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Definition of lightlike separated events in the Wolfram model?

Posted 25 days ago
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Causal Cones are well defined in the technical introduction and in Gorard's paper on relativity: given an event (or a set of events) in the causal graph, its causal cone is the set of all the events that can be connected to it by a directed path in the causal graph. Causal cones in the Wolfram model can be directly related the concept of light cones in the theory of relativity, and this is nice.

In relativity there is a distinction between the interior of the cone, where the events are said timelike separated, and the surface of the cone, where the events are said to be lightlike separated, although still causally connected. I cannot find the same distinction in the Wolfram model.

Gorard's paper on relativity defines lightlike separated events using a discrete Minkowski norm defined on a "Minkowski lattice", but, as I have pointed out in this post, a Minkowski lattice may not be suited for the purpose because the spacetime separation of two events would be dependent on the embedding.

Even the more recent Gorard's paper Algorithmic Causal Sets and the Wolfram Model isn't (in my opinion) satisfactory regarding the concept of lightlike separation:

Two updating events may consequently be said to be causally related (i.e. connected by a directed path in the causal network) if and only if they are either lightlike-separated or timelike-separated (with lightlike separation corresponding to directed paths which lie on the boundary of discrete light cones, and all other such paths yielding timelike separations)

The problem is that I can't find an obvious way to define the boundary of a discrete light cone. To clarify what I mean, I copy here figure 8: light cone

the caption says:

[...] the discrete future light cone [...] of a given updating event in the causal network [...] the two lightlike paths are highlighted in green and blue, with all other (timelike) paths shown in red.

It seems to me that the fact that the blue and green paths are on the boundary, is only due to the particular way the graph is drawn. I find no reason for which the two paths shown must necessary lie on the boundary and thus be called lightlike.

I think that if we want the causal graph to have a lorentzian manifold as limiting structure, it is important to find a good correspondence with the concept of lightlike separation in the Wolfram model. Are you aware of any way to properly define lightlike separation of events in a causal graph?

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At the beginning of the Wolfram Physics Project, there was the following identification, corresponding to the assumption that spacetime is as Minkowski described it (a continuous manifold),

light cones: causal cones in the causal graph

link

but later S. Wolfram developed a subtle distinction of these concepts in his theory of faster-than-light travel (this theory predicts that the interior of black holes may interact with the exterior environment via faster-than-light interactions across microscopic space tunnels... but this is another subject):

The “causal cone” of affected events is very well defined. But now the question is: how does this relate to what happens in space and time?

When one thinks about the propagation of effects in space and time one typically thinks of light cones. Given a light source somewhere in space and time, where in space and time can this affect?

And one might assume that the causal cone is exactly the light cone. But things are more subtle than that. The light cone is normally defined by the positions in space and time that it reaches. And that makes perfect sense if we’re dealing with a manifold representing continuous spacetime, on which we can, for example, set up numerical coordinates. But in our models, there’s not intrinsically anything like that. Yes, we can say what element in a hypergraph is affected after some sequence of events. But there’s no a priori way to say where that element is in space. That’s only defined in some limit, relative to everything else in the whole hypergraph.

link

Therefore, if I am not mistaken, the notion of light-like separated events is extrinsic, depending on the embedding of the causal graph into a Minkowski space. Indeed, in Tommaso Bolognesi's paper Algorithmic Causal Sets for a Computational Spacetime, page 4, we can see an extrinsic definition of light-like separated events. This is the reason why faster-than-light interactions are possible in the Wolfram Model without violating special relativity, although they may be microscopic in practice (the detection of such interactions in a particle accelerator or a black hole would be evidence for the Wolfram Model).

I am not particularly knowledgeable in this part of the Wolfram Model, but I had some ideas which may help define light-like, space-like and time-like events, while avoiding any imbedding (I suspect some of them may be already be present in Gorard's paper, but I haven't read all of it).

In special relativity, events inside the light cone can be reached by several light-like curves, while events on the boundary can be reached by only one light-like curve. The light-like events in WM could be defined in a similar way: two events A and B are light-like connected, if there exists only one unidirected causal chain connecting them (A->...->B or B->...->A) (two paths are considered equal, if they contain the same sequence of nodes). Similarly, two events are space-like connected, if there is no unidirected chain that would connect them; and two events are time-like connected, if there are more than one unidirected chain connecting them.

Do you think this would work?

Thank you both for sharing your thoughts. I think I like the idea of a light cone that emerges fully only when taking the limit of a big causal graph embedded in a manifold.

Pavlo, interesting idea. I hadn't thought about that. However it may have some issues, because the causal graph could become rapidly so connected that there are no lightlike paths traversing it from top to bottom. For example, looking at the picture from Gorard's paper, there is only one lightlike path connecting the first event of the causal graph with the bottom events (the one I have drawn in violet) lightlike paths

In other words, only one of the bottom events can be reached in only one way. I suspect that doing some further step in the evolution, it may happen that no lightlike path traversing the whole graph exists anymore. This will definitely be the case if the system is confluent.

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