What the light cones alone can decide

The transformation that never mentions light derived the Lorentz transformation from four assumptions and found that light was not among them. The habit it left behind was to ask, of every assumption in a derivation, how much of the result survives without it. The first of its four, homogeneity, did the most work: it is what made the transformation linear, sending straight worldlines to straight worldlines.

That essay also found why the third possible world, the one with a negative constant, has to go: in it, a chain of ordinary boosts can reverse the order of two events that one of them could have caused. The ordering of events by cause and effect was the thing the physics refused to give up. So the question becomes sharper. Throw away every assumption except that one. Keep no clocks, no rulers, no notion of a straight line — only the light cones, the record of which events can influence which. How much geometry is left?

The answer depends on how many dimensions space has, and the difference between one and more than one is total.

A future that is a quadrant #

In one dimension of space the light cone of an event is a pair of lines at 45°, and everything about causal order can be read off two numbers. Define $u = t - x$ and $v = t + x$, in units where light has speed one. A light ray moving right keeps $u$ fixed and one moving left keeps $v$ fixed, so the lines of constant $u$ and constant $v$ are the two families of light rays.

An event lies in another’s future exactly when it is later in both: $\Delta u \ge 0$ and $\Delta v \ge 0$. The cone is a quadrant in these coordinates, and the causal order of the whole plane is nothing but two separate orderings, one along each family of light lines, taken together. Two axes and one speed drew the cone as the fixed feature every observer agrees on; in these coordinates it is not even tilted.

The quadrant also separates what is a matter of fact from what is a matter of convention. The speed that cannot be measured one way found that the speed of light in one direction, as opposed to its round-trip average, depends on how distant clocks are synchronised, and that no experiment picks the synchronisation. A different choice relabels which events are simultaneous: it slides the horizontal lines of the grid up on one side and down on the other. It cannot move an event from one quadrant to another, because the quadrants are fixed by light signals, which every synchronisation agrees on. The causal order is precisely the part of the diagram that survives every convention — which is what makes it a candidate for the whole of the physics.

That observation is already the whole result for one dimension. Take any function that increases, and apply it to $u$; take any other increasing function, and apply it to $v$. Each ordering is preserved separately, so the order of every pair of events is preserved, so every light cone is carried to a light cone. The first figure does exactly this, with $u + 0.45\tanh 1.5u$ and $\sinh(0.45v)/0.45$, and checks the order of four thousand pairs before and after. None changes.

What does change is everything else. The straight vertical worldlines of observers at rest come out curved, which is to say the map turns inertial observers into accelerated ones. The lines of simultaneity bend. And the quantity nobody argues about, the interval, is argued about after all.

The same order, and every interval changed #

The scatter in that figure is the measure of how little the order says in one dimension. A pair of events whose proper time was one second can come out at one second or at nearly two, depending on where in the plane it sits, and no statement of the form “this event could have influenced that one” can tell the difference. An observer moving inertially and an observer shaking back and forth are indistinguishable to someone who knows only the cones.

They are not indistinguishable to the observers. An accelerometer carried along the bent worldline registers a push, and a clock carried along it runs at a rate set by its path — the clock that does not feel the turn is the account of how proper time depends on the whole worldline and not on the acceleration at any instant. But an accelerometer is a physical object with a rest mass, a spring and a scale, and none of that is in the causal order. The statement is about how much the cones alone contain, and in one dimension of space the answer is that they cannot tell a free fall from a shake.

So in one dimension of space the transformation of the earlier essay needed its homogeneity assumption in full. Without it the four assumptions that remain, causality included, leave a family of maps as large as two arbitrary functions. That family is not a curiosity. Physics in one space and one time dimension has a symmetry group this large whenever nothing in it sets a length: a string sweeping out a surface, or a critical system at a phase transition, is described by field theories invariant under exactly these independent stretches of the two light-cone directions, and the size of that symmetry is why such theories can be solved exactly when their four-dimensional counterparts cannot.

A second dimension makes the cones rigid #

Add one more dimension of space and the situation reverses. The same kind of map — stretch $u$ by an increasing function, stretch $v$ by another, leave the new direction $y$ alone — no longer preserves the order.

