The Conventionality of Simultaneity

The relativity of simultaneity tells us that observers in relative motion disagree about which distant events are simultaneous. A subtler and more contested question lurks beneath it: even within a single inertial frame, is there a fact about which distant events are simultaneous — or is that too, in part, a matter of convention? This is the problem of the conventionality of simultaneity, raised by Einstein himself in his 1905 paper and pressed into a full philosophical thesis by Hans Reichenbach and Adolf Grünbaum. It is the temporal twin of the conventionality of geometry: the claim that the standard definition of simultaneity rests on a free stipulation about the one-way speed of light, which no experiment can verify.

This page states the problem, Reichenbach's ε, the arguments on both sides, and Malament's theorem; it presupposes the relativity of simultaneity and parallels the geometric conventionalism page.

Einstein's definition and the one-way speed of light

To coordinate clocks at separated locations $A$ and $B$ in a single frame, Einstein proposed a synchronization procedure. Send a light signal from $A$ at time $t_{1}$ (by $A$ 's clock); let it reflect off $B$ and return to $A$ at time $t_{3}$ . We set $B$ 's clock so that the reflection event at $B$ is simultaneous with the moment at $A$ given by

$t_{2} = t_{1} + ε (t_{3} - t_{1}), 0 < ε < 1.$

Standard synchrony takes $ε = \frac{1}{2}$ : the light is assumed to take equal times on the outward and return legs, so the reflection is simultaneous with the midpoint of the round trip. This builds in the assumption that the one-way speed of light is the same in both directions.

But here is the difficulty. To measure the one-way speed of light from $A$ to $B$ , we would need already-synchronized clocks at $A$ and $B$ — and synchronizing them is exactly what we are trying to do. Every actual measurement of the speed of light is a measurement of the round-trip (two-way) speed, which is unambiguously $c$ . The one-way speed cannot be measured without first assuming a synchronization, so the choice $ε = \frac{1}{2}$ appears to be a stipulation, not a discovery. This is the circularity at the heart of the problem.

Reichenbach's thesis

Reichenbach (and Grünbaum) concluded that simultaneity within a frame is conventional: any value of $ε$ in the open interval $(0, 1)$ yields a self-consistent way of assigning simultaneity, all empirically equivalent, and none is the true one. The only constraint physics imposes is that no signal outrun the round-trip light connection — i.e. that the synchronization respect the causal (light-cone) structure, which forbids $ε = 0$ or $ε = 1$ (these would make emission and a later distant event simultaneous, allowing causal order to be reversed). Within that open interval, Reichenbach held, the choice is free; $ε = \frac{1}{2}$ is merely the simplest and most convenient convention, chosen for elegance (it makes the laws isotropic), not because it is uniquely correct.

On this view the factual content of relativistic time is exhausted by the frame-invariant causal structure — the light cones, the timelike/spacelike distinction — and the assignment of a definite simultaneity within that structure adds a conventional element, just as Grünbaum's metrically amorphous continuum has its metric imposed from outside. The parallel is exact: spatial metric and temporal simultaneity are both, on the conventionalist reading, stipulations layered on a factual causal/topological skeleton.

The case against: standard synchrony is not arbitrary

The strong conventionalist thesis has been forcefully challenged. Several considerations suggest $ε = \frac{1}{2}$ is privileged after all:

Alternative operational definitions agree. Standard synchrony can be reached by procedures that make no reference to light and no assumption about its one-way speed — for instance, slow clock transport: synchronize two clocks at the same place, then carry one infinitely slowly to $B$ . In the limit of zero transport velocity the time-dilation effect vanishes to first order, and the transported clock agrees with $ε = \frac{1}{2}$ synchrony. That an independent method converges on the same simultaneity suggests it tracks something real, not a mere choice.
Symmetry and isotropy. In an inertial frame space is isotropic; $ε = \frac{1}{2}$ is the unique choice that respects this isotropy (any other value would build a preferred spatial direction into the time coordinate, treating $A \to B$ differently from $B \to A$ for no physical reason). Non-standard $ε$ is not wrong so much as gratuitously anisotropic.
Malament's theorem (1977). The decisive technical result. David Malament proved that standard simultaneity is the unique non-trivial equivalence relation definable from the causal structure of Minkowski spacetime (specifically, from the relation of causal connectibility, together with the worldline of a single inertial observer, requiring the relation to be invariant under the causal automorphisms that fix that observer). In other words, given only the light-cone structure and one inertial observer, the standard $ε = \frac{1}{2}$ simultaneity is the only frame-independent simultaneity relation you can define. This is widely taken to show that standard synchrony is not a free convention: it is forced by the causal structure plus minimal symmetry requirements, so simultaneity is factual after all, relative to a frame.

Where the debate rests

The literature remains divided, but the weight of opinion has shifted against strong conventionalism, largely because of Malament's theorem and the slow-clock-transport argument. The nuanced consensus:

The one-way speed of light genuinely cannot be measured without a synchronization convention; that circularity is real, and in that narrow sense simultaneity involves a conventional choice of coordinates.
But standard simultaneity is not arbitrary: it is uniquely singled out by the causal structure (Malament) and by isotropy and slow transport, so it is not on a par with arbitrary $ε$ . The "convention" is no more troubling than the conventional-yet-natural choice of orthonormal coordinates.

The upshot mirrors the verdict on geometric conventionalism: there is a thin conventional element in setting up the coordinate framework, but the substantive temporal structure — the causal order, and the standard simultaneity it uniquely determines — is factual, not stipulated. Reichenbach was right that measurement of the one-way speed is circular, wrong that this makes all $ε$ equally legitimate.

Where this sits

The conventionality of simultaneity is the temporal counterpart of geometric conventionalism, and, like it, a test case for the method question of how to separate the factual from the conventional in a physical theory. It refines the relativity of simultaneity: where that result denies a frame-independent present, this debate asks whether even the frame-relative present is fully factual — and Malament's theorem answers, largely, yes. Because it locates the objective content of relativistic time in the causal structure, it also connects to the block universe (whose invariant skeleton is precisely that causal structure) and looks ahead to quantum gravity, where causal structure is often taken as more fundamental than metric or time. This completes the relativity subfolder; the physics pages take up time's direction, its fate in quantum gravity, and its beginning.