Geometry and Regularity of Moving Punctures

TL;DR

This paper investigates the geometry and regularity of moving punctures in black-hole simulations, revealing how the puncture's geometry evolves and demonstrating that numerical results align with an analytical stationary solution.

Contribution

It provides an analytic understanding of the puncture geometry evolution, confirming numerical findings and clarifying the nature of the coordinate singularity.

Findings

Puncture geometry evolves from asymptotically flat to cylindrical during simulation.

Numerical results match the derived stationary analytical solution.

The evolution is not dominated by numerical artefacts at the puncture.

Abstract

Significant advances in numerical simulations of black-hole binaries have recently been achieved using the puncture method. We examine how and why this method works by evolving a single black hole. The coordinate singularity and hence the geometry at the puncture are found to change during evolution, from representing an asymptotically flat end to being a cylinder. We construct an analytic solution for the stationary state of a black hole in spherical symmetry that matches the numerical result and demonstrates that the evolution is not dominated by artefacts at the puncture but indeed finds the analytical result.