6 Cauchy-Characteristic Extraction of Waveforms

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 [200254] treatment of the perturbed metric, as reviewed in [177], and also by calculating the Newman-Penrose Weyl component Ψ 4, as first done for the binary black hole problem in [171956618]. 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 Ψ4 (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 [167Jump To The Next Citation Point] 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 [46255256] 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 [44Jump To The Next Citation Point], 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 Ψ4 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 [197Jump To The Next Citation Point].

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 Φ, the accuracies of the circular-stereographic and cubed-sphere methods were compared [14Jump To The Next Citation Point]. For equivalent computational expense, the cubed-sphere error in the scalar field ℰ(Φ ) was ≈ 1 3 the circular-stereographic error but the advantage was smaller for the higher ∂-derivatives (angular derivatives) required in gravitational waveform extraction. The cubed-sphere error ¯ 2 ℰ (∂∂ Φ ) was 4 ≈ 5 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 ℓ = 1∕r, so that future null infinity + ℐ is given by ℓ = 0 [237Jump To The Next Citation Point]. In the resulting μ A x = (u,ℓ,x ) Bondi coordinates, where u is the retarded time defined on the outgoing null hypersurfaces and xA are angular coordinates along the outgoing null rays, the physical space-time metric gμν has conformal compactification ˆgμν = ℓ2gμν of the form

ˆg dx μdxν = − αdu2 + 2e2βdudℓ − 2h U BdudxA + h dxAdxB, (49 ) μν AB AB
where α, β, U A and hAB are smooth fields at ℐ+.

The news function and Weyl component Ψ4, which describe the radiation, are constructed from the leading coefficients in an expansion of ˆg μν in powers of ℓ. The requirement of asymptotic flatness imposes relations between these expansion coefficients. In terms of the Einstein tensor ˆG μν and covariant derivative ˆ∇ μ associated with gˆμν, the vacuum Einstein equations become

2 α α − ℓ Gˆμν = 2ℓ(ˆ∇ μˆ∇ νℓ − ˆgμν∇ˆ ∇ˆα ℓ) + 3ˆgμν(ˆ∇ ℓ)ˆ∇ αℓ. (50 )
Asymptotic flatness immediately implies that ˆgℓℓ = (ˆ∇ αℓ)ˆ∇ ℓ = O(ℓ) α so that ℐ+ is a null hypersurface with generators in the ˆμ ∇ ℓ direction. From (50View Equation) there also follows the existence of a smooth trace-free field ˆΣ μν defined on ℐ+ by
ˆΣ := lim 1-(ˆ∇ ˆ∇ ℓ − 1ˆg ˆΘ ), (51 ) μν ℓ→0 ℓ μ ν 4 μν
where ˆΘ := ˆ∇ μˆ∇ ℓ μ is the expansion of ℐ+. The expansion ˆΘ depends upon the conformal factor used to compactify + ℐ. In an inertial conformal Bondi frame, tailored to a standard Minkowski metric at + ℐ, ˆΘ = 0. 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 ˆΣ μν, which in turn depends on the leading terms in the expansion of ˆgμν:

2 A A hAB = HAB + ℓcAB + O(ℓ ), β = H + O (ℓ), U = L + O (ℓ). (52 )
In an inertial conformal Bondi frame, HAB = QAB (the unit sphere metric), H = LA = 0 and the Bondi news function reduces to the simple form
1 N = --QAQB ∂ucAB, (53 ) 4
where QA is a complex polarization dyad on the unit sphere, i.e. QAB = Q (A ¯QB ). The spin rotation freedom Q β → e−iγQ β is fixed by parallel propagation along the generators of ℐ+, so that the real and imaginary parts of N correctly describe the ⊕ and ⊗ polarization modes of inertial observers at + ℐ.

