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

Asymptotic flatness immediately implies that so that is a null hypersurface with generators in the direction. From Equation (70) there also follows the existence of a smooth trace-free field defined on by where is the expansion of . The expansion depends upon the conformal factor used to compactify . In an inertial conformal Bondi frame, tailored to a standard Minkowski metric at , . But this is not the case for the computational frame used in characteristic evolution, which is determined by conditions on the inner extraction worldtube.The gravitational waveform depends on , which in turn depends on the leading terms in the expansion of :

In an inertial conformal Bondi frame, (the unit-sphere metric), and the Bondi news function reduces to the simple form where is a complex polarization dyad on the unit sphere, i.e., . The spin rotation freedom is fixed by parallel propagation along the generators of , so that the real and imaginary parts of correctly describe the and polarization modes of inertial observers at .However, in the computational frame the news function has the more complicated form

where is the conformal factor relating to the unit-sphere metric, i.e., . The conformal factor obeys the elliptic equation governing the conformal transformation relating the metric of the cross-sections of to the unit-sphere metric, where is the curvature scalar and the covariant derivative associated with . By first solving Equation (75) at the initial retarded time, can then be determined at later times by evolving it according to the asymptotic relation All of these procedures introduce numerical error, which presents a challenge for computational accuracy, especially because of the appearance of second angular derivatives of in the news function (74).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

which corresponds in Newman–Penrose notation to . The main calculational result in [13] is that which is independent of the freedom in the choice of . In inertial Bondi coordinates, this reduces to which is related to the Bondi news function by so that with up to numerical error.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

where and . In the same approximation, the news function is given by (The relationship (80) still holds in the linearized approximation but in the nonlinear case, the derivative along the generators of is and Equation (80) must be modified accordingly.)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].

Living Rev. Relativity 15, (2012), 2
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