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Characteristic Evolution and Matching

Jeffrey WinicourDepartment of Physics and Astronomy,

University of Pittsburgh, Pittsburgh, PA 15260, [email protected]

Published 8 May 1998

Living Reviews in Relativity

Published by the Max Planck Institute for Gravitational Physics

(Albert Einstein Institute), Potsdam, Germany

AbstractWe review the development of numerical evolution codes for general relativity

based upon the characteristic initial value problem. Progress is traced from the earlystage of 1D feasibility studies to current 3D black hole codes that run forever. A primeapplication of characteristic evolution is Cauchy-characteristic matching, which is alsoreviewed.

1 Introduction

I review here the status of numerical relativity based upon characteristic evolution.In the spirit of the name “Living Reviews” I encourage readers to provide me withfeedback at [email protected]. This will help me update the review toinclude overlooked material. I also invite authored contributions on items of relevancethat I may have omitted because of lack of expertise. Occasionally, I will point outwhere such contributions would be valuable and would make this review more usefulthan I could achieve by myself.

We are entering an era in which Einstein’s equations can effectively be consideredsolved at the local level. Several groups, as reported in these Reviews, have developed3D codes which are stable and accurate in some sufficiently local setting. Globalsolutions are another matter. In particular, there is no single code in existence todaywhich purports to be capable of computing the waveform of gravitational radiationemanating from the inspiral and merger of two black holes, the premier problem inclassical relativity. Just as several coordinate patches are necessary to describe a

c© Max Planck Society and the author(s)http://www.livingreviews.org

Jeffrey Winicour 2

space-time with nontrivial topology, the most effective attack on the binary blackhole problem is likely to involve patching together regions of space-time handled bydifferent codes.

Most work in numerical relativity is based upon the Cauchy “3 + 1” formalism[108], with the gravitational radiation extracted by perturbative Cauchy methodswhich introduce an artificial Schwarzschild background [1, 3, 2, 5]. These wave ex-traction methods have not been tested in a fully nonlinear 3D setting. Anotherapproach specifically tailored to study radiation can be based upon the characteristicinitial value problem. In the 1960’s, Bondi [22, 23] and Penrose [79] pioneered the useof null hypersurfaces to study radiation. This new approach has flourished in generalrelativity. The standard description of the “plus” and “cross” polarization modes ofgravitational radiation is in terms of the real and imaginary parts of the Bondi newsfunction at future null infinity (I+).

From a computational standpoint, the major drawback of the characteristic ap-proach arises from the formation of caustics in the light rays generating the nullhypersurfaces. In the most ambitious scheme proposed at the theoretical level suchcaustics would be treated “head-on” as part of the dynamical problem [98]. This is aprofoundly attractive idea. Only a few structural stable caustics can arise in numer-ical evolution, and their geometrical properties are well enough understood to modeltheir singular behavior numerically [44]. However, a computational implementationof this approach has not been achieved. It seems to be a great idea that is ahead ofits time.

In the typical setting for a characteristic initial value problem, the domain ofdependence of a single nonsingular null hypersurface is empty. In order to obtaina nontrivial evolution problem, the null hypersurface must either be completed toa caustic-crossover region where it pinches off, or an additional boundary must beintroduced. So far, the only caustics that have been successfully evolved numeri-cally in general relativity are pure point caustics (the complete null cone problem).When spherical symmetry is not present, it turns out that the stability conditionsnear the vertex of a nonsingular light cone place a strong restriction on the allowedtime step [65]. Point caustics in general relativity have been successfully handledthis way for axisymmetric space-times [52], but the computational demands for 3Devolution would be prohibitive using current generation supercomputers. This is un-fortunate because, away from the caustics, the characteristic evolution offers myriadcomputational and geometrical advantages.

As a result, at least in the near future, the computational application of charac-teristic evolution is likely to be restricted to some mixed form, in which boundaryconditions are also set on a non-singular but incomplete initial null hypersurfaceand on a second nonsingular hypersurface (or perhaps several), which together withthe initial null hypersurface present a nontrivial domain of dependence. This sec-ond hypersurface may itself be either (i) null, (ii) timelike or (iii) spacelike. Thesepossibilities give rise to the (i) the double null problem, (ii) the nullcone-worldtubeproblem or (iii) the Cauchy-characteristic matching (CCM) problem.

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3 Characteristic Evolution and Matching

In CCM, it is possible to choose the matching interface between the Cauchy andcharacteristic regions to be a null hypersurface, but it is more practical to match acrossa timelike worldtube. CCM combines the advantages of characteristic evolution intreating the outer radiation zone in spherical coordinates which are naturally adaptedto the topology of the worldtube with the advantages of Cauchy evolution in Cartesiancoordinates in the region where spherical coordinates would break down.

In this review, we trace the development of characteristic algorithms from model1D problems to a 3D code for black holes that attains the Holy Grail of numericalrelativity, as specified 12 years ago by Shapiro and Teukolsky [91]. And we trace thedevelopment of CCM from early feasibility studies through current attempts to treatthe binary black hole problem.

2 The Characteristic Initial Value Problem

Characteristics have traditionally played an important role in the analysis of hyper-bolic partial differential equations. However, the use of characteristic hypersurfacesto supply the foliation underlying an evolution scheme has been mainly restricted torelativity. This is perhaps natural because in curved space-time there is no longer thepreferred Cauchy foliation provided by the Euclidean 3-spaces allowed in Galilean orspecial relativity. The method of shooting along characteristics is a standard tech-nique in many areas of computational physics, but evolution based upon characteristichypersurfaces is quite uniquely limited to relativity.

Bondi’s initial use of null coordinates to describe radiation fields [22] was followedby a rapid development of other null formalisms. These were distinguished either asmetric based approaches, as developed for axisymmetry by Bondi, Metzner and vanden Burg [23] and generalized by Sachs [87], or as null tetrad approaches in which theBianchi identities appear as part of the set of equations, as developed by Newmanand Penrose [76].

At the outset, null formalisms were applied to construct asymptotic solutionsat null infinity by means of 1/r expansions. Soon afterwards, Penrose devised theconformal compactification of null infinity I (“scri”), thereby reducing to geometrythe asymptotic description of the physical properties of radiative space-times, mostnotably the Bondi mass and news function [79]. The characteristic initial value prob-lem rapidly became an important tool for the clarification of fundamental conceptualissues regarding gravitational radiation and its energy content. It laid bare and geo-metricized the far field “radiation zone” of the gravitational field.

The initial focus on asymptotic solutions clarified the kinematic properties ofradiation fields but could not supply the waveform from a specific source. It wassoon realized that instead of carrying out a 1/r expansion, one could reformulate theapproach in terms of the integration of ordinary differential equations along the char-acteristics (null rays). The integration constants supplied on some inner boundarythen determined the specific waveforms obtained at infinity. In the double-null initial


Jeffrey Winicour 4

value problem of Sachs [88], the integration constants are supplied at the intersectionof outgoing and ingoing null hypersurfaces. In the worldtube-nullcone formalism, thesources inside a worldtube were represented by integration constants on the worldtube[99]. These early formalisms have gone through much subsequent revamping. Somehave been reformulated to fit the changing styles of modern differential geometry.Some have been reformulated in preparation for implementation as computationalalgorithms. See the articles in [37] for a representative sample. Rather than includ-ing here a review of the extensive literature on characteristic formalisms in generalrelativity, I will concentrate here on those approaches which have been (or are inthe process of being) implemented as computational evolution schemes. I also regretomission of the topic of well-posedness of the underlying boundary value problems.This topic has obvious relevance to the success of numerical simulations but wouldrequire a separate Living Review to do it justice.

All characteristic evolution schemes share the same skeletal form. The fundamen-tal ingredient is a foliation by null hypersurfaces u = const which are generated by a2-dimensional set of null rays, labeled by coordinates xA, with a coordinate λ varyingalong the rays. In (u, λ, xA) null coordinates, the main set of Einstein equations takethe schematic form

F,λ = HF [F,G] (1)

andG,uλ = HG[F,G]. (2)

Here F represents a set of hypersurface variables; G, a set of evolution variables; andHF and HG are nonlinear hypersurface operators, i.e. they operate locally on thevalues of F and G intrinsic to a single null hypersurface. In addition to these mainequations, there is a subset of four Einstein equations which are satisfied by virtue ofthe Bianchi identities, provided that they are satisfied on a hypersurface transverse tothe characteristics. These equations have the physical interpretation as conservationlaws. Mathematically they are analogous to the constraint equations of the canonicalformalism. But they are not elliptic, since they are imposed upon null or timelikehypersurfaces, rather than spacelike.



3 Characteristic Evolution Codes

Computational implementation of characteristic evolution may be based upon differ-ent versions of the formalism (i.e. metric or tetrad) and different versions of the initialvalue problem (i.e. double null or worldtube-nullcone). The performance and com-putational requirements of the resulting evolution codes can vary drastically with theparticular choice. However, all characteristic evolution codes have certain commonadvantages:

(i) There are no elliptic constraint equations. This eliminates the need for timeconsuming iterative methods to enforce constraints or to otherwise test the challengeof constraint free evolution.

(ii) No second time derivatives appear so that the number of basic variables is halfthe number for the corresponding version of the Cauchy problem.

(iii) The main Einstein equations form a system of coupled ordinary differentialequations with respect to the parameter λ varying along the characteristics. Thisallows construction of an evolution algorithm in terms of a simple march along thecharacteristics.

(iv) In problems with isolated sources, the radiation zone can be compactifiedinto a finite grid boundary using Penrose’s conformal technique. Because the Penroseboundary is a null hypersurface, no extraneous outgoing radiation condition or otherartificial boundary condition is required.

(v) The grid domain is exactly the region in which waves propagate, which isideally efficient for radiation studies. Since each null hypersurface of the foliationextends to infinity, the radiation is calculated immediately (in retarded time) with noneed to propagate it across the grid.

(vi) In black hole space-times, a large redshift at null infinity relative to internalsources is an indication of the formation of an event horizon and can be used to limitthe evolution to the exterior region of space-time.

