The contraction no photograph shows

The Lorentz transformation was published in 1905, and the first correct statement of what a fast-moving object would look like appeared in 1959. In between, every picture of the subject showed a squashed object, drawn by people who had the transformation in front of them and had asked the wrong question of it.

The wrong question was “where are all the parts of the object at one moment?” — which is what length contraction answers. The right question for a photograph is “where were all the parts of the object at the moments the light now arriving left them?”, and those are different moments, because the parts are at different distances from the camera.

What length contraction actually says #

A metre stick moving at 0.9c has its ends, measured at one instant in the laboratory, 43.6 cm apart. That statement is true, it is what “length” means for a moving object, and it is not what a photograph shows. The distinction between the two is the whole of this essay, and the number above is the one that does not appear in the picture.

Length is defined as the distance between two simultaneous positions of the ends. For a stationary object the word “simultaneous” is doing no work; for a moving one it is doing all of it, because simultaneity is a choice of slicing and different observers slice differently.

What one frame calls “the two ends at the same moment” is, to the other, the two ends at different moments — and the difference is exactly what produces the contraction. So length contraction is a statement about a simultaneity convention as much as about lengths, which is the first hint that a camera, which has no access to any such convention, is measuring something else.

So the contraction is real in the same sense the definition is real. It is what a set of synchronised clocks and markers laid along the laboratory would record, it is what makes a fast muon’s atmosphere thinner in its own frame, and it is not negotiable. It is also not what a camera measures.

Two things follow that are worth separating carefully, because the essay’s whole subject is a confusion between them. The first is that the contraction is not an illusion and not a distortion of a measuring apparatus — the moving object’s ends genuinely are that distance apart, at one laboratory time, and every consequence that depends on it is real. A fast-moving lattice really is more densely charged; a muon really does cross an atmosphere that is thinner in its own frame; a heavy-ion collision really does begin as two flattened discs, and the models that describe it would fail if it did not. The second is that none of those is a statement about a picture, and no argument from any of them tells anybody what a camera would record.

The light that left earlier #

A camera records photons that arrive together. Photons that arrive together did not leave together unless they came from the same distance — the same bookkeeping that makes a jet appear to outrun light, applied to an object small enough to photograph.

Write it out. For a distant camera looking along $\hat z$, with the object moving along $\hat x$ at speed $\beta$ and $c = 1$, a point at rest-frame coordinates $(x', y', z')$ has its light reach the camera at a moment set by $z'$, so its apparent transverse position is

$$x_{\text{app}} = \frac{x'}{\gamma} + \beta z', \qquad y_{\text{app}} = y'.$$

The first term is the contraction. The second is the light delay. And together they are the projection of the same object, unshrunk, rotated by an angle $\alpha$ with

$$\sin\alpha = \beta, \qquad \cos\alpha = \frac1\gamma.$$

Those two are consistent because $\sin^2 + \cos^2 = \beta^2 + (1-\beta^2) = 1$, which is not a coincidence: it is the Lorentz factor’s defining relation, wearing the clothes of a trigonometric identity. The generator does not assume any of this. It computes the apparent positions from the delay, computes a rotated copy independently, and refuses to draw if they differ by more than a part in a million million.

The contraction does enter physical arguments, and one of them is worth naming because it is not about measurement at all. A lattice of charge seen from a moving frame has its spacing contracted, and that contraction is the reason a wire carrying a current is neutral in one frame and charged in another — which is magnetism as electricity seen sideways. There the contraction is doing real work, because nothing is being photographed.

The transformation that does it #

The conformal argument sketched below is worth backing with the formula it rests on, because that formula is the whole of how directions behave under a change of frame and it appears three times in this essay wearing different names.

A ray arriving from direction $\theta$ in one frame arrives from $\theta'$ in another moving at $\beta$, with

$$\cos\theta' = \frac{\cos\theta - \beta}{1 - \beta\cos\theta}.$$

That is stellar aberration, and it is a map of the sphere of directions onto itself. Its effect is to crowd directions toward the forward pole: a set of rays spread evenly over the sky in one frame is bunched into a cone ahead in a frame moving through it, and the faster the frame the tighter the cone.

Three consequences that are usually taught separately are this one map used three ways. Applied to starlight it is the annual aberration Bradley found in 1728 and mistook for parallax. Applied to emitted light it is the forward beaming of a fast source, which is why a relativistic particle radiates into a narrow cone. And applied to the directions from which a photograph collects light, it is the apparent rotation this essay is about.

The reason the last of those comes out as a rotation rather than as something messier is a property of the map itself: it takes circles on the sphere to circles, and preserves the angles between them. Locally, a map with those two properties can only be a rotation and a magnification. So a small object, which subtends a small patch, can only ever be recorded turned and rescaled — and the detailed calculation supplies the angle rather than the shape.

A sphere that is a sphere at every speed #

The rotation reading makes one prediction immediately, and it is the one that convinced people.

