Introduction to Isolated Horizons in Numerical Relativity

TL;DR

The paper tackles the problem of extracting black hole parameters from numerical spacetimes without relying on global constructs like the event horizon. It develops an isolated-horizon based approach to compute $M_\Delta$ and $J_\Delta$ from data on the apparent horizon, using an axial vector $\varphi^a$ found via a horizon Killing-transport procedure. It validates the method with boosted Kerr-Schild tests, showing convergence and agreement with analytic values, and argues it is robust, coordinate-independent, and applicable in both early and late-time regimes, with potential benefits for initial data, wave extraction, and cross-simulation comparisons.

Abstract

We present a coordinate-independent method for extracting mass (M) and angular momentum (J) of a black hole in numerical simulations. This method, based on the isolated horizon framework, is applicable both at late times when the black hole has reached equilibrium, and at early times when the black holes are widely separated. We show how J and M can be determined in numerical simulations in terms of only those quantities which are intrinsic to the apparent horizon. We also present a numerical method for finding the rotational symmetry vector field (required to calculate J) on the horizon.

Figures (7)

• Figure 1: The figure shows an apparent horizon $S$ embedded in a spatial slice $\Sigma$. $T^a$ is the unit timelike normal to $\Sigma$ and $R^a$ is the outward pointing unit spatial normal to $S$ in $\Sigma$; $\ell^a$ and $n^a$ are the outgoing and ingoing null vectors, respectively. The vector $m^a$ (not shown in the figure) is tangent to $S$. $H$ is the world tube of apparent horizons.
• Figure 2: Plots of the real and imaginary parts of the eigenvalues of the matrix $\mathbf{M}$ (defined in eqn. (\ref{['eq:matrix']})) versus a large range of the spin parameter $a$.
• Figure 3: Plots of the real and imaginary parts of the eigenvalues of the matrix $\mathbf{M}$ versus the spin parameter $a$, when $a$ is small.
• Figure 4: The numerically computed angular momentum of the black hole at different boosts for a black hole with mass $M=1$ and spin $a=1/2$. Three different resolutions $d\phi$ are shown.
• Figure 5: Resolution tests for the angular momentum $J_\Delta$ of the horizon. The scenarios II -- IV are explained in table \ref{['tab:table3']}.