core.ac.uk · THE GLOBAL EXISTENCE PR OBLEM IN GENERAL RELA TIVITY LARS ANDERSSON 1 Abstra ct. W e...

THE GLOBAL EXISTENCE PROBLEM

IN GENERAL RELATIVITY

Lars ANDERSSON

H

Institut des Hautes �Etudes Scienti�ques

35, route de Chartres

91440 { Bures-sur-Yvette (France)

Avril 2000

IHES/M/00/18

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2000

-198

01/

04/

2000

THE GLOBAL EXISTENCE PROBLEM IN GENERAL

RELATIVITY

LARS ANDERSSON1

Abstract. We survey some known facts and open questions concerningthe global properties of 3+1 dimensional space{times containing a com-pact Cauchy surface. We consider space{times with an `{dimensionalLie algebra of space{like Killing �elds. For each ` � 3, we give somebasic results and conjectures on global existence and cosmic censorship.For the case of the 3+1 dimensional Einstein equations without symme-tries, a new small data global existence result is announced.

Contents

1. Introduction 12. The Einstein equations 33. Bianchi 144. U(1)�U(1) 215. U(1) 266. 3+1 297. Concluding remarks 40Appendix A. Basic Causality concepts 41References 42

1. Introduction

In this review, we will describe some results and conjectures about theglobal structure of solutions to the Einstein equations in 3+1 dimensions.We consider space{times ( �M; �g) with an `{dimensional Lie algebra of space{like Killing �elds. We may say that such space{times have a (local) isometrygroup G of dimension ` with the action of G generated by space{like Killing�elds.For each value ` � 3 of the dimension of the isometry group, we state the

reduced �eld equations as well as attempt to give an overview of the mostimportant results and conjectures. We will concentrate on the vacuum case.

Date: November 9, 1999.1Supported in part by the Swedish Natural Sciences Research Council (SNSRC), con-

tract no. F-FU 4873-307 and ITP, Santa Barbara. Based on a talk given at the ArthurL. Besse Table Ronde de G�eom�etrie Pseudo-Riemannienne Globale, Nancy, June 1998.

1

2 L. ANDERSSON

In section 2, we present the Einstein equations and give the 3+1 decompo-sition into constraint and evolution equations, cf. subsection 2.2. Due to thegauge freedom in the Einstein equations, questions on the global propertiesof solutions to the Einstein equations must be posed carefully. We intro-duce the notions of vacuum extension and maximal Cauchy development andstate the uniqueness theorem of Choquet{Bruhat and Geroch, for maximalvacuum Cauchy developments. We also collect here a few basic facts aboutKilling �elds on globally hyperbolic space{times. In section 2.1 a version ofthe cosmic censorship conjecture appropriate for vacuum space{times, withcompact Cauchy surface, is stated and in subsection 2.3 we discuss a fewgauge conditions which may be of use for the global existence problem forthe Einstein equations. Section 2 is ended by a discussion of a few PDEaspects of the Einstein equations which are relevant for the topic of thispaper, cf. subsection 2.4.In the cases ` = 3 (Bianchi, cf. section 3) and a special case of ` =

2 (polarized Gowdy, cf. section 4), the global behavior of the Einsteinequations is well understood, both with regard to global existence of theevolution equations and the cosmic censorship problem. For the general` = 2 case (local U(1)�U(1) symmetry, cf. section 4), there are only partialresults on the global existence problem and the cosmic censorship problemremains open, although conjectures supported by numerical evidence give agood indication of what the correct picture is.In the cases ` = 1 (U(1) symmetry, cf. section 5) and ` = 0 (no symmetry,

i.e. full 3+1 dimensional Einstein equations, cf. section 6), the large dataglobal existence and cosmic censorship problems are open. In the U(1) caseconjectures supported by numerical evidence give a good idea of the genericbehavior, and there is a recent small data semi{global existence result forthe expanding direction due to Choquet{Bruhat and Moncrief [34].For 3+1 Einstein gravity without symmetries the only global existence

results are theorem on nonlinear stability of Minkowski space of Christo-doulou and Klainerman, and the semi{global existence theorem of Friedrichfor the hyperboloidal initial value proble. Both of these are small data re-sults, and deal with the asymptotically at case, see section 6 for discussion.In section 6 we also announce a new small data semi{global existence resultin the expanding direction, for a class of spatially compact 3+1 space{times(joint work with Vince Moncrief) [7].Due to the high degree of complexity of the numerical solution of the

Einstein equations in 3+1 dimensions it is too early to draw any conclusionsrelevant to the asymptotic behavior at the singularity for the full 3+1 di-mensional Einstein equations, from the numerical studies being performed.However, an attractive scenario is given by the so called BKL picture, cf.section 7 for some remarks and references.The Einstein equations are derived from a variational principle, and can

be formulated as a Hamiltonian system (or time{dependent Hamiltonian sys-tem, depending on the gauge), and therefore the Hamiltonian aspect of the

