The diagram a ruler cannot read

Two axes and one speed sets up the drawing and the quantity nobody argues about gives it a metric — the interval, which every observer computes the same. Put those together and a difficulty appears that neither raises: the diagram is drawn on paper, paper measures lengths with Pythagoras, and the interval does not.

So the drawing is a map of a geometry it cannot faithfully represent, and every quantitative reading taken off it with a ruler is wrong.

The hyperbola is the ruler #

The circle is the wrong figure. On a Euclidean plane the points at unit distance from the origin form a circle, and the tick marks on any pair of rotated axes are where those axes cross it. In spacetime the invariant is $t^2 - x^2$ rather than $t^2 + x^2$, so the set of events at unit interval is a hyperbola, and there are two of them: one for timelike separations and one for spacelike.

Every observer’s unit tick is where their own axis meets the appropriate branch. That is the whole of how to calibrate a spacetime diagram, and it is what the figure checks — the drawn ticks are verified to lie on the hyperbolae rather than placed by eye.

The consequence is immediate. A Lorentz boost is a hyperbolic rotation, and hyperbolic rotations do not preserve Euclidean length, so the boosted axes get longer on the page even though nothing about them has changed invariantly. The apparent stretching is a property of the sheet of paper.

An ordinary rotation has the same character in reverse: rotate a pair of axes on a page and their unit ticks stay on a circle, so a ruler reads both frames correctly. The reason relativity’s diagrams need warning labels and Euclidean geometry’s do not is exactly that the invariant and the page agree in one case and not the other.

How wrong the ruler is #

The page length of the unit tick works out to $\sqrt{(1+\beta^2)/(1-\beta^2)}$, and it is worth naming that expression only to be able to say that it appears nowhere else in relativity.

It is not the Lorentz factor $1/\sqrt{1-\beta^2}$, though it looks enough like it to be mistaken for it, and the figure draws both so the difference is visible rather than argued. It is not the Doppler factor. It is not a ratio of times or lengths that any observer measures. It is the length of a line segment on a piece of paper, and the only reason to compute it is to know how much to distrust the drawing.

This is the source of a specific and common confusion. Somebody reads a diagram, sees the moving frame’s second-tick sitting further out, and concludes that the moving observer’s seconds are longer — which sounds like time dilation and has the sign of it. It is not time dilation, it does not have the size of time dilation, and it would be present in a diagram of two observers whose clocks agreed.

Two sticks, two slices #

The mutual contraction is where the diagram earns its keep, because it is the one place where the drawing makes something clear that the algebra makes murky.

The apparent paradox is that each observer measures the other’s metre stick short. If length were a property of a stick, that would be a contradiction — one of them would have to be right. The diagram shows immediately that it is not a contradiction, because the two measurements are not measurements of the same thing.

A stick occupies a two-dimensional region of the diagram: a band, not a segment. To assign it a length, an observer takes a slice across that band — the set of events they call simultaneous — and measures the width of the intersection. The two observers take different slices, because simultaneity is frame-dependent, and different slices across the same band have different widths.

The disagreement is about which events are the two ends of the stick at one time, and it is the length that depends on when drawn rather than argued. With the picture in front of the reader the symmetry stops being surprising: two bands crossing at an angle, each sliced by the other’s notion of now, and by the symmetry of the arrangement each slice must come out the same fraction short.

The figure computes both fractions independently — one by transforming the far end of the stationary stick onto the moving observer’s slice, the other by reading the moving stick off the stationary slice — and requires them to be equal. They come out at $1/\gamma$ both times.

The same trap, at a higher speed #

Redrawing the first figure at a higher speed shows how quickly the page becomes useless as a quantitative instrument.

At six-tenths of light speed the moving observer’s second tick is about one and a half times as long on the page. At eight-tenths it is about two. Past nine-tenths the axes crowd so close to the light cone that a legible diagram is not available at all, and every published one stops well short of that — which is itself worth noticing, because it means the diagrams a reader has seen are all drawn at speeds where the distortion is mild enough to be overlooked and severe enough to mislead.

