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

386 Numerical relativityto a very accurate determination of the location of the EH at earlier times. Notethat it is the earlier time when the black hole is under highly dynamical evolutionthat we are really interested in.Although one can integrate individual null geodesics backwards in time,we find that there are various advantages to integrating the entire null surfacebackwards in time. A null surface, if defined by f (t, x i ) = 0 satisfies theconditiong µν ∂ µ f ∂ ν f = 0. (18.24)Hence the evolution of the surface can be obtained by a simple integration,√∂ t f = −gti ∂ i f + (g ti ∂ i f ) 2 − g tt g ij ∂ i f ∂ j fg tt . (18.25)The inner and outer boundary of the horizon containing region when integratedbackwards in time, will rapidly converge to practically a single surface to withinthe resolution of the numerically constructed spacetime, i.e. a small fraction ofa grid point. An accurate location of the event horizon is hence obtained. Wehenceforth shall represent the horizon surface as the function f H (t, x i ). Asidefrom the simplicity of this method, there are a number of technical advantagesas discussed in [103]. One particularly noteworthy point is that this method iscapable of giving the caustic structure of the event horizon if there is any; fordetails see [103].The function f H (t, x i ) provides the complete coordinate location of theEH through the spacetime (or a very good approximation of it, as shown in[104]). This function by itself directly gives us the topology and location of theEH. When combined with the induced metric function on the surface, which isrecorded throughout the evolution, it gives the intrinsic geometry of the EH. Whenfurther combined with the spacetime metric, all properties of the EH including itsembedding can be obtained. Moreover, as the normal of f H (t, x i ) = 0givesthe null generators of the horizon, it is an easy further step to determine the nullgenerators, and hence the complete dynamics of the horizon in this formulation.As described in a series of papers, the event horizon, once found with such amethod, can be analysed to provide important information about the dynamics ofblack holes in a numerically generated spacetime [103, 104, 110–113].18.4.4 Wave extractionThe gravitational radiation emitted is one of the most important quantities ofinterest in many astrophysical processes. The radiation is generated in regionsof strong and dynamic gravitational fields, and then propagated to regions faraway where it will someday be detected. We take the approach of computing thegeneration and evolution of the fields in a fully nonlinear way, while analysing theradiation with a perturbation formulation in the regions where it can be so treated.