Characteristic schemes also share as a common disadvantage the necessity eitherto deal with caustics or to avoid them altogether. The scheme to tackle the causticshead on by including their development as part of the evolution is perhaps a greatidea still ahead of its time, one that should not be forgotten. There are only a handfulof structurally stable caustics, and they have well known algebraic properties. Thismakes it possible to model their singular structure in terms of Pade approximants.The structural stability of the singularities should in principle make this possible,and algorithms to evolve the elementary caustics have been proposed [35, 96]. Inthe axisymmetric case, cusps and folds are the only stable caustics, and they havealready been identified in the horizon formation occurring in simulations of head-oncollisions of black holes and in the temporarily toroidal horizons occurring in collapseof rotating matter [74, 92].


Jeffrey Winicour 6

3.1 One+One Dimensional Codes

It is now often said that the solution of the general ordinary differential equationis essentially known, in light of the success of computational algorithms and presentday computing power. Perhaps this is an overstatement because investigating singularbehavior is still an art. But, in the same vein, it is fair to say that the general system ofhyperbolic partial differential equations in one spatial dimension is a solved problem.At least, it seems to be true in general relativity.

One of the earliest characteristic evolution codes was constructed by Corkill andStewart [35, 95] to treat space-times with two Killing vectors. The grid was basedupon a double null coordinate system, with the null hypersurfaces intersecting in thesurfaces spanned by the Killing vectors. This allowed simulation of colliding planewaves (as well as the Schwarzschild solution). They were able to evolve the Khan-Penrose [70] collision of impulsive (δ-function curvature) plane waves to within a fewnumerical zones from the singularity that forms after the collision, with extremelyclose agreement with the analytic results. Their simulations of collisions with moregeneral waveforms, for which exact solutions are not known, have provided input tothe theoretical understanding of singularity formation in this problem.

Most of the 1+1 dimensional applications of characteristic methods have beenfor spherically symmetric systems. Here matter must be included in order to makethe system non-Schwarzschild. Except for some trivial applications to evolve dustcosmologies by characteristic evolution, matter has been represented by a masslessKlein-Gordon field. This allows simulation of radiation effects in the simple contextof spherical symmetry. Highly interesting results have been found this way.

On the analytic side, working in a characteristic initial value formulation basedupon outgoing null cones, Christodoulou made a penetrating study of the existenceand uniqueness of solutions to this problem. [29, 28, 31, 30] He showed that weakinitial data evolve to Minkowski space asymptotically in time, but that sufficientlystrong data form a horizon, with nonzero Bondi mass. In the latter case, he showedthat the geometry is asymptotically Schwarzschild in the approach to I+ (futuretime infinity) from outside the horizon, thus establishing a rigorous version of theno-hair theorem. What this analytic tour-de-force did not reveal was the remarkablecritical behavior in the transition between these two regimes, which was discoveredby Choptuik [26, 27] using computational simulation.

A characteristic evolution algorithm for this problem centers about the evolutionscheme for the scalar field, which constitutes the only dynamical field. Given thescalar field, all gravitational quantities can be determined by integration along thecharacteristics of the null foliation. But this is a coupled problem, since the scalarwave equation involves the curved space metric. It provides a good illustration ofhow null algorithms lead to a hierarchy of equations which can be integrated alongthe characteristics to effectively decouple the hypersurface and dynamical variables.

In a Bondi coordinate system based upon outgoing null hypersurfaces u = const,



the metric is

ds2 = −e2βdu(V

rdu+ 2dr) + r2(dθ2 + sin2 θdφ2). (3)

Smoothness at r = 0 allows the coordinate conditions

V (u, r) = r +O(r3) and β(u, r) = O(r2). (4)

The field equations consist of the wave equation 2Φ = 0 for the scalar field and twohypersurface equations for the metric functions:

β,r = 2πr(Φ,r)2 (5)

V,r = e2β . (6)

The wave equation can be expressed in the form

2(2)g − (V

r),re−2βg

r= 0, (7)

where g = rΦ and 2(2) is the D’Alembertian associated with the two dimensionalsubmanifold spanned by the ingoing and outgoing null geodesics. Initial null data forevolution consists of Φ(u0, r) at initial retarded time u0.

Because any two dimensional geometry is conformally flat, the surface integral of2(2)g over a null parallelogram Σ gives exactly the same result as in a flat 2-space,and leads to an integral identity upon which a simple evolution algorithm can bebased [54]. Let the vertices of the null parallelogram be labeled by N , E, S and Wcorresponding, respectively, to their relative locations North, East, South and West inthe 2-space. Upon integration of (7), curvature introduces an area integral correctionto the flat space null parallelogram relation between the values of g at the vertices:

gN − gW − gE + gS = −1

2

∫Σ

dudr(V

r),rg

r. (8)

This identity, in one form or another, lies behind all of the null evolution algo-rithms that have been applied to this system. The prime distinction between thedifferent algorithms is whether they are based upon double null coordinates or Bondicoordinates as in Eq. (3). When a double null coordinate system is adopted, thepoints N , E, S and W can be located in each computational cell at grid points, sothat evaluation of the left hand side of Eq. (8) requires no interpolation. As a result,in flat space, where the right hand side of Eq. (8) vanishes, it is possible to formulatean exact evolution algorithm. In curved space, of course, there is truncation errorarising from the approximation of the integral by evaluating the integrand at thecenter of Σ.

The identity (8) gives rise to the following explicit marching algorithm. Let thenull parallelogram lie at some fixed θ and φ and span adjacent retarded time levels


Jeffrey Winicour 8

u0 and u0 + ∆u. Imagine for now that the points N , E, S and W lie on the spatialgrid, with rN − rW = rE − rS = ∆r. If g has been determined on the initial coneu0, which contains the points E and S, and radially outward from the origin to thepoint W on the next cone u0 + ∆u, then Eq. (8) determines g at the next radial gridpoint N in terms of an integral over Σ. The integrand can be approximated to secondorder, i.e. to O(∆r∆u), by evaluating it at the center of Σ. To this same accuracy,the value of g at the center equals its average between the points E and W , at whichg has already been determined. Similarly, the value of (V/r),r at the center of Σ canbe approximated to second order in terms of values of V at points where it can bedetermined by integrating the hypersurface equations (5) and (6) radially outwardfrom r = 0.

After carrying out this procedure to evaluate g at the point N , the procedure canthen be iterated to determine g at the next radially outward grid point on the u0 +∆ulevel. Upon completing this radial march to null infinity, in terms of a compactifiedradial coordinate such as x = r/(1 + r), the field g is then evaluated on the nextnull cone at u0 + 2∆u, beginning at the vertex where smoothness gives the startupcondition that g(u, 0) = 0.

In the compactified Bondi formalism, the vertices N , E, S and W of the nullparallelogram Σ cannot be chosen to lie exactly on the grid because, even in Minkowskispace, the velocity of light in terms of a compactified radial coordinate x is notconstant. As a consequence, the fields g, β and V at the vertices of Σ are approximatedto second order accuracy by interpolating between grid points. However, cancellationsarise between these four interpolations so that Eq.(8) is satisfied to fourth orderaccuracy. The net result is that the finite difference version of (8) steps g radiallyoutward one zone with an error of fourth order in grid size, O((∆u)2(∆x)2). Inaddition, the smoothness conditions (4) can be incorporated into the startup for thenumerical integrations for V and β to insure no loss of accuracy in starting up themarch at r = 0. The resulting global error in g, after evolving a finite retarded time,is then O(∆u∆x), after compounding errors from 1/(∆u∆x) number of zones.

Because of the explicit nature of this algorithm, its stability requires an analogue ofthe Courant-Friedrichs-Lewy (CFL) condition that the physical domain of dependencebe contained in the numerical domain of dependence. In the present sphericallysymmetric case, this condition requires that the ratio of the time step to radial stepbe limited by (V/r)∆u ≤ 2∆r, where ∆r = ∆[x/(1−x)]. This condition can be builtinto the code using the value V/r = e2H , corresponding to the maximum of V/r atJ +. The strongest restriction on the time step then arises just before the formationof a horizon, where V/r → ∞ at J +, This infinite redshift provides a mechanismfor locating the true event horizon “on the fly” and restricting the evolution to theexterior space-time. Points near J + must be dropped in order to evolve across thehorizon in this gauge.

Such algorithms have been applied to many interesting problems. A characteristicalgorithm based upon double null coordinates was used by Goldwirth and Piran in astudy of cosmic censorship [46]. Their early study lacked the sensitivity of adaptive



mesh refinement which later enabled Choptuik to discover the critical phenomenaappearing in this problem. The most accurate global treatment of this problem using aCauchy code has been achieved by Marsa and Choptuik [73] using ingoing Eddington-Finklestein coordinates. This use of a null based time slicing enabled them to avoidproblems with the singularity by excising the black hole exterior and to construct a1D code that runs forever.

Gomez and Winicour [54] constructed a characteristic code for this problem basedupon the compactified Bondi formalism outlined above. Studies with the code re-vealed interesting high amplitude behavior under the rescaling Φ→ aΦ. As a→∞,the red shift creates an effective boundary layer at I+ which causes the Bondi massMB and the scalar field monopole moment Q to be related by MB ∼ π|Q|/

√2, rather

than the quadratic relation of the weak field limit [54]. This can also be establishedanalytically so that the high amplitude limit provides a check on the code’s abilityto handle strongly nonlinear fields. In the small amplitude case, this work incorrectlyreported that the radiation tails from black hole formation had an exponential decaycharacteristic of quasinormal modes rather than the polynomial 1/t or 1/t2 falloffexpected from Price’s [82] work on perturbations of Schwarzschild black holes. Inhindsight, the error here was not having confidence to run the code sufficiently longto see the proper late time behavior.

Gundlach, Price and Pullin [57, 58] subsequently reexamined the issue of powerlaw tails using a double null code similar to that developed by Goldwirth and Pi-ran. Their numerical simulations verified the existence of power law tails in the fullnonlinear case, thus establishing consistency with analytic perturbative theory. Theyalso found normal mode ringing at intermediate time, again with properties in ac-cord with the perturbative results. The consistency they found between perturbationtheory and numerical evolution of the nonlinear system is very reassuring. Thereis a region of space-time where the results of linearized theory are remarkably reli-able even though highly nonlinear behavior is taking place elsewhere. These resultshave led to a methodology that has application beyond the confines of sphericallysymmetric problems, most notably in the “close approximation” for the binary blackhole problem [83]. Power law tails and quasinormal ringing were also confirmed usingCauchy evolution [73].

