Symmetry-seeking spacetime coordinates

TL;DR

This paper reviews methods for choosing spacetime coordinates aligned with approximate symmetries to improve numerical simulations of phenomena like black hole inspirals and critical collapse, testing these methods in spherical models.

Contribution

It summarizes existing and introduces new approaches for finding symmetry-adapted coordinates, and evaluates their effectiveness in toy models with spherical symmetry.

Findings

Coordinate choices can be optimized for better numerical stability.

Methods tested successfully in spherical symmetry models.

Improved understanding of symmetry-adapted coordinates in critical collapse.

Abstract

In numerically constructing a spacetime that has an approximate timelike Killing vector, it is useful to choose spacetime coordinates adapted to the symmetry, so that the metric and matter variables vary only slowly with time in these coordinates. In particular, this is a crucial issue in numerically calculating a binary black hole inspiral. An approximate homothetic vector plays a role in critical gravitational collapse. We summarize old and new suggestions for finding such coordinates from a general point of view. We then test some of these in various toy models with spherical symmetry, including critical fluid collapse and critical scalar field collapse.