In my last post, I talked about Lorentz invariance. I got some great feedback. (Thank you Keir.) And from that, it seems pretty clear that relativity is not something I can pass over lightly. I’m going to go over the rest of how to capture special relativity in networks as carefully as I can.
Last time, I suggested that you could duplicate relativistic effects by creating a hidden, rolled-up dimension to capture the notion of subjective time. One of the comments I got was that this seemed to imply that time was going round in a tiny loop, which isn’t what we experience. Fair point. What I was aiming to say was that the act of traversing the hidden dimension produces the sensation of subjective time, not that the hidden direction was actually a compact time axis. A fine-grained distinction, I grant you.
In fact, whichever way you cut it, having to have this little extra dimension isn’t very satisfactory. We’d like to have a way of capturing the experience of subjective time that’s not dependent on it. Not least because creating networks that contain extra compact dimensions is complicated. So how can we do better?
We can do better by making the extra direction s be a feature of particles, rather than a feature of spacetime itself. In other words, if a particle’s not there, the extra direction isn’t there. And only particles that have mass can create this extra direction.
For those of you familiar with the idea of the Higgs boson, this might sound familiar. For the Higgs field, we imply that there’s a special field everywhere in space, except where a particle happens to be. The gap in that field creates wiggle-room that the the particle can use to create the phenomenon of mass. The way we currently understand physics, the mass that’s endowed by the Higgs field has nothing to do with the mass endowed by relativistic effects. But wouldn’t it be nice if we could achieve both kinds of mass with a single mechanism? Maybe we can.
If we’re implying, though, that particles carry the extra direction around with them, how can that possibly work? How can a particle have a dimension inside it? What would that even mean?
It turns out we don’t need an extra dimension. We just need the particle to create some wiggle-room, the same as for the Higgs field. We can imagine this by creating a particle inside another particle. The way we do this is by creating a relation between the inner particle and the outer one that people don’t usually use in physics, but which is very easy to do with networks.
Let’s call the inner particle the ant. The ant is always racing about at fixed speed. The outer particle, we’re going to call the pipe-elf. The job of the pipe-elf is to make sure that the ant has something to walk on (some wriggle-room). Whenever the ant reaches the front of the pipe, the pipe-elf builds a new piece of pipe and sticks it on the front so that the ant has somewhere to go.
At each time-step in our simulation, the ant either reaches the front of the pipe, or it does not. If it doesn’t reach the front, the elf has some time on his hands. He can do things like receive phone-calls or clear up the old bits of pipe he’s left lying around. However, while the ant is keeping him busy, doing these things is impossible.
Now, let’s think about the different possible paths the ant can take. If it’s travelling straight down the pipe, the elf will never have any free-time. He’s going to be building new pipe-segments as fast as he can. However, if the ant is just racing around and around near the front of the pipe like a hamster on a wheel, the elf can do whatever he likes. He has all the time in the world. In other words, so far as the elf is concerned, he’s either experiencing lots of free time, moving very fast, or something in between.
Let’s call the phone-calls that the elf gets photons, or messenger particles. Let’s call the amount of old pipe left hanging about the relativistic mass of the particle. And let’s say that the ant is the one who’s really in charge. Stopping this particle means you have to find and bump into the ant. When you do that, and only then, you collapse all the elf’s pipe-segments down on top of you. Unless you meet the ant, the pipe sections are like so much smoke. You can walk through them without knowing that they’re there.
This pretty much covers the bases of what we need for special relativity. The set of angles that the ant can walk at exactly corresponds to the set of possible directions we might need to cover to model special relativity. The ant is a particle constrained by its context, just as for the Higgs field, and so travelling on a helical path. The only wacky thing here is the notion that the elf can only interact with the rest of the universe when it’s not building pipe segments. But that nicely covers the relation between velocity and time. And we don’t need a special network for the ant-elf pair to travel around on. A perfectly ordinary spatial network will do.
Hence, we can imagine a universe filled with lengths of invisible, untouchable pipe arcing through the void, each filled with whizzing ants. Do I think that the universe actually looks this way? No. This isn’t a theory, it’s a model. But what it does give us is the behavior described by special relativity happening against a discrete background, without a hair of Minkowski space in sight.
Not everyone may be cheering just yet, I admit. Anyone familiar with special relativity may in fact be writing in their chair by now because I haven’t mentioned Lorentz-contraction–the effect that special relativity has on distance. The way that we’re used to thinking about relativity, the length of objects in their direction of travel is affected just as much as the time they experience.
But this omission is on purpose. In this model, you don’t need Lorentz-contraction. It’s not there. That may sound counter-intuitive, but I assure you, the math works out. The observed contraction is the same. And the quantization of the background doesn’t even give you any problems when you change reference frame. Next time, I’ll try to explain why. I may even get round to telling you how quantum mechanics might fit in this picture.