The five places infinity turns out to be

A spacetime diagram is a map, and like any map of something unbounded it has to stop at its edges. Two axes and one speed draws a few seconds and a few light-seconds; every diagram of flat spacetime ever printed shows a square window onto a plane that goes on for ever, and everything interesting about the far future and the far distance happens off the page.

What the light cones alone can decide found a freedom that makes the edge negotiable. With one dimension of space, any increasing function applied to the light-cone coordinate $u = t - x$, and any other applied to $v = t + x$, keeps every light cone and the causal order of every pair of events, while changing all the distances. There that freedom was a warning about how little the cones contain. Here it becomes a tool. Pick functions that carry the whole infinite range into a finite one, and the whole of spacetime fits on the page with every cone intact.

The construction is due to Roger Penrose, in the early 1960s, and to Brandon Carter, who used it to draw the spacetimes of black holes. It shows something no finite window can: what infinity looks like, and that it is not one place.

A squeeze that keeps the order #

The function used is arctan, applied to each light-cone coordinate separately. It increases everywhere, so it keeps the order of events along each family of light rays, and it takes the whole real line into the interval between $-\pi/2$ and $\pi/2$.

The squeeze is gentle near zero and ruthless far away. The first unit of $u$ takes up a quarter of the available range; the nine units after it take up a little over a fifth more; everything beyond ten units, all the way to infinity, fits into the last six per cent, and everything beyond a hundred into a little over half of one per cent. So the middle of the new diagram looks much like the ordinary one, and its edges hold an infinite amount of spacetime compressed to nothing.

With $U = \arctan u$ and $V = \arctan v$, the new time and space coordinates are their average and half their difference, $T = (U+V)/2$ and $X = (V-U)/2$, exactly as the old ones were built from $u$ and $v$. Because $U$ and $V$ are each confined to a band, $T$ and $X$ are confined to a diamond, $|T| + |X| < \pi/2$. Because the new coordinates are built from the squeezed light-cone coordinates in the same way the old ones were built from the unsqueezed ones, a line of constant $U$ is still a line at 45° in the new picture, and so is a line of constant $V$. Every light ray is still drawn at 45°, and the figure checks it along the rays through the origin to a part in a billion.

Where the edges are #

The first figure is the whole of spacetime, and its boundary can be read the way a map’s coastline can: by following things out to it.

Follow an observer at rest. As $t$ grows without limit, both $u$ and $v$ grow without limit, so both are squeezed to $\pi/2$, and the observer arrives at the top corner of the diamond, called $i^+$ — future timelike infinity. Running time backwards takes it to the bottom corner, $i^-$. Every one of the drawn observers, wherever it stands, goes from the same bottom point to the same top point — the same corners which came first, and who decides would call the absolute past and future of every event, pushed out to their limit; the figure checks that each is within a millionth of those corners at ten million units of time.

Follow an instant instead, the line of constant $t$, out to large distance. Now $u$ goes to minus infinity and $v$ to plus infinity, or the reverse, and the instant ends at one of the side corners, $i^0$ — spatial infinity. Every instant ends at the same two points, whatever time it is.

Follow a light ray. A ray moving right keeps its $u$ fixed while $v$ grows without limit, so it ends somewhere on the upper-right edge of the diamond, at a height set by its $u$. That edge is future null infinity, written $\mathscr{I}^+$ and pronounced “scri plus”, and it is not a point but a whole line. The upper-left edge is its mirror image for rays moving left, and the two lower edges, $\mathscr{I}^-$, are where light rays come from.

So the boundary of flat spacetime has distinct parts, and each is reached by a different kind of journey: $i^+$ and $i^-$ by anything slower than light, $i^0$ by going far in space at fixed time, and $\mathscr{I}^\pm$ by light. With three dimensions of space the two side points merge into one, and the count is five.

Every inertial observer ends at one point #

The observers in that figure set out from the same event at speeds from zero to ninety-nine per cent of light’s. On an ordinary diagram they fan out and never meet again. In the squeezed one they fan out, bend, and all converge on $i^+$. Speed changes the route, not the destination: a worldline slower than light has both $u$ and $v$ growing without limit, and that is the only thing that decides where it ends.

