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From Geometry to Numerics: interdisciplinary

aspects in mathematical and numerical relativity

Jose Luis Jaramillo1,3‡ Juan Antonio Valiente Kroon2§ and

Eric Gourgoulhon3 ‖1 Instituto de Astrofısica de Andalucıa, CSIC, Apartado Postal 3004, Granada18080, Spain2 School of Mathematical Sciences, Queen Mary, University of London, MileEnd Road London E1 4NS U.K.3 Laboratoire Univers et Theories, UMR 8102 du C.N.R.S., Observatoire deParis, Universite Paris Diderot, F-92190 Meudon, France

Abstract. This article reviews some aspects in the current relationship betweenmathematical and numerical General Relativity. Focus is placed on the descriptionof isolated systems, with a particular emphasis on recent developments in thestudy of black holes. Ideas concerning asymptotic flatness, the initial valueproblem, the constraint equations, evolution formalisms, geometric inequalitiesand quasi-local black hole horizons are discussed on the light of the interactionbetween numerical and mathematical relativists.

PACS numbers: 04.20.-q, 04.25.D

‡ Email address: [email protected]§ Email address: [email protected]‖ Email address: [email protected]

http://arxiv.org/abs/0712.2332v2

From Geometry to Numerics 2

1. Introduction

1.1. Objectives

In this review article we focus on specific problems which are of interest both fornumerical relativists and for geometers. A number of review articles has been devotedin the last years to the technical aspects and state of the art of each respective domain—e.g. Andersson [12], Ashtekar & Krishnan [52], Bartnik & Isenberg [65], Baumgarte& Shapiro [71], Berger [88], Booth [117], Cook [187], Friedrich [260], Friedrich &Rendall [258], Gourgoulhon [287, 289], Gundlach & Martın-Garcıa [298], Klainerman& Nicolo [359], Krishnan [370], Lehner [374] , Lehner & Reula [373], Rendall [436],Reula [438], Shinkai & Yoneda [466], Thornburg [490], Winicour [515], and also [2].More theme specific reviews will be referred to in the corresponding sections. The aimhere is not the exhaustive description of the respective specific tools, but rather toidentify and discuss conceptual or structural challenges in General Relativity whichrepresent good candidates for a close collaboration between geometers and numericalrelativists.

This review is inspired in a workshop with a similar name that took place inthe context of the General Relativity Trimester held at the Institut Henri Poincare,Paris, from November 20 to 24, 2006 [1], which brought together specialists from boththe mathematical and numerical ends of General Relativity (GR). Due to the broadrange of topics covered in the workshop we have taken the methodological decisionto restrict our attention to numerical and mathematical aspects of the descriptionof isolated systems in GR. This somehow reflects the underlying ultimate interestsof the organisers, biased towards astrophysically motivated problems not includingCosmology. Unavoidably, not all the topics discussed at the time are covered in thisreview and on the other hand, some of the topics discussed in the review did not havea counterpart talk in the workshop.

1.2. General thoughts

1.2.1. Why from Geometry to Numerics? General Relativity is a theory thatdescribes the gravitational interaction as a manifestation of the geometry of spacetime.It is therefore natural to expect that a geometrical perspective may be helpful, andeven sometimes fundamental, for the further development of the theory at bothconceptual and practical levels —the latter understood as the explicit constructionof solutions. Does this point of view live up to the expectations raised?

In our opinion, the answer to this last question has two facets. On the one hand,as discussed in [260], understanding GR can be seen as tantamount to understandingthe properties of its solutions, and in this sense “getting qualitative and quantitative(respectively analytical, theoretical, and numerical) control on the long time evolutionof gravitational fields under general assumptions is still the most important openproblem in classical general relativity”. In this context there seems to be plenty ofoccasion for the successful interaction between geometry and numerics, not only inthe direction suggested by the title of the review but also on the converse one, fromnumerics to geometry.

Regarding the second facet, arguably, the other great challenge for the theoryis to make full contact with observations —in particular, nowadays, in the areaof gravitational waves physics. The nature of this specific endeavour requires theefficient calculation of spacetimes describing isolated systems. This calculation, to


be physically realistic has to be performed using numerical simulations of someparticular formulation of the Einstein equations. In this context, the fruitfulness ofusing geometry as a guideline in numerical applications seems much harder to assess,since the understanding of a particular analytical aspect of a geometric problem doesnot ensure its successful numerical implementation. Furthermore, it should not beforgotten that one is talking about the (potential) interaction of two communitieswith a different history, traditions, languages and in great measure, objectives.

1.2.2. Numerical Relativity and Relativistic Astrophysics. As somehow hinted in theprevious paragraphs, it is natural to assume that a “generic” relativist would haveinterest in both studying the nature of solutions of theory —and hence improving hisor her understanding of the theory— and also of making contact with the observation—which then would help to set the context and validity of the theory. Given thebroad spectrum of research topics that relativists deal with, it is to be expected thatthe particular choice of research problem will set the emphasis on either conceptualor observational aspects. In particular, this applies to the case of a numericalrelativist. For example, from a simplified point of view a “gravitational collapsecode”, can be potentially read as a tool to either understand conceptual issues likeCosmic Censorship or as a tool to predict astrophysical phenomena such as supernovaeexplosions. In the first case, it is natural for the numerical relativist to gain insight andeven try to simplify the code through a geometric view point. In the second case, sheor he has no option but to attempt to include all physical ingredients that are believedto play a role. Therefore, geometric insights may be comparatively less relevant. Inany case, whether a geometry point of view is used or not reflects, ultimately, in theway the code is constructed and read. Summarising, a given problem in numericalrelativity can be labelled as either geometric or as astrophysical numerical relativity.The tenet of this review is that the effects of the interaction between mathematicaland numerical relativists is bound to be more fruitful in the former case, while in thelatter the rewards would be more indirect. This duality will be a recurrent theme inthe review.

1.3. Tensions and synergy between Geometry and Numerics

As a first contact with the topic of the review we present some general reflexions whichillustrate the possible tensions and synergies to be expected as result of the interactionbetween geometry and numerics.

Global versus local. Global issues dominate modern mathematics. On the other hand,most numerical simulations are local in time. This leads to a tension between the toolsand goals in each community. An example of how different the points of view could beis given by the notion of black hole. The classical definition of a black hole makes useof global ingredients —e.g. [177]. On the contrary, everyday numerical simulationsmake use of quasi-local characterisations related to the notion of trapped region. Thistension reaches its climax in statements like there are no results about the existence ofasymptotically flat vacuum black hole spacetimes with radiation —see the discussion in2.2— by a mathematical relativist that can provoke the smile of a numerical relativist.However, this tension is not as dramatic as it appears at first sight. On the one hand,it is very possible that numerical simulations of binary black holes could provide themissing clues for an eventual global existence proof of the existence of dynamical


black hole spacetimes [427]. On the other hand, the global framework provided bythe Cosmic Censorship Conjecture acts precisely as guarantor of some of the technicaltools used in numerical simulations —for example excision or punctures.

Good geometric or analytic properties as a hope for good numerics. To a geometerwho is inexperienced in the field of numerics, one of the first reality checks he or shehas to confront is that analytic well-posedness and convergence of a numerical solutionto the analytic solution, do not guarantee the long term stability of the simulation—this issue has been clearly discussed in [53]. This last point raised before the bigbreak-throughs in the simulation of black hole binaries is still valid as regards to therealistic and accurate extraction of physics [424, 141, 59]. For this, it is possible thatan analytic or geometry insight may provide the crucial ingredient.

Geometry as a way of prescribing and extracting physics. Numerics relies on theuse of coordinates. This results, unavoidably, in ambiguities in the extraction ofphysics if geometric notions are not employed. However, these methods are globaland hard to implement numerically. In practical applications, one could use, forexample, perturbative or post-Newtonian notions to extract the physics. Nevertheless,this approach already relies in the acceptance of a certain global behaviour of thespacetime. But more importantly, the recent advances in the numerical simulations ofspacetimes could provide the opportunity for the reassessment of geometrical objectsas tools for the extraction of physics.

We conclude the section presenting some examples which give a taste of both thedifficulties and rewards of the interaction between geometry and numerics:

- Application of ideas from dynamical horizons to the extraction of physicalparameters. The development of some quasi-local approaches to black horizons—see section 6.2— has been directly inspired by the needs of the numericalimplementations. In particular Dynamical Horizons have already provided firstexamples of successful interaction in the calculation of mass, angular momentumand associated parameters —cf. Krishnan’s IHP talk and [57, 454, 455, 371, 146].

- Semiglobal evolution from small data. The existence results of the developmentof hyperboloidal initial data close to Minkowski by Friedrich [247] has beenillustrated by the numerical simulations of Hubner —see [328].

- Hawking mass. The straight-forward implementation of some natural geometricnotions meets often with numerical difficulties. This has been exemplified for thecase of the Hawking mass in Schnetter’s IHP talk [1].

- Calculation of high order derivatives. Any numerical simulation contains noisewhich tend to be amplified in the numerical evaluation of derivatives —see againSchnetter’s IHP talk [1]. This, for instance, complicates the implementation ofconstructions involving derivatives of the curvature tensor —like in the case ofthe metric equivalence problem in GR [355, 356].

- Handling of coordinates. Sometimes the use of coordinates which are easy tohandle from a numerical point of view, could lead to complications like coordinatesingularities and/or singular equations —see e.g. Schnetter’s IHP talk [1]. In somecircumstances a detailed geometric analysis does suggest a reformulation of theproblem which would solve this particular tension [28, 29].


- Penrose inequality. Classical bounds on energy loss in the head-on collision ofblack holes using the geometric Penrose inequality —see e.g. [276]— have beenshown to be overtly optimistic by numerical simulations. On the other hand, linesof thoughts based on the Penrose inequality have been useful in the discussion ofinitial data [209].

- Geometry-inspired initial data sets. Attractive geometric properties have beenemployed and prescribed in the construction of initial data sets for black holespacetimes —see among others e.g. [196, 198, 437, 349]. However, the numericalusefulness of these geometric inputs is still to be assessed.

- Geometry-inspired evolution systems. The close interaction between geometryand numerics is illustrated in the application of geometry as a guideline in thedevelopment and implementation of certain GR evolution systems —e.g. Z4formalism in [107] and also [441, 444, 523, 524]. However, the role of Geometryis not so evident in most numerical codes —see section 5.

It may be of interest for the reader to look at the list of successes given in [12].

1.4. Cautionary notes

Although this review stems from a workshop with a similar name [1], it should not beregarded as a proceedings or a systematic account of the former. We have used thediverse contributions to inspire a reflection on the interaction between geometric oranalytic methods and numerical tools in the context of GR. In this sense, unavoidably,it projects our personal prejudices and it is certainly not comprehensive. We areindebted to all the participants for the effort put on their presentations and apologisebeforehand in case of any misrepresentation —or omission— of their ideas. Thepresentations during the workshop will be referenced in a bundle as IHP talks.

It is clear —although it is already evident from the introductory paragraphs—that the word Geometry has been used in a loose sense that encompasses more thanDifferential Geometry and includes, for example, (global and local) analysis, PDEtheory or Group theory.

As already mentioned, our discussion will be centred in numerical andmathematical aspects of isolated systems in GR. There are other streams of Relativitywhich have experienced a fruitful interaction between the analytical and numericalcommunities: namely, on the one hand higher dimensional spacetimes and on theother the study of Cosmological spacetimes. In the latter case, this interaction hasbeen reported in the review [88] —see also the discussion in [12]. Without going intomuch detail, we just mention that the analysts (geometers) working in mathematicalCosmology have gained much from insights obtained through numerical experimentsin which, for example, the behaviour of Gowdy and Bianchi IX spacetimes is exploredtowards the initial singularity —for a recent work on the subject see [93]. In particular,it was using this sort of numerical experiments that the existence of the so-called spikes—non-smooth behaviour and big gradients of certain field quantities as one approachesto the singularity— was firstly attested in the Gowdy spacetimes [88]. The analyticalexistence and nature of these spikes has since been thoroughly analysed. This is aprime example of the use of numeric techniques as a tool to explore the nonlinearbehaviour of the gravitational field.

We start from the premise that Mathematical Relativity and Numerical Relativityare research areas on their own right. This review is addressed to those researchers


working in any of these two communities who feel that there is place for a fruitfulinteraction. We assume a broad general knowledge of GR. We will not be extremelyrigorous in our mathematical presentation and will refer the reader to the specialisedliterature.

The article is structured as follows. Section 2 reviews some conceptual issuesconcerning the notion of isolated bodies in GR, including asymptotic flatness andthe black holes. In section 3 the initial value problem in GR is presented as theappropriate approach to the generic construction of spacetimes. In particular, 3.1reviews the different types of initial value problem. Section 4 explores some aspectsof the construction of solutions to the constraint equations which we believe are, orcould be, of interests for numerical applications —the topology of initial hypersurfaces,the conformal method, the thin sandwich and the recent developments in gluingconstructions. Section 5 discusses various aspects of evolutions formalisms. Here,particular emphasis is given to initial value problems of Cauchy type, although someobservations concerning the characteristic and hyperboloidal problems are made.There is, also, a discussion of the conformal field equations. Section 6 reviewssome aspects of the modern description of black holes including recent results aboutuniqueness and rigidity of the Kerr spacetime, quasi-local definitions of black holes,geometric inequalities, helical Killing vectors and aspects of “puncture” black holeevolutions. Section 7 discusses miscellaneous topics that did not find an adequateplace in the structure of the review, but that we believe could not go unmentioned.Finally, 8 provides some conclusions and a brief list of topics where, we believe, theinteraction between Geometry and Numerics will acquire particular relevance in thecoming years.

2. Isolated systems in General Relativity

As mentioned before, this review will be concerned with solutions of General Relativitydescribing isolated systems. GR is formulated as a gauge theory of gravity, usually interms of the Einstein field equations (EFE)

Rµν − 1

2Rgµν = 8πTµν , (1)

where gµν denotes the metric of the spacetime, Rµν its corresponding Ricci tensor,R the Ricci scalar and Tµν is the so-called stress-energy tensor —in geometric unitsc = G = 1. Solutions to the EFE correspond to geometries rather than to metrics givenin a specific coordinate system, since metrics related by spacetime diffeomorphismsare physically equivalent. As a consequence of this, equation (1) understood as apartial differential equations (PDE) system for the metric components gµν constitutesan underdetermined (incomplete) system that cannot produce a unique solution. Anextra ingredient, namely a gauge fixing procedure, must be provided to establish astandard PDE problem —note this remark applies to any geometrically introducedPDE. Here we adopt a broad notion of gauge which includes not only a coordinatechoice but could also contemplate, for example, a tetrad choice and even perhaps achoice of conformal factor.

We are interested in solving equation (1) subject to boundary conditionsdescribing an isolated system for which the effects of the Cosmological expansionare neglected. A first way of approaching this problem is the construction of exactsolutions to equation (1) by making assumptions on the symmetries of the spacetime


and/or on the algebraic structure of their curvature tensors —the most comprehensivetreatise on exact solutions is [477]; reviews concerned with the physical aspects ofexact solutions can be found in [97, 98]. This way of doing things has provided deepconceptual and physical insight, as it has supplied the tools of most of the observationalpredictions of GR. Furthermore, it is the basis of perturbative analysis.

However, if one is interested in making statements about generic spacetimeshaving the right kind of boundary conditions and on how to construct them in asystematic manner, then one is led to consider an initial value problem or moregenerally and initial boundary value problem. Initial value problems can be of Cauchytype —that is, the initial data is prescribed on a Cauchy hypersurface† — or not —like in the case of the characteristic and the hyperboloidal [244] initial value problems—for a more complete discussion on these ideas see subsection 3.1. Only Cauchyinitial value problems allow, in principle, to reconstruct the whole spacetime, whilethe characteristic and hyperboloidal problems allow at most to reconstruct the domainof influence of the initial hypersurface.

The overall strategy is to prescribe suitable initial data on the initial hypersurfaceand then to evolve them by means of the EFE and the appropriate equations for thematter part. This is very much in the spirit of the approach taken by numericalrelativists. There are, however, a number of caveats to this approach —like existenceand uniqueness of solutions, stability and global issues including Cosmic Censorship—which are the concern of mathematical relativists (see [449] for a review addressingsome open problems in mathematical relativity and articulated around the notion ofCauchy hypersurface). In particular, the satisfactory resolution of these caveats by themathematical community is the guarantor of the consistency of the strategy employedby the numerical relativists.

Very often, the needs of the numerical community are well ahead of thedevelopments of the mathematical community. This applies, in particular, to thestudy of the motion of isolated bodies in GR, a problem of clear interest forastrophysicists which has been dealt numerically for a long time now —see e.g.[76]. The existence issue, that is taken for granted in the numerical community, isa challenging mathematical problem. We emphasise that this last point represents acrucial aspect in the overall coherence of the problem. The afore-discussed state ofaffairs is exemplified by the recent work on the existence of solutions describing themotion of isolated bodies —dust balls in this case— by Y. Choquet-Bruhat and H.Friedrich [161]. See also Choquet-Bruhat’s IHP talk [1]. Finally it must be said thata converse phenomenon is also true. The numerics for many mathematical questionsis still awaiting to be developed —for example the well-posed, constraint preservinginitial boundary value problem constructed by Friedrich & Nagy [257].

2.1. Asymptotic flatness

The physical intuition suggests that the spacetime describing an isolated body faraway from it, should be Minkowskian in some appropriated sense. Such a behaviouris generically referred to as asymptotic flatness. As already mentioned, checks on theconsistency of the theory requires a rigorous formulation of this physical intuition.

† For a discussion of the relationship between global hyperbolic spacetimes and the existence andproperties of smooth Cauchy surfaces, in particular the existence of Cauchy time functions, see[90, 91, 92].


The metric of an asymptotically flat spacetime, when expanded in terms of someconvenient distance parameter, should coincide with the flat Minkowski metric atleading order. The physics of the problem is, on the other hand, encoded in the loworder terms. In particular, one should expect to find there information about the massand radiative properties of the system. Now, it is natural to ask, firstly, whether itis possible to rephrase the above statements in a coordinate independent way; andsecondly, if one can make generic statements about the properties of the low terms inthe asymptotic expansions of the metric of an isolated system. To this end, Penroseintroduced the more specific notion of asymptotic simplicity —see e.g. [412, 418] for aprecise definition. Roughly speaking, a spacetime (M, gµν) is said to be asymptoticallysimple if there exists a positive scalar Ω (the conformal factor) and a spacetime withboundary (M, gµν) —the compactified, unphysical spacetime— with gµν = Ω2gµνsuch that the boundary of M, defined by Ω = 0, has a structure similar to that of thestandard compactification of Minkowski spacetime —that is, the conformal boundaryconsists, at least, of a null hypersurface (null infinity) I = I −∪I +. In the particularcase of the Minkowski spacetime, the conformal boundary consists, in addition, ofthree points: i0 —spatial infinity— and i− and i+ —respectively, past and futuretimelike infinity. In general, these additional points will not be present in the conformalcompletion of a generic asymptotically simple spacetime. However, if the spacetime isconstructed as the development of asymptotically Euclidean initial data —see equation(2)— then i0 will be part of the conformal boundary by construction. The situationwith respect to i− and i+ is more complicated. If the relevant spacetimes containblack holes, then there is no reason to expect that i± be present in the conformalcompletion, or if they are, that their structure should be that of a point —this isa difficult open question which will benefit from input from numerical simulations.In any case, and to summarise, the definition of asymptotic simplicity gets aroundthese issues by being only concerned with null infinity. The definition of asymptoticsimplicity is geometric in what it avoids the use of coordinates. Penrose’s originaldefinition requires the boundary of the unphysical spacetime to be smooth (C∞).Penrose’s original idea was to use the notion of asymptotic simplicity as a way ofproviding a characterisation of isolated systems in GR: the far fields of spacetimesdescribing isolated systems should be asymptotically simple in the sense that they admita smooth conformal compactification to null infinity. The latter suggestion is knownin the literature by the name of Penrose’s proposal.