The study of the radiation tail decay was subsequently extended by Gomez,Schmidt and Winicour [53] using a code based upon the nullcone-worldtube versionof the Bondi formalism. They showed that the Newman-Penrose constant [78] forthe scalar field is the factor which determines the exponent of the power law (notthe static monopole moment as often stated). When this constant is non-zero, thetail decays as 1/t, as opposed to the 1/t2 decay for the vanishing case. (There arealso ln t/tn corrections in addition to the exponentially decaying contributions of thequasinormal modes). This code was also used to study the instability of a topologi-cal kink in the configuration of the scalar field [12]. The kink instability provides thesimplest example of the turning point instability [64, 94] which underlies gravitationalcollapse of static equilibria.


Jeffrey Winicour 10

Hamade and Stewart [61] have applied a double null code to the problem of criticalphenomena. In order to obtain the accuracy necessary to confirm Choptuik’s resultsthey developed the first example of a characteristic grid with adaptive mesh refine-ment (AMR). They did this with both the standard Berger and Oliger algorithm andtheir own simplified version. These different versions of AMR gave indistinguishablenumerical results. Their simulations of critical collapse of a scalar field agree withChoptuik’s values for the universal parameters governing mass scaling and displaythe echoing associated with discrete self-similarity. Hamade, Horne and Stewart [60]extended this study to the spherical collapse of an axion/dilaton system and found inthis case that the self-similarity was a continuous symmetry of the critical solution.

The Southampton group has also constructed a 1+1 dimensional characteristiccode for space-times with cylindrical symmetry [34, 40]. Their motivation was notto produce a stand-alone code for the study of cylindrically symmetric relativity butrather to use it as a test case for combining Cauchy and characteristic codes intoa global scheme. Their work will be discussed later in this review under Cauchy-characteristic matching.

3.2 Two-D Codes

One dimensional characteristic codes enjoy a very special simplicity due to the two pre-ferred sets (ingoing and outgoing) of characteristic null hypersurfaces. This eliminatesa source of gauge freedom that otherwise exists in either two or three dimensionalcharacteristic codes. However, the manner in which characteristics of a hyperbolicsystem determine domains of dependence and lead to propagation equations for shockwaves is exactly the same as in the one dimensional case. This makes it desirable forthe purpose of numerical evolution to enforce propagation along characteristics asextensively as possible. In basing a Cauchy algorithm upon shooting along character-istics, the infinity of characteristic rays (technically, bicharacteristics) at each pointleads to an arbitrariness which, for a practical numerical scheme, makes it necessaryeither to average the propagation equations over the sphere of characteristic direc-tions or to select out some preferred finite subset of propagation equations. The latterapproach has been successfully applied by Butler [25] to the Cauchy evolution of 2-dimensional fluid flow but there seems to have been very little follow-up along theselines.

The formal ideas behind the construction of two or three dimensional characteristiccodes are the same, although there are some additional technical complications inthree dimensions associated with a nonsingular choice of angular coordinates for thenull rays. Historically, most characteristic work graduated first from 1D to 2D becauseof the available computing power.



3.3 The Bondi Problem

The first characteristic code based upon the original Bondi equations for a twist-freeaxisymmetric space-time was constructed by Isaacson, Welling and Winicour [66].The space-time was foliated by a family of null cones, complete with point verticesat which regularity conditions were imposed. The code accurately integrated thehypersurface and evolution equations out to compactified null infinity. This allowedstudies of the Bondi mass and radiation flux on the initial null cone, but it could notbe used as a practical evolution code because of an instability at the vertices of thenull cones.

To a group uninitiated at the time to the pitfalls of numerical work, these instabil-ities came as a rude shock and led to a retreat to the simpler problem of axisymmetricscalar waves propagating in Minkowski space, with the metric

ds2 = −du2 − 2dudr + r2(dθ2 + sin2 θdφ2) (9)

in outgoing null cone coordinates. A null cone code for this problem was constructedusing an algorithm based upon Eq.(8), with the angular part of the flat space Lapla-cian replacing the curvature terms in the integrand on the right hand side. Thissimple setting allowed the instability to be traced to a subtle violation of the CFLcondition near the vertices of the cones. In terms of grid spacings ∆xα, the CFLcondition in this coordinate system take the explicit form

∆u/∆r < −1 + [K2 + (K − 1)2 − 2K(K − 1) cos ∆θ]1/2, (10)

where the coefficient K, which is of order 1, depends on the particular startup proce-dure adopted for the outward integration. Far from the vertex, this condition (10) onthe time step ∆u is quantitatively similar to the CFL condition for a standard Cauchyevolution algorithm in spherical coordinates. But the condition (10) is strongest nearthe vertex of the cone where (at the equator θ = π/2) it implies that

∆u < K∆r(∆θ)2. (11)

This is in contrast to the analogous requirement

∆u < K∆r∆θ, (12)

for stable Cauchy evolution near the origin of a spherical coordinate system. Theextra power of ∆θ is the price that must be paid near the vertex for the simplicityof a characteristic code. Nevertheless, the enforcement of this condition still allowedefficient global simulation of axisymmetric scalar waves. Global studies of backscatter-ing, radiative tail decay and solitons have been carried out for nonlinear axisymmetricwaves [66], but 3-dimensional simulations extending to the vertices of the cones wouldbe impractical on current machines.


Jeffrey Winicour 12

Aware now of the subtleties of the CFL condition near the vertices, the Pittsburghgroup returned to the Bondi problem, i.e. evolve the metric [23]

ds2 = (V

re2β − U2r2e2γ)du2 + 2e2βdudr + 2Ur2e2γdudθ

−r2(e2γdθ2 + e−2γ sin2 θdφ2), (13)

by means of the three hypersurface equations

β,r =1

2r (γ,r)

2 (14)

[r4 e2(γ−β)U,r],r = 2r2[r2(β

r2),rθ −

(sin2 θ γ),rθ

sin2 θ+ 2 γ,r γ,θ)] (15)

V,r = −1

4r4e2(γ−β)(U,r)

2 +(r4 sin θU),rθ

2r2 sin θ

+e2(β−γ)[1− (sin θβ,θ),θsin θ

+ γ,θθ + 3 cot θγ,θ − (β,θ)2 − 2γ,θ(γ,θ − β,θ)](16)

and the evolution equation

4r(rγ),ur = [2rγ,rV − r2(2γ,θ U + sin θ(U

sin θ),θ)],r − 2r2 (γ,rU sin θ),θ

sin θ

+1

2r4e2(γ−β)(U,r)

2 + 2e2(β−γ)[(β,θ)2 + sin θ(

β,θsin θ

),θ]. (17)

The beauty of the Bondi set of equations is that they form a clean hierarchy.Given γ on an initial null hypersurface, the equations can be integrated radially todetermine β, U , V and γ,u on the hypersurface (in that order) in terms of integrationconstants determined by boundary or smoothness conditions. The initial data γ isunconstrained except by smoothness conditions. Because γ represents a spin-2 field,it must be O(sin2 θ) near the poles of the spherical coordinates and must consist ofl ≥ 2 spin-2 multipoles.

In the computational implementation of this system [52], the null hypersurfaceswere chosen to be complete null cones with nonsingular vertices, which (for simplicity)trace out a geodesic worldline r = 0. The smoothness conditions at the vertices wereformulated by referring back to local Minkowski coordinates. The computationalalgorithm was designed to preserve these smoothness conditions.

The vertices of the cones did not turn out to be the chief source of difficulty. A nullparallelogram marching algorithm, similar to that used in the scalar case, gave rise toan instability that sprang up throughout the grid. In order to reveal the source of theinstability, physical considerations suggested looking at the linearized version of theBondi equations, since they must be related to the wave equation. If this relationshipwere sufficiently simple, then the scalar wave algorithm could be used as a guide in



stabilizing the evolution of γ. A scheme for relating γ to solutions Φ of the waveequation had been formulated in the original paper by Bondi, Metzner and van denBurgh [23]. However, in that scheme, the relationship of the scalar wave to γ wasnonlocal in the angular directions and not useful for the stability analysis.

A local relationship between γ and solutions of the wave equation was found[52]. This provided a test bed for the null evolution algorithm similar to the Cauchytest bed provided by Teukolsky waves [100]. More critically, it allowed a simple vonNeumann linear stability analysis of the finite difference equations, which revealedthat the evolution would be unstable if the metric quantity U was evaluated on thegrid. For a stable algorithm, the grid points for U must be staggered between thegrid points for γ, β and V . This unexpected feature emphasizes the value of linearstability analysis in formulating stable finite difference approximations.

These considerations led to an axisymmetric code for the global Bondi problemwhich ran stably, subject to a CFL condition, throughout the regime in which causticsand horizons did not form. Stability in this regime was verified experimentally byrunning arbitrary initial data until it radiated away to I+. Also, new exact solutionsas well as the linearized null solutions were used to perform extensive convergence teststhat established second order accuracy. The code generated a large complement ofhighly accurate numerical solutions for the class of asymptotically flat, axisymmetricvacuum space-times, a class for which no analytic solutions are known. All results ofnumerical evolutions in this regime were consistent with the theorem of Christodoulouand Klainerman [32] that weak initial data evolve asymptotically to Minkowski spaceat late time.

An additional global check on accuracy was performed using Bondi’s formularelating mass loss to the time integral of the square of the news function. The Bondimass loss formula is not one of the equations used in the evolution algorithm butfollows from those equations as a consequence of a global integration of the Bianchiidentities. Thus it not only furnishes a valuable tool for physical interpretation, butit also provides a very important calibration of numerical accuracy and consistency.

An interesting feature of the evolution arises in regard to compactification. Byconstruction, the u-direction is timelike at the origin where it coincides with the world-line traced out by the vertices of the outgoing null cones. But even for weak fields,the u-direction generically becomes spacelike at large distances along an outgoing ray.Geometrically, this reflects the property that I is itself a null hypersurface so that allinternal directions are spacelike, except for the null generator. For a flat space time,the u-direction picked out at the origin leads to a null evolution direction at I, butthis direction becomes spacelike under a slight deviation from spherical symmetry.Thus the evolution generically becomes “superluminal” near I. Remarkably, thereare no adverse numerical effects. This fortuitous property apparently arises from thenatural way that causality is built into the marching algorithm so that no additionalresort to numerical techniques, such as “causal differencing” [4], is necessary.