THE GLOBAL EXISTENCE PROBLEM IN GENERAL RELATIVITY 3

dynamics should not be ignored, see eg. the work by Fischer and Moncriefon the Hamiltonian reduction of the Einstein equations, [52] and referencestherein. In fact, the Hamiltonian point of view on the Einstein equations hasplayed a vital role as motivation and guide in the development of the resultsdiscussed here. The notion that the Einstein evolution equations in terms ofcanonical variables can be viewed as the geodesic spray for a metric on thephase space (deWitt metric) modi�ed by a curvature potential, is naturalfrom the Hamiltonian point of view, and this picture has been relevant tothe development of ideas on asymptotic velocity dominance, see sections 4and 5.In this review, however, we will concentrate exclusively on the di�erential

geometric and analytical point of view. Even with this restriction, manyimportant topics are left out and we make no claim of complete coverage.See also [112] and [92] for recent, related surveys.Acknowledgements: I am grateful to Vince Moncrief for numerous con-versations on the topics covered here and for detailed comments on an earlyversion. Thanks are due to H�akan Andr�easson, Piotr Chru�sciel, Jim Isenbergand Alan Rendall and others for helpful comments. I am happy to acknowl-edge the hospitality and support of the Institute of Theoretical Physics,UCSB, where part of the writing was done, and Institut des Hautes �EtudesScienti�ques where the paper was �nished.

2. The Einstein equations

Let ( �M; �g) be a smooth 4-dimensional Lorentz manifold1 of signature�+++. The Lorentzian metric �g de�nes a causal structure on �M . Forthe convenience of the reader we give a quick review of the basic causalityconcepts in appendix A, see [128, Chapter 8],[21, 68, 106] for details.We will here consider only the vacuum case, i.e. the case when �g is Ricci

at,

�Rab = 0: (2.1)

LetM � �M be a space{like hypersurface, i.e. a hypersurface with time{likenormal T . We let ei be a frame on M and use indices i; j; k for the framecomponents. Let g; k be the induced metric and second fundamental form ofM � �M , where kij = h �riej ; T i. Let t be a time function on a neighborhoodof M . Then we can introduce local coordinates (t; xi; i = 1; 2; 3) on �M sothat xi are coordinates on the level setsMt of t. This de�nes the coordinatevector �eld @t of t. Alternatively we can letMt = i(t;M) where i : R�M !�M is a 1-parameter family of imbeddings of an abstract 3{manifoldM . Then@t = i�d=dt where d=dt is the coordinate derivative on R.De�ne the Lapse N and Shift X w.r.t. t by @t = NT +X . Assume that

T is future oriented so that N > 0. A 3+1 split of equation (2.1) gives the

1We denote the covariant derivative and curvature tensors associated to ( �M; �g) by�r; �Rabcd etc. All manifolds are assumed to be Haussdor�, second countable and C1, andall �elds are assumed to be C1 unless otherwise stated.

4 L. ANDERSSON

Einstein vacuum constraint equations

R� jkj2 + (trk)2 = 0; (2.2a)

ritrk �rjkij = 0; (2.2b)

and the Einstein vacuum evolution equations

L@tgij = �2Nkij + LXgij; (2.3a)

L@tkij = �rirjN +N(Rij + trkkij � 2kimkmj ) + LXkij : (2.3b)

where L@t denotes Lie{derivative w.r.t. @t. In case [@t; ei] = 0, L@t can bereplaced by @t.A triple (M; g; k) consisting of a 3{manifold M , a Riemannian metric g

on M and a symmetric covariant 2{tensor k is a vacuum data set for theEinstein equations if it solves (2.2).

De�nition 2.1. Let (M; g; k) be a vacuum data set.

1. A vacuum space{time ( �M; �g) is called a vacuum extension of (M; g; k)if there there is an imbedding i with time{like normal T of (M; g; k)into ( �M; �g) so that g = i��g and k = �i�( �rT ).

2. A globally hyperbolic vacuum space{time ( �M; �g) is called a vacuumCauchy development of (M; g; k) if there is an imbedding i withtime{like normal T of (M; g; k) into ( �M; �g) so that i(M) is a Cauchysurface in ( �M; �g), g = i��g and k = �i�( �rT ). If ( �M; �g) is maximalin the class of vacuum Cauchy developments of (M; g; k) then ( �M; �g)is called the maximal vacuum Cauchy development (MVCD) of(M; g; k). In the following, when convenient, we will identify M withi(M).

The Einstein vacuum equations are not hyperbolic in any standard sensedue to the coordinate invariance (\general covariance") of the equation�Rab = 0. Nevertheless, the Cauchy problem for the Einstein vacuum equa-tion is well posed in the following sense.

Theorem 2.2 (Choquet{Bruhat and Geroch [33]). Let (M; g; k) be a vac-uum data set. Then there is a unique, up to isometry, maximal vacuumCauchy development (MVCD) of (M; g; k). If � : M ! M is a di�eomor-phism, the MVCD of (M;��g; ��k) is isometric to the MVCD of (M; g; k).