The conceptual and practical advantages of the use of the compactified pictureto describe isolated systems have been discussed at length in the literature —see e.g.[36, 249, 250, 252, 253, 275]. Here we just mention that: i) the use of the compactifiedpicture allows to rephrase questions concerning the asymptotic decay of fields byquestions of their differentiability at the conformal boundary; ii) if the spacetime isasymptotically simple, then it is possible to deduce a certain asymptotic behaviour forthe components of the Weyl tensor known as the peeling behaviour —see e.g. [236]. Ifthe spacetime is such that it would not admit a smooth compactification —say, justCk instead of being C∞— then part of this formalism can still be recovered, but onehas to be more careful —see e.g. [183].

The idea of asymptotic flatness and the notion of asymptotic simplicity areinspired by the analysis of exact solutions to the EFE. If one wants to establish moregeneral properties of this class of spacetimes one has to resort to a formulation of theproblem based on an initial value problem and then make use of analytic techniques toobtain qualitative information about the solution or construct the solution numerically.


Here, again, intuition suggests that the right initial conditions for obtaining anasymptotically flat spacetime are asymptotically Euclidean ones. More precisely, onecould require an initial hypersurface Σ with at least one asymptotic end (a regionwhich is topologically R

3 minus a ball) in which one can introduce coordinates xi suchthat

γij =

(

1 +2m

|x|

)

δij +O(

1

|x|2)

, Kij = O(

1

|x|2)

, (2)

where |x| =√

(x1)2 + (x2)2 + (x3)2, and γij and Kij are, respectively, the first andsecond fundamental forms of Σ —see section 4 for more on this. More generalnotions of asymptotic Euclideanity have been considered in the literature —see e.g.[152, 153, 173]. The one given here, has a well defined ADM mass at the asymptoticend in question.

The first results concerning the existence of solutions to the vacuum EFE underthe above boundary conditions can be traced back to [231] in which local existencein time was shown to hold —well-posedness. Choquet-Bruhat and Geroch introducedthe notion of a maximal future development of an initial data set, and showed that foreach given initial data set there is a unique maximal future development [162]. Thework by Penrose has given a negative answer to the question of completeness in hissingularity theorem [413]: given initial data where the initial Cauchy hypersurface isnon-compact and complete, if the hypersurface contains a closed trapped surface —see section 6.2.2— then the corresponding maximal future development is geodesicallyincomplete. A further milestone was the resolution of the so-called boost problem inGR given in [160, 173]. Along these lines, the existence results of Friedrich [247]where it has been shown that the development of initial sufficiently small data givenon an hyperboloid is complete. It should be noted that a hyperboloid is not a Cauchyhypersurface, and in this sense the results of [247] are semiglobal. The analysis ofthe non-linear stability of the Minkowski spacetime of [172] has shown, among otherthings, the existence of vacuum radiative spacetimes which are asymptotically flat.More precisely, they show that every asymptotically flat initial data which is globallyclose to the trivial data gives rise to a solution which is a complete spacetime tendingto the Minkoswki spacetime along any geodesic. There are no additional restrictionson the data. The result gives a precise description of the asymptotic behaviour atnull infinity. Remarkably, however, the analysis is insufficient to guarantee the peelingbehaviour of the solutions. An alternative proof has been provided by Lindblad &Rodnianski [378]. This provides a global stability result of Minkowski spacetimefor the vacuum EFE in wave coordinate gauge —see section 5.1. The initial datacoincides with the Schwarzschild solution in the neighbourhood of spacelike infinity.This result is less precise as far as the asymptotic behaviour is concerned. Morerecently, following the spirit of Christodoulou & Klainerman’s proof, Bieri —see herIHP talk [1]— has considered more general asymptotically Euclidean initial data —i.e.with less decay and one less derivative— and shown the existence of a solution whichis a geodesically complete spacetime, tending to the Minkowski spacetime at infinityalong any geodesic. The solutions have finite ADM energy, but the angular momentumon the initial hypersurface is not necessarily well defined. The question answered byBieri’s analysis was what is the most general non-trivial asymptotically flat initialdata set giving rise to a maximal development that is complete. The remaining openquestion in this programme is to find sharp criteria for non-trivial asymptotically flatinitial data sets to give rise to complete maximal developments.


In what concerns the existence of spacetimes which satisfy the peeling behaviour,Klainerman & Nicolo [360, 361] have shown that the peeling behaviour is satisfied bythe development of a big class of asymptotically Euclidean spacetimes. This result is,however, not sharp for it can be seen that initial data for the Kerr spacetime —whichsatisfies the peeling behaviour as it is a stationary spacetime, see e.g. [212, 197]— isnot contained in the class of initial data sets they considered.

Regarding the related question of the existence of asymptotically simplespacetimes, it is now known that there is indeed a big class of spacetimes with thisproperty —see [179]. The essential ingredients for the latter result are the semiglobalresults of [247], and a refined version of the initial data sets which can be constructedusing the Corvino-Schoen method. This construction allows to glue fairly arbitraryasymptotically Euclidean initial data sets to a Schwarzschild asymptotic end —formore on this gluing construction see section 4.5. It has to be mentioned that theclass of spacetimes obtained in [179] are very special because by construction, theydo not contain any radiation near spatial infinity. The fact that the initial datais Schwarzschildean near infinity, implies that the Newman-Penrose constants of thespacetime vanish —the Newman-Penrose constants are absolutely conserved quantitiesdefined in the cuts on null infinity [404, 405]. This value propagates along thegenerators of null infinity, and it allows us to conclude that the Weyl tensor vanishesat future timelike infinity i+ —see [259, 256].

Given the state of affairs described in the last paragraph, the challenge is nowto find the sharp criteria on an initial data set to obtain an asymptotically simplespacetime. Seminal work in this direction is [251] —see also [255]— where a convenientframework to discuss the behaviour of the region of spacetime where null and spatialinfinity meet —the so-called cylinder at spatial infinity— has been introduced. It isimportant to point out that this work and related ones —see e.g. [498, 497, 500, 501]—are carried out in the conformally rescaled spacetime (unphysical spacetime) andmake use of the so-called conformal field equations, a generalisation of the EFE whichexploits the extra gauge freedom —a conformal one— that has been introduced viathe notion of asymptotic simplicity. The programme started in [251] and continuedin [255, 498, 497, 500, 501] has showed that there is a hierarchy of obstructions to thesmoothness of infinity which control the smoothness of the spacetimes at null infinity—see also Valiente Kroon’s IHP talk [1]. These obstructions are expressible in terms ofparts of the initial data of the spacetime. A further analysis of the obstructions pointstowards the following conjecture: if the development of asymptotically Euclideaninitial data admits a smooth conformal compactification (in the sense of Penrose)at both past and future null infinity, then the initial set is stationary near infinity—see [498, 497, 500, 501]. If shown to be true, the latter conjecture would give aprominent role to the initial data sets which can be constructed by means of theCorvino-Schoen gluing construction. It is worth mentioning that the developments ofconformally flat initial data sets like the Brill-Lindquist [129], Misner [400], Bowen-York [124] and Brandt-Brugmann [125], as well as the data used systematically in thenumerical simulations of binary black hole mergers —see the end of section 6.4.1—will have a null infinity (if any) which is not smooth. However the differentiability ofthe conformal boundary seems to be enough to guarantee the peeling behaviour [502].

Given an arbitrary asymptotically Euclidean initial data set (Σ, γij ,Kij),using the Corvino-Schoen construction one could construct another initial data set(Σ′, γ′

ij ,K′ij) which coincides with the former inside a compact set, but which is exactly

stationary near infinity. The development of (Σ, hij ,Kij) will possess a non-smooth


but possibly peeling null infinity. On the other hand, the development of (Σ′, h′ij ,K

′ij)

will be smooth. A natural question is the following: is there any substantial differencebetween the physics of the two developments? This is a complicated question whichwill only be answered in a satisfactory manner by a close collaboration betweennumerical and mathematical relativists. It is very likely that the latter issue will nothave any relevance for numerical relativists interested in astrophysical applications.On the other hand, they provide a whole unexplored area for those who want to usethe numerical methods to understand the nature of GR.

2.2. Black holes

A black hole is a region of no escape which does not extend out to infinity. Traditionally—see e.g. [310]— the definition is made using the conformal compactification ofspacetime. Namely, if the spacetime admits a suitable null infinity and the causal pastof null infinity is globally hyperbolic —i.e. it admits a Cauchy hypersurface— thenone defines the black hole region like the spacetime minus the causal past of future nullinfinity. The boundary of the black hole is the event horizon. A spacetime containinga single black hole contains two natural boundaries, one is null infinity and the other isthe event horizon —an inner boundary. The original picture arose from an analysis ofspherical symmetry. According to the so-called weak Cosmic Censorship picture —cf.section 6.3— the end state of any generic asymptotically Euclidean initial data set willbe a particular stationary black hole spacetime: the Kerr solution. Naked singularitiescan actually form, but the resulting spacetime are unstable [157, 170, 171] and non-generic. The rigorous definition of a black hole makes use of global ingredients: inparticular it requires the existence of a complete I +. As already mentioned, there areno global existence proofs of dynamical black hole vacuum spacetimes. The latter is apoint deserving further clarification. There are results concerning the global existenceof dynamical black holes in, for example, the spherically symmetric Einstein-scalarfield theory —e.g. [193] for the formation of black holes from a regular past— or inthe vacuum 5-dimensional setting. Regarding the latter example, see [194] where theorbital non-linear stability —i.e. with respect to perturbations respecting symmetry—of the Schwarzschild-Tangherini black hole is discussed, and thus the existence of aglobal, non-stationary black hole is established. Unfortunately, these examples are notrelevant for astrophysics. The available examples of 4-dimensional vacuum dynamicalblack holes suffer from some pathology —like the Robinson-Trautman spacetimesdiscussed in [174] with a complete future null infinity, but where past null infinityis incomplete— or are local —like in the discussion of “multi-black hole” spacetimesof [184], or that of spacetimes with isolated horizons of [375].

Among other things, a proof of the nonlinear stability of the Kerr spacetime wouldshow the existence of a wide class of spacetimes with non-stationary black holes. Theidea behind such a proof is to show that initial data which are in some sense close toKerr initial data will give rise to a black hole spacetime whose global structure will besimilar to that of the Kerr spacetime. In addition to proving the existence of dynamicalblack hole spacetimes, such a result would be an important step towards a proof ofthe (strong) Cosmic Censorship Conjecture —see section 3. A particular situationin which the latter conjecture has been proved in an asymptotically flat context isthe spherically symmetric Einstein-scalar field system studied by Christodoulou —see [166, 167, 168, 169, 171]. A crucial property of stationary black holes is theso-called rigidity: that is, the fact that stationarity together with a couple of further


assumptions on the behaviour of the horizon imply further symmetries in the spacetime—i.e. axial symmetry. The ideas behind the rigidity theorems have been recentlyrevisited by Isenberg & Moncrief with the aim of obtaining generalisations which arevalid for higher dimensions —cf. Isenberg’s IHP talk [1] and section 6.1.1.

2.3. Critique and practicalities

There has been a number of critiques to the notion of asymptotic flatness, mostnotoriously the ones articulated by Ellis [222, 223]. This critique has Cosmologicalmotivations and arose from the desire of analysing the nature of the interactionbetween local physics and boundary conditions in the expanding Universe. Inparticular, there was a desire to understand what difference does the use of particularboundary conditions make on results depending in an essential manner on asymptoticflatness, like the peeling behaviour, the positivity of mass —cf. section 6.3— and blackhole uniqueness. Ellis argues, for example, that it is unrealistic when considering localgravitational collapse to worry about observations with infinite life-times. The latteris what is implied in the usual definition of a black hole —see however section 6.2.The alternative to null infinity as a natural boundary of spacetime given in [222] is toconsider a world-tube —finite infinity— with a timelike boundary located at a spatialradius which is sufficiently far away from the sources. If brought to the context ofan initial value problem, Ellis proposal implies the use of an initial boundary valueproblem to describe the physics of isolated systems in which the effects of the Universeare put in by appropriate boundary conditions. The well possedness of the initialboundary value problem has been firstly discussed in [257]. The analysis given thereshows that if one makes use of a finite infinity then there is no covariant way ofprescribing the boundary information. This, in turn, would lead to ambiguities in theextraction of physics. One of the rationales behind the introduction of the null infinityformalism was precisely to avoid this sort of problems. For a neat discussion of theseissues, with an emphasis to the extraction of radiation see [234].

This critique is remarkable in what that if one takes it away from its Cosmologicalcontext it could also be employed when addressing the relevance of global issues fornumerical relativity. Most numerical simulations make an implicit use of the notion offinite infinity by calculating numerically the solution to an initial boundary problem.In this approach there is always the potential of “messing up” the physics of theproblem by setting boundary conditions which are not appropriate —see section 5.4.1.There are, however, also simulations which compactify in space, see [427]. Moreover,the numerical simulations do not cover an infinite time although given the currentadvances in the field there seems to be no essential difficulty for letting the simulations“to run as long as it is necessary”.

It is worth noticing that there have been some semiglobal numerical calculationsusing the conformal field equations which exemplify the hyperboloidal existence resultsof [246] —see [326, 327, 329, 328]. There are some further attempts to calculateportions of the Schwarzschild from hyperboloidal data —[330] unpublished. For acritical review of the successes and problems of this programme see [332]. Morerecently, there have been some global calculations of the Schwarzschild spacetime[523, 524]. These calculations make use of a more general version of the conformal fieldequations and conformal Gauss coordinates. These numerical simulations exemplifythe analysis of this type of coordinates given in [254]. We also mention the recentfully pseudo-spectral scheme [318] for solving hyperbolic equations on conformally


compactified spacetimes.The discussion in the previous paragraphs leads to a situation where besides

a number of simulations tailor-made for exemplifying some analytical aspects ofsolutions of the EFE, a majority of the state-of-the-art numerical codes used tosimulate spacetimes containing dynamical black holes are not designed for addressingglobal issues in GR. So, what is the relevance for the numerical community of thework on global issues carried out by mathematical relativists? As mentioned in theintroductory section, the role is to ensure that some of the approaches implementedin numerical simulations are justified. For example, if the non-linear stability of theSchwarzschild turned out to be false, then it could happen that the development ofan asymptotically Euclidean black hole initial data would not have the asymptoticstructure of the Kerr spacetime, and thus measurements of gravitational radiationat finite radius would be rethought. Similarly, if the no-hair theorems were notvalid, there would be no justification for employing perturbative methods on a Kerrbackground to analyse the late stages of the evolution of a black hole binary merger.These examples are extreme but highlight the need of having the theoretical frameworkon a sound ground.

Finally, we would like to retake the point raised in [12] that the problem of thestability of the Kerr spacetime opens a natural arena for the interaction of numericaland mathematical relativity. Indeed, it is easy to imagine that tailor-made simulationscould provide crucial insights concerning the asymptotic decay of the gravitational fieldnear the horizon. This interaction would be, notwithstanding, on the lines of fromNumerics to Geometry.

3. Initial value problems

If one wants to discuss generic properties of solutions to the Einstein field equationswhich describe in some appropriate sense isolated systems, one has to find asystematic way of obtaining these solutions. The only approach to the construction ofsolutions which seems general enough is that of an initial value problem (IVP). Thismathematically sound approach is perhaps not ideal from a physical point of view,and indeed a number of objections have been raised. One could argue that all majorobservational predictions of GR have been obtained by means of the analysis of exactsolutions.

A first (conceptual) objection consists in making sure that one is able toreconstruct, starting from initial data, the whole maximal extension of the spacetime—the strong Cosmic Censorship [416, 417, 11]. This has implications on causalityviolations and predictability of the theory —see e.g. [449].

Another objection questions to what extent an approach to GR based on aninitial value problem is appropriate to obtain predictions which can be contrastedwith measurements. One of the crucial problems behind is that for a certain system ofphysical interest, there may be many possible ways of constructing physically plausibleinitial data. This is particularly the case when attempting to construct initial data forspacetimes containing dynamical black holes. In order to construct data for a blackhole spacetime, one should be able to find a way of identifying the black hole in theinitial data. For these purposes, the notion of event horizon is of no use for it is aglobal quantity which is only known once the whole spacetime —up to null infinity— isknown. Instead, one can use the notion of apparent horizon: for if there is an apparenthorizon on the data it must be contained in the black hole region —see section 6.2. In


other words, the presence of an apparent horizon indicates the existence of a black hole—if the weak Cosmic Censorship is true. Furthermore, data with an apparent horizonwill be geodesically incomplete as a consequence of Penrose’s singularity theorem [413]—thus, indicating the formation of a singularity. It should be emphasised that a blackhole data does not need to contain an apparent horizon, but if it contains one, it mustbe a black hole. Similarly, it is not clear whether data with two apparent horizons willcontain two black holes. However, this is a reasonable assumption to begin with. Tomake things worse, note that many of the approaches for constructing initial data forblack holes are explained more in terms of mathematical advantages, like the simplicityof the equations —like in the case of conformal flatness— than in physically basedconsiderations —see however section 6.2 for an account of the ideas introduced bythe quasi-local approach to black holes. To summarise, physics seems to be hard toencode in initial data sets for the EFE.

There has been a number of works devoted to compare the results of numericalsimulations in the search of robust aspects —that is, structures in the numericallycalculated spacetimes which are independent of the particular type of initial databeing used —see e.g. [60, 474, 491] and IHP talks by Hannam and O’Murchadha.Crucially, the usefulness of numerical simulations as a source of wave form templatesfor the detection of gravitational radiation in the current generation of interferometricgravitational wave observatories depends on the assumption of the robustness. Fromthe point of view of a mathematical relativist or a geometer, this issue is essentiallyopen and unexplored. The problem is certainly complicated, and possibly one will haveto wait several years before some qualitative statement can be made. The resolution ofthe non-linear stability of the Kerr spacetime may provide first insights and rigorousanswers to the question of robustness of initial data sets.

3.1. On the different types of initial value problems

Initial value problems can be classified according to the nature of the hypersurfaceon which the initial data is prescribed. Usually, the initial hypersurface Σ is takento be spacelike. As mentioned in section 2.1, when discussing initial value problemsfor isolated systems the spacelike initial hypersurface Σ is naturally assumed to beasymptotically Euclidean. This particular type of initial value problem in GR will bereferred to as a Cauchy initial value problem as it would allow, in principle, to recoverthe whole spacetime —see references [310, 504] for a rigorous discussion of these issues,and [258] for a recent review on the subject. The seminal work of Choquet-Bruhat [231]has shown the well-posedness of the Cauchy initial value problem in GR: an existencetheorem local in time —that is, existence is guaranteed for a small time interval, ina neighbourhood of the initial hypersurface. The portion of spacetime recovered bythis local existence theorem is called the development of the initial data. At this levelis not possible to discuss global existence of solutions without more assumptions orinformation on the initial data —for further discussion on this, see for example thereview [359]. Nevertheless, it has been shown that the development has a uniquemaximal extension [162].

Initial value problems on spacelike hypersurfaces are not necessarily Cauchy initialproblems. For example, the hyperboloidal initial value problem first introduced by H.Friedrich in [244], on which initial data are prescribed on a spacelike hypersurface—which could be thought of as intersecting null infinity— is not of Cauchy type. Aninitial value problem on a hyperboloidal hypersurface would not allow us to recover


the whole spacetime. Moreover, it is not possible to find out where the developmentof hyperboloidal data lies in a globally hyperbolic spacetime, more specifically, onecannot relate the hyperboloidal data with a hypothetical Cauchy data.

As pointed out in [478], the Cauchy problem in GR is in some sense natural fortheoretical discussions, but strictly speaking it does not necessarily correspond to theway things are done in numerical relativity, where one usually aims to solve the EFEonly within a compact domain —cf. sections 2.3 and 5.4.1. The latter requires thesolution of an initial boundary value problem, where in addition to the data to beprescribed on a spacelike domain, one also has to prescribe extra data on a timelikeboundary. The presence of this extra data —the boundary data— raises the questionof which data can and must be specified on the timelike boundary in order to havea solution —and further, whether there are any compatibility conditions requiredbetween the initial data and the boundary data. A first rigorous discussion of theboundary value problem in GR, including a local existence result has been providedby Friedrich & Nagy [257]. A recent IVBP well-posedness result in the context of theharmonic formulation —see section 5.1— has been presented in [368, 367].

