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 . Because the
transformation from null coordinates
to Cauchy coordinates
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
error, this error accumulates over many cycles to produce an error of order
in the
phase.
Anderson and Hobill proposed that their method be extended to general relativity by matching a
numerical solution to an analytic 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, 255
, 251
, 214
].
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 [240] and Zerilli [309
] equations using
traditional spacelike Cauchy hypersurfaces. At one level, these approaches extract the radiation from a
numerical solution in a region with outer boundary
by using data on an inner worldtube
to
construct the perturbative solution. Ambiguities are avoided by use of Moncrief’s gauge invariant
perturbation quantities [212]. For this to work,
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
be sufficiently far outside
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
instead of a perturbative solution, i.e., Cauchy-characteristic extraction [56
] (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., , where
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.,
, 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 [246
, 220
] 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 .
Cauchy-perturbative matching [255, 251] has been implemented using the same modular structure
described for CCM in Section 5.2. Nagar and Rezzolla [214
] 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.
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
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