However, in the computational frame the news function has the more complicated form

( ) N = 1-Q αQβ ˆΣ − ω ˆ∇ ˆ∇ 1-+ 1-(∂ ˆg )(∇ˆμℓ)ˆ∇ ω , (54 ) 2 αβ α βω ω ℓαβ μ
where ω is the conformal factor relating H AB to the unit sphere metric, i.e. Q = ω2H AB AB. The conformal factor obeys the elliptic equation governing the conformal transformation relating the metric of the cross-sections of ℐ+ to the unit sphere metric,
ℛ = 2(ω2 + HABDADB log ω), (55 )
where ℛ is the curvature scalar and DA the covariant derivative associated with HAB. By first solving (55View Equation) at the initial retarded time, ω can then be determined at later times by evolving it according to the asymptotic relation
1 ˆnα∂ αlogω = − -e−2H DALA, ˆnα = ˆ∇α ℓ. (56 ) 2
All of these procedures introduce numerical error which presents a a challenge for computational accuracy, especially because of the appearance of second angular derivatives of ω in the news function (54View Equation).

Similar complications appear in Ψ4 extraction. Asymptotic flatness implies that the Weyl tensor vanishes at ℐ+, i.e. Cˆμνρσ = O (ℓ). This is the conformal space statement of the peeling property [187]. Let (ˆnμ, ˆℓμ, ˆmμ ) be an orthonormal null tetrad such that ˆnμ = ˆ∇μℓ and ˆℓμ∂μ = ∂ℓ at ℐ+. Then the radiation is described by the limit

1- 1- μ ν ρ σ Ψˆ := − 2 liℓ→m0 ℓ ˆn ˆm ˆn mˆ Cˆμνρσ, (57 )
which corresponds in Newman-Penrose notation to 0 − (1∕2 )¯ψ4. The main calculational result in [14Jump To The Next Citation Point] is that
1 μ ν ρ( ) ˆΨ = -nˆ ˆm ˆm ˆ∇ μˆΣ νρ − ∇ˆνˆΣ μρ , (58 ) 2
which is independent of the freedom mˆμ → ˆm μ + λˆnμ in the choice of m μ. In inertial Bondi coordinates, this reduces to
Ψˆ = 1-QAQB ∂2cAB, (59 ) 4 u
which is related to the Bondi news function by
ˆΨ = ∂ N (60 ) u
so that
∫ u ˆ N Ψ = N |u=0 + Ψdu, (61 ) 0
with N Ψ = N up to numerical error.

As in the case of the news function, the general expression (58View Equation) for ˆΨ 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 ˆ Ψ on + ℐ is given by the fairly complicated expression

ˆ 1- 2 1- 1- 1- 2 ¯ ¯ 2 Ψ = 2∂ u∂ℓJ − 2∂uJ − 2∂L − 8 ∂ (∂L + ∂L ) + ∂u∂ H, (62 )
where A B J = Q Q hAB and A L = Q LA. In the same approximation, the news function is given by
N = 1∂ ∂ J + 1∂2 (ω + 2H ). (63 ) 2 u ℓ 2
(The relationship (60View Equation) still holds in the linearized approximation but in the nonlinear case, the derivative along the generators of + ℐ is μ −2H A ˆn ∂μ = e (∂u + L ∂A) and (60View Equation) must be modified accordingly.)

These linearized expressions provide a starting point to compare the advantages between computing the radiation via N or N Ψ. The troublesome gauge terms involving L, H and ω all vanish in inertial Bondi coordinates (where ω = 1). One difference is that ˆ Ψ contains third order angular derivatives, e.g. ∂3¯L, as opposed to second angular derivatives for N. This means that the smoothness of the numerical error is more crucial in the ˆΨ approach. Balancing this, N contains the 2 ∂ ω term, which is a potential source of numerical error since ω must be evolved via (56View Equation).

The accuracy of waveform extraction via the Bondi news function N and its counterpart NΨ constructed from the Weyl curvature has been compared in a linearized gravitational wave test problem [14Jump To The Next Citation Point]. The results show that both methods are competitive, although the Ψ4 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 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 O (1∕r) 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 [14Jump To The Next Citation Point], the news consisted of the sum of three terms, N = A + B + C, where because of cancellations N ≈ A ∕24. The individual terms A, B and C had small fractional error but the cancellations magnified the fractional error in N.

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 ℐ+ 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 + ℐ. Perturbative waveform extraction suffers the same problem. Lehner and Moreschi [167] have shown that the delicate issues involved at + ℐ have counterparts in Ψ4 extraction of radiation on a finite worldtube. They show that some of the techniques used at ℐ+ can also be used to reduce the effect of some of these ambiguities, in particular the ambiguity arising from the conformal factor ω. The analogue of ω 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 Ψ4 analogous to the conventions at + ℐ 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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