The eth formalism in numerical relativity

TL;DR

This paper introduces a finite difference eth formalism using overlapping stereographic patches to handle tensor fields in spherical coordinates, avoiding singularities and improving numerical relativity simulations.

Contribution

It develops a new computational method employing overlapping patches and interpolation for tensor fields in spherical coordinates, enhancing numerical relativity tools.

Findings

Successfully tested with wave evolution in 3D

Calculated curvature scalar on spherical manifolds

Revealed new features of gravitational waveforms

Abstract

We present a finite difference version of the eth formalism, which allows use of tensor fields in spherical coordinates in a manner which avoids polar singularities. The method employs two overlapping stereographic coordinate patches, with interpolations between the patches in the regions of overlap. It provides a new and effective computational tool for dealing with a wide variety of systems in which spherical coordinates are natural, such as the generation of radiation from an isolated source. We test the formalism with the evolution of waves in three spatial dimensions and the calculation of the curvature scalar of arbitrarily curved geometries on topologically spherical manifolds. The formalism is applied to the solution of the Robinson-Trautman equation and reveals some new features of gravitational waveforms in the nonlinear regime.