How to move a black hole without excision: gauge conditions for the numerical evolution of a moving puncture

TL;DR

The paper addresses the challenge of evolving moving black holes without excision by analyzing gauge choices within the $3+1$ BSSN formulation, focusing on $1+log$ slicing and $\Gamma$-driver shift to avoid numerical pathologies. Through analytic eigenmode analysis and targeted moving-puncture simulations, it demonstrates that certain advection terms in the lapse and shift equations are essential to eliminate slow- and zero-speed modes that can destabilize the puncture. It identifies two robust gauges—the 'shifting-shift' and 'non-shifting-shift'—which produce smooth, stable puncture motion, especially as the damping parameter $\eta$ is reduced toward zero, approaching Gamma-freezing behavior; a practical substitution for the auxiliary field $B^i$ further simplifies implementation without sacrificing stability. Together, these results provide a principled route to reliable, long-duration moving-puncture evolutions suitable for accurate gravitational-wave modeling of binary black holes.

Abstract

Recent demonstrations of unexcised black holes traversing across computational grids represent a significant advance in numerical relativity. Stable and accurate simulations of multiple orbits, and their radiated waves, result. This capability is critically undergirded by a careful choice of gauge. Here we present analytic considerations which suggest certain gauge choices, and numerically demonstrate their efficacy in evolving a single moving puncture black hole.

Figures (13)

• Figure 1: Results of the non-moving puncture gauge $\partial_t\alpha = -2\alpha K$, $\partial_t\beta^i = \frac{3}{4}\alpha \Psi_0^{-n}B^i$, $\partial_tB^i = {\dot{\tilde{\Gamma}}^i}-\eta B^i$ at time $t=17M$. The conformal metric becomes singular by $t=18M$.
• Figure 2: Results of the gauge $\partial_t\alpha = -2\alpha K$, $\partial_t\beta^i = \frac{3}{4}\alpha B^i$, $\partial_tB^i = {\dot{\tilde{\Gamma}}^i}-\eta B^i$ at time $t=17M$. Sharp features around $x=-3M$ fail to propagate. The conformal metric becomes singular by $t=18M$.
• Figure 3: Results of the gauge $\partial_t\alpha = -2\alpha K$, $\partial_t\beta^i = \frac{3}{4}B^i$, $\partial_tB^i = {\dot{\tilde{\Gamma}}^i}-\eta B^i$ at times $t=4.5M$ (top panel) and $t=30M$ (bottom panel). The top panel clearly shows the lapse lagging behind the puncture, while the bottom panel exhibits noise propagating from the puncture.
• Figure 4: Results for the gauge ${\partial_t \alpha} = -2\alpha K + \beta^j\partial_j\alpha$, ${\partial_t \beta^i} = \frac{3}{4}\alpha B^i$, ${\partial_t B^i} = {\partial_t{\tilde{\Gamma}}^i} -\eta B^i$ (Case #1 in Table \ref{['table']} with p=1) at time $t=14M$. Non-propagating features are evident around $x=-3M$. The conformal metric becomes singular by $t=15M$
• Figure 5: Results for the gauge ${\partial_t \alpha} = -2\alpha K + \beta^j\partial_j\alpha$, ${\partial_t \beta^i} = \frac{3}{4}B^i$, ${\partial_t B^i} = {\partial_t{\tilde{\Gamma}}^i} -\eta B^i$ (Case #1 in Table \ref{['table']} with p=0) at time $t=30M$. The evolution appears smooth.