We are entering an era in which Einstein's equations can effectively be considered solved at the local level. Several groups, as reported in these Reviews, have developed 3D codes which are stable and accurate in some sufficiently local setting. Global solutions are another matter. In particular, there is no single code in existence today which purports to be capable of computing the waveform of gravitational radiation emanating from the inspiral and merger of two black holes, the premier problem in classical relativity. Just as several coordinate patches are necessary to describe a space-time with nontrivial topology, the most effective attack on the binary black hole problem is likely to involve patching together regions of space-time handled by different codes.
Most work in numerical relativity is based upon the Cauchy ``3 + 1'' formalism , with the gravitational radiation extracted by perturbative Cauchy methods which introduce an artificial Schwarzschild background [2, 3, 4, 5]. These wave extraction methods have not been tested in a fully nonlinear 3D setting. Another approach specifically tailored to study radiation can be based upon the characteristic initial value problem. In the 1960's, Bondi [6, 7] and Penrose  pioneered the use of null hypersurfaces to study radiation. This new approach has flourished in general relativity. The standard description of the ``plus'' and ``cross'' polarization modes of gravitational radiation is in terms of the real and imaginary parts of the Bondi news function at future null infinity ().
From a computational standpoint, the major drawback of the characteristic approach arises from the formation of caustics in the light rays generating the null hypersurfaces. In the most ambitious scheme proposed at the theoretical level such caustics would be treated ``head-on'' as part of the dynamical problem . This is a profoundly attractive idea. Only a few structural stable caustics can arise in numerical evolution, and their geometrical properties are well enough understood to model their singular behavior numerically . However, a computational implementation of this approach has not been achieved. It seems to be a great idea that is ahead of its time.
In the typical setting for a characteristic initial value problem, the domain of dependence of a single nonsingular null hypersurface is empty. In order to obtain a nontrivial evolution problem, the null hypersurface must either be completed to a caustic-crossover region where it pinches off, or an additional boundary must be introduced. So far, the only caustics that have been successfully evolved numerically in general relativity are pure point caustics (the complete null cone problem). When spherical symmetry is not present, it turns out that the stability conditions near the vertex of a nonsingular light cone place a strong restriction on the allowed time step . Point caustics in general relativity have been successfully handled this way for axisymmetric space-times , but the computational demands for 3D evolution would be prohibitive using current generation supercomputers. This is unfortunate because, away from the caustics, the characteristic evolution offers myriad computational and geometrical advantages.
As a result, at least in the near future, the computational application of characteristic evolution is likely to be restricted to some mixed form, in which boundary conditions are also set on a non-singular but incomplete initial null hypersurface and on a second nonsingular hypersurface (or perhaps several), which together with the initial null hypersurface present a nontrivial domain of dependence. This second hypersurface may itself be either (i) null, (ii) timelike or (iii) spacelike. These possibilities give rise to the (i) the double null problem, (ii) the nullcone-worldtube problem or (iii) the Cauchy-characteristic matching (CCM) problem.
In CCM, it is possible to choose the matching interface between the Cauchy and characteristic regions to be a null hypersurface, but it is more practical to match across a timelike worldtube. CCM combines the advantages of characteristic evolution in treating the outer radiation zone in spherical coordinates which are naturally adapted to the topology of the worldtube with the advantages of Cauchy evolution in Cartesian coordinates in the region where spherical coordinates would break down.
In this review, we trace the development of characteristic algorithms from model 1D problems to a 3D code for black holes that attains the Holy Grail of numerical relativity, as specified 12 years ago by Shapiro and Teukolsky . And we trace the development of CCM from early feasibility studies through current attempts to treat the binary black hole problem.
|Characteristic Evolution and Matching
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