The proof relies on the fact that in space{time harmonic coordinates,��gx

� = 0, the Ricci tensor is of the form

�R(h)�� = �1

2��g�g�� + S��[�g; @�g]; (2.4)

where ��g is the scalar wave operator in ( �M; �g). Hence the Einstein vacuumequations in space{time harmonic coordinates is a quasi{linear hyperbolic

system and therefore the Cauchy problem2 for �R(h)�� = 0 is well posed and

2Note that the Einstein equations in space{time harmonic gauge should be viewed asan evolution equation for (g; k;N;X).


standard results give local existence. One proves that if the constraints andgauge conditions are satis�ed initially, they are preserved by the evolution.This together with a Zorn's lemma argument gives the existence of a MVCD.Uniqueness is proved using the �eld equations to get a contradiction to theHaussdor� property, given a pair of non{isometric vacuum Cauchy develop-ments, which are both maximal w.r.t. the natural partial ordering on theclass of Cauchy developments.We end this subsection with by stating a few facts about Killing �elds.

Proposition 2.3 ([53]). Let (M; g; k) be a vacuum data set with MVCD( �M; �g). Let Y be a Killing �eld on ( �M; �g) and let Y = Y?T + Yk be thesplitting of Y into its perpendicular and tangential parts atM . Then (Y?; Yk)satisfy the conditions

1. Y? = 0, LYkg = 0, LYkk = 0, in case g is non{ at or k 6= 0.2. Y? is constant and LYkg = 0 if g is at and k = 0.

On the other hand, given Y?; Yk on M satisfying conditions 1, 2 above, there

is a unique Killing �eld Y on �M , with Y = Y?T + Yk on M .

A space{time ( �M; �g) is said to satisfy the time{like convergence con-dition (or strong energy condition) if

�RabVaV b � 0; for all V with �gabV

aV b � 0: (2.5)

Globally hyperbolic space{times with compact Cauchy surface and satisfyingthe time{like convergence condition are often called \cosmological space{times" in the literature, following [20]. Here we will use the term spatiallycompact to refer to the existence of a compact Cauchy surface.A spacelike hypersurface (M; g) in ( �M; �g) has constant mean curvature if

ritrgk = 0, cf. subsection 2.3 below.

Proposition 2.4. Let ( �M; �g) be a globally hyperbolic space{time.

1. Assume that ( �M; �g) satis�es the time{like convergence condition andcontains a compact Cauchy surface M with constant mean curvature.Then either ( �M; �g) is a metric product M �R or any Killing �eld Yon ( �M; �g) is tangent to M . In particular, if ( �M; �g) is vacuum and hasa nonzero Killing not tangent to M , then ( �M; �g) is at.

2. Assume a compact group G acts by isometries on ( �M; �g). Then the ac-tion of G is generated by space{like Killing �elds and ( �M; �g) is foliatedby Cauchy surfaces invariant under the action of G.

3. Assume that ( �M; �g) is 3+1 dimensional. Let M be a Cauchy surfacein �M , let Y be a Killing �eld on �M and assume Y is strictly spacelike,�g(Y; Y ) > 0, on M . Then Y is strictly spacelike on �M .

Proof. Point 1 is a well known consequence of the uniqueness result forconstant mean curvature hypersurfaces of Brill and Flaherty [31], cf. [99].Point 2 is essentially [24, Lemma 1.1]. For the proof, note that as G iscompact we can construct a G invariant time function on �M by averagingany global time function t on �M w.r.t. the G action, cf. the proof of [24,

6 L. ANDERSSON

Lemma 1.1]. The level sets of the averaged time function are Cauchy surfacesand are invariant under the action of G. The result follows.The following argument for point 3 is due to Alan Rendall3. Let N =

fp 2 �M : �g(Y; Y ) = 0g and assume for a contradiction N is nonempty.Choose p 2 N and a time function t on �M so that t(M) = 0 and t(p) > 0.Let A denote the intersection of the past of p with the future of M and lett1 = infft(q) : q 2 A\Ng. The set N is closed and by global hyperbolicity Ais compact and hence t1 > 0 and there is a q 2 A\N with t(q) = t1. If Y (q)is nonzero and null, then using the equation Y ara(Y

bYb) = 0 which holdssince Y is Killing, gives a null curve of points in N where Y is null. Followingthis into the past, shows that there is a q 2 N with t(q) < t1, which givesa contradiction. In case Y (q) = 0, the linearization of Y acts by isometrieson Tq �M , and as the sphere of null directions in Tq �M is two dimensionalit leaves a null direction �xed. Using the exponential map shows that theaction of Y near q leaves a null geodesic invariant, along which Y must bezero or null. This leads to a contradiction as above.

2.1. Cosmic Censorship. Theorem 2.2 proves uniqueness of the MVCDof a given data set (M; g; k). However, examples show that the MVCD mayfail to be maximal in the class of all vacuum extensions, i.e. there existexamples of vacuum data sets (M; g; k) with vacuum extensions ( �M; �g) suchthat the MVCD of (M; g; k) is a strict subset of ( �M; �g).

