1 Introduction

Living Reviews in Relativity

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 in Relativity, 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 may involve patching together pieces of spacetime handled by a combination of different codes and techniques.

Most of the effort in numerical relativity has centered about the Cauchy {3 + 1} formalism [226 ], with the gravitational radiation extracted by perturbative methods based upon introducing an artificial Schwarzschild background in the exterior region [1

, 4

, 2

, 3

, 181

, 180

, 156

]. These wave extraction methods have not been thoroughly 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 [45

, 46

] and Penrose [166

] pioneered the use of null hypersurfaces to describe gravitational waves. This new approach has flourished in general relativity. It led to the first unambiguous description of gravitational radiation in a fully nonlinear context. It yields the standard linearized description of the “plus” and “cross” polarization modes of gravitational radiation in terms of the Bondi news function $N$ at future null infinity $I+$ . The Bondi news function is an invariantly defined complex radiation amplitude $N = No + + iNo x$ , whose real and imaginary parts correspond to the time derivatives $@th o+$ and $@th ox$ of the “plus” and “cross” polarization modes of the strain $h$ incident on a gravitational wave antenna.

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 evolution problem [205

]. 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 [87

], 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 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 [136 ]. Point caustics in general relativity have been successfully handled this way for axisymmetric spacetimes [106

], but the computational demands for 3D evolution would be prohibitive using current generation supercomputers. This is unfortunate because, away from the caustics, characteristic evolution offers myriad computational and geometrical advantages.

As a result, 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 N and on a second boundary hypersurface B, which together with the initial null hypersurface determine a nontrivial domain of dependence. The hypersurface B 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

Figure 1:

The three applications of characteristic evolution with data given on an initial null hypersurface N and boundary B. The shaded regions indicate the corresponding domains of dependence.

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 and 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.

This material includes several notable developments since my last review. Most important for future progress have been two Ph.D. theses based upon characteristic evolution codes. Florian Siebel’s thesis work [191

], at the Technische Universität München, integrates an axisymmetric characteristic gravitational code with a high resolution shock capturing code for relativistic hydrodynamics. This coupled general relativistic code has been thoroughly tested and has yielded state-of-the-art results for the gravitational waves produced by the oscillation and collapse of a relativistic star (see Sections 5.1 and 5.2). In Yosef Zlochower’s thesis work [228

], at the University of Pittsburgh, the gravitational waves generated from the post-merger phase of a binary black black hole is computed using a fully nonlinear three-dimensional characteristic code [229

] (see Section 3.8). He shows how the characteristic code can be employed to investigate the nonlinear mode coupling in the response of a black hole to the infall of gravitational waves.

A further notable achievement has been the successful application of CCM to the linearized matching problem between a 3D characteristic code and a 3D Cauchy code based upon harmonic coordinates [208

] (see Section 4.7). 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.

The problem of computing the evolution of a self-gravitating object, such as 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 in Relativity on “Numerical Hydrodynamics in General Relativity” by Font [80

]. 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, state-of-the-art axisymmetric studies of the oscillation and gravitational collapse of relativistic stars have been achieved (see Section 5.2) and progress has been made in the 3D simulation of a body in close orbit about a Schwarzschild black hole (see Sections 5.3 and 5.3.1).

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 makes this impractical to continue. A separate Living Review in Relativity on boundary conditions is certainly warranted and is presently under consideration. In view of this, I will not attempt to keep this subject up to date except for material of direct relevance to CCM, although I will for now retain the past material.

Animations and other material from these studies can be viewed at the web sites of the University of Canberra [217 ], Louisiana State University [148

], Pittsburgh University [218 ], and Pittsburgh Supercomputing Center [145