The dimension a list of befores can count

What the light cones alone can decide took away the metric of spacetime and kept only which events could influence which. It found that in more than one dimension of space, the light cones fix nearly all the geometry: the only maps that keep every cone are the Lorentz transformations with an overall scale. So the order of events is almost everything, and what is missing is one number per region, a conformal factor. That essay ended by naming the idea that follows from this. If only a scale is missing, and if counting could supply the scale, then perhaps spacetime at bottom is nothing but a set of events with an order on it, and the smooth manifold is what it looks like from far away.

The proposal, due to Luca Bombelli, Joohan Lee, David Meyer and Rafael Sorkin in 1987 and named a causal set, is easy to state. A spacetime is a collection of elements. Each pair is either related, with one before the other, or not. Nothing else is given. The number of elements in a region of the spacetime is its volume, in units of a fundamental volume near the Planck scale. Whether that is the correct theory of space and time is not known, and it may not be. But the mathematics of it can be examined with no theory of quantum gravity at all, and with a computer. It turns out that a bare list of befores contains a very surprising amount of what geometry contains. Some of this does not need the proposal to be true to be interesting.

Sprinkling and forgetting #

The way to construct such a set corresponding to a chosen spacetime is called sprinkling: place points at random, uniformly by volume, with a density of one per unit of volume. Random is required, and not merely a convenience, for a reason that will appear. The next step is to discard the coordinates, retaining only the relation.

The region drawn is a causal diamond, the set of events that lie both in the future of one event and in the past of another. It is the natural region for this purpose, the same in every frame, and every question about the list can be asked of the diamond between two of its elements. A reader looking at the figure sees a square rotated by 45 degrees with spots in it. The list that remains is only an enormous table of yes or no, one entry per pair. What can that table possibly recall?

How many directions there are #

One number is almost trivially available from the table, the fraction of pairs that are related. Take two events chosen at random within the diamond. What is the chance that one lies within the other’s cone? In a spacetime with more directions of space, there are more ways of being out of touch. A cone occupies a smaller and smaller share of the region, as the number of dimensions grows, because the volume of a sphere in many dimensions sits in its outer shell.

Jacob Myrheim worked out the answer in 1978 and David Meyer gave the general form a decade later. In a diamond in flat $d$-dimensional spacetime, the expected fraction of related pairs is

$$f(d) = \frac{\Gamma(d+1)\,\Gamma(d/2)}{2\,\Gamma(3d/2)},$$

which gives exactly one half for $d = 2$ and a tenth for $d = 4$. It depends on the dimension and on nothing else. The diamond’s size has gone from it, since a larger diamond merely has more points.

So here is a dimension estimator that uses no geometry at all. Given a table of befores, count the fraction of yeses, and read off the dimension from the curve. Thanks to the curve’s steep fall, a modest number of events resolves it easily. Six trials of 800 events have an agreement that is better than a per cent at two dimensions, and about ten per cent at five, which has the fewest related pairs to count. Nothing about the formula knows anything of the construction used here. In it, a dimension need not even be a whole number. So a causal set that came from no manifold could be given a non-integer dimension, or a different value on large and small scales, and this has been a real suggestion within research on quantum gravity. Several approaches report the dimension of spacetime falling towards two at the shortest distances.

It also matters that this particular quantity, the dimension of spacetime, is often taken for a choice of coordinates. In this construction it is a number that can be measured, and it might come out differently from what was put in.

Time as the longest thing to count #

A second number needs a little more work. Between two related events, look for the longest possible chain of elements, each before the next. It is a question that can be answered from the table alone, by dynamic programming on the order. What does that length correspond to in the spacetime?

In flat spacetime, the longest path between two timelike-separated points in the sense of proper time is the straight line. That is the reversed triangle inequality, which the longest way round is the shortest clock presented as the true content of the twin paradox. So a chain as long as possible hugs the straight path through the diamond, with as many elements as it can pick up. The more proper time there is, the more of them. In two dimensions, this question reduces to one in pure combinatorics, thanks to light-cone coordinates. Each event has two numbers $u$ and $v$, and one is before another if both numbers increase. So a chain is precisely an increasing subsequence of a random permutation, and in a diamond with $N$ events the longest such subsequence has been studied for a long time. This was Ulam’s problem from around 1961. Vershik and Kerov, and independently Logan and Shepp, showed in 1977 that its length tends to $2\sqrt{N}$. Later, a famous result of Baik, Deift and Johansson gave the fluctuations, the same Tracy–Widom distribution that describes the largest eigenvalue of a random matrix.

