3 Characteristic Evolution CodesCharacteristic Evolution and Matching1 Introduction

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 space-time there is no longer the 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 [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 [7Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline1515 (``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 [17Jump To The Next Citation Point In The Article]. 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 tex2html_wrap_inline1521, with a coordinate tex2html_wrap_inline1523 varying along the rays. In tex2html_wrap_inline1525 null coordinates, the main set of Einstein equations take the schematic form




Here F represents a set of hypersurface variables; G, a set of evolution variables; and tex2html_wrap_inline1531 and tex2html_wrap_inline1533 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.

3 Characteristic Evolution CodesCharacteristic Evolution and Matching1 Introduction

image Characteristic Evolution and Matching
Jeffrey Winicour
© Max-Planck-Gesellschaft. ISSN 1433-8351
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