Jeffrey Winicour 14

3.3.1 The Conformal-Null Tetrad Approach

J. Stewart has also implemented a characteristic evolution code which handles theBondi problem by a null tetrad, as opposed to metric, formalism [97]. The geometri-cal algorithm underlying the evolution scheme, as outlined in [98, 44], is H. Friedrich’s[43] set of conformal-null equations. This casts the Einstein equations for a compact-ified space-time into a first order system of partial differential equations. The setof variables include the metric, the connection, and the curvature, as in a Newman-Penrose formalism, but in addition it includes the conformal factor (necessary forcompactification of I) and its first derivatives. As a result, without assuming anysymmetry, there are more than 7 times as many variables as in a metric based nullscheme, and the corresponding equations do not decompose into as clean a hierar-chy. This disadvantage, compared to the metric approach, is balanced by severaladvantages: (1) The equations form a symmetric hyperbolic system so that standardtheorems can be used to establish that the system is well-posed. (2) Standard evolu-tion algorithms can be invoked to ensure numerical stability. (3) The extra variablesassociated with the curvature tensor are not completely excess baggage, since theysupply essential physical information. (4) The regularization necessary to treat I isbuilt-in as part of the formalism so that no special numerical regularization techniquesare necessary as in the metric case. (This last advantage is somewhat offset by thenecessity of having to locate I by tracking the zeroes of the conformal factor.)

The code was intended to study gravitational waves emanating from an axisym-metric star. Since only the vacuum equations are evolved, the outgoing radiation fromthe star is represented in terms of data (Ψ4 in Newman-Penrose notation) on an ingo-ing null cone (an advanced time null hypersurface), which forms the inner boundaryof the evolved domain. The inner boundary data is supplemented by Schwarzschilddata on a initial outgoing null cone, which models an initially quiescent state of anisolated star. This provides the necessary data for a double-null initial value prob-lem. The evolution would normally break down where the ingoing null hypersurfacedevelops caustics. But by choosing a scenario in which a black hole is formed, it ispossible to evolve the entire region exterior to the horizon. An obvious test bed is theSchwarzschild space-time for which a numerically satisfactory evolution was achieved(convergence tests were not reported).

Physically interesting results were obtained by choosing data corresponding to anoutgoing quadrupole pulse of radiation with various waveforms. By increasing theinitial amplitude of the data Ψ4, it was possible to evolve into a regime where theenergy loss due to radiation was large enough to drive the total Bondi mass negative.Although such data is too grossly exaggerated to be consistent with an astrophysicallyrealistic source, the formation of a negative mass is an impressive test of the robustnessof the code.



3.3.2 Twisting Axisymmetry

The Southampton group, as part of its long range goal of combining Cauchy andcharacteristic evolution, has begun development of a code which extends the Bondiproblem to full axisymmetry [38]. By dropping the requirement that the Killingdirection φ be twist-free, they are able to include rotational effects, including radia-tion in the “cross” polarization mode (only the “plus” mode is allowed by twist-freeaxisymmetry).

They have completed the preliminary theoretical work [39]. The null equations andvariables have been recast into a suitably regularized form to allow compactificationof null infinity. Regularization at the vertices or caustics of the null hypersurfaces isnot necessary, since they anticipate matching to an interior Cauchy evolution across afinite worldtube. The Southampton program offers promise of some highly interestingresults which I hope will be reported in a not too future version of this review.

3.4 The Bondi Mass

Numerical calculations of asymptotic quantities such as the Bondi mass must over-come severe technical difficulties arising from the necessity to pick off nonleadingterms in an asymptotic expansion about infinity. For example, in an asymptoticallyinertial frame (called a Bondi frame at I+), the mass aspectM(u.θ.φ) must be pickedoff from the asymptotic expansion of Bondi’s metric quantity V , (see Eq. (16)) ofthe form V = r − 2M + O(1/r). This is similar to the experimental task of deter-mining the mass of an object by measuring its far field. What makes the job muchmore difficult is that, in most computational setups, (the global null problem or theCauchy-characteristic matching problem), the gauge choice does not correspond to aBondi frame at I+. One must deal with an arbitrary foliation of I+ into retardedtime slices which are determined by the details of the interior geometry. As a result,V has the more complicated asymptotic behavior (given in the axisymmetric case)

V − r = r2(L sin θ),θ/ sin θ + re2(H−K)[(1− e−2(H−K))

+ 2(H,θ sin θ),θ/ sin θ +K,θθ + 3K,θ cot θ + 4(H,θ)2

− 4H,θK,θ − 2(K,θ)2]− 2e2HM+O(r−1), (18)

where L, H and K are gauge dependent functions of (u, θ) which would vanish in aBondi frame [99, 66]. The calculation of the Bondi mass requires regularization ofthis expression by numerical techniques so that the coefficient M can be picked off.The task is now similar to the experimental determination of the mass of an objectby using non-inertial instruments in a far zone which contains O(1/r) radiation fields.But it can be done!

It was accomplished in Stewart’s code by reexpressing the formula for the Bondimass in terms of the well-behaved fields of the conformal formalism [97]. In the


Jeffrey Winicour 16

Pittsburgh code, it is accomplished by re-expressing the Bondi mass in terms ofrenormalized metric variables which regularize all calculations at I+ and make themsecond order accurate [47]. The calculation of the Bondi news function (which pro-vides the waveforms of both polarization modes) is an easier numerical task than theBondi mass. It has also been implemented in both of these codes, thus allowing theimportant check of the Bondi mass loss formula.

Mainstream astrophysics is couched in Newtonian concepts, some of which haveno well defined extension to general relativity. In order to provide a sound basis forrelativistic astrophysics, it is crucial to develop general relativistic concepts whichhave well defined and useful Newtonian limits. Mass and radiation flux are mostfundamental in this regard. The results of characteristic codes show that the energyof a radiating system can be evaluated rigorously and accurately according to therules for asymptotically flat space-times, while avoiding the deficiencies that plaguedthe “pre-numerical” era of relativity: (i) the use of coordinate dependent conceptssuch as gravitational energy-momentum pseudotensors; (ii) a rather loose notion ofasymptotic flatness, particularly for radiative space-times; (iii) the appearance ofdivergent integrals; and (iv) the use of approximation formalisms, such as weak fieldor slow motion expansions, whose errors have not been rigorously estimated.

The above codes have extended the role of the Bondi mass from that of a geomet-rical construct in the theory of gravitational radiation to that of a highly accuratecomputational tool. The Bondi mass loss formula provides an important global checkon the preservation of the Bianchi identities. The mass loss rates themselves haveimportant astrophysical significance. The numerical results demonstrate that com-putational approaches, rigorously based upon the geometrical definition of mass ingeneral relativity, can be used to calculate radiation losses in highly nonlinear pro-cesses where perturbation calculations would not be meaningful.

Numerical calculation of the Bondi mass has been used to explore both the Newto-nian and the strong field limits of general relativity [47]. For a quasi-Newtonian systemof radiating dust, the numerical calculation joins smoothly on to a post-Newtonianexpansion of the energy in powers of 1/c, beginning with the Newtonian mass andmechanical energy as the leading terms. This comparison with perturbation theoryhas been carried out to O(1/c7), at which stage the computed Bondi mass peels awayfrom the post-Newtonian expansion. It remains strictly positive, in contrast to thetruncated post-Newtonian behavior which leads to negative values.

Another subtle numerical problem associated with the Bondi mass stems fromits role as one component of the total energy-momentum 4-vector, whose calculationrequires identification of the translation subgroup of the Bondi-Metzner-Sachs group[86]. This introduces boost freedom into the problem. Identifying the translationsubgroup is tantamount to knowing the conformal transformation to a conformalBondi frame [99] in which the time slices of I have unit sphere geometry. An importantfeature providing flexibility to both Stewart’s code and the Pittsburgh code is thatthe coordinates are adopted to simplify the description of the interior sources. Thisresults in an arbitrary foliation of I. The determination of the conformal factor which



relates the 2-metric hAB of a slice of I to the unit sphere metric is an elliptic problemequivalent to solving the second order partial differential equation for the conformaltransformation of Gaussian curvature. In the axisymmetric case, the PDE reducesto an ODE with respect to the angle θ, and is straightforward to solve [47]. Theintegration constants determine the boost freedom along the axis of symmetry.

The non-axisymmetric case is more complicated. Stewart [97] proposes a promis-ing approach based upon the dyad decomposition

hABdxAdxB = mAdx

AmBdxB . (19)

The desired conformal transformation is obtained by first relating hAB conformallyto the flat metric of the complex plane. Denoting the complex coordinate of the planeby ζ, this relationship can be expressed as dζ = efmAdx

A. The conformal factor fcan then be determined from the integrability condition

m[A∂B]f = ∂[AmB]. (20)

This is equivalent to the classic Beltrami equation for finding isothermal coordinates.It would appear to be a more effective scheme than tackling the second order PDEdirectly, but numerical implementation has not yet been carried out.

3.5 Three-D Characteristic Evolution

The Binary Black Hole Grand Challenge has fostered striking progress in developinga 3D characteristic code. At the outset of the Grand Challenge, the Pittsburgh grouphad just completed calibration of their axisymmetric characteristic code. Now, thishas not only been extended to a full 3D code which calculates waveforms at infinity [20,14], it has also been supplied with a horizon finder to successfully move distorted blackholes [51, 49] on a computational grid. This has been accomplished with unlimitedlong term stability and demonstrated second order accuracy, in the harshest nonlinearphysical regimes corresponding to radiation powers of a galactic rest mass per second,and with the harshest gauge conditions, corresponding to superluminal coordinaterotation.

The waveforms are initially calculated in arbitrary coordinates determined by the“3+1” gauge conditions on an inner worldtube. An important feature for the binaryblack hole problem is that these coordinates can be rigidly rotating, so that theevolution near infinity is highly superluminal, without affecting code performance.The waveforms are converted to the standard “plus and cross” inertial polarizationmodes by numerically carrying out the transformation to an inertial frame at infinity.