The initial data must not necessarily be prescribed on spacelike hypersurfaces.One could also prescribe it on null hypersurfaces, in which case we speak of acharacteristic initial value problem. However, as the domain of influence of a non-singular null hypersurface —i.e. a null hypersurface without caustics— is empty, oneneeds, for example, to consider data prescribed on two intersecting null hypersurfacesin order to obtain a non-trivial development. A local existence theorem for this sortof problem has been given in [435]. A variant of the problem is to prescribe data ona light cone. There is no existence result for the Einstein equations in this case —seehowever [246].

The characteristic initial value problem in GR can be traced back to the seminalwork of Bondi and collaborators on gravitational radiation in the early 1960’s —see [116, 447]. An early systematic discussion of the characteristic problem can befound in [448]. From the point of view of asymptotics, one can formulate the so-called asymptotic characteristic initial value problem in which initial data is prescribedon both an outgoing light cone and on null infinity. The portion of spacetime tobe recovered from this initial problem lies at the past of these null hypersurfaces.Existence results for this type of characteristic initial value problem have been givenin [242, 241, 243, 350] in the context of the conformal field equations. An analysisof the well-posedness of the problem more in the lines of the original work of Bondiand collaborators can be found in [262]. Another type of characteristic initial valueproblem is the 2 + 2 initial value problem —see e.g. [217]— and the initial valueproblems with data prescribed at past null infinity of [246, 248].

On a characteristic initial value problem, a part of the EFE reduce to interior(transport) equations on the null hypersurface which can be recast as ordinarydifferential equations along the generators of the null hypersurface. Further, thedata are free —i.e. not subject to elliptic constraints like in the case of spacelikehypersurfaces, see section 4. This feature was crucially exploited in the seminal workon gravitational radiation. The characteristic problem is naturally adapted to thediscussion of gravitational radiation, however, it tends to run into problems whencaustics form. Further, physics is hard to prescribe on null hypersurfaces.

Finally it is noted that an IVP of a mixed type on which data is prescribed ona combination of spacelike and null hypersurfaces has been considered: the Cauchy-characteristic or matching problem —see [94] and the review [515].


3.2. Gauge reduction

As it has been discussed at the beginning of section 2, the formulation of the EFEas a standard PDE problem requires the adoption of a gauge choice. This reductionprocess —see e.g. [260, 258]— involves four different differential systems:

i) The main evolution —also, reduced or relaxed— system, whose solution providesthe spacetime geometry in a certain coordinate system.

ii) The system of constraints, which controls the permanence in the submanifold ofsolutions of the theory.

iii) The gauge system, which allows the fixation of a coordinate system and thereforecasts the main evolution system as a genuine PDE system.

iv) The subsidiary system, consisting in the evolution equations of the auxiliary(subsidiary) constraints introduced in the previous steps and guaranteeing theoverall consistency.

The different manners of dealing with the previous points define a variety of evolutionformalisms in GR, where relative stress to be placed on systems i)-iv) depends on thespecific aspect of the EFE one wants to address. In section 5 we will briefly reviewsome of the more relevant evolution formalisms considered in numerical relativity witha focus on formulations based in the GR Cauchy IVP. Prior to this, one must discussthe system of constraints on a spacelike hypersurface.

4. Initial data for the GR Cauchy problem

The initial data to be prescribed in this IVP problem are given by a pair of symmetrictensor fields (γij ,Kij) on the initial Σ, respectively corresponding to the inducedRiemannian metric and the extrinsic curvature under the embedding of Σ into(M, gµν). Such an interpretation of Σ as a spacelike hypersurface of (M, gµν) demandsthe fulfillment of the so-called Hamiltonian and momentum constraints, respectively

(3)R−KijKij +K2 = 16πρ (3)

Dj

(

Kij − γijK)

= 8πJ i, (4)

where (3)R and Di are, respectively, the Ricci scalar and the Levi-Civita connectionassociated with γij , K is the trace of Kij , ρ is the energy density and J i the currentvector. An initial data set is said to be maximal if K = 0. If in addition Kij = 0,then it is called time symmetric.

We proceed now to discuss some aspects concerning the way the Einsteinconstraints (3) and (4) are solved, putting some emphasis on the relation of these ideasto numerical implementations —a major review on general aspects of the constraintequations can be found in [65]; see also [187, 420, 289] for a numerical counterpart. Forconcreteness we shall concentrate our discussion on the vacuum constraint equations.

4.1. Topology of the initial hypersurface

As mentioned already in section 2, from a Cauchy initial value problem point of view,the natural requirement to obtain a spacetime describing an isolated body is to havean initial hypersurface Σ that has at least one asymptotically Euclidean end —seeequation (2). The theory of hyperboloidal hypersurfaces is less developed, althoughsome of the essential ideas of the asymptotically Euclidean setting can be retaken —see


again [65, 16, 13, 15, 14]. Initial hypersurfaces with more than one asymptotic end aresaid to have a non-trivial topology. Note however, that the introduction of multipleasymptotic regions is not the only way of having initial data sets with non-trivialtopology. The introduction of a wormhole connecting two points in a hypersurfacewould also result in a non-trivial topology. This was firstly exemplified in [399]; withthe use of the Isenberg-Mazzeo-Pollack (IMP) gluing techniques —see section 4.4— itis now possible to construct this type of initial data in a systematic way.

Initial hypersurfaces with non-trivial hypersurfaces have been routinely used inthe mathematical literature as a way of manufacturing initial data sets for blackhole collisions. The simplest example of the latter are the usual time symmetricinitial data for the Schwarzschild spacetimes in isotropic coordinates, which containtwo asymptotic regions connected by a so-called Einstein-Rosen bridge, throat orwormhole. This example can be readily be generalised to contain n + 1 asymptoticregions rendering the so-called Brill-Lindquist initial data [129]. Another approach,which renders time symmetric initial data on a hypersurface Σ containing 2 asymptoticregions and n wormholes and can be interpreted as the data for n black holes —theMisner initial data— was given in [400].

A first step towards the construction of non-time symmetric data for blackholes was given by Bowen & York in [124], where an analytic solution of themomentum constraint in the conformally flat case was presented, the Bowen-Yorksecond fundamental form —a discussion of the general solution of the so-calledEuclidean momentum constraint can be found in e.g. [205, 85]. The so-called Brandt-Brugmann puncture initial data [125] can be in some ways regarded as the non-timesymmetric generalisation of the Brill-Lindquist data. It is important to mention thatall the initial data sets discussed in this last paragraph are conformally flat —seesection 4.2. But this is merely a simplifying assumption and in no way essential.

The fall-off conditions for γij and Kij to be asymptotically Euclidean havealready been discussed in section 2.1 —see equation (2). As in the discussion ofboundary conditions for the whole spacetime, the discussion of the fall-off conditionson Σ can be geometrically reformulated in terms of a conformally compactified 3-dimensional manifold Σ —see [83, 84, 246, 251]. The compactification procedure is,in this circumstance, analogous to that induced by the introduction of stereographiccoordinates to compactify R

2 to render the 2-sphere S2. From this point of view, the

compactified initial hypersurface Σ contains, say, m singled out points i1, . . . , imcorresponding to the m asymptotic ends of the physical initial hypersurface. Forexample, the standard Schwarzschild initial data render a Σ with the topology of the3-sphere S

3 and two singled out points. The Brill-Lindquist data for 2 black holeswould render a Σ with 3 singled out points, while in case of the Misner data theresulting Σ has the topology of a torus with 2 singled out points.

From a mathematical point of view, working with Σ instead of Σ has severaltechnical advantages. It is simpler to prove existence of solutions for an ellipticequation on a compact manifold than on a non-compact one. The price one hasto pay is that the equations will be singular at the points at infinity i1, . . . , im.However, these singularities are mild. It is also simpler to analyse the fields interms of local differentiability in a neighbourhood of the points at infinity than interms of fall-off expansions at infinity in Σ. This approach to discussing the theoryof the constraint equations has been introduced and used in various applications in[246, 251, 196, 197, 198, 82].

The puncture initial data used in many of the simulations of the coalescence of


black holes, as introduced in [125], is a sort of compromise between using the physicalhypersurface Σ and the compactified one Σ. For punctured data, one of the asymptoticends —the so-called reference, or physical asymptotic end— is not compactified. Theresulting initial hypersurface Σ will have the topology of S3\i1. The singular behaviourof the constraints near the punctures reflects in a singular behaviour of their solutionsthat is mild and well understood —it is related to the singular behaviour of the Greenfunctions of the elliptic operators appearing in the constraint equations —see [83, 84].Numerical relativists interpret this phenomenon as a coordinate singularity at thepuncture —see [83, 84, 305, 306]. The attractive of puncture initial data sets fornumerical relativists is that they represent black holes in R

3 without excision and itis well understood how to construct puncture initial data for any number of boosted,spinning black holes [125, 199].

The relevance for numerical relativists of initial data sets with non-trivial topologystems from the fact that they allow to model black hole spacetimes by only makinguse of the vacuum EFE: indeed, if there is a wormhole in the initial data, then it canbe concluded that the spacetime will be geodesically incomplete [265]. Note however,that in all this argument one is indirectly assuming that Cosmic Censorship holds andthat the resulting spacetime is really a black hole spacetime. In particular, from aphysical point of view one is not interested in what happens in the extra asymptoticends —see however the discussions in section 6.4.2 on why the puncture methods work.From the point of view of a physicist, it is fair to say that the use of initial data setswith non-trivial topology is a trick —there have been, however, some works on tryingto analyse the possible physical differences between initial data sets with differenttopology but describing the same system —see e.g. [283, 284, 496, 467, 147].

4.2. The Conformal Method.

A standard approach to study the space of solutions of the constraint equations is theso-called conformal method —or conformal Ansatz. The method not only providesa systematic approach to make statements about the existence of solutions to theconstraint equations, but also gives a starting point for the numerical computation ofthe solutions to the equations. A recent account of what is known about solutions tothe constraint equations is given in [65] —see also [187, 420, 289].

The conformal method is based on the conformal rescaling of the metric

γij = Ψ4γij , (5)

and on the combined conformal rescaling and tensor splitting of the traceless partAij of the extrinsic curvature Kij into a transverse-traceless and longitudinal part.Different —non-equivalent— constructions follow from the non-commutativity of theconformal rescaling and tensor splitting process. A key point in all of them is theneed of a specific rescaling for the (conformal) traceless part of the extrinsic curvaturein order to set the constraint equations in a form which is optimal for their analyticstudy. In particular, if K is constant, the conformal method reduces the problem ofsolving the constraint equations to a decoupled underdetermined elliptic problem.

Much has been learned about solutions of the Einstein constraint equations usingthe conformal method. However, many other interesting theoretical questions cannotbe addresses using these methods. The main problem being that the conformalmethod produces solutions of the constraint equations, essentially from scratch. Itturns out that it is very difficult to encode physics into the conformal metric γij


and in the symmetric transverse-traceless tensor ATTij —which act as seeds of the

method. For example, there seems to be no way of constructing systematically initialdata for stationary solutions, although there is an a posteriori check for the caseof time symmetric data [200]. Most of the natural simplifying assumptions thatare directly suggested by the structure of the conformal method happen to have noessential physical or geometrical content —most notably the assumption of conformalflatness. This is bluntly illustrated by results showing that there are no conformallyflat hypersurfaces in the Kerr spacetime [267, 499, 498]. Some examples of solutionsto the Einstein constraints where it has been possible to put some desirable physicalproperties with some level of success can be found in for example [196, 197, 209]. Inthese last references either a non-flat metric or a different choice of the Bowen-YorkAnsatz for the extrinsic curvature —see [124]— have been used. Another approachto prescribe physics is the use of tensors γij and Aij motivated by post-Newtoniananalysis as in [408].

It is noted that given the relation of Kij with the canonical momentum in theHamiltonian formulation of the theory [35]

πij =√γ(Kγij −Kij) , (6)

this view of the constraints can be referred to as a Hamiltonian point of view [520, 421].No reference to the slicing is present and in particular no reference to the lapse functionor shift vector emerges in the formulation of the constraints. This formulation reflectsthe character of the initial data problem as a one-hypersurface embedding problem.In this context, in particular in the numerical community, the conformal Ansatz isalso referred to as the extrinsic curvature approach.

4.3. The conformal thin sandwich method.

The idea of a thin-sandwich approach to GR —in the spirit of the Jacobi principle inMechanics— seems to have been firstly introduced by Baierlein, Sharp and Wheeler(BSW) [56]. They conjectured the possibility of specifying the spacetime metric gµνon two hypersurfaces and then of recovering the spacetime in between. In particular,when applied to two infinitesimally closed spatial slices —the BSW Thin Sandwich(TS) conjecture— this approach would imply a manner of providing initial data fora Cauchy evolution of the Einstein system. The TS conjecture, through its two-hypersurfaces Ansatz, proposes a different perspective to the constraint problem ascompared to the one-hypersurface embedding of the extrinsic curvature approach. Inthis setting one aims at prescribing the induced metric and its time derivative (γij , γij),in what can be seen as a Lagrangian formulation of the problem. However, althoughthis conjecture can be seen to work for particular examples, the analysis in [67] showsthat the conjecture fails in generic cases.

More recently —see also previous works [341, 340, 514, 189] and section 6.4.1— anew interpretation of the Einstein constraint equations, partly on the spirit of the TSview point, has been given in [520]. In this approach, the freely specifiable data on theinitial hypersurface Σ consists of a conformal metric γij and its time variation uij ≡ ˙γij

—which is imposed to be traceless. The latter can be shown to amount to the choiceof the conformal metric in two infinitesimally close slices. Using the kinematic relationbetween the time derivative of the physical 3-metric and the extrinsic curvature, thetraceless part Aij can be written explicitly in terms of the lapse function N and

shift vector βi —cf. notation in section 5.2. A function N conformally related to


the physical lapse is identified as the naturally freely specifiable parameter. Thisconformal rescaling together with the absence of rescaling of the shift vector, permitsto derive the key conformal rescaling for Aij introduced by Lichnerowicz [376] andleading to the decoupling of Hamiltonian and momentum constraints in the constantK case. The specification of the free data (γij , ˙γij ;K, N) transforms the constraints

into an elliptic system on Ψ and βi similar in form to the one obtained in the extrinsiccurvature approach [520]. In this Conformal TS approach (CTS) the role of every partof the metric is explicit. The formulation has been extended and improved in [421],which in particular reconciles the Hamiltonian and Lagrangian points of view. The newkey feature is the introduction of a weight function σ which permits to recover thecommutativity between conformal rescalings and tensor splittings. Moreover, withthe choice σ = 2N , the vanishing of the tranverse-traceless part of the extrinsiccurvature Aij

TT characterises stationarity, in agreement with the general (intuitive)

understanding of AijTT as the piece encoding the radiative —or dynamical— degrees of

freedom. Regarding the free data, the approach in [421] pushes forward the Lagrangiananalogy. Understanding K as a configuration variable, it is proposed the prescriptionof K instead of N at the (first) initial slice. This translates into a fifth elliptic equationfor the conformal lapse N which together with the CTS equations gives rise to theExtended Conformal Thin Sandwich (XCTS) system —see e.g. [421, 420, 70] for thelist of the equations.

One of the advantages of the CTS approach is that using it, it seems easierto prescribe physics than in the standard approach based on the extrinsic curvatureapproach. In particular, as discussed in [187, 289], the CTS method motivates definiteAnsatze for the free part in the construction of stationary initial data —at least in theaxisymmetric case. More generally, this CTS approach has been used also to constructbinary initial data sets in quasi-equilibrium —see e.g. [69, 114, 291, 495] or Uryu’sIHP talk [1] for the case of neutron stars; [290, 295, 517, 188, 151] for the case ofbinary black holes; and [72, 487, 294] for the mixed black hole-neutron star binarysystems.

4.3.1. Uniqueness issues in the XCTS system. In contrast to the CTS and closelyrelated elliptic systems —see e.g. [410, 411, 518] and also further references in[520, 421]— there is to date a lack of systematic analyses of the mathematicalproperties of XCTS system. In particular, no definite results are available aboutexistence conditions in this system, and very little is known about uniqueness.Numerical evidence of the non-uniqueness of solutions of the XCTS elliptic systemhas been found [422]. This evidence is of relevance as the XCTS system or somewhatrelated ideas are used to construct initial data for binary neutron star and black holebinary systems —cf. the literature listed in the previous paragraph. Moreover, italso has implications for the groups implementing constrained evolution schemes —cf.section 5.2.2.

A first analysis of the non-uniqueness issue, looking in particular at the questionof genericity, has been given in [70]. The most complete analysis available of the non-uniqueness of the solutions to the XCTS system, using Lyapunov-Schmidt bifurcationtheory, has been given in [507] —see also Walsh’s IHP talk [1]. It is found that if thelinearised system has a Kernel for sufficiently large data, then there exists a parabolicbranching of the solutions as the one happening in [422]. The prosaic reason forthe existence of the non-trivial Kernel is the presence of a wrong sign in one of the


equations of the XCTS system —more precisely, in the equation for the lapse. Thus,standard methods based on the use of a maximum principle cannot be employed.The numerical evidence shows that the assumption of a nontrivial Kernel is certainlynot far-fetched, and actually occurs in numerical implementations. The situation getseven worse if a treatment of boundary conditions is required —see e.g. [345].

The issue of the non-uniqueness of solutions to the XCTS equations is an exampleof the tension underlying the relation between numeric and mathematical relativists:a successful method of prescribing physics into numerical simulations with non-desired mathematical properties. In the absence of better alternatives, the numericalcommunity seems prepared to take the risk.

4.4. Construction of initial data sets using the Isenberg-Mazzeo-Pollack gluing.

In recent years it has been possible to address a number of issues concerning propertiesof initial data sets using a tool of geometric analysis: gluing —cf. Pollack’s IHP talk.While the conformal method is a procedure to construct solutions to the constraintequations from scratch, gluing constructions allow to obtain new solutions by meansof direct sums of two already known solutions. Analytic gluing techniques have playeda prominent role in many areas of differential geometry: some notable applicationsinclude the study of the topology of 4-manifolds and the study of minimal and constantmean curvature surfaces in Euclidean 3-space.

In [337, 338, 182] a particular gluing construction for solutions to the Einsteinconstraint equations was developed under the assumption of a constant K —theIsenberg-Mazzeo-Pollack (IMP) gluing. This construction allows to demonstrate howspacetimes can be joined by means of a geometric connected sum, or how a wormholecan be added between two points in a given spacetime —at the level of the initialdata. This construction makes it possible to address a number of issues concerningthe relation of the spatial topology to the geometry of solutions of the constraints andthe constructibility of multi-black hole solutions [184].

The sharpest possible gluing theorem for the Einstein constraint equations hasbeen developed in [182]. Using it, it is possible to show that for a generic solutionof the constraint equations and any pair of points in this solution, one can add awormhole connecting these points to the solution with no change in the data awayfrom a neighbourhood of each of these points. Further, one can show that for almostany pair of initial data sets —including, say, a pair of black hole data sets, or acosmological data set paired with a set of black hole data— one can construct a newset which joins them.

In order to assess the relevance of the IMP gluing construction for numericalrelativity, we present a rough overview on how the IMP gluing method works. Considertwo initial data sets (Σ1, γ

1ij ,K

1ij) and (Σ2, γ

2ij ,K

2ij) for the EFE. Let p1 ∈ Σ1 and

p2 ∈ Σ2 be two arbitrary points. The idea of the IMP construction is to find an initialdata set (Σ12, γ

12ij ,K

12ij ) such that Σ12 has the topology of Σ1#Σ2, where # denotes the

connected sum of manifolds ‡. The pair (γ12ij ,K

12ij ) is such that on Σ1\ball around p1

it coincides with (γ1ij ,K

1ij) and on Σ2 \ ball around p2 with (γ2

ij ,K2ij).

