The Simplicial Ricci Tensor

Abstract

The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-linear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincare conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher-dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area -- an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimension.

19 pages, 2 figures

References in corpus (10)

• Ricci flow with surgery on three-manifolds
• Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
• Ricci flow and black holes
• Some Applications of Ricci Flow in Physics
• Entanglement entropy and the Ricci flow
• A geometric construction of the Riemann scalar curvature in Regge calculus
• Geometric flows and black hole entropy
• The Modelling of Degenerate Neck Pinch Singularities in Ricci Flow by Bryant Solitons
• A Kirchhoff-like conservation law in Regge calculus
• Ricci Flow of Biaxial Bianchi IX Metrics

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• Introducing Quantum Ricci Curvature
• Simplicial Ricci Flow
• A combinatorial Yamabe problem on two and three dimensional manifolds
• A cosmological solution of Regge calculus
• On the deformation of ball packings
• Combinatorial Ricci Curvature for Polyhedral Surfaces and Posets
• Distributed mean curvature on a discrete manifold for Regge calculus
• Approximate Killing symmetries in non-perturbative quantum gravity
• Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I
• Equivalence of Simplicial Ricci Flow and Hamilton's Ricci Flow for 3D Neckpinch Geometries
• Conformal variations and quantum fluctuations in discrete gravity