Example 2.5. Consider the n+ 1{dimensional Minkowski space Rn;1 withmetric � = �dt2 + (dx1)2 + � � �+ (dxn)2, let I+(f0g) be the interior of thefuture light cone. I+(f0g) is globally hyperbolic with the hyperboloids asCauchy surfaces, and with the mantle of the light cone as Cauchy horizon.Let � be a cocompact discrete subgroup of the Lorentz group SO(n; 1). Thenthe quotient space �M = �nI+(f0g) is a globally hyperbolic, spatially compactspace{time. By choosing � to be the Lorentzian distance from the origin,we get �g = �d�2 + �2 where is the standard hyperbolic metric on thecompact quotient M = �nHn. ( �M; �g) is the MVCD of the vacuum data set(M; ;� ).In case n = 1, H1 = R, and a fundamental domain for � can be found

which intersects the null boundary of I+(f0g) in an open interval. Thereforeif n = 1, there is a nontrivial extension of �M , which is still at, but whichfails to be globally hyperbolic, cf. �gure 1. This space{time is known as theMisner universe. The maximal extension is unique in this case.If n > 1, the ergodicity of the geodesic ow on (M; ) can be shown to

prevent the existence of a space{time extending ( �M; �g) [84].Higher dimensional examples of at globally hyperbolic spatially compact

space{times which admit nontrivial non{globally hyperbolic extensions canbe constructed by taking products of the n = 1 Misner universe with the attorus.

3private communication, 1999


Figure 1. The 1+1 dimensional Misner universe, showinga few orbits of SO(1; 1) (dashed lines) and orbits of a dis-crete subgroup � � SO(1; 1) (circles). The boundary of afundamental domain for � is plotted with solid lines.

The maximal vacuum extension may be non{unique, as is shown by theTaub-NUT example, cf. [42]. If the MVCD of a vacuum data set is notthe maximal vacuum extension, any extension of it must fail to satisfy theintuitively reasonable causality requirement of global hyperbolicity.According to physical intuition, causality violations should be rare. This

leads to the idea of cosmic censorship, essentially due to Penrose, see [107] fordiscussion. One way of stating this, relevant to the class of space{times weare concerned with here, is the following form of the strong cosmic censorshipconjecture.

Conjecture 1 (Strong Cosmic Censorship). Let M be a compact manifoldof dimension 3. Then for generic vacuum data sets (M; g; k), the maximalvacuum Cauchy development of (M; g; k) is equal to the maximal vacuumextension of (M; g; k).

In the case of asymptotically at space{times (describing isolated sys-tems in general relativity), the so{called weak cosmic censorship conjecturestates that naked singularities (i.e. singularities which can be seen by anobserver at in�nity) should not occur generically, see the review paper byWald [129] for a discussion of the status of the weak cosmic censorship con-jecture. Recent work of Christodoulou, see [38] and references therein, seealso the discussion in [129, x5], establishes weak cosmic censorship in theclass of spherically symmetric Einstein{scalar �eld space{times, but alsogives examples of initial data such that the Cauchy development has a nakedsingularity. For earlier surveys on the strong cosmic censorship conjecture,see [77] and [40].The Penrose inequality, giving a lower bound on the ADM mass in terms

of the area of a horizon in black hole space{times, was derived by a heuristicargument assuming the validity of the weak cosmic censorship conjecture.

8 L. ANDERSSON

The recent proof of the Riemannian version of the Penrose inequality byHuisken and Ilmanen [73, 74], gives indirect support for the conjecture.Let ( �M; �g) be a space{time and let M � �M be a space{like hypersurface.

The Cauchy horizon H(M) is the boundary of the domain of dependenceD(M), cf. Appendix A for de�nition. If M is compact without boundary,then every point of H+(M) lies on a past inextendible null geodesic andevery point of H�(M) lies on a future inextendible null geodesic, whereH+(M) and H�(M) are the future and past components of H(M), respec-tively.Let ( �M; �g) be a maximal vacuum extension of a vacuum data set (M; g; k)

withM compact, and let D(M) � �M be the MVCD of (M; g; k). If D(M) 6=�M , then the Cauchy horizon H(M) is nonempty. One approach to SCC isto study the geometry of Cauchy horizons in vacuum space{times and toprove rigidity theorems as a consequence of extendibility of D(M).Isenberg and Moncrief proved for analytic vacuum or electrovac space{

times, with analytic Cauchy horizon H(M), that under the additional as-sumption that H(M) is ruled by closed null geodesics, there is a nontrivialKilling �eld which extends to D(M), see [81, 104]. This result was recentlygeneralized to the C1 case by Friedrich et. al. [55]. As space{times withKilling �elds are non{generic, this may be viewed as supporting evidencefor the SCC.In the class of Bianchi space{times (i.e. spatially locally homogenous

space{times, cf. section 3), it has been proved by Chru�sciel and Rendall[44], generalizing work by Siklos [116] in the analytic case, that any C1Bianchi space{time which contains a compact locally homogenous Cauchyhorizon is a Taub space{time, cf. section 3, (3.7) for de�nition. This resultmay be viewed as a version of SCC in the class of Bianchi space{times.In this context, it is worth mentioning that recent work by Chru�sciel andGalloway [41] gives examples which indicate that Cauchy horizons may benon{di�erentiable, generically.

