aa r X i v : . [ g r- q c ] N ov Geometric Asymptotics and Beyond
Helmut FriedrichMax-Planck-Institut f¨ur GravitationsphysikAm M¨uhlenberg 114476 Golm, GermanyOctober 20, 2018
Enormous progress has been made in the last hundred years in the investigation of Ein-stein’s field equations, their solutions, their solution manifold, and thus the content ofthe theory. Research on the equations has been largely guided by physical ideas but un-foreseen mathematical results repeatedly asked for revisions, a process that may not havecome to an end yet [55]. Mathematical analysis including approximations and formal ex-pansions led Einstein to an approximate notion of gravitational radiation and preparedthe way to the invariant concept available today. Explicit solutions, mainly dealing withhighly symmetric and idealized situations, revealed physical phenomena on global scales(horizons, black holes, singularities) which had not been anticipated and which sometimestook years to be absorbed into a coherent world view. The abstract analysis of the fieldequations, dealing with the existence of solutions and their parametrization in terms ofboundary data, evolved rather slowly, even after the breakthrough marked by the workof Choquet-Bruhat [21]. More recently, it became a tool to establish the existence andinvestigate the details of solutions with distinguished properties. Numerical techniquesallowed researchers to foray into domains of the solution manifold hardly accessible by an-alytic methods [20] and the calculation of quantitative data for comparisons with physicalobservations. In fact, certain data calculated for astrophysical processes seem to have ledto new observational results [74]. These activities went through layers of generalizationsand refinements, discoveries of unexpected features, deepening of physical insights andever more sophisticated methods of mathematical analysis. In the following this processwill be illustrated by a particular line of research. Originally motivated by the quest to‘understand’ gravitational radiation, it opened new views on the global space-time struc-tureBecause there exists now a vast literature on the subject I will have to focus in thefollowing on particular aspects of it. To keep the list of references at a reasonable lengthI shall often cite articles which appeared at later stages in the development of some topic nd give detailed refererence to earlier work. It should also be noted that the statementsbelow which begin with a key word written in bold letters followed by some references maynot always be found verbatim in these references but follow immediately from the resultsand arguments given there.The second half of the 1950’s saw an intense activity concerned with attempts to finda covariant concept of gravitational radiation in non-linear general relativity that had noneed of approximation arguments. It is difficult to give a ‘correct’ historical account of thisprocess and the contributions of the various researchers. I just point out a few highlightsand refer to the literature for a more complete picture.Pirani raises the question and tries to characterize radiative fields in terms of thePetrov structure of their curvature tensor [88]. He observes that pointwise algebraic con-siderations hardly suffice and proposes to study the evolution of the Petrov structure byanalyzing the Bianchi identity ∇ µ R µ νλρ = 0. Emphasizing the role of asymptotic do-mains in radiation problems, Trautman brings a global aspect into the discussion [98],[99]. Defining coordinate systems x µ near space-like infinity that satisfy conditions anal-ogous to Sommerfeld’s ‘Ausstrahlungsbedingungen’, he discusses on the space-like slices x = const. a total energy-momentum formula which may be considered as a precursorof a formula introduced later at null infinity. Trying to integrate these considerationsinto a coherent picture, Sachs shifts the point of view and considers the field along outgo-ing null geodesic congruences extending to infinity . Analysing the Bianchi equations in apseudo-orthonormal frame, he is led to suggest that the curvature tensor shows at infinitya characteristic peeling-off behaviour of components related to the Petrov types [92].A decisive step is taken when Bondi et al [19], Sachs [93], and Newman and Penrose[80] consider what in the language of today amounts to the asymptotic characteristic initialvalue problem , where data are prescribed on an outgoing null hypersurface and at futurenull infinity . They specify the asymptotic behaviour of the fields in terms of distinguishednull coordinates whose hypersurfaces are ruled by null geodesics extending to future nullinfinity. The behaviour of the fields is analyzed in terms of formal expansion in powersof 1 /r as r → ∞ , where r is a suitable parameter along the null geodesics. In [80] theBianchi equation is included into the system and the equations are expressed in terms ofa pseudo-orthonormal (more precisely, a spin) frame. The required asymptotic behaviouris expressed in terms of the conformal Weyl tensor and includes the peeling property. In[19] and [93] are considered mass terms and mass-loss relations which are interpreted asindication that the mass must decrease if the system is radiating. This result stronglysupports the view of being on the right track.This development reached an apex when Penrose introduces an idea that makes therole of the conformal structure, which features in the analysis above in terms of nullhypersurfaces and null geodesics, quite explicit and offers at the same time a new view onthe asymptotic and, in particular, on the global structure of gravitational fields [84]. Let( ˆ M , ˆ g ) denote the space-time of a self-gravitating isolated system that is so far away fromother such systems that one can essentially ignore the influence of the latter, possibly withthe exception of their gravitational radiation effects. Penrose proposes that the asymptoticbehaviour of ( ˆ M , ˆ g ) is characterized by the following property: Definition 1.1
The space-time ( ˆ
M , ˆ g ) is said to be asymptotically simple if there exists space-time ( M, g ) with boundary J 6 = ∅ such that ˆ M can be diffeomorphically identifiedwith the interior ˆ M = M \ J of M so that g µν = Ω ˆ g µν on ˆ M ,with a conformal factor Ω which is a boundary defining function on M that satisfies Ω > on ˆ M ,
Ω = 0 , d Ω = 0 on J . The requirements only concern the asymptotic properties of the conformal structureof ˆ g since they are invariant under rescalings of the form (Ω , ˆ g ) → ( θ Ω , θ ˆ g ) with positivefunctions θ . Conditions which restrict this freedom will be referred to as conformal gaugeconditions . Usually there are also stated conditions which ensure that the conformalboundary J is as complete as possible. This is also related to the question of the uniquenessof the conformal boundary. We shall not discuss these questions here. In the situationsconsidered below, the conformal boundary will be uniquely determined by the evolutionprocess defined by the field equations once suitable initial data are prescribed and be itwill independent to the conformal gauge employed.The degree of smoothness with which the conditions of the definition can be achievedreflects in a precise way the fall-off behaviour of the physical fields. It will be seen that itcan pose some of the most delicate problems. Unless remarked to the contrary, however,it will be supposed that Ω and g (and the functions θ above) are of class C ∞ on M .It follows then that ˆ g -null geodesics are complete in those directions in which theyapproach J . The set J thus represents a null infinity for the ‘physical’ space-time ( ˆ M , ˆ g ),its points being endpoints of ˆ g -null geodesics. In general one would add conditions whichensure that no essential boundary points are left out in the construction of M and for sometypes of systems there would be derived consequences such as a splitting J = J − ∪ J + ofthe boundary into two components J ± which represent the past/future endpoints and thuspast/future null infinity respectively. This will be seen later in concrete situations. Theimpression that the definition disburdens us from considering distinguished coordinates isnot quite correct. To decide whether a given space-time is asymptotically simple one has toextend its differentiable structure, which amounts to singling out (if possible) coordinatesystems which can be extended consistent with the conditions above.Statements about asymptotic simplicity gain significance if Einstein’s field equationsare involved. Penrose extended the range of application of Definition 1.1 beyond that ofisolated systems and also considered solutions with non-vanishing cosmological constant λ [85]. In general a smooth solution ( ˆ M , ˆ g ) to the vacuum equations with cosmologicalconstant λ does not even satisfy the conditions with an extension ( M, g,
Ω) of finite dif-ferentiability. But if it does, with sufficient smoothness, it follows that the field equationsdetermine the causal nature of the boundary. The set J will be time-like, null, or space-like depending on λ being negative, zero, or positive (assuming the signature ( − , + , + , +)).While this is a simple consequence of the field equations, it shows that the sign of the cos-mological constant has far-ranging effects on the overall behaviour of the solutions. Todaythis is taken for granted, but at a time when local coordinates still dominated the way oflooking at space-times, it must have come as a revelation.The conformal Weyl tensor C µ νκσ [ˆ g ] of ˆ g goes to zero at J if there exists a confor-mal extension ( M, g,
Ω) of sufficiently high differentiability [85]. In fact, observing that µ νκσ [ˆ g ] = C µ νκσ [ g ] on ˆ M , one can easily specify precise smoothness conditions on( M, g,
Ω) which imply that ∇ µ Ω C µ νκσ [ g ] = 0 on J . If λ = 0, so that g ( d Ω , d Ω) = 0,pointwise algebra shows that C µ νκσ [ g ] = 0 on J . If λ = 0 the conclusion is more subtle.The argument given in [85] contains an implicit assumption on the smoothness of theconformal extension ( M, g,
Ω) which requires difficult global considerations for its justifi-cation (in the review of the argument given in [52] this assumption is taken as the startingpoint). We shall see below, that the situation is not completely understood yet. Thereforeit will sometimes be convenient to use the notation J ∗ to refer to null infinity simply as aunstructured set of fictitious endpoints of null geodesics.Suppose there does exist a smooth conformal extension satisfying the requirements ofDefinition 1.1. Then many questions concerning the asymptotic structure which requireddelicate limits before can conveniently be studied in terms of local differential geometry.The conditions then comprise fall-off properties of the curvature tensor which are suchthat W µ νκσ = Ω − C µ νκσ [ˆ g ] extends smoothly to the boundary J and it follows by purealgebra that the Sachs peeling conditions are satisfied [86]. In the vacuum case with λ = 0the setting provides a precise notion of radiation field . It is given on J + (say) by thecomplex-valued function ψ = ψ ABCD ι A ι B ι C ι C where ψ ABCD is the symmetric spinorfield which represents the limit of the tensor W µ νκσ to J + and ι A is a two-index spinorfield so that ι A ¯ ι A ′ is tangential to the null generators of the null hypersurface J + . For adiscussion of the relations between the radiation field, the mass, and the mass loss along J + we refer to [86].The situation described in Definition 1.1 will be referred to by saying that ( ˆ M , ˆ g ) admits a smooth conformal extension at null infinity . We emphasize that this comprisesthe smooth extendibility of the metric g as well as that of the conformal factor Ω withthe required properties. In the context of the Einstein equations both requirements areimportant, because they relate in a precise way to the decay behaviour of the gravitationalfield indicated above. That these things should not be taken for granted is illustrated bythe following example. The space-time with manifold ˆ M = R × S and metric ˆ g = − dt + dr + cosh r h S is geodesically complete. That it admits a conformal extension is seenas follows [17]. With the transformation defined by cosh τ cos ψ = tanh r , cosh τ sin ψ =cosh t cosh − r , sinh τ = sinh t cosh − r and the conformal factor Ω = cosh − r one obtainsΩ ˆ g = g ≡ − dτ + cosh τ dψ + h S . The metric g is the Nariai solution which lives infact on M = R × S × S . The conformal embedding of ( ˆ M , ˆ g ) into ( M, g ) so obtainedcovers only the domain where | cosh τ cos ψ | <
1, 0 < ψ < π . This domain is boundedby null hypersurfaces on which | cosh τ cos ψ | = 1. These can be understood as defining aconformal boundary for the space-time ( ˆ M , ˆ g ). While the metric Ω ˆ g extends smoothlyto this boundary, the function Ω, which can be written Ω = p − cosh τ cos ψ , onlyextends continuously with Ω = 0 but d Ω divergent on the boundary. Moreover, theconformal Weyl tensor C µ νλρ [ g ] does not go to zero on this boundary. The early studies of the notion of asymptotic simplicity focus on geometrical and physicalissues, the existence of solutions satisfying the conditions is discussed mainly in terms of xplicit, usually static or stationary ones, which have vanishing radiation field [63], [69],[85]. There have been found explicit vacuum solutions which admit pieces of a smoothconformal boundary with non-vanishing radiation field [14], but non of them arises fromsmooth, asymptotically flat, geodesically complete data on a Cauchy hypersurface andadmits a conformal boundary that satisfies reasonable completeness conditions [64]. Atthe time, anyone toiling at the abstract analysis of the Cauchy problem for the non-linearEinstein equations must have registered with surprise that the complicated global problemof characterizing the asymptotic behaviour of solutions should allow, in any generality, ananswer in so simple and clean geometric terms as used in Definition 1.1. In fact, it hasbeen argued from early on that the assumptions on the asymptotic behaviour underlyingthe work referred to above might be too stringent and that consistent formal expansionsat null infinity can also obtained under more general assumptions [25], [33].Many questions remained open and we shall address here only a few of them. Phys-ically a particularly interesting one, which will obviously become most important oncegravitational radiation can be measured directly, refers to fields of isolated self-gravitatingsystems: What kind of information on the structure of the sources can be extracted fromthe radiation field ?
