Generalized harmonic formulation in spherical symmetry

TL;DR

This paper presents a generalized harmonic formulation of Einstein's equations in spherical symmetry, focusing on gauge choices and their effectiveness in numerical simulations of scalar field collapse.

Contribution

It introduces a regularized generalized harmonic formulation in spherical symmetry and evaluates gauge driver performance, revealing stability issues not seen in Cartesian coordinates.

Findings

Certain gauge drivers are unstable in spherical symmetry.

Optimal gauge parameters depend on the specific scenario.

The formulation is regular at the origin, suitable for collapse simulations.

Abstract

In this pedagogically structured article, we describe a generalized harmonic formulation of the Einstein equations in spherical symmetry which is regular at the origin. The generalized harmonic approach has attracted significant attention in numerical relativity over the past few years, especially as applied to the problem of binary inspiral and merger. A key issue when using the technique is the choice of the gauge source functions, and recent work has provided several prescriptions for gauge drivers designed to evolve these functions in a controlled way. We numerically investigate the parameter spaces of some of these drivers in the context of fully non-linear collapse of a real, massless scalar field, and determine nearly optimal parameter settings for specific situations. Surprisingly, we find that many of the drivers that perform well in 3+1 calculations that use Cartesian coordinates, are considerably less effective in spherical symmetry, where some of them are, in fact, unstable.