But what about an explicit particle of the kind we discussed in the previous section that exists as a structure in a network? Given two nodes in a network, one can always identify a shortest path from one to the other that goes along a sequence of individual connections in the network. But in a sense a structure that corresponds to a particle will normally not fit through this path. For usually the structure will involve many nodes, and thus typically require many connections going in more or less the same direction in order to be able to move across the network.
But if one assumes a certain uniformity in networks—and in particular in the causal network—then it still follows that particles of the kind that we discussed in the previous section will tend to move along geodesics. And whereas in traditional general relativity the idea of motion along geodesics is essentially an assumption, this can now in principle be derived explicitly from an underlying network model.
One might have thought that in the absence of matter there would be little to say about gravity—since after all the Einstein equations then say that there can be no curvature in space, at least of the kind described by the Ricci tensor. But it turns out that there can still be other kinds of curvature—described for example by the so-called Riemann tensor—and these can in fact lead to all sorts of phenomena. Examples include familiar ones like inverse-square gravitational fields around massive objects, as well as unfamiliar ones like gravitational waves.
But while the mathematical structure of general relativity is complicated enough that it is often difficult to see just where in spacetime effects come from, it is usually assumed that matter is somehow ultimately required to provide a source for gravity. And in the full Einstein equations the Ricci tensor need not be zero; instead it is specified at every point in space as being equal to a certain combination of energy and momentum density for matter at that point. So this means that to know what will happen even in phenomena primarily associated with gravity one typically has to know all sorts of properties of matter.
But why exactly does matter have to be introduced explicitly at all? It has been the assumption of traditional physics that even though gravity can be represented in terms of properties of space, other elements of our universe cannot. But in my approach everything just