6.1 Waveforms at null infinity

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 [288Jump To The Next Citation Point]. In the resulting x μ = (u,ℓ,xA) 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
μ ν 2 2β B A A B ˆgμνdx dx = − αdu + 2e dudℓ − 2hABU dudx + hABdx dx , (69 )
where α, β, U A and h AB 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 ˆgμν 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 gˆ μν, the vacuum Einstein equations become

− ℓ2Gˆμν = 2ℓ(ˆ∇ μˆ∇ νℓ − ˆgμν∇ˆα∇ˆα ℓ) + 3ˆgμν(ˆ∇ αℓ)ˆ∇ αℓ. (70 )
Asymptotic flatness immediately implies that ℓℓ ˆ α ˆ ˆg = (∇ ℓ)∇ αℓ = O(ℓ) so that + ℐ is a null hypersurface with generators in the ˆ∇μ ℓ direction. From Equation (70View Equation) there also follows the existence of a smooth trace-free field ˆΣ μν defined on ℐ+ by
ˆ 1- ˆ ˆ 1- ˆ Σμν := liℓ→m0 ℓ(∇ μ∇ νℓ − 4ˆgμνΘ ), (71 )
where ˆΘ := ˆ∇ μˆ∇ μℓ 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, ˆ Θ = 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 ˆg μν:

h = H + ℓc + O(ℓ2), β = H + O (ℓ), U A = LA + O (ℓ). (72 ) AB AB AB
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
N = 1-QAQB ∂ c , (73 ) 4 u AB
where QA is a complex polarization dyad on the unit sphere, i.e., QAB = Q (A ¯QB ). The spin rotation freedom β −iγ β Q → e Q 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

1 ( 1 1 ) N = --Q αQβ ˆΣ αβ − ω ˆ∇ αˆ∇ β-+ --(∂ ℓˆgαβ )(∇ˆμℓ)ˆ∇ μω , (74 ) 2 ω ω
where ω is the conformal factor relating HAB to the unit-sphere metric, i.e., QAB = ω2HAB. 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,
2 AB ℛ = 2(ω + H DADB log ω), (75 )
where ℛ is the curvature scalar and DA the covariant derivative associated with HAB. By first solving Equation (75View 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. (76 ) 2
All of these procedures introduce numerical error, which presents a challenge for computational accuracy, especially because of the appearance of second angular derivatives of ω in the news function (74View Equation).

Similar complications appear in Ψ4 extraction. Asymptotic flatness implies that the Weyl tensor vanishes at ℐ+, i.e., Cˆ = O(ℓ) μνρσ. This is the conformal space statement of the peeling property [227]. Let μ μ μ (ˆn , ˆℓ , ˆm ) be an orthonormal null tetrad such that μ μ ˆn = ˆ∇ ℓ and μ ˆℓ ∂μ = ∂ℓ at + ℐ. Then the radiation is described by the limit

ˆ 1- 1- μ ν ρ σ ˆ Ψ := − 2 liℓ→m0 ℓ ˆn ˆm ˆn mˆ C μνρσ, (77 )
which corresponds in Newman–Penrose notation to ¯0 − (1∕2)ψ4. The main calculational result in [13Jump To The Next Citation Point] is that
( ) ˆΨ = 1nˆμ ˆm ν ˆm ρ ˆ∇ ˆΣ − ∇ˆ ˆΣ , (78 ) 2 μ νρ ν μρ
which is independent of the freedom mˆμ → ˆm μ + λˆnμ in the choice of m μ. In inertial Bondi coordinates, this reduces to
ˆ 1- A B 2 Ψ = 4 Q Q ∂ucAB, (79 )
which is related to the Bondi news function by
ˆΨ = ∂uN (80 )
so that
∫ u N Ψ = N |u=0 + ˆΨdu, (81 ) 0
with N Ψ = N up to numerical error.

As in the case of the news function, the general expression (78View 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, (82 )
where A B J = Q Q hAB and A L = Q LA. In the same approximation, the news function is given by
1 1 2 N = -∂u ∂ℓJ + -∂ (ω + 2H ). (83 ) 2 2
(The relationship (80View Equation) still holds in the linearized approximation but in the nonlinear case, the derivative along the generators of ℐ+ is ˆnμ∂μ = e− 2H (∂u + LA ∂A) and Equation (80View 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 Ψˆ 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 Equation (76View 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 [13Jump To The Next Citation Point]. The results show that both methods are competitive, although the Ψ 4 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 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 [13Jump 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 [13Jump To The Next Citation Point] 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 + ℐ 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 ℐ+, 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 + ℐ have counterparts in Ψ4 extraction of radiation on a finite worldtube. They show how some of the analytic techniques used at ℐ+ can also be used to reduce the effect of these ambiguities on a finite worldtube, in particular the ambiguity arising from the conformal factor ω. The analogue of ω 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 Ψ4 analogous to the conventions at ℐ+ 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].

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