Light is different in exactly the way the cones require. A light ray has one of its coordinates fixed, so it cannot reach the corner; it ends partway along an edge, and parallel rays that set out from different places end at different points of it. The figure’s four parallel rays land at four distinct heights. The far future of everything with mass is a single point, and the far future of light is a line, which keeps track of where each ray came from.

Two consequences follow from reading the picture as a map of what can influence what. The past light cone of $i^+$ is the whole diamond: every event in flat spacetime has a light ray to the upper edges, and every inertial observer’s worldline eventually crosses every such ray on its way to $i^+$. An inertial observer who waits long enough can receive a signal from every event there is. Flat spacetime has no horizons for observers who do not accelerate.

An observer who accelerates forever does not share that fate. A worldline of constant proper acceleration is a hyperbola, and its two ends go to infinity in speed as well as in time: one of its light-cone coordinates stays bounded, so in the diamond it ends on $\mathscr{I}^+$ rather than at $i^+$. The past of a point on that edge is only half the diamond, and the other half is events from which no signal will ever arrive. That is the wall of silence behind a rocket, and in the conformal diagram it is not a special construction but a reading of where a curve ends. The size of the silent region is not exotic: for an acceleration of one standard gravity the horizon trails 0.97 light-years behind the start of the journey, and anything that happens beyond it stays unseen for as long as the acceleration lasts.

Cones kept, distances lost #

The figure makes the trade explicit. At each of four events on the same worldline the light cone opens at exactly 45°, because the map keeps light rays as light rays. The hyperbola marking half a unit of proper time does not keep its size. At the origin it sits a visible distance ahead of its event; by $t = 3$ it is less than a tenth as far; by $t = 10$ it is a hundredth, and the infinite remainder of the worldline’s proper time is packed into the short segment of curve left before $i^+$.

The shrinking is predictable, which makes it a check rather than a curiosity. Along the worldline $x = 0$ both light-cone coordinates equal $t$, arctan compresses each of them by a factor $1/(1+t^2)$ near that point, and so a very short interval of proper time should appear shrunk by that factor. It gives one half at $t = 1$, a tenth at $t = 3$ and one part in 101 at $t = 10$; the figure’s half-unit hyperbolae, which are not very short, come out at 0.426, 0.094 and 0.010, closing in on the local factor as the squeeze grows and the half-unit becomes small by comparison.

This is the precise sense in which the diagram is a map of causal structure and not of geometry. The diagram a ruler cannot read found that on an ordinary spacetime diagram the ticks on a moving observer’s axis sit at distances that mean nothing on the page; on the conformal diagram even the stationary observer’s ticks mean nothing, and only the angles of light rays and the question of which region lies inside which cone can be read. Mathematically, the interval in the new coordinates is the old one multiplied by a factor that varies from place to place and goes to zero at the boundary. Maps of that kind, which multiply all distances at a point by the same factor and so keep angles, are called conformal, and the diagram is named after them.

For light itself the loss costs nothing. The equations of electromagnetism in empty space do not contain any length scale, and they are unchanged by a conformal rescaling of spacetime, so an electromagnetic wave can be followed in the squeezed picture all the way to the boundary and beyond it without the equations breaking down. The boundary that was infinitely far away becomes an ordinary place where a light wave’s behaviour can be written down. That is what the construction was originally for.

The same trick does not work for a field with mass. A wave of a massive field travels slower than light, so its energy goes where massive observers go, to $i^+$, not out along $\mathscr{I}^+$, and its equations contain the mass as a length scale that a conformal rescaling would change. The boundary edges of the diagram belong to massless fields — light, and gravitational waves — and the corners to everything else, which is the physical content of the difference between a point and an edge at infinity.

Three dimensions of space #

With three dimensions of space and spherical symmetry, the drawing keeps only time and the distance from a centre, and every point of it stands for a sphere. Distance cannot be negative, so the diamond is cut in half along its middle, and the left edge of the triangle is the centre itself. The two side corners become one, $i^0$, at the far right, and the count of distinct places at infinity is five: $i^+$, $i^-$, $i^0$, $\mathscr{I}^+$ and $\mathscr{I}^-$.