3.5.1 The eth Module

Spherical coordinates and spherical harmonics are standard analytic tools in the de-scription of radiation, but, in computational work, spherical coordinates have mainly


Jeffrey Winicour 18

been used in axisymmetric systems, where polar singularities may be regularized bystandard tricks. In the absence of symmetry, these techniques do not generalize andwould be especially prohibitive to develop for tensor fields. A crucial ingredient of the3D characteristic code is a module which allows use of spherical coordinates by imple-menting a computational version of the Newman-Penrose eth formalism [77]. The ethmodule covers the sphere with two overlapping stereographic coordinate grids (Northand South). It provides everywhere regular, second order accurate, finite differenceexpressions for tensor fields on the sphere and their covariant derivatives [50].

3.5.2 Computer Algebra Scripts

Although the eth calculus both simplifies the equations and avoids spurious coordinatesingularities, there is a large proliferation of angular derivatives of vector and tensorfields in reexpressing Einstein’s equations in eth form. MAPLE scripts have beendeveloped which greatly facilitate reliable coding of the curvature and other tensorsentering the problem. The translation of a formula from tensor to eth formalism isideal for computer algebra: It is straightforward and algorithmic - but lengthy.

3.5.3 Code Tests

The code ran stably in all regimes of radiating, single black hole space-times, in-cluding extremely nonlinear systems with the Bondi news as large as N = 400 (indimensionless geometric units). This means that the code can cope with an enormouspower output N2c5/G ≈ 1060W in conventional units. This exceeds the power thatwould be produced if, in 1 second, the whole Galaxy were converted to gravitationalradiation.

Code tests verified second order accuracy of the 3D code in an extensive numberof testbeds:

• Linearized waves on a Minkowski background in null cone coordinates

• Boost and rotation symmetric solutions

• Schwarzschild in rotating coordinates

• Polarization symmetry of nonlinear twist-free axisymmetric waveforms

• Robinson-Trautman waveforms from perturbed Schwarzschild black holes.

The simulations of nonlinear Robinson-Trautman space-times showed gross qual-itative differences with perturbative waveforms once radiative mass losses rose above3% of the initial energy.



3.5.4 Nonlinear Scattering Off a Schwarzschild Black Hole

The chief physical application of the code has been to the nonlinear version of theclassic problem of scattering off a Schwarzschild black hole, first solved perturbativelyby Price [82]. Here the inner worldtube for the initial value problem consists of theingoing r = 2m surface (the past horizon), where Schwarzschild data is prescribed.The nonlinear problem of a gravitational wave scattering off a Schwarzschild blackhole is then posed in terms of data on an outgoing null cone consisting of an incomingpulse with compact support.

The news function for this problem was studied as a function of incoming pulseamplitude. Here the computational eth formalism smoothly handles the complicatedtime dependent transformation between the non-inertial computational frame at I+

and the inertial (Bondi) frame necessary to obtain the standard “plus” and “cross”polarization modes. In the perturbative regime, the news corresponds to the backscat-tering of the incoming pulse off the effective Schwarzschild potential. However, forhigher amplitudes the waveform behaves quite differently. Not only is its ampli-tude greater, but it also reveals the presence of extra oscillations. In the very highamplitude case, the mass of the system is dominated by the incoming pulse, whichessentially backscatters off itself in a nonlinear way.

3.5.5 Moving Black Holes

The 3-D characteristic code was extended to handle evolution based upon a foliationby ingoing null hypersurfaces [49]. This code incorporates a null hypersurface versionof an apparent horizon finder, which is used to excise black hole interiors from thecomputation. The code accurately evolves and tracks moving, distorted, radiatingblack holes. Test cases include moving a boosted Schwarzschild black hole across a 3Dgrid. A black hole wobbling relative to an orbiting characteristic grid has been evolvedand tracked for over 10, 000M , corresponding to about 200 orbits, with absolutely nosign of instability. These results can be viewed online. [56]. The surface area ofdistorted black holes is calculated and shown to approach the equilibrium value ofthe final Schwarzschild black hole which is built into the boundary conditions.

The code excises the singular region and evolves black holes forever with secondorder accuracy. It has attained the Holy Grail of numerical relativity as originallyspecified by Teukolsky and Shapiro. [91]

This exceptional performance opens a promising new approach to handle the innerboundary condition for Cauchy evolution of black holes by the matching methodsreviewed below.


Jeffrey Winicour 20

4 CCM

4.1 Computational Boundaries

Boundary conditions are both the most important and the most difficult part of atheoretical treatment of most physical systems. Usually, that’s where all the physicsis. And, in computational approaches, that’s usually where all the agony is. Com-putational boundaries for hyperbolic systems pose special difficulties. Even with ananalytic form of the correct physical boundary condition in hand, there are seeminglyinfinitely more unstable numerical implementations than stable ones. In general,a well posed problem places more boundary requirements on the finite differenceequations than on the corresponding partial differential equations. Furthermore, themethods of linear stability analysis are often more unwieldy to apply to the boundarythan to the interior evolution algorithm.

The von Neumann stability analysis of the interior algorithm linearizes the equa-tions, while assuming a uniform infinite grid, and checks that the discrete Fouriermodes do not grow exponentially. There is an additional stability condition that aboundary introduces into this analysis. Consider the one dimensional case. Normallythe mode ekx, with k real, is not included in the von Neumann analysis. However, ifthere is a boundary to the right on the x-axis, one can legitimately prescribe such amode (with k > 0) as initial data so that its stability must be checked. In the case ofan additional boundary to the left, the Ryaben’kii-Godunov theory allows separateinvestigation of the interior stability and the stability of each individual boundary[93].

The correct physical formulation of any asymptotically flat, radiative Cauchy prob-lem requires boundary conditions at infinity. These conditions ensure not only thatthe total energy and the energy loss by radiation are both finite, but also ensure theproper 1/r asymptotic falloff of the radiation fields. However, when treating radiativesystems computationally, an outer boundary must be established artificially at somelarge but finite distance in the wave zone, i.e. many wavelengths from the source.Imposing an accurate radiation boundary condition at a finite distance is a difficulttask even in the case of a simple radiative system evolving on a fixed geometric back-ground. The problem is exacerbated when dealing with Einstein’s equation.

Nowhere is the boundary problem more acute than in the computation of grav-itational radiation produced by black holes. The numerical study of a black holespace-time by means of a pure Cauchy evolution involves both inner and outer gridboundaries. The inner boundary is necessary to avoid the topological complicationsand singularities introduced by a black hole. For multiple black holes, the innerboundary consists of disjoint pieces. W. Unruh (see [102]) initially suggested whatis now the accepted strategy for Cauchy evolution of black holes. An inner bound-ary located at (or near) an apparent horizon is used to excise the interior region.Below, we discuss an alternative version of this strategy, based upon matching to acharacteristic evolution in the inner region.



First, consider the more universal outer boundary problem, in which Cauchy-characteristic matching has a natural application. In the Cauchy treatment of sucha system, the outer grid boundary is located at some finite distance, normally manywavelengths from the source. Attempts to use compactified Cauchy hypersurfaceswhich extend to spatial infinity have failed because the phase of short wavelengthradiation varies rapidly in spatial directions [67]. Characteristic evolution avoids thisproblem by approaching infinity along phase fronts.

Assuming that the system is nonlinear and not amenable to an exact solution, afinite outer boundary condition must necessarily introduce spurious physical effectsinto a Cauchy evolution. The domain of dependence of the initial Cauchy data in theregion spanned by the computational grid shrinks in time along ingoing characteris-tics, unless data on a worldtube traced out by the outer grid boundary is included aspart of the problem. In order to maintain a causally sensible evolution, this worldtubedata must correctly substitute for the missing Cauchy data which would have beensupplied if the Cauchy hypersurface had extended to infinity. In a scattering prob-lem, this missing exterior Cauchy data might, for instance, correspond to an incomingpulse initially outside the outer boundary. In a problem where the initial radiationfields are confined to a compact region inside the boundary, this missing Cauchy datais easy to state when dealing with a constraint free field, such as a scalar field Φ wherethe Cauchy data outside the boundary would be Φ = Φ,t = 0. However, the determi-nation of Cauchy data for general relativity is a global elliptic constraint problem sothat there is no well defined scheme to confine it to a compact region. Furthermore,even if the data were known on a complete initial hypersurface extending to infinity,it would be a formidable nonlinear problem to correctly represent it by counterfeitdata on the outer boundary.

Instead, in practice, some artificial boundary condition (ABC), such as an out-going radiation condition, is imposed upon the boundary in an attempt to approxi-mate the proper data for the exterior region. This ABC may cause partial reflectionof an outgoing wave back into the system [71, 67, 63, 84], which contaminates theaccuracy of the interior evolution and the calculation of the radiated waveform. Fur-thermore, nonlinear waves intrinsically backscatter, which makes it incorrect to tryto entirely eliminate incoming radiation from the outer region. The errors introducedby these problems are of an analytic origin, essentially independent of computationaldiscretization. In general, a systematic reduction of this error can only be achievedby simultaneously refining the discretization and moving the computational bound-ary to larger and larger radii. This is computationally very expensive, especially forthree-dimensional simulations.

A traditional outer boundary condition for the wave equation is the Sommerfeldcondition. For a 3D scalar field this takes the form g,t + g,r = 0, where g = rΦ. Thiscondition is exact for a linear wave with spherically symmetric data and boundary. Inthat case, the exact solution is g = f1(t− r) +f2(t+ r) and the Sommerfeld conditioneliminates the incoming wave f2.

Much work has been done on formulating boundary conditions, both exact and


Jeffrey Winicour 22

approximate, for linear problems in situations that are not spherically symmetric andin which the Sommerfeld condition would be inaccurate. These boundary conditionsare given various names in the literature, e.g. absorbing or non-reflecting. A varietyof successful implementations of ABC’s have been reported for linear problems. Seethe recent articles [45, 84, 105, 85, 18] for a general discussion of ABC’s.

