From: baez@rosencrantz.stcloudstate.edu () Subject: Re: Asymptotics of 6j symbols Date: 11 Sep 2000 23:22:59 GMT Newsgroups: sci.physics.research Summary: [missing] In article <8pfoa0$2fm$1@nnrp1.deja.com>, wrote: >What is the original Regge calculus about? Take spacetime and chop it into 4-simplices - which are like tetrahedra, only 4-dimensional. Instead of using a metric tensor, describe the geometry of spacetime by assigning lengths to the edges of these 4-simplices. Then take the action for general relativity and figure out an analogous expression that you can compute starting from the edge-lengths of these 4-simplices. That's the Regge calculus in a nutshell. For more details go back and reread "week120": ..................................................................... In 1961 Regge came up with a discrete analog of the usual formula for the action in classical general relativity. It works best in signature ++++. His formula applies to a triangulated 4-manifold whose edges have specified lengths. In this situation, each triangle has an "angle deficit" associated to it. It's easier to visualize this two dimensions down, where each vertex in a triangulated 2-manifold has an angle deficit given by adding up angles for all the triangles having it as a corner, and then subtracting 2 pi. No angle deficit means no curvature: the triangles sit flat in a plane. The idea works similarly in 4 dimensions. Here's Regge's formula for the action: take each triangle in your triangulated 4-manifold, take its area, multiply it by its angle deficit, and then sum over all the triangles. Simple, huh? In the continuum limit, Regge's action approaches the integral of the Ricci scalar curvature --- the usual action in general relativity. For more see: T. Regge, General relativity without coordinates, Nuovo Cimento 19 (1961), 558-571.