Gauge conditions for long-term numerical black hole evolutions without excision

TL;DR

The paper tackles the long-standing problem of slice stretching in three-dimensional numerical relativity evolutions without singularity excision by developing a robust gauge system that combines $K$-freezing/$1+\log$ lapse with hyperbolic/elliptic Gamma-driver shifts. By incorporating puncture data within the BSSN framework and regularizing the lapse and shift near punctures (including a $ abla^2$-type lapse equation and a fall-off for the shift), the authors achieve stable evolutions for thousands of $M$ and accurate gravitational wave extraction. They demonstrate the approach on Schwarzschild punctures, a distorted BH, and head-on Brill–Lindquist puncture collisions, showing near-stationary late-time behavior, convergence near punctures, and clear quasi-normal mode ringdowns. The methods significantly extend runnable timescales without excision, improving reliability and efficiency for waveform generation in binary black hole spacetimes and enabling more realistic long-term simulations in numerical relativity.

Abstract

Numerical relativity has faced the problem that standard 3+1 simulations of black hole spacetimes without singularity excision and with singularity avoiding lapse and vanishing shift fail after an evolution time of around 30-40M due to the so-called slice stretching. We discuss lapse and shift conditions for the non-excision case that effectively cure slice stretching and allow run times of 1000M and more.

Figures (14)

• Figure 1: Schematic representation on the Kruskal diagram of the effect of the different boundary conditions on the slices obtained. The first panel shows the case of an odd lapse at the throat, the second panel the case of an even lapse at the throat, and the last panel the case of a lapse with zero gradient at the puncture. The dashed lines show the singularities and the dotted lines the event horizon.
• Figure 2: Schwarzschild black hole evolved for $t = 1000M$. Shown are lapse $\alpha$ and shift component $\beta^x$ along the x-axis, which are (anti-)symmetric about $x = 0$. By that time lapse and shift are approximately static. The lapse has collapsed to zero at the puncture and approaches one in the outer region. The shift crosses zero at the puncture, pointing away from the puncture and thereby halting the infall of points towards the puncture.
• Figure 3: Schwarzschild black hole evolved for $t = 1000M$. Shown are the BSSN variables $\phi$, $K$, $\tilde{\gamma}_{xx}$, $\tilde{A}_{xx}$, and $\tilde{\Gamma}^x$ along the x-axis, and also the Hamiltonian constraint $H$.
• Figure 4: Schwarzschild black hole evolved for $t = 1000M$. Shown is a comparison along the x-axis between two versions of the hyperbolic Gamma-driver for the shift, Eq. (\ref{['gamma2']}) (dashed line) and Eq. (\ref{['gamma0']}) (solid line).
• Figure 5: Schwarzschild black hole evolved for $t = 3000M$. Shown are the maximum of the shift and the root-mean-square value of the Hamiltonian constraint as a function of time, again for two versions of the hyperbolic Gamma-driver for the shift, Eq. (\ref{['gamma2']}) (dashed line) and Eq. (\ref{['gamma0']}) (solid line), with diffusion parameter $\eta = 2.0/M$ and $\eta = 2.8/M$, respectively. After a short time interval during which lapse and shift adjust themselves dynamically, the evolution slows down significantly.