Which came first, and who decides

Two events happen: a lamp is switched on in one place, and a bell rings in another. In one laboratory the lamp comes first by a second. In a rocket passing through, the bell comes first. Neither observation is a mistake, neither observer is moving in any way that would spoil an instrument, and there is no experiment that would settle which of them is right — because both are.

The question this essay answers is why that does not destroy the notion of cause, and the answer is a single inequality.

How negotiable an order is depends on how far the pair sits from the light cone, and the dependence is sharp rather than gradual. The quantity that does add is what keeps a boost from ever reaching the cone.

What a frame is doing when it disagrees #

Simultaneity is a slicing rather than a fact: each observer draws a family of surfaces through spacetime and calls each one an instant, and observers in relative motion draw different families.

On a diagram with position across and time up, a line of constant $t'$ in a frame moving at $\beta$ has slope $\beta$. Since $|\beta| < 1$, those lines can be tilted up to the diagonal and no further. Everything that follows is a consequence of that one restriction.

Take two events separated by $\Delta x$ and $\Delta t$. The moving observer assigns them a time difference

$$\Delta t' = \gamma\left(\Delta t - \frac{\beta \Delta x}{c}\right),$$

which changes sign when $\beta$ passes $c\,\Delta t/\Delta x$. So the order reverses if and only if there is a legal $\beta$ doing that, which requires

$$\left|\frac{c\,\Delta t}{\Delta x}\right| < 1, \qquad\text{that is,}\qquad |c\,\Delta t| < |\Delta x|.$$

The order of two events is negotiable exactly when light could not have got from one to the other.

Three kinds of separation, and only three #

That condition is not a special case; it is one of the three possibilities the invariant interval allows.

For a timelike pair, $|c\Delta t| > |\Delta x|$, the required boost exceeds the speed of light and no frame achieves it. Every observer agrees which came first. These are exactly the pairs one of which could have influenced the other.

For a spacelike pair, the required boost is legal, and observers disagree. These are exactly the pairs neither of which could have influenced the other.

For a lightlike pair the required boost is exactly $c$ and is not attained: the order is agreed, marginally.

So the two properties line up perfectly. The pairs whose order is a matter of opinion are precisely the pairs where the question of cause does not arise, and the pairs where cause could operate have an order everyone agrees about. Causality is not protected in spite of the relativity of simultaneity but by the same inequality that produces it.

Drawn as a cone the same statement is easier to hold. Everything inside an event’s future cone can be reached from it and happens after it for everybody; everything inside its past cone can reach it; everything outside is elsewhere, and its time relative to the event is a matter of frame. The cone is the same cone for every observer — that is what makes it usable as a boundary at all, and it is the only structure in the diagram that no boost distorts.

What goes wrong if the boundary is crossed #

The prohibition on faster-than-light signalling is often stated as though it were a speed limit — as though the objection to a tachyon were that it is too quick. It is not. The objection is that it makes a contradiction available.

One such signal is odd. Two are fatal. Send a superluminal message from A to B in the laboratory frame; have a confederate on a rocket send a superluminal reply from B back to A, using their own frame’s simultaneity. The reply arrives at A before the original message was sent. Nothing in that construction requires the signals to travel backwards in anybody’s time — each is forward-going in its own sender’s frame — and the round trip lands in the sender’s past regardless.

That is why the constraint is absolute and why it is a constraint on information rather than on motion. A great many things move faster than light: the intersection of two closing scissor blades, the spot cast by a rotating searchlight on a distant wall, the peak of a pulse in an amplifying medium. None of them can carry a message, because each is a pattern whose parts were arranged in advance rather than a disturbance propagating.

The construction spelled out #

The two-signal argument deserves to be written down with its numbers, because it is short and because it is the whole of the case.

Let A be at the origin of the laboratory frame and B three light-seconds away, at rest with respect to A. At $t = 0$, A sends a signal to B at twice the speed of light; it arrives at $t = 1.5$ s. Now let a rocket pass B at $0.6c$, travelling from B toward A. In the rocket’s frame the arrival at B happened at $t' = \gamma(1.5 - 0.6\times 3) = -0.375$ s — before A sent anything.

Have B’s confederate on the rocket send a reply at twice the speed of light in the rocket’s frame, back toward A. That reply is forward-going in the rocket’s own time, so nothing about sending it is peculiar from on board. Transform its arrival back into the laboratory frame and it lands at A at a laboratory time of about $-0.94$ s: nearly a second before A’s original message was sent.

Every step is legal in the frame it is performed in. Neither signal goes backwards in its own sender’s time. Both senders are ordinary observers with ordinary clocks. And the composition is a message A receives before A decides to send it, which is not a paradox about time travel but a plain contradiction — A can be arranged to send the first message only if the reply does not arrive, and the reply arrives only if the first message is sent.

This is why the constraint is stated as a prohibition on signalling outside the light cone rather than as a speed limit. A limit would be a fact about mechanisms, and a mechanism can always be imagined to be better. The contradiction is a fact about logic, and no improvement in engineering touches it.

