But first we must consider the concept of motion.

To say that one is not moving means that one imagines one is in a sense sampling the same region of space throughout time. But if one is moving—say at a fixed speed—then this means that one imagines that the region of space one is sampling systematically shifts with time, as illustrated schematically in the simple pictures below.

But as we have seen in discussing causal networks, it is in general quite arbitrary how one chooses to match up space at different times. And in fact one can just view different states of motion as corresponding to different such choices: in each case one matches up space so as to treat the point one is at as being the same throughout time.

Motion at a fixed speed is then the simplest case—and the one emphasized in the so-called special theory of relativity. And at least in the context of a highly regular causal network like the one in the picture on page 518 there is a simple interpretation to this: it just corresponds to looking at slices at different angles through the causal network.

Successive parallel slices through the causal network in general correspond to successive states of the underlying system at successive moments in time. But there is nothing that determines in any absolute way the overall angle of these slices in pictures like those on page 518. And the point is that in fact one can interpret slices at different angles as corresponding to motion at different fixed speeds.

If the angle is so great that there are connections going up as well as down between slices, then there will be a problem. But otherwise it will always be the case that regardless of angle, successive slices must correspond to possible evolution histories for the underlying system.

One might have thought that states obtained from slices at different angles would inevitably be consistent only with different sets of underlying rules. But in fact this is not the case, and instead the exact same rules can reproduce slices at all angles. And this is a consequence of the fact that the substitution system on page 518 has the property of causal invariance—so that it gives the same causal network independent of the scheme used to apply its underlying rules.

It is slightly more complicated to represent uniform motion in causal networks that are not as regular as the one on page 518. But Graphical representation in space and time of motion at fixed speeds.

From Stephen Wolfram: A New Kind of Science [citation]
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