Early this year, I won a book contract for a series of science fiction novels. I’m having a huge amount of fun with it, but am plagued by a peculiar anxiety. That people will have issues with my book to do with Lorentz Invariance.
My books are intended to be thinking person’s space opera. There are all those things that people enjoy about science fiction: starships, robots, alien worlds, etc. However, they’re also intended to be at least slightly realistic in the way that they deal with social and scientific themes. And one of the themes that’s used heavily in the books is warp drive.
No matter that SF writers have used warp drive for years, and no matter that the kind of warp drive I use is very similar to the sort that NASA is investigating right now. Still I am plagued with the notion that someone will call me out for apparent causality violations and thus consider the work implausible. Eyebrows will be raised. Readers will flee. Scorn will descend. Etc.
Is this the kind of neurotic thought process that happens when one spends years doing scientific research where you have to justify your every choice, who’s then segueing back into fiction? Absolutely. But here’s the thing: how many people think about Lorentz Invariance is just wrong, and how my books cover it is right and proper. I am filled with shining righteous glee on this subject because Lorentz invariance is a topic that I care about and have researched well beyond the limits of common sense.
The standard argument against faster than light travel goes something like this: travel faster than light in your reference frame and you’re going backward in time in someone else’s. Thus if you travel faster than light, you’ve broken causality. No. This is what drives me crazy. Wrong. Bad. A conclusion based on false assumptions. The person who believes this gets ten minutes in the naughty corner. With a novelty hat on.
This belief gets such a strong reaction from me because there are many science fiction writers who believe it is true. Including some very notable ones who have worked in physics. They pat themselves on the back for being science-savvy and diligently write books that preclude FTL. Gah!
It is true that if you travel faster than light, something about your experience of the universe breaks, but it doesn’t have to be causality. There is another, perfectly natural way that our experience of spacetime might change which is in perfect keeping with the math. It is this: travel faster than light, and you break Lorentz Invariance. In other words, all reference frames don’t look the same any more.
This is my preferred model, not only because it works, but because I think there’s evidence that this is what would actually happen. Why? For starters, there is one reference frame that Nature has pulled out and made screamingly special for us already: the one defined by the CMB. While this fact doesn’t interfere with how we do physics, it reveals that the observable universe started with a specific frame. Furthermore, there is no evidence that bits of the universe far away from us are traveling wildly, randomly fast compared to us, suggesting that the entire universe shares that same frame.
Given this, in order for Lorentz invariance to be strictly true, the vast majority of possible reference frames would have to be ones in which the universe hasn’t started yet and is totally flat, i.e.: two-dimensional. This is because no matter how close you get to the speed of light, you can always go closer. This means that for almost all possible frames, nothing can have possibly happened, as the duration of the universe to date is less than the Planck length. Can we honestly say that those frames exist if the universe hasn’t started in them yet?
Most of the available frames are ones that we could never even reach, because even if you totaled up all the energy in the universe and used it to push a single particle to some absurdly high speed, there would still be an endless spread of reference frames beyond it, all exactly equivalent and immaculately unreachable. Thus, even if you go for an infinite universe model such as eternal inflation, almost all possible frames will never be used. The local energy density at any point will never be high enough to make things pan out otherwise.
So the simple fact that the universe has a starting frame means that Lorentz invariance can only ever be measured to be locally true. It is also true that finite, discrete universe models (my favorites) only work if Lorentz invariance does not strictly hold. That’s true even if you build your discrete universe out of some nice Minkowski-metric compatible structure such as causal sets. Something can only be truly Lorentz invariant if it has infinite size, and exists for infinite time.
So given that strict Lorentz invariance is outlandish enough that we could never even prove that it held were it true, all possible models that can encompass local Lorentz invariance must be considered equally valid. Thus, holding physical reality to the absurd requirement of resembling Minkowski-space simply because it’s where we do most of the math that people are used to seems ludicrous to me.
There is a lovely upside to all this. While we have no evidence that anything in nature can go faster than light, there is also nothing in relativity that rules it out. Which means that NASA’s experiments with an Alcubierre drive may yet bear fruit. And that’s something worth being truly optimistic about.
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.