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  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  and generalized to three dimensions by Sachs , or as null tetrad approaches in which the Bianchi identities appear as part of the system of equations, as developed by Newman and Penrose .
At the outset, null formalisms were applied to construct asymptotic solutions at null infinity by means of expansions. Soon afterward, Penrose  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) . 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 , 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 . 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  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
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:
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 .
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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