What Is Space?

Let’s look at a?particular, simple rule from our infinite collection:

{{x,?y,?y}, {z,?x,?u}}?→?{{y,?v,?y}, {y,?z,?v}, {u,?v,?v}}

Here’s what it does:

And after a while this is what happens:

It’s basically making us a very simple “piece of space”. If we keep on going longer and longer it’ll make a finer and finer mesh, to the point where what we have is almost indistinguishable from a piece of a continuous plane.

Here’s a different rule:

{{x,?x,?y}, {z,?u,?x}}?→?{{u,?u,?z}, {v,?u,?v}, {v,?y,?x}}

It looks it’s “trying to make” something 3D. Here’s another rule:

{{x,?y,?z}, {u,?y,?v}}?→?{{w,?z,?x}, {z,?w,?u}, {x,?y,?w}}

Isn’t this strange? We have a rule that’s just specifying how to rewrite pieces of an abstract hypergraph, with no notion of geometry, or anything about 3D space. And yet it produces a hypergraph that’s naturally laid out as something that looks like a 3D surface.

Even though the only thing that’s really here is connections between points, we can “guess” where a surface might be, then we can show the result in 3D:

If we keep going, then like the example of the plane, the mesh will get finer and finer, until basically our rule has grown us—point by point, connection by connection—something that’s like a continuous 3D surface of the kind you might study in a calculus class. Of course, in some sense, it’s not “really” that surface: it’s just a hypergraph that represents a bunch of abstract relations—but somehow the pattern of those relations gives it a structure that’s a closer and closer approximation to the surface.

And this is basically how I think space in the universe works. Underneath, it’s a bunch of discrete, abstract relations between abstract points. But at the scale we’re experiencing it, the pattern of relations it has makes it seem like continuous space of the kind we’re used to. It’s a bit like what happens with, say, water. Underneath, it’s a bunch of discrete molecules bouncing around. But to us it?seems like a continuous fluid.

Needless to say, people have thought that space might ultimately be discrete?ever since antiquity. But in modern physics there was never a way to make it work—and anyway it was much more convenient for it to be continuous, so one could use calculus. But now it’s looking like the idea of space being discrete is actually crucial to getting a fundamental theory of physics.

The Dimensionality of Space

A very fundamental fact about space as we experience it is that it is three-dimensional. So can our rules reproduce that? Two of the rules we just saw produce what we can easily recognize as two-dimensional surfaces—in one case flat, in the other case arranged in a certain shape. Of course, these are very bland examples of (two-dimensional) space: they are effectively just simple grids. And while this is what makes them easy to recognize, it also means that they’re not actually much like our universe, where there’s in a sense much more going on.

So, OK, take a case like:

If we were to go on long enough, would this make something like space, and, if so, with how many dimensions? To know the answer, we have to have some?robust way to measure dimension. But remember, the pictures we’re drawing are just visualizations; the underlying structure is a bunch of discrete relations defining a hypergraph—with no information about coordinates, or geometry, or even topology. And, by the way, to emphasize that point, here is the same graph—with exactly the same connectivity structure—rendered four different ways:

But getting back to the question of dimension, recall that the area of a circle is?πr2; the volume of a sphere is:

In general, the “volume” of the?d-dimensional analog of a sphere is a constant multiplied by?rd. But now think about our hypergraph. Start at some point in the hypergraph. Then follow?r?hyperedges in all possible ways. You’ve effectively made the analog of a “spherical ball” in the hypergraph. Here are examples for graphs corresponding to 2D and 3D lattices:

And if you now count the number of points reached by going “graph distance?r” (i.e. by following?r?connections in the graph) you’ll find in these two cases that they indeed grow like?r2?and?r3.

So this gives us a way to measure the effective dimension of our hypergraphs. Just start at a particular point and see how many points you reach by going?r?steps:

Now to work out effective dimension, we in principle just have to fit the results to?rd. It’s?a bit complicated, though, because we need to avoid small?r?(where every detail of the hypergraph is going to matter) and large?r?(where we’re hitting the edge of the hypergraph)—and we also need to think about how our “space” is refining as the underlying system evolves. But in the end we can generate a series of fits for the effective dimension—and in this case these say that the effective dimension is about 2.7:

If we do the same thing for

it’s limiting to dimension 2, as it should:

What does the fractional dimension mean? Well, consider?fractals, which our rules can easily make:

{{x,?y,?z}}?→?{{x,?u,?w}, {y,?v,?u}, {z,?w,?v}}

If we measure the dimension here we get 1.58—the usual fractal dimension?log2(3)?for a Sierpiński structure:

Our rule above doesn’t create a structure that’s as regular as this. In fact, even though the rule itself is completely deterministic, the structure it makes?looks quite random. But what our measurements suggest is that when we keep running the rule it produces something that’s like 2.7-dimensional space.