In order to construct the pair (γ12ij ,K

12ij ) satisfying the Einstein constraints, one

considers a conformal metric γ12ij with the property that on Σ1 \ B(p1) and Σ2 \ B(p2)

‡ The connected sum is performed by removing B(p1) = ball around p1 and B(p2) =ball around p2 and then joining S1 \ B(p1) and S2 \ B(p2) at ∂B(p1) and ∂B(p2) with a cylinder,I, of topology ∂B(p1)× [0, 1].


coincides with γ1ij and γ2

ij , respectively, and on the cylinder I it interpolates between

them. Similarly, one constructs a γ12ij -traceless tensor A12

ij which again interpolates

between A1ij and A2

ij . By construction, the resulting pair renders a solution to theconstraints on Σ1\B(p1) and Σ2\B(p2) but not on the cylinder I. The machinery of theconformal method —see section 4.2— is used then to find a solution to the constraintson Σ1#Σ2 using the pair (γ12

ij , A12ij ) as seeds for the procedure. Some amount of

technical work has to be invested, once the solution (γ12ij ,K

12ij ) has been obtained to

guarantee that the solution coincides with the original (γ1ij ,K

1ij) and (γ2

ij ,K2ij) outside

the gluing neighbourhoods B(p1) and B(p2).The conditions on the initial data sets to be glued by the afore-discussed method

seem, at first sight, relatively mild. Essentially, it is only required that the initial datasets are such that the domain of influence of the neighbourhoods B(p1) ⊂ Σ1 andB(p2) ⊂ Σ2 have no Killing vectors. The latter condition can be reformulated in anintrinsic way in terms of the so-called Killing Initial Data (KID) —see e.g. [79]. Moreprecisely, the condition for gluings is that there are no KIDs in B(p1) and B(p2) —inparticular, the latter implies that there are no Killing vectors in the neighbourhoods.The condition on the non-existence of KIDs has been shown to be generic in a veryspecific sense —see [80].

Every step in the procedure as described in, say, references [338, 182] is completelyconstructive. However, it would require to solve elliptic equations on manifolds withnontrivial topology. This complication is aggravated by the fact that, because ofthe requirement of the non-existence of Killing vectors in the initial data sets to beglued, it is not possible to use initial data sets with some symmetry —like axialone. Furthermore, the standard simplifying assumption of conformal flatness maycomplicate matters further. The reason is that the only conformally flat initial datawith trivial topology is Minkowski data. Thus, the use of conformally flat initial datato glue would require gluing data which already have non-trivial topology.

4.5. The Corvino-Schoen gluing method.

An alternative gluing method for the construction of solutions to the constraintequations, which may be of relevance for numerical applications, is the so-calledCorvino-Schoen (CS) gluing construction. This method has been developed in[191, 192, 179, 180] and it allows to smoothly glue any interior region of anasymptotically Euclidean initial data to the asymptotic region of a slice of a stationaryspacetime —e.g. Kerr initial data. For time symmetric asymptotically Euclideansolutions of the constraints, this method glues any interior region to an exterior regionof a slice of the Schwarzschild spacetime.

The procedure is best explained by considering the time symmetric situation inwhich one tries to glue a time symmetric solution of the constraints to an Schwarzschildexterior. As it is well known, for time symmetric initial data sets the constraintequations reduce to

(3)R[γij ] = 0, (7)

where [γij ] in the last expression highlights the fact that in what follows the Ricciscalar will be regarded as a mapping between the spaces of 3-metrics on Σ and scalarson Σ. Under certain circumstances this mapping happens to be an isomorphism. Thatis, given a metric γij and a scalar f on Σ such that

(3)R[γij ] = f, (8)


then if a further scalar field g happens to be close enough to f —in a functional sense—then there is a metric γ′

ij close to γij such that

(3)R[γ′

ij ] = g. (9)

The CS method makes use of precisely this property of the Ricci scalar mapping.Let γij be a metric defined on a asymptotically Euclidean manifold 3-manifold S

satisfying the time symmetric constraints. Consider a subset K ⊂ Σ which is obtainedby removing from Σ one of the asymptotic ends. The boundary of ∂K regarded as asubset of S will be required to lie in the asymptotic region and to have the topologyof the 2-sphere. Next, introduce some radial coordinate r so that the locus of ∂K isgiven by r = r0, with r0 a constant. It is noted that K is not necessarily compact, asit could well be the case that there is another asymptotic end inside K. Denote by Athe asymptotic end of time symmetric Schwarzschildean data. This asymptotic end ischaracterised by four parameters. Namely the mass, and the location of the centre ofmass, ci:

γSij =

(

1 +m

2|x− c|

)4

δij . (10)

As part of the gluing construction, one connects the regions K and A by means of anannular region C to obtain a new asymptotically Euclidean 3-manifold Σ′. A positivedefinite symmetric tensor γij is defined on the resulting 3-manifold by requiring it to beidentical to γij on K and to γS

ij on A. On C, it is chosen so that it interpolates smoothly

between γij and γSij . Now, if

(3)R[γij ] is close enough to the constant vanishing scalarfield —this can be achieved by moving ∂K, regarded as a subset of Σ, suitably furtherinto the asymptotic region— the result about the isomorphism of the Ricci Scalarmapping guarantees that there is a small —again in a functional sense— tensor δγijsuch that γij + δγij is a 3-metric and more importantly

(3)R[γij + δγij ] = 0, (11)

on Σ′. The theorems ensuring the possibility of performing the above constructionallow to show that the support of δγij is C, so that γij + δγij coincides with γij on Kand with γS

ij on A.The precise details of the gluing construction require the linearisation of the

Ricci scalar mapping. In order to obtain a differential equation to be solved, one hasto consider the composition of this linearisation and its adjoint to render a fourthorder elliptic problem. The non-linear problem is then solved by a Newton iterationmethod. It turns out that in a similar way to what happens in the IMP gluing, thepresence of KIDs in the gluing region C is an obstruction to the construction. Thesituation here is actually more delicate as the mere presence of asymptotic KIDs couldbe enough to cause problems. Note that as one moves the gluing region further andfurther into the asymptotic region, the problem could worsen. In order to get aroundthese complications, one uses the freedom in the choice of the Schwarzschild dataone is gluing to. Indeed, it has been shown in [191] that by a clever choice of theparameters m and ci one can get around the problems with the asymptotic KIDs.Unfortunately, it seems that precisely this procedure, is non-constructive. Thus, anumerical implementation of the CS gluing method would have to devise an alternativeway of determining the parameters of the Schwarzschild exterior.

Although the above discussion has been limited to time symmetric initial datasets which are to be glued to a Schwarzschild exterior, it has been shown that generic


metrics can be glued to a wide class of exteriors —including stationary ones. In thiscase, instead of just considering the Ricci scalar mapping, one has to consider thegeneral constraint mapping which sends a pair of symmetric tensors on S to a scalarand a 3-vector —see [180, 192].

The gluing constructions for initial data sets are remarkable geometric andanalytical results which have provided new insights into the structure of solutions tothe Einstein constraint equations. Important to analyse and discuss is the hypotheticaluse of these constructions in the simulation of black hole spacetimes. There arevarious issues at stake here: of course one has the conceptual ones; but there arealso the computational aspects. Is it possible to calculate everything one requires? Asmentioned above, a straight-forward implementation of the CS gluing technique maynot be possible as it contains non-constructive aspects. Nevertheless, the existence ofthe gluing constructions has, at least, an indirect connection with the discussion of therobustness of the simulations of black hole spacetimes. In theory, one could use theCS techniques on the Brill-Lindquist (BL) initial data, and construct from it a newinitial data set which will be exactly BL inside a compact set, but in addition it willbe exactly Schwarzschildean in the reference asymptotic end. As already mentionedin section 2.1, a set of conserved constants for the Schwarzschild spacetime —theNewman-Penrose constants— are zero. On the other hand, for BL data these arenon-zero —see [206]. As these constants are conserved along null infinity, what onefinds is that each spacetime will have a different radiation content close to timelikeinfinity. This could be interpreted saying that each spacetime approaches in a differentway the asymptotic state —i.e. the Schwarzschild spacetime.

5. Evolution formalisms

As discussed in section 3.2, different strategies in the gauge reduction process leadto distinct evolution formalisms. A natural classification of evolution formalismsin GR is according to the type of initial value problem they solve. Following thediscussion in section 3.1, they can be broadly classified as being Cauchy, hyperboloidalor characteristic. In this review we focus our attention on the Cauchy-type formalismsand will briefly touch on some aspects of hyperboloidal and characteristic schemes.

Following reference [107] we will distinguish between evolution formalisms,understood as the schemes devised to deal with the reduction process —involvinge.g. the choice of variables or the strategy of resolution of the constraints— andevolution systems, as the concrete PDE systems set in a particular gauge and actuallyexplicitly solved during the evolution. PDE evolution systems must include thereduced (or main evolution) system of section 3.2, but can also incorporate additionalPDEs —e.g. fixing the gauge system. In general terms, a line can be drawnbetween i) formalisms addressing specific mathematical issues —such as existence,uniqueness, completeness— and ii) schemes aiming at the numerical construction ofexplicit solutions. We focus on some evolution formalisms where interaction betweenanalysts and numerical relativists has been, or is expected to be, particularly fruitful.Excellent reviews on different aspects of this problem can be found in the literature[258, 466, 373, 260, 438, 439, 466, 427]. The following two sections are devoted toCauchy schemes: manifestly covariant schemes in section 5.1 and to the so-called3+ 1 formalisms in section 5.2. Non-Cauchy approaches are considered in section 5.3.


5.1. Cauchy manifestly covariant schemes: generalised harmonic (or general wavegauge) formulations and Z4 formalism

A classical approach to the study of the EFE consists in casting it as a wave equationfor which there exists a well developed theory. Using the deDonder expression forthe Ricci tensor [219, 229] and keeping in the left hand side only the principal part,equation (1) is written as

− 1

2gρσ∂ρ∂σgµν + ∂(µΓν) = Sµν , (12)

where Γµ ≡ gρσΓµρν and Sµν does not contain second derivatives of the metric.

Prescribing the gauge Γµ = 0, equation (12) becomes a wave equation. Since2xµ = −Γµ (with 2 denoting the scalar wave operator associated with the metricgµν) this choice corresponds to the use of harmonic coordinates. These coordinateswere early used in the first results on well-posedness [231, 228]. More recently, theelucidation of the role of Γµ [245] —see also [269]— led to the introduction of generalgauge source functions Hµ, leading to Generalised Harmonic (GH) or general wavegauge conditions

2xµ = −Γµ = −Hµ . (13)

In the language of section 3.2, this constitutes the gauge system in this covariantevolution formalism, and the reduced system follows from inserting (13) into equation(12). The local Cauchy problem is well-posed for this reduced system when appropriateinitial conditions consistent with the constraints (3)-(4) are employed —see e.g. [258]and references therein for details and rigour. Consistency between the reduced andgauge systems —namely propagation of the gauge condition Qµ ≡ Γµ −Hµ = 0 alongthe evolution— is controlled by the subsidiary system. This follows from Bianchiidentities, which imply the hyperbolic equation for Qµ

∇µ∇ρQρ +RρνQρ = 0 . (14)

Imposing the gauge condition Qµ = 0 and the constraints (3)-(4) on the initial sliceΣ —which imply ∂tQµ = 0, cf. [258]— the vanishing of Qµ along the evolution andthe overall consistency are guaranteed.

This subsidiary system issue exemplifies the different nature of the problemsan analyst and a numerical relativist must address. In the analytical setting, thereasoning above shows the consistency of the whole scheme. In particular one no longerneeds to consider the gauge system (13) while studying the properties of the reducedsystem. The situation gets more complicated in the numerical setting, since numericalerrors trigger constraint violation modes that invalidate the analytic argument —seealso [260, 261] for a discussion of the consequences of the intrinsic non-linearities inequation (14). The Z4 formalism proposed by Bona et al. [107, 108, 109, 111] isparticularly interesting in this context. Developed as a full spacetime generalisationof previous numerically motivated 3+1 schemes [110, 112, 106], its manifest covariantformulation permits a general analysis in the lines of the harmonic schemes. Thekey elements in this formalism are the introduction of a spacetime form Zµ as a newdynamical field, and the modification of the EFE to

Rµν +∇µZν +∇νZµ = 8π

(

Tµν −1

2Tgµν

)

. (15)

The solution space of this system extends that of GR, which is recovered for Zµ = 0—in fact, the EFE is recovered under the milder (non-generic) condition of Zµ being


a Killing vector. The evolution of the new field is driven by the Hamiltonian andmomentum GR constraints (3)-(4). Consequently, the Zµ field can be used as ameasurement of GR constraint violations —see section II.B. in [107] and [109] for adiscussion involving the Bianchi identity and the constraint structure of the theory. Asa further and key development for numerical applications, reference [299] introducesadditional damping terms in the Z4 system —and also in the GH scheme— to drivethe solution towards the GR solution submanifold, Zµ = 0.

Together with the early and ongoing application in analytic studies of GRequations —see e.g. the work of Lindblad & Rodnianski in section 2.1, and alsothe Kerr uniqueness proof by Klainerman & Ionescu discussed in section 6.1.1— thesecovariant formalisms have led to evolution systems with a successful application innumerical implementations. Harmonic coordinates have been employed in numericalstudies of the IBVP in GR [484, 485, 53] and related evolution issues [53, 54], or inthe study generic singularities [269] —the latter also discusses GH coordinates. In thecurrent numerical relativity scenario, it is specially relevant the evolution of binaryblack holes through the merger with gravitational wave extraction by Pretorius [424],for which he employed a generalised harmonic scheme [426] —see also [427]— usingthe damping terms based on the scheme in reference [299]. First-order symmetrichyperbolic forms of the generalised harmonic codes have been developed in [380] —cf.also [228, 7]— where it is studied the appropriate choice of gauge source functions Hµ

for preserving hyperbolicity of the whole evolution system. The latter includes nownot only the reduced system, but also the Hµ evolution equations.

5.2. Cauchy 3+1 formalisms

The Cauchy IVP involves the use of 3+1 spacetime spatial slicings. This appliesin particular to the case of the harmonic systems in the previous section. We nowconsider a family of evolution formalisms developed in the 3 + 1 framework of GR[214, 376, 231, 35, 509], which provides the natural setting for the canonical orHamiltonian approach to GR —notably inspired by GR quantisation considerations[89, 218, 35, 216, 37, 488]— and is very extended in its application to numericalrelativity —e.g. [519, 468, 518, 287]. In this 3 + 1 framework, in addition to thesplitting of spacetime in terms of a foliation by spatial hypersurfaces (Σt), the basicvariables are fields naturally living on those slices Σt. In this section, we use theexpression “3 + 1” in this restricted sense, customary in numerical relativity. Wedenote the lapse function, measuring the proper distance between slices Σt in thenormal nµ direction, by N and the shift vector, governing the choice of coordinatesystem on each slice, by βµ. The evolution vector is then written as tµ = Nnµ + βµ.In this 3 + 1 decomposition, the EFE splits in two sets. The first one consists of theconstraints in equations (3)-(4) whereas the second one —together with the kinematicrelationship between the 3-metric γij and the extrinsic curvature Kij— is frequentlyreferred to as the ADM evolution system (but is actually due to York [518]):

(∂t − Lβ) γij = −2NKij ,

(∂t − Lβ)Kij = −DiDjN

+N(

(3)Rij +KKij − 2KikKkj + 4π [(S − E)γij − 2Sij ]

)

. (16)

The ADM system is not uniquely defined, since terms homogeneous in the constraints(3)-(4) can be added. Moreover, it becomes an actual PDE evolution system once


prescriptions for the lapseN and shift βi are actually supplied. The resulting evolutionsystems have suffered from the presence of numerical instabilities and the closelyrelated violation of the constraints. A variety of 3+1 formalisms have been developedalong the years with the aim of coping with these problems —see [466] for a (non-updated) review. Depending on the strategy adopted to handle the constraints,evolution formalisms can be broadly classified as: i) free evolution formalisms ifconstraints (3)-(4) are not enforced during the evolution, and ii) constrained evolutionformalisms, that can be partially- or fully-constrained, according to the prescriptionof some or all the constraints, respectively.

The underlying intrinsic hyperbolicity [260] of the EFE is an ingredient exploitedin the task of obtaining a specific PDE evolution system. However, this does notmean that all the resulting evolution systems are necessarily hyperbolic. A clearexample is provided by constrained schemes that, due to the elliptic nature of theconstraints, lead to mixed elliptic-hyperbolic systems —see however [274]. Among themany possible criteria, in the present general discussion we will classify 3+1 evolutionsystems according to the presence or absence of elliptic equations in the system.

5.2.1. Evolution systems not containing elliptic equations. A significant effort hasbeen devoted to casting the EFE—under the appropriate gauge conditions— as a PDEhyperbolic system. The reason behind is the analytic control these systems provide onexistence, uniqueness and stability issues [438, 439] —see LeFloch’s IHP talk for someexamples illustrating the richness of this approach in the interplay between non-linearPDEs and geometry. This is the case of strongly hyperbolic systems —with their well-posedness results on the IVP— and of symmetric-hyperbolic systems, where energyestimates enhance the analytic control particularly in the IBVP permitting the use ofmaximally dissipative boundary conditions. In the generalised harmonic formalismsdiscussed in the previous section 5.1 hyperbolicity is manifest. With regards to systemsderived from the ADM formulation [466], hyperbolicity analysis is more complex dueto the loss of covariance. This also applies to certain 3+1 formalisms, that can bederived from covariant formalisms by performing a 3+1 splitting of the fields —e.g.3+1 versions of the Z4 scheme in [107]. A first class of systems in the 3+1 contextconsists of first order —in time and space— evolution systems. Among these, the KSTsystem [358] provides a very general class from which, for example, the Fritelli-Reula[263, 264] or the Einstein-Christoffel [8] systems can be recovered. The Bona-Massosystems [110, 112] include some additional variables —expressible in terms of theconnection coefficients of the three metric, as the BSSN system below— and can berelated by variable reduction to KST systems [106]. A second class of 3+1 systemshas been studied in which the second-order-in-space aspect of the ADM system isexplicitly maintained. In this sense, reference [297] presents a very general second-order evolution system that includes the successful BSSN [464, 75] or the NOR [403]systems. In this context of second-order systems, we also highlight the work in [106],as a predecessor of the Z4 formalism discussed in section 5.1. We refer the reader toreference [466] for the presentations of a third subclass, the asymptotically constrainedsystems aiming at the construction of an evolution system where the constraint surfaceis an attractor —cf. in particular the discussion of the λ-system and adjusted systemapproaches. Many evolution systems can be obtained from the previous systems —or closely related ones— by adopting distinct prescriptions for the lapse and shift.An exhaustive account of all the resulting 3+1 systems is beyond the scope of thisarticle. Assessment of analytic well-posedness —strongly dependent of gauge choices—


and of the relations between first and second-order formulations, can be found in[106, 403, 451, 297].