Conjecture 2 (Bartnik [20, Conjecture 2]). Let ( �M; �g) be a spatially com-pact globally hyperbolic space{time satisfying the time{like convergence con-dition (2.5). Then if ( �M; �g) is time{like geodesically complete, ( �M; �g) splitsisometrically as a product (R�M;�dt2 + g).

If the Bartnik conjecture 2 is true, then any vacuum, globally hyperbolic,spatially compact space{time, is either at and covered by R� T 3 or is hasan inextendible time{like geodesic which ends after a �nite proper time, i.e.it is time{like geodesically incomplete. A sequence of points approachingthe \end" of a �nite length inextendible geodesic is often thought of asapproaching a singularity. See [62] for a recent discussion of the status ofthe Bartnik conjecture.Inextendibility of D(M) can be detected by monitoring the asymptotic

behavior of curvature invariants such as the Kretschmann scalar �, de�nedby � = �Rabcd

�Rabcd. If � blows up along causal geodesics, then D(M) fails to


be extendible, and therefore proving blowup for � for generic space{times isan approach to proving SCC. This is the method used in the proof of cosmiccensorship for the class of polarized Gowdy space{times [43], cf. section 4,and is likely to be important also in the cases with less symmetry. Thestructure of the horizon and extensions in the polarized Gowdy class canbe very complicated as shown by the work of Chru�sciel et. al., see [40] fordiscussion, see also [42]. It was recently proved, cf. Theorem 3.1, that forvacuum Bianchi space{times of class A, either the space{time is Taub, cf.section 3, or � blows up at the singularity.

2.2. The evolution equations. A solution to the vacuum Einstein evolu-tion equations with initial data is a curve t 7! (g(t); k(t); N(t); X(t)) de�nedon some interval (T0; T1), satisfying (2.3).Every regular solution (g; k;N;X) to the vacuum evolution equations

(2.3) with initial data solving the vacuum constraint equations, gives a vac-uum space{time. This is due to the fact that the constraint quantities

B = R+ (trk)2 � jkj2Di = ritrk � 2rjkij

evolve according to a symmetric hyperbolic system and energy estimatestogether with an application of the Gronwall inequality allow one to showthe the constraints are satis�ed during the time of existence of the solutioncurve. Now the fact that the Einstein vacuum equation is equivalent tothe system of constraint and evolution equations shows that the space{time( �M; �g) constructed from the curve (g; k;N;X) by letting �M = (T0; T1)�M ,and setting

�g = �N2dt2 + gij(dxi +X idt)(dxj +Xjdt)

is a solution to the Einstein vacuum equations (2.1).Note that in order for the solution to be well de�ned, it is necessary

to specify the Lapse and Shift (N;X), either as functions on space{timeM � (T0; T1) or as functions of the data, N = N [g; k];X = X [g; k]; this maybe viewed as a gauge �xing for the Einstein equations.The choice of Lapse and Shift is crucial for the behavior of the solution

curve. In particular, a foliation constructed for a particular choice of N;Xmay develop singularities which are not caused by any singular or irreg-ular nature of the Cauchy development. Consider for example the Gaussfoliation condition N = 1; X = 0. Then the hypersurface M ows in thedirection of its unit normal and Mt is simply the level set of the Lorentziandistance function t(p) = d(M; p). The foliation fMtg will develop singulari-ties precisely at the focal set ofM , which in general will be nonempty, evenin Minkowski space.Many authors have considered hyperbolic reformulations of the Einstein

equations, see the paper by Friedrich [58] for discussion, see also [59, 75, 76]for recent work. The development of singularities for hyperbolic systems

10 L. ANDERSSON

presents a serious obstacle to the numerical treatment of the Einstein evolu-tion equations using hyperbolic reformulations, see [1, 2] for discussion andexamples. It is therefore necessary to consider also gauges which make (2.3)into an elliptic{hyperbolic system.

