Jason (jcreed) wrote,

You don't have to be Stephen Wolfram to recognize that discrete models of physics are pretty cool. (Of course this is a common criticism of Wolfram...)

Ari Stern looks like he has done some cool things with them. Despite the fact that I currently totally don't understand GR, I have a feeling that maybe some day I might at least understand it a little bit from the angle of the Regge Calculus, which got invented nearly 50 years ago in a paper by Tullio Regge. I found Friedberg and Lee's "Derivation of Regge's action from Einstein's theory of general relativity" (behind an academic paywall, sorry) to be pretty helpful too.

The basic idea is it replaces Einstein's field equation (or, equivalently, the action minimization principle it corresponds to) with an action on 4-d graphs with their edges labelled by lengths. And it's rather simple: the Regge action is the sum (over all triangles in the graph) of the triangle's area times its "angle deficit". Trouble is I don't presently know how to compute (or more importantly, how to think about) the angle deficit of a triangle in a 4d lattice, although I certainly do know the angle deficit of a point in 2d lattice: look at a polyhedron, look at a vertex, count up the angles around it, and measure how much they fall short of 360.

Anyway the punchline is supposed to be that the concept angle deficit corresponds to the sqrt(|g|) R (where g is the metric tensor and R the Ricci tensor I think?) that you integrate to get the Einstein GR action. So I find this very encouraging, that I only need to generalize to 4d a single concept that I already understand, instead of grokking what the metric and Ricci tensors are, and what the determinant of the metric tensor really means, etc. etc.

eta: This looks like a less ancient paper by Regge.
Tags: physics

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