The reason is in the condition for causal contact. With two space directions an event lies in another’s future when $\Delta t^2 \ge \Delta x^2 + \Delta y^2$ with $\Delta t$ positive, which in the light-cone coordinates reads

$$\Delta u\,\Delta v \ge \Delta y^2.$$

It is no longer two orderings side by side. It couples $u$ and $v$ through their product and ties that product to the distance in the other direction. Stretching $u$ and $v$ separately changes the product by different amounts in different places while $\Delta y$ stays put, so some pairs that were just outside each other’s cones are pushed inside and some just inside are pushed out. The figure finds violations at every nonzero strength it tries, growing steadily with the strength of the bend. The shares grow more slowly than the strength of the bend: doubling it from 0.05 to 0.1 does not quite double the violations, and from 0.4 to 0.8 raises them by only seven-tenths. Only pairs that were close to each other’s cones can be pushed across, and the band of such pairs a bend can reach widens with its strength; but the bend used here, $\tanh$, stops growing for large separations, so the widest pairs are the least affected and the growth flattens. The shape of the curve is a property of this particular bend. That it leaves zero only at zero strength is not.

The control is as important as the violations: a Lorentz boost at half the speed of light followed by a uniform stretch by two changes the relation of none of the four thousand pairs.

That contrast is a theorem. In 1964 Christopher Zeeman proved that in a spacetime with two or more space dimensions, every one-to-one map of the whole of spacetime onto itself that preserves the causal order is a Lorentz transformation combined with a shift and a uniform stretch. Nothing is assumed about the map beyond preserving the order: not linearity, not continuity, not that it sends straight lines to straight lines. Linearity, which the earlier derivation took from homogeneity, comes out of causality alone.

The theorem also works as a constraint on speculation. Proposals for physics beyond relativity sometimes add symmetries to spacetime or remove some, and Zeeman’s result says what either move costs. A transformation of four-dimensional spacetime that is not a Lorentz transformation, a shift or a stretch cannot keep every light cone; so a theory with any larger symmetry group acting on events must change which events can influence which, and a theory in which the Lorentz transformations are only approximate must allow some process that tells a preferred frame by its causal structure. Experiments that test Lorentz symmetry to parts in $10^{20}$ are, by this route, also tests of the geometry of cause and effect. Alfred Robb had built the geometry of flat spacetime on the single relation “after” as early as 1914; Zeeman’s theorem is the statement that, given more than one dimension of space, that relation was always enough.

The size of what survives is easy to count. In three dimensions of space there are four independent shifts, one in time and three in space; three rotations; three boosts, one along each direction; and the single uniform stretch. That is eleven numbers, against a one-dimensional family that needed two whole functions to describe. Going from one space dimension to two does not remove a few maps from the list. It removes all but a handful of parameters from an infinite supply.

Why the cone gives way #

The proof runs through the light rays, and the picture shows where one dimension escapes it. From the causal order alone a light ray can be recognised: two events are separated by a light ray exactly when one is in the other’s future but only just, in the sense that no third event fits strictly between them in both orders. So an order-preserving map must send light rays to light rays. In one space dimension there are only two directions of light ray, and a map can slide events along each family independently, which is the warp. In two or more there is a whole circle of light-ray directions at every event, the families cross each other in every orientation, and a map that keeps every ray straight and every cone a cone has no room to stretch one direction without distorting the others. From there, a classical theorem about maps that send lines to lines forces the map to be linear.

The figure shows the failure directly. The events on the dashed circle are unreachable from the event at the centre. The bend leaves the two on the $x$ axis outside, because for them $\Delta y$ is zero and the one-dimensional argument applies. Everywhere else it moves them, and half of them land inside the cone: events that could not be influenced now can be. In one dimension only those two axis points exist, and the escape is complete.

With three dimensions of space the circle of light-ray directions becomes a sphere: the sky, as seen by an observer at the event, is the set of light rays arriving there. A Lorentz transformation does something specific to that sphere. The sky that crowds into a cone follows the stars as an observer speeds up, crowding forwards; what is preserved in that crowding is that every circle on the sky goes to a circle, and every angle between two nearby directions is kept, while areas and distances on the sky are not. Those circle-preserving maps of a sphere form exactly the group that the boosts and rotations form. The rigidity of the cones in three dimensions is therefore the rigidity of the sky: a map that kept every light ray a light ray but distorted the sky in any other way would have to break some cone somewhere.

