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Is causal relation dependent on planar embedding?

Posted 3 months ago
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I am stuck reading a passage from Jonathan Gorard's paper about relativity in the Wolfram model. At page 11, there is the definition of a "downward planar embedding".

Definition 11 A “downward planar embedding” is an embedding of a directed, acyclic graph in the Euclidean plane, in which edges are represented as monotonic downwards curves without crossings.

Then, he considers to embed a causal graph into the discrete “Minkowski lattice” $ \mathbb{Z}^{n,1}$ and to label the updating events according to their integer coordinates on the lattice.

There are many ways to embed a planar graph into the lattice, therefore I expect that these coordinates depend on the chosen embedding.

In the paper, follows the definition of the discrete Minkowski norm and the timelike/lightlike/spacelike separation of two events, and then, after a few sentences, the passage I cannot understand.

From our definition of the discrete Minkowski norm and the properties of layered graph embedding, we can see that a pair of updating events are causally related (i.e.connected by a directed edge in the causal graph) if and only if the corresponding vertices are timelike-separated in the embedding of the causal graph into the discrete Minkowski lattice $ \mathbb{Z}^{n,1}$, as required.

I don't understand how this statement could be true. Consider for example a simple causal graph constituted only of two events, where event 1 has a directed edge towards event 2 (i.e. event 1 is "the cause" of event 2). This graph is planar and so it is possible to perform a downward planar embedding in the Minkowski lattice $ \mathbb{Z}^{1,1}$ (just one spatial dimension to make things easier). Here I have drawn two possible embeddings of such graph.

enter image description here

enter image description here

In the first case, the events are labelled by the coordinates (0,0) and (1,0) and therefore their spacetime separation is -1 and they are timelike separated.

But in the second case, which is still a perfectly fair "downward planar embedding" by the definition given, the events have coordinates (0,0) and (1,3) and are therefore spacelike separated with norm 8.

The fact that the sign of the spacetime separation is dependent on the chosen embedding seems in contradiction with the statement quoted.

Can someone explain what am I missing here? Thank you

3 Replies

Before proceeding, I would like to mention two complementary materials to the paper that you are mentioned, where you can see significative a diversity of ideas concerning causality.

(1) Gorard, Jonathan. Algorithmic Causal Sets and the Wolfram Model. arXiv preprint arXiv:2011.12174 (2020). ,

(2) Max Piskunov's Bulletin Confluence and Causal Invariance.

The first remark is that in reference (1), which is basically a paper about causality, the phrases Minkowski lattice and planar embedding are not used anymore. We see some evolution in the development of the theory. Personally, I do not like Minkowski lattice.

A yes/no answer to your question is "no, causal relation does not dependent on planar embedding". Indeed, we could imagine a spacetime that is the surface of a torus, where one dimension is a one-dimensional space (circumference) and the other dimension is a cyclic time (circumference). Locally, this spacetime looks like ordinary spacetime and the local causal graph is acyclic. Nevertheless, the surface of a torus cannot be embedded in a plane and the global causal graph contains cycles.

Even considering this question only locally, the combinatorial structure of the (local) causal graph should remain invariant under the composition of the embedding with a Lorentz transformation: this is the essence of special relativity.

The core of the Wolfram Model is what was written by S. Wolfram in A New Kind of Science and in the Technical Introduction to the project and complemented in his writings. The ways in which S. Wolfram's ideas are mathematically formalized and explored by other researchers may change with time. Therefore, if you are interested in learning about causality in the Wolfram Model, my advice is to read S. Wolfram first, e.g.,

The Phenomenon of Causal Invariance

The Role of Causal Graphs

Event Horizons, Singularities and Other Exotic Spacetime Phenomena

Faster than Light in Our Model of Physics: Some Preliminary Thoughts

and then, after you are familiar with S. Wolfram's original ideas, you could read other authors, like references (1) and (2). This is just a personal opinion, other people may have other opinions.

Of course the answer is no, it would have been a problem if it were yes. Thank you for clarifying it.

I was indeed looking for a more formal exposition of Wolfram's ideas. I am not familiar with your first reference, thank you for telling me about it.

I would like to add to this thread mentioning that Gorard does similar imbedding in his paper on quantum mechanics, where the definition of branch-like or entanglement-like separation depends on the particular way one imbeds the multiway system graph.

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