Schwarzschild black hole as moving puncture in isotropic coordinates

TL;DR

This paper analyzes stationary 1+log slices of Schwarzschild spacetime in isotropic coordinates to understand the coordinate singularity at the puncture, aiding numerical black hole simulations.

Contribution

It introduces an alternative integration method for isotropic coordinates that simplifies calculations and provides insights into the local behavior near the puncture.

Findings

Certain quantities are well approximated linearly near the puncture

The isotropic radius may have an exponent close to but not exactly one

The method improves understanding of coordinate singularities in black hole simulations

Abstract

The success of the moving puncture method for the numerical simulation of black hole systems can be partially explained by the properties of stationary solutions of the 1+log coordinate condition. We compute stationary 1+log slices of the Schwarzschild spacetime in isotropic coordinates in order to investigate the coordinate singularity that the numerical methods have to handle at the puncture. We present an alternative integration method to obtain isotropic coordinates that simplifies numerical integration and that gives direct access to a local expansion in the isotropic radius near the puncture. Numerical results have shown that certain quantities are well approximated by a function linear in the isotropic radius near the puncture, while here we show that in some cases the isotropic radius appears with an exponent that is close to but unequal to one.