Riemann Normal Coordinates, Smooth Lattices and Numerical Relativity

TL;DR

This paper introduces a novel lattice-based numerical relativity scheme that applies Einstein's equations directly using Riemann normal coordinates, achieving second-order accuracy and emphasizing the importance of Bianchi identities.

Contribution

The paper presents a new method combining lattice data with Riemann normal coordinates to directly solve Einstein's equations in numerical relativity.

Findings

Achieves second-order accuracy in metric and curvature estimates.

Successfully applies to Schwarzschild initial data.

Highlights the role of Bianchi identities in data construction.

Abstract

A new lattice based scheme for numerical relativity will be presented. The scheme uses the same data as would be used in the Regge calculus (eg. a set of leg lengths on a simplicial lattice) but it differs significantly in the way that the field equations are computed. In the new method the standard Einstein field equations are applied directly to the lattice. This is done by using locally defined Riemann normal coordinates to interpolate a smooth metric over local groups of cells of the lattice. Results for the time symmetric initial data for the Schwarzschild spacetime will be presented. It will be shown that the scheme yields second order accurate estimates (in the lattice spacing) for the metric and the curvature. It will also be shown that the Bianchi identities play an essential role in the construction of the Schwarzschild initial data.