5.2.2. Mixed elliptic-hyperbolic systems. Constrained formalisms are not the onlyavenue to mixed elliptic-hyperbolic systems. There are geometric or physically wellmotivated choices for N and βi which are elliptic in nature and thus lead to a mixedsystem. We present some of the elliptic-hyperbolic PDE systems discussed in theliterature:

a) Free systems. They involve elliptic gauge choices for the slicing and/or theshift. An example of this is provided by the first of the axisymmetric 2+1+1 systemsconsidered in reference [441]. A paradigmatic example is presented by Anderssonand Moncrief in [19], where the authors establish the well-posedness of the IVP fora system containing elliptic equations for the lapse and the shift. These equationsfollow, respectively, from a constant mean curvature condition and a spatial harmonicgauge. To the best of our knowledge, no numerical counterpart has been implemented.Other recent analytic studies can be found in references [163, 165, 164].

b) Partially constrained systems. The enforcement the Hamiltonian constraintas an equation for a conformal factor has been widely used in axisymmetric codessince the eighties [64, 476, 227]. More recently, maximal slicing and the Hamiltonianconstraint have been employed in [271], whereas the third constrained 2+1+1 systemin [441] implements maximal slicing and imposes the momentum constraint —all this,still in the axisymmetric setting.

c) Fully constrained systems. Early systems of this kind were implemented in 2Dcodes [226], both for non-rotating [463] and rotating spacetimes [3]. More recently,axisymetric 2+1+1 codes [159, 158] have been used in the analysis of critical collapse.In this line, reference [441] provides again an example of constrained system of thistype, where ill-posedness is concluded after a maximum principle analysis of theinvolved scalar elliptic equations —see discussion in section 4.3.1. Regarding thefull 3D case, fully-constrained schemes have been discussed in [113, 9] and Moncrief’sIHP talk [1]. The fully-constrained scheme proposed by the Meudon group [113]makes use of a 3+1 conformal decomposition of spacetime based on slices reachingspatial infinity i0. The elliptic part of the PDE system includes equations for thelapse, the conformal factor and the shift, following from maximal slicing K = 0,the Hamiltonian constraint and a combination of the momentum constraint and aDirac-like gauge, respectively. Only two hyperbolic scalar modes are evolved, andthe rest of the components are reconstructed by using the Dirac gauge —cf. Novak’sIHP talk [1]— and a unimodular condition on the conformal metric. The schemehas been numerically tested for gravitational wave spacetimes. Regarding the schemeproposed in Moncrief’s IHP talk [1], it is ultimately devised to bypass the limitationsin the extraction of gravitational radiation at finite distances by using instead thenatural boundary of the problem —i.e. future null infinity. Even though importantfeatures are shared with [113] —use of a conformal 3+1 decomposition, resolution ofHamiltonian and momentum constraints along the evolution, elliptic gauges for theshift— it also represents a sound shift. Namely, it is not a Cauchy formalism butrather of hyperboloidal type: hyperboloidal 3-slices reaching up to I + are chosen bymeans of a constant mean curvature K condition, implemented as an elliptic equationfor the lapse. Finally, it is worth noting an interesting proposal within this schemefor the determination of the conformal class representative γµν by solving an ellipticYamabe equation.


5.2.3. Current status of 3+1 systems. Most numerical groups make use of codesbased on 3 + 1 formalisms derived from the ADM scheme. Codes based on harmonicformulations have produced excellent results [424, 483], but their use is still limitedto a smaller part of the community. Among the 3+1 formalisms, the mainstreamis represented by free systems not involving the resolution of elliptic equations, andthey have provided the longest lived evolutions —in particular BSSN. In comparison,mixed elliptic-hyperbolic systems have offered limited applicability or are still at apreliminary stage —cf. [113] and Moncrief et al. scheme [1]. On the one hand,constrained systems are in principle expected to provide a better control on instabilitiesrelated to violation of the constraints. On the other hand, well-posedness in mixedsystems is difficult to establish and, in particular, characteristic fields in the hyperbolicpart are difficult to determine since part of the dynamics is encoded in fields solvedthrough the (non-causal) elliptic part. In addition, non-uniqueness issues in the XCTSsystem discussed in section 4.3.1 could have strong implications on the well-posednessof some fully-constrained evolutions schemes enforcing a maximal condition for theslicing. The numerical consequences of this non-uniqueness in the elliptic sector arestill unclear and behaviour near the critical value of the parabolic branching discussedin section 4.3.1 can be very dependent on the details of the numerical implementation.As a significant example, authors in reference [444] have renounced to the use of mixedsystems in [441], opting for a 2+1+1 version of the Z4 formalism. The lesson herecontained should certainly not be underestimated. At the same time, it should notpreclude the development of such an alternative line of research that can only resultin overall benefits.

5.3. Other approaches to evolution

We proceed to discuss some alternative approaches to the Cauchy formulation. Wefocus our attention on some aspects of characteristic evolutions and later on aparticular version of the hyperboloidal problem.

5.3.1. Evolution formalisms based on a Characteristic approach to GR. Thecharacteristic initial value problem presented in section 3.1 provides an avenue to thenumerical construction of spacetimes —alternative to the Cauchy approach— thatis particularly well suited for the study of gravitational radiation. Since the pioneerwork by Bondi and Penrose [115, 116, 412] on the characterisation of gravitationalradiation in terms of null hypersurfaces, developments based on the use of Bondi-Sachs metric [116, 447] have resulted in successful numerical evolution —e.g. [96]—and radiation extraction —e.g. [55]— in a number of non-trivial cases, includingsingle black hole spacetimes. An excellent account can be found in the articlereview by Winicour [515]. The main drawback of this approach is the need tocope with caustics formation during the evolution. A second obstacle lies in thedifficulty of encoding physics. Alternatively, mixed Cauchy-characteristic formulations[94] (Cauchy-characteristic matching) offer a compromise between stable evolutionsand accurate treatment of gravitational radiation. Full application to astrophysicalscenarios seems currently limited as compared to the standard Cauchy approach. Onthe other hand, regarding applications in more geometric settings —in particular, thestudy of global solutions— characteristic or mixed Cauchy-characteristic approachesprovide a stimulating framework for the fruitful collaboration between geometers andnumerical relativists. In this sense, we highlight the ongoing activity in this line of


research —see e.g. [95, 434, 285] and references therein.

5.3.2. Conformal field equations. § The conformally regular approach in numericalrelativity is based on analytic studies by Friedrich [242, 244, 247] and started withnumerical studies by Hubner and Frauendiener [326, 327, 329, 328, 233, 232, 238,234, 235]. In this approach one solves numerically a hyperboloidal initial valueproblem for the conformally regular field equations —for reviews see [235, 236, 332].The use of hyperboloidal foliations is promising as hyperboloidal surfaces combineadvantages of Cauchy and characteristic surfaces. Instead of approaching spatialinfinity as Cauchy surfaces do, they reach null infinity which makes them suitablefor radiation extraction. Contrary to characteristic surfaces, these spacelike surfacesare as flexible as Cauchy surfaces and can be used in numerical calculations withinthe 3+1 approach. A difficulty with the conformally regular field equations is thatequations are significantly larger than usual formulations of Einstein equations. Dueto the large number of constraint equations, numerical errors require a stronger controlon constraint propagation properties of the system. As there is not enough numericalexperience with the equations, one cannot use established methods easily to deal withthe encountered instabilities. Another difficulty is that the equations include, amongothers, evolution equations for the conformal factor. The representation of null infinitydepends, in general, on the solution, so that the numerical boundary does not coincidewith the conformal boundary. As a consequence, a numerical boundary treatmentoutside the physical spacetime is required and the calculation of the unphysical partof the conformal extension wastes computational resources. While these problemsare not of a principal nature, they have made progress in the conformally regularapproach difficult. The conformally regular approach as described above has beenbased on the hyperboloidal initial value problem. As such, spatial infinity is not partof the computational domain and one cannot calculate the maximal development ofCauchy data. This has various drawbacks. In the hyperboloidal approach it is not clearhow the cut of the initial hyperboloidal surface at null infinity is related to timelikeor spatial infinity. One would also like to be able to relate asymptotic quantities suchas mass or momentum defined at null infinity to corresponding quantities at spatialinfinity. An alternative conformally regular system was proposed by Friedrich thatallows one to construct a regular finite initial value problem at spatial infinity [251].The study of this problem led to analytic results concerning the applicability of thePenrose proposal and smoothness properties of null infinity [255, 498, 497, 500, 501].A major advantage of the underlying system, called the reduced general conformalfield equations, is that the representation of the conformal factor is known a prioriin terms of initial data and that the system consists mainly of ordinary differentialequations except the Bianchi equation admitting a symmetric hyperbolic reduction.The finite initial value problem at spatial infinity has been studied numerically in[524]. The Cauchy problem could be solved for the entire Schwarzschild-Kruskalsolution including timelike, spatial and null infinity [523]. Also, certain radiativespacetimes could be studied near spatial and null infinity. This approach is currentlythe only approach that allows the numerical study of both spatial and null infinity ina single finite picture so that one has, in principle, numerical access to the maximaldevelopment of Cauchy data and can calculate global spacetimes including their entire

§ The content of his section is essentially due to A. Zenginoglu. We we are thankful for his enthusiasticinput.


asymptotics. A further development of this approach might give valuable input fromnumerics to geometry as most open problems in mathematical relativity concern globalquestions.

A valuable input from geometry to numerics is the idea of null infinity. Having nullinfinity in the computational domain would solve both the outer boundary problemand the radiation extraction problem in numerical relativity. Unfortunately, this ideacould not yet be implemented in astrophysically motivated numerical calculationsbased on the Einstein equations. The main difficulty is due to the appearance offormally singular terms arising from conformal compactification of the metric. Areasonable approach is to choose a gauge for the Einstein equations in which eachformally singular term attains a regular limit in a spacetime that admits a smoothconformal compactification at null infinity. The construction of such a gauge in thecontext of the characteristic approach has been known for a long time [486]. Thismethod has been quite successful within the characteristic approach. The underlyingcoordinates, however, do not allow the simulation of highly dynamical strong fieldsdue to formation of caustics in the light rays generating the coordinate hypersurfaces[515]. The implementation of this idea within the 3+1 approach has turned out tobe exceptionally difficult numerically as well as analytically. In view of the recentbreakthrough within numerical relativity, it seems clear that yet another formulationof the Einstein equations to deal with this problem cannot be regarded as a practicablesolution. What is needed is a novel numerical treatment of null infinity that does notalter the successful simulation of the sources in the interior.

An attempt to use a common reduction of the Einstein equations to study nullinfinity has been made in [402, 10]. A scri-fixing gauge has been suggested in thecontext of a hyperboloidal initial value problem for the ADM reduction of the Einsteinequations. In this gauge the spatial coordinate location of null infinity is independentof time. Such a gauge has first been constructed in [232] in the context of a frame-based conformally regular field equations. A similar approach is followed in Moncrief’sIHP talk [1]. A scri-fixing gauge is based on a mixed elliptic-hyperbolic system. Aconstant mean curvature condition fixes the hyperboloidal foliation by an elliptic gaugecondition for the lapse. In such attempts, an important question is how to fix theconformal factor. It has been shown that the representation of the conformal factorin terms of coordinates can be given a priori [524]. In this case, the well-posednessof the scri-fixing gauge has been proven. Further, each formally singular term arisingfrom conformal compactification attains a regular limit at I . Preliminary numericalexperiments suggest that the method has some promise to be used in astrophysicallymotivated calculations. In [525] scri-fixing gauges have been studied in Minkowski andSchwarzschild spacetimes. A discussion of the tail behaviour of Schwarzschild is usedto demonstrate the astrophysical relevance of the notion of null infinity for numericalapplications.

5.4. Some specific issues regarding evolution formalisms

5.4.1. Outer boundaries and the Initial Boundary Value Problem. The treatmentof infinity poses diverse geometrical, numerical and physical challenges. Morespecifically, among the different manners of coping numerically with this issue wecan mention compactifications of spatial infinity [424], use of hyperboloidal slicestogether with compactification of null infinity —[326, 327, 329, 328, 233, 232, 238,234, 235], Moncrief’s IHP talk [1] and previous section 5.3.2— or implementation


of a characteristic scheme. Alternatively, one can remove infinity from the problemand consider the evolution of the EFE in a region bounded by an outer timelikeboundary. This leads to the discussion of the well-posedness of the IBVP presented insection 3. A part of the community no longer regards at this issue as a fundamentalproblem in practise [2], since current simulations can push outer boundaries sufficientlyfar away. However, the relevance of the topic is in particular reflected in thequantity of works in the subject —also manifest in the number IHP talks [1] dealingdirect or indirectly with this topic; cf. IHP talks by Buchman, Rinne, Tiglio andWinicour. The work by Friedrich and Nagy [257] provided the first formalism inwhich well-posedness has been fully shown. However, its formulation in terms oftetrads and the Weyl tensor makes it difficult to implement using standard numericaltechniques/infrastructures. More recently Kreiss & Winicour [368] have presented asystem based on the harmonic formalism for which the IBVP is well-posed in thegeneralised sense, and the result has been extended to well-posedness in the classicalsense in [367]. Generally speaking, symmetric hyperbolic systems play a critical role inthe analysis of the well-posedness in the IBVP problem, through the use of maximallydissipative boundary conditions for getting rid of possible constraint violations —cf.[299] for an alternative approach making use of point-like damping terms. In thiscontext, pseudo-differential theory of strongly well-posed systems in the generalisedsense and techniques from semi-group theory have been used to analyse IBVP well-posedness —cf. [478, 140, 139, 452, 425, 109, 442, 380, 443] for some other recentreferences on the subject. Absorbing boundary conditions have been intensely studiedin the last years [409, 136, 137, 446] —cf. the review [450].

5.4.2. Gauge conditions. The discussion about numerically and physicallyappropriate choices of coordinates is a vast topic. Here, we limit ourselves to brieflyaccount for some recent developments. First, regarding sources functions Hµ in GHformalisms, choices must be done in such a way that the hyperbolicity of the GHformulation is not spoiled. References [424, 425, 426] make use of a wave evolutionequation for the determination of Hµ, but more research is needed on this specifictopic [427]. In this sense, reference [379] has provided new gauge drivers preservinghyperbolicity in GH systems, and including a wider class of conditions motivatedby successful 3+1 implementations. The further discussion of stability as well asthe study of intrinsically GH-motivated slicings constitute an open line of research.In a different setting, the appropriate choice of coordinates has been crucial in thesuccessful binary black holes implementations using the BSSN formalism [141, 59]. Inparticular, modifications of the 1+log slicings and Gamma-driver condition for theshift, have proved to be fundamental ingredients in the moving puncture approachto black hole evolution, where the punctures are advected in the integration domain—cf. section 6.4.2. An analysis of the well-posedness of the resulting evolution systemhas been carried out in [297], and in particular has given indications of a breakdownfor large shifts. This binary black hole case shows the relevance of using symmetryseeking coordinates [272]. Recent contributions in this sense are the almost-Killingcondition derived from a variational principle in [105, 111], and its compatibility withthe adoption of singularity avoiding coordinates in schemes not implementing excision[104].


5.5. Final discussion on evolution formalisms

Research in evolution formalisms for the Einstein equation provides a fruitful areafor the collaboration between mathematical and numerical relativists. However, theanalytic issues on which we have focused in this section should not shadow the fullcomplexity of the problem, in particular the fact that current numerical successes area direct consequence of decades of steady and systematic numerical experimentation.The most successful codes —e.g. generalised harmonic or BSSN— represent acompromise between well-posed but complicate schemes —like the Friedrich-Nagysystem— and formalisms easy to implement but potentially ill-behaved, like the ADMapproach. The situation could be deceiving for the analyst, but it is just a reflectionof the overall complexity here involved. Analytic well-posedness is not a guarantyof numerical stability —cf. discussion in [53]— but rather a necessary condition fornumerical convergence, once consistency of the scheme and numerical stability aresatisfied. In sum, analytic issues are only one aspect in the problem of choosing theappropriate numerical evolution scheme. Together with them, the numerical relativistmust cope with computational and physical issues [53]. In particular, a presentationof the formalisms employed in numerics cannot be complete without a discussion ofthe numerical techniques involved, a task that would require a full review in itself.

For all these reasons, the discussion about evolution formalisms is perceived by agood part of the numerical community as an essentially technical discussion that, ina good measure, has lost its critical character after the breakthroughs in the binaryblack hole problem. After decades of struggle, analytic and computational issuesare considered to be under reasonable control and, following this line of thought,focus should consequently be shifted to the more physical aspects of the problem—cf. special New Frontiers in Numerical Relativity issue [2] and in particular thetranscription of the final discussion. As far as astrophysical aspects in numericalrelativity are concerned, this attitude seems to be an appropriate one —althoughsurprises should be expected when addressing other physical problems. However,regarding the application of numerical techniques to deepen our understanding of thegeometric structures of GR —what we have called geometric numerical relativity insection 1.2.2— it seems reasonable to consider that further research in this subject isnot only fully justified, but will probably be needed as new geometric and/or physicalgoals are set.

6. State of the art of black holes

If one wants to make use of numerical methods to calculate dynamical black holespacetimes then, in one way or the other, one has to resort to an initial valueformulation —see section 3. It is not obvious at all that an initial value formulationof the merge of two black holes is the most adequate way of discussing this problem‖—one is dealing with an idealised astrophysical problem whose initial conditions cannot be set in a laboratory. The whole approach hinges very heavily on the assumptionthat there are certain robust aspects of the output of the numerical simulations —essentially the wave forms— which are not dependent on the detailed way the initialdata are constructed. The only real dependence of the wave forms should come fromthe physical parameters of the problem —again, the masses and spin of the blackholes— but not from things like the initial separation —as the holes rather approach

‖ We thank S. Dain for discussion on this point.


from infinity undergoing an adiabatic process— or if the initial data is conformallyflat or not.

The issue of the robustness of certain features in the simulation of dynamicalblack hole spacetimes provides a rich arena for the interaction between Geometry andNumerics. This is a very challenging mathematical problem whose resolution probablywill take long. For the time being one will have to content oneself with the evidencecoming from the numerical simulations. Still, the assessment of this evidence is initself also a challenge, and may also provide further ground for interaction.

In defence of the use of an initial value point of view, one can say that it offersa framework to look at things. By framework it is being understood a repertoire ofmathematical techniques and theorems which for example allows to state the well-posedness of the differential equations governing the problem, so one knows thatthe problem has been formulated in a consistent way. From a strict mathematicalpoint of view, it is not known if there exist spacetimes with a black hole andgravitational radiation —this is essentially the problem of the nonlinear stability ofSchwarzschild/Kerr, cf. the discussion in section 2.2.

6.1. Mathematical black holes

6.1.1. Rigidity and uniqueness of black holes. A cornerstone of what it has beencalled the establishment’s point of view on black holes —see e.g. [415]— is the evidenceshowing that the Kerr solution describes all stationary, vacuum solutions to the EFEdescribing black holes. This uniqueness assertion is also called a no hair theorem.The relevance of this result stems from the fact that it characterises all possibleasymptotic states of the general evolution of isolated systems in vacuum. Thus, forthe community of numerical relativists it justifies the analysis of the late stages ofblack hole spacetimes by means of perturbation theory on a Kerr background —seee.g. [363].

The problem of the uniqueness of black holes has been resolved in two differentmanners. Starting from a stationary, asymptotically flat black hole spacetime it followsthat the stationary Killing vector field must be tangent to the event horizon, E , of theblack hole. If the Killing vector is null at the horizon —i.e. it is a Killing horizon—then it is possible to conclude that the spacetime is actually static [479]. For staticblack hole spacetimes, the uniqueness results state that the spacetime has to be aSchwarzschild spacetime —see e.g. [175, 342, 138]. If the stationary Killing vector isspacelike on the horizon then from Hawking’s area law one can conclude the existenceof another vector, ηµ, tangent to the generators of the horizon which is a Killing vectoron the horizon [310, 339, 181]. This extra Killing vector field of the horizon can beused to define the notion of surface gravity which is of great relevance in the discussionof the thermodynamics of black holes. In particular if the surface gravity is non-zero,then the horizon is said to be non-degenerate. In the case of a non-degenerate horizonit is customary to assume that the horizon is a smooth null hypersurface consisting oftwo components —a bifurcate horizon— E = E+ ∪ E− intersecting at a 2-surface withthe topology of the 2-sphere —this assumption is supported by [430].

Given that it is known that the Kerr solution with 0 ≤ a = J/M ≤ M isthe only stationary, axially symmetric, vacuum black hole with non-degenerate andconnected horizon —see [150, 445]—, it is suggestive to try to extend the vector ηµ inthe horizon to a Killing vector of the region exterior to the black hole —the domainof outer communication. The problem with this approach is that it requires posing


a boundary value problem on characteristic hypersurfaces (the horizon of the blackhole): one constructs a wave equation for the vector ηµ with data prescribed on E . Thisproblem is ill posed —i.e. it is not possible to establish local existence and uniquenessby standard methods— and only admits a solution if the spacetime is taken to beanalytic. Thus, under the hypothesis of analyticity it is possible to show that ηµ

extends to a Killing vector on the domain of outer communication —see [310, 176].This type of result is known in the literature as a rigidity theorem for it shows that acertain assumption on the symmetry of the spacetime —stationarity— together withsome further technical assumptions imply further symmetries —axial symmetry. Therigidity of black holes has been recently revisited with the particular aim of extendingit from the 3 + 1 dimensional to the n + 1 setting —cf. Isenberg’s IHP talk [1] andalso [323].