This question does not only ask for qualitative control on the farfields and the (massive) sources, but also for quantitative results. These can, in particularin the case of strong and highly dynamical fields, only be obtained by numerical methods.In this article we shall mainly be interested in a more basic problem:
How rich is the classof solutions satisfying Definition 1.1 ?
Besides providing mathematical results on thestructure of the field equations and the large scale nature of the solutions (which shouldalso help with the first question above) the complete answer should include a discussionof the physical significance of possible obstructions to asymptotic simplicity. We shall beinterested in the following in solutions of four space-time dimensions which are general inthe sense that they are not required to have symmetries.Answers to the second question above can only be obtained by methods of generalabstract analysis and global or semi-global existence results. Provided the extension issmooth, asymptotic simplicity makes a statement about the asymptotic behaviour that isabsolutely sharp. This makes the concept most delicate from the PDE point of view andharbours the danger of leading to undesired restrictions. On the other hand, it may givedeeper insight into the mathematical structure and the physical meaning of solutions. Inthe following sections we will have various opportunities to discuss this dichotomy.In the early 1970’s there existed a huge gap between the formal expansion type studiesat null infinity and the abstract existence theory for Einstein’s field equations. At thetime, the latter supplied existence results local in time for the wave equation obtainedfrom Einstein’s equations in harmonic coordinates. The two approaches to Einstein’s fieldequations were completely unrelated. An effort to understand the field equations in theconformal setting combined with a search for alternative ways to exploit the intrinsichyperbolicity of the Einstein equations showed, however, that Penrose’s emphasis on theconformal structure might lead to new methods for the existence problem [40].If the vacuum equation R µν [ˆ g ] = 0 for the ‘physical’ metric ˆ g µν is expressed in termsof a conformal factor Ω and the conformal metric g µν = Ω ˆ g µν , it reads R µν [ g ] = − − ∇ µ ∇ ν Ω − g µν g αβ (cid:0) Ω − ∇ α ∇ β Ω − − ∇ α Ω ∇ β Ω (cid:1) . (2.1) he right hand side becomes singular where Ω →
0. It is one of the remarkable featuresof the Einstein equations that (2.1) can be included into a larger system of equations, the conformal field equations discussed below , which avoids this type of singularity. By itselfthis might not help much but it turns out that the resulting system admits hyperbolicreductions [41]. It generalizes the Einstein equations by being equivalent to them whereΩ = 0 but still implying hyperbolic evolution equations where Ω = 0.It is clear that in controlling the large scale behaviour of their solutions in terms ofestimates on the physical field ˆ g and fields derived from it, the conformal properties ofthe equations exhibited here must play implicitly an important role. However, the explicituse of these properties in terms of the conformal field equations seemed to offer ways tocalculate global or semi-global solutions to Einstein’s field equations numerically on finitegrids. Formulating corresponding initial or initial-boundary value problems and workingout their qualitative consequences for the global space-time structure analytically thereforesuggested itself as a way to also obtain quantitative results and thus answers to the firstquestion asked above as well. Einstein’s equations with cosmological constant λ and energy-momentum tensor ˆ T µν ,ˆ R µν −
12 ˆ R ˆ g µν + λ ˆ g µν = κ ˆ T µν , (2.2)coupled to suitable matter field equations, are reexpressed in terms of a conformal factorΩ, the metric g µν = Ω ˆ g µν , suitably transformed matter fields, and the derived fields s ≡ ∇ ρ ∇ ρ Ω + Ω R, L µν = (cid:0) R µν − R g µν (cid:1) , W µ ρνλ ≡ Ω − C µ ρνλ , T ∗ ρµ ≡ ˆ T ρµ − ˆ T ˆ g ρµ , ˆ ∇ ρ ˆ L µν ≡ κ ˆ ∇ ρ (cid:16) ˆ T µν − ˆ T ˆ g µν (cid:17) ,where ˆ ∇ µ , ∇ µ and ˆ C µ ρνλ , C µ ρνλ denote the covariant derivative operators and the con-formal Weyl tensors of the metrics ˆ g and g respectively, L µν is the Schouten tensor of g ,and ˆ T the ˆ g -trace of ˆ T µν . Equations (2.2) imply the conformal field equations ([40], [41],[48]) 6 Ω s − ∇ ρ Ω ∇ ρ Ω − λ = − κ T , (2.3) ∇ µ ∇ ν Ω + Ω L µν − s g µν = κ T ∗ µν , (2.4) ∇ µ s + ∇ ρ Ω L ρµ = κ ∇ ρ Ω T ∗ ρµ − κ
24 Ω ˆ ∇ µ ˆ T , (2.5) ∇ ν L λρ − ∇ λ L νρ − ∇ µ Ω W µ ρνλ = 2 ˆ ∇ [ ν ˆ L λ ] ρ , (2.6) ∇ µ W µ ρνλ = 2Ω ˆ ∇ [ ν ˆ L λ ] ρ , (2.7)where all contractions are performed with the metric g . The first two equations are justa rewrite of (2.2). They imply the differential identity (2.5). Equation (2.7) is obtainedfrom the contracted Bianchi identity for ˆ g with the conformally covariant relations µ ρνλ = ˆ C µ ρνλ , ∇ µ (Ω − C µ ρνλ ) = Ω − ˆ ∇ µ ˆ C µ ρνλ .With (2.7) equation (2.6) is obtained from the Bianchi identity for g . These equationsmust be supplemented by equations which relate the tensorial unknownsΩ , s, L µν , W µ ρνλ , (2.8)to the metric g and the connection ∇ . One possibility to do this is by introducing a g -orthonormal frame field { e k } k =0 ,... and suitable coordinates x µ . In terms of the framecoefficients e µ k and the connection coefficients Γ i k j defined by the relations e k = e µ k ∂ x µ and ∇ i e j = Γ i k j e k so that g ( e i , e j ) = g µν e µ i e ν j = η ij and Γ ijk = − Γ ikj , where ∇ i = ∇ e i and Γ ijk = Γ i l k η lj , the structural equations then take the form of the torsion-free condition e µ i, ν e ν j − e µ j, ν e ν i = (Γ j k i − Γ i k j ) e µ k , (2.9)and the Ricci identity Γ l i j, µ e µ k − Γ k i j, µ e µ l + 2 Γ [ k i p Γ l ] pj − [ k p l ] Γ p i j (2.10)= Ω W i jkl + 2 { g i [ k L l ] j + L i [ k g l ] j } . If equations (2.3) to (2.7) are expressed in terms of the frame and combined with thestructural equations, they are equivalent (ignoring subtleties which may arise in the caseof very low differentiability) to Einstein’s vacuum equations where Ω > < , s, W ijkl ) → ( − Ω , − s, − W ijkl ) and the other fieldsunchanged the equations remain satisfied as they stand).If (2.3) holds on an initial slice, which can always be arranged, it will be satisfied asa consequence of the other equations. In the vacuum case ˆ T µν = 0 the right hand sides ofequations (2.4) to (2.7) vanish and all factors 1 / Ω in the equations are gone. If the Ricciscalar R is prescribed as a function of the coordinates (which can locally be done in anarbitrary way) and the system (2.4) to (2.10) is written with respect to a suitable choice ofcoordinates and frame field, it implies reduced equations which are hyperbolic even where Ω changes sign . Moreover, the evolution by the reduced system preserves the constraintsand the gauge conditions. Various versions of hyperbolic systems are given in [41], [46],[48], [52]. The case of zero rest-mass fields, for which the energy momentum tensor istrace free and the matter fields admit regular conformal representations is similar. Forother matter models the situation is more difficult and will in general result in equationswhere the 1 / Ω terms cannot be removed. A remarkable case of non-zero rest-mass fieldswith non-vanishing trace of the energy momentum tensor where this can be done will bediscussed in section 3.While the search for hyperbolic reductions which could be adapted to various par-ticular situations led to quite general reduction procedures, in fields of isolated systemsa precise analysis of the equations remained extremely difficult in the neighbourhood ofspace-like infinity. Admitting besides conformal rescalings also transitions to Weyl connec-tions, one obtains, however, an extended version of the conformal field equations. It allowsone to employ a type of geometric gauge , based on conformal geodesics which has not beenused in the context of the Einstein equations before. Apart from some freedom on theinitial slice, the gauge is then defined completely in terms of the conformal structure [49], general conformal field equations . Theexistence results discussed in the following sections have been obtained with the conformalfield equations above or such generalizations.In the following years were studied modifications of the equations above, in particularsystems of wave equations obtained by taking further derivatives [23], [83], and versionsof conformal field equations that apply to other situations. In the asymptotically flatcase the static and stationary vacuum Einstein equations with λ = 0 were shown toadmit conformal representations on the 3-dimensional quotient space which imply thatthe fields extend as real analytic fields to spatial infinity [8], [9], [76]. Tod discussesbig bang like isotropic cosmological singularities that are characterized by the existenceof conformal rescalings which, in contrast to the asymptotically simple case, blow up aneighborhood of the singularity so as to represent the big bang by a hypersurface smoothlyattached to the original space-time [95]. These solutions were studied in the following byversions of conformal field equations which include suitable matter fields and deal withsome remaining singular terms [82], [96], [97].Though the conformal field equations above do admit regular generalizations to higherdimensions, the space-time dimension four is special for them [52]. Only in that dimensiondo they supply hyperbolic evolutions systems of first order in the unknowns (2.8). Thisraises the question whether there exist generalizations. Various authors studied formalexpansion at the (space- or time-like) conformal boundary J of solutions to Einstein’sequations with or without matter fields and with cosmological constant λ = 0 in space-time dimensions n ≥ J . Thelogarithmic terms not to occur on the even dimensional boundary J if the Fefferman-Graham obstruction tensor O jk of the free data vanishes on J . This tensor, defined by anoperator of order n for any metric g on a manifold of even dimension n ≥
