Bondi's initial use of null coordinates to describe radiation fields [6] 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 [7] and generalized by Sachs [14], or as null tetrad approaches in which the Bianchi identities appear as part of the set of equations, as developed by Newman and Penrose [15].

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 radiative space-times,
most notably the Bondi mass and news function [8]. 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 far field ``radiation zone''
of the gravitational 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). 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 [16], the integration constants are supplied at the intersection of
outgoing and ingoing null hypersurfaces. In the
worldtube-nullcone formalism, the sources inside a worldtube were
represented by integration constants on the worldtube [17]. 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 [18] for a representative sample. Rather than including here a
review of the extensive literature on characteristic formalisms
in general relativity, I will concentrate here on those
approaches which have been (or are in the process of being)
implemented as computational evolution schemes. I also regret
omission of the topic of well-posedness of the underlying
boundary value problems. This topic has obvious relevance to the
success of numerical simulations but would require a separate
Living Review to do it justice.

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 2-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*
and
*G*
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.

Characteristic Evolution and Matching
Jeffrey Winicour
http://www.livingreviews.org/lrr-1998-5
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