2.3. Constant mean curvature foliations. A particularly interestingchoice of gauge condition for the Lapse function is given by the constantmean curvature (CMC) condition

ritrk = 0;

i.e. the level setsMt of the time function t are assumed to be hypersurfaces ofconstant mean curvature in ( �M; �g). If ( �M; �g) is globally hyperbolic, spatiallycompact and satis�es the time{like convergence condition (2.5), then for� 2 R, either there is at most one Cauchy surface with mean curvaturetrk = � or ( �M; �g) splits as a product, cf. [31].This indicates that the mean curvature trk may be useful as a time func-

tion on ( �M; �g) in the spatially compact case. Setting t = trk leads, using(2.2{2.3) to the Lapse equation

��N + jkj2N = 1: (2.6)

Equation (2.6) supplemented by a suitable Shift gauge makes the Einsteinevolution equations (2.3) into an elliptic{hyperbolic system of evolutionequations, cf. subsection 6.1.The maximal slicing condition trk = 0 is of interest mainly for the asymp-

totically at case. This was used in the proof of the nonlinear stability ofMinkowski space by Christodoulou and Klainerman [39], cf. the discussionin section 6. Due to the \collapse of the Lapse" phenomenon, see [22], themaximal foliation is not expected to cover the whole MVCD except in thesmall data case. See [99] for a discussion of maximal slices. Asymptotically at space{times satisfying certain restrictions on the causal structure areknown to contain maximal hypersurfaces [19].The mean curvature operator satis�es a geometric maximum principle, see

[6] for a proof of this under weak regularity. This allows one to use barriersto prove existence of constant mean curvature hypersurfaces. A space{timeis said to have crushing singularities if there are sequences of Cauchysurfaces with mean curvature trk tending uniformly to �1. Gerhard [63]proved, using a barrier argument, that any space{time satisfying (2.5) withcrushing singularities is globally foliated by CMC hypersurfaces. These factsindicate that the CMC foliation condition is an interesting time gauge forthe Einstein evolution equation.Let R[g] be the scalar curvature. A 3{manifoldM is said to be of Yamabe

type �1 if it admits no metric with R[g] = 0 (and hence no metric withnonnegative scalar curvature), of Yamabe type 0 if it admits a metric withR[g] = 0 but no metric with R[g] = 1 and of Yamabe type +1 if it admits ametric with R[g] = 1, cf. [52, De�nition 9].


If the Cauchy surface M is of Yamabe type �1, it follows from the con-straint equation that ( �M; �g) cannot contain a maximal (i.e. trk = 0) Cauchysurface and therefore one expects that (if the dominant energy conditionholds) the maximal time interval of existence for (2.3) in CMC time is ofthe form (after a time orientation) (�1; 0) with � % 0 corresponding toin�nite expansion.If M is of Yamabe type 0, then one expects that either the maximal

CMC time interval is (�1; 0) (possibly after a change of time orientation)or ( �M; �g) splits as a product, and therefore in the vacuum case is coveredby R� T 3 with the at metric. Finally in case M is of Yamabe type +1,one expects the maximal CMC time interval to be (�1;1), i.e. the space{time evolves from a \big bang" to a \big crunch". This is formalized in the\closed universe recollapse conjecture" of Barrow, Galloway and Tipler [18].

Conjecture 3 (Constant mean curvature foliations). Let M be a compact3{manifold and let (M; g; k) be a vacuum data set on M , with constantmean curvature. The Cauchy problem for the Einstein vacuum evolutionequations with data (M; g; k) have global existence in the constant meancurvature time gauge, i.e. there is a CMC foliation in the MVCD ( �M; �g) of(M; g; k), containing M , with mean curvature taking all values in (�1;1)in case M has Yamabe type +1 and in case M has Yamabe type 0 or �1,taking all values in (�1; 0) (possibly after a change of time orientation).

Remark 2.1. Conjecture 3 has been stated essentially this form by Rendall[109, Conjecture 1], see also Eardley and Moncrief [103, Conjecture C2] fora closely related statement.Note that as � & �1, the past focal distance of the (unique) CMC surface

with mean curvature � tends to zero, and hence the foliation exhausts thepast of M . It follows that in case M has Yamabe type +1, then if Conjecture3 is true, ( �M; �g) is globally foliated by CMC hypersurfaces. In case M hasYamabe type 0 or �1 on the other hand, there is the possibility that the CMCfoliation does not cover all of �M , due to the fact that as the mean curvature� % 0, the CMC hypersurfaces are expected to avoid black holes, by analogywith the behavior of CMC and maximal hypersurfaces in the Schwarzschildspace{time. See [109] for further remarks and conjectures related to this.

If one were able to prove Conjecture 3, then as remarked in [103], thiswould give the possibility of attacking the Cosmic Censorship Conjectureusing PDE methods. There are no known counterexamples to Conjecture3 for vacuum space{times. However, Isenberg and Rendall [83] give anexample of dust space{times, not covered by a CMC foliation. Bartnik[20] gave an example of a spatially compact, globally hyperbolic space{time satisfying the time{like convergence condition, which contains no CMCCauchy surface. It is an interesting open question whether or not similarcounter examples are possible in the vacuum case.The CMC conjecture 3 has been proved in a number of cases for space{

times with symmetry, in the sense of the existence of a group G of isometries

12 L. ANDERSSON

acting (locally) on ( �M; �g) by isometries and with space{like orbits. In thecase of Bianchi IX, cf. section 3, which has Yamabe type +1, the closeduniverse recollape conjecture and consequently Conjecture 3, was proved byLin and Wald [97]. In the case of 2+1{dimensional vacuum space{timeswith cosmological constant, the conclusion of Conjecture 3 is valid [9]. Inthe 2+1 case, the CMC foliations are global.The work of Rendall and Burnett, see [32] and references therein, proves

under certain restrictions on the matter that a maximal, globally hyperbolic,spherically symmetric space{time, which contains a CMC Cauchy surfacedi�eomorphic to S2 � S1, is globally foliated by CMC hypersurfaces withmean curvature taking on all real values.We end this subsection by mentioning the harmonic time gauge condition,

