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 [227].
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 (78) 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
Equation (76
).
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 [13
]. 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 in test beds with
analytic data, there was still considerable error, of the order of 5% 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 [13
], 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 [13] 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 [242], which, in
addition, confirmed the expectation that fourth-order finite-difference approximations for the
-operator
gives improved accuracy. As demonstrated recently [134], 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 [48] 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 , but they are of numerical origin and can be reduced with
increased accuracy. Perturbative waveform extraction suffers the same gauge effects but in this
case they are of analytic origin and cannot be controlled by numerical accuracy. Lehner and
Moreschi [199] have shown that the delicate gauge issues involved at
have counterparts in
extraction of radiation on a finite worldtube. They show how some of the analytic techniques used
at
can also be used to reduce the effect of these ambiguities on a finite worldtube, in
particular the ambiguity arising from the conformal factor
. The analogue of
on a finite
worldtube can reduce some of the non-inertial effects that enter the perturbative 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 reduce
gauge ambiguities in the perturbative calculation of momentum recoil in the merger of black
holes [125].
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
![]() This work is licensed under a Creative Commons License. E-mail us: |