The spot that outruns light and carries nothing

Speeds that refuse to add found the rule that replaces simple addition of velocities in relativity: two speeds below that of light always combine into a third below it, and the speed of light combined with anything gives the speed of light. It is the rule that makes $c$ a ceiling. The motion that measures faster than light found a first way round the ceiling that does not break it: a jet of plasma moving towards an observer at nearly the speed of light, whose apparent sideways motion on the sky can exceed $c$ because the light from its later positions has less far to travel. The jet was real; only its apparent speed was an artefact of the light’s travel time.

There is a second way round, which does not involve an illusion at all. Some things that move really do move faster than light, measured carefully with clocks and rulers in one frame, and relativity has no objection to them, because they are not things. The spot a turning beam makes on a distant screen is the standard example. Working out what such a spot does, carefully, shows how much the ceiling depends on the word “thing”, and why the composition of velocities — the very law that makes the ceiling — has nothing to say about it.

A beam sweeping a screen #

Stand a distance $D$ from a long flat screen and turn a narrow beam of light across it at angular speed $\omega$. Where the beam points straight at the screen, the spot it makes moves along the screen at $\omega D$. Nothing stops $\omega D$ from exceeding $c$: a laser pointer flicked through a radian in a tenth of a second sweeps the face of the Moon, four hundred thousand kilometres away, at more than ten times the speed of light, and the beam of a pulsar turning thirty times a second sweeps a nebula a light-year away at millions of times that speed.

None of this needs exotic equipment. The spot’s speed is a ratio of a distance on the screen to the time between lightings, and both are perfectly well measured in one frame: put photodetectors on the screen a metre apart, each with a clock synchronised to the others in the screen’s rest frame, and record when each is lit. If the second clicks a nanosecond after the first, the spot crossed a metre in a nanosecond, three times faster than light, and no correction for anyone’s line of sight is involved. That is what distinguishes the spot from the jet whose motion only looks faster than light: the jet’s apparent speed shrinks below $c$ once the light-travel delay to the observer is removed, and the spot’s does not, because there is no observer’s delay in it.

What needs care is where the spot is and when, because each point of the screen is lit by light that left the source at a different time and travelled a different distance. The light reaching the point $x$ along the screen left when the beam pointed there, at time $\theta/\omega$ with $\theta = \arctan(x/D)$, and then took $\sqrt{D^2 + x^2}/c$ to arrive. So the spot is at $x$ at the time

$$t(x) = \frac{\arctan(x/D)}{\omega} + \frac{\sqrt{D^2 + x^2}}{c}.$$

That curve, drawn on a spacetime diagram, is the spot’s whole history.

Born in the middle, as a pair #

The curves have a feature that the simple picture of a spot “moving across” the screen does not: each has a lowest point. Light reaches that point of the screen before it reaches any other. Before that moment the screen is dark; at that moment a spot appears at that point; and after it, the light reaches points on both sides. The spot is born in the middle of the screen as a pair, one running back towards where the beam came from and one running forwards.

The reason is the screen’s obliqueness. Far along the screen the beam meets it at a grazing angle and its sweep along the screen is much faster than $\omega D$ — it rises as the square of the distance — so far enough out it outruns light whatever $\omega$ is. Light sent earlier towards those far points has further to go than light sent a little later towards nearer points, and arrives later. The birth point is where the two effects balance, where the spot’s speed along the screen passes through infinity. For a beam that sweeps the centre at half the speed of light it is a screen distance and a quarter off-centre; for a sweep at three times the speed of light it is only a third of a screen distance off.

The two spots are not symmetric. The forward spot slows as it runs on: three times light’s speed at the centre, barely faster than light at two and a half screen distances, approaching $c$ from above. For any sweep faster than twice light’s speed at the centre, it never drops below $c$ at all, however far out the screen extends. The backward spot runs into ever more oblique light and also stays superluminal. Neither carries anything from its birth point to anywhere.

