Introduction to dynamical horizons in numerical relativity

TL;DR

The paper develops and applies a dynamical-horizon framework for quasi-local analysis of dynamical black holes in numerical relativity, extending isolated horizons to time-dependent regimes. By tracking MOTSs, constructing MTTs, and computing horizon mass, angular momentum, multipole moments, and fluxes, it provides a coordinate-invariant set of tools to study horizon dynamics and validate black-hole settles-to-Kerr behavior. The authors demonstrate the approach with three canonical simulations—head-on BH collisions, non-axisymmetric mergers, and axisymmetric neutron-star collapse—revealing horizon growth, inner-outer MTT dynamics, and convergence of multipole moments toward Kerr values, along with exact balance laws for area and angular momentum. These diagnostics offer deeper physical insight into horizon geometry and dynamics, with implications for waveform interpretation and numerical-relativity diagnostics, while highlighting current limitations and avenues for refinement.

Abstract

This paper presents a quasi-local method of studying the physics of dynamical black holes in numerical simulations. This is done within the dynamical horizon framework, which extends the earlier work on isolated horizons to time-dependent situations. In particular: (i) We locate various kinds of marginal surfaces and study their time evolution. An important ingredient is the calculation of the signature of the horizon, which can be either spacelike, timelike, or null. (ii) We generalize the calculation of the black hole mass and angular momentum, which were previously defined for axisymmetric isolated horizons to dynamical situations. (iii) We calculate the source multipole moments of the black hole which can be used to verify that the black hole settles down to a Kerr solution. (iv) We also study the fluxes of energy crossing the horizon, which describes how a black hole grows as it accretes matter and/or radiation. We describe our numerical implementation of these concepts and apply them to three specific test cases, namely, the axisymmetric head-on collision of two black holes, the axisymmetric collapse of a neutron star, and a non-axisymmetric black hole collision with non-zero initial orbital angular momentum.

Figures (16)

• Figure 1: A dynamical horizon $H$ bounded by MOTSs $S_1$ and $S_2$. $\ell^a$ is the outgoing null normal, $n^a$ is the ingoing null normal, $\hat{r}^a$ is the unit spacelike normal to the cross-sections, and $\hat{\tau}^a$ is the unit timelike normal to $H$. $\Sigma$ is a Cauchy surface intersecting $H$ in a 2-sphere $S$. $T^a$ is the unit timelike normal to $\Sigma$ and $R^a$ is the unit space-like outward pointing vector normal to $S$ and tangent to $\Sigma$.
• Figure 2: Two MOTSs $S_{(1)}$ and $S_{(2)}$ surrounded by a common MOTS $S_{out}$. Spheres lying just inside these FMOTSs must have negative outgoing expansion. Thus, there must be a inner trapped horizon $S_{in}$ inside $S_{out}$ which encloses $S_{(1)}$ and $S_{(2)}$.
• Figure 3: Coordinate shapes of the horizons at $t=1$ in the $xz$ plane. A common horizon has formed, and the inner and outer common horizons have already separated. Compare figure \ref{['fig:multibh']}.
• Figure 4: Determinant of the horizon world tube's three-metric vs. latitude $\theta$ at $t=0.6$ and $t=1$. The individual MTTs are null, i.e., $\mathrm{det}\; \tilde{q} =0$ (up to numerical errors). The common outer MTT is spacelike (i.e., $\mathrm{det}\; \tilde{q} > 0$) and it tends to null at late times. The inner common MTT is partially timelike at $t=0.6$; later it becomes completely timelike.
• Figure 5: Irreducible mass vs. time for the individual and the common MTTs. The outer common MTT grows and accretes mass, while the inner MTT shrinks and loses mass.