The above argument for the uniqueness of the Kerr black hole has been criticisedon the grounds that the analyticity assumption is overly restrictive: general spacetimesare at most smooth, and it is not obvious why a smooth spacetime will become analyticafter it has emitted gravitational radiation. In order to get around with this probleman alternative approach to uniqueness has been given in [334] —see also Klainerman’sIHP talk [1]. The idea behind this approach is that although the boundary valueproblem for wave-like equations on E is ill posed, if one knows that solutions doexists, then it is possible to show uniqueness [335]. In [390, 391] a certain tensor —the Mars-Simon tensor— whose vanishing implies that the vacuum spacetime underconsideration is locally isometric to the Kerr solution has been discussed. This tensorrequires, for its definition the existence of a stationary Killing vector. In [334] it hasbeen shown that the Mars-Simon tensor obeys a wave equation and that it vanisheson E under a further technical assumption —which does not require analyticity. Ingeneral, one does not know whether a solution to the boundary problem for thewave equation satisfied by the Mars-Simon tensor with vanishing data on E exists.However, if a solution exists, then it is possible to argue that the Mars-Simon tensorvanishes not only in a neighbourhood of the horizon but on the whole domain of outercommunication —this is equivalent to show uniqueness of the ill posed boundary valueproblem. It is important to stress that this is a global result and not only local tothe horizon, like the one based on the hypothesis of analyticity. Note further, thatthis proof bypasses completely the use of rigidity results, and renders the existence ofaxial symmetry as part of the main result.

6.2. Quasi-local black holes

In spite of the success and strength of the traditional formalisation of the black holenotion in the asymptotic flatness setting —cf. section 2.2— black hole characterisationin terms of elements not involving global aspects of the spacetime has attractedsignificant efforts. This area of research is particularly well suited for a fruitfulcollaboration between mathematical and numerical relativists.

6.2.1. Motivations for quasi-local Black Holes. Quasi-local characterisations of blackholes have been approached from different communities addressing distinct problems.Here, we highlight geometric motivations from the mathematical community, onthe one hand, and “practical” needs for numerical relativists, on the other hand.Regarding the former, the difficulties discussed in section 2.2 related to theunderstanding of the full implications of asymptotic flatness, and illustrated in the


absence of known examples of non-stationary “strict” black holes spacetimes, depictsa state of affairs which is non-satisfactory at a conceptual level. More dramatic arethe issues raised from a numerical perspective, where the required global asymptoticinformation is simply not accesible during the evolution. Quasi-local characterisationsof black holes are needed both for “technical” reasons —e.g. need of tracking the blackholes along the evolution— and for physical interpretative reasons —e.g. the need todisentangle information about “individual black holes” in an interacting scenario.

Other motivations have played a crucial role in the historical development ofquasi-local notions of black holes: i) characterisation of black holes in cosmologicalsettings where asymptotic flatness notions do not apply; ii) extension of black holethermodynamical results to situations where global notions are not under control;iii) studies of black holes in the context of quantum gravity, such as the microscopicevaluation of black hole entropy; and iv) conceptual compliance with the consensus’snotion of “black hole” in the astrophysical community which is completely foreign toglobal issues. In a general sense, it is fair to say that numerical and thermodynamicalissues have been the main drivers in the development of quasi-local black holes as aline of research on its own.

6.2.2. Seminal concepts and objects: trapped surfaces. As discussed in section 2.2,the idea of a black hole as a region of no escape can be formalised in terms of theset points not being able of sending a signal to infinity —this, as long as we have asensible notion of infinity. Local convergence of all the light rays emitted from givensets of points provides an alternative approach to encapture the notion of no-escape.This second approach does not resort to a notion of infinity and provides the rationalefor the quasi-local approaches to black holes to be discussed in this section.

The seminal idea behind this characterisation is the concept of trapped surface.This notion has played a fundamental role in the context of the singularity theorems[413, 312]. Given a surface S which is spacelike orientable and closed —i.e. compactand without boundary— one can unambiguously define at each point p ∈ S two nulldirections ℓµ and kµ, which span the normal plane at p. That is, vector fields ℓµ andkµ span the normal bundle T⊥S. The idea behind the notion of a trapped surface S isthat either all light rays emitted in the normal null directions do locally converge or, atleast, light rays which would be naturally expected to expand, do actually converge. Inthe first case, the trapped character is an intrinsic property of the surface S, whereasin the second case some extra structure is needed to make sense of the intuition ofnaturally expanding light rays. Let us define the expansions θ(ℓ) and θ(k), respectivelyassociated with the null congruences ℓµ and kµ, as

θ(ℓ) = qµν∇µℓν , θ(k) = qµν∇µkν , (17)

where qµν is the spacelike metric induced on S from the ambient spacetime metric.Penrose [413] presents an intrinsic notion of trapped surface irrespective of anyembedding. A closed surface S is a Trapped Surface (TS) if and only if

θ(ℓ) < 0 , θ(k) < 0 . (18)

More generally, in order to cover Cosmological settings, a TS is defined as satisfyingθ(ℓ)θ(k) > 0, but we only discuss here the cases related to black holes. The limitingcase in which one the expansions —say θ(ℓ)— vanishes, characterises S as a MarginallyTrapped Surface (MTS):

θ(ℓ) = 0 , θ(k) < 0 . (19)


In the context of isolated systems it is natural to consider surfaces S embedded in anasymptotically flat (Euclidean) spacelike hypersurface Σ. In this case, we can naturallydefine the outer null direction, say ℓµ, as the one “pointing” towards infinity. FollowingHawking [310] we say that a surface S is an Outer Trapped Surface (OTS) if and onlyif:

θ(ℓ) < 0 . (20)

The vanishing limiting case defines a Marginally Outer Trapped Surface (MOTS):

θ(ℓ) = 0 . (21)

Dropping the global condition this last condition characterises the marginal surfacesdefined by Hayward [315]. In order simplify the discussion, we will refer genericallyto (21) as the MOTS condition, and the context will make it clear if global conditionsare being taken into account or not —this is, in fact, the common practise in theliterature, e.g. in [18]. These are not the only attempts to classify and characterisethe idea that a surface is trapped. See in this sense [393, 460] for an alternativenomenclature, [307] for the introduction of average trapped surfaces —cf. in this sensealso [384]— and [461] for a geometric formulation of the hoop conjecture in terms ofthe notion of trapped circle.

Trapped surfaces open an avenue to control quasi-locally the notion of black hole.Given a spacelike hypersurface Σ, the trapped region TΣ is defined in [310] as the setof points p ∈ Σ belonging to some OTS, S ⊂ Σ. The Apparent Horizon (AH) isdefined as the outer boundary of the trapped region TΣ. It attempts to encapturequasi-locally the notion of black hole horizon at the hypersurface Σ. A crucial resultis the characterisation, under the appropriate regularity conditions, of the AH asthe outermost MOTS [310]. Even though these quasi-local notions were originallydeveloped in the mathematical studies of black holes, particularly in the context ofthe gravitational collapse and singularity theorems, they were very early employed innumerical implementations.

First numerical implementations using geometric quasi-local black hole ideas. Usingtheir characterisation as MOTS, AHs have played a key role in the relation betweenmathematical and numerical relativists. Firstly, for numerical relativists, condition(21) represents a tentative characterisation of “black holes” that can be efficientlycalculated in terms of data on a compact region of a 3-dimensional slice. Important inthis context, is the development of AH-finders, i.e. algorithms to locate AHs in spatialslices by solving equation (21) —see [490] for a review and [377] for a recent work; cf.also [397] for related work with a strong geometric input. Moreover, under appropriateenergy conditions and assuming the weak Cosmic Censorship conjecture, AHs aregeometrically defined surfaces that are guaranteed to lie inside the event horizon —seee.g. [310, 186]— and therefore are causally disconnected from the rest of the spacetime.Although the extrapolation of this last feature from the continuum description to thediscretised level is not straightforward —see e.g. [131]— this geometric idea has actedas a positive criterion in the development of black hole evolution codes —see section6.4.2 for a brief discussion of the different manners of exploiting this idea.

Together with the numerical control of the black hole during the evolution, AHswere early applied as inner boundary conditions in the construction of initial data forblack hole spacetimes —following an idea proposed by Unruh. In this case, condition(21) is used to complete the elliptic system defined by the constraints (3)-(4) when


an interior sphere has been removed —cf. section 6.4.2 for further comments on thisexcision approach. This idea were first numerically implemented by Thornburg [489],later becoming a standard technique. This problem has also received attention fromthe mathematical community [207, 396, 469] but unfortunately the interaction betweenthe communities seems to be scarce. More recently, the idea of AHs inner boundarieshas also been proposed in the evolution context by Eardley [221]. The difficulties inimplementing these ideas exemplify the needs of interaction between geometric andnumerical communities. See also [344, 343] for an alternative geometric solution tothe inner excision problem in evolutions using, instead of AHs, locally area-preservingevolutions of a given initial trapped surface.

The need of a spacetime point of view. In spite of their conceptual and practicalinterest, AHs do not provide a spacetime characterisation of black holes. The AHnotion depends on the spacelike slice Σ and this feature limits its applicability. Thelatter has been illustrated in examples given by Wald & Iyer [506] of Schwarzschildslicings where Cauchy hypersurfaces come arbitrarily close to the singularity but donot contain any OTS. In particular, if we understand the evolution of an AH, sayS0, as the “hypersurface” formed by piling up the AHs —denoted by St— foundin the 3-slices Σt used in a Cauchy evolution —that is, Σt ⊃ St— the resultingAH-worldtube

⋃

t St explicitly depends on the chosen 3+1 slicing. Furthermore thegeometric, dynamical and thermodynamical properties of this AH-worldtube are notunder control —in particular non-continuous jumps can occur. This contrasts withevent horizons, that are smooth null hypersurfaces under the appropriate conditions[310].

An intrinsic spacetime formulation can be obtained by defining the trapped regionwithout any reference to a particular slice Σ. In this context, the trapped region Tof a spacetime M is the set of points p ∈ M belonging to some TS, S ⊂ M. Theouter boundary of this region is sometimes also referred in the literature as the AH, butfollowing [313] we will denote it as the trapping boundary. This offers an intrinsic quasi-local notion for the horizon where no reference to asymptotic quantities is needed. Anatural question to address consists in clarifying its relation with the event horizon—if the latter can be defined. As in the AH case, assuming Cosmic Censorship, thetrapping boundary do not extend beyond the event horizon. Finding out if it actuallyextends up to it, or if it remains strictly in the interior, is an open issue which canbe of relevance in the understanding of Cosmic Censorship Conjecture and in whichmathematical and numerical collaboration can prove to be very useful —cf. ideasand results in [313, 221, 369, 454, 87]. However, in spite of its conceptual interestthis notion also entails “practical” difficulties. Namely, the trapping boundary is acomplicate object to locate, in particular in a numerical evolution. And secondly,regularity issues may arise on this horizon [313].

6.2.3. Evolution of marginally trapped surfaces: quasi-local black holes. The needof an operational quasi-local notion of black hole horizon in an evolution has led tothe formalisation of the idea of world-tubes of AHs or, more generally, world-tubes ofMOTS —understood here as marginal surfaces in the sense of [313]. This is done atthe price of loosing uniqueness.

First steps where taken by Haicek in [301, 302, 303] were he introduced perfecthorizons as non-expanding null hypersurfaces devised to model stationary black hole


horizons. These totally geodesic null hypersurfaces [301] were the first examples ofworld-tubes of AHs where not only the expansion θ(ℓ) vanishes but also the shearof the null congruence does. A more systematic and general approach was initiatedby Hayward in [313], where he introduced future outer trapping horizons (FOTH)to model quasi-local black holes in generic situations. A parallel line of researchin quasi-local black holes as AH world-tubes has been developed by Ashtekar andcollaborators. This has lead to isolated horizons (IH) [43, 47, 49, 44, 50, 45, 46, 48]and, together with Krishnan, to dynamical horizons (DH) [40, 41]. FOTHs, onthe one hand, and IHs together with DHs, on the other hand, offer complementaryapproaches to the quasi-local black hole problem. Whereas FOTHs have the virtueof providing a general single characterisation which encompasses both stationaryand dynamical situations, the asset of IHs and DHs lies in their adaptation to thespecific and rather different geometric structures associated with null (stationary) andspacelike (dynamical) AH-worldtubes. Both approaches underline different aspects ofthe problem and conceptual compatibility is guaranteed since equivalence have beenshown in generic conditions [51, 18, 123] —see also [365] and later in this section.Finally, Booth & Fairhurst [120, 123, 357] have introduced the notion of slowly evolvingdynamical horizons to address the physical issues happening in the transition fromthe stationary to the dynamical regimes. General review articles on these quasi-localapproaches can be found in [52, 117, 292, 370]. We briefly review the ideas containedin them.

Trapping horizons. A trapping horizon is (the closure of) a hypersurface H foliatedby MOTS [313] —this notion has applicability beyond the context of black holes, inCosmological contexts. Here the MOTS condition θ(ℓ) = 0 must be read in its localsense as a marginal surface. White and black hole situations are distinguished bythe sign of θ(k), whereas the (local) outer- or innermost character is captured by thesign of δkθ

(ℓ) —where δk denotes formally a variation in the direction of the vectorkµ. Trapping horizons are originally formulated in a double-null foliation of spacetime[313], where δ becomes a Lie derivative. In this sense, trapping horizons are:

i) Future if θ(k) < 0, or past if θ(k) > 0.

ii) Outer if δkθ(ℓ) < 0, or inner if δkθ

(ℓ) > 0.

Note that the extremal case δkθ(ℓ) = 0 is not treated —see section 6.3. Inner trapping

horizons are of interest in Cosmological scenarios. But in quasi-local black hole horizonsettings, inner light are expected to converge —that is, θ(k) < 0— whereas outerlight rays should converge just inside the black hole and diverge just outside —that is, δkθ

(ℓ) < 0. Therefore, the adequate characterisation is in terms of futureouter trapping horizons, hypersurfaces H foliated by MTSs St on which the stabilitycondition δkθ

(ℓ) < 0 holds.We comment on some fundamental features of FOTHs. Under the dominant

energy condition, the MTS slices St of a FOTH have spherical topology —the topologylaw. Under the null energy condition: i) the vector tangent to the FOTH and normalto the MOTS slices can be either spacelike or null —the point-wise signature law— thelatter case happening if both the shear σ(ℓ) and T (ℓ, ℓ) vanish, and ii) the area of theMTS slices remains constant if H is null and grows otherwise (area law). Property i)follows from δkθ

(ℓ) < 0 and formalises the idea of AH-worldtubes as being stationary(null) if nothing falls into the horizon and superluminal (spatial) otherwise, whereasproperty ii) is the quasi-local counterpart of Hawking’s area theorem [308, 309].


A main application of FOTHs has been the derivation of balance laws for physicalquantities on the horizon, and its subsequent application to black hole dynamics —see Hayward’s IHP talk [1] and below. Regarding the conceptual characterisation ofquasi-local black holes, Hayward has also shown that under a regularity assumption—the existence of foliation of H by limit sections [313]— the trapping boundary isactually a trapping horizon, in fact the outermost.

Isolated horizons and dynamical horizons. The important geometric differencesbetween null and spatial hypersurfaces suggest the use of different strategies foraddressing specific issues in the stationary and dynamical regime.

The framework of isolated horizons —see [52] for a very comprehensive account—provides a hierarchy of geometric structures built on a null hypersurface, and devisedin order to capture different stationarity levels of a black hole placed in an otherwisedynamical environment.

a) Non-expanding horizons (NEH) provide the minimal notion of stationarity.They are defined as null hypersurfacesH of S2×R topology where: i) the null generatorexpansion θ(ℓ) vanishes, and ii) the vector −T µ

ρℓρ is future directed and causal. Using

the (null) Raychaudhuri equation, a NEH H is a hypersurface foliated by MOTS andcharacterised by the vanishing of both σ(ℓ) = 0 and T (ℓ, ℓ). The geometry of a NEHis given in terms of (qµν , ∇), where qµν is the spatial metric induced on any compact

slice S and ∇ is a connection uniquely induced from the ambient spacetime connection∇. The NEH condition expresses the evolution invariance of the intrinsic geometry:Lℓqµν = 0, and essentially coincides with the notions of perfect horizons and nulltrapping horizons.

b) Weakly isolated horizons (WIH) are NEHs with some additional structureneeded for the analysis of equilibrium black hole dynamics: an equivalence class ofnull normals [ℓ] for which the surface gravities κℓ, defined from ℓρ∇ρℓ

µ = κℓℓµ, are

constant on H —the zeroth law. A Hamiltonian analysis of the symplectic space ofsolutions of the EFE containing a WIH leads to a (Gibbs-like) quasi-local first lawof black hole thermodynamics, providing in particular a quasi-local expression for themass of a black hole with a stationary horizon.

c) Isolated horizons (IH) are NEHs where the extrinsic geometry ∇ is alsoinvariant long the evolution: [Lℓ, ∇] = 0. They represent the strongest quasi-localnotion of stationarity and crucially allow the definition of mass and angular momentum—in the axisymmetric case— multipoles [48]. The richness of the Isolated Horizonframework has led applications in very diverse areas of black hole physics, such as themicroscopic evaluation of entropy in the context of Loop Quantum Gravity [38, 42, 39],or the study of properties of hairy black holes and solitonic solutions in theoriesinvolving the coupling with additional fields —e.g. Yang-Mills, dilaton, Higgs, Procaand Skyrme fields [52].

Regarding the dynamical case, Ashtekar & Krishnan introduced in [40, 41] thenotion of dynamical horizons as spatial hypersurfaces H that can be foliated by MTSs—surfaces St where (19) holds. Note the difference with the FOTH definition, wherethe local outermost condition δkθ

(ℓ) < 0 has been substituted by the condition onthe spatial character of H. FOTHS and DHs are equivalent in generic circumstances[51, 123]. A FOTH for which δℓθ

(ℓ) 6= 0 for at least one point on each MTS section,is a DH. For the converse, a genericity condition —namely δℓθ

(ℓ) 6= 0 everywhere onH— and the null energy condition are required. Examples of DHs not satisfying the


genericity condition, and thus failing to be FOTHs, are presented in [459] —see alsocomments in section 6.3 in the context of extremal DHs. In contrast with the doublenull formalism of FOTHs, DHs are formulated in a 3+1 framework. Therefore they arespecially suited for their use in numerical relativity. Although black hole mechanicsrelations can and have been derived in this setting, a strong emphasis is put in theapplication of the associated balance equations to the analysis and test of numericalsimulations in the strong field regime.

Quasi-local Black Hole thermodynamics. The extension of classical black holethermodynamical results [63] from stationary spacetimes to more generic situationshas been one of the main motivations for the development of quasi-local black holes—see Hayward’s IHP talk [1]. Different approaches and definitions of the physicalparameters have led to distinct existing versions of the generalised thermodynamicslaws, whose detailed review goes beyond the scope of this article. We commentin general terms: i) Zeroth law: constancy of the surface gravity, defined as thenon-affinity parameter associated with the generating null vector in the context ofisolated horizons, it is shown to characterise a WIH [52] —and therefore holds onan IH. A generalised zeroth law in terms of an inequality bounding the mean ofthe (trapping) gravity is proposed in [313], where the constancy of κ is characterisedby the saturation of the inequality —see also [407] for a general discussion on thesurface gravity in the quasi-local context. ii) The second law, non-decrease of thearea, follows from the spatial (or null character) of the MOTS-worldtubes, togetherwith the condition θ(k) < 0. Therefore, the second law is built into the definition ofquasi-local black holes. iii) The discussion of the first law, as a Gibbs-like expressionrelating the variation of the energy, area and angular momentum of the horizon, ismore problematic due to the ambiguities in defining quasi-local physical parameters—cf. section 7.2. In the spherically symmetric and axisymmetric IH cases, this hasbeen successfully addressed —see [50, 45]— and has lead to a first law in terms ofvariations in the space of physical states —Gibbs’ version. However, the attempt ofderiving a first law relating time variations of physical quantities (Clausius’ version)has led to different versions according to the distinct choices of evolution vector orquasi-local energy. For these reasons, the status of such a version of the first law in thedynamical regime is unclear. In this context, the ensemble of balance or conservationlaws obtained from the restriction of different components of the Einstein equationon the MOTS-worldtubes H acquire a particular relevance. We highlight the balanceequation relating the variation of the area and angular momentum to the flux of energy[313, 40, 41, 315, 314, 120, 121] —see [123] for a discussion of the comparison betweensome of them. A set of evolution equations for the area [313, 41, 52, 120, 293], angularmomentum [40, 40, 120, 121, 288, 316] or the charge [317] have also been derived.In particular, references [288, 292, 293] discuss a quasi-local version of the membraneparadigm [210, 211, 429, 493] based on a hydrodynamical analogy between horizonsand viscous fluids — whose full analysis will plausibly involve the use of concepts andtechniques from PDE theory of hyperbolic systems.

