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A Discrete Representation of Einstein's Geometric Theory of Gravitation: The Fundamental Role of Dual Tessellations in Regge Calculus by McDonald, Jonathan R.; Miller, Warner A. is a scholarly article available to read on EtoBox.

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In 1961 Tullio Regge provided us with a beautiful lattice representation of Einstein's geometric theory of gravity. This Regge Calculus (RC) is strikingly different from the more usual finite difference and finite element discretizations of gravity. In RC the fundamental principles of General Relativity are applied directly to a tessellated spacetime geometry. In this manuscript, and in the spirit of this conference, we reexamine the foundations of RC and emphasize the central role that the Voronoi and Delaunay lattices play in this discrete theory. In particular we describe, for the first time, a geometric construction of the scalar curvature invariant at a vertex. This derivation makes use of a new fundamental lattice cell built from elements inherited from both the simplicial (Delaunay) spacetime and its circumcentric dual (Voronoi) lattice. The orthogonality properties between these two lattices yield an expression for the vertex-based scalar curvature which is strikingly similar to the corresponding and more familiar hinge-based expression in RC (deficit angle per unit Voronoi dual area). In particular, we show that the scalar curvature is simply a vertex-based weighted averag

Author

McDonald, Jonathan R.; Miller, Warner A.

Published

2008

Language

EN