Improved Cauchy-characteristic evolution system for high-precision numerical relativity waveforms

TL;DR

This paper improves the Cauchy-characteristic evolution method in numerical relativity to produce more accurate gravitational waveforms at null infinity, addressing gauge issues and enhancing numerical stability.

Contribution

It introduces a comprehensive gauge handling method, reformulates characteristic evolution equations, and ensures asymptotically inertial angular coordinates for better waveform extraction.

Findings

Enhanced waveform accuracy at null infinity.

Reduced gauge-related numerical issues.

Improved compatibility of Weyl scalars with coordinate systems.

Abstract

We present several improvements to the Cauchy-characteristic evolution procedure that generates high-fidelity gravitational waveforms at $\mathcal{I}^+$ from numerical relativity simulations. Cauchy-characteristic evolution combines an interior solution of the Einstein field equations based on Cauchy slices with an exterior solution based on null slices that extend to $\mathcal{I}^+$. The foundation of our improved algorithm is a comprehensive method of handling the gauge transformations between the arbitrarily specified coordinates of the interior Cauchy evolution and the unique (up to BMS transformations) Bondi-Sachs coordinate system of the exterior characteristic evolution. We present a reformulated set of characteristic evolution equations better adapted to numerical implementation. In addition, we develop a method to ensure that the angular coordinates used in the volume during the characteristic evolution are asymptotically inertial. This provides a direct route to an expanded set of waveform outputs and is guaranteed to avoid pure-gauge logarithmic dependence that has caused trouble for previous spectral implementations of the characteristic evolution equations. We construct a set of Weyl scalars compatible with the Bondi-like coordinate systems used in characteristic evolution, and determine simple, easily implemented forms for the asymptotic Weyl scalars in our suggested set of coordinates.