(ed.). Gravitational waves (IOP, 2001)(422s).

Tools for analysing the numerical spacetimes 383that the position of an EH can only be found if the whole history of the spacetimeis known. For numerical simulations of black hole spacetimes in particular, thisimplies that in order to locate an EH one needs to evolve sufficiently far into thefuture, up to a time where the spacetime has basically settled down to a stationarysolution. Recently, methods have been developed to locate and analyse blackhole horizons in numerically generated spacetimes, with a number of interestingresults obtained [103, 104, 110–113].In contrast, an apparent horizon (AH) is defined locally in time as the outermostmarginally trapped surface [114], i.e. a surface on which the expansion ofout-going null geodesics is everywhere zero. An AH can therefore be defined ona given spatial hypersurface. A well-known result [114] guarantees that if an AHis found, an EH must exist somewhere outside of it and hence a black hole hasformed.18.4.2 Locating the apparent horizonsThe expansion of a congruence of null rays moving in the outward normaldirection to a closed surface can be shown to be [20] =∇ i s i + K ij s i s j − tr K, (18.17)where ∇ i is the covariant derivative associated with the 3-metric γ ij , s i is thenormal vector to the surface, K ij is the extrinsic curvature of the time slice, andtr K is its trace. An AH is then the outermost surface such that = 0. (18.18)This equation is not affected by the presence of matter, since it is purely geometricin nature. We can use the same horizon finders without modification for vacuumas well as non-vacuum spacetimes. The key is to find a closed surface with normalvector s i satisfying this equation.18.4.2.1 Minimization algorithmsAs apparent horizons are defined by the vanishing of the expansion on a surface,a fairly obvious algorithm to find such a surface involves minimizing a suitablenorm of the expansion below some tolerance while adjusting test surfaces.Minimization algorithms for finding apparent horizons were among the firstmethods developed [115, 116]. More recently, a 3D minimization algorithmwas developed and implemented by the Potsdam/NCSA/Wash U group, appliedto a variety of black hole initial data and 3D numerically evolved black holespacetimes [117–121]. Essentially the same algorithm was also implementedindependently by Baumgarte et al [122].The basic idea behind a minimization algorithm is to assume the surface canbe represented by a function F(x i ) = 0, expand it in terms of some set of basis