There is the clock. The number of elements in a region is its volume, the square root of which in two dimensions is a length, so the chain’s length is a proper time in Planck units, up to a known constant. Between any two related elements, the table therefore gives the elapsed time. In this way, the table holds a kind of distance for every pair of related elements, and that is the Lorentzian metric’s main content for timelike separations. Spacelike distances can also be extracted, with more trouble. A related but stronger result is due to Brightwell and Gregory, who showed in 1991 that the constant in this relation depends on the dimension in general. In higher dimensions the best constants are not known exactly. Its asymptotics in dimension four have been bounded only.

The two figures together make a claim. A list of befores tells both what dimension it lives in and how long time has passed between its elements. In a truly literal sense, geometry has been computed from counting.

Why the points must be random #

The obvious first idea for a discrete spacetime is a lattice, like a crystal. This is how quantum chromodynamics is done on computers. Its great problem is that a lattice is not invariant under a boost. The rows of a crystal are rows in one frame and not in another, so a regular discrete spacetime carries a preferred frame with it.

A boost is an area-preserving squeeze along the light rays, the map described in the five places infinity turns out to be. A lattice squeezed that way looks very different; it has been stretched into long diagonal chains. A Poisson process has the very useful property that an area-preserving map carries it into the same process. So a sprinkling has no frame of its own. If a sufficiently accurate observer could see a lattice, that observer could find a preferred rest frame, which would produce a difference of photons’ speed with energy. The photons that raced for seven billion years reported the very stringent limits on any such thing. So a theory with a regular lattice at the Planck scale would have been in trouble from that alone, and a random one avoids this. It is one of the best arguments for the proposal: Bombelli, Henson and Sorkin showed in 2009 that a sprinkling does not select any frame at all, in a strong mathematical sense, even for any one realisation.

The randomness has a side-effect, and it may be a prediction. A count of elements in a volume $V$ has a Poisson fluctuation of order $\sqrt{V}$. Sorkin argued, before 1998, that if the cosmological constant is conjugate to the volume of the universe in the way a time is conjugate to energy, then a fluctuation in that volume would give a fluctuation in the cosmological constant of the order of $1/\sqrt{V}$ in Planck units. Since the visible universe has about $10^{240}$ elements, in this counting, that gives a value of about $10^{-120}$, which is the observed order. That is a tantalising result, and it needs a measure of caution. It was published as a prediction before dark energy’s discovery. The estimate that misses by a hundred and twenty describes the problem. No complete dynamical theory of causal sets has yet delivered this value, and the argument depends on assumptions that are not themselves established.

The rest of the lists #

The cost of the approach is displayed in the last figure, and it is severe. Consider all partial orders on $N$ elements, every possible table of befores that does not contradict itself. Which is typical?

The answer comes from a theorem by Daniel Kleitman and Bruce Rothschild in 1975. As $N$ grows, almost all orders have three layers. They contain a quarter of the elements at the bottom, half in the middle and a quarter at the top, with each element linked to about half of the next layer. Every element is at most two steps from any other. Such a thing has no extended time, no space and no resemblance to anything. Worse, they outnumber the manifold-like orders by a factor of about $2^{N^2/4}$. The sprinkled orders that look like a spacetime are a very tiny and special subset of what is possible. A theory that sums over all orders with equal weight, in an obvious analogue of a path integral, would produce those three layers with overwhelming probability, and it would have no spacetime in it.

This is known as the entropy problem of the causal-set programme. Rideout and Sorkin found in 1999 a family of growth laws, with elements added one at a time and with the choice of what relates to what subject to causal conditions, which do produce order that looks more like a growing universe. And it has been shown that for a certain gravitational action added as a weight, the three-layer orders are suppressed. It has not been shown that this leaves a four-dimensional spacetime as the winner. Whether any natural rule produces that is the central open problem.

The one thing an order cannot say #

Why does the proposal insist on counting at all? The reason is an old theorem. David Malament and, in slightly different form, Hawking, King and McCarthy showed in the 1970s that the causal order of a suitable spacetime determines its metric up to an overall local scale. That is the theorem in the background of what the light cones alone can decide. What light cones cannot tell is a conformal factor: a spacetime with every distance doubled, or stretched by an amount that varies smoothly from place to place, has exactly the same cones. A light ray has no length, and an order made from light rays has nothing to say about length either.

