Wormholes and trumpets: the Schwarzschild spacetime for the moving-puncture generation

TL;DR

This paper analyzes the Schwarzschild spacetime using moving-puncture methods, deriving stationary foliations, transforming coordinates, and demonstrating the robustness of locating stationary geometries in numerical simulations.

Contribution

It provides a derivation of stationary 1+log foliations and shows how the moving-puncture method reliably finds stationary solutions in numerical relativity.

Findings

Derivation of analytic stationary 1+log foliations.

Transformation to isotropic-like coordinates.

Robustness of the moving-puncture method in locating stationary geometries.

Abstract

We expand upon our previous analysis of numerical moving-puncture simulations of the Schwarzschild spacetime. We present a derivation of the family of analytic stationary 1+log foliations of the Schwarzschild solution, and outline a transformation to isotropic-like coordinates. We discuss in detail the numerical evolution of standard Schwarzschild puncture data, and the new time-independent 1+log data. Finally, we demonstrate that the moving-puncture method can locate the appropriate stationary geometry in a robust manner when a numerical code alternates between two forms of 1+log slicing during a simulation.