Puncture Evolution of Schwarzschild Black Holes

TL;DR

This paper analyzes the moving puncture method for simulating Schwarzschild black holes, revealing that the puncture region is unresolved and evolves into a finite-radius cylinder, with implications for numerical relativity techniques.

Contribution

It demonstrates that the moving puncture method effectively acts as a form of natural excision by showing the unresolved puncture region's evolution.

Findings

Puncture region is not resolved by current codes.

Geometry near puncture evolves into an infinite cylinder.

Puncture remains at spacelike infinity during evolution.

Abstract

The moving puncture method is analyzed for a single, non-spinning black hole. It is shown that the puncture region is not resolved by current numerical codes. As a result, the geometry near the puncture appears to evolve to an infinitely long cylinder of finite areal radius. The puncture itself actually remains at spacelike infinity throughout the evolution. In the limit of infinite resolution the data never become stationary. However, at any reasonable finite resolution the grid points closest to the puncture are rapidly drawn into the black hole interior by the Gamma-driver shift condition. The data can then evolve to a stationary state. These results suggest that the moving puncture technique should be viewed as a type of "natural excision".