We must emphasise that some of the ambiguities present in the fully dynamicalregime can be controlled when considering slight deviations from equilibrium, inthe setting of Booth & Fairhurst’s slowly evolving dynamical horizons. Finally,regarding the impossibility of reaching extremal black holes (third law), a discussionof extremality in this quasi-local context has been recently developed in [122, 118].


6.2.4. Numerical implementations of dynamical trapping horizons. The implementa-tion of ideas from dynamical trapping horizons offers an example of a fruitful relationbetween Geometry and Numerics. The application of some prescriptions derived fromthe framework of dynamical trapping horizon have been used to extract physical infor-mation about numerically constructed black holes. Conversely, numerical experimentshave provided key insights into the geometric structure of MOTS-worldtubes.

DHs and “a posteriori” analyses. The main application of DHs in numerics isthe physical and geometric analysis of MOTS-worltubes located by an AH-finderalong a numerical evolution. Krishnan’s IHP presentation [1] provides an excellentaccount of this kind of analysis. Regarding the extraction of physics, DHs havebeen used to extract individual black hole masses and spins in black hole spacetimes[57, 455, 146, 371]. More specifically, deformation of black holes can be analysed bycomputing mass and angular momentum DH multipoles [455]. DHs can be employedto characterise the rate of approach to Kerr [455] or to study spin-orbit effects inbinary black hole simulations [146]. Heuristic approaches can be formulated in thissetting for determining a quasi-local linear momentum of a black hole, very relevantin the astrophysical study of recoil velocities after mergers [371]. From a geometricpoint of view, the results in [455] support the picture that AHs jumps occurring inblack hole evolutions correspond to a situation in which the involved DHs are actuallyconnected by a world-tube of MOTS which violates some of the FOTH conditions.The resulting single MOTS-worldtube presents timelike or signature-mixed sectionswhere topology change is possible. This picture is also supported by analytic resultsin [119] on the Tolman-Bondi collapse —see also [406]. A prediction of this picture isthat in binary black hole evolutions modelled by two MOTS-worldtubes H1 and H2,right after the formation of the outer common horizon So, the latter should bifurcateinto two MOTS-worldtubes: an exterior spacelike one Houter of growing area, and aninterior one Hinner of mixed signature evolving into a timelike MOTS-worltube. Thisis actually seen in numerical experiments [455]. Under evolution, this Hinner shouldannihilate with the ghost horizons H1 and H2, but the details of this process areunknown. In this context, the assessment of the recent results in [483] showing theoverlap of the inner horizons H1 and H2 is of special relevance.

DHs as “a priori” ingredient. An alternative application of quasi-local black holesto numerics is their use as a constitutive element in the PDE evolution system.This is exemplified by the use of NEHs to prescribe inner boundary conditionsin the construction of initial data describing spacetimes containing black holes ininstantaneous equilibrium —see e.g. [188, 28, 151, 348, 208, 292, 345]. A schemefor using MOTS —and not MTS— as inner boundary conditions in the context ofconstrained evolution formalisms has been presented in [347] and at Gourgoulhon& Jaramillo IHP talk [1]. This essentially recasts Eardley programme [221] in thedynamical trapping horizon framework. Its feasibility must still be assessed.

6.2.5. Geometric Analysis. A fundamental shift in the research on quasi-localhorizons has occurred with the application of tools of geometric analysis to theunderstanding of dynamical horizons properties. Besides the continuing effortsto characterise DH physical parameters [366] and derive appropriate conservationequations for them, some effort has been put into the use of maximum principles and


related notions to the study of the properties of elliptic equations defined on horizonsections.

The methods of geometric analysis has been put into work to show that, underappropriate conditions, FOTHs can be fully partitioned into NEH and DH sections —see [18, 123]. This result rules out the possibility of finding MOTS sections in FOTHsthat are partially null and partially spacelike —a possibility which was not discarded inearly works. This means that the transition from equilibrium to the dynamical regimehappens all at once. This shows the complete equivalence, in generic circumstances,between the two main approaches to quasi-local black holes —at least in what regardsequilibrium and the dynamical stages of the quasi-local horizons.

Two results deserve special mention. Firstly, the foliation uniqueness theorem fordynamical horizons by Ashtekar & Galloway [51]: given a dynamical horizon H, itsfoliation by MTSs is unique. Secondly, a local existence theorem by Andersson, Marsand Simon [18] stating that, given a 3+1 slicing (Σt) and an initial MTS So ⊂ Σo

satisfying a stability condition closely related to the outer condition of FOTHs, thereexists a unique worldtube foliated by MTSs St, such that St ⊂ Σt —at least as longas the stability condition is satisfied.

Some more recent developments in the geometric study of dynamical trappinghorizons include the refinement of previous work [18] on the stability and existenceanalysis of MOTS-worldtubes by Andersson, Mars and Simon in [17]; the derivationof estimates for the curvature in MOTS with application on DHs [20]; studies byBartnik & Isenberg [66] of necessary and sufficient conditions for the existence ofDHs in spherically symmetric spacetimes; the analysis of the asymptotics of MOTS-worldtubes in spherically symmetric spacetimes [513]; the formulation of a conjectureabout the peeling behaviour of DHs [461] in the context of the geometric discussionof the hoop conjecture; or the study of area estimates for outermost MOTS andthe characterisation of the boundary of the trapped region [21] —see also [398, 148].Finally, we mention an approach towards a general proof of Penrose inequality a laHuisken-Ilmanen [331] using a spacetime generalisation of the inverse mean curvatureflows in terms of uniformly expanding flows [128]. These developments are very closein spirit to some of the possible applications of dynamical trapping horizons [12].

6.2.6. General perspective. The recent numerical and geometric insights haveenriched the research in quasi-local horizons, which was previously focused mainly onblack hole thermodynamical aspects. From the DH existence and foliation uniquenesstheorems [51, 18] it follows that the question about the evolution of a given initial MTS,So, is not well-defined in generic situations. The initial So can evolve into differentDHs depending on the chosen foliation —i.e. on the chosen lapse. Although this canbe completely harmless in most “practical” numerical situations, it is conceptuallyimportant and was not sufficiently stressed in the numerical community. In numericalevolutions, black holes characterised by MOTSs world-tubes are treated as otherstandard compact objects. In particular, this assumes a well-defined unique evolutionindependent of the observer. The clarification of this point permits to turn theargument around, opening the possibility of setting a preferred choice of lapse functionin terms of a geometrically single out DH—first attempts in this direction are discussedin [293]. In a different line of work, numerical results by Schnetter and Krishnan —seealso [119]— have shed light on the global structure of tubes of MOTS, and on thechanges of signature along the evolution and the geometric criteria to control them.Other issues, such as the study of generic dynamical trapping horizon asymptotics


towards the event horizon, and its possible application to fundamental problems suchas Kerr stability [12] or Penrose inequality will probably require the combination ofanalytic, geometric and numerical skills.

6.3. Geometric inequalities involving black hole horizons

As in the case of other physical theories, geometric inequalities in GR are often thereflect of a fundamental underlying physical principle. A prime example of this is thepositivity of mass in GR [456, 457, 516]. The fundamental nature of this result ismanifest from its role in many crucial developments in GR. Furthermore, its failurewould put the physical consistency of the theory under question.

In this section we will briefly comment on some geometric inequalities in thecontext of black hole spacetimes which, in particular, constraint the gravitationalcollapse process. Our present understanding of the gravitational collapse is based ona chain of results and conjectures. First, the singularity theorems [413, 311, 312, 310]guarantee that the appearance of a trapped surface during the gravitational collapseleads to the development of a singular spacetime. Second, the singularity should behidden behind a black hole event horizon so as to avoid a lack of predictability —thisphysically motivated hypothesis excluding the formation of naked singularities wasproposed by Penrose [414] and is known as the weak Cosmic Censorship Conjecture.Third, the black hole spacetime should reach a stationary state —this assumptionis justified by the finiteness of the amount of radiation that an isolated system canemit. And fourth, assuming that all fields have fallen into the black hole after somefinite time, the black hole uniqueness theorems [321] then guarantee that the spacetimesettles down to a Kerr black hole. Thus, barring some technical assumptions, if enoughmatter concentrates in a sufficiently compact region, then the system evolves to a finalKerr provided weak Cosmic Censorship Conjecture holds and a final stationary stateis reached.

Using a chain of heuristic arguments based on the previous standard picture —the so-called establishment picture— Penrose proposed [415] a lower bound for thetotal (ADM) mass of a black hole spacetime in terms of the square root of the areaof the black hole. This Penrose inequality provides a lower bound for the black holecontribution to the total mass. It conjectures, in particular, a significant strengtheningof positive mass theorems in the black hole context. In its first version —that can bereferred to as global— this Penrose inequality conjectures that the area AE of anysection of the event horizon E satisfies M

ADM≥

√

AE/16π. Remarkably, it can beformulated as a problem for initial data on a Cauchy surface Σ, providing a versionlocal in time of the Penrose inequality. Given complete, asymptotically flat Cauchydata on Σ satisfying the dominant energy condition, the Penrose inequality —in theformulation of [325]— conjectures

Amin ≤ 16πM2ADM

, (22)

where Amin is the minimal area enclosing the apparent horizon —cf. [86] for an explicitconstruction illustrating the need of using the area of that minimal surface, rather thanthe outermost MOTS in Σ. There is a rigidity side to the conjecture. Namely, thatthe equality is only attained in the spherically symmetric (Schwarzschild) case. ThePenrose inequality was proposed in an attempt to provide evidence of the violationof weak Cosmic Censorship. Other inequalities involving minimal surfaces in [276]—employed to provide lower bounds for areas of event horizon sections, as well as


upper bounds for the efficiency in the emission of gravitational radiation— were alsoconstructed with the aim of providing evidence against Cosmic Censorship. However,growing evidence has accumulated in time on the generic validity of weak CosmicCensorship [505] and the effort has shifted to the construction of a proof of Penroseinequality. Although the latter clearly does not imply the correctness of the standardgravitational collapse picture, it would actually provide a strong support for it. Thespherically symmetric case has been proved in [388]. Huisken & Ilmanen [331] and Bray[127] have provided independent proofs for the so-called Riemannian case —whereKij = 0— but a general result has not yet been obtained. The Penrose inequality hasevolved into a problem in its own right, becoming one of the important challenges inGR and Differential Geometry. Its alternative name as the isoperimetric inequalityfor black holes [281, 277, 280] underlines its intrinsic geometric relevance and might“lead to the importation into black hole theory of further useful ideas and techniquesfrom global analysis” [277] —see e.g. [100, 99, 384, 382, 383, 300] for references relatedto isoperimetric inequalities in this context. Reviews discussing the original Penroseargument, historical developments, main results and open questions can be found in[68, 392]. For early attempts to probe numerically the Riemannian case see [351, 353],and for some more recent numerical studies see, for example, [209, 354, 352, 346].

There exist some generalisations of the Penrose inequality which involve linearmomentum [387], charges inside the apparent horizon [279, 320, 508] —cf. also[389] and related works [388, 336]— or a cosmological constant —cf. [185, 278] forasymptotically anti-de Sitter spacetimes. Here we comment further on a particularsharpened version involving the angular momentum J in axially symmetric spacetimes[209, 309]:

A ≤ 8π(

M2ADM

+√

M4ADM

− J2)

. (23)

A rigidity property —Dain’s rigidity conjecture— has been explicitly formulated in[209]: the equality holds if and only if the (exterior) initial data corresponds to(exterior) Kerr data. Defining ǫ

A:= A/

[

8π(

M2ADM

+√

M4ADM

− J2)]

, inequality (23)is expressed as ǫ

A≤ 1 and Dain’s conjecture, ǫ

A= 1, characterises Kerr data by the

evaluation of a single real number —see [209, 346] for possible numerical applications.In case of being true, this would strengthen current results on characterisations ofKerr [390, 391] and Schwarzschild [268] involving the evaluation of tensor quantities.The non-triviality of Dain’s proposal can be appreciated in inequality (37) of [392].The latter would provide the counterpart for a variational characterisation of Reissner-Nordstrom data, but has been found to be false [508].

A necessary condition for inequality (23) to make sense, is the positivity of thequantity under the square root symbol. This leads to the consideration of mass-angularmomentum inequalities in vacuum —in presence of matter such an inequality is easilyviolated. The data has to be subextremal. In a series of articles [201, 202, 203, 195, 204]Dain has proved the validity of |J | ≤ M2

ADMfor maximal, vacuum, asymptotically flat,

axisymmetric initial data. Equality is only reached for extremal Kerr. This theoremcan be seen as a first step in the study of the non-linear stability of Kerr. Theseresults have been discussed by Chrusciel in his IHP talk, and further developed in[178]. In spite of its characterisation in terms of initial data, these mass-angularmomentum inequalities are spacetime properties. There has also been an interest tostudy some quasi-local versions of these inequalities involving, in particular, the localcharacterisation of extremality. In a first attempt, Ansorg and Petroff [31, 419, 27] haveconsidered the substitution of the ADM mass by the Komar mass M

Komarevaluated


on the black hole horizon of stationary and axially symmetric spacetimes. Thishas led to the numerical construction of stationary configurations of a black holesurrounded by a matter torus where the quotient |J |/M2

Komarcould reach arbitrary

high values [31, 419] or even MKomar

could become negative [27]. Instead, and inorder to refer only to intrinsic quantities on the horizon, one could consider usingthe irreducible mass. This has led Petroff and Ansorg —in Ansorg’s IHP talk [1]—to conjecture an inequality for axisymmetric stationary spacetimes only involvingthe area: 8π|J | ≤ A. This conjecture has been further developed —including thecharged case— in [32], where use is made of the Christodoulou-Ruffini mass and itis analytically shown that extremality is characterised by the saturation of the area-angular momentum-charge inequality. In parallel, Booth & Fairhurst [122, 118] haveundertaken an analysis of the local characterisation of extremality of black holes basedon the dynamical trapping horizon framework of section 6.2. After also considering theuse of geometric inequalities involving the area A —or alternatively, the vanishing of alocally defined surface gravity— they have opted for a characterisation of extremalityin terms of the absence of trapped surfaces inside the apparent horizon. In terms of theouter/inner trapping horizon characterisation [313], this reads δkθ

(ℓ) = 0, and leads tothe introduction of a quasi-local parameter e satisfying e ≤ 1, such that extremalitycorresponds to e = 1. DHs satisfying the genericity conditions [51, 123] referred toin section 6.2, are found to be subextremal in this sense. In fact, dropping these DHgenericity conditions is equivalent to the extremal characterization by δkθ

(ℓ) = 0 —precisely the feature exploited in [459] to construct examples of spacetimes containingDHs but without trapped surfaces.

Before concluding this section on geometric inequalities involving black holes, wemust briefly comment on the so-called hoop conjecture. It proposes that black holehorizons happen whenever matter gets sufficiently compacted in all spatial directions.Stated in an intentionally vague manner, “black holes with horizons form when,and only when, a mass M gets compacted into a region whose circumference C inevery direction satisfies C . 4πM” [401, 492]. In particular, the hoop conjectureoffer an interesting example of interaction between geometry and numerics —see e.g.[156, 462, 154, 155, 522] for some numerical studies. Recently, a reformulation ofthis conjecture as a genuine and mathematically sound —see also [384]— geometricinequality has been presented in [461] —see also references therein for a review of theoriginal conjecture.

Black hole geometric inequalities offer a link between conceptual issues in GRsuch as Cosmic Censorship, positivity of mass or black hole dynamics —the latter,through the role of the area in the second law. Furthermore, its constraining role inthe gravitational collapse process is of potential interest in numerical constructions.

6.4. Binary black holes

The binary black hole problem has been the main challenge for the numerical relativistsin the last decades. The relevance of this problem lies, on the one hand, on itsconceptual richness —this two-body problem provides a probe into the strong fieldnon-linear regime of General Relativity— and, on the other hand, on its astrophysicalinterest —stellar and supermassive black holes constitute some of the main candidatesfor the detection of gravitational waves signals by interferometric antennae. Here, weemphasise the role of the binary black hole problem as a laboratory for the developmentand test of new numerical, analytical and geometrical ideas in GR. The developments


in the numerical evolutions of binary black hole spacetimes through the inspiral [135],merger and ringdown phases together with the extraction of gravitational radiation[424, 141, 59] have constituted a milestone. Since then, quite a large number of groupshave succeeded in developing binary black hole evolution codes. The last couple ofyears have witnessed a rush in scientific activity. Given the volume of the associatedliterature, we refer the reader to the review [427]. We limit here ourselves to discusssome points of interest in the interaction between Geometry and Numerics.

6.4.1. Helical Killing vectors and binary black hole initial data. Gravitationalradiation reaction drives relativistic systems into inspiral motion, circularising theorbits very efficiently at least for comparable mass systems [102]. Therefore, for twobodies sufficiently separated, it is natural to approximate the spiral orbits by closedcircular ones. This physical image is geometrically en-captured by the existenceof a one-parameter helical symmetry χλ : M → M [240] of the spacetime whose

infinitesimal generator will be denoted by hµ —that is,dχ

µλ

dλ= hµ. Near the binary

system, this helical Killing vector hµ is timelike. Sufficiently far away hµ becomesspacelike [240], but a number T > 0 exists such that the separation between a givenpoint p ∈ M and its image by the isometry flow χT (p), is timelike —see also [287] fordetails. The integral lines associated with hµ are helices.

Helical symmetry is exact in theories with no gravitational radiation, likeNewtonian gravity, (second order) post-Newtonian gravity and the Isenberg-Wilson-Mathews approximation to GR [341, 340, 514]. Helical symmetry can be exact infull GR for non-axisymmetric systems with standing gravitational waves [101, 215],although the spacetime cannot be asymptotically flat, in the sense that the ADMmass cannot be defined [282] —more precisely, it cannot have a smooth null infinityif there is no additional stationary Killing vector close to null infinity. Helical Killingvectors have been used in the numerical literature to model slow-motion adiabaticconfigurations of binary systems [512, 511, 510, 428]. This leads to the study of mixed-type PDEs. The light cylinder, characterised by the null character of the helical vectorhµ, separates an inner domain where the PDE is elliptic from an outer one where itbecomes hyperbolic —cf. also the so-called periodic standing-wave approximation[23, 130, 78, 372, 77] and other recent numerical works [239, 521]. Further theoreticaldevelopments making use of the helical Killing symmetry are given in [362] and [81].In particular, the work by Klein —cf. his IHP talk [1]— aims at setting a consistentframework for numerics by taking full advantage of the presence of a Killing vector.He studies the EFE in the presence of a helical Killing vector for a vacuum spacetimewith two disconnected Killing horizons of spherical topology. The use of a projectionformalism on the space of orbits of the Killing vector, permits then to cast the problemin terms of the 3-dimensional gravity equations with a SL(2,R)/SO(1, 1) sigma modelas material source.

