The Simplicial Ricci Tensor

TL;DR

This paper introduces a geometric discretization of the Ricci tensor using Regge calculus, enabling higher-dimensional discrete Ricci flow applications and providing a new foundation for numerical relativity and geometric analysis.

Contribution

It provides the first geometric discretization of the Ricci tensor within Regge calculus, applicable to higher dimensions and dual lattices, advancing numerical methods in geometric and gravitational research.

Findings

Derived a new edge-based expression for Ricci tensor

Proved equivalence of Ricci tensor on simplicial and dual lattices

Applied Ricci tensor to derive Einstein tensor in arbitrary dimensions

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.