The Simplicial Ricci Tensor
Paul M. Alsing, Jonathan R. McDonald, Warner A. Miller

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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