4, is trace-free,symmetric, conformally invariant, and vanishes for metrics conformal to Einstein metrics[66]. Anderson uses the equation O jk [ g ] = 0 as a conformal vacuum Einstein equation [2]. In recent years various observations suggested an accelerating expansion of the universeand all of them seem to be consistent with the assumption of a positive cosmological con-stant in Einstein’s field equations [37]. There exist neither convincing theoretical expla-nations for the acceleration nor for the cosmological constant. Therefore it is of particularinterest to understand the manifold of solutions to Einstein’s field equations with positive λ . In the vacuum case ˆ T µν = 0 we write equation (2.2) in the form Ric [ˆ g ] = λ ˆ g. (3.1)The simply connected, conformally flat prototype solutions to (3.1) with λ = 3 is deSitter space ( ˆ M = R × S , ˆ g = − dt + cosh t h S ) where h S n denotes the standard metricon the unit n -sphere S n . With the coordinate transformation t → τ = 2 arctan e t − π and the conformal factor Ω = cosh − t = cos τ one obtains the conformal representationˆ M = ] − π , π [ × S , g = Ω ˆ g = − dτ + h S . The metric on the right hand side and the onformal factor extend smoothly to the manifold with boundary M = [ − π , π ] × S .Their extensions will be denoted again by g and Ω. The conformal boundary J splits intothe components J ± = {± π , } × S which are space-like for g and represent for ˆ g futureand past null and time-like infinity respectively.In the following we shall be interested in globally hyperbolic generalizations of thissolution which either admit a smooth conformal extension with conformal boundary J + in the future (which maps onto a J − under time reversal), or conformal extensions in thepast as well as in the future, with conformal boundaries J ± . These solutions will be suchthat each null geodesics acquires precisely one future endpoint on J + in the first caseand precisely one past endpoint on J − and one future end point on J + in the secondcase. The hypersurfaces J + , resp. J − and J + will constitute Cauchy hypersurfacesfor the conformal extension ( M, g ) and initial hypersurfaces suitable for the conformalfield equations. To avoid misunderstandings arising from the fact that the sign of thecosmological constant only becomes meaningful if the signature of the metric is fixed, weshall refer to such space-times as de Sitter type space-times . To construct such solutions, suitable initial data are needed. Because there seem to be nonatural boundary conditions for solutions with λ >
0, we shall assume in both types ofinitial data sets discussed below that the -manifold S is compact . Moreover, only smoothinitial data will be considered. Let λ denote a fixed positive number.A standard initial data set for Einstein’s vacuum field equations (3.1) with cosmologicalconstant λ consists of an orientable 3-dimensional Riemannian manifold ( S, ˆ h ab ) and asymmetric tensor field ˆ χ ab which satisfy on S the Hamiltonian constraint R [ˆ h ] = ˆ ψ ab ˆ ψ ab +2 λ − ˆ χ and the momentum constraint ˆ D a ˆ ψ ab = ˆ D b ˆ χ , where ˆ D and R [ˆ h ] denotethe Levi-Civita operator and the Ricci scalar of ˆ h and we use the decomposition ˆ χ ab =ˆ ψ ab + ˆ χ h ab into ˆ h -trace-free part and trace.The set of such data will be denoted by D S . Any data set ( S, ˆ h ab , ˆ χ ab ) in D S deter-mines a unique (up to diffeomeorphisms) maximal, globally hyperbolic solution ( ˆ M , ˆ g ) to(3.1) which contains a Cauchy hypersurface ˆ S that is, together with the first and secondfundamental form induced on it by ˆ g , diffeomorphic( S, ˆ h ab , ˆ χ ab ) [22]. We note that in thecase ˆ χ < λ the Hamiltonian constraint implies R [ˆ h ] > S, ˆ h ). From the initial data ( λ, S, ˆ h ab , ˆ χ ab ) can be derived initialdata for the conformal field equations.An asymptotic initial data set for Einstein’s vacuum field equations (3.1) with cos-mological constant λ > S, h ab ) and a symmetric tensor field w ab on S that satisfies w a a = 0 and D a w ab = 0,where D denotes the Levi-Civita operator of h . Semi-global existence [45]:
An asymptotic initial data set ( λ, S, h ab , w ab ) for the vacuumfield equations (3.1) determines a unique, maximal, globally hyperbolic solution ( ˆ M , ˆ g ) to Ric [ˆ g ] = λ ˆ g with ˆ M ∼ R × S which admits a smooth conformal future extension ˆ M → M = ˆ M ∪ J + , ˆ g → g = Ω ˆ g and a diffeomorphism j : S → J + = { Ω = 0 } that identifies h with the metric induced by g on J + and w with the J + -electric part of the extension of he rescaled conformal Weyl tensor W µ λρν to J + . The result follows from an analysis of the constraints induced by the conformal fieldequations on a hypersurface { Ω = 0 } and by solving a Cauchy problem backwards in timefor the hyperbolic reduced conformal fields equations. It is a semi-global result becausethe solutions are null-geodesically future complete. All de Sitter-type vacuum solutionswhich admit smooth conformal extensions in the future (or past) are characterized here.No further restrictions on the topology of S , no smallness conditions on the data, andno restrictions on the conformal class of ( S, h ) need to be imposed. The last propertyfollows from the observation that the Hamiltonian constraint becomes trivial on the set { Ω = 0 } . This is related to the fact that ( λ, S, θ h ab , θ − w ab ), with θ > S , is also an asymptotic initial data set which determines the same ‘physical’solution as ( λ, S, h ab , w ab ). Because the solutions are obtained by solving Cauchy problemswith data prescribed on J + = { Ω = 0 } the smooth conformal extensibility of the solutionsis built in here by construction .Let ˆ S denote a Cauchy hypersurface of a solution ( ˆ M , ˆ g ) as above and let ( λ, ˆ S, ˆ h ab , ˆ χ ab )be the standard initial data induced by ˆ g on ˆ S . We denote the set of such data by A + S .Thus A + S denotes the set of standard initial data sets which develop into de Sitter-typesolutions that admit smooth conformal extensions in the future. Similarly, A ± S denotesthe set of standard initial data sets which develop into de Sitter-type solutions that admitsmooth conformal extensions in the past as well as in the future. Strong future stability [46]:
The set A + S is open in D S (in suitable Sobolev norm). ‘Strong’ has been added here to emphasize that not only geodesic (future) null com-pleteness but also smooth (future) conformal extensibility is preserved under small per-turbations. The result is obtained as follows. Let ( λ, ˆ S, ˆ h ab , ˆ χ ab ) be an initial data set in A + S and denote by ( ˆ M , ˆ g ) the maximal, globally hyperbolic vacuum solution determinedby it. Then ( ˆ M , ˆ g ) admits after a rescaling with a suitable conformal factor a smoothextension with boundary J + in the future. Using the induced asymptotic data on J + , it can be smoothly extended as a solution to the conformal field equations into a domainwhich contains a Cauchy hypersurface on which Ω = const. < . If ( λ, ˆ S, ˆ h ∗ ab , ˆ χ ∗ ab ) aredata in D S that are close (with respect to suitable Sobolev norms) to ( λ, ˆ S, ˆ h ab , ˆ χ ab ) thendata for the conformal field equations associated with these two data sets can be arrangedso as to be also close to each other. The result then follows by Cauchy stability for the(symmetric) hyperbolic equations (see [72]) induced by the conformal field equations andthe fact that equation (2.3) ensures that the sets { Ω = 0 } are space-like hypersurfaces. Strong global stability [46]:
The set A ± S is open in D S . This follows by similar arguments. It implies that the de Sitter solution or solutionsobtain from it by factoring out suitable symmetries of ( S , h S ) are strongly globally stable.We note that in the two stability results the smooth conformal extensibility of the solutionsclose to the reference solution is derived as a consequence of the conformal properties ofEinstein’s field equations. Generalizations including matter fields [48]:
With suitably generalized definitionsof A + S , A ± S , D S to include matter fields, the results above generalize to Einstein’s fieldequations (2.2) with λ > coupled to ‘conformally well behaved’ matter field equations . e shall not try to characterize ‘conformally well behaved’ here because it may requirecomplicated transformations to arrive at a set of unknowns so that no 1 / Ω terms appearon the right hand sides (2.3) to (2.7) or in the transformed matter field equations and theequations imply hyperbolic reduced systems. The statement certainly applies to matterequations like the Maxwell and the Yang-Mills equations, discussed in detail in [48], whichin four space-time dimensions are conformally invariant in the most direct sense. Othercases of matter fields with trace-free energy momentum tensor and conformally covariantequations have recently been worked out in [77], [78]. Further below we shall discuss aless obvious case.
Generalizations to higher dimensions [2]:
The vacuum results above generalize to alleven space-time dimension larger that four.