Why nothing accelerates through the barrier either #

The prohibition applies to signals, and it happens that ordinary matter cannot cross the boundary either, for a separate reason.

There is a second reason nothing crosses the boundary and it is worth keeping separate from the first, because the two are different kinds of argument. The factor by which time, length and inertia are scaled rises without limit as the speed approaches $c$, so the energy needed to accelerate a massive body grows without limit too. That is arithmetic. The causal argument above is logic: relativity forbids signals outside the cone because they produce a contradiction, and forbids massive bodies from reaching $c$ because the bill is infinite. Neither implies the other.

In energy the same statement reads more directly: a body’s total energy diverges as its speed approaches that of light, so no finite energy takes it there. A massless particle sidesteps the whole argument by having no rest energy to scale — there is nothing for the divergent factor to multiply — and travels at exactly $c$ in every frame, which is the only speed a massless thing is allowed.

Nor can the barrier be reached in stages. Two boosts of three-quarters of the speed of light do not compose to one and a half; they compose to 0.96, and no finite chain of legal boosts reaches one. What adds is not the speed but a quantity related to it by a hyperbolic tangent, and the boundary is a value that sum cannot attain rather than a wall anything strikes.

The quantity everyone does agree on #

The disagreements above are disagreements about coordinates. Underneath them sits a quantity that does not vary.

The calibration those tilted axes need comes from the same curves. A unit of time on a tilted axis is longer on the page than a unit on an upright one, and the hyperbola of constant interval is what says by how much — two events on one hyperbola are separated by the same interval however they are sliced. Without them a spacetime diagram shows the tilt and hides the scale, and no number can be read off it.

For a timelike pair the interval measures the time actually lived along a path between them, and it is largest for the straight path and shorter for every bent one. That is the quantity nobody argues about, and the fact that observers who disagree about durations agree about it exactly is what makes the disagreement bookkeeping rather than chaos.

All of it comes from one construction: a clock made of light, and one postulate about the speed of light. The tilted slices, the reversible orders, the invariant interval and the unreachable barrier are consequences of that single arrangement, and none of them needs any assumption about matter, or forces, or what an apparatus is made of. That is why the conclusions are so hard to escape — there is almost nothing in the premises to deny.

The order that survives everything, and what it is good for #

Because timelike order is absolute, it can be used as a foundation rather than as a consequence, and that is a more powerful move than it first appears.

Everything an observer can ever measure about an event’s relations to other events is contained in which events are in its past cone and which in its future. The metric structure — distances, durations, angles — can be reconstructed from that ordering together with a single volume measure, a result of Hawking, King and McCarthy and independently of Malament: the causal order determines the geometry up to an overall scale. A spacetime is very nearly nothing more than a partially ordered set.

That is a striking amount of content for a relation with three cases. It says that no experiment measuring only what could have affected what can distinguish two spacetimes with the same causal order and different geometries, because there are none. It also explains why relativity is so much easier to reason about with cones than with coordinates: the cones carry nearly all the information and none of the arbitrariness, while the coordinates carry the same information plus a choice.

The practical form of the same point is the rule every relativistic calculation obeys. Ask only questions whose answers are the same in every frame — an interval, a proper time, a rest mass, whether one event is in another’s past — and no frame-dependent quantity need ever be computed. Ask a question about “when” or “how long” or “how far”, and the answer arrives with a frame attached, and half the labour of the subject is remembering which.

The theorem that entanglement cannot signal #

The remark that entangled correlations do not violate the prohibition is often made as though it were a lucky escape, and it is a theorem — worth stating, because the theorem is stronger than the escape.

Two parties share an entangled pair. One of them chooses what to measure and gets a result. The question is whether the other party, holding their half, can detect which choice was made — because if they could, the choice would be a signal.

They cannot, and the reason is structural. The statistics of any measurement made on one half are computed from that half’s reduced description, obtained by summing over everything the other half might be doing. An operation performed on the far half acts only on the far half’s part of the description, and summing over a system after acting on it gives the same answer as summing over it before. So the near half’s statistics are literally unchanged, whatever is done to its partner, and whether anything is done at all.

That is the no-signalling theorem, and it holds for every state, every measurement and every operation permitted by the theory. It is not an inequality that happens to be satisfied; it is an identity.

What makes it interesting is how tight the fit is. Quantum mechanics produces correlations stronger than any pre-arranged instruction list could — that is what a Bell inequality violation means — and it produces them while remaining exactly incapable of sending a message. Stronger-than-quantum correlations are imaginable that also do not signal, and weaker ones obviously exist, so the theory is not sitting at either edge of what no-signalling permits. It is sitting somewhere inside, for reasons nobody has reduced to a principle.

The practical form of the theorem is the one every quantum communication protocol relies on. Entanglement alone conveys nothing; entanglement plus an ordinary classical message conveys something neither could alone. The classical message travels at ordinary speed, so the whole procedure respects the light cone at every step, and the correlations do their work only once the two halves are compared.

What a tachyon in a field theory actually means #

There is a term in the technical literature that reads as though it contradicts everything on this page, and the resolution is worth having because it is a good example of a word meaning something other than it appears to.

