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

Still newer formulations: towards a stable evolution system 379at some finite radius, and the null slices can be carried out to null infinity. 3Dcharacteristic evolution codes have progressed dramatically in recent years, andalthough the full 3D matching remains to be completed, tests of the schemein specialized settings show promise [11]. One can also use the hyperbolicformulations of the Einstein equations to find eigenfields, for which outgoingconditions can in principle be applied [32] in 1D. In 3D this technique is stillunder development, but it shows promise for future work. Less sophisticatedapproaches that seem nonetheless rather successful are discussed in [63]. Finally,there is another hyperbolic approach which uses conformal rescaling to movethe boundary to infinity [12–15]. These methods have different strengths andweaknesses, but all promise to improve boundary treatments significantly, helpingto enable longer evolutions than are presently possible.18.3.3 Special difficulties with black holesThe techniques described so far are generic in their application in numericalrelativity. However, in this section we describe a few problems that arecharacteristic of black holes, and special algorithms under development to handlethem. Black hole spacetimes all have in common one problem: singularities lurkwithin them, which must be handled numerically. Developing suitable techniquesfor doing so is one of the major research priorities of the community at present.If one attempts to evolve directly into the singularity, infinite curvature will beencountered, causing any numerical code to break down.Traditionally, the singularity region is avoided by the use of ‘singularityavoiding’ time slices, that wrap up around the singularity. Consider the evolutionshown in figure 18.1. A star is collapsing, a singularity is forming, and time slicesare shown which avoid the interior while still covering a large fraction of thespacetime where waves will be seen by a distant observer. However, these slicingconditions by themselves do not solve the problem; they merely serve to delay theonset of instabilities. As shown in figure 18.1, in the vicinity of the singularitythese slicings inevitably contain a region of abrupt change near the horizon, and aregion in which the constant time slices dip back deep into the past in some sense.This behaviour typically manifests itself in the form of sharply peaked profilesin the spatial metric functions [80], ‘grid stretching’ [101] or large coordinateshift [73] on the BH throat, etc. Numerical simulations will eventually crash dueto these pathological properties of the slicing.18.3.3.1 Apparent horizon boundary conditions (AHBC)Cosmic censorship suggests that in physical situations, singularities are hiddeninside BH horizons. Because the region of spacetime inside the horizon is causallydisconnected from the region of interest outside the horizon, one is temptednumerically to cut away the interior region containing the singularity, and evolveonly the singularity-free region outside, as originally suggested by Unruh [102].