We must briefly comment on current constructions of binary black hole initialdata. In the context of the XCTS construction —see section 4.3— the choice of theevolution vector as an (approximate) helical Killing vector, i.e. tµ = hµ, has beenused in the literature for motivating the Ansatz ˙γµν = 0 = K for the free XCTSdata. The helical Killing vector idea was applied for the first time to the constructionof binary black hole initial data in [290, 295] and has subsequently been used in[188, 28, 29, 151]. All these data make use of an excision technique to deal withthe singularity —see section 6.4.2. Problems arise when combining XCTS and helical


symmetry with a puncture approach to initial data, as shown in [304]. For this reason,punctured initial data make use of the extrinsic curvature approach —see section 4.2—which turns out to be difficult to reconcile with a helical symmetry idea. However,due probably to their relatively large initial separations, linear superpositions of twoBowen-York solutions —firstly used in [73, 58, 30]— have led to good results in recentpunctured binary black evolutions in [59, 62, 503] and [141, 142, 143, 144].

6.4.2. On the issue of moving punctures. A key issue in numerical black holesimulations is the manner in which black holes are modelled in the calculations. Thetwo main techniques employed are: excision, where a spatial neighbourhood of thesingularity is removed from the numerical grid on the initial hypersurface and thensubsequently, also on each spacelike hypersurface constructed during the evolution[489, 458, 26, 424, 475, 453, 483] —see also [385, 386] for particular analyses of globalexistence in this setting; and punctures, where one begins with punctured initial datain the sense discussed in section 4.1 and then evolves the data including the singularpoint —see for example [134, 141, 319, 473, 286, 146, 61, 491, 494]. A third alternativereplaces the singular black hole interior at each spacelike slice by a regular one, leadingto the idea of stuffed black holes [33, 34] that has been recently brought back —see[131, 225]. All these approaches —excision, punctures and stuffing— rely on theintuitive idea that no information escapes from the black hole interior, and thus,assume from the onset some sort of Cosmic Censorship. In particular, a detailedanalysis of this last —non-trivial— point has been undertaken in [131].

There has been a recent interest in understanding some geometric aspects of themoving punctures picture —see e.g. [305, 306, 491, 133]. However there seems to beno fully consensed view on why the puncture method works, and what happens withthe punctures during the evolution. In particular, it is of interest to know if they stillrepresent a compactified infinity. More importantly, it is crucial to know whether themethod relies on numerical errors near an under-resolved puncture, in which case itcould fail when probed at higher resolutions or if the evolutions are let to run muchlonger than up to now.

Assuming the establishment point of view on black holes, the asymptotic stateof the evolution will be described in some sense by a Kerr spacetime. So, after thesystem has evolved long enough it is natural to expect the existence of an approximatestationary Killing vector that could be use to drive the evolution to an eventuallystationary slice. A crucial element in the evolution of punctured data is the use ofthe so-called symmetry-seeking choices of lapse and shift —the 1+log lapse and theGamma freezing shift [5, 503, 297]. A symmetry-seeking gauge is one that tends toalign itself with this approximate Killing vector as the evolution proceeds. In thiscontext, for a stationary slice it is understood a hypersurface such that for the chosengauge-drivers, the evolution vector is parallel to the stationary Killing vector —so thatthe evolution of the spacetime effectively freezes. The discussion of stationary slicesin the Schwarzschild spacetime has been elaborated further in [224, 433, 305, 306, 74].Clearly, it is of interest to carry out an analogous analysis for the Kerr spacetime.

The symmetry-seeking nature of the gauges used in the evolution of dynamicalblack hole spacetimes is supported by numerical evidence. According to this evidence,the evolution using symmetry-seeking gauges seems to pile pointwise in almoststationary slices of the gauge. It would be of great interest, both practical andtheoretical to have a rigorous analysis of this symmetry seeking-behaviour of particulargauges. The resolution of the problem of the non-linear stability of the Kerr spacetime


is likely to clarify this issue¶.General results on the topology of hypersurfaces in asymptotically flat spacetimes

—see e.g. [265, 266]— preclude the change of topology during the evolution, unless theslices touch the singularity —see also the discussion in [401], section 31.6. Numericalevolutions —together with analytical considerations— of punctured Schwarzschildinitial data using symmetry-seeking gauges [305, 306] show indeed that the solutionapproaches pointwise a stationary slice that does not hit the physical singularity.Most importantly, the limiting slice does not reach the inner asymptotically flat spatialinfinity, but rather ends on a cylinder of finite areal radius whose throat has an infiniteproper distance as measured from the apparent horizon. This is interpreted as a changeof the geometry of the puncture along the evolution —from a 1/r behaviour of theconformal factor near the puncture to 1/

√r— that suggests the use of asymptotically

cylindrical data to represent black holes —see also the discussion in the IHP talks byHannam and O’Murchadha. These results have also been discussed in [133], whereit is pointed out that the puncture actually remains at the inner spacelike infinityalong the whole evolution and, in particular, the evolution using positive lapse isnever stationary in the limit of infinite resolution. Both apparently contradictoryinterpretations can be reconciled in terms of the differences between uniform andpointwise convergence: the limit to the final cylindrical stationary slice seems to bepointwise, but not uniform. In actual numerical simulations, the punctured regionis under-resolved and there is no practical need to distinguish between both kindsof convergence. In particular, this lack of resolution is not likely to cause problemsfor finite difference codes at any reasonable —practically realistic— resolution. Inthis interpretation, the moving puncture evolution can be seen as an effective naturalexcision method [133]. It is expected that the mechanisms operating in the case ofsimulations for Schwarzschildean initial data are in essence the same ones workingin numerical evolutions of dynamical black hole spacetimes —see comments in [133]about the presence of spin and momentum.

6.4.3. Discussion. The recent intense activity in numerical evolutions of binary blackholes has meant a very rapid advance in the understanding of the physics of binaryblack holes —with a particular emphasis on astrophysical applications. Althoughmore systematic studies are needed, a large number of results are already available,for example concerning the effect that different configurations of physical parameters—such as mass ratios, spins or orbit eccentricities— have on the final state of theresulting black hole, in particular on the recoil velocities of the final black hole. Againwe refer to [427] for a bibliographic account of these advances —cf. also the IHP talksby Lousto, Lindblom and Van Meter.

In what concerns astrophysical applications, one should mention the complemen-tarity of numerical approaches with others of analytic nature. On the one hand, giventhe computational cost of the numerical simulations it is natural to explore analyticor semi-analytical methods to cover the whole parameter space of the problem. On

¶ It is perhaps of interest to note that in [268] a particular prescription for a lapse and a shift —in terms of 3+1 quantities— has been calculated such that if the spacetime is the Schwarzschildsolution, then these lapse and shift coincide with those implied by the timelike Killing vector. Ifone is analysing, say, the evolution of a head-on collision of black holes where one knows that theasymptotic state will be the Schwarzschild solution, then it is natural to expect these lapse and shift—which can be evaluated in any spacetime— to be also symmetry seeking. This particular analysisdraws ideas from the invariant characterisation of spacetimes which has been extensively developedin the study of exact solutions to the Einstein field equations —see e.g. [477].


the other hand, analytical approximations often offer a (quick) hint on the physics ofthe problem. The need of such a synergy between numerical and analytical efforts wasadvocated by Damour and Blanchet in their respective IHP talks [1], where differentpost-Newtonian approaches were confronted with numerical results [213, 103]. Blackhole perturbation theory offers a further instance of potentially fruitful combinationof analytical and numerical approaches, for example in the analysis of the extrememass ratio case —cf. Nagar’s IHP talk [1].

Astrophysical aspects of the binary black hole problem offer plenty of room for thecollaboration between analysts and numerical relativists. In addition, the very successof the simulations already offer some indirect evidence of geometric issues like CosmicCensorship or Kerr stability. However, the numerical control of this non-linear problemopens an outstanding possibility of gaining insights into some of the key geometric andconceptual problems of GR, if specific efforts are geared in this direction. Geometricideas and results, such as Cosmic Censorship or black hole uniqueness theorems, havebuilt a solid conceptual framework for the study of gravitational collapse. In turn, itis to be expected that strong numeric tests to some of these geometric ideas could bedevised, offering a window to the assessment of non-linear features in GR. In currentsimulations, non-linear effects in the binary black hole have resulted unexpectedlymild. But, as discussed for example in Lehner’s IHP talk [1], “unchartered trails”remain in the road and much should be learnt from studies of generic configurations—cf. in this sense [381, 145] where multi-black hole encounters are studied.

7. Miscellaneous topics

There are some important topics in the relation between geometry and numericsthat have not been discussed in previous sections. Here we briefly present —non-exhaustively— some of those aspects and lines of research.

7.1. Critical collapse

The study of critical phenomena+ in gravitational collapse is one of the paradigms ofthe interaction between numerical and mathematical relativity: a type of phenomenawhich was discovered by means of numerical experiments [157] which then, in turn,have been understood by analytical methods —see [298, 88] for a review. In 1992Choptuik [157] found that it is possible to form arbitrarily small black holes inthe process of gravitational collapse of a spherical scalar field, by fine tuning anyparameter which affects the self-gravity of the initial configuration. Before the blackhole is formed the spacetime becomes selfsimilar and universal, forgetting the initialcondition except for a single length scale which determines the mass of the blackhole to be formed. This interesting behaviour has been found in many other mattersystems, and perturbative arguments suggest that it could be also present withoutspherical symmetry, being therefore intrinsic to the strong regime of GR dynamics.Finally, perfect fine tuning generates a naked singularity with infinite curvature, visibleto observers at infinity, with important consequences for Cosmic Censorship andQuantum Gravity. These critical phenomena in gravitational collapse were completelyunexpected before 1992 and are now considered the best example of how numericalrelativity can contribute new physics to GR —see Aichelburg’s and Harada’s IHP

+ The content of his section is essentially due to J.M. Martın-Garcıa. We are thankful for hisenthusiastic input.


talks for examples of ongoing research. A posteriori, they can be understood as thediscovery of codimension 1 exact solutions in the infinite-dimensional phase spaceof General Relativity —Minkowski, black holes and stars begin global attractors ofcodimension 0— opening a new way of interpreting GR from the point of view of thetheory of dynamical systems.

7.2. Quasi-local physical parameters

The determination of physical parameters in a finite region of spacetime is of clearpractical importance in numerical implementations. More generally, it is of conceptualrelevance in the theory as a whole and, more concretely, in some particular geometricconstructions such as certain geometric flows [277, 278, 68, 331, 128] or in the studyof geometric inequalities —cf. section 6.3. The determination of the amount ofenergy in a compact domain is a classical problem in GR and an exhaustive listof proposals regarding this goes beyond the scope of this article —see [480]. Therelevance of this subject as a boost for the interaction between geometry and numericsis illustrated, for example, in the use of quasi-local dynamical trapping horizons —section 6.2— for the extraction of physics in current numerical simulations. Wehighlight the attempts of defining a quasi-local angular momentum by means ofan integral on a closed surface S involving an axial vector φi defined on S —e.g.[132, 45, 480, 41, 52, 121, 288, 316, 481, 366], see also the related work in [482]—and in particular the recent efforts for deriving an unambiguous prescription for φi

[220, 190, 366, 481]. Note that the divergence-free character of φi in not guaranteed inall the latter schemes, a relevant point if a slice-independent definition —i.e. intrinsicto S— of the angular momentum is desired. In addition to determining the magnitudeof the spin, this vector φi can be used to estimate the direction of the angularmomentum vector —cf. [146] for preliminary results.

7.3. Spacetime singularities and extensions of GR

The study of the limits and extensions of GR offers stimulating perspectives ofresearch. One the one hand, numerical studies of singularities [88, 423, 322, 270] —e.g.Cauchy horizon instabilities and mass inflation near the Cauchy horizon, naked andnull singularities, approach to spatial singularities and analysis of BKL conjecture—explore the internal consistency of the theory and offer insight into new conceptualdevelopments in it. Numerical implementations of higher dimensional spacetimesas well as the coupling of the geometry to additional fundamental fields —e.g. inEinstein-Yang-Mills theory— probe extensions of the theory that can be relevant inthe quantum context.

7.4. Some astrophysical considerations

Ultimately motivated by the astrophysical study of relativistic compact objects, thisarticle has focused on isolated systems in GR. Moreover, we have centred the discussionon black holes, since they offer a particularly rich context for the dialogue betweengeometers and numerical relativists. This methodological choice could shadow theextraordinary developments in astrophysical numerical relativity regarding systemswith matter. Examples of the latter are the results relative to compact objects, suchas neutron or strange stars — notably the first computations of merger of binaryneutron stars by Shibata and Uryu [465]; see also Uryu’s and Saijo’s IHP talks [1]—


and other advances in relativistic hydrodynamics [22, 230, 394]. This constitutespresumably an extremely active area of research in next future. Moreover, the studyof problems such as the Einstein-Vlasov system presents an analytic as well as numericinterest [24, 25].

7.5. Geometrical spacetime discretisations.

In section 3, initial value problem formulations of GR have been presented as aparticularly powerful approach to the construction of generic spacetimes. A radicallydifferent approach consists in adopting a formulation of GR which avoids the useof coordinates in the spirit of the Regge calculus —see e.g. [432, 273]. For a currentprogramme of research in these lines, see [237, 440] —and also Frauendiener’s IHP talk[1]— which is based on Cartan’s method of frames and the use of discrete differentialforms.

7.6. Numerical techniques

As we have commented in section 5, a comprehensive presentation of the evolutionformalisms used in numerical relativity should be complemented with a discussionof the employed numerical techniques. Topics offering plenty of occasion for thecollaboration between analysts and numerical relativists are, among others, adaptativemesh refinement, high order methods (cf. Tiglio’s IHP talk) spectral methods [296]—including their application to evolutions [318]— and spectral elements (see alsoMaday’s IHP talk), finite elements (e.g. [324, 471, 472, 470, 364, 4]) finite volumes(e.g. [6]) or advances in high-resolution shock capturing methods.

7.7. Algebraic symbolic calculus

The increase in computational capabilities and the development of powerful newcomputer algebra systems has had a great impact in many areas of GR. Many problemswhich for a long time seemed to be out reach merely on the grounds of computationalcomplexities are becoming feasible. Algebraic symbolic methods were firstly usedin GR as a tool in the general research area of exact solutions. Besides its evidentutility in the derivation of exact solutions, computer algebra systems had a significantapplication in the metric equivalence problem —which consists in deciding whethertwo metrics given in arbitrary coordinates are locally isometric to each other. Theequivalence of two metrics is a classical problem in Differential Geometry, and asolution was given by Cartan [149]. Early discussions of the equivalence problemin GR can be found in [126] —see also [355, 356]. A particular implementation ofthese ideas is the system CLASSI —see [431]. Considerations involving the equivalenceproblem can be of utility in the comparison of numerical simulations.

In recent years, algebraic symbolic calculus has been increasingly used as asystematic tool for analysing the analytic properties of evolution systems —e.g. inthe construction of symmetric systems, analysing characteristics, in the constructionof propagation systems, construction of asymptotic expansions and in perturbativeanalyses. Among the systems explicitly constructed with this purpose in mind onehas xAct [395]. Of relevance is also the use of computer algebra systems for theautomatic generation of computer codes like in the case of Kranc [333].


8. Conclusions

The extraordinary results in the numerical evolution of black hole binaries have hadand will have a deep impact on the relation between mathematical and numericalrelativists. In particular, it forces a reflection on the long term scientific objectives ofthe research of both communities. On the one hand it questions the potential pay-offs of certain lines of investigation, and on the other hand it offers the possibility ofaddressing problems which for technological reasons were considered out of reach. Inparticular, theory will have the unique opportunity to confront observation by meansof the accurate simulation of relativistic astrophysical systems. Previsively, a bigproportion of the numerical community will be involved in this endeavour. Arguably,the implementation of the latter will not require many geometrical insights, althoughthe invariant extraction of physical content would certainly benefit from it —e.g.through the development of more efficient AH-finders, quasi-local characterisationsof physical quantities, or invariant algorithms for the extraction of gravitationalradiation. However, if one wants to study the other fundamental aspect of the theory,namely the structural ones —exploiting the state-of-the-art numerical possibilities—then evidently a geometrical perspective is fundamental. Even more, one could arguethat the study of certain crucial geometrical aspects of GR will require numericalapproaches to come to fruition.

In this review we have focused on the latter geometric aspects of numericalrelativity and also on genuinely mathematical results which, we believe, are ofrelevance for numerical applications. There is an intrinsic aesthetic appeal in thestudy of intrinsic geometric aspects of GR, but the benefit goes much further. Wehave tried to emphasise the role —and necessity— of analytical studies as guarantorsof the internal consistency of the theory. Geometric-oriented lines research can anddo offer general conceptual frameworks in which physically motivated problems canbe unambiguously formulated. The gravitational collapse paradigm is an exampleof this. On a second stage, it is also undeniable that Geometry also providespowerful tools and insights into calculating things. Following the premise that acrucial aspect of “understanding a theory consists in understanding its solutions”,geometrical numerical relativity is one of the most powerful tools available to studythe space of solutions of GR. Numerical GR can, potentially —i.e. assuming enoughcomputer resources— solve to any desired finite precision any well formulated problemconcerning a property of a specific solution in a concrete problem. When the probleminvolves an infinite number of solutions, then only the generic behaviour can beanalysed. This is however where numerical GR can be most useful, finding new andunexpected results ∗.

We finalise by naming a number of problems for which, we believe, the interactionbetween numerical and mathematical relativists is of particular relevance —this, ofcourse, notwithstanding the other issues that have been suggested in the main text.Cosmic Censorship remains in the eyes of most relativists the most important openproblem in classical GR. It is to be expected that numerical investigations of globalspacetimes will prove of utmost value in the strive towards a proof of the conjectures.In the closely related issue of the non-linear stability of the Kerr spacetime, numericalinvestigations have already provided information about the decays of various fields.This interaction is bound to become even closer in the future. A related topic,

∗ We acknowledge J. M. Martın-Garcıa for bringing out this point.


the discussion of the robustness aspects —with respect to changes in the initialdata— in the production of gravitational radiation by isolated systems will requirestrong input from numerics —even to obtain a rigorous formulation of the problem.The assessment of the physical relevance of certain aspects of the characterisationof isolated systems through the notion of asymptotic simplicity —like the peelingbehaviour— will also require close numerical examination. Something similar can besaid about the relevance of the construction of initial data sets by means of gluing.Close interaction between analytics and numerics will be required in the study ofdynamical trapping horizons and their asymptotic behaviour close to the event horizonand in the relation between the global and local characterisation of the black holenotion —extension of the trapping boundary to the event horizon. Finally, a study ofthe solution space of GR —in the spirit of the theory of dynamical systems— usingideas developed in the study of critical phenomena should also be based on a closeinteraction between numerics and geometry.

Acknowledgements

We would like to thank all participants of the “From Geometry to Numerics”workshop, part of the “General Relativity Trimester” at the Institut Henri Poincarefor the effort put in preparing their presentations. We also thank T. Damour and N.Deruelle for having organised the General Relativity Trimester. We are particularlyindebted to J.M. Martın-Garcıa and A. Zenginoglu for their contributions to sections7.1 and 5.3.2, respectively. All misunderstandings or omissions in these sectionsare, nevertheless, our responsibility. We also would like to thank V. Aldaya, M.Ansorg, C. Barcelo, N. Bishop, S. Bonazzola, C. Bona, I. Cordero-Carrion, S. Dain, P.Grandclement, B. Krishnan, S. Husa, J.M. Ibanez, L. Lehner, E. Malec, M. Mars, G.Mena-Marugan, V. Moncrief, J. Novak, T. Pawlowski, M. Sanchez, J. Seiler, J.M.M.Senovilla, L. Szabados, N. Vasset and D. Walsh for very valuable discussions andcomments. JLJ acknowledges the support of the Marie Curie contract MERG-CT-2006-043501 in the 6th European Community Framework Program and wish to thankthe School of Mathematical Sciences of the Queen Mary for its hospitality. JAVKis supported by an EPSRC Advanced Research Fellowship and wish to thank theInstituto de Astrofısica de Andalucıa (CSIC) for its hospitality. EG acknowledgessupport from the ANR grant 06-2-134423 Methodes mathematiques pour la relativitegenerale.

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