cone and asks how the number of events that lie within the cone increases as one goes to successive slices. After t steps, the number of events reached will be proportional to td. But there is then a correction proportional to td + 2, that has a coefficient that is a combination of the spacetime Ricci scalar and a projection of the spacetime Ricci tensor along what is in effect the time direction defined by the sequence of spacelike slices chosen.
So how does this relate to general relativity? It turns out that when there is no matter present the Einstein equations simply state that the spacetime Ricci tensor—and thus all of its projections—are exactly zero. There can still for example be higher-order curvature, but there can be no curvature at the level described by the Ricci tensor.
So what this means is that any causal network whose behavior obeys the Einstein equations must at the level of counting nodes in a cone have the same uniform structure as it would if it were going to correspond to ordinary flat space. As we saw a few sections ago [12, 13], many underlying replacement rules end up producing networks that are for example too extensively connected to correspond to ordinary space in any finite number of dimensions. But I suspect that if one has replacement rules that are causal invariant and that in effect successfully maintain a fixed number of dimensions they will almost inevitably lead to behavior that follows something close to the Einstein equations.
Probably the situation is somewhat analogous to what we saw with fluid behavior in cellular automata in Chapter 8—that at least if there are underlying rules whose behavior is complicated enough to generate significant effective randomness, then almost whenever the rules lead to conservation of total particle number and momentum something close to the ordinary Navier–Stokes equation behavior emerges.
So what about matter?
As a first step, one can ask what effect the structure of space has on something like a particle—assuming that one can ignore the effect of the particle back on space. In traditional general relativity it is always assumed that a particle which is not interacting with anything else will move along a shortest path—or so-called geodesic—in space.