Causal Sets and Leaning Towers

An article freshly transplanted from my digital physics blog:

Last year I had the incredible good fortune to spend a couple of months collaborating with Tommaso Bolognesi at CNR-ISTI, in Pisa, Italy. Tommaso runs his own research program into the interface between computation and physics and is a champion of the Digital Physics cause. He hired me to see if together we could answer a very specific question:

Is it possible to build networks that have the same properties as spacetime using simple algorithms, and if so, how?

I’ve had plenty to say on the subject of modeling space before this. However, what Tommaso was looking for was very specific. He wanted us to find ways to build causal sets. Causal set theory is probably the point of closest approach between digital physics and more mainstream quantum gravity research and it’s a fascinating subject. In a nutshell, causal set theorists believe that spacetime is most usefully thought of as a discrete structure and that the way to model it is to try to mimic the kinds of relationships between events that we see in relativity. To achieve this, they connect nodes using something called a partial order—a set of relationships that define which nodes must come before others, but which falls short of providing an exact numbering for all nodes.

Broadly speaking, the Causal Set Program uses two methods to build their sets. The first, called sprinkling, is to deposit nodes at random onto a surface, and hook them together based on the geometry of that surface. The other way, called percolation dynamics, is to add nodes one by one to a set, and randomly add links from existing members of that set to each new node.

Sprinkling is useful for exploring how causal sets behave but it has a huge problem: in order to construct the discrete structure of spacetime, you have to deposit your points onto a smooth spacetime first! Clearly, if we want to come up with a background-independent theory of physics, we need to build the sets some other way. On the other hand, percolation dynamics has all the nice statistical properties that physicists would like to see and doesn’t need a background, but sadly doesn’t actually produce graphs that look like spacetime (though people are working on that).

The right solution would seem to be to come up with a third way: a process that produces the right structures without needing a background surface. However, this comes with problems. The key features that differentiate spacetime-like causal sets from others are dimensionality and Lorentz invariance.

Dimensionality essentially says that we should expect the graph that we build to have some consistent number of dimensions, rather than just being a tangled mess. Lorentz invariance is a little trickier. What it implies is that if you build your network first and then lay the nodes onto a flat surface afterward, the positions of the nodes should appear random. There should be no way you can stretch or squish the network to make it look otherwise. This is vitally important because in order to treat every relativistic reference frame the same way, as special relativity says we must, we need about the same number of links between nodes in each frame.

Another way to say this is that, thanks to Einstein, we know that no matter how fast we’re moving, space will always feel the same to us. The way a causal set works is that each link corresponds to a step through time and space taken at a certain speed. So, if for some speed of travel, our network doesn’t have enough links, it’s definitely not going to feel the same to someone traveling through it. If this happens, our model has failed. The only way that people have ever found to make Lorentz-invariant causal sets is to have the network be random.

My collaboration with Tommaso was founded on a neat way around this problem that works like this:

  • Because any causal set we can build is finite, it can only ever approximate perfect randomness.
  • Furthermore, for a finite network of given size, we can always find some algorithm that can approximate that level of randomness through a deterministic process.
  • Thus, no matter how big our network needs to be, we should still always be able to find an algorithm that could give rise to it.
  • This will always be true so long as we believe that spacetime is discrete, that the universe has finite size, and that it has existed for finite time.

In essence, what this tells us is that just because the network we want to build needs to look random, that doesn’t mean that we can’t use a completely non-random method for building it. This is all great as far as it goes, but it leaves us with an enormous problem: how to find an algorithm that can build spacetime.

In the two months we had, Tommaso and I didn’t manage to crack this problem (otherwise you would have heard about it on the news by now) but we learned some fascinating things along the way. I hope to share some of them with you in my later posts.

However, in the mean time, there are plenty of really excellent introductory papers on causal sets that are very approachable for those who’re interested. While my favorite approach to discrete physics is a little different from the causal set methodology, I can recommend this field very highly to anyone interested in learning more about quantum gravity without taking on a full-time career as a string theorist.

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