We have entered an era in which Einstein's equations can effectively be considered solved at the local level. Several groups, as reported here and in other Living 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 spacetime with nontrivial topology, the most effective attack on the binary black hole problem is likely to involve patching together regions of spacetime handled by different codes.

Much work in numerical relativity is based upon the Cauchy {3 + 1} formalism [156], with the gravitational radiation extracted by perturbative Cauchy methods which introduce an artificial Schwarzschild background [1, 3, 2, 142]. These wave extraction methods have not been tested in a nonlinear 3D setting. A different approach which is specifically tailored to study radiation is based upon the characteristic initial value problem. In the 1960's, Bondi [34, 35] and Penrose [111] pioneered the use of null hypersurfaces to describe gravitational waves. This new approach has flourished in general relativity. It yields the standard description of the ``plus'' and ``cross'' polarization modes of gravitational radiation 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 [136]. 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 [62]. However, a computational implementation of this approach has not yet been achieved. It is a great idea that is perhaps ahead of its time.

In the typical setting for the 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 light cone place a strong restriction on the allowed time step [91]. Point caustics in general relativity have been successfully handled this way for axisymmetric spacetimes [74], 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 data is prescribed on a non-singular but incomplete initial null hypersurface N and on a second boundary hypersurface B, which together with the initial null hypersurface determine a nontrivial domain of dependence. This second hypersurface may itself be either (i) null, (ii) timelike or (iii) spacelike. These possibilities give rise to (i) the double null problem, (ii) the nullcone-worldtube problem or (iii) the Cauchy-characteristic matching (CCM) problem, in which the Cauchy and characteristic evolutions are matched transparently across a worldtube W, as schematically depicted in Fig. 1 .

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 designed to calculate the waveform emitted in the merger to ringdown phase of a binary black hole. And we trace the development of CCM from early feasibility studies through current attempts to treat the binary black hole problem.

There have been several notable developments since my last review. Most important for future progress has been the award of three new doctorates, with theses based upon characteristic evolution codes: Denis Pollney [115] at the the University of Southampton, Luis Lehner [97] and Bela Szilágyi [137] at the University of Pittsburgh. Lehner received the Nicholas Metropolis award of the American Physical Society for his thesis research, recognizing characteristic evolution as an important computational technique. In the area of scientific application, light cone evolution codes for hydrodynamics have been used to study astrophysical processes. In addition, a 3D characteristic vacuum code developed at the University of Canberra has been successfully applied to the scattering of waves off a Schwarzschild black hole (see Sec. 3.5) and the Pittsburgh 3D code is being applied to obtain the postmerger to ringdown waveform from binary black holes (see Sec. 3.6). Simulations from these studies can be viewed at the Canberra [149] and Pittsburgh [150] web sites.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-2001-3
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |