It is my pleasure to review progress in numerical relativity based upon characteristic evolution. In the spirit of Living Reviews in Relativity, I invite my colleagues to continue to send me contributions and comments at winicour@pitt.edu.

We are now in 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 Cauchy evolution codes, which are stable and accurate in some sufficiently-bounded domain. The pioneering works [235] (based upon a harmonic formulation) and [78, 21] (based upon BSSN formulations [268, 38]) have initiated dramatic progress in the ability of these codes to simulate the inspiral and merger of binary black holes, the premier problem in classical relativity. Global solutions of binary black holes are another matter. Characteristic evolution codes have been successful in treating the exterior region of asymptotically-flat spacetimes extending to future null infinity. Just as several coordinate patches are necessary to describe a spacetime with nontrivial topology, the most effective attack on the binary black-hole waveform might involve a global solution patched together from pieces of spacetime handled by a combination of different codes and techniques.

Most of the effort in numerical relativity has centered about Cauchy codes based upon the {3 + 1} formalism [308], which evolve the spacetime inside an artificially-constructed outer boundary. It has been common practice in Cauchy simulations of binary black holes to compute the waveform from data on a finite extraction worldtube inside the outer boundary, using perturbative methods based upon introducing a Schwarzschild background in the exterior region [1, 3, 2, 4, 255, 251, 214]. In order to properly approximate the waveform at null infinity the extraction worldtube must be sufficiently large but at the same time causally and numerically isolated from errors propagating in from the outer boundary. Considerable improvement in this approach has resulted from efficient methods for dealing with a very large outer boundary and from techniques to extrapolate the extracted waveform to infinity. However, this is not an ideally efficient approach and is especially impractical to apply to simulations of stellar collapse. A different approach, which is specifically tailored to study radiation at null infinity, can be based upon the characteristic initial-value problem. This eliminates error due to asymptotic approximations and the gauge effects introduced by the choice of a finite extraction worldtube.

In the 1960s, Bondi [62, 63] and Penrose [227] pioneered the use of null hypersurfaces to describe gravitational waves. The characteristic initial-value problem did not receive much attention before its importance in general relativity was recognized. Historically, the development of computational physics has focused on hydrodynamics, where the characteristics typically do not define useful coordinate surfaces and there is no generic outer boundary behavior comparable to null infinity. But this new approach has flourished in general relativity. It has led to the first unambiguous description of gravitational radiation in a fully nonlinear context. By formulating asymptotic flatness in terms of characteristic hypersurfaces extending to infinity, it was possible to reconstruct, in a nonlinear geometric setting, the basic properties of gravitational waves, which had been developed in linearized theory on a Minkowski background. The major new nonlinear features were the Bondi mass and news function, and the mass loss formula relating them. The Bondi news function is an invariantly-defined complex radiation amplitude , whose real and imaginary parts correspond to the time derivatives and of the “plus” and “cross” polarization modes of the strain incident on a gravitational wave antenna. The corresponding waveforms are important both for the design of detection templates for a binary black-hole inspiral and merger and for the determination of the resulting recoil velocity.

The recent success of Cauchy evolutions in simulating binary black holes emphasizes the need to apply global techniques to accurate waveform extraction. This has stimulated several attempts to increase the accuracy of characteristic evolution. The Cauchy simulations have incorporated increasingly sophisticated numerical techniques, such as mesh refinement, multi-domain decomposition, pseudo-spectral collocation and high-order (in some cases eighth-order) finite difference approximations. The initial characteristic codes were developed with unigrid second-order accuracy. One of the prime factors affecting the accuracy of any characteristic code is the introduction of a smooth coordinate system covering the sphere, which labels the null directions on the outgoing light cones. This is also an underlying problem in meteorology and oceanography. In a pioneering paper on large-scale numerical weather prediction, Phillips [229] put forward a list of desirable features for a mapping of the sphere to be useful for global forecasting. The first requirement was the freedom from singularities. This led to two distinct choices, which had been developed earlier in purely geometrical studies: stereographic coordinates (two coordinate patches) and cubed-sphere coordinates (six patches). Both coordinate systems have been rediscovered in the context of numerical relativity (see Section 4.1). The cubed-sphere method has stimulated two new attempts at improved codes for characteristic evolution (see Section 4.2.4). An ingenious third treatment, based upon a toroidal map of the sphere, was devised in developing a characteristic code for Einstein equations [36] (see Section 4.1.3).

Another issue affecting code accuracy is the choice between a second or first differential order reduction of the evolution system. Historically, the predominant importance of computational fluid dynamics has favored first-order systems, in particular the reduction to symmetric hyperbolic form. However, in acoustics and elasticity theory, where the natural treatment is in terms of second-order wave equations, an effective argument for the second-order form has been made [187, 188]. In general relativity, the question of whether first or second-order formulations are more natural depends on how Einstein’s equations are reduced to a hyperbolic system by some choice of coordinates and variables. The second-order form is more natural in the harmonic formulation, where the Einstein equations reduce to quasilinear wave equations. The first-order form is more natural in the Friedrich–Nagy formulation [118], which includes the Weyl tensor among the evolution variables, and was used in the first demonstration of a well-posed initial-boundary value problem for Einstein’s equations. Investigations of first-order formulations of the characteristic initial-value problem are discussed in Section 4.2.3.

The major drawback of a stand-alone 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 evolution problem [283]. 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 [120], although a computational implementation has not yet been attempted.

