5.4 Perturbative matching schemes

In numerous analytic studies outside of general relativity, matching techniques have successfully cured pathologies in perturbative expansions [215]. Matching is a strategy for obtaining a global solution by patching together solutions obtained using different coordinate systems for different regions. By adopting each coordinate system to a length scale appropriate to its domain, a globallyconvergent perturbation expansion is sometimes possible in cases where a single coordinate system would fail.

In general relativity, Burke showed that matching could be used to eliminate some of the divergences arising in perturbative calculations of gravitational radiation [70]. Kates and Kegles further showed that use of an exterior null coordinate system in the matching scheme could eliminate problems in the perturbative treatment of a scalar radiation field on a Schwarzschild background [179]. The Schwarzschild light cones have drastically different asymptotic behavior from the artificial Minkowski light cones used in perturbative expansions based upon a flat space Green function. Use of the Minkowski light cones leads to nonuniformities in the expansion of the radiation fields, which are eliminated by the use of true null coordinates in the exterior. Kates, Anderson, Kegles, and Madonna extended this work to the fully general-relativistic case and reached the same conclusion [10]. Anderson later applied this approach to the slow motion approximation of a binary system and obtained a derivation of the radiation-reaction effect on the orbital period, which avoided some objections to earlier approaches [6]. The use of the true light cones was also essential in formulating as a mathematical theorem that the Bondi news function satisfies the Einstein quadrupole formula to leading order in a Newtonian limit [304]. Although questions of mathematical consistency still remain in the perturbative treatment of gravitational radiation, it is clear that the use of characteristic methods pushes these problems to a higher perturbative order.

One of the first computational applications of characteristic matching was a hybrid numerical-analytical treatment by Anderson and Hobill of the test problem of nonlinear 1D scalar waves [7, 8, 9]. They matched an inner numerical solution to a far field solution, which was obtained by a perturbation expansion. A key ingredient is that the far field is solved in retarded null coordinates (u,r). Because the transformation from null coordinates (u,r) to Cauchy coordinates (t,r) is known analytically for this problem, the matching between the null and Cauchy solutions is quite simple. Causality was enforced by requiring that the system be stationary prior to some fixed time. This eliminates extraneous incoming radiation in a physically correct way in a system, which is stationary prior to a fixed time but it is nontrivial to generalize, say, to the problem of radiation from an orbiting binary.

Later, a global, characteristic, numerical study of the self-gravitating version of this problem confirmed that the use of the true null cones is essential in getting the correct radiated waveform [145]. For quasi-periodic radiation, the phase of the waveform is particular sensitive to the truncation of the outer region at a finite boundary. Although a perturbative estimate would indicate an 𝒪 (M ∕R ) error, this error accumulates over many cycles to produce an error of order π in the phase.

Anderson and Hobill proposed that their method be extended to general relativity by matching a numerical solution to an analytic 1∕r expansion in null coordinates. Most perturbative-numerical matching schemes that have been implemented in general relativity have been based upon perturbations of a Schwarzschild background using the standard Schwarzschild time slicing [1, 3, 2, 4, 255Jump To The Next Citation Point, 251Jump To The Next Citation Point, 214Jump To The Next Citation Point]. It would be interesting to compare results with an analytic-numeric matching scheme based upon the true null cones. Although the full proposal by Anderson and Hobill has not been carried out, characteristic techniques have been used [207, 77, 174] to study the radiation content of numerical solutions by treating the far field as a perturbation of a Schwarzschild spacetime.

Most metric-based treatments of gravitational radiation are based upon perturbations of the Schwarzschild metric and solve the underlying Regge–Wheeler [240Jump To The Next Citation Point] and Zerilli [309Jump To The Next Citation Point] equations using traditional spacelike Cauchy hypersurfaces. At one level, these approaches extract the radiation from a numerical solution in a region with outer boundary ℬ by using data on an inner worldtube 𝒲 to construct the perturbative solution. Ambiguities are avoided by use of Moncrief’s gauge invariant perturbation quantities [212]. For this to work, 𝒲 must not only be located in the far field, i.e., many wavelengths from the source, but, because of the lack of proper outer boundary data, it is necessary that the boundary ℬ be sufficiently far outside 𝒲 so that the extracted radiation is not contaminated by back-reflection for some significant window of time. This poses extreme computational requirements in a 3D problem. This extraction strategy has also been carried out using characteristic evolution in the exterior of 𝒲 instead of a perturbative solution, i.e., Cauchy-characteristic extraction [56Jump To The Next Citation Point] (see Section 6).

A study by Babiuc, Szilágyi, Hawke, and Zlochower carried out in the perturbative regime [15] shows that CCE compares favorably with Zerilli extraction and has advantages at small extraction radii. When the extraction worldtube is sufficiently large, e.g., r = 200 λ, where λ is the characteristic wavelength of the radiation, the Zerilli and CCE methods both give excellent results. However, the accuracy of CCE remains unchanged at small extraction radii, e.g., r = 10 λ, whereas the Zerilli approach shows error associated with near zone effects. This flexibility to apply CCE to small extraction radii has proved advantageous in the simulations of stellar collapse [246Jump To The Next Citation Point, 220Jump To The Next Citation Point] discussed in Section 6.3.

The contamination of the extracted radiation by back-reflection can only be completely eliminated by matching to an exterior solution, which injects the physically-appropriate boundary data on 𝒲. Cauchy-perturbative matching [255, 251] has been implemented using the same modular structure described for CCM in Section 5.2. Nagar and Rezzolla [214Jump To The Next Citation Point] have given a review of this approach. At present, perturbative matching and CCM share the common problem of long term stability of the outer Cauchy boundary in 3D applications.

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