The birth of a pair has been filmed. In 2016 a group led by Daniele Faccio at Heriot-Watt University swept a short laser pulse obliquely across a scattering screen and recorded the scattered light with a camera fast enough to resolve its arrival, frame by frame, at a trillion frames a second. Where the pulse’s intersection with the screen moved faster than light, the camera saw a pair of images appear together and run apart, and, in another arrangement, two images run together and vanish — the time-reverse, an annihilation of the pair. The geometry is the one in the figures, with the camera’s own line of sight adding a further delay.

This birth of a pair has a close cousin. The binary star that would be seen twice showed what would happen if light from a moving star travelled at different speeds at different times: the light from later in the orbit could overtake the light from earlier, and the star would be seen in two places at once. The spot’s pair is the same overtaking — light from a later moment of the sweep arriving before light from an earlier one — and it is perfectly real, because here the light paths genuinely differ in length.

Why the ceiling does not apply #

Relativity forbids anything that carries energy, momentum, matter or information from travelling faster than light, and the reason is causality. A signal faster than light could, in some frame, arrive before it was sent, and the order of events too far apart for light to connect them is not absolute.

The spot carries none of these from one point to another. The light at each point on the screen came from the source, along its own path, at the speed of light. No part of the light that lit one point goes on to light the next. Nothing that happens at one point of the screen can affect whether the spot appears at the next — not a mirror placed there, not a detector, not a person who sees the spot and wants to pass a message along it. To send a message using the spot, the sender has to be at the source, deciding where the beam points, and the message then travels to the screen at the speed of light like any other.

The spacetime diagram puts that in one picture. Draw the future light cone of any event on the spot’s track — the region that anything leaving that event could reach — and the rest of the track lies outside it. The spot’s successive positions are pairwise spacelike-separated: no event on the track can influence any other. The spot is not a worldline, in the sense of the history of an object. It is a pattern drawn across spacetime by many separate causes.

What a moving observer sees #

The composition of velocities is a statement about how one observer’s measurement of a moving object’s speed relates to another’s. For a real object the answer, $(u - v)/(1 - uv/c^2)$, never takes a speed below $c$ to one above it, never reverses a direction of motion for an observer moving slower than the object, and never makes a speed infinite. The spot does all three.

Applied to the spot’s successive positions, the same formula gives a speed that becomes infinite when the observer moves at $c^2/u$, and reverses beyond it. For an observer moving along the screen at a third of the speed of light, a spot running at three times the speed of light crosses the screen instantaneously: all its positions are simultaneous. For a faster observer it runs the other way, starting at the end it would otherwise have finished at.

None of that is paradoxical, because the order of spacelike events is a matter of frame, and the spot’s positions are spacelike events. A succession whose direction can be reversed by a change of observer is not a succession of causes and effects, and that is the test for whether something is a thing. A real object’s speed transforms by the composition law and stays below $c$ for everyone. A spot’s “speed” transforms by the same formula, applied to events that happen not to be linked, and goes anywhere.

Other things that go faster than light #

The crossing point of two straight edges is another. Close a pair of scissors with long blades and the point where the blades cross runs along them faster than the blades themselves move, by a factor of one over the tangent of the angle between them.

For blades moving at a hundredth of the speed of light, the crossing point exceeds $c$ once the blades are within about half a degree of parallel. The calculation assumes the blade stays straight while it moves, which no real blade can: the push that moves it travels along it at the speed of sound, and nothing is allowed to be rigid. A real blade bends, and its crossing point’s motion is set by the bending wave rather than by the geometry of straight lines. But even for bent blades, nothing travels with the crossing point; each piece of each blade moves only sideways.

Phase velocities are a third family. The crests of a wave in a waveguide, or of light passing through a medium near an absorption line, can travel faster than light, because the crests are a pattern formed by the superposition of components rather than a thing in themselves, and a wave is a shape that travels, not a substance. Even the peak of a pulse can be made to advance faster than light, as when a particle tunnels through a barrier or light passes through an amplifying medium; what cannot advance faster than light is the front of the pulse, the first arrival of anything at all, which is what carries the news that the pulse was sent. In each case the faster-than-light motion belongs to a pattern, and the pattern’s speed is not limited because the pattern is not carrying its own causes along with it.