Of course, 2.7 is not 3, and presumably this particular rule isn’t the one for our particular universe (though it’s not clear what effective dimension it’d have if we ran it 10100?steps). But the process of measuring dimension shows an example of how we can start making “physics-connectable” statements about the behavior of our rules.

By the way, we’ve been talking about “making space” with our models. But actually, we’re not just trying to make space; we’re trying to make everything in the universe. In standard current physics, there’s space—described mathematically as a manifold—and serving as a kind of backdrop, and then there’s everything that’s in space, all the matter and particles and planets and so on.

But in our models there’s in a sense?nothing but space—and in a sense everything in the universe must be “made of space”. Or, put another way, it’s the exact same hypergraph that’s giving us the structure of space, and everything that exists in space.

So what this means is that, for example, a particle like an electron or a photon must correspond to some local feature of the hypergraph, a bit like in?this toy example:

To give a sense of scale, though, I have an estimate that says that 10200?times more “activity” in the hypergraph that represents our universe is going into “maintaining the structure of space” than is going into maintaining all the matter we know exists in the universe.

Curvature in Space & Einstein’s Equations

Here are a few structures that simple examples of our rules make:

But while all of these look like surfaces, they’re all obviously different. And one way to characterize them is by their local curvature. Well, it turns out that in our models, curvature is a concept closely related to dimension—and this fact will actually be critical in understanding, for example, how gravity arises.

But for now, let’s talk about how one would measure?curvature on a hypergraph. Normally the area of a circle is?πr2. But let’s imagine that we’ve drawn a circle on the surface of a sphere, and now we’re measuring the area on the sphere that’s inside the circle:

This area is no longer?πr2. Instead it’s?π *?

, where?a?is the radius of the sphere. In other words, as the radius of the circle gets bigger, the effect of being on the sphere is ever more important. (On the surface of the Earth, imagine a circle drawn around the North Pole; once it gets to the equator, it can never get any bigger.)

If we generalize to?d?dimensions, it turns out the?formula for the growth rate?of the volume is?

, where?R?is a mathematical object known as the Ricci scalar curvature.

So what this all means is that if we look at the growth rates of spherical balls in our hypergraphs, we can expect two contributions: a leading one of order?rd?that corresponds to effective dimension, and a “correction” of order?r2?that represents curvature.

Here’s an example. Instead of giving a flat estimate of dimension (here equal to 2), we have something that dips down, reflecting the positive (“sphere-like”) curvature of the surface:

What is the significance of curvature? One thing is that it has implications for?geodesics. A geodesic is the shortest distance between two points. In ordinary flat space, geodesics are just lines. But when there’s curvature, the geodesics are curved:

In the case of positive curvature, bundles of geodesics converge; for negative curvature they diverge. But, OK, even though geodesics were originally defined for continuous space (actually, as the name suggests, for paths on the surface of the Earth), one can also have them in graphs (and hypergraphs). And it’s the same story: the geodesic is the shortest path between two points in the graph (or hypergraph).

Here are geodesics on the “positive-curvature surface” created by one of our rules:

And here they are for a more complicated structure:

Why are geodesics important? One reason is that in?Einstein’s general relativitythey’re the paths that light (or objects in “free fall”) follows in space. And in that theory gravity is associated with curvature in space. So when something is deflected going around the Sun, that happens because space around the Sun is curved, so the geodesic the object follows is also curved.

General relativity’s description of curvature in space turns out to all be based on the Ricci scalar curvature?R?that we encountered above (as well as the slightly more sophisticated Ricci tensor). But so if we want to find out if our models are reproducing Einstein’s equations for gravity, we basically have to find out if the Ricci curvatures that arise from our hypergraphs are the same as the theory implies.

There’s quite a bit of?mathematical sophistication involved?(for example, we have to consider curvature in space+time, not just space), but the bottom line is that, yes, in various limits, and subject to various assumptions, our models do indeed?reproduce Einstein’s equations. (At first, we’re just reproducing the vacuum Einstein equations, appropriate when there’s no matter involved; when we discuss matter, we’ll see that we actually get the?full Einstein equations.)

It’s a big deal to reproduce Einstein’s equations. Normally in physics, Einstein’s equations are what you start from (or sometimes they arise as a consistency condition for a theory): here they’re what comes out as an emergent feature of the model.