The axes closing on the cone is not an artefact. It is the honest statement that as a frame’s speed approaches light’s, its time axis and its space axis approach the same null line, and the space of simultaneous events approaches the cone. That is a real degeneracy of the transformation and is why there is no frame moving at light speed to be found by taking a limit.

Nothing about the physics is singular at any speed below light’s, and the diagram gets steadily harder to draw anyway. That mismatch is the surest sign that what is failing is the representation.

The drawing that does not pick a favourite #

There is a way to remove the distortion, at the cost of removing something else.

Draw the diagram from the frame that sees the two observers moving equally and oppositely. Their axes then tilt by equal and opposite amounts, their unit ticks are the same length on the page by symmetry, and the ruler works. That is a Loedel diagram, and it is the right drawing whenever the point being made is about the symmetry between two frames rather than about how one of them sees the world.

What it costs is the vertical worldline. In the ordinary drawing one observer is at rest and everything is described in their coordinates, which is what makes it easy to read a particular frame’s answers off it. In the symmetric drawing neither observer is at rest and the coordinates belong to a third party nobody is interested in.

The two drawings contain identical information, which is worth stating plainly: no experiment distinguishes them and no calculation comes out differently. The choice is about which misreading is more dangerous — the ordinary diagram invites the reader to think one frame is special, and the symmetric one invites nothing but is harder to read a coordinate off.

What the diagram is actually good for #

Given all that, it is fair to ask what a spacetime diagram is for, and the answer is that it is good at the things a ruler is not needed for.

It is good at ordering. Whether one event is in another’s past, future, or neither is read off the light cone directly, and no calibration is required to see it. Every causality argument on these diagrams is of that kind.

It is good at counting crossings. Whether two worldlines meet, how many times, and in what order is a topological question the drawing answers exactly. The twin paradox is settled this way — the returning twin’s worldline is bent and the stationary one is not — and the twin who comes back younger needs nothing quantitative from the picture beyond that.

It is good at slicing, which the previous figure is entirely about. Any statement of the form “which events does this observer call simultaneous” is a line on the diagram and is exact.

It is good at settling disputes about order that are not really disputes. Two events outside each other’s light cones can be put in either order by choosing a frame, and the diagram shows exactly which pairs those are: the ones separated by a line steeper than 45°, which is now is a choice of slicing made visible. Two events inside each other’s cones cannot be reordered by any frame, and the drawing shows that too, by the fact that no permitted slice crosses them in the other sense. Deciding which case a pair falls into requires nothing but the cone, and which came first and who decides is that decision procedure applied.

It is bad at magnitudes, and every one of the confusions above is somebody reading a magnitude off it. The honest way to use one is to get the structure from the picture and the numbers from the interval — which is the same division of labour the diagram was invented for.

What the geometry actually is #

The stretched axes are the visible symptom of something worth naming, because it turns up again in general relativity in a form that cannot be drawn away.

Minkowski space is a metric space with a metric that is not positive: the interval between two distinct events can be positive, negative or zero, and events at zero interval are not the same event. No positive-definite geometry has that property, and a sheet of paper is positive-definite, so no drawing on paper can represent Minkowski distances by paper distances. The failure is not a matter of a bad projection that a cleverer one would fix; it is a theorem.

That is the same reason a flat map of the Earth must distort something. The difference is instructive: a map’s distortion is a matter of degree and can be made small over a small region, while the Minkowski case is a difference in signature and does not become small anywhere. A spacetime diagram is more like a graph than like a map, and its axes are labelled quantities rather than directions in a space the reader is looking at.

The light cone survives the projection intact, which is why it is the one feature that can be trusted absolutely. Null separations are zero interval and zero Euclidean length along the cone’s own direction only in the sense that both agree on which lines are cones — the 45° lines are the same lines in every frame, and that is the single fixed feature every diagram shares.

