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

374 Numerical relativityEinstein did not specify these quantities; they are up to the numerical relativist tochoose at will!LapseThe choice of lapse corresponds to how one chooses 3D spacelike hypersurfacesin 4D spacetime. The ‘lapse’ of proper time along the normal vector of one sliceto the next is given by α dt, where dt is the coordinate time interval betweenslices. As α(x, y, z) can be chosen at will on a given slice, some regions ofspacetime can be made to evolve farther into the future than others.There are many motivations for particular choices of lapse. A primaryconcern is to ensure that it leads to a stable long-term evolution. It is easy tosee that a naive choice of the lapse, for example, α = 1, the so-called geodesicslicing, suffers from a strong tendency to produce coordinate singularities [79,80].A related concern is that one would like to cover the region of interest in anevolution, say, where gravitational waves generated by some process could bedetected, while avoiding troublesome regions, say, inside black holes wheresingularities lurk (the so-called ‘singularity avoiding’ time slicings). Anotherimportant motivation is that some choices of α allow one to write the evolutionequations in forms that are especially suited to numerical evolution. Finally,computational considerations also play an important role in the choice of thelapse; one prefers a condition for α that does not involve great computationalexpense, while also providing smooth, stable evolution.Some ‘traditional’ choices of the lapse used in the numerical constructionof spacetimes are [81]: (1) Lagrangian slicing, in which the coordinates arefollowing the flow of the matter in the simulation. This choice simplifies thematter evolution equations, but it is not always applicable, for example, in avacuum spacetime or when the fluid flow pattern becomes complicated. (2)Maximal slicing, [79, 80] in which the trace of the extrinsic curvature is requiredto be zero always, i.e, K (t = 0) = 0 = ∂ t K . The evolution equations of theextrinsic curvature then lead to an elliptic equation for the lapseD a D a α − α(R + K 2 ) = 0. (18.15)The maximal slicing has the nice property of causing the lapse to ‘collapse’ toa small value at regions of strong gravity, hence avoiding the region where acurvature singularity is forming. It is one of the so-called ‘singularity avoidingslicing conditions’. Maximal slicing is easily the most studied slicing conditionin numerical relativity. (3) Constant mean curvature, where we let K = constantdiffer from zero, a choice often used in constructing cosmological solutions.(4) Algebraic slicing, where the lapse is given as an algebraic function of thedeterminant of the 3-metric. Algebraic slicing can also be singularity avoiding[82]. As there is no need to solve an elliptic equation as in the case of maximalslicing, algebraic slicing is computationally efficient. Some algebraic slicings(e.g., the harmonic slicing in which α is set proportional to the square root of