The solutions which develop from data in A + S have been characterized indirectly in termsof the asymptotic data induced on J + by the conformal extension of the solutions. Adirect characterization of the standard data in A + S is not known. That A + S is a propersubset of D S is illustrated by the analytically extended Schwarzschild-de Sitter solutions,which admit only patches of smooth conformal extensions in the future and past [65]. Amore extreme case is that of the standard Nariai space-time, given byˆ M = R × ( S × S ), ˆ g = − dt + cosh t h S + h S ,which solves (3.1) with λ = 1. It is globally hyperbolic and geodesically complete. In [12]has been used a topological argument to show that the standard Nariai solution does noteven admit a patch of a smooth conformal boundary . Observing that C µνρλ [ˆ g ] C µνρλ [ˆ g ] =Ω C µνρλ | g ] C µνρλ | g ] if g µν = Ω ˆ g µν , where the contractions are performed with ˆ g on theleft and with g on the right hand side, and using the result C µνρλ [ˆ g ] C µνρλ [ˆ g ] = const. = 0of a calculation, it follows directly that there cannot exist a piece of J ± of class C .Similarly, one would like to characterize the data in A + S which are in fact in A ± S . Thatthe latter is a proper subset of A + S is shown by the following examples. The space-timeˆ M = R × T , ˆ g = − dt + e t k ,with k an Euclidean metric on T , solves (3.1) with λ = 3. It only admits a smoothextension in the future. Its causal geodesics are future complete but only the causalgeodesics t → ( t, p ), p ∈ T are past complete. Another case is the space-timeˆ M = R × ( S × S ), ˆ g = − dt + sinh t h S + cosh t h S which solves (3.1) with λ = 3, admits smooth conformally extensions at the ends where | t | → ∞ but becomes singular as t →
0. These solutions illustrate a general phenomenon:
Obstructions to smooth conformal extensibility in the past [6]:
A solution ( ˆ
M , ˆ g ) to (3.1) with λ > which develops from data in A + S does not even admit a patch of asmooth conformal extension in the past if the fundamental group of S is not of finite orderor if the asymptotic data ( h, w ) induced by ˆ g on J + are such that the conformal structuredefined by ( J + , h ) is not of positive Yamabe type. The result gives little information about what exactly prevents the solution fromextending smoothly in the past. Further, there exist situations in which smooth conformal xtensibility fails for reasons different from those given above. Let { α a } a =1 , , denote abasis of 1-forms on S such that δ ab α a α b = h S and dα a = − ǫ bc a α b ∧ α c . Thenˆ M = R × S , ˆ g = − t v dt + (1 + t )( α α + α α ) + v t α α ,with v = (1 + t ) − α t , α ∈ R , denote members of the λ -Taub-NUT family whichsolve (3.1) with λ = 3. Their conformal structures extend smoothly in the past and in thefuture. With a suitable conformal scaling the metric g induces on J + ∼ S the asymptoticinitial data h = h S , w = − α ( δ ab α a α b − α α ), so that the obstructions pointed outabove are not present. If | α | < α ∗ ≡ √ · /
9, then v > τ ∈ R , the solutions aregeodesically complete and imply Cauchy data belonging to A ± S . If | α | ≥ α ∗ the solutionsare no longer globally hyperbolic. If α = α ∗ the function v has a double zero on thehypersurface { τ = τ ∗ ≡ / √ } , which represents a smooth compact Cauchy horizon thatcontains closed null curve. If α > α ∗ the function v has two simple zeros at values τ ± with0 < τ − < τ ∗ < τ + . The hypersurfaces { τ = τ ± } are Cauchy horizons which sandwich adomain that contains closed time-like curves.For given asymptotic initial data on S or S × S Beyer investigates solutions to thebackward Cauchy problem for the conformal field equations numerically [10]. This allowshim in particular to calculate solutions determined by data in A ± S from J + all the waydown to J − . Among other solutions, he studies a class of λ -Taub-NUT solutions largerthan the one given above. The investigation of the stability properties of solutions withCauchy horizons (numerically a delicate adventure since uniqueness of local extensions failsat Cauchy horizons) suggests that the solutions develop curvature singularities instead ofCauchy horizons if the asymptotic data on J + are slightly perturbed [11]. The argument which give the stability results uses the fact that solutions which admitsmooth conformal extensions to J + can in fact be smoothly extended as solutions to theconformal field equations into domains beyond J + where Ω <
0. The extension definesthere another solution to the Einstein equations. The conformal representation of deSitter space given by Ω = cos τ and g = − dτ + h S extends analytically to the manifold R × S , where Ω defines an infinite sequence of domains on which Ω = 0, separated byhypersurfaces where Ω = 0, d Ω = 0. On any such domain Ω − g is isometric to the deSitter metric. Denote by X ∼ S a hypersurface which separates two of these domainsand let k > X . Then, by Cauchystability, the solution to the conformal field equations determined by these data will extendover k different domains where Ω = 0 if the data are sufficiently close to the asymptoticde Sitter data induced on X by the metric g . In this case the ‘physical metrics’ inducedon the different domains will not necessarily be isometric. It is an interesting questionwhether a solution to the conformal field equations which extends over an infinite numberof domains where Ω = 0 must necessarily be locally conformally flat.So far the extensibility was used as a technical device. The ‘physical solutions’ consid-ered above, can hardly be considered as cosmological models, they expand exponentiallyin both time directions while current wisdom expects that the universe starts with a bigbang. One may still ask why the ‘physical world’ should end at J + if extensions acrossconformal boundaries are a natural consequence of the field equations. At this stage we ecall that matter fields with non-zero rest mass have been ignored so far. There existslittle precise information about the behaviour of the conformal structure near J + ∗ if suchfields are coupled to Einstein’s equations. If the conformal structure can be controlled atall, it may depend in subtle ways on the specific nature of the matter model.A transition process across conformal boundaries is at the basis of the conformal cycliccosmology proposed by Penrose [87]. The underlying picture is that of a smooth, timeoriented conformal structure of signature ( − , + , + , +) (which we shall refer to as the longconformal structure ) on a 4-dimensional manifold M ∼ R × S with compact 3-manifold S ,into which an infinite sequence of aeons , i.e. time oriented ‘physical’ solutions to Einstein’sfield equations with cosmological constant λ >
0, are conformally embedded so that anytwo consecutive aeons are separated by a crossover -surface X ∼ S which is space-likewith respect to the conformal structure. Each aeon starts with a big bang that ‘touches’the preceding crossover surface and ends in the future with an exponentially expandingphase for which the following crossover surface defines a smooth conformal boundary.This scenario asks for a global PDE result which establishes the existence of a long con-formal structure. The transition process through the crossover surfaces must be controlledwith suitable versions of conformal field equations. It must be clarified what happens atthe prospective conformal boundaries in the presence of fields with non-vanishing rest-masses. In [87] it is assumed that only zero rest-mass fields will be present in some pastneighbourhoods of the crossover surfaces. No justification is known for this requirement.Finally, it will not suffice to be able to glue in specific cases the future conformal boundaryof a given aeon to the hypersurface that conformally represents the isotropic singularity inthe past of the subsequent aeon. A general mechanism is needed that forces the equationsto take the route from a expanding phase to big bang phase instead of simply using thesmooth transitions across the conformal boundaries discussed above.Most likely, the last two problems are not independent of each other and it will beworthwhile to have a closer look at the asymptotic behaviour of solutions involving non-zero-rest mass fields. Ringstr¨om obtained quite general results on the Einstein-scalar-fieldsystem. Consider the de Sitter metric ˆ g together with a function φ = 0 on ˆ M = R × S asa special solution to the Einstein’s equations (2.2) with energy momentum tensorˆ T µν = ˆ ∇ µ φ ˆ ∇ ν φ − (cid:16) ( ˆ ∇ ρ φ ˆ ∇ ρ φ + m φ ) + U ( φ ) (cid:17) ˆ g µν , coupled to the scalar field equationˆ ∇ µ ˆ ∇ µ φ = m φ + U ′ ( φ ),with rest-mass m > U . Global stability [90]:
If the potential is such that U = O ( | φ | ) as φ → , then thede Sitter solution (with φ ≡ ) is non-linearly stable in the sense that sufficiently smallperturbations of de Sitter Cauchy data on a hypersurface ∼ S develop into solutions to thecoupled system whose causal geodesics are past and future complete and which are, withrespect to suitable close norms, close to the de Sitter solution. The work referred to above gives detailed estimates in terms of the physical metric.Getting precise ideas about the asymptotic behaviour of the conformal structure willrequire more, however. In studying these questions it may be useful to take into accountthe possibility that a loss of differentiability of the long conformal structure at the crossoversurfaces may be admissible as long as uniqueness of the extensions is assured. The loss ay depend on various specific features of the matter model. The following, somewhatunexpected, result shows that there are possibilities which are not obvious if one just looksat the Einstein equations in their standard form. Strong global stability [61]:
If the mass and the cosmological constant are related inthe Einstein-scalar-field system by m = 2 λ and the potential satisfies U = O ( | φ | ) as φ → , then Cauchy data sufficiently close to de Sitter data with φ = 0 evolve into globallyhyperbolic solutions that admit smooth conformal extensions in the past and in the future. If the scalar field is replaced by the function ψ = Ω − φ , the conformal field equationsand the transformed scalar field equation contain under the assumptions above no 1 / Ωterms and imply in fact hyperbolic reduced systems. With asymptotic data on a slice { Ω = 0 } that generalize the asymptotic vacuum data considered earlier, a local existenceresult follows and arguments similar to the ones given earlier imply the result.This raises various questions. To which extent can the smoothness assumptions andresults be relaxed and the matter models be generalized ? Can extensions of low smooth-ness and suitable matter models help initiate the transition process from expanding to bigbang phases ? Do there exist physical fields satisfying the condition ? Does the relationbetween the matter field and the cosmological constant shed any light on the origin andthe role of the cosmological constant ? These questions will have to be discussed elsewhere. The calculation of the gravitational radiation generated by spatially localized processeslike the encounters of stars or merges of black holes is one of the main motivations foranalyzing asymptotically flat solutions. The simply connected, conformally flat model caseis Minkowski-space, given in spatial spherical coordinates byˆ M = R , ˆ g = − dt + dr + r h S .Coordinates τ and χ and a conformal factor Ω satisfying t = Ω − sin τ, r = Ω − sin χ, Ω = cos τ + cos χ, ≤ χ, | τ ± χ | < π ,give Ω ˆ g = h ≡ − dτ + dχ + sin χ h S , whence a smooth conformal embedding ofMinkowski space into the Einstein cosmos ( M ∗ = R × S , g ∗ = − dτ + h S ) [85].The metric g and the conformal factor Ω extend smoothly to the range 0 ≤ χ , | τ ± χ | ≤ π of the coordinates. This extension adds to ˆ M the sets J ± = { τ = ± ( π − χ ) , < χ < π } ,which are null hypersurfaces for the extended metric g referred to as future and past nullinfinity. The sets i = { τ = 0 , χ = π } and i ± = { τ = ± π, χ = 0 } represent regular pointsof the conformal extension, it holds there