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

Still newer formulations: towards a stable evolution system 381out to sufficiently late times [103, 104]. Hence they are useless in providingboundaries as one integrates forward in time. On the other hand the apparenthorizon, if it exists, can be found on any given slice by searching for closedtwo-surfaces with zero expansion. Although one should worry that in a genericBH collision, one may evolve into situations where no apparent horizon actuallyexists, let us cross that bridge if we come to it! Methods for finding apparenthorizons will also be discussed below, but for now we assume that such a methodexists.Given these considerations, there are two basic ideas behind theimplementation of the apparent horizon boundary condition (AHBC), also knownas black hole excision:(a) It is important to use a finite-differencing scheme which respects thecausal structure of the spacetime. Since the horizon is a one-way membrane,quantities on the horizon can be affected only by quantities outside but not insidethe horizon: all quantities on the horizon can in principle be updated solely interms of known quantities residing on or outside the horizon. There are varioustechnical details and variations on this idea, which is called ‘causal differencing’[105] or ‘causal reconnection’ [106], but here we focus primarily on the basicideas and results obtained to date.(b) A shift is used to control the motion of the horizon, and the behaviour ofthe grid points outside the BH, as they tend to fall into the horizon if uncontrolled.An additional advantage to using causal differencing is that it allows oneto follow the information flow to create grid points with proper data on them,as needed inside the horizon, even if they did not exist previously. (Rememberabove that we have cut away a region inside the horizon, so in fact we have no datathere.) One example is to let a BH move across the computational grid. If a BHis moving physically, it may also be desirable to have it move through coordinatespace. Otherwise, all physical movement will be represented by the ‘motion’ ofthe grid points. For a single BH moving in a straight line, this may be possible(though complicated), but for spiralling coalescence this will lead to hopelesslycontorted grids. The immediate consequence of this is that as a BH moves acrossthe grid, regions in the wake of the hole, now in its exterior, must have previouslybeen inside it where no data exist! However, with AHBC and causal differencingthis need not be a problem.Does the AHBC idea work? Preliminary indications are very promising. Inspherical symmetry (1D), numerous studies show that one can locate horizons, cutaway the interior, and evolve for essentially unlimited times (t ∝ 10 3−4 M, whereM is the black hole mass). The growth of metric functions can be completelycontrolled, errors are reduced to a very low level, and the results can be obtainedwith a large variety of shift and slicing conditions, and with matter falling in theBH to allow for true dynamics even in spherical symmetry [105, 107–109].In 3D, the basic ideas are similar but the implementation is much moredifficult. The first successful test of these ideas to a Schwarzschild BH in 3D usedhorizon excision and a shift provided from similar simulations carried out with a