Spacetime as a Network, and Entropy as a Network
Dr. Stephen Wolfram is perhaps best known as the CEO of Wolfram Research, the company that runs Wolfram Alpha and Mathematica. However, in his spare time, Dr. Wolfram studies physics. Dr. Wolfram has developed quite an interesting idea, which is that space could be simply a network and that all of the laws of physics could be simply emergent behavior of a set of simple network update rules. This network evolution, he claims, could actually be what underpins the universe’s behavior. That would mean particles are just specific local network structures, and condensed matter is just structures built from many of those smaller local structures. For the update rules, the location where the update rules start must not change the outcome of the update. In other words, if you apply your update rules from location A, it must have the same affect as starting the update from location B in the network or else the universe would have a sort of “center” from which all laws of physics propagate. Because of this so called “causal invariance” rule, which is necessitated by any network model of spacetime, special relativity results, according to Dr. Wolfram. It is possible to determine the effective dimension of a network by looking at the number of nodes within r connections of that node. The way the number of nodes within r connections scales with the dimension so a 2 dimensional network would have the number of connections scale as the square of r, just like area scales with the square of its extent in continuous space. If one assumes that the dimension of the network for large r limits to the correct value (which can be defined as true since the point is to build a network model of physics), and that there is statistical randomness at micro scales, then general relativity is also implied by a network model, or at least it does at large scales. This is an incredibly powerful result because it means one of the greatest triumphs of physics could be simply the result of a much more fundamental rule about edge evolution in a network. To get the network model to correctly describe all of physics, not just general relativity, the correct update rules must be found within the constraints already described, if such a set of rules exist. So far, no such rules have been found but it has not been proven that such a set of rules do not exist. A network model of spacetime is perhaps still very much within the realm of speculation, but it does show quite a bit of promise.
Here you can see how connectivity can help determine local and limiting dimension
Here you can see the affect of curvature
Networks can also be used to make a fun little model of time. I often ask “what is time, and why does it move monotonically?” Generally the idea of entropy is induced in order to break the symmetry of time, but at its most fundamental, entropy is an argument about the statistics of observable behavior and unknowable states, and it still relies on time itself to work even if it does do a good job of breaking its symmetry. Thus the argument becomes somewhat circular. Lets expand that argument given our understanding of networks. Imagine a phase space plot of dimension equal to the number of states required to fully describe our observable universe. At every location in that phase space where the universe could be, we put a node. Due to [Frampton 2008], we can say the entropy of the universe is about ~10^102. That corresponds to ~exp(10^125) micro states, and thus that is approximately the number of nodes in the plot. Now draw an edge between all nodes that correspond to states that can be reached in one “update rule increment”. According to the laws of physics, not every node can be reached from every other node, so the graph is not complete. Because of the symmetries of the universe, the graph isn’t even connected, although that isn’t importiant to our discussion. Now lets consider a random walk on that graph. A random walk should not violate any laws of physics because we already defined the edges of the graph to be such that only nodes that are connected by the laws of physics are immediately accessible from any one node. The randomness of the walk would presumably come from the fundamental unknowingness of quantum states, although I am trying to avoid overcomplicating the model at first. A path defined by the random walk would thus define the flow of time. Re-tracing one’s steps would be like having time reverse and nearly re-tracing one’s steps would also appear to be like reversing time as well (since very close nodes in phase space correspond to very close observable states of the universe), even if it wasn’t a perfect reversal. However, neither of these situations are likely since the statistically averaged net flow of random walks on a network are determined by the number of edges leading in difforent directions. Since every node has lots of edges leading out of it and most nodes do not connect to most other nodes, even ones close by in phase space, it is essentially certain that the random walk will lead away from the point of origin, and it is exceedingly unlikely that any given node will be passed through more than once (corresponding to a closed loop “groundhog day” type scenario), especially since it is exceedingly unlikely that any node will be passed through in the first place (this is slightly difforent to the random walk on a surface but there is some correspondance). If the universe worked in this way, asking about the arrow of time would mean nothing, and time itself might not even have much meaning because at best it would be just the path the random walk took. Assuming there are no flaws in this understanding of time, on of the only remaining mysteries about time would be what caused the sort of “update increments”, although one should notice that the update increments don’t have to correspond to the passage of some other more fundamental time. This understanding of time is quite flawed, but I believe it does offer some possibilities for where time comes from (statistics of random walks on a network of specific topology) and what time fundamentally means (little or nothing).
http://blog.stephenwolfram.com/2015/12/what-is-spacetime-really/
https://arxiv.org/pdf/0801.1847.pdf