v 2 1 J an 2 00 6 Event and Apparent Horizon Finders for 3 + 1 Numerical Relativity

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’s a global property of the entire spacetime, and must be found in a separate post-processing phase after (part of) the spacetime has been numerically computed. There are 3 basic algorithms for finding event horizons, based respectively 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 so can be (and usually is) found “on the fly” 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, Nakamura et al.’s algorithm 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 ellipticPDE algorithm’s robustness. Flow algorithms are generally quite slow, but can be very robust in their convergence.