Idea alert! Wouldn't it be nice, if SmoothLife were Lorentz invariant?
Like Konrad Zuse said in his paper "rechnender Raum" (calculating space), the first objection from physicists against cellular automaton based physics is always, that it isn't Lorentz invariant. But then, he says, we have to think of ways how we could MAKE it Lorentz invariant, and not simply write it off. See, that's the attitude I like. Maybe now with the SmoothLife model where we have a PDE, and no grid (at least theoretically), this would help to make a step in this direction.
So the question is: how can an equation of the form
d_t f(x,t) = S f(x,t) (1)
be made Lorentz invariant?
Well, here is one idea, I had: think of the Maxwell equations where we too have an ordinary time derivative on the left hand side and something else on the right hand side. So what's the trick, why are the Maxwell equations Lorentz invariant, and (1) is not? The trick is that we have another Maxwell equation with a time derivative on the left hand side and a different "something else" on the right hand side. And these two "something else"s can transform into each other! So maybe if we replicate this structure we could get Lorentz invariance. That was my idea today. We would have something like
d_t f(x,t) = S g(x,t) and
d_t g(x,t) = T f(x,t)
i.e. two fields instead of only one. And maybe two different nonlinear operators. (But maybe as well only one, who knows: S=T ?) Maybe we would have to make f and g complex or vectors. But we would still get gliders with the right nonlinear operators S and the right values for the snm function. Maybe these gliders then would have a phase just like photons. Imagine that, gliders with a phase! You think I'm crazy? Well, actually there's a whole chapter in a book already about this topic: "Game of Life Cellular Automata" (Adamatzky). They look at a quantum mechanical phase for the cells of a cellular automaton. But relativistic cellular automata? Has anyone ever considered this?
As you know, SmoothLife on a sphere works, as there is a metric involved. It can be used for correcting the distortion of square arrays when projected onto a sphere. This underlying metric could be coupled to the field. Or to emphasize it more: it could be MADE coupling to the field. So this sounds like general relativity even. This gives me hope somehow that there could be something going in this direction. There could be length contraction of the disk and ring kernels for example. In case of something moving with the speed of light these shapes could even be flattened, i.e. have one dimension less, and still be meaningful, as you can see with SmoothLife working equally well in 3D, 2D and even 1D.
So, all in all, for me, there are so many tantalizing hints that it could work, that I just can't stop thinking about it. But, as always, having the idea is the easy part. Now there would be some work to do. And I have done nothing as of yet, I'm just talking out of the blue here. I'm feeling a bit tired. There are so many other things to do, so many other ideas.
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