Local ABC’s have been extensively applied to linear problems with varying success[71, 42, 13, 104, 63, 21, 68]. Some of these conditions are local approximations to exactintegral representations of the solution in the exterior of the computational domain[42], while others are based on approximating the dispersion relation of the so-calledone-way wave equations [71, 104]. Higdon [63] showed that this last approach isessentially equivalent to specifying a finite number of angles of incidence for whichthe ABC’s yield perfect transmission. Local ABC’s have also been derived for thelinear wave equation by considering the asymptotic behavior of outgoing solutions[13], which generalizes the Sommerfeld outgoing radiation condition. Although suchABC’s are relatively simple to implement and have a low computational cost, theirfinal accuracy is often limited because the assumptions made about the behavior ofthe waves are rarely met in practice [45, 105]. In general, systematic improvementcan only be achieved with local conditions by moving the computational boundary toa larger radius, which is computationally expensive - especially for three-dimensionalsimulations.

The disadvantages of local ABC’s have led some workers to consider the exactnonlocal boundary conditions based on integral representations of the infinite domainproblem [103, 45, 105]. Even for problems where the Green’s function is known andeasily computed, such approaches were initially dismissed as impractical [42]; however,the rapid increase in computer power has made it possible to implement exact nonlocalABC’s even for the 3D linear wave equation [36]. If properly implemented, this kindof method can yield numerical solutions which converge to the exact infinite domainproblem in the continuum limit, keeping the artificial boundary at a fixed distance.However, due to nonlocality, the computational cost per time step usually grows at ahigher power of grid size (O(N4) per time step in three dimensions) than in a localapproach [45, 36, 105], which in multidimensional problems may be very demandingeven for today’s supercomputers.

The extension of ABC’s to nonlinear problems is much more difficult. The problemis normally treated by linearizing the region between the outer boundary and infinity,using either local or nonlocal linear ABC’s [105, 85]. The neglect of the nonlinearterms in this region introduces an unavoidable error at the analytic level. But evenlarger errors are typically introduced in prescribing the outer boundary data. Thisis a subtle global problem because the correct boundary data must correspond tothe continuity of fields, and their normal derivatives have to be extended across theboundary into the linearized exterior. This is a clear requirement for any consistentboundary algorithm, since discontinuities in the field or its derivatives would otherwiseact as spurious sheet source on the boundary, thereby contaminating both the interiorand the exterior evolutions. But the fields and their normal derivatives constitute an



overdetermined set of data for the linearized exterior problem. So it is necessary tosolve a global linearized problem, not just an exterior one, in order to find the properdata. The designation “exact ABC” is given to an ABC for a nonlinear system whoseonly error is due to linearization of the exterior. An exact ABC requires the use ofglobal techniques, such as the difference potential method, to eliminate back reflectionat the boundary [105].

To date there have been only a few applications of ABC’s to strongly nonlinearproblems [45]. Thompson [101] generalized a previous nonlinear ABC of Hedstrom[62] to treat 1D and 2D problems in gas dynamics. These boundary conditions per-formed poorly in some situations because of their difficulty in adequately modelingthe field outside the computational domain [101, 45]. Hagstrom and Hariharan [59]have overcome these difficulties in 1D gas dynamics by a clever use of Riemann in-variants. They proposed a heuristic generalization of their local ABC to 3D, but thishas not yet been implemented.

In order to reduce the level of approximation at the analytical level, an artificialboundary for an nonlinear problem must be placed sufficiently far from the strong-fieldregion. This sharply increases the computational cost in multidimensional simulations[42]. There seems to be no numerical method which converges (as the discretizationis refined) to the infinite domain exact solution of a strongly nonlinear wave problemin multidimensions, while keeping the artificial boundary fixed.

Cauchy-characteristic matching is a strategy that eliminates this nonlinear sourceof error. In CCM, Cauchy and characteristic evolution algorithms are pasted togetherin the neighborhood of a worldtube to form a global evolution algorithm. The char-acteristic algorithm provides an outer boundary condition for the interior Cauchyevolution, while the Cauchy algorithm supplies an inner boundary condition for thecharacteristic evolution. The matching worldtube provides the geometric frameworknecessary to relate the two evolutions. The Cauchy foliation of the space-time cutsthe worldtube into spherical slices. The characteristic evolution is based upon theoutgoing null hypersurfaces emanating from these slices, with the evolution proceed-ing from one hypersurface to the next by the outward radial march described earlier.There is no need to truncate space-time at a finite distance from the sources, sincecompactification of the radial null coordinate makes it possible to cover the wholespace-time with a finite computational grid. In this way, the true waveform may bedirectly computed by a finite difference algorithm. Although characteristic evolutionhas limitations in regions where caustics develop, it proves to be both accurate andcomputationally efficient in the treatment of exterior regions.

CCM evolves a mixed spacelike-null initial value problem in which Cauchy datais given in a spacelike region bounded by a spherical boundary S and characteris-tic data is given on a null hypersurface emanating from S. The general idea is notentirely new. An early mathematical investigation combining space-like and charac-teristic hypersurfaces appears in [41]. The three chief ingredients for computationalimplementation are: (i) a Cauchy evolution module, (ii) a characteristic evolutionmodule and (iii) a module for matching the Cauchy and characteristic regions across


Jeffrey Winicour 24

an interface. The interface is the timelike worldtube which is traced out by the flow ofS along the worldlines of the Cauchy evolution, as determined by the choice of lapseand shift. Matching provides the exchange of data across the worldtube to allowevolution without any further boundary conditions, as would be necessary in either apurely Cauchy or a purely characteristic evolution.

CCM may be formulated as a purely analytic approach, but its advantages areparamount in the solution of nonlinear problems where analytic solutions would beimpossible. One of the first applications of CCM was a hybrid numerical-analyticalversion, initiated by Anderson and Hobill for the 1D wave equation [7] (see below).There the characteristic region was restricted to the far field where it could be handledanalytically by a linear approximation.

The full potential of CCM lies in a purely numerical treatment of nonlinear systemswhere its error converges to zero in the continuum limit of infinite grid resolution[15, 16, 33]. For high accuracy, CCM is also by far the most efficient method. Forsmall target target error ε, it has been shown that the relative amount of computationrequired for CCM (ACCM ) compared to that required for a pure Cauchy calculation(AC) goes to zero, ACCM/AC → O as ε → O [20, 19]. An important factor hereis the use of a compactified characteristic evolution, so that the whole space-timeis represented on a finite grid. From a numerical point of view this means thatthe only error made in a calculation of the radiation waveform at infinity is thecontrolled error due to the finite discretization. Accuracy of a Cauchy algorithmwhich uses an ABC requires a large grid domain in order to avoid error from nonlineareffects in the exterior. The computational demands of matching are small because theinterface problem involves one less dimension than the evolution problem. Becausecharacteristic evolution algorithms are more efficient than Cauchy algorithms, theefficiency can be further enhanced by making the matching radius as small as possibleconsistent with avoiding caustics.

At present, the purely computational version of CCM is exclusively the tool ofgeneral relativists who are used to dealing with novel coordinate systems. A discussionof its potential appears in [15]. Only recently [33, 34, 40, 17] has its practicability beencarefully explored. Research on this topic is currently being stimulated and guidedby the requirements of the Binary Black Hole Grand Challenge Alliance. CCM is oneof the strategies being pursued to provide the boundary conditions and determinethe radiation waveforms for the binary black hole problem. But I anticipate that itsuse will eventually spread throughout computational physics because of its inherentadvantages in dealing with hyperbolic systems, particularly in 3-dimensional problemswhere efficiency is necessary. A detailed study of the stability and accuracy of CCMfor linear and non-linear wave equations has been presented in Ref. [18], illustratingits potential for a wide range of problems.



4.2 Perturbative CCM

In numerous purely analytic applications outside of general relativity, matching tech-niques have successfully cured pathologies in perturbative expansions [75]. Matchingis a strategy for obtaining a global solution by patching together solutions obtainedusing different coordinate systems for different regions. By adopting each coordinatesystem to a length scale appropriate to its domain, a globally convergent perturba-tion expansion is sometimes possible in cases where a single coordinate system wouldfail. Burke showed that matching could be used to eliminate some of the divergencesarising in perturbative calculations of gravitational radiation [24]. Kates and Keglesfurther showed that the use of a null coordinate system for the exterior region isessential in the perturbation expansions of curved space radiation fields [69]. Theyinvestigated the perturbative description of a scalar field on a Schwarzschild back-ground, in which case the asymptotic behavior of the Schwarzschild light cones differsdrastically from that of the artificial Minkowski light cones used in the perturbativeexpansions based upon the flat space Green function. Use of the Minkowski lightcones leads to nonuniformities in the expansion of the radiation fields which are elim-inated by the use of exterior coordinates based upon the true light cones. Kates,Anderson, Kegles and Madonna extended this work to the fully general relativisticcase and reached the same conclusion [10].

Anderson later applied this approach to the slow motion approximation of a binarysystem and obtained a derivation of the radiation reaction effect on the orbital periodwhich avoided some objections concerning earlier approaches [6]. The use of thetrue light cones was also essential in formulating as a mathematical theorem thatthe Bondi news function satisfies the Einstein quadrupole formula to leading orderin a Newtonian limit [107]. Although questions of mathematical consistency stillremain in the perturbative treatment of gravitational radiation, it is clear that theuse of characteristic methods pushes these problems to a higher perturbative order.Characteristic techniques are used in present day codes for the Zerilli equation [72].

4.3 Analytic-Numerical Matching for Waves

One of the first computational applications of null coordinates in a matching schemewas a hybrid numerical-analytical version, initiated by Anderson and Hobill for thetest problem of 1D scalar waves [7, 8, 9]. They were motivated by the circumstancesthat, although the nonlinear near field cannot be treated analytically in a general rel-ativistic radiation problem, the far field can be handled by a perturbative expansion.Their strategy was to match an inner numerical solution to an approximate outeranalytic solution of the problem.

The initial conditions for the exterior solution were fixed by the requirement thatthe interior sources be stationary prior to some fixed time, As a result, the exterioranalytic solution is causal in the sense that it is stationary in the past of some nullcone. This effectively introduces a condition that eliminates extraneous incoming


Jeffrey Winicour 26

radiation from the system in a physically plausible way and determines the exteriorsolution uniquely. Their strategy was to introduce an overlap region between thenumerical interior and the analytic exterior. In the overlap, the numerical solution ismatched to the causal analytic solution, resulting in an evolution that is everywherecausally meaningful.