Reading a transformation off the picture #

There is one quantitative use the diagram supports honestly, and it is worth setting out because it is the technique that makes the calibration pay for itself.

To find a moving observer’s coordinates for an event, do not measure anything. Draw the two lines through the event parallel to that observer’s axes, find where they cut the axes, and count unit ticks — the ticks placed by the hyperbolae, not by the ruler. That is a projection along the axes rather than a perpendicular drop, which is the second thing Euclidean habit gets wrong: in a Minkowski diagram the coordinate lines are oblique, and dropping a perpendicular from an event to an axis lands in the wrong place.

Done that way the reading is exact and matches the Lorentz transformation exactly, because the construction is the transformation drawn. Both errors — perpendicular projection and ruler measurement — come from the same source, and both disappear once the hyperbolae are on the page.

The same technique gives the time dilation and the contraction without algebra. The moving observer’s clock reading at an event is a count of ticks along their time axis, and comparing it with the count along the stationary axis at the same event gives $\gamma$ directly. The picture computes the factor rather than illustrating it, which is what a diagram ought to do and what the ruler-and-perpendicular version cannot.

Where the model stops #

Everything here is one space dimension. A real boost mixes time with one spatial direction and leaves the perpendicular ones alone, so a two-dimensional diagram is exact for motion along a line and silent about everything else. Aberration, the rotation two boosts leave behind, and the appearance of a passing object all live in the dimensions the drawing has dropped.

The diagram is flat. Gravity curves spacetime and the picture becomes a curved surface with no global inertial coordinates at all, so the axes-and-ticks construction survives only locally. What survives globally is the light cone structure, which is why causal diagrams remain useful in general relativity when coordinate diagrams stop being.

The calibration hyperbolae are drawn for unit interval. Nothing distinguishes one second from one year, so the drawing has a scale that has to be chosen and stated, and a diagram with no stated unit cannot be read quantitatively at all.

And “what an observer sees” is a further step. Every line on these diagrams is about coordinates assigned after light travel time has been subtracted. What reaches an eye is a different construction, and the contraction no photograph shows is what happens when the two are confused.

And the construction assumes the two frames share an origin. A boost is a hyperbolic rotation about a fixed event, so both observers’ axes are drawn through the same point and both count their ticks from there. A translation of the origin adds four constants that no amount of calibration will supply, and comparing two diagrams drawn about different events is a mistake the picture gives no warning of — the axes look the same either way.

What the pictures cannot show #

The hyperbolae are drawn as curves and read as rulers, and a reader who has spent years with Euclidean diagrams will keep seeing them as curves. Nothing in the drawing can convey that the hyperbola is playing the part the circle plays in a rotation — that has to be said, and the figure’s caption says it.

The symmetric diagram cannot show that it is symmetric. Its symmetry is a fact about the two tilts being equal, which is visible, and about the two page units being equal, which is not — that has to be computed, and the figure computes it rather than leaving the reader to trust the drawing.

One further thing the drawing cannot carry is the difference between the two hyperbolae. They are drawn as two curves of the same kind, and they are not: one is a set of events every observer agrees lies in the future, and the other a set every observer agrees lies neither in the past nor the future. Their branches are separated by the cone and no continuous motion of the axes takes a tick from one to the other, which is the geometric form of the statement that a timelike separation cannot be boosted into a spacelike one. On the page they are simply two curves.

Where the ladder goes next #

The spacetime-diagram ladder began with two axes and one speed, which is the construction, and continued with the quantity nobody argues about, which is its metric. This rung asks how to read the picture without being misled by the page it is drawn on. The rungs after it: the causal structure as a subject in its own right, where the cone is the only geometry retained; the momentum-space version of the same diagram, in which the hyperbola is the mass shell; and the conformal diagrams that compress infinity into a finite drawing, which trade every distance for the one thing that survives.

The habit worth carrying away is that a representation has properties of its own that the thing represented does not. The stretch on the page is real and means nothing, and telling those two apart is most of the skill in using any diagram at all.