Bondi's initial use of null coordinates to describe radiation fields [34] 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 den Burg [35] and generalized by Sachs [122], or as null tetrad approaches in which the Bianchi identities appear as part of the set of equations, as developed by Newman and Penrose [106].

At the outset, null formalisms were applied to construct
asymptotic solutions at null infinity by means of 1/
*r*
expansions. Soon afterwards, Penrose devised the conformal
compactification of null infinity
(``scri''), thereby reducing to geometry the asymptotic
description of the physical properties of the radiation zone,
most notably the Bondi mass and news function [111]. 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 geometricized 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) [139]. 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 [123], 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 the worldtube [139]. 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. See
the articles in [50] for 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 topic of well-posedness of the underlying boundary
value problems, which has obvious relevance to the success of
numerical simulations, is treated in a separate Living Review on
Local and Global Existence Theorems by A. Rendall [119].

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
*F*
represents a set of hypersurface variables,
*G*
a set of evolution variables, and
and
are nonlinear hypersurface operators, i.e. they operate locally
on the values of
*F*,
*G*
and
intrinsic to a single null hypersurface. 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
elliptic, since they are imposed upon null or timelike
hypersurfaces, rather than spacelike Cauchy hypersurfaces.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-2001-3
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