In the typical setting for the characteristic initial-value problem, the domain of dependence of a single smooth 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 inner 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, the stability conditions near the vertex of a light cone place a strong restriction on the allowed time step [146]. Nevertheless, point caustics in general relativity have been successfully handled for axisymmetric vacuum spacetimes [142]. Progress toward extending these results to realistic astrophysical sources has been made by coupling an axisymmetric characteristic gravitational-hydro code with a high-resolution shock-capturing code for the relativistic hydrodynamics, as initiated in the thesis of Siebel [269]. This has enabled the global characteristic simulation of the oscillation and collapse of a relativistic star in which the emitted gravitational waves are computed at null infinity (see Sections 7.1 and 7.2). Nevertheless, computational demands to extend these results to 3D evolution would be prohibitive using current generation supercomputers, due to the small timestep required at the vertex of the null cone (see Section 3.3). This is an unfortunate feature of present-day finite-difference codes, which might be eliminated by the use, say, of a spectral approach. Away from the caustics, characteristic evolution offers myriad computational and geometrical advantages. Vacuum simulations of black-hole spacetimes, where the inner boundary can be taken to be the white-hole horizon, offer a scenario where both the timestep and caustic problems can be avoided and three-dimensional simulations are practical (as discussed in Section 4.5). An early example was the study of gravitational radiation from the post-merger phase of a binary black hole using a fully-nonlinear three-dimensional characteristic code [311, 312].

At least in the near future, fully three-dimensional computational applications of characteristic evolution are likely to be restricted to some mixed form, in which data is prescribed on a non-singular but incomplete initial null hypersurface and on a second inner boundary , which together with the initial null hypersurface determines a nontrivial domain of dependence. The hypersurface may be either (i) null, (ii) timelike or (iii) spacelike, as schematically depicted in Figure 1. The first two possibilities give rise to (i) the double null problem and (ii) the nullcone-worldtube problem. Possibility (iii) has more than one interpretation. It may be regarded as a Cauchy initial-boundary value problem where the outer boundary is null. An alternative interpretation is the Cauchy-characteristic matching (CCM) problem, in which the Cauchy and characteristic evolutions are matched transparently across a worldtube W, as indicated in Figure 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 treating the inner region in Cartesian coordinates, where spherical coordinates would break down.

In this review, we trace the development of characteristic algorithms from model 1D problems to a 2D axisymmetric code, which computes the gravitational radiation from the oscillation and gravitational collapse of a relativistic star, 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 to successful implementation in the linear regime and through current attempts to treat the binary black-hole problem.

CCM eliminates the need of outer boundary data for the Cauchy evolution and supplies the waveform at null infinity via a characteristic evolution. At present, the only successful 3D application of CCM in general relativity has been to the linearized matching problem between a 3D characteristic code and a 3D Cauchy code based upon harmonic coordinates [287] (see Section 5.8). Here the linearized Cauchy code satisfies a well-posed initial-boundary value problem, which seems to be a critical missing ingredient in previous attempts at CCM in general relativity. Recently, a well-posed initial-boundary value problem has been established for fully nonlinear harmonic evolution [192] (see Section 5.3), which should facilitate the extension of CCM to the nonlinear case.

Cauchy-characteristic extraction (CCE), which is one of the pieces of the CCM strategy, also supplies the waveform at null infinity by means of a characteristic evolution. However, in this case the artificial outer Cauchy boundary is left unchanged and the data for the characteristic evolution is extracted from Cauchy data on an interior worldtube. Since my last review, the most important development has been the application of CCE to the binary black-hole problem. Beginning with the work in [243], CCE has become an important tool for gravitational-wave data analysis (see Section 6.2). The application of CCE to this problem was developed as a major part of the PhD thesis of Reisswig [241].

In previous reviews, I tried to include material on the treatment of boundaries in the computational mathematics and fluid dynamics literature because of its relevance to the CCM problem. The fertile growth of this subject has warranted a separate Living Review on boundary conditions, which is presently under construction and will appear soon [261]. In anticipation of this, I will not attempt to keep this subject up to date except for material of direct relevance to CCM. See [260, 250] for independent reviews of boundary conditions that have been used in numerical relativity.

The well-posedness of the associated initial-boundary value problem, i.e., that there exists a unique solution, which depends continuously on the data, is a necessary condition for a successful numerical treatment. In addition to the forthcoming Living Review [261], this subject is covered in the review [119] and the book [185].

If well-posedness can be established using energy estimates obtained by integration by parts with respect to the coordinates defining the numerical grid, then the analogous finite-difference estimates obtained by summation by parts [191] provide guidance for a stable finite-difference evolution algorithm. See the forthcoming Living Review [261] for a discussion of the application of summation by parts to numerical relativity.

The problem of computing the evolution of a neutron star in close orbit about a black hole is of clear importance to the new gravitational wave detectors. The interaction with the black hole could be strong enough to produce a drastic change in the emitted waves, say by tidally disrupting the star, so that a perturbative calculation would be inadequate. The understanding of such nonlinear phenomena requires well-behaved numerical simulations of hydrodynamic systems satisfying Einstein’s equations. Several numerical relativity codes for treating the problem of a neutron star near a black hole have been developed, as described in the Living Review on “Numerical Hydrodynamics in General Relativity” by Font [109]. Although most of these efforts concentrate on Cauchy evolution, the characteristic approach has shown remarkable robustness in dealing with a single black hole or relativistic star. In this vein, axisymmetric studies of the oscillation and gravitational collapse of relativistic stars have been achieved (see Section 7.2) and progress has been made in the 3D simulation of a body in close orbit about a Schwarzschild black hole (see Sections 4.6 and 7.3).

Living Rev. Relativity 15, (2012), 2
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