It’s worth saying a little about how the derivation works. It’s actually somewhat analogous to the?derivation of the equations of fluid flow?from the limit of the underlying dynamics of lots of discrete molecules. But in this case, it’s the structure of space rather than the velocity of a fluid that we’re computing. It involves some of the same kinds of mathematical approximations and assumptions, though. One has to assume, for example, that there’s enough?effective randomness generated?in the system that statistical averages work. There is also a whole host of subtle mathematical limits to take. Distances have to be large compared to individual hypergraph connections, but small compared to the whole size of the hypergraph, etc.

It’s pretty common for physicists to “hack through” the mathematical niceties. That’s actually happened for nearly a century in the case of deriving fluid equations from molecular dynamics. And we’re definitely guilty of the same thing here. Which in a sense is another way of saying that there’s lots of nice mathematics to do in actually making the derivation rigorous, and understanding exactly when it’ll apply, and so on.

By the way, when it comes to mathematics, even the setup that we have is interesting. Calculus has been built to work in ordinary continuous spaces (manifolds that locally approximate Euclidean space). But what we have here is something different: in the?limit of an infinitely large hypergraph, it’s like a continuous space, but ordinary calculus doesn’t work on it (not least because it isn’t necessarily integer-dimensional). So to really talk about it well, we have to invent something that’s kind of a generalization of calculus, that’s for example capable of dealing with curvature in fractional-dimensional space. (Probably the closest current mathematics to this is what’s been coming out of the very active field of?geometric group theory.)

It’s worth noting, by the way, that there’s a lot of subtlety in the precise tradeoff between changing the dimension of space, and having curvature in it. And while we think our universe is three-dimensional, it’s quite possible according to our models that there are at least local deviations—and most likely there were actually?large deviations in the early universe.

Time

In our models, space is defined by the large-scale structure of the hypergraph that represents our collection of abstract relations. But what then is time?

For the past century or so, it’s been pretty universally assumed in fundamental physics that time is in a sense “just like space”—and that one should for example lump space and time together and talk about the “spacetime continuum”. And certainly the theory of relativity points in this direction. But if there’s been one “wrong turn” in the history of physics in the past century, I think it’s the assumption that space and time are the same kind of thing. And in our models they’re not—even though, as we’ll see, relativity comes out just fine.

So what then is time? In effect it’s much as we experience it: the inexorable process of things happening and leading to other things. But in our models it’s something much more precise: it’s the progressive application of rules, that continually modify the abstract structure that defines the contents of the universe.

The version of time in our models is in a sense very computational. As time progresses we are in effect seeing the results of more and more steps in a computation. And indeed the phenomenon of?computational irreducibility?implies that there is something definite and irreducible “achieved” by this process. (And, for example, this irreducibility is what I believe is responsible for the “encrypting” of initial conditions that is associated with the?law of entropy increase, and the?thermodynamic arrow of time.) Needless to say, of course, our modern computational paradigm did not exist a century ago when “spacetime” was introduced, and perhaps if it had, the history of physics might have been very different.

But, OK, so in our models time is just the progressive application of rules. But there is a subtlety in exactly how this works that might at first seem like a detail, but that actually turns out to be huge, and in fact turns out to be the key to both relativity and quantum mechanics.

At the beginning of this piece, I talked about the rule

{{x,?y}, {x,?z}}?→?{{x,?z}, {x,?w}, {y,?w}, {z,?w}}

and showed the “first few steps” in applying it

But?how exactly did the rule get applied? What is “inside” these steps? The rule defines how to take two connections in the hypergraph (which in this case is actually just a graph) and transform them into four new connections, creating a new element in the process. So each “step” that we showed before actually consists of several individual “updating events” (where here newly added connections are highlighted, and ones that are about to be removed are dashed):

But now, here is the crucial point: this is not the only sequence of updating events consistent with the rule. The rule just says to find two adjacent connections, and if there are several possible choices, it says nothing about which one. And a crucial idea in our model is in a sense just to do all of them.

We can represent this with a?graph that shows all possible paths:

For the very first update, there are two possibilities. Then for each of the results of these, there are four additional possibilities. But at the next update, something important happens: two of the branches merge. In other words, even though we have done a different sequence of updates, the outcome is the same.

Things rapidly get complicated. Here is the graph after one more update, now no longer trying to show a progression down the page:

So how does this relate to time? What it says is that in the basic statement of the model there is not just one path of time; there are many paths, and many “histories”. But the model—and the rule that is used—determines all of them. And we have seen a hint of something else: that even if we might think we are following an “independent” path of history, it may actually merge with another path.

It will take some more discussion to explain how this all works. But for now let me say that what will emerge is that time is about causal relationships between things, and that in fact, even when the paths of history that are followed are different, these causal relationships can end up being the same—and that in effect, to an observer embedded in the system, there is still just a single thread of time.