Ω = 0, d Ω = 0, and 0 = Hess g Ω ∼ g . For agiven space-like slice ˆ S t o = { sin τ = t o Ω } , t o ∈ R , one may consider i as a point addedat spatial infinity which makes the slice into a sphere ∼ S . The point i defines, howeveran endpoint (in both directions) for all space-like geodesics and thus represents space-likeinfinity for ( ˆ M , ˆ g ). The points i ± are approached by the time-like geodesics in the futureand the past and thus represent for ( ˆ M , ˆ g ) future and past time-like infinity. The set J + isruled by the past directed null geodesics through the point i + , which coincide with futuredirected null geodesics through i . Similar relations hold for J − , i − , and i . here are of course no stars or black holes around here but since the registrationof gravitational radiation takes place at large distances from the sources, generalizationsof the situation above to the far fields of non-trivial vacuum solutions are of particularinterest. It is sometimes said that it were too extreme an idealization to put the measuringdevice at null infinity or, in other words, to read off the radiation field there. This is in factone of the questions we are interested in when we try to control the field near null infinity.If the field turns out to admit a smooth conformal extension at null infinity then the sizeand structure of the field will hardly be affected if the location of the ideal measuring deviceat null infinity is shifted slightly, in terms of the g adapted conformal coordinates, into thespace-time. In terms of the physical metric ˜ g such a shift covers an infinite distance, whichputs the measuring device at a reasonable distance to the sources. This is the situationconsidered in most numerical calculations. What is lost in this procedure, however, is theunique tangent space of null infinity which serves to define the radiation field and givesan automatable prescription for the numerical calculation of radiation fields.Quite early the requirements of Definition 1.1 were shown to be met by some of themost important explicit vacuum solutions that admit time-like Killing fields [69], [85].These were special cases of a general fact. Static and stationary fields [35]:
Asymptotically flat static or stationary vacuum solu-tions admit smooth conformal extensions at future and past null infinity .This result establishes the existence of a fairly large class of asymptotically simplevacuum solutions. The main purpose of introducing Definition 1.1 is, however, to discussgravitational radiation and all the solutions above have vanishing radiation fields. Thereremains the question to what extent Definition 1.1 applies to dynamical solutions.In hindsight the PDE problem whose analysis paved the way to the notion of asymp-totic simplicity can understood as the asymptotic characteristic initial value problem wheredata are prescribed on an outgoing null hypersurface N which is supposed to intersect fu-ture null infinity in a 2-dimensional space-like slice Σ and the part J + ′ of future nullinfinity in the past of Σ. A detailed formulation of this problem for the conformal fieldequations which specifies the freedom to prescribe data has been given in [41]. Its elabo-ration gives: Well-posedness of the asymptotic characteristic initial value problem [70]:
Forgiven smooth null data on N and J + ′ and certain smooth functions given on Σ , thereexists a smooth solution to the conformal vacuum field equations in a past neighborhood U of Σ which induces the given data on U ∩ ( N ∪ J + ′ ) . It induces on ˆ U = U \ J + ′ a uniquesolution ˆ g to Einstein’s vacuum field equation ˆ R µν = 0.The smooth conformal extensibility of ( ˆ U , ˆ g ) has been built in by the way the PDEproblem is formulated. The freedom to prescribe null data (two components of W µ νρλ oneach hypersurface) is similar to that in characteristic initial value problems for Einstein’svacuum field equations with data given on two intersecting null hypersurfaces N , N which are thought as lying in the physical space-time [40]. Only some differences in thefreedom to prescribe data on Σ resp. N ∩ N indicates that J + ′ is geometrically a specialhypersurface. The null datum on J + ′ is in fact the radiation field ψ . If the data on N and Σ are trivial, the solution is completely determined by ψ . In the time reversedsituation it is thus quite natural to identify ψ with the incoming radiation field on past ull infinity. Since the radiation field can be prescribed freely we have solutions of thetype we are looking for. There also exists a real analytic version of this result [42]. Thesolution to the conformal field equations then extends analytically into the future of nullinfinity to a domain where Ω < purely radiative space-times is of particular interest [47]. These space-timesare defined be the requirement that they posses a smooth past conformal boundary J − whose null generators are complete in a conformal gauge that makes J − expansion free andwhich admit a smooth conformal extension containing a point i − so that N i − = J − ∪{ i − } is the cone generated by the future directed null geodesics emerging from i − . Some au-thors say ‘purely radiative’ under much weaker assumptions. It should be noted, however,that only with sufficient regularity at i − the solutions will be determined uniquely by theradiation field, a counter example being given by the Schwarzschild solution. The simplestway to create purely radiative space-times is to assume that the radiation field vanishesin a neighbourhood of i − , so that the solution will be Minkowskian near past time-likeinfinity. If one wants to exploit the full freedom to prescribe data, however, one has toface problems arising from the non-smoothness of the initial set at the vertex. One needsan appropriate notion of smoothness for the free data on N i − and it must be shown thatfield equations themselves then ensure the smoothness of the solution in the future of N i − [59]. The most difficult part, the existence problem near i − , has been solved only recently. Existence for the pure radiation problem near past time-like infinity i − [29]: For a given radiation field on the cone N i − that satisfies appropriate smoothness condi-tions, there exists near i − a unique (up to diffemorphisms) smooth solution to the vacuumequations in the future of N i − . For this solution N i − represents a smooth conformal pastboundary with regular vertex i − on which the solution induces the given data. Characteristic problems are important in various arguments and are being used as thebasis of numerical calculations extending to null infinity. They will not be considered hereany further. Being ruled by null geodesics, null hypersurfaces have an intrinsic tendencyto develop caustics that can cause extreme technical difficulties.
We shall consider initial value problems based on space-like hypersurfaces. Two classes ofsuch hypersurfaces are of interest to us. These are standard Cauchy hypersurfaces like thesets ˆ S t o considered above, that extend to space-like infinity, and hyperboloidal hypersurfaces like the sets { τ = τ o , ≤ χ < π − | τ o |} , 0 < | τ o | = const. < π , in the conformally extendedMinkowski space, which extend smoothly to null infinity as space-like slices . The prototypeexample, obtained for τ o = π , is the unit hyperboloid H + = {− t + r = − , t > } which motivates the name [43]. In the conformally extended Minkowski space it is easilyseen that the future null cone with vertex at the origin, future null infinity J + , and theextension of the hypersurface H + intersect each other transversally. Referring to the latteras ‘asymptotically null’, as it is done sometimes, is thus easily misleading.Hyperboloidal initial data ( ˆ S, ˆ h ab , ˆ χ ab ) for Einstein’s vacuum field equations ˆ R µν = 0have in common with asymptotically flat Cauchy data that the underlying Riemannianspace ( ˆ S, ˆ h ab ) is required to be orientable and geodesically complete, the vacuum con- traints R [ˆ h ] = ˆ χ ab ˆ χ ab − ( ˆ χ a a ) and ˆ D b ˆ χ a b = ˆ D a ˆ χ b b must be satisfied, and the meanextrinsic curvature ˆ κ = ˆ χ a a can be assumed to be constant. This will be done in thefollowing to simplify the discussion. They differ, however, in their asymptotic behaviour.In the hyperboloidal case one must then have ˆ κ = 0 whereas ˆ κ = 0 in the asymptoticallyflat case. Moreover, if hyperboloidal data are supposed to lead to a asymptotically smoothsituation, they must admit a smooth conformal completionˆ S → S = ˆ S ∪ Σ , ˆ h ab → h ab = Ω ˆ h ab , ˆ χ ab → χ ab = Ω ( ˆ χ ab − ˆ κ ˆ h ab ),where ( S, h ab ) is a Riemannian space with compact boundary ∂S = Σ, Ω a smooth definingfunction of Σ with Ω > S , and W µ νλρ = Ω − ˆ C µ νλρ extends smoothly to Σ on S where ˆ C µ νλρ denotes then conformal Weyl tensor determined by the metric ˆ h ab and thesecond fundamental form ˆ χ ab . Such data will be called smooth . Existence for hyperboloidal problems [43]:
Smooth hyperboloidal initial data developinto a unique smooth, maximal, globally hyperbolic solution to Einstein’s vacuum fieldequations ˆ R µν = 0 which admits in the future of the embedded hypersurface ˆ S a smoothconformal extension at future null infinity, that approaches Σ in its past end, and whichpossesses a Cauchy horizon in the past of ˆ S that approaches Σ in its future end. While in the solution above the future directed null geodesics which approach futurenull infinity are future complete, the smooth hyperboloidal hypersurfaces that connect thetwo future null infinities of the conformally extended Schwarzschild-Kruskal space-time[69] show that further assumptions are required to ensure future completeness for all nullgeodesics.Hyperboloidal initial data for which this completeness requirement is satisfied are givenby the data induced on a space-like hypersurface ˆ S in Minkowski space with the followingproperties: Any past inextendible causal curve starting in the future of ˆ S intersects ˆ S pre-cisely once and ˆ S has a (unique) smooth space-like extension S in the conformally extendedMinkowski space so that the surface Σ = S ∩ J + is diffeomorphic to S . Data of this typewill be referred to as Minkowskian hyperboloidal data and their future development as a
Minkowskian hyperboloidal development . Strong future stability for Minkowskian hyperboloidal developments [46]:
Let ( ˆ S, ˆ h ∗ ab , ˆ χ ∗ ab ) denote Minkowskian hyperboloidal data with smooth conformal extension tothe -manifold S = ˆ S ∪ Σ . Then any smooth hyperboloidal vacuum initial data set ( ˆ S, ˆ h ab , ˆ χ ab ) which is sufficiently close (in suitable Sobolev norms) to the Minkowskiandata set develops into a solution which is null geodesically future complete and admits asmooth conformal extension at future null infinity with conformal boundary J + ′ . More-over, the conformal extension contains a regular point i + such that the past directed nullgeodesics emanating from i + generate the set J + ′ and approach the boundary Σ attachedto ˆ S in their past. The proof uses again the possibility to extend solutions to the conformal field equationsthrough sets where Ω vanishes. We don’t go into any details but point out the remarkableconsequence of equations (2.4) and (2.5) that the null generators of the set J + ′ are forcedto meet under the given conditions at exactly one point i + , where the conformal factorhas a non degenerate singularity with Hess g Ω = s g , s = s | i + = 0. Generalizations including matter fields [48]:
The vacuum results above can be gen- ralized to include conformally well behaved matter fields coupled to Einstein’s equations. Generalization to higher dimensions [3]:
The vacuum results generalize to even space-time dimensions larger than four .Hyperboloidal initial data can be constructed by a suitable adaption of the conformalmethod known from the construction of asymptotically flat Cauchy data. Certain seed dataconsistent with the required fall-off behaviour at space-like infinity are prescribed freely andthen elliptic equations are solved to determine correction terms so that the corrected datawill satisfy the constraints.
