In [16] 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
. This approach
is dual to the technique of using
to extract outgoing gravitational waves. For a linear
perturbation of the Schwarzschild metric, this
condition eliminates incoming radiation
crossing the initial null hypersurface. Since
consists of a second radial derivative of the
characteristic data, the condition allows both continuity of
at the extraction worldtube
and the desired asymptotic falloff of
at infinity. In the linearized limit, setting
reduces to
, in terms of the compactified radial coordinate
. In terms of
the compactified grid coordinate
(where
is the Cartesian radius of
the extraction worldtube defined by the Cauchy coordinates), the corresponding solution is
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 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 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
and does not explore the additional benefits of the more gauge invariant
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.
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
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