A rotated sphere is the same sphere. So a photograph of a sphere passing at any speed shows a circular disc of the proper radius, with its surface features rotated toward the observer as though the sphere had turned to show its back. This is what Penrose pointed out in 1959, in a note two paragraphs long; Terrell published the general result for an arbitrary small object independently in the same year.

It is worth noting how strong the statement is. The outline is a circle of the proper radius — the radius the sphere has at rest — not the contracted one and not some intermediate value. A sphere passing at 0.999c, whose simultaneous shape is a disc a twentieth as thick as it is wide, photographs as a circle exactly the size it would be if it were sitting still. There is no speed at which a camera detects any change in its size at all, and the only clue that anything unusual is happening is that the far side of it has come into view.

Which raises the obvious question of how everybody missed it for fifty-four years. Part of the answer is that nobody was looking: the objects relativity was applied to were particles and reference frames, not visible bodies, and the question “what would it look like” was regarded as a schoolroom illustration rather than a calculation. The rest of the answer is vocabulary. The subject’s whole language was contraction, and a rotation is not a smaller contraction or a corrected one — it is a different kind of answer, and nobody was going to arrive at it by refining the first one.

There is a satisfying way to see that the answer had to be a rotation, without doing the arithmetic. A camera collects light along a set of directions, and a change of frame transforms directions among themselves — the aberration formula maps the sphere of directions onto itself. A map of a sphere onto itself that preserves angles and maps circles to circles is a conformal transformation, and for a small patch that is a rotation combined with a scaling. Since a small object subtends a small patch, its image can only be rotated and rescaled, never sheared or squashed. The detailed calculation supplies the angle; the general argument supplies the fact that there is an angle, and it applies to any shape whatever.

What a camera is measuring instead #

The rotation is not the only thing a photograph gets that the simultaneous picture does not.

A fast object in a photograph is not merely in the wrong place — it is the wrong colour, and the shift varies across it because different parts of it are approaching and receding. So even the appearance this essay reconstructs is monochrome fiction: a real image would carry a colour gradient along the object, and the gradient is a Doppler factor rather than a contraction.

The aberration that crowds the forward sky into a cone reorganises the directions of emitted light, and it reorganises the directions of received light in the same way, and the two are the same transformation applied at the two ends. That is why the visual effects of high speed — the rotation, the beaming, the colour shift — all come out of one piece of geometry, and why treating them separately makes them look like a collection of surprises.

There is a further subtlety the figures suppress. The exact statement — object photographs as rotated — is for a small object viewed from far away, where every part of it is at nearly the same distance and nearly the same direction. A large object subtends a range of angles, the rotation angle differs across it, and what is recorded is a shear rather than a rotation: a long rod passing close by appears bent. That is the visual cousin of the string stretched between two rockets, where an extended object is the thing the frame cannot treat as a point. Terrell’s theorem is a statement about the limit of small angular size, and it is exact there and approximate everywhere else.

One more thing a camera gets that a ruler does not is the time of what it records. A photograph of a passing object records different parts of it at different proper times of the object — the far side younger than the near side, in the sense of having its own clock read less. For a rigid cube that means nothing, since nothing on it is changing. For an object with a process going on inside it, the photograph records a spread of stages of that process across its body, and the spread is real: a fast-moving clock face would be photographed showing several different readings at once, one on each part of its dial, and the spread grows with the object’s depth along the line of sight. A photograph is a slicing of spacetime too — just a cone-shaped one rather than a flat one.

The same delay, in a quasar #

Nobody has photographed a fast cube, and the light-travel argument has an astronomical consequence that has been observed for fifty years and is startling in a different way.

Watch a jet of material emerging from an active galaxy over a few years and measure how far a bright knot in it has moved across the sky. Convert the angular motion into a transverse speed using the galaxy’s distance, and the answer comes out larger than the speed of light — routinely by a factor of several, and in the most extreme cases by tens.

Nothing is moving faster than light. The cause is exactly the effect this essay is about: light from a later position of the knot has less far to travel than light from an earlier position, because the knot has moved toward the observer in between, so the two arrivals are closer together in time than the emissions were. For a knot moving at speed $\beta$ at angle $\theta$ to the line of sight, the apparent transverse speed is

$$\beta_{\text{app}} = \frac{\beta\sin\theta}{1 - \beta\cos\theta},$$

and the denominator is what does it. At small angles and speeds near light the denominator is tiny, and the ratio exceeds one.

Maximising over the angle gives $\beta_{\text{app}} = \gamma\beta$, so an apparent transverse speed of ten times light requires a Lorentz factor of at least ten. That inverts into a measurement: an observed apparent speed puts a lower bound on how relativistic the flow is, obtainable from an angular displacement and a distance and nothing else. It is one of the few direct handles on the bulk Lorentz factor of a jet, and it comes from the same sentence as the cube — light arriving together did not leave together.