Existence of smooth hyperboloidal vacuum data [4], [5]:
Seed data on a -manifold S with boundary Σ which extend smoothly to Σ determine solutions to the vacuum con-straints. In general these admit at Σ only asymptotic expansions in terms of powers of x and log x where x is a local coordinate with x = 0 on Σ and x > on ˜ S = S \ Σ (polyhomogenous expansions). The hyperboloidal data are smooth if and only if the seeddata satisfy certain conditions at Σ . It is important here to note that conditions on the seed data need only be imposedat the boundary at infinity to ensure the smoothness of the resulting hyperboloidal data.The result suggests a generalization.
Existence of polyhomogeneous hyperboloidal vacuum data [4]:
Seed data whichare smooth on ˆ S and admit certain polyhomogenous expansions at Σ determine hyper-boloidal vacuum data which admit polyhomogenous expansions at Σ . The fact that the data are ‘physical’ only on the open set ˆ S leaves a large ambiguity ofhow to specify the data at Σ if some roughness is admitted . It is a critical open question,which will come up again in the standard Cauchy problem, whether there exist physicalsituations of interest which can only be modeled by using non-smooth data.For the calculation of radiation fields at null infinity the hyperboloidal initial valueproblem is as good as the standard Cauchy problem and it has the advantage of avoidingthe difficulties at space-like infinity discussed below. P. H¨ubner pioneered the numericalcalculation of solutions including parts of future null infinity from hyperbolical data [67].He developed a code for the conformal vacuum field equations which enabled him to calcu-late numerically, without assuming any symmetries, the entire future of the hyperboloidalinitial slice as considered in the strong stability result above. The calculation includesthe radiation field on the asymptotic region J + ′ and its limit at i + [68]. A survey onthis and other numerical developments involving the conformal field equations is given byFrauendiener [38]. Once it had been established that certain classes of hyperboloidal datadevelop into solutions with a smooth asymptotic structure it became less daring to tryother equations. There are now being developed numerical codes based on equations thatare more directly related to the singular conformal representation (2.1) [91]. For asymptotically flat space-times the standard Cauchy problem for Einstein’s field equa-tions is more fundamental than the hyperboloidal problem, its solutions cover the past aswell as the future in a unified way while a solution to a hyperboloidal problem is thought f as part of an ambient asymptotically flat space-time. One may wonder whether theproblem raised above about non-smooth hyperboloidal initial data could be answered byanalyzing the developments of standard Cauchy data. The picture of the conformallycompactified Minkowski space and the results on the hyperboloidal initial value problemthen suggest that the field equations decide on the asymptotic smoothness at null infinityin any neighbourhood of space-like infinity. There is again an ambiguity concerning thefall-off behaviour.One has to decide between conflicting requirements. Too much generality can obscureimportant features by irrelevant noise while overly stringent conditions aiming at sharpcontrol of physical concepts may cause a loss of physically relevant input. To make a usefulchoice one needs to understand a reasonably broad spectrum of possibilites.Existence results are usually formulated in terms of suitable function spaces whichencode in a precise way the fall-off behaviour and other properties of the solutions. Inthe following comparisons we shall ignore all these important technicalities and just useorder symbols to indicate the asymptotic behaviour of the fields which is essential to ourdiscussion. For the full precision we refer to the original articles. In the following weconsider smooth initial data sets, i.e. solutions ( ˆ S, ˆ h ab , ˆ χ ab ) to the vacuum constraints,on the manifold ˆ S = R with standard Euclidean coordinates ˆ x a , which are in a suitablesense close to the Minkowski data ( ˆ S, δ ab , Stability of Minkowski space A [15]:
Smooth vacuum Cauchy data ( ˆ S, ˆ h ab , ˆ χ ab ) whichare sufficiently close to the Minkowski data and which are asymptotically flat so that ˆ h ab = δ ab + o ( | ˆ x | − / ) , ˆ χ ab = o ( | ˆ x | − / ) ,develop into solutions to Einstein’s vacuum equations ˆ R µν = 0 , for which all causalgeodesics are complete and whose curvature tensor approaches zero asymptotically in alldirections. A global stability result based on such weak fall-off conditions is a remarkable mathe-matical achievement. One has to pay a price, however. The analysis gives little informationon the precise behaviour of the solution near null infinity. In particular, a concept of radi-ation field is no longer available [16]. Christodoulou and Klainerman obtained their resultunder stronger requirements.
Stability of Minkowski space B [24]:
Smooth vacuum Cauchy data ( ˆ S, ˆ h ab , ˆ χ ab ) whichare sufficiently close to the Minkowski data and which are asymptotically flat so that ˆ h ab = (cid:0) m | ˆ x | − (cid:1) δ ab + o ( | ˆ x | − / ) , ˆ χ ab = o ( | ˆ x | − / ) ,with some constant m > develop into solutions to Einstein’s vacuum equations ˆ R µν = 0 ,for which all causal geodesics are complete and whose curvature tensor approaches zeroasymptotically in all directions. Null infinity is complete. A concept of radiation field canbe defined .Some information about null infinity, such as being approximated by complete nullgeodesics, is obtained, but sharp fall-off statements are missing. The results obtained onthe fall-off behaviour of the conformal Weyl tensor at null infinity are weaker than thoserequired by the Sachs peeling behaviour. If they are sharp the solutions do not admitsmooth conformal extensions. This raises doubts whether there exist asymptotically flatvacuum data at all that develop into solutions which are null geodesically complete and dmit smooth conformal extensions at null infinity [24]. The characterization of such dataclearly requires a detailed analysis of the evolution near space-like infinity. It is reasonableto base the investigation on a choice of data with clean fall-off behaviour at all orders. Existence of Cauchy data with prescribed asymptotic behaviour [36]:
Thereexists a large class of asymptotically flat vacuum Cauchy data on R so that in suitablecoordinates ˆ x a near space-like infinity the data behave as ˆ h ab = (cid:0) m | ˆ x | − (cid:1) δ ab + O ( | ˆ x | − ) , ˆ χ ab = O ( | ˆ x | − ) as | ˆ x | → ∞ ,and admit asymptotic expansions in terms of powers of | ˆ x | − as | ˆ x | → ∞ with smooth,bounded coefficients. In [36] are prescribed seed data possessing this property, where the seed metric isrequired in addition to admit a smooth conformal compactification at space-like infinity(for possible generalizations see [35]). It is then shown that the elliptic equations givecorrection terms with the desired asymptotic expansion if and only if | ˆ x | ˆ χ ab is boundedon R . Otherwise one obtains data which only admit polyhomogeneous expansions. It isinteresting to note that the latter are admitted in [15] but excluded in [24]. They will alsobe excluded in the following discussion .The smooth conformal extension of Minkowski space contains the point i that repre-sents space-like infinity for the space-time as well as for any Cauchy data. If one stipulatesa similar picture for general asymptotically flat vacuum solutions with mass m >
0, onefinds that the data for the conformal field equations induced on Cauchy hypersurfaces arestrongly singular at the point i . In the following it will become clear that a detailed andgeneral analysis of the field equations and their evolution properties is impossible if thestructure near space-like infinity exhibited below is compressed into one point.Surprisingly, the Einstein equations allow us to define a setting which explains why astability result as strong and unrestricted as in the de Sitter case cannot be obtained inthe case λ = 0. The regular finite Cauchy problem [51]:
With asymptotically flat initial dataas above the Cauchy problem for ˆ R µν = 0 is equivalent to an initial value problem forthe general conformal field equations with smooth initial data on a 3-manifold S withboundary I ∼ S . These data develop smoothly on a manifold M diffeomorphic to anopen neighborhood of S ≡ { } × S in R × S which, assuming a suitable ‘time’-coordinate t taking values in the first factor, has I = ] − , × I as a boundary. It is a consequence of the assumptions that the data extend in a suitable conformalscaling smoothly to the boundary I , which represents space-like infinity on the initialslice S . What may look strange, is that the Cauchy problem has been replaced by aninitial-boundary value problem. The boundary is, however, of a very peculiar nature.It is ‘totally characteristic’ in the sense that at points of the boundary the hyperbolicreduced equations contain only differential operators tangential to the boundary. On theboundary the unknowns are thus evolved by inner equations and boundary data cannotbe prescribed. The set I , which is generated in the given gauge by the evolution processfrom I , defines a smooth extension of the physical manifold which represents space-likeinfinity for the solution space-time.The setting has further remarkable features. In the given gauge the conformal factorΩ is an explicitly known function of the coordinates and there are sets J ± = { Ω = 0 , d Ω = } ± with known finite (gauge dependent) coordinate location in the future and the past of S respectively so that Ω > M \ I . If the solution to the conformalfield equations extends to these sets with sufficient smoothness they will in fact define theconformal boundary at null infinity so that J ± = J ± .The sets J ± approach the boundary I transversely at the critical sets I ± = {± }× I ,which can be understood as defining a boundary of I . While the reduced equations willbe hyperbolic along I as well as between J − and J + (as long as the gauge does not breakdown), and even on the latter sets if the solution extends smoothly, their hyperbolicitydegenerates in a specific way at the critical sets. The decision which initial data evolveinto solutions that admit a smooth conformal structure at null infinity takes place preciselyat these sets .Data which are static in a neighborhood of I in S evolve into solutions which are realanalytic and extend with that property across I ± and J ± [54]. If the data are stationarythe setting works as well and one gets a similar result [1]. For more general data the lossof hyperbolicity at I ± can entail, however, a loss of smoothness at these sets. This is a fundamental difference with the hyperboloidal initial value problem. While inthe latter smoothness of the initial data ensures smoothness of the conformal boundarynear the initial slice, this is not the case in the standard Cauchy problem. Here, the fieldequations define a hierarchy of conditions, involving all orders of differentiabilty, which gobeyond simple smoothness requirements on the initial slice.