Counting supplies that missing factor. A number of elements in a region is a volume, and in a continuum there is nothing to count, so the factor has to be brought in as extra information. In a discrete set it is present automatically. Bombelli and his colleagues took a slogan from this: order plus number equals geometry. It is a pleasing feature of the idea that the two pieces are exactly what the theorem left out, without any addition. Riemann had already hinted at something like it in 1854, in his lecture on the foundations of geometry, when he noted that a discrete manifold carries its measure within itself while a continuous one has to have it imposed.

The result also changes how one should think about the quantity nobody argues about. There, the interval is a thing that observers agree about because it is built from their disagreements. Here, the interval between two related events is a count that nobody could disagree about, because it contains no observers: the number of elements in the longest chain. Any two people given the table would produce the same number. It also gives a concrete sense to how big now is, which asked what is left of a simultaneous present in a universe without one. In a causal set there is a natural replacement for a present: a maximal collection of mutually unrelated elements, called an antichain. It is a slice with no special status, and any number of different ones can be drawn through a set. One could describe the growth of the universe one element at a time. Whether a list of “which was added first” has any physical meaning is one of the conceptual questions the approach has to confront, and its authors’ answer is that a physically meaningful statement must not depend on the labelling of the elements, which is a discrete version of general covariance.

A warning about scale comes from the length no experiment can resolve. If the fundamental density is set near the Planck scale, nothing here is available to direct measurement. Any effect would have to accumulate over a very great many elements.

What this picture does not claim #

The figures all show what can be recovered if a list came from flat spacetime in the first place. They say nothing on whether space and time are actually made in this way. No experiment has seen discreteness, and the evidence for this picture comes from internal coherence and a very weak prediction about dark energy. Geometry recovered from a sprinkling is, by construction, the geometry that went in. The proposal that the list is fundamental has to be argued separately.

The sprinkling density is taken as one per unit of volume, and the calculations were made with very few elements by the standards of anything physical. A region of a universe that contained $10^{240}$ elements would have far smaller fluctuations than these figures do. A real test of dimension at five dimensions with 800 events shows a ten per cent error, and for curved spacetime there are further corrections, as the fraction depends on the curvature within the diamond. Estimates for curved regions exist and they depend on the region’s size. So the clean Myrheim–Meyer curve has been drawn at the limit of an infinitely small diamond with plenty of elements in it.

And nothing here has discussed the rest of physics. Fields on a causal set, particles and quantum dynamics are all subject to continuing work. Local operators cannot simply be defined by nearest neighbours because in a sprinkling of a Lorentzian manifold, an element has an infinite number of nearest neighbours, connected by long thin chains close to the light cone. That non-locality is a major difficulty, and it is built into the thing that preserves Lorentz invariance. Causal sets have to pay for their boost symmetry with a kind of action at a distance, confined to the light cones.

Still open: whether anything in the sky remembers the grain #

A theory with a fundamental random discreteness might predict a small Brownian diffusion of particles in momentum, a “swerve” from the kicks of a random microstructure. Philpott, Dowker and Sorkin computed what this would do. Light from a distant source should change in polarisation and spectrum in small ways, and the limits are now strong. It is natural to ask whether anything else could reveal discreteness at this level, for instance in the shape of the early universe’s fluctuations, or the very faint energy loss of a cosmic ray over a long path. So far nothing has been detected, with limits on the diffusion strength that are near the natural value for some forms of the model. On the theory side, the search for a rule that makes four dimensions out of a growing order goes on, with results for two-dimensional versions of the problem, in which simple actions do generate two-dimensional spacetimes from the combinatorics. In four dimensions the answer is not known.

Events sprinkled at random and reduced to a table of befores still hold their dimension, in the fraction of related pairs, 1/2 in two and 1/10 in four, and their proper time, in the longest chain at about $2\sqrt{N}$; a random set has no preferred frame where a lattice does, and yet almost all possible tables have three layers and no resemblance to any spacetime. Geometry can be counted, once the right list has been found. The habit worth keeping is to ask of any structure which weaker structure already implies it, and then, more cautiously, how common among all possibilities the weaker structure’s successful examples really are.