A spherical pulse of light shows the whole shape of a light ray’s history in one line: it comes in from $\mathscr{I}^-$ as a sphere of enormous size, shrinks to a point at the centre, and passes through to expand back out to $\mathscr{I}^+$. The figure checks that both ends land on the midpoints of those edges.

This is where the diagram earns its keep in physics. Radiation is energy that escapes to infinity, and the precise sense of “escapes to infinity” is “arrives at $\mathscr{I}^+$”. A charge that turns must glow computes the power an accelerated charge radiates by looking at its field on a very large sphere; in the conformal picture that sphere is carried out to $\mathscr{I}^+$, where the part of the field that falls off as one over the distance survives and everything else vanishes. In 1962 Hermann Bondi, with Mariolina van der Burg, Adrian Metzner and Rainer Sachs, used exactly this boundary to settle whether gravitational waves carry energy: the mass of an isolated system, defined at $\mathscr{I}^+$, decreases whenever waves that stretch one way and squeeze the other cross it. Until then, whether such waves were physical or an artefact of coordinates had been argued for decades.

What the diagram is for #

Flat spacetime is the simplest case and the least surprising. The construction becomes indispensable when spacetime is curved, because then there is no global inertial frame to draw an ordinary diagram in, and the conformal diagram is often the only picture that shows the whole causal structure at once.

For a black hole it draws the horizon as a line at 45°, which is what it is: a surface made of light rays that neither escape nor fall in. The surface that only lets things in describes that one-way character; in the diagram the one-way character is visible as the fact that every future cone inside the line points away from $\mathscr{I}^+$. The definition of a black hole’s region itself uses the boundary this essay has drawn: it is the set of events that have no light ray to future null infinity. Without a boundary to reach, the definition could not be stated.

Where the model stops #

The boundary is not part of spacetime. The corners and edges of the diamond are added by the construction; no event lies on them and no observer reaches them. They are a precise way of talking about limits, and the diagram treats them as places only because the squeeze gives them positions.

The distances have been thrown away. Nothing measured in seconds or metres can be read off the diagram, and a worldline that looks short near the boundary may contain an infinite amount of proper time.

The drawings use one or two dimensions of what is four. The diamond is exact for one space dimension; the triangle is exact only for spherically symmetric situations, where the angular directions can be suppressed. Anything without that symmetry needs a separate picture for each direction, or none.

And the point $i^0$ is not a smooth part of the boundary. The squeezed geometry is well-behaved along the edges $\mathscr{I}^\pm$ but degenerate at the spatial corner, where all the instants meet, and much of the care in the theory of isolated systems goes into handling it.

What the pictures cannot show #

None of the figures shows anything happening. A conformal diagram is a picture of the whole history at once, and the sense of a pulse “coming in” or an observer “arriving” is supplied by reading a line from bottom to top; the drawing itself has no time in it, and every event from the beginning of the pulse to its end is on the page simultaneously.

Nor can the triangle show what a point stands for. Each point of it is a whole sphere of events, and two points at the same height stand for spheres of different sizes; a drawing that turns a sphere into a dot cannot also show the sphere.

Still open: what the edge at infinity is really for #

Once $\mathscr{I}^+$ is treated as a place, it can be asked what symmetries it has. The expected answer was the symmetries of flat spacetime: translations, rotations and boosts. Bondi, van der Burg, Metzner and Sachs found in 1962 that the transformations that preserve the structure at null infinity form a much larger group, containing an infinite family of angle-dependent translations as well, now called supertranslations. For decades that looked like an embarrassment of the formalism.

Since the 2010s the larger group has been tied to two other results — theorems about how very low-energy radiation is emitted in collisions, and the memory effect, a permanent displacement of freely floating test masses left behind by a burst of gravitational waves — as three faces of one structure, a connection developed by Andrew Strominger and collaborators. Whether the infinite symmetry at $\mathscr{I}^+$ constrains what happens to information falling into a black hole, and whether a quantum theory of gravity in flat spacetime can be written as a theory living on that edge, are active and unsettled questions, and the memory effect itself, predicted to be tiny, is still waiting to be measured cleanly.

The habit worth keeping is the one the squeeze teaches. When a picture has to lose something, choose to lose the quantity the question does not need. A question about what can influence what does not need distances, and giving them up buys a finite picture of an infinite spacetime in which every such question can be answered at a glance.