Denote by u the set of unknowns in the conformal field equations, by r a local coordinatenear I with r = 0 on I and r > M \ I , and by t a ‘time’ coordinate on I with t = 0on I and t = ± I ± . Then the peculiar nature of the boundary I has the consequencethat the functions ∂ pr u , p ≥
1, evolve from the initial data ∂ pr u | I as solutions to linear,hyperbolic transport equations in I . Explicit calculations show that these functions, whichare smooth on I , do in general not extend smoothly to I + but only admit asymptoticexpansion in terms of ( t − k log( t − l as t → I and with exponents k which are increasing with p so that the singularbehaviour gets milder at higher order. A similar behaviour is found at I − . In which way does this affect the smoothness at the sets J ±∗ ? This question has notbeen answered yet. But that it will have an effect is seen if the setting is linearized atMinkowski space. The resulting problem is then controlled by the Bianchi equation forthe linearized rescaled conformal Weyl tensor W ′ µ νρλ . It turns out that if the initial dataare such that some quantity ∂ pr W ′ µ νρλ on I develops a logarithmic term at I + then thelogarithmic singularity spreads along the null generators of null infinity [53]. The situationcan hardly be expected to improve in the non-linear case. Which conditions on the initial data on S ensure that the functions ∂ pr u extend smoothlyto I ± ? This question is technically quite difficult. A first family of such conditions havebeen derived in [51], which concentrates on the time reflection symmetric case so that thephysical data on S \ I are given by the 3-metric ˆ h ab . Let B ab denote he (dualized) Cottontensor of the rescaled 3-metric h ab . It has been shown in [51] that the symmetrized h -covariant derivatives of B ab necessarily vanish at all orders at space-like infinity if the ∂ pr u extend smoothly to I ± . Since the sequence of these conditions is conformally invariantthey define a condition on the asymptotic conformal structure of ˆ h ab . If the metric ˆ h ab s conformally flat near space-like infinity these conditions are satisfied. In that case themetric is determined near space-like infinity only the by scaling factor of the flat metric,which is restricted by the Hamiltonian constraint. It has been shown that even in thatcase there might arise obstructions to the smoothness of the ∂ pr u at I ± [100].In the time reflection symmetric case the complete set of necessary conditions is notknown yet and the situation is even less clear in general. In the time reflection symmetriccase there are, however, strong indications which suggest that a necessary and sufficientcondition for the smoothness of the ∂ pr u , p ∈ N , is that the datum ˆ h ab is asymptoticallystatic at space-like infinity . We refer to [58] for a detailed discussion. The discussion above raises questions about the possibilities to construct solutions to thevacuum constraints which satisfy at space-like infinity additional asymptotic conditionsat all orders. Cutler and Wald considered the even more difficult problem of constructingdata with a complete metric on R that are exactly Schwarzschild in a neighbourhoodof space-like infinity. They managed to construct for the Einstein-Maxwell equationsa smooth family of such data which includes the Minkowski data [34]. This allowedthem to show for the first time the existence non-trivial solutions to the Einstein-Maxwellequations which are null geodesically complete and admit smooth conformal extensions atnull infinity with complete J ± and regular points i ± . In fact, the control on the asymptoticbehaviour of the solutions in the static region near space-like infinity allows them toconclude that the solutions contain hyperboloidal hypersurfaces with induced data thatcan be arbitrarily close to Minkowskian hyperboloidal data. Invoking the strong stabilityresult on Minkowskian hyperboloidal developments then gives the result.The data so constructed look highly special but the work initiated by Corvino showsthat there exist in fact large classes of data with specialized ends. Complete vacuum data with Schwarzschild ends. [30]:
A given time reflectionsymmetric, asymptotically flat vacuum data set can be deformed outside some prescribedcompact domain so as to obtain vacuum data which are exactly Schwarzschild in someneighbourhood of spatial infinity.
This has been generalized by Corvino and Schoen ([32]) and Chru´sciel and Delay([27]), who obtained vacuum data which agree on prescribed compact sets with givenasymptotically flat data and which are stationary near spatial infinity. (It follows inparticular that in contrast to static or stationary data, which are determined completelyby their multipoles at space-like infinity, such quantities are of little significance in thecontext of more general data, where they hardly carry relevant information about thestructure of the data in the interior.) This work also led to generalizations of the Cutler -Wald idea.
Vacuum solutions with smooth asymptotic structure [31], [26]:
There exist largeclasses of vacuum solutions that are null geodesically complete and static or stationarynear space-like infinity, which admit smooth and complete conformal boundaries J ± atnull infinity and regular points i ± at future and past time-like infinity. t appears that we are getting close to obtaining conditions which are necessary andsufficient for asymptotically flat solutions to admit a smooth conformal extension in aneighborhood of the critical set or, with suitable smallness assumptions on the data, alongcomplete null infinities. But closing the gap still requires some complicated analysis. Thereare good arguments why it would be worth the effort.In the time reflection symmetric case it has been shown in [62] that, given enoughsmoothness, the completion of the picture reduces many questions about the asymptoticsand the associated physical concepts which have been discussed in the literature for along time to straightforward (though possibly lengthy) calculations. In particular, thesettings of [80], [81] are related to the setting of [51] near space-like infinity, it is shownhow quantities of physical interest like the Bondi-mass or the NP conserved quantitiesnear the critical set on J + can be related explicitly to quantities on the initial slice nearspatial infinity like the ADM-mass and higher order expansion coefficients etc. The settingalso allows us to single out in a unique way the Poincare group as a subgroup of the BMS-group. It would be interesting to understand the weakest smoothness conditions underwhich this can still be done and under which the BMS-group can still be defined. Somegeneralizations of these results are discussed in [101].The precise identification of the data content which needs to be supressed to achievea certain amount of asymptotic smoothness may help understand the role and meaning ofthat content (if there is any). Referring to it as ‘radiation near space-like infinity’ explainsvery little. It needs to be decided whether there exist physical systems of interest whichrequire this content for their adequate modelling.The setting proposed in [51] offers the possibility to calculate numerically entire asymp-totically flat solutions, including their asymptotics and radiation fields, on finite grids.First steps to develop adequate numerical methods are being taken in [13] and [39]. Thework in [13] even includes a discussion of the logarithmic terms exhibited in [53] and itsuggests that the difficulties arising from the loss of hyperbolicity of the equation at thecritical sets can be compensated by the information supplied by the transport equationson the cylinder I at space-like infinity.An important open problem is the numerical calculation of Corvino-type data. In thetime reflection symmetric case there has been proposed a general method how to obtainsuch data by solving PDE systems [7]. But this needs further analytical and numericalstudy. If such data can be provided numerically, it will help analyzing the effect of Corvino-type deformations on the developments of solutions in time, a subject of greatest interestfrom our point of view. In the conformal picture the solutions seem to be modified by suchdeformations only in some ‘thin’ neighbourhood of J − ∪ i ∪ J + . But to what extent dothey affect the structure of the radiation field in the causal future of that part of the initialhypersurface where the essential physical processes take place ? What is the differencein the radiation fields resulting from two different Corvino-type deformations ? If theradiation field on J + in the causal future of a merger process would hardly be affected bydeformations that are performed sufficiently close to space-like infinity, there would be noreason to worry about the data asymptotics. If it would be affected in an essential way,however, choosing the ‘correct’ asymptotics of the Cauchy data would become a delicatematter in any case. Anti-de Sitter-type solutions.
The AdS/CFT correspondence proposed by Maldacena [79] triggered an enormous interestin solutions to Einstein’s field equations with cosmological constant λ <
0. There exists,however, no observational evidence which would motivate the choice λ < λ < anti-de Sitter covering space , short AdS, which is given byˆ M = R × R , ˆ g = − cosh r dt + dr + sinh r h S ,where r ≥ R . It solves (3.1) with λ = −
3. A clear pictureof its global and asymptotic structure is obtained by combining the coordinate transfor-mation ρ = 2 arctan( e r ) − π with a rescaling by the conformal factor Ω = cosh − r = cos ρ to obtain the conformal representation g = Ω ˆ g = − dt + dρ + sin ρ h S , ≤ ρ < π .The metric g and the conformal factor Ω extend smoothly as ρ → π and then live on themanifold M = R × S / S . The boundary J = { ρ = π } ∼ R × S attached by this process to ˆ M is time-like for g and its points can be understood as endpoints of the space-like and null geodesics of ˆ g so that J represents space-like and null infinity for AdS.Any solution of Einstein’s equations (3.1) with λ < J which represents space-like and null infinity will be referred to as an AdS-type space-time.
