Characteristics have traditionally played an important role in the analysis of hyperbolic partial differential equations. However, the use of characteristic hypersurfaces to supply the foliation underlying an evolution scheme has been mainly restricted to relativity. This is perhaps natural because in curved spacetime there is no longer a preferred Cauchy foliation provided by the Euclidean 3-spaces allowed in Galilean or special relativity. The method of shooting along characteristics is a standard technique in many areas of computational physics, but evolution based upon characteristic hypersurfaces is quite uniquely limited to relativity.

Bondi’s initial use of null coordinates to describe radiation fields [62] was followed by a rapid development of other null formalisms. These were distinguished either as metric based approaches, as developed for axisymmetry by Bondi, Metzner, and van der Burg [63] and generalized to three dimensions by Sachs [258], or as null tetrad approaches in which the Bianchi identities appear as part of the system of equations, as developed by Newman and Penrose [216].

At the outset, null formalisms were applied to construct asymptotic solutions at null infinity by means of expansions. Soon afterward, Penrose [227] devised the conformal compactification of null infinity (“scri”), thereby reducing to geometry the asymptotic quantities describing the physical properties of the radiation zone, most notably the Bondi mass and news function. The characteristic initial-value problem rapidly became an important tool for the clarification of fundamental conceptual issues regarding gravitational radiation and its energy content. It laid bare and geometrized the gravitational far field.

The initial focus on asymptotic solutions clarified the kinematic properties of radiation fields but could not supply the dynamical properties relating the waveform to a specific source. It was soon realized that instead of carrying out a expansion, one could reformulate the approach in terms of the integration of ordinary differential equations along the characteristics (null rays) [288]. The integration constants supplied on some inner boundary then played the role of sources in determining the specific waveforms obtained at infinity. In the double-null initial value problem of Sachs [259], the integration constants are supplied at the intersection of outgoing and ingoing null hypersurfaces. In the worldtube-nullcone formalism, the sources were represented by integration constants on a timelike worldtube [288]. These early formalisms have gone through much subsequent revamping. Some have been reformulated to fit the changing styles of modern differential geometry. Some have been reformulated in preparation for implementation as computational algorithms. The articles in [97] give a representative sample of formalisms. Rather than including a review of the extensive literature on characteristic formalisms in general relativity, I concentrate here on those approaches, which have been implemented as computational evolution schemes.

All characteristic evolution schemes share the same skeletal form. The fundamental ingredient is a foliation by null hypersurfaces , which are generated by a two-dimensional set of null rays, labeled by coordinates , with a coordinate varying along the rays. In null coordinates, the main set of Einstein equations take the schematic form

and Here represents a set of hypersurface variables, a set of evolution variables, and and are nonlinear hypersurface operators, i.e., they operate locally on the values of , and intrinsic to a single null hypersurface. In the Bondi formalism, these hypersurface equations have a hierarchical structure in which the members of the set can be integrated in turn in terms of the characteristic data for the evolution variables and the computed values of prior members of the hierarchy. In addition to these main Einstein equations, there is a subset of four subsidiary Einstein equations, which are satisfied by virtue of the Bianchi identities, provided that they are satisfied on a hypersurface transverse to the characteristics. These equations have the physical interpretation as conservation laws. Mathematically they are analogous to the constraint equations of the canonical formalism. But they are not elliptic since they may be intrinsic to null or timelike hypersurfaces, rather than spacelike Cauchy hypersurfaces.Computational implementation of characteristic evolution may be based upon different versions of the formalism (i.e., metric or tetrad) and different versions of the initial value problem (i.e., double null or worldtube-nullcone). The performance and computational requirements of the resulting evolution codes can vary drastically. However, most characteristic evolution codes share certain common advantages:

- The characteristic initial data is free, i.e., there are no elliptic constraints on the data. This eliminates the need for time consuming iterative constraint solvers with their accompanying artificial boundary conditions. This flexibility and control in prescribing initial data has the trade-off of limited experience with prescribing physically realistic characteristic initial data.
- The coordinates are very“rigid”, i.e., there is very little remaining gauge freedom.
- The constraints satisfy ordinary differential equations along the characteristics, which force any constraint violation to fall off asymptotically as .
- No second time derivatives appear so that the number of basic variables is at most half the number for the corresponding version of the Cauchy problem.
- The main Einstein equations form a system of coupled ordinary differential equations with respect to the parameter varying along the characteristics. This allows construction of an evolution algorithm in terms of a simple march along the characteristics.
- In problems with isolated sources, the radiation zone can be compactified into a finite grid boundary with the metric rescaled by as an implementation of Penrose’s conformal boundary at future null infinity . Because is a null hypersurface, no extraneous outgoing radiation condition or other artificial boundary condition is required. The analogous treatment in the Cauchy problem requires the use of hyperboloidal spacelike hypersurfaces asymptoting to null infinity [116]. For reviews of the hyperboloidal approach and its status in treating the associated three-dimensional computational problem, see [171, 110].
- The grid domain is exactly the region in which waves propagate, which is ideally efficient for radiation studies. Since each null hypersurface of the foliation extends to infinity, the radiation is calculated immediately (in retarded time).
- In black-hole spacetimes, a large redshift at null infinity relative to internal sources is an indication of the formation of an event horizon and can be used to limit the evolution to the exterior region of spacetime. While this can be disadvantageous for late time accuracy, it allows the possibility of identifying the event horizon “on the fly”, as opposed to Cauchy evolution where the event horizon can only be located after the evolution has been completed.

Perhaps most important from a practical view, characteristic evolution codes have shown remarkably robust stability and were the first to carry out long term evolutions of moving black holes [139].

Characteristic schemes also share as a common disadvantage the necessity either to deal with caustics or to avoid them altogether. The scheme to tackle the caustics head on by including their development and structure as part of the evolution [283, 120] is perhaps a great idea still ahead of its time but one that should not be forgotten. There are only a handful of structurally-stable caustics, and they have well-known algebraic properties. This makes it possible to model their singular structure in terms of Padé approximants. The structural stability of the singularities should in principle make this possible, and algorithms to evolve the elementary caustics have been proposed [92, 280]. In the axisymmetric case, cusps and folds are the only structurally-stable caustics, and they have already been identified in the horizon formation occurring in simulations of head-on collisions of black holes and in the temporarily toroidal horizons occurring in collapse of rotating matter [209, 267]. In a generic binary black-hole horizon, where axisymmetry is broken, there is a closed curve of cusps, which bounds the two-dimensional region on the event horizon where the black holes initially form and merge [197, 173].

Living Rev. Relativity 15, (2012), 2
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