��gt = 0 or

�gab��0ab = 0

In case X = 0, this is equivalent to the condition N =pdet g=

pdet e

where eij is some �xed Riemannian metric on M . The Einstein evolutionequations with X = 0 were proved to be hyperbolic with this time gaugeby Choquet{Bruhat and Ruggeri [37]. This time gauge also appears in thework of Gowdy and is used in the analysis of the Gowdy space{times as wellas in the numerical work of Berger, Moncrief et. al. on Gowdy and U(1)space{times, cf. sections 4 and 5.The papers of Smarr and York [117, 118] contain an interesting discus-

sion of gauge conditions for the Einstein equations. See also section 6 fora discussion of the spatial harmonic coordinate gauge and the survey ofKlainerman and Nicolo [92] for further comments on gauges.

2.4. The Einstein equations as a system of quasi{linear PDE's. Asmentioned above, the Einstein vacuum equations in space{time harmoniccoordinates form a quasi{linear hyperbolic system of the form

�1

2��g�g�� + S��[�g; @�g] = 0: (2.7)

The system (2.7) is a quasi{linear wave equation, quadratic in the �rstorder derivatives @�g and with top order symbol depending only on the �eld�g itself. Standard results show that the Cauchy problem is well posed inSobolev spaces Hs � Hs�1, s > n=2 + 1. This was �rst proved for theEinstein equations by Hughes, Kato and Marsden [72]. It is also possible toprove this for elliptic{hyperbolic systems formed by the Einstein evolutionequations together with the CMC{spatial harmonic coordinates gauge, see[7], see also section 6 for a discussion of the spatial harmonic coordinatesgauge.Recent work using harmonic analysis methods [124, 16] has pushed the

regularity needed for systems of the above type on Rn;1 to s > n=2+3=4 forn � 3. In order to get well posedness for s lower than the values given above,it is likely one needs to exploit some form of the null condition [122, 120].The null condition for equations of the form ��u = F [u; @u] on Minkowski


space states roughly that the symbol of the non{linearity F cancels nullvectors. For a discussion of the null condition on a curved space background,see [121, 119]. Counter examples to well posedness for quasi{linear waveequations with low regularity data are given by Lindblad, see [98].The standard example of an equation which satis�es the null condition

is �u = @�u@�u�� where � is the wave operator w.r.t. the Minkowski

metric � on Rn;1. This equation is well posed for data in Hs � Hs�1 withs > n=2 [93], and has global existence for small data for n � 3. On the otherhand, the equation �u = (@tu)

2 which does not satisfy the null conditioncan be shown to have a �nite time of existence for small data in the 3+1dimensional case.For quasi{linear wave equations which satisfy an appropriate form of the

null condition [71], global existence for small data is known in 3+1 dimen-sions. The Einstein equations, however, are not known to satisfy the nullcondition in any gauge. In particular, it can be seen that in space{time har-monic coordinates, the Einstein equations do not satisfy the null condition.However, the analysis by Blanchet and Damour [30] of the expansion of so-lutions of Einstein equations in perturbation series around Minkowski spaceindicates that the logarithmic terms in the gravitational �eld in space{timeharmonic coordinates, arising from the violation of the null condition, canbe removed after a (nonlocal) gauge transformation to radiative coordinateswhere the coordinate change depends on the history of the �eld. A similaranalysis can be done for the Yang{Mills (YM) equation in Lorentz gauge. Itmay further be argued that the small data, global existence proof of Christo-doulou and Klainerman for the Einstein equations exploits properties of theEinstein equations related to the null condition.Global existence is known for several of the classical �eld equations such

as certain nonlinear Klein{Gordon (NLKG) equations and the YM equationon R3;1 (proved by Eardley and Moncrief, [49, 50]). The proofs for NLKGand the proof of Eardley and Moncrief for YM use light cone estimatesto get apriori L1 bounds. The proof of Eardley and Moncrief used thespecial properties of YM in the radial gauge. This method was also used inthe global existence proof for YM on 3+1{dimensional, globally hyperbolicspace{times by Chru�sciel and Shatah [45]. Klainerman and Machedon [91]were able to prove that the YM equations on R3;1 in Coloumb gauge satisfya form of the null condition and are well posed in energy space H1 � L2.They were then able to use the fact that the energy is conserved to proveglobal existence for YM. See also [90] for an overview of these ideas andsome related conjectures.An important open problem for the classical �eld equations is the global

existence problem for the wavemap equation (nonlinear �{model, hyperbolicharmonic map equation). This is an equation for a map Rn;1 ! N , whereN is some complete Riemannian manifold,

