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

Tools for analysing the numerical spacetimes 385where ḡ ij is the flat background metric associated with the coordinates (r,θ,φ),andαA =l max (l max + 1) + β, B = β (18.23)αwith α = c and β = c/2. Here c is a variable step size, with a typical valueof c ∼ 1. l max is the maximum value of l one chooses to use in describingthe surface. The iteration procedure is a finite-difference approximation to aparabolic flow, and the adaptive step size is chosen to keep the finite-differenceapproximation roughly close to the flow limit to prevent overshooting of the trueapparent horizon. The adaptive step size is determined by a standard method usedin ODE integrators: we take one full step and two half steps and compare theresulting a lm . If the two results differ too much one from another, the step size isreduced.Other methods for finding apparent horizons, based directly on computingthe Jacobian of the finite differenced horizon equation, have been developed[125, 126] and successfully used in 3D. For details, see these references.18.4.3 Locating the event horizonsThe AH is defined locally in time and hence is much easier to locate than theevent horizon (EH) in numerical relativity. The EH is a global object in time;it is traced out by the path of outgoing light rays that never propagate to futurenull infinity, and never hit the singularity. (It is the boundary of the causal pastof future null infinity Â˙− (Á + ).) In principle, one needs to know the entire timeevolution of a spacetime in order to know the precise location of the EH. However,in spite of the global properties of the EH, hope is not lost for finding it veryaccurately, even in a numerical simulation of finite duration. Here we discussa method to find the EH, given a numerically constructed black hole spacetimethat eventually settles down to an approximately stationary state at late times. Inprinciple, as the EH is a null surface, it can be found by tracing the path of nullrays through time. Outward going light rays emitted just outside the EH willdiverge away from it, escaping to infinity, and those emitted just inside the EHwill fall away from it, towards the singularity. In a numerical integration it isdifficult to follow accurately the evolution of a horizon generator forward in time,as small numerical errors cause the ray to drift and diverge rapidly from the trueEH. It is a physically unstable process. However, we can actually use this propertyto our advantage by considering the time-reversed problem. In a global sense intime, any outward going photon that begins near the EH will be attracted to thehorizon if integrated backward in time [103, 117]. In integrating backwards intime, it turns out that it suffices to start the photons within a fairly broad regionwhere the EH is expected to reside. Such a horizon-containing region, as we callit, is often easy to determine after the spacetime has settled down to approximatestationarity. The crucial point is that when integrated backward in time along nullgeodesics, this horizon-containing region shrinks rapidly in ‘thickness’, leading