The one freedom left #

Zeeman’s theorem leaves one thing unfixed, and the last figure is it. Stretching spacetime uniformly about any event, every length and every duration multiplied by the same factor, keeps every cone exactly where it was and every causal relation intact, and it multiplies every interval by the factor. The hyperbola that marked one second of proper time now marks two. Nothing about which events can influence which can tell those two pictures apart.

So the causal order fixes the geometry of flat spacetime up to a scale, and the scale has to be supplied by something that is not an order: a clock, a vibrating atom, a rod. The diagram a ruler cannot read found that the scale on a spacetime diagram’s axes is not the scale on the page; this is the deeper version of the same point, that the cones do not carry a scale at all. Two further things the theorem excludes are worth noticing. Reversing time maps future cones onto past ones, so it reverses the order rather than preserving it, and it is excluded; causality as used here has a direction. And the transformations that preserve every light cone locally without being Lorentz transformations — the special conformal transformations under which Maxwell’s equations in empty space are invariant, as Harry Bateman and Ebenezer Cunningham found in 1909 — each send some events off to infinity, so none is a map of spacetime onto itself, and a theorem about the whole of spacetime rules them out.

Combined with boosts, the rotations of space come along for free: they preserve every cone, and the theorem’s group includes them. What the turn that two pushes leave behind computes — that two boosts in different directions combine into a boost and a rotation — is a property of that group, and here the group itself arrives from nothing but the cones.

Where the model stops #

Spacetime is flat. The theorem is about Minkowski spacetime, where there is one global family of inertial frames. In a curved spacetime the cones tilt from place to place and there are no global Lorentz transformations to find; the corresponding result, stated in which came first, and who decides, is that the causal order determines the geometry up to a scale that may now vary from event to event, and that a measure of volume supplies the rest.

The map is a bijection of the whole of spacetime. A map defined only on a region, or one allowed to miss some events, has more freedom, which is how the special conformal transformations escape the rigidity above. The theorem’s force comes partly from its global reach.

Events are points. The order is a relation between ideal events with no extent. Real measurements localise events to within some spread, and a relation between smeared events is only approximately an order.

And the checks sample. The figures classify thousands of pairs, which is evidence that the maps behave as the theorem says and not a proof; the proof is a page of synthetic geometry about light rays.

What the pictures cannot show #

With one space dimension a cone is two lines and can be drawn exactly. The case the theorem is about, with three dimensions of space, has cones that are three-dimensional surfaces in a four-dimensional spacetime, and every figure here that shows the rigidity uses two space dimensions and a slice through them. The argument generalises without change, but the picture of it does not.

Nor can the figures show the direction of the logic. They start from maps and check the order; the theorem starts from the order and constructs the maps. That every order-preserving map is a Lorentz transformation, and not merely that Lorentz transformations preserve the order, is the content, and only the proof establishes it.

Still open: whether spacetime is an order at bottom #

If the causal order plus a scale is enough to recover the geometry of spacetime, it is natural to ask whether spacetime at its smallest scales is nothing but an order — a discrete set of events with a relation of before and after, from which the smooth geometry emerges on average. That is the programme of causal sets, proposed by Luca Bombelli, Joohan Lee, David Meyer and Rafael Sorkin in 1987, in which the missing scale is supplied by counting: the number of elements in a region stands for its volume.

The kinematics works: a set of events scattered at random in flat spacetime, keeping only their order, recovers the dimension, the proper times along the longest chains, and the geometry. What is not settled is the dynamics — whether any rule for growing such a set produces, from among the overwhelming majority of orders that look like nothing of the kind, one that resembles a smooth four-dimensional spacetime, and whether the discreteness would leave a trace in any measurement. In the flat, continuous setting of this essay the question has an answer; at the scale where gravity and quantum mechanics meet, it has a programme.

The habit worth keeping is the one the derivation began with, carried one step further. When a structure is assumed, ask whether a weaker one already implies it — and in how many dimensions. Homogeneity looked indispensable, and with enough dimensions of space causality alone delivers it; with one, it delivers almost nothing, and the difference is a count of directions in which light can travel.