This is the physically correct approach to a system which is stationary prior toa fixed time but is nontrivial to generalize, say, to the problem of radiation froman orbiting binary. Anderson and Hobill first tackled the 1D model problem of anoscillator coupled to a spherically symmetric, flat space, massless scalar field. Thenumerical results were in excellent agreement with the exact analytic solution (whichcould be obtained globally for this problem).

They extended the model to include spherical scalar waves propagating in a spheri-cally symmetric curved background space-time. This introduces backscattering whichobscures the concept of a purely outgoing wave. Also, no exact solution exists to thisproblem so that an approximation method is necessary to determine the exterior an-alytic solution. The approximation was based upon an expansion in terms of a scaleparameter that controls the amount of backscatter. In the 0th approximation, thescale parameter vanishes and the problem reduces to the flat space case which can besolved exactly. The flat space Green function is then used to generate higher ordercorrections.

This is a standard perturbative approach, but a key ingredient of the scheme isthat the wave equation is solved in retarded null coordinates (u, r) for the curvedspace metric, so that causality is built into the Green function at each order of ap-proximation. The transformation from null coordinates (u, r) to Cauchy coordinates(t, r) is known analytically for this problem. This allows simple matching betweenthe null and Cauchy solutions at the boundary of the Cauchy grid. Their scheme isefficient and leads to consistent results in the region that the numerical and analyticsolutions overlap. It is capable of handling both strong fields and fast motions.

Later, a global, characteristic, numerical study of the self-gravitating version ofthis problem confirmed that the use of the true null cones is essential in getting thecorrect radiated waveform [55]. For quasi-periodic radiation, the phase of the wave-form is particular sensitive to the truncation of the outer region at a finite boundary.Although a perturbative estimate would indicate an O(M/R) error, this error accu-mulates over many cycles to produce an error of order π in the phase.

Anderson and Hobill proposed that their method be extended to general relativityby matching a numerical solution to an analytic 1/r expansion in null coordinates.However, the only analytic-numerical matching schemes that have been implementedin general relativity have been based upon perturbations of a Schwarzschild back-ground using the standard Schwarzschild time slicing [1, 3, 2, 5]. It would be inter-esting to compare results with an analytic-numeric matching scheme based upon thetrue null cones. However the original proposal by Anderson and Hobill has not beencarried out.



4.4 Numerical Matching for 1D Gravitational Systems

The first numerical implementations of CCM were 1D feasibility studies. These modelproblems provided a controlled situation for the development of CCM, in which eitherexact or independent numerical solutions were known. The following studies showedthat CCM worked like a charm in a variety of 1D applications - the matched evolutionswere essentially transparent to the presence of the interface.

4.4.1 Cylindrical Matching

The Southampton group chose cylindrically symmetric systems as their model prob-lem for developing matching techniques. It provides a good testbed, although theinfinite extent of a cylindrically symmetric source is unphysical. In preliminary work,they showed how CCM could be consistently carried out for a scalar wave evolvingin Minkowski space-time but expressed in a nontrivial cylindrical coordinate system[33].

They then tackled the gravitational problem. In their first paper in a series,they set up the machinery necessary for investigating cylindrically symmetric vac-uum space-times [34]. Although the problem involves only one spatial dimension,there are two independent modes of polarization. The Cauchy metric was treated inthe Jordan-Ehlers-Kompaneets canonical form, using coordinates (t, r, φ, z) adaptedto the (φ, z) cylindrical symmetry. The advantage here is that u = t− r is then a nullcoordinate which can be used for the characteristic evolution. They successfully recastthe equations in a suitably regularized form for the compactification of I+ in termsof the coordinate y =

√1/r. The simple analytic relationship between Cauchy coor-

dinates (t, r) and characteristic coordinates (u, y) facilitates the translation betweenCauchy and characteristic variables on the matching worldtube, given by r = const.

The next paper in this series implements the scheme as a numerical code [40]. Theinterior Cauchy evolution is carried out using an unconstrained leap frog scheme. Itis notable that they report no problems with instability, which have arisen in otherattempts at unconstrained leapfrog evolution in general relativity. The characteris-tic evolution also uses a leap frog scheme for the evolution between retarded timelevels u, while numerically integrating the hypersurface equations outward along thecharacteristics.

The matching interface is located at points common to both the Cauchy and char-acteristic grids. In order to update these points by Cauchy evolution, it is necessary toobtain field values at the Cauchy “guard” points which lie outside the worldtube in thecharacteristic region. These values are obtained by interpolation from characteristicgrid points (lying on three levels of null hypersurfaces in order to ensure second orderaccuracy). Similarly, the boundary data for starting up the characteristic integrationis obtained by interpolation from Cauchy grid values inside the worldtube.

The code was tested using exact solutions for cylindrical waves, which come in fromI−, pass through the symmetry axis and expand out to I+. The numerical errors are


Jeffrey Winicour 28

oscillatory with low growth rate, and second order convergence was confirmed. Ofspecial importance, little numerical noise was introduced by the interface.

Comparisons of CCM were made with Cauchy evolutions using a standard out-going radiation boundary condition [80]. At high amplitudes the standard conditiondeveloped a large error very quickly and was competitive only for weak waves witha large outer boundary. In contrast, the matching code performed well even with asmall matching radius.

Some interesting simulations were presented in which an outgoing wave in onepolarization mode collides with an incoming wave in the other mode, a problemstudied earlier by pure Cauchy evolution [81]. The behavior of the collision wasqualitatively similar in these two studies.

4.4.2 Spherical Matching

A joint collaboration between Pennsylvania State University and the University ofPittsburgh applied CCM to the Einstein-Klein-Gordon (EKG) system with sphericalsymmetry [48]. This model problem allowed simulation of black-hole formation aswell as wave propagation.

The geometrical setup is analogous to the cylindrically symmetric problem. Initialdata are specified on the union of a spacelike hypersurface and a null hypersurface.The evolution used a 3-level Cauchy scheme in the interior and a 2-level characteristicevolution in the compactified exterior. A fully constrained Cauchy evolution wasadopted because of its earlier success in providing accurate calculations of scalarwave collapse [27]. Characteristic evolution was based upon the null parallelogrammarching algorithm Eq. (8). The matching between the Cauchy and characteristicfoliations was achieved by imposing continuity conditions on the metric, extrinsiccurvature and scalar field variables, ensuring smoothness of fields and their derivativesacross the matching interface. The extensive analytical and numerical studies of thissystem in recent years aided the development of CCM in a non-trivial geometricalsetting without exact solutions by providing basic knowledge of the expected physicaland geometrical behavior.

The CCM code exhibited accurate simulation of wave propagation and black holeformation for all values of M/R at the matching radius, with no symptoms of insta-bility or back reflection. Second order accuracy was established by checking energyconservation.

4.4.3 Excising 1D Black Holes

In further developmental work on the EKG model, the Pittsburgh group used CCM toformulate a new treatment of the inner Cauchy boundary for a black hole space-time[51]. In the conventional approach, the inner boundary of the Cauchy evolution islocated at an apparent horizon, which must lie inside (or on) the event horizon [106].Thus truncation of the interior space-time at the apparent horizon cannot causally



affect the gravitational waves radiated to infinity. This is the physical rationale behindthe apparent horizon boundary condition. However, instabilities reported with theconventional approach motivated an alternative treatment.

In the CCM strategy, the interior black hole region is evolved using an ingoingnull algorithm whose inner boundary is a marginally trapped surface and whose outerboundary lies outside the black hole and forms the inner boundary of a region evolvedby the Cauchy algorithm. In turn, the outer boundary of the Cauchy region is handledby matching to an outgoing null evolution extending to I+. Data is passed betweenthe inner characteristic and central Cauchy regions using a CCM procedure similarto that already described for an outer Cauchy boundary. The main difference is that,whereas the outer Cauchy boundary is matched to an outgoing null hypersurface, theinner Cauchy boundary is matched to an ingoing null hypersurface which enters theevent horizon and terminates at a marginally trapped surface.

The translation from an outgoing to an incoming null evolution algorithm can beeasily carried out. The substitution β → β + iπ/2 in the 3D version of the Bondimetric Eq. (3) provides a simple formal recipe for switching from an outgoing to aningoing null formalism.

In order to ensure that trapped surfaces exist on the ingoing null hypersurfaces,initial data is chosen which guarantees black hole formation. Such data can be ob-tained from initial Cauchy data for a black hole. However, rather than extendingthe Cauchy hypersurface inward to an apparent horizon, it is truncated sufficientlyfar outside the apparent horizon to avoid computational problems with the Cauchyevolution. The initial Cauchy data is then extended into the black hole interior asinitial null data until a marginally trapped surface is reached. Two ingredients areessential in order to arrange this. First, the inner matching surface must be chosento be convex, in the sense that its outward null normals uniformly diverge and itsinner null normals uniformly converge. (This is trivial to satisfy in the sphericallysymmetric case). Given any physically reasonable matter source, the focusing theo-rem then guarantees that the null rays emanating inward from the matching spherecontinue to converge until reaching a caustic. Second, the initial null data must leadto a trapped surface before such a caustic is encountered. This is a relatively easyrequirement to satisfy because the initial null data can be posed freely, without anyelliptic or algebraic constraints other than continuity with the Cauchy data.

A code was developed which implemented CCM at both the inner and outerboundaries [51]. Its performance showed that CCM provided as good a solution tothe black hole excision problem in spherical symmetry as any previous treatment[89, 90, 73, 11]. CCM was computationally more efficient than these pure Cauchyapproaches (fewer variables) and much easier to implement. Achieving stability witha purely Cauchy scheme in the region of an apparent horizon is trickier, involvingmuch trial and error in choosing difference schemes. There were no complicationswith stability of the null evolution at the marginally trapped surface.

