analysis.

What Einstein said about it #

The history is short and unusually clear, because the person who introduced the convention said it was one.

The 1905 paper defines simultaneity by the radar procedure and then adds, in as many words, that the assumption of equal times out and back is neither a supposition nor a hypothesis about the physical nature of light, but a stipulation made of free will in order to arrive at a definition of simultaneity. It is a definition, offered as such, in the paper that made it famous.

What happened afterwards is that the stipulation was absorbed into the physics and stopped being visible. Textbooks state the constancy of the speed of light as a postulate, without separating the round-trip fact from the one-way convention, and generations of readers have taken both to be measured. The conventionality was rediscovered by Reichenbach in the 1920s and has been argued about ever since, mostly by philosophers of physics rather than by physicists — which is appropriate, since nothing measurable turns on it.

What is not conventional is the cone and what is inside it. Everything Einstein’s postulate does that has content is a statement about that structure; everything the stipulation does is a statement about how to draw lines across it. Separating the two is the whole of the episode, and the reason it took twenty years is that nobody had written the second half down as a separate sentence.

The episode is a good example of a general hazard. A definition that is adopted for excellent reasons, used universally, and never restated eventually becomes indistinguishable from a discovery — and the way to tell them apart is to ask what experiment would come out differently.

The ladder from here #

Later rungs on this anchor: Reichenbach’s and Grünbaum’s arguments in their own terms, and the objections to them; the general ε-dependent formulation of the Lorentz transformation, and what it does to the metric; the Sagnac effect as the case where no consistent choice exists; and the modern test-theory framework, in which the conventional and the physical parts of anisotropy are separated explicitly and bounded independently.

The neighbouring ladders are now as a choice of slicing, which is the frame-dependence this convention sits on top of, which came first and who decides, which is the causal question that stays invariant, and the quantity nobody argues about, which is what is left when every convention has been stripped out.