A pattern that radiates like a charge #

A spot carries nothing, but it can still make something. Suppose the screen is not passive: each point, when lit, emits a little light of its own, as a phosphor does, or the lit points are not points of light at all but points where a current is switched on in a long antenna, one segment after another, by a timing signal sent from elsewhere. Then the moving pattern is a moving source, and a source moving faster than the waves it emits does what a charge moving faster than light in water does: the waves it leaves behind pile up on a cone, the cone the source leaves behind, and the radiation along the cone arrives all at once.

Viktor Ginzburg and Boris Bolotovskii pointed out in the 1970s that nothing in electrodynamics forbids such superluminal sources, since a current pattern switched on in sequence is no more a moving object than the spot is, and that they radiate as a superluminal charge would. Laboratory antennas built as long arrays of independently timed elements have been made to produce exactly this kind of focused, cone-shaped burst. The pattern carries no message from one segment to the next, but the radiation it produces is a perfectly ordinary electromagnetic wave travelling at $c$, and it carries the news of the pattern, as every signal must, at the speed of light.

The question every faster-than-light claim needs #

Faster-than-light claims recur, and the spot gives the question that sorts them. It is not “is something moving faster than light?” but “could the motion carry a message from one place to the other?” — which is the same as asking whether events at the earlier place can affect events at the later one. For the spot, the scissors and the crests, the answer is no, because each point of the pattern has its own cause elsewhere. For a real signal, the answer is yes by definition, and every such claim examined so far has turned out, on inspection, to be a pattern rather than a signal, or a mistake.

The question has a precise form in terms of what can be measured one way: a speed measured between two places requires synchronised clocks at both, and the synchronisation is a convention. For a real signal, the convention cannot make it faster than light in any frame. For a pattern, the convention can make it anything, including infinite, because the pattern’s arrival times at the two places are not connected by anything a convention must respect.

What the pictures cannot show #

The figures treat the beam as infinitely narrow and the screen as infinitely long and perfectly flat. A real beam has a width, so a real spot is a smear whose leading and trailing edges move at slightly different speeds; a real screen is finite, and a sweep that ends before the screen does never reaches the far region where the spot stays superluminal. And no figure shows what an observer standing somewhere would see: light from each lit point has to travel to the eye too, adding another delay that depends on where the observer stands, so that the spot a person sees can be born at a different place, split differently, or move at a different apparent speed from the spot the screen actually carries.

The domain of the argument is any pattern formed by separately caused events — lit points, crossing points, wave crests. Inside it, speeds faster than light are allowed and frame-dependent in every way the composition law forbids for real objects. Outside it — anything that carries energy, momentum or a message from point to point — the ceiling holds.

Still open: whether nature has any genuine exception #

Every faster-than-light signal claimed so far has dissolved: superluminal jets into projection, tunnelling times into reshaped pulses, a 2011 neutrino measurement into a loose cable. The ceiling is supported by every test, and by the consistency of causality with the rest of physics. What is not settled is whether the ceiling is exact or approximate in theories that go beyond the ones tested. Some approaches to quantum gravity allow tiny violations of Lorentz symmetry at very high energies, and photons racing for seven billion years have been used to bound them; others make the structure of spacetime at the smallest scales uncertain, so that the light cone itself is fuzzy. Whether any of these permits a genuine signal faster than light, even in principle, and what it would do to causality if it did, are questions for which no experiment yet has an answer.

The spot itself is settled. A beam turning at ω sweeps a screen at distance D faster than light wherever ωD or the obliqueness of the screen makes it so; its spot is born as a pair at the point light reaches first, runs at any speed up to infinity, and seen from a frame moving at $c^2/u$ crosses the screen instantaneously — because its positions are separately lit points, no one of which can reach another. The composition of velocities caps things, and a spot is not a thing.