The two effects are the same effect used at opposite ends of the geometry. The cube’s far side is photographed displaced backwards because its light left earlier; the jet’s earlier position is recorded later than it should be because its light had further to come. In both cases the contraction has nothing to do with it, and in both cases the intuition that a picture shows a moment is what fails.

Why nobody has taken the photograph #

It is worth being plain about the observational status of everything above, because the essay is about a picture and no such picture exists.

Nothing macroscopic has ever been moved fast enough. The largest speeds given to anything visible are a few kilometres a second, at which $\beta$ is $10^{-5}$ and the apparent rotation is two thousandths of a degree. The particles that do move at relativistic speeds are individually invisible and are detected by their energy and their tracks rather than photographed as bodies.

So the figures here are computed rather than observed, and the check the generator applies — that the positions derived from light delay coincide with a rotated copy — is a check on arithmetic rather than on nature. What supports the result is that its ingredients are separately confirmed to great precision: the aberration of starlight, the relativistic Doppler shift, the beaming of radiation from fast particles, and the apparent superluminal motion above. The theorem assembles them; none of the assembly is in doubt, and none of it has been assembled into a photograph.

What has been done is the next best thing. The scene can be rendered — the light-delay calculation done for every point of a model, with the Doppler shift and the beaming applied — and such renderings are now routine. They are simulations of an observation nobody will make, and they are the only images of the effect there are.

Where the model stops #

The camera is at infinity. Every result here is for parallel rays reaching a distant observer. A camera at a finite distance sees different parts of the object at different angles, and the apparent shape depends on where it is; the clean rotation is the limit.

The exposure is instantaneous. A photograph is taken over a finite time, and an object crossing the field at 0.9c moves during it. Everything drawn here is the idealised zero-exposure case, and the correction is a motion blur that has nothing relativistic in it.

The object is opaque and non-luminous. The figures cull the surface that faces away from the camera after the rotation, which is the correct visibility criterion — but for a transparent or self-luminous object all of it contributes, and the appearance is different again. The sphere figure would be a full grid rather than a hemisphere.

The object is assumed rigid, and there is no such thing. A rigid body in relativity is impossible, since rigidity would let a push at one end move the other end instantly. For the purposes here that does not matter — the cube is in steady motion and has been for ever, so every part of it is moving at the same speed and no signal has to propagate along it — but it matters the moment anything accelerates, and the accelerating case is where the interesting version of this problem lives.

And the whole calculation is about appearance. Nothing here modifies the transformation, the contraction, the twin’s clocks or any measurement made with rulers and synchronised clocks. It is an essay about what an instrument that collects light reports, and light-collection is one particular measurement with one particular systematic in it.

Everything here is read off one diagram, and it is worth saying which line does what. Length contraction is read off the tilted lines of constant time; the photograph is read off the light cone, which is the set of events whose light arrives together. The two families of lines are not parallel, they answer different questions, and the entire confusion this essay is about comes from using the answer to one as though it were the answer to the other.

There is a last limit that is really a strength. Nothing in the derivation used the object’s shape, its size, its material or its internal structure — only that its parts are at rest with respect to one another and that they are at different distances from the camera. That is why the result is a theorem rather than a calculation about cubes, and why it applies unchanged to a spacecraft, a galaxy or a coin. The one thing it does need is that the object be small in angular size, and that is not a property of the object at all but of how far away the camera is.

What the pictures cannot show #

The cube figures draw wireframes and no surfaces, so hidden-line removal has been left out and the drawings show edges the camera could not see. The sphere figure does cull correctly, and the difference between the two is a reminder that “what a camera records” contains a visibility calculation that is separate from the relativity and easy to get backwards — culling on the rest-frame hemisphere rather than the rotated one draws the half of the sphere the camera cannot see.

Nothing here is in colour, and the Doppler shift is the largest visible effect at high speed. A true rendering of a cube at 0.9c would be dominated by the fact that its leading face is ultraviolet and its trailing face is infrared, with the geometry a secondary matter. The figures draw the geometry because the geometry is the argument, and the argument is not what a traveller would notice first.

Where the ladder goes next #

This ladder began with the length that depends on when and continued with the pole that fits and does not, which is the same care about which events are being compared applied to a paradox. This rung applies it to a camera. The rungs after it: Born rigidity, and the horizon behind a uniformly accelerating rod; the rod-and-hole problem, where a rod falls through a slot and is guillotined in one frame and bent in the other; and relativistic ray-tracing in earnest, where the Doppler shift, the beaming and the geometry are computed together and the result looks like nothing in any textbook.

The habit worth carrying away is about instruments. An instrument does not report a quantity; it reports what reaches it, and turning that into a quantity requires a model of the journey. For fifty-four years the model of the journey was left out of a subject whose entire content is that journeys take time — and the omission produced a picture that was drawn in every book and was wrong.