We know that the Wolfram model rules do not depend on the event ordering function. In particular, that means that they are not affected by the choice of a foliation of a causal graph, or, in other words, by a reference frame.
That, in turn, would imply Lorentz invariance of the rules as a special case of the invariance under any change of the reference frame.
However, the invariance of the rules under a coordinate-frame change does not imply that the evolution they generate is invariant as well (one might say the init is not invariant, although it's not clear what it means given that it usually consists of a single expression). And we are yet to find a Lorentz-invariant evolution.
This issue concerns with finding such an evolution.
Possible solution
Let's say we have constructed a causal graph together with its layout (in 2D, x- and t-coordinates corresponding to each vertex). If the graph is Lorentz invariant, any distribution over its properties will have to be invariant under boosts.
Suppose then the vertex out degrees are finite (and all the same, which would be the case in the Wolfram model). Also, suppose that there is a nonzero probability the outgoing causal edges are timelike (i.e., not lightlike). In that case, there will always be an average "velocity" of the outgoing edges, and boosting is going to change it, which means this kind of graph will not be Lorentz invariant.
Therefore, all causal edges have to be lightlike. What matters then is the distribution of their lengths. This distribution must not change after boosting. In other words,
p(t') dt' = p(t) dt, where dt=dx is the distance of the causal edges, and a prime corresponds to boosted coordinates.
Solving this equation, we get p(t) = const / t, which, of course, diverges. So, we do need a cutoff. It diverges as a log(t), though, so the cutoff can be large (and, inversely, the other cutoff, small).
If we do introduce a cutoff, it seems reasonable we can construct a causal graph with this distribution (although I didn't actually do it yet, so I cannot be sure).
All of this does not apply to the multiway causal graph, where outdegrees can be infinite in Wolfram models. This, in fact, seems to be a more promising direction to investigate, given that we expect models corresponding to physics to be multiway.
The problem
We know that the Wolfram model rules do not depend on the event ordering function. In particular, that means that they are not affected by the choice of a foliation of a causal graph, or, in other words, by a reference frame.
That, in turn, would imply Lorentz invariance of the rules as a special case of the invariance under any change of the reference frame.
However, the invariance of the rules under a coordinate-frame change does not imply that the evolution they generate is invariant as well (one might say the init is not invariant, although it's not clear what it means given that it usually consists of a single expression). And we are yet to find a Lorentz-invariant evolution.
This issue concerns with finding such an evolution.
Possible solution
Let's say we have constructed a causal graph together with its layout (in 2D, x- and t-coordinates corresponding to each vertex). If the graph is Lorentz invariant, any distribution over its properties will have to be invariant under boosts.
Suppose then the vertex out degrees are finite (and all the same, which would be the case in the Wolfram model). Also, suppose that there is a nonzero probability the outgoing causal edges are timelike (i.e., not lightlike). In that case, there will always be an average "velocity" of the outgoing edges, and boosting is going to change it, which means this kind of graph will not be Lorentz invariant.
Therefore, all causal edges have to be lightlike. What matters then is the distribution of their lengths. This distribution must not change after boosting. In other words,
p(t') dt' = p(t) dt, where dt=dx is the distance of the causal edges, and a prime corresponds to boosted coordinates.
Solving this equation, we get p(t) = const / t, which, of course, diverges. So, we do need a cutoff. It diverges as a log(t), though, so the cutoff can be large (and, inversely, the other cutoff, small).
If we do introduce a cutoff, it seems reasonable we can construct a causal graph with this distribution (although I didn't actually do it yet, so I cannot be sure).
All of this does not apply to the multiway causal graph, where outdegrees can be infinite in Wolfram models. This, in fact, seems to be a more promising direction to investigate, given that we expect models corresponding to physics to be multiway.