Lorentz Invariance

In my last post, I showed off an algorithm that could create nicely irregular networks with integer dimension two, without using geometric information. In other words, I made lumpy spheres.

While I’m proud of this result, it doesn’t look much like the kind of spacetime networks that are used in almost all branches of discrete physics. That’s because the dimension of time is missing. And, as anyone who’s read a little Einstein will tell you, time and space are part of the same thing. They can’t really be uncoupled.

Except, of course, they can be uncoupled. It’s dead easy. It’s just that for most of the math that physicists do, it makes more sense to wedge them together.

I’ve briefly outlined in previous posts the way in which space and time can be unpacked from each other. However, after a fun conversation with a very math literate friend the other day revealed to me that I haven’t really done a good enough job of explaining. In this post, I’m going to try to set that straight.

Perhaps the easiest way of describing the relation between space and time that Einstein uncovered is using Minkowski space. In other words, space and time are connected by the following equation:

s^2 = t^2 – x^2 – y^2 – z^2

Where t is time, x, y, and z are the familiar dimensions of space, and s is the ‘spacetime interval’. What this relation says is that if someone is moving toward or away from you, how fast they’re doing it is going to affect how you both perceive time to be passing.

To illustrate this, we can invoke a classic example from science fiction. If you get on a spaceship and make a very fast trip to a distant star (let’s call it Distantia) and back at almost light speed, when you return almost no time will have passed for you. However, for us, it will seem as if decades have gone by. What matters here is who does the accelerating. If you go to Distantia, and a week later I join you there, then we will both perceive the same amount of elapsed time.

What’s chewy about this is that we know from experiment that we have to treat all reference frames as the same. Consider the following scenario: we discover a very fast moving planet in the heavens–called Speedia. Aliens from Speedia decide to visit the Earth. They show up, chat for a while, and then head home. From the perspective of the people of Speedia, the same relation should hold as did for our trips to Distantia. In other words, their travelers should stay young while the homebodies get creaky. (They should also see themselves as still, and everyone else as travelling fast.)

The way to fix this is to invoke some high-school algebra and move the terms in our equation about. In other words, we reframe the relation between space and time as follows:

t^2 = s^2 + x^2 + y^2 + z^2

At first, it’s not clear that we’ve done anything here. It’s the same formula as before, just the other way around. But let’s ask ourselves what the terms in this expression actually mean. What is a spacetime interval? What is the Minkowski relation actually saying?

It’s saying that for a pair of travelers approaching me from the same place, that the subjective experience of time needed to reach me will depend on how fast they’re going. In other words, what the spacetime interval defines is the experience of subjective time for those travelers, as taken from my perspective.

For relativity specialists out there, this may seem obvious. It may seem like I haven’t said anything yet. But here’s the kicker–once you’ve framed things this way round, you can pick a frame of reference for t and describe all the others in terms of it. In other words, so long as we have a way of encoding distance traveled in the s direction and if we maintain a fixed relation between the distance traveled in the s direction and the distance travelled in x, y, or z, we can describe everything in ordinary Euclidean space. (Note that the fixed relation here is key!)

Another way to think about this is that by turning the normal formula for spacetime around, we’ve created an external reference frame. Let’s call it Father Christmas’s reference frame. Nobody in the universe has access to FC’s frame. As far as they’re concerned, Minkowski space works are usual. All frames of reference are still equal and the math is exactly the same. Only FC can see this special view of the universe, which is handy as he needs to visit a very large number of chimneys very quickly and surprise everyone at Christmas.

The awesome thing for Father Christmas is that the universe has an unambiguous, objective geometry that encompasses everything that’s going on, and has this natty extra dimension s. For FC, creating a discrete model of spacetime is a breeze. He just divides everything up into a locally connected network. End of problem.

But hang on, we can’t see a direction s. And we haven’t detected it in any experiments! So how can I claim that this is a solution to the problem of relativity? It turns out that the physicists solved this for us years ago when they came up with something called Kaluza-Klein theory. The trick is that we roll the s direction up very tight into a little circle so that it’s invisible to us, but important at small scales. Sound familiar? It should do, this is exactly the trick that they use to make String Theory work. In fact, there’s exactly nothing new here. What I’m describing is old physics. If you can’t believe in a compact direction for s, you have to throw String Theory away too!

From a discrete physics perspective, this trick is super-useful. This is because it means that so long as an object can only travel a fixed distance with each objective time step, special relativity will hold as long as we add a hidden Euclidean dimension. I’ve tested this and it works. For those of you who like videos, here’s a small demonstration. The flashing of each blob represents the time its experiencing. Note how slow blobs flash quickly, and fast blobs hardly flash at all. If you make the blobs send messages to each other at the speed of light, everything pans out just as Einstein would have predicted. (The video is a special superluminal Christmas treat, because you’re viewing it from FC’s reference frame.)

Note that this only works for simulations that are isotropic (the same in all directions). This means that, unless you’re being super clever, the same trick can’t work for cellular automata.

So where does this leave us? With a really nice tool for making thorny spacetime problems go away. However, we still need to build networks that have the extra magic direction s, and it still needs to magically relate to subjectively experienced time. The network we started off doesn’t have that direction, and we don’t have a way to encode the experience of particles, so lots more work is needed. But in the next post in this series, I’ll show you how to pull these tricks off too.

(By the way, if this post still doesn’t make a shred of sense, somebody please let me know and I’ll try again.)

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