6.6 Initial characteristic data for CCE

Data on the initial null hypersurface must be prescribed to begin the characteristic evolution. This data consist of the conformal 2-metric h AB of the null hypersurface. Because of the determinant condition (35View Equation), this data can be formulated in terms of the spin-weight 2 variable J given in (47View Equation). In the first applications of CCE, it was expedient to set J = 0 on the initial hypersurface outside some radius. This necessitated a transition region to obtain continuity with the initial Cauchy data, which requires non-zero initial characteristic data at the extraction worldtube.

In [16Jump To The Next Citation Point] the initialization was changed by requiring that the Newman–Penrose component of the Weyl tensor intrinsic to the initial null hypersurface vanish, i.e., by setting ψ0 = 0. This approach is dual to the technique of using ψ4 to extract outgoing gravitational waves. For a linear perturbation of the Schwarzschild metric, this ψ0 condition eliminates incoming radiation crossing the initial null hypersurface. Since ψ 0 consists of a second radial derivative of the characteristic data, the condition allows both continuity of J at the extraction worldtube and the desired asymptotic falloff of J at infinity. In the linearized limit, setting ψ0 = 0 reduces to (∂ℓ)2J = 0, in terms of the compactified radial coordinate ℓ = 1∕r. In terms of the compactified grid coordinate x = r∕(RE + r) (where RE is the Cartesian radius of the extraction worldtube defined by the Cauchy coordinates), the corresponding solution is

J = J|xE(1-−-x)xE-, (90 ) (1 − xE)x
where J|xE is determined by Cauchy data at the extraction worldtube. Since this solution also implies J |ℐ+ = J|x=1 = 0, ℐ+ has unit-sphere geometry so that the conformal gauge effects discussed in Section 6.1 are minimized at the outset of the evolution.

Besides the extraneous radiation content in the characteristic initial data there is also extraneous “junk” radiation in the initial Cauchy data for the binary black hole simulation. Practical experience indicates that the effect of this “junk” radiation on the waveform is transient and becomes negligible by the onset of the plunge and merger stage. However, another source of waveform error with potentially longer time consequences can arise from a mismatch between the initial characteristic and Cauchy data. This mismatch arises because the characteristic data is given on the outgoing null hypersurface emanating from the intersection of the extraction worldtube and the initial Cauchy hypersurface. Since in CCE the extraction worldtube cannot be located at the outer Cauchy boundary, part of the initial null hypersurface lies in the domain of dependence of the initial Cauchy data. Thus, a free prescription of the characteristic data can be inconsistent with the Cauchy data.

The initial characteristic data ψ0 = 0 implies the absence of radiation on the assumption that the geometry of the initial null hypersurface is close to Schwarzschild. This assumption becomes valid as the extraction radius becomes large and the exterior Cauchy data can be approximated by Schwarzschild data. Thus, this mismatch could in principle be reduced by a sufficiently large choice of extraction worldtube. However, that approach is counter productive to the savings that CCE can provide.

An alternative approach developed in [58] attempts to alleviate this problem by constructing a solution linearized about Minkowski space. The linearized solution is modeled upon binary black-hole initial Cauchy data. By evaluating the solution on the initial characteristic null hypersurface, this solves the compatibility issue up to curved space effects. A comparison study based upon this approach shows that the choice of J = 0 initial data does affect the waveform for time scales, which extend long after the burst of junk radiation has passed. Although this study is restricted to CCE extraction radii R > 100 M and does not explore the additional benefits of the more gauge invariant ψ0 = 0 initial data implemented in [16], it emphasizes the need to control potential long terms effects, which might result from a mismatch between the Cauchy and characteristic initial data.

Ideally, this mismatch could be eliminated by placing the extraction worldtube at the artificial outer boundary of the Cauchy evolution by means of a transparent interface with the outer characteristic evolution. This is the ultimate goal of CCM, although a formidable amount of work remains to develop a stable implementation.

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