Two basic featuresdistinguish the global causal resp. conformal structure of AdS from that of de Sitter- orMinkowski-space. The obvious one is the fact that AdS (in fact any AdS-type space-time)fails to be globally hyperbolic. A less obvious one will be discussed in the context of theAdS stability problem, where it will become as important as the presence of a time-likeboundary.If more general AdS-type solutions to (3.1) are to be constructed by solving PDEproblems, one needs to analyze the freedom to prescribe boundary data. The formalexpansions at the the conformal boundary found in the literature amount to analyzingCauchy problems with data on J . This may be of interest in some contexts but will notsuffice for us. Cauchy problems for hyperbolic equations with data on time-like hypersur-faces are known to be ill-posed. Moreover, if the formal expansions can be shown to definereal analytic solutions in some neighborhood of J (see [73]) the analytic extension intothe interior will, more likely than not, end in a singularity and it remains unclear whetherthese solutions admit any extension at all which is regular in the sense that it contains acomplete (in the induced metric) space-like hypersurface that intersects J in a space-likesurface. .1 An existence result The natural problem to consider is the initial-boundary value problem with boundary dataprescribed on J and Cauchy data given on a space-like slice that extends to the boundary. Existence of AdS-type soutions local in time [49]:
Suppose λ is a negative numberand ( ˆ S, ˆ h ab , ˆ χ ab ) is a smooth Cauchy data set for (3.1) with ˆ κ = ˆ h ab ˆ χ ab = const. = 0 sothat ˆ S is an orientable -manifold and ( ˆ S, ˆ h ab ) is a complete Riemannian space. Let thesedata admit a smooth conformal completion ˆ S → S = ˆ S ∪ Σ , ˆ h ab → h ab = Ω ˆ h ab , ˆ χ ab → χ ab = Ω ( ˆ χ ab − ˆ κ ˆ h ab ) ,where ( S, h ab ) is a Riemannian space with compact boundary ∂S = Σ , Ω a smooth definingfunction of Σ with Ω > on ˆ S , and W µ νλρ = Ω − ˆ C µ νλρ extends smoothly to Σ on S where ˆ C µ νλρ denotes then conformal Weyl tensor determined by the metric ˆ h ab and thesecond fundamental form ˆ χ ab .Consider the boundary J = R × ∂S of M = R × S and identify S with { } × S ⊂ M and Σ with { } × ∂S = S ∩ J . Let on J be given a smooth 3-dimensional Lorentzianconformal structure which satisfies in an adapted gauge together with the Cauchy data thecorner conditions implied on Σ by the conformal field equations, where it is assumed thatthe normals to S are tangent to J on Σ .Then there exists for some t o > on the set ˆ W = ] − t o , t o [ × ˆ S ⊂ R × ˆ S ⊂ M a uniquesolution ˆ g to (3.1) which admits with some smooth boundary defining function Ω on M (that extends the function Ω on S above) a smooth conformal extension ˆ W → W =] − t o , t o [ × S, ˆ g → g = Ω ˆ g ,that induces (up to a conformal diffeomorphism) on S and J o =] − t o , t o [ × ∂S the givenconformal data. This is the first and still the only well-posed initial boundary value problem for Ein-stein’s field equations which is general in the sense that no symmetries are required andwhich admits a covariant formulation [57]. It supports the view that the setting of asymp-totic simplicity is natural for Einstein’s equations. The idea of a smooth conformal ex-tension is basic for the formulation of the PDE problem which leads to this result. Allpossible AdS-type solutions local in time (with | t o | sufficiently small) are obtained. In thefollowing we point out the particular features of AdS-type vacuum solutions which allowone to obtain this result, and comment on a particular choice of boundary condition.In [49] has been observed a correspondence between Cauchy data sets as requiredabove and hyperboloidal Cauchy data for Einstein’s equations (3.1) with λ = 0. This hasbeen worked out in detail in [71]. It is not known whether the existence result local in timecan be extended to hyperboloidal data which only admit poly-homogeneous expansions atinfinity. Most likely the non-smoothness will spread into the physical manifold.Because the data on J are not subject to constraints the discussion of the boundaryconditions looks simple. The covariant formulation rests, however, on specific properties ofAdS-type solutions and is obtained in two steps. In the first step boundary conditions/dataare considered which relate directly to a well-posed PDE problem. AdS-type solutionshave the special feature that the second fundamental form κ ab induced on J can be madeto vanish on J in a suitable conformal gauge. As a consequence there exists a certain eometric gauge in which the location of the boundary J is known, the conformal factorΩ with Ω = 0 and d Ω = 0 on J is known explicitly, and on J the gauge is determined interms of the inner metric induced on J . In this gauge the conformal field equations implyhyperbolic evolution equations which assume in Newman-Penrose notation the form ∂ τ u = F ( u, ψ, x µ ), (1 + A ) ∂ τ ψ + A α ∂ α ψ = G ( u, ψ, x µ ),where τ = x is a time coordinate, x α are spatial coordinates, and the unknown u com-prises besides the pseudo-orthonormal frame coefficients ( e µ k ) k =0 ,.., also the connectioncoefficients and the Schouten tensor L jk = ( R jk − R g jk ) of the metric g with respectto this frame. The matrices A µ depend on the frame coefficients and the coordinates and ψ = ( ψ , . . . ψ ) represents the essential components of the symmetric spinor field ψ ABCD that corresponds to the tensor field W i jkl . The frame is chosen so that the future directedtime-like vector field e + e is tangential to J and the space-like vector field e − e isnormal to J and inward pointing. Then e , e are tangential to J , e is inward and e is outward pointing on J . General results on initial-boundary value problems then giveboundary conditions of the form ψ − a ψ − c ¯ ψ = d = d + i d , | a | + | c | ≤ J ,where the smooth complex-valued function d on J denotes the essential free boundarydatum and the smooth complex-valued functions a and c on J can be chosen freely withinthe indicated restrictions.Given the conformal Cauchy data on S in the gauge used above, the formal expansionof the unknowns u , ψ in terms of the coordinate τ is determined at all orders on S , inparticular on Σ. On J let be given smooth functions a and c as above. Using their formalexpansion in terms of τ on Σ, the formal expansion of the term on the left hand side of ofthe boundary condition on Σ is obtained. The corner conditions consist in the requirementthat this formal expansion coincides on Σ with the formal expansion of the free boundarydatum d . Borel’s theorem guarantees that there always exist smooth functions d on J which satisfy this requirement. Away from Σ they are essentially arbitrary.With Cauchy data as stated in the theorem and boundary conditions as above where d satisfies the corner conditions one obtains a well-posed initial-boundary value problemwhich preserves the constraints and the gauge conditions . This implies the existence anduniqueness of smooth solutions on a domain as indicated in the theorem.The formulation so obtained has the drawback that the boundary condition dependsimplicitly on the choice of of the time-like vector field n e + e . This reflects a generaldifficulty with initial boundary-value problems for Einstein’s equations [57]. In the case ofAdS-type solutions it can be overcome by using the observation that such solutions mustsatisfy the relation w ∗ ab = p / | λ | B ab on J , where w ∗ ab denotes the J -magnetic part of W i jkl , obtained be contracting the right dual of W i jkl twice with the inward pointingunit normal of J , and B ab is the (dualized) Cotton tensor of the metric k ab induced on J .With the particular choice ψ − ¯ ψ = d + i d of the boundary condition this allows one toexpress certain components of the Cotton tensor in terms of the real-valued functions d , d . If these components are given, the structural equations of the normal conformal Cartanconnection defined by the conformal structure on J provides in our gauge a hyperbolicdifferential system on J which determines, with the data given on Σ, the inner metric k ab uniquely in terms of d and d . Conversely, the function ψ − ¯ ψ , whence the free data d , an be calculated in the given gauge uniquely from the inner metric k ab on J .Irrespective of the functions a and c , any boundary condition with d = 0 on J canbe considered as a reflective boundary condition. With an additional requirement onΣ = S ∩ J it implies the relation B ab [ k ] = 0 of local conformal flatness on J . This willbe referred to as the reflective boundary condition . It appears so natural to many authorsthat they refer to it as the AdS-boundary condition . There seems to be no particularreason, however, why it should be preferred. Moreover, when conditions are imposed onthe boundary data, the clean separation between the evolution problem and the problemof the constraints we are used to from the standard Cauchy problem, is not maintained anylonger. If reflective boundary conditions are imposed on J , consistency with the cornerconditions requires that the Cauchy data on S satisfy besides the underdetermined ellipticconstraint equations on S an infinite number of differential conditions at Σ [60].The results obtained in four space-time dimensions in the cases λ ≥ λ <
0. This requires, however, a new study of the problem ofconstraint propagation and different arguments to obtain covariant boundary conditions.
Given the local existence result, there arises the question whether the solution can becontrolled globally in time or, more modestly, whether AdS is non-linearly stable. It turnsout that this stability problem is more challenging than the corresponding ones in thecases λ ≥
0. There is the technical difficulty of controlling the evolution for an arbitrarylength of time, but it is as already problematic to say what should be meant by ‘stability’in the present context.Bizo´n and Rostworowski [18] recently presented a first study of the AdS-stabilityproblem by using mainly numerical methods. Their work raises some extremely inter-esting questions concerning solutions to Einstein’s equations (3.1) with λ < J at space-like and null infinity. They analyse thespherically symmetric Einstein-massless-scalar field system with homogeneous Dirichletasymptotics and Gaussian type initial data and observe the formation of trapped surfacesfor (numerically) arbitrarily small initial data. They supply numerical evidence that thedevelopment of trapped surfaces results from an energy transfer from low to high frequencymodes. They perform a perturbative analysis, which points into the same direction butalso exhibits small one-mode initial data which develop into forever smooth solutions. It isnot so clear whether there are also small neighborhoods of these data which develop into(numerically) forever smooth solutions. The results led them to suggest: AdS is unstableagainst the formation of black holes for a large class of arbitrarily small perturbations .While it has been formulated in the context of a particular model this conjecture mayeasily be understood as applying more generally. In the following we wish to point outthat the situation considered here is extremely special and may obscure the view onto ascenery which is much richer in possibilities. he setting considered in [18], mainly motivated by questions of technical feasibility,involves reflecting boundary conditions. These are convenient because they lead to cleaninitial boundary value problems but they are also very restrictive from a physical pointof view. Such systems cannot interact with an ambient universe and thus certainly donot represent observable objects as suggested by some of the names given to them in theliterature.For the stability problem a second feature of the global conformal structure of AdSis of equal importance as the existence of the time-like boundary. Contrary to what isoccasionally suggested in the literature and what does hold in the cases λ ≥
0, AdSdoes not admit conformal rescalings that put past or future time-like infinity, representedby points or more general sets, in a finite coordinate location and extend smoothly . Inthis sense AdS is always of infinite length in time, even in ‘conformal time’ (this can besupported by rigorous arguments).Some of the observations in [18] may then not be too surprising. By the non-linearityof the Einstein equations the spherically symmetric field perturbation can be expectedto experience a focussing effect when it travels through the center and the boundarycondition leads to a reflection and refocussing of the perturbation at the boundary. Becauseconformal time is potentially unlimited, this process can repeat itself arbitrarily often andthe focussing effects, however tiny at each step, may eventually add up to produce acollapse. The occurrence of islands of stability then appears in fact more surprising thanthe tendency to develop a collapse.The relatives sizes of the ocean of data of instability and the tiny islands of stabilitymay change drastically if the full freedom to impose boundary conditions is taken intoaccount. As long as physical considerations do not tell us what kind of objects should berepresented by solutions with λ < λ > J − and outgoing information can be controlled on J + . If the information entering at J − istrivial, the solution is trivial. Potential difficulties arising from spatial compactness arecompensated by the exponential expansion of the solutions.If initial data are prescribed in the case λ = 0 on a Cauchy hypersurface, the require-ment that they be close to Minkowskian data leaves ambiguities near space-like infinitywhich may affect the smoothness at J ±∗ . But under fairly general assumptions we findagain that there is a clear separation between incoming information, specified in terms ofthe radiation field on J − , and outgoing information, registered by the radiation field on J + . If the incoming information is trivial the solution is trivial.In the AdS-type case λ < J . Refusing information toenter the space-time through J by imposing reflecting boundary conditions does not leadto trivial solutions. There do exist non-trivial vacuum initial data consistent with theseboundary conditions [28]. Moreover, under more general assumptions on the boundaryconditions/data it is far from obvious how to control in- and outgoing radiation separatelyto achieve a balance along J which avoids the development of a gravitational collapse. eferences [1] A. E. Ace˜na, J. A. Valiente Kroon. Conformal extensions for stationary space-times. Class. Quantum Grav.
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