Field theories are written with a mass parameter that enters as a square. Occasionally a theory is written down in which that parameter is negative — an imaginary mass — and the field is called tachyonic. Taken literally, an imaginary mass in the relativistic energy relation gives a particle that always travels faster than light.

That is not what such a field describes. A negative mass-squared term means the state the field was expanded about is not a minimum of the energy but a maximum: the field is sitting on top of a hill, and the “imaginary frequency” is the exponential growth of a small displacement rather than an oscillation. The theory is not describing superluminal particles; it is announcing that the configuration it was written about is unstable.

What happens next is that the field rolls off the hill to a genuine minimum, and expanded about that the mass-squared is positive and the excitations are ordinary particles moving at less than light speed. The best-known instance is the field responsible for giving masses to the elementary particles: its potential has a maximum at the symmetric point and a ring of minima around it, and the tachyonic parameter in the symmetric description is the statement that the symmetric state is not the one the world is in.

Nothing propagates outside the light cone at any stage. The field equation is hyperbolic with the same characteristic speed whatever the sign of the mass term, so a disturbance still spreads at exactly the speed of light and no faster — which is the mathematical form of the statement that this essay’s prohibition is built into the equations rather than imposed on their solutions.

The vocabulary is unfortunate and it has stuck. A “tachyonic instability” is an instability, and the fact that it is named after a particle nobody believes in is an accident of how the algebra was first read.

Where the model stops #

Flat spacetime, everywhere. The whole argument is local in the sense that it assumes a single global family of inertial frames. In general relativity the light cones still exist at every point and the local statements survive unchanged, but the global structure need not be so tidy: spacetimes with closed timelike curves are solutions of the field equations, and whether nature contains any is an open question rather than a settled one.

Signals, not correlations. Entangled particles produce correlations across spacelike separations which no shared instruction list can reproduce. That does not violate anything here, because the correlations cannot be used to send a message: each side’s own results look random until the two are compared, and the comparison travels at ordinary speed. The prohibition is on signalling, and it is exactly strong enough to be consistent with entanglement and no stronger.

Sharp events. Every argument treats an event as a point. Real events have extent, and two overlapping processes need not have a well-defined order even in one frame — which is a separate ambiguity that has nothing to do with relativity and is easily confused with it.

Two events with a number attached #

It helps to put the abstraction on a scale, because the sizes involved are what make the effect invisible in ordinary life.

Take two events a metre apart. Light crosses a metre in 3.3 nanoseconds, so their order is negotiable only if they are separated by less than that in time. Any pair of events in a room whose timing differs by more than a few nanoseconds has an order every observer agrees on, which is why the whole business never arises at human scales — not because the speeds involved are small, but because the distances are.

Now take two events on opposite sides of the Earth, 12,700 km apart, separated by ten milliseconds. Light needs 42 milliseconds to cross, so the pair is spacelike and the order is frame-dependent. The boost required to reverse it is $\Delta t/\Delta x$ in units of $c$, which is about 0.24 — a quarter of the speed of light, and no aircraft or satellite comes anywhere near it. So the disagreement is available in principle and unavailable in practice, which is the usual situation.

The one everyday context where it is neither is a global navigation system. The satellites are 20,000 km up and moving at 3.9 km/s, and the timing precision needed is nanoseconds. There the slicing genuinely matters, and the system’s clocks are synchronised in a single stated frame — the Earth-centred inertial one — precisely because “simultaneous” has to be defined rather than assumed before the arithmetic can proceed.

What the pictures cannot show #

Every figure here draws one spatial dimension. Two events separated in the other two directions have the same classification, and the cone becomes a cone in three dimensions and then a hypercone, none of which can be drawn honestly.

Nor do the diagrams show anybody making a measurement. An observer in relativity is a coordinate system, not a person looking; the finite time light takes to reach an eye is a separate effect entirely, and it produces its own distortions that these figures deliberately leave out.

And no figure can show two orders at once. Each diagram is drawn in the coordinates of one frame, with the other’s axes superimposed, so the eye reads one of them as “the truth” and the other as tilted. The symmetry between the two is real and the drawing cannot express it — which is the most persistent source of confusion in the subject and the one thing about it a picture makes worse rather than better.

Where the ladder goes next #

Simultaneity began as a choice of slicing and has become a statement about which questions have observer-independent answers. Two rungs follow. One is the general structure of causal sets: what remains of spacetime if only the ordering relation is kept and all the geometry thrown away, which turns out to be almost everything. The other is what happens to all of this when gravity is present and the cones tilt from place to place — where a horizon becomes definable as the boundary beyond which every future cone points inward.

The habit worth carrying away is about how a symmetry protects a principle. Causality could have been protected by a rule — no signals faster than light, by fiat — and it is not. It is protected by the same transformation that makes simultaneity relative, which reverses exactly those orders that carry no causal content and leaves the rest alone. When a theory makes something observer-dependent, the question worth asking is what it has made observer-independent at the same time, because the two are usually the same statement read from opposite ends.