Event and apparent horizons are key diagnostics for the presence and properties of black holes. In this article I review numerical algorithms and codes for finding event and apparent horizons in numerically-computed spacetimes, focusing on calculations done using the 3 + 1 ADM formalism. The event horizon of an asymptotically-flat spacetime is the boundary between those events from which a future-pointing null geodesic can reach future null infinity and those events from which no such geodesic exists. The event horizon is a (continuous) null surface in spacetime. The event horizon is defined nonlocally in time: it is a global property of the entire spacetime and must be found in a separate post-processing phase after all (or at least the nonstationary part) of spacetime has been numerically computed. There are three basic algorithms for finding event horizons, based on integrating null geodesics forwards in time, integrating null geodesics backwards in time, and integrating null surfaces backwards in time. The last of these is generally the most efficient and accurate. In contrast to an event horizon, an apparent horizon is defined locally in time in a spacelike slice and depends only on data in that slice, so it can be (and usually is) found during the numerical computation of a spacetime. A marginally outer trapped surface (MOTS) in a slice is a smooth closed 2-surface whose future-pointing outgoing null geodesics have zero expansion Θ. An apparent horizon is then defined as a MOTS not contained in any other MOTS. The MOTS condition is a nonlinear elliptic partial differential equation (PDE) for the surface shape, containing the ADM 3-metric, its spatial derivatives, and the extrinsic curvature as coefficients. Most "apparent horizon" finders actually find MOTSs. There are a large number of apparent horizon finding algorithms, with differing trade-offs between speed, robustness, accuracy, and ease of programming. In axisymmetry, shooting algorithms work well and are fairly easy to program. In slices with no continuous symmetries, spectral integral-iteration algorithms and elliptic-PDE algorithms are fast and accurate, but require good initial guesses to converge. In many cases, Schnetter's "pretracking" algorithm can greatly improve an elliptic-PDE algorithm's robustness. Flow algorithms are generally quite slow but can be very robust in their convergence. Minimization methods are slow and relatively inaccurate in the context of a finite differencing simulation, but in a spectral code they can be relatively faster and more robust.

中文翻译：

---

3 + 1数值相对论的事件和视在地平线查找器。

事件和视界是黑洞的存在和性质的关键诊断。在本文中，我将回顾用于在数字计算时空中查找事件和视域的数值算法和代码，重点介绍使用3 +1 ADM形式主义进行的计算。渐近平坦时空的事件范围是指那些指向未来的零大地测量可以到达未来零无穷大的事件与不存在这种大地测量的事件之间的边界。事件范围是时空中的（连续）空表面。事件范围是在时间上非局部定义的：它是整个时空的全局属性，在对所有时空（至少是非平稳部分）进行数值计算后，必须在单独的后处理阶段中找到它。有三种基本的算法可用于查找事件视界，它们是基于在时间上向前集成空测地线，在时间上向后集成空测地线以及在时间上向后集成空曲面。这些中的最后一个通常是最有效和最准确的。与事件视界不同，视界在时间上像空间切片中局部定义，并且仅取决于该切片中的数据，因此可以（通常）在时空的数值计算过程中找到它。切片中的边缘外陷表面（MOTS）是光滑的封闭2面，其指向未来的外向零大地测地线具有零膨胀Θ。然后将视域定义为任何其他MOTS中都不包含的MOTS。MOTS条件是表面形状的非线性椭圆偏微分方程（PDE），包含ADM 3-metric，其空间导数和外部曲率作为系数。大多数“视界”发现者实际上都找到MOTS。有许多明显的视界发现算法，它们在速度，鲁棒性，准确性和易于编程之间具有不同的折衷。在轴对称中，射击算法工作良好且易于编程。在没有连续对称性的切片中，光谱积分迭代算法和椭圆PDE算法快速准确，但需要很好的初始猜测才能收敛。在许多情况下，Schnetter的“预跟踪”算法可以大大提高椭圆PDE算法的鲁棒性。流算法通常很慢，但是收敛性非常强。