When an artificial finite outer boundary is introduced there are two broad sources of error:
CCM addresses both of these items. Cauchy-characteristic extraction (CCE), which is one of the pieces
of the CCM strategy, offers a means to avoid the second source of error introduced by extraction at a finite
worldtube. In current codes used to simulate black holes, the waveform is extracted at an interior worldtube
which must be sufficiently far inside the outer boundary in order to isolate it from errors introduced by the
boundary condition. There the waveform is extracted by a perturbative scheme based upon the
introduction of a background Schwarzschild spacetime. This has been carried out using the
Regge–Wheeler–Zerilli [200, 254] treatment of the perturbed metric, as reviewed in [177], and also by
calculating the Newman-Penrose Weyl component , as first done for the binary black hole
problem in [17, 195, 66, 18]. In this approach, errors arise from the finite size of the extraction
worldtube, from nonlinearities and from gauge ambiguities involved in the arbitrary introduction of
a background metric. The gauge ambiguities might seem less severe in the case of
(vs
metric) extraction, but there are still delicate problems associated with the choices of a preferred
null tetrad and preferred worldlines along which to measure the waveform (see [167
] for an
analysis).
CCE offers a means to avoid this error introduced by extraction at a finite worldtube. In CCE, the inner
worldtube data supplied by the Cauchy evolution is used as boundary data for a characteristic evolution to
future null infinity, where the waveform can be unambiguously computed in terms of the Bondi news
function. By itself, CCE does not use the characteristic evolution to inject outer boundary data for the
Cauchy evolution, which can be a source of instability in full CCM. A wide number of highly nonlinear
tests involving black holes [46, 255, 256] have shown that CCE is a stable procedure which
provides the gravitational waveform up to numerical error which is second order convergent.
Nevertheless, in astrophysical applications which require high resolution, such as the inspiral of matter
into a black hole [44], numerical error has been a troublesome factor in computing the news
function. The CCE modules were developed in a past period when stability was the dominant
issue and second order accuracy was considered sufficient. Only recently have they begun to be
updated to include the more accurate techniques now standard in Cauchy codes. There are two
distinct ways, geometric and numerical, that the accuracy of CCE might be improved. In the
geometrical category, one option is to compute
instead of the news function as the primary
description of the waveform. In the numerical category, some standard methods for improving
accuracy, such as higher order finite difference approximations, are straightforward to implement
whereas others, such as adaptive mesh refinement, have only been tackled for 1D characteristic
codes [197
].
A major source of numerical error in characteristic evolution arises from the intergrid interpolations
arising from the multiple patches necessary to coordinatize the spherical cross-sections of the outgoing null
hypersurfaces. More accurate methods have been developed to reduce this interpolation error, as
discussed in Section 4.1. In a test problem involving a scalar wave , the accuracies of the
circular-stereographic and cubed-sphere methods were compared [14
]. For equivalent computational
expense, the cubed-sphere error in the scalar field
was
the circular-stereographic error
but the advantage was smaller for the higher
-derivatives (angular derivatives) required in
gravitational waveform extraction. The cubed-sphere error
was
the stereographic
error.
In order to appreciate why waveforms are not easy to extract accurately it is worthwhile to review the
calculation of the required asymptotic quantities. A simple approach to Penrose compactification is by
introducing an inverse surface area coordinate , so that future null infinity
is given by
[237
]. In the resulting
Bondi coordinates, where
is the retarded time defined
on the outgoing null hypersurfaces and
are angular coordinates along the outgoing null rays, the
physical space-time metric
has conformal compactification
of the form
The news function and Weyl component , which describe the radiation, are constructed from the
leading coefficients in an expansion of
in powers of
. The requirement of asymptotic flatness
imposes relations between these expansion coefficients. In terms of the Einstein tensor
and covariant derivative
associated with
, the vacuum Einstein equations become
The gravitational waveform depends on , which in turn depends on the leading terms in the
expansion of
:
However, in the computational frame the news function has the more complicated form
where Similar complications appear in extraction. Asymptotic flatness implies that the Weyl tensor
vanishes at
, i.e.
. This is the conformal space statement of the peeling property [187].
Let
be an orthonormal null tetrad such that
and
at
. Then the
radiation is described by the limit
As in the case of the news function, the general expression (58) for
must be used. This challenges
numerical accuracy due to the large number of terms and the appearance of third angular derivatives. For
instance, in the linearized approximation, the value of
on
is given by the fairly complicated
expression
These linearized expressions provide a starting point to compare the advantages between computing the
radiation via or
. The troublesome gauge terms involving
,
and
all vanish in inertial
Bondi coordinates (where
). One difference is that
contains third order angular
derivatives, e.g.
, as opposed to second angular derivatives for
. This means that the
smoothness of the numerical error is more crucial in the
approach. Balancing this,
contains
the
term, which is a potential source of numerical error since
must be evolved via
(56
).
The accuracy of waveform extraction via the Bondi news function and its counterpart
constructed from the Weyl curvature has been compared in a linearized gravitational wave test
problem [14
]. The results show that both methods are competitive, although the
approach has an
edge.
However, even though both methods were tested to be second order convergent, there was still
considerable error, of the order of for grids of practical size. This error reflects the intrinsic difficulty
in extracting waveforms because of the delicate cancellation of leading order terms in the underlying metric
and connection when computing the
radiation field. It is somewhat analogous to the experimental
task of isolating a transverse radiation field from the longitudinal fields representing the total mass, while in
a very non-inertial laboratory. In the linearized wave test carried out in [14
], the news consisted of the sum
of three terms,
, where because of cancellations
. The individual terms
,
and
had small fractional error but the cancellations magnified the fractional error in
.
The tests in [14] were carried out with a characteristic code using the circular-stereographic patches.
The results are in qualitative agreement with tests of CCE using a cubed-sphere code [201], which in
addition confirmed the expectation that fourth-order finite difference approximations for the -operator
gives improved accuracy. As demonstrated recently [111], once all the necessary infrastructure for
interpatch communication is in place, an advantage of the cubed-sphere approach is that its shared
boundaries admit a highly scalable algorithm for parallel architectures.
Another alternative is to carry out a coordinate transformation in the neighborhood of to inertial
Bondi coordinates, in which the news calculation is then quite clean numerically. This approach was
implemented in [41] and shown to be second order convergent in Robinson–Trautman and Schwarzschild
testbeds. However, it is clear that this coordinate transformation also involves the same difficult numerical
problem of extracting a small radiation field in the presence of the large gauge effects that are present in the
primary output data.
These underlying gauge effects which complicate CCE are introduced at the inner extraction worldtube
and then propagate out to . Perturbative waveform extraction suffers the same problem. Lehner and
Moreschi [167] have shown that the delicate issues involved at
have counterparts in
extraction of
radiation on a finite worldtube. They show that some of the techniques used at
can also be used to
reduce the effect of some of these ambiguities, in particular the ambiguity arising from the conformal factor
. The analogue of
on a finite worldtube can eliminate some of the non-inertial effects that might
enter the radiation waveform. In addition, use of normalization conventions on the null tetrad defining
analogous to the conventions at
can avoid other spurious errors. This approach can also be
used to correct gauge ambiguities in the calculation of momentum recoil in the merger of black
holes [103].
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