And essentially the reason for this is that—as I discussed above—each individual connection in any causal network must almost by definition represent propagation of effects at the speed of light. The overall structure of space that emerges may be complicated, and there may be objects that end up moving at all sorts of speeds. But at least locally the individual connections basically define the speed of light as a fixed maximum rate of propagation of any effect. And the point is that they do this regardless of how fast the source of an effect may be moving.
So from this one can use essentially standard arguments to derive all the various phenomena familiar from ordinary relativity theory. A typical example is time dilation, in which a fixed time interval for a system moving at some speed seems to correspond to a longer time interval for a system at rest. The picture on the next page shows schematically how this at first unexpected result arises.
The basic idea is to consider what happens when a system that can act as a simple clock moves at different speeds. At a traditional physics level one can think of the clock as having a photon of light bouncing backwards and forwards between mirrors a fixed distance apart. But more generally one can think of following criss-crossing connections that exist in some fixed fragment of a causal network.
In the picture on the next page time goes down the page. The internal mechanism of the clock is shown as a zig-zag black line—with each sweep of this line corresponding to the passage of one unit of time.
The black line is always assumed to be moving at the speed of light—so that it always lies on the surface of a light cone, as indicated in the top row of pictures. But then in successive pictures the whole clock is taken to move at increasing fractions of the speed of light.
The dark gray region in each picture represents a fixed amount of time for the clock—corresponding to a fixed number of sweeps of the black line. But as the pictures indicate, it is then essentially just a matter of geometry to see that this dark gray region will correspond to progressively larger amounts of time for a system at rest—in just the way predicted by the standard formula of relativistic time dilation.