Test-beds and applications for apparent horizon finders in numerical relativity

TL;DR

The paper tackles the robust detection of apparent horizons in 3D numerical relativity by deploying three independent horizon finders (axisymmetric, 3D minimization, and 3D fast flow) across a broad suite of test spacetimes, including toy bowl geometries, time-symmetric and non-time-symmetric black-hole data, and Brill wave initial data. It formalizes the local horizon condition via the expansion $H$, develops and compares different numerical strategies for solving $H=0$ on complex surfaces, and demonstrates that some previously published results were incorrect by applying cross-checks among the three methods. The study provides detailed, quantitative benchmarks (critical separations, horizon areas, ADM masses) and shows strong cross-method agreement in many cases while also revealing where and why particular algorithms may fail, especially near horizon formation. The findings lay groundwork for robust horizon finding in evolutions, inform the selection of AHFs for AHBC implementations, and offer valuable validation data for the wider numerical relativity community. The work thus advances the reliability and utility of horizon finding as a tool for exploring black-hole dynamics and gravitational-wave spacetimes in numerical simulations.

Abstract

We present a series of test beds for numerical codes designed to find apparent horizons. We consider three apparent horizon finders that use different numerical methods: one of them in axisymmetry, and two fully three-dimensional. We concentrate first on a toy model that has a simple horizon structure, and then go on to study single and multiple black hole data sets. We use our finders to look for apparent horizons in Brill wave initial data where we discover that some results published previously are not correct. For pure wave and multiple black hole spacetimes, we apply our finders to survey parameter space, mapping out properties of interesting data sets for future evolutions.

Figures (21)

• Figure 1: We show Fermi bowl embeddings for values of $\lambda$ between $0$ and $1.6$ every $0.1$, with $r_0 = 1.5$ and $\sigma = 5.0$. The "bag of gold" feature of the higher $\lambda$ metrics is clear from the embedding.
• Figure 2: Position of the AH on the $x-z$ plane for a prolate Gaussian bowl with $\lambda=1.5$, $r_0=2.5$, $\sigma=1$ and $d_z=1.2$. The solid line is the AH, and the dotted lines are the zeroes of the expansion $H$.
• Figure 3: Comparison of the AHs found with our three finders for a prolate Gaussian bowl with $\lambda=1.5$, $r_0=2.5$, $\sigma=1$ and $d_z=1.2$. The solid line corresponds to the 2D finder, the dotted line to the fast flow finder, and the dashed line to the minimization finder. All three lines lie on top of each other.
• Figure 4: Position of the AH on the $x-y$, $x-z$ and $y-z$ planes for a Fermi bowl with $\lambda=4.25$, $r_0=2.5$, $\sigma=1$, $d_y=1.2$ and $d_z=0.8$. The solid line is the AH, and the dotted lines are the zeroes of the expansion $H$.
• Figure 5: Comparison of the AHs found with our two 3D finders for a Fermi bowl with $\lambda=4.25$, $r_0=2.5$, $\sigma=1$, $d_y=1.2$ and $d_z=0.8$, using $l_{max}=8$. The solid lines correspond to the exact position of the horizon, the dashed lines to the position found with the fast flow finder, and the dotted lines to that found with the minimization finder.