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2 The Characteristic Initial Value Problem

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 [45] 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 [46Jump To The Next Citation Point] and generalized to 3 dimensions by Sachs [184Jump To The Next Citation Point], or as null tetrad approaches in which the Bianchi identities appear as part of the system of equations, as developed by Newman and Penrose [158].

At the outset, null formalisms were applied to construct asymptotic solutions at null infinity by means of 1/r expansions. Soon afterward, Penrose devised the conformal compactification of null infinity I (“scri”), thereby reducing to geometry the asymptotic quantities describing the physical properties of the radiation zone, most notably the Bondi mass and news function [166]. 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 geometrised the gravitational far field.

The initial focus on asymptotic solutions clarified the kinematic properties of radiation fields but could not supply the waveform from a specific source. It was soon realized that instead of carrying out a 1/r expansion, one could reformulate the approach in terms of the integration of ordinary differential equations along the characteristics (null rays) [209Jump To The Next Citation Point]. The integration constants supplied on some inner boundary then determined the specific waveforms obtained at infinity. In the double-null initial value problem of Sachs [185Jump To The Next Citation Point], 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 [209Jump To The Next Citation Point]. 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 [72] 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. The existence and uniqueness of solutions to the associated boundary value problems, which has obvious relevance to the success of numerical simulations, is treated in a separate Living Review in Relativity on “Theorems on Existence and Global Dynamics for the Einstein Equations” by Rendall [179].

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

F,c = HF [F, G] (1)
G,uc = HG[F, G, G,u]. (2)
Here F represents a set of hypersurface variables, G a set of evolution variables, and H F and H G are nonlinear hypersurface operators, i.e. they operate locally on the values of F, G and G,u intrinsic to a single null hypersurface. In the Bondi formalism, these hypersurface equations have a hierarchical structure in which the members of the set F can be integrated in turn in terms of the characteristic data for the evolution variables and prior members of the hierarchy. In addition to these main equations, there is a subset of four 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 necessarily 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:

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 [102Jump To The Next Citation Point].

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 as part of the evolution 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 [69Jump To The Next Citation Point202]. 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 [151189]. 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 horizon where the black holes initially form and merge [144Jump To The Next Citation Point133Jump To The Next Citation Point].

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