Black holes are characterized by the horizons surrounding them. Clearly, then, the numerical simulation of black holes requires the ability to locate and analyze black hole horizons in numerically generated spacetimes. In this chapter we first review different concepts of horizons in asymptotically flat spacetimes, and then discuss how these horizons can be probed numerically.
Several different notions of horizons exist in general relativity. The defining property of a black hole is the presence of an event horizon (Section 7.2), but, as we will see, apparent horizons (Section 7.3) also play an extremely important role in the context of numerical relativity. In addition, the concepts of isolated and dynamical horizons (Section 7.4) serve as useful diagnostics in numerical spacetimes containing black holes.
A black hole is defined as a region of spacetime from which no null geodesic can escape to infinity. The surface of a black hole, the event horizon, acts as a one-way membrane through which light and matter can enter the black hole, but once inside, can never escape. It is the boundary in spacetime separating those events that can emit light rays that can propagate to infinity and those which cannot. More precisely, the event horizon is defined as the boundary of the causal past of future null infinity. It is a 2 + 1 dimensional hypersurface in spacetime formed by those outward-going, future-directed null geodesics that neither escape to infinity nor fall toward the center of the black hole.