Geometry and Regularity of Moving Punctures

TL;DR

The paper investigates why moving-puncture evolutions of black-hole binaries are stable by analyzing a single Schwarzschild puncture within the BSSN framework using $1+\log$ slicing and a Gamma-freezing shift; it shows that the shift can be decomposed into $β_{adv}$ and $β_{sls}$ to separate advection from gauge stabilization. By constructing a local stationary expansion near the puncture and explicit stationary Schwarzschild slices, the authors demonstrate that the conformal factor evolves from $ψ ∼ 1/r$ to $ψ ∼ O(1/\sqrt{r})$ and that the slice ends at a finite areal radius $R_0 ≈ 1.31241 M$. The stationary slices are shown to exist globally, with a cylinder throat and an asymptotically cylindrical geometry, explaining the observed behavior of the puncture and the convergence of simulations. The results validate the moving-puncture approach, clarify the puncture geometry, and point to future work on perturbations, constraint stability, and the construction of asymptotically cylindrical initial data.

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.

Figures (2)

• Figure 1: Numerical evolutions of a spherically symmetric puncture result in specific kinks at the puncture (left). The figure on the right represents a true numerical experiment. We compare two spherical slices by plotting $\alpha$ as a function of $K$. The dots are the data from the 3D numerical evolution, the solid line was obtained by independently integrating up Eq. (9). At the puncture, $\alpha\approx 0.0$ and $K\approx 0.3$.
• Figure 2: Schwarzschild coordinate $R$ versus proper distance from the (outer) horizon. The left panel shows the slices at $t = 0,1,2,3M$, and the right panel shows the slice at $t = 50M$. The final numerical slice terminates at $R \approx 1.3M$. The vertical line indicates the horizon at $R = 2M$, and the six points represent $x/M = 1/40,1/20,1/8,2,5,8$ on each slice.