The Cauchy evolution was carried out in ingoing Eddington-Finklestein (IEF)coordinates. The initial Cauchy data consisted of a Schwarzschild black hole with


Jeffrey Winicour 30

an ingoing Gaussian pulse of scalar radiation. Since IEF coordinates are based oningoing null cones, it is possible to construct a simple transformation between theIEF Cauchy metric and the ingoing null metric. Initially there was no scalar fieldpresent on either the ingoing or outgoing null patches. The initial values for theBondi variables β and V were determined by matching to the Cauchy data at thematching surfaces and integrating the hypersurface equations (5) and (6).

As the evolution proceeds, the scalar field passes into the black hole, and themarginally trapped surface (MTS) grows outward. The MTS could be easily locatedin the spherically symmetric case by an algebraic equation. In order to excise thesingular region, the grid points inside the marginally trapped surface were identifiedand masked out of the evolution. The backscattered radiation propagated cleanlyacross the outer matching surface to I+. The strategy worked smoothly, and secondorder accuracy of the approach was established by comparison with an independentnumerical solution obtained using a second order accurate, purely Cauchy code[73].As discussed below, this inside-outside application of CCM has potential applicationto the binary black hole problem.

4.5 CCM for 3D Scalar Waves

CCM has been successfully implemented in the fully 3D problem of nonlinear scalarwaves evolving in a flat space-time. [18, 17] The main purpose of the study wasto demonstrate the feasibility of matching between Cartesian Cauchy coordinatesand spherical null coordinates. This is the setup required to apply CCM to thebinary black hole problem. Unlike the previous examples of matching, the Cauchyand characteristic patches do not now share a common coordinate which can beused to define the matching interface. This introduces a major complication intothe matching procedure, resulting in extensive use of inter-grid interpolation. Theaccompanying short wavelength numerical noise presents a new challenge in obtaininga stable algorithm.

The nonlinear waves were modeled on the equation

c−2∂ttΦ = ∇2Φ + F (Φ) + S(x, y, z, t), (21)

with self-coupling F (Φ) and external source S. The initial Cauchy data Φ(x, y, z, t0)and ∂tΦ(x, y, z, t0) are assigned in a spatial region bounded by a spherical matchingsurface of radius Rm.

The characteristic initial value problem (21) is expressed in standard sphericalcoordinates (r, θ, ϕ) and retarded time u = t− r +Rm:

2∂urg = ∂rrg −L2g

r2+ r(F + S), (22)

where g = rΦ and L2 is the angular momentum operator

L2g = −∂θ(sin θ∂θg)

sin θ− ∂ϕϕg

sin2 θ. (23)



The initial null data is g(r, θ, ϕ, u0), on the outgoing characteristic cone u0 = t0emanating from the matching worldtube at the initial Cauchy time.

CCM was implemented so that, in the continuum limit, Φ and its normal deriva-tives would be continuous across the interface r = Rm between the regions of Cauchyand characteristic evolution. The use of a Cartesian discretization in the interior anda spherical discretization in the exterior complicated the treatment of the interface.In particular, the stability of the matching algorithm required careful attention to thedetails of the inter-grid matching. Nevertheless, there was a reasonably broad rangeof discretization parameters for which CCM was stable.

Two different ways of handling the spherical coordinates were used. One was basedupon two overlapping stereographic grid patches and the other upon a multiquadricapproximation using a quasi-regular triangulation of the sphere. Both methods gavesimilar accuracy. The multiquadric method showed a slightly larger range of stabil-ity. Also, two separate tactics were used to implement matching, one based uponstraightforward interpolations and the other upon maintaining continuity of deriva-tives in the outward null direction (a generalization of the Sommerfeld condition).Both methods were stable for a reasonable range of grid parameters. The solutionswere second order accurate and the Richardson extrapolation technique could be usedto accelerate convergence.

The performance of CCM was compared to traditional ABC’s. As expected, thenonlocal ABC’s yielded convergent results only in linear problems, and convergencewas not observed for local ABC’s, whose restrictive assumptions were violated in allof the numerical experiments. The computational cost of CCM was much lower thanthat of current nonlocal conditions. In strongly nonlinear problems, matching appearsto be the only available method which is able to produce numerical solutions whichconverge to the exact solution with a fixed boundary.

4.6 The CCM Gravitational Module

The most important application of CCM is anticipated to be the binary black holeproblem. The 3D Cauchy codes now being developed to solve this problem employa single Cartesian coordinate patch [4]. A thoroughly tested and robust 3D charac-teristic code is now in place [14], ready to match to the boundaries of this Cauchypatch. Development of a stable implementation of CCM represents the last majorstep necessary to provide a global code for the binary problem.

From a cursory view, the application of CCM to this problem might seem routine,tantamount to translating into finite difference form the textbook construction of anatlas consisting of overlapping coordinate patches. In practice, it is an enormousproject, which would not have been possible to undertake without the impetus andsupport of the Binary Black Hole Grand Challenge.

A CCM module has been constructed and interfaced with Cauchy and character-istic evolution modules. It provides a model of how two of the best current codes totreat gravitation, the Grand Challenge ADM and characteristic codes, can be pieced


Jeffrey Winicour 32

together as modules to form a single global code. The documentation of the under-lying geometrical algorithm is given in Ref. [19]. The main submodules of the CCMmodule are:

• The outer boundary module which sets the grid structures. This definesmasks identifying which points in the Cauchy grid are to be evolved by theCauchy module and which points are to be interpolated from the characteris-tic grid; and vice versa. The reference base for constructing the mask is thematching worldtube, which in Cartesian coordinates is the “Euclidean” spherex2 +y2 +z2 = R2. The choice of lapse and shift for the Cauchy evolution governthe actual dynamical and geometrical properties of the matching worldtube.

• The extraction module whose input is Cauchy grid data in the neighborhoodof the worldtube and whose output is the inner boundary data for the exteriorcharacteristic evolution. This module numerically implements the transforma-tion from Cartesian “3+1” coordinates to spherical null coordinates. The al-gorithm makes no perturbative assumptions and is based upon interpolationsof the Cauchy data to a set of prescribed points on the worldtube. The metricinformation is then used to solve for the null geodesics normal to the slices ofthe worldtube. This provides the Jacobian for the transformation to null co-ordinates in the neighborhood of the worldtube. The characteristic evolutionmodule is then used to propagate the data from the worldtube to null infinity,where the waveform is calculated.

• The injection module which completes the interface by using the exteriorcharacteristic evolution to supply the outer boundary condition for Cauchy evo-lution. This is the inverse of the extraction procedure but must be implementedoutside the worldtube to allow for the necessary overlap between Cauchy andcharacteristic domains. (Without overlap, the domain of dependence of the ini-tial value problem would be empty.) The overlap region is constructed so thatit shrinks to zero in the continuum limit. As a result, the inverse Jacobian canbe obtained to a prescribed accuracy in terms of an affine parameter expansionof the null geodesics about the worldtube.

The CCM module has been calibrated to give a second order accurate interfacebetween Cauchy and characteristic evolution modules. When its long term stabilityhas been established, it will provide an accurate outer boundary condition for aninterior Cauchy evolution by joining it to an exterior characteristic evolution whichextracts the waveform at infinity.

4.7 The Binary Black Hole Inner Boundary

It is clear that the 3-dimensional inspiral and coalescence of black holes challenges thelimits of present computational know-how. CCM offers a new approach for excising



an interior trapped region which might provide the enhanced flexibility to solve thisproblem. In a binary system, there are major computational advantages in posingthe Cauchy evolution in a frame which is co-rotating with the orbiting black holes.Such a description seems necessary in order to keep the numerical grid from beingintrinsically twisted and strangled. In this co-orbiting description, the Cauchy evo-lution requires an inner boundary condition inside the black holes and also an outerboundary condition on a worldtube outside of which the grid rotation is likely to besuperluminal. An outgoing characteristic code can routinely handle such superlumi-nal gauge flows in the exterior [14]. Thus, successful implementation of CCM wouldsolve the exterior boundary problem for this co-orbiting description.

CCM also has the potential to handle the inner black hole boundaries of theCauchy region. As described earlier, an ingoing characteristic code can evolve a mov-ing black hole with long term stability [51, 49]. This means that CCM would also beable to provide the inner boundary condition for Cauchy evolution once stable match-ing has been accomplished. In this approach, the interior boundary of the Cauchyevolution is located outside the apparent horizon and matched to a characteristicevolution based upon ingoing null cones. The inner boundary for the characteristicevolution is a trapped or marginally trapped surface, whose interior is excised fromthe evolution.

In addition to restricting the Cauchy evolution to the region outside the blackholes, this strategy offers several other advantages. Although, finding a marginallytrapped surface on the ingoing null hypersurfaces remains an elliptic problem, there isa natural radial coordinate system (r, θ, φ) to facilitate its solution. Moving the blackhole through the grid reduces to a 1-dimensional radial motion, leaving the angulargrid intact and thus reducing the computational complexity of excising the innersingular region. (The angular coordinates can even rotate relative to the Cauchycoordinates in order to accommodate spinning black holes.) The chief danger inthis approach is that a caustic may be encountered on the ingoing null hypersurfacebefore entering the trapped region. This is a gauge problem whose solution lies inchoosing the right location and geometry of the surface across which the Cauchy andcharacteristic evolutions are matched. There is a great deal of flexibility here becausethe characteristic initial data can be posed without constraints.

This global strategy is tailor-made to treat two black holes in the co-orbitinggauge. Two disjoint characteristic evolutions based upon ingoing null cones wouldbe matched across worldtubes to a central Cauchy region. The interior boundary ofeach of these interior characteristic regions would border a trapped surface. At theouter boundary of the Cauchy region, a matched characteristic evolution based uponoutgoing null hypersurfaces would propagate the radiation to infinity.

Present characteristic and Cauchy codes can handle the individual pieces of thisproblem. Their unification appears to offer the best chance for simulating the inspiraland merger of two black holes. The CCM module is in place and calibrated foraccuracy. The one missing ingredient is the long term stability of CCM, which wouldmake future reviews of this subject very exciting.


Jeffrey Winicour 34

5 Acknowledgments

This work was supported by NSF PHY 9510895, by NSF INT 9515257 and by theBinary Black Hole Grand Challenge Alliance, NSF PHY/ASC 9318152 (ARPA sup-plemented). I thank the Alliance members for their support of characteristic methods.I also want to thank the many people who have supplied me with references. Pleasekeep me updated.

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