I finally understand how a discrete model of spacetime can be Lorentz invariant.
To see the problem, note that a spacetime divided into little Planck-length parts would look different in a boosted reference frame — the lengths would differ. (A proposed workaround, “doubly-special relativity,” has been severely challenged at a very basic level.)
A recent review gives the clearest explanation I’ve seen of “what a causal set is and how it succeeds in being both discrete and Lorentz invariant.” Causal sets are part of the ongoing search for a theory of quantum gravity that explains spacetime, rather than assuming it.
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Hmm… do you think this makes Wolfram’s New Kind of Science slightly more plausible? I’d always dismissed his idea of cellular automata physics in the past, primarily because I couldn’t work out how to get Lorentz invariance out of it.
I’m not a professional physicist though, so my intuitions may be wonky.
I noticed that causal sets have inherent relative frequency ratios, and that these can serve physics as energy ratios, in accord with Planck’s E=hf. In that case, the causal link is the quantum of energy. I’ve reconstructed physics on that basis, so that the most common particles are modeled as causal sets, and their mass ratios to one another are calculable. The other causal set theorists are unaware of this development, due to a certain lack of faith in the meaning of a standalone causal set. — Carey
on this subject you might find my recent book ‘causal space-time’ (available on amazon.com) interesting. quantum mechanics (including the exact Dirac equation) and relativity arise from the simplest causal network.
Richard D. Bateson
UCL, London