��uA + �ABC(u)@�u

B@�uC�� = 0: (2.8)

14 L. ANDERSSON

Here � is the Christo�el symbol on N .The wave map equation satis�es the null condition and hence we have

small data global existence for n � 3. Small data global existence is alsoknown for n = 2. Further, scaling arguments provide counterexamples toglobal existence for n � 3, whereas n = 2 is critical with respect to scaling.For n � 2, global existence for \large data" is known only for symmetricsolutions, and in particular, the global existence problem for the wave mapequation (2.8) is open for the case n = 2. For the case n = 1, global existencecan be proved using energy estimates or light cone estimates. See [115] fora survey, see also [54] for recent results in the 2+1 dimensional case.The above discussion shows that the situation for the wave map equation

is reminiscent of that for the Einstein equations, cf. sections 4, 5. In partic-ular, it is interesting to note that equations of the wave map type show up inthe reduced vacuum Einstein equations for the Gowdy and U(1) problems.

3. Bianchi

Let ( �M; �g) be a 3+1 dimensional space{time with 3{dimensional localisometry group G. Assume the action of G is generated by space{like Killing�elds and that the orbits of G in the universal cover of �M are 3{dimensional.This means there is a global foliation of �M by space{like Cauchy surfacesMwith locally homogeneous induced geometry. Such space{times are knownas Bianchi space{times, see [51, 126]. The assumption of local homogeneityof the 3{dimensional Cauchy surfaces means that from a local point of viewa classi�cation is given by the classi�cation of 3{dimensional Lie algebras.Let ea, a = 0; : : : ; 3 be an ON frame on �M , with e0 = u, a unit time{like

normal to the locally homogeneous Cauchy surfaces, let cab be the commu-tators of the frame, [ea; eb] = cabec. Let the indices i; j; k; l run over 1; 2; 3.We may without loss of generality assume that [ea; �i] = 0 where f�ig3i=1 isa basis for the Lie algebra g of G.Choose a time function t so that t;au

a = 1, i.e. the level sets of t coincidewith the group orbits. Restricting to a level setM of t, the spatial part of thecommutators kij are the structure constants of g. These can be decomposed

into a constant symmetric matrix nkl and a vector ai,

kij = �ijlnkl + ai�

kj � aj�

ki :

We will brie y describe the classi�cation used in the physics literature, cf.[51], [126, x1.5.1].The Jacobi identity implies nijaj = 0 and by choosing the frame feig to

diagonalize nij and so that a1 6= 0 we get

nij = diag(n1; n2; n3); ai = (a; 0; 0): (3.1)

The 3{dimensional Lie algebras are divided into two classes by the conditiona = 0 (class A) and a 6= 0 (class B). The classes A and B correspondin mathematical terminology to the unimodular and non{unimodular Lie


Table 1. Bianchi geometries

Class A

Type n1 n2 n3I 0 0 0II 0 0 +VI0 0 - +VII0 0 + +VIII - + +IX + + +

Class B

Type n1 n2 n3V 0 0 0IV 0 0 +VIh 0 - +VIIh 0 + +

algebras. Let the scalar h be de�ned by

a2 = hn2n3: (3.2)

Table 1 gives the classi�cation of Bianchi geometries, following [126, x1.5.1],[51]. Note that the invariance of the Bianchi types under permutations andsign changes of the frame elements has been used to simplify the presenta-tion. Here the notation VI0, VII0, VIh, VIIh refers to the value of h de�nedby (3.2). In the list of Bianchi types I{IX, the missing type III is the sameas VI�1.Due to the local homogeneity of the Cauchy surfaces M in a Bianchi

space{time, the topologies of the spatially compact Bianchi space{timescan be classi�ed using the classi�cation of compact manifolds admittingThurston geometries.

The eight Thurston geometries S3, E3,H3, S2�R,H2�R, Nil, fSL(2;R),Sol, are the maximal geometric structures on compact 3{manifolds, see [114,125] for background. Each compact 3{manifold with a Bianchi (minimal)geometry also admits a Thurston (maximal) geometry, and this leads to aclassi�cation of the topological types of compact 3{manifolds with Bianchigeometry, i.e. compact manifolds of the form X=� where X is a complete,simply connected 3{manifold with a Bianchi geometry and � is a cocompactsubgroup of the isometry group of X . It is important to note that � is notalways a subgroup of the 3{dimensional Bianchi group G.The relation between the Bianchi types admitting a compact quotient and

the Thurston geometries is given by Table 2. For each Bianchi type we giveonly the maximal Thurston geometries corresponding to it, see [60, 94] forfurther details and references. We make the following remarks

Remark 3.1. (i). Let (M; g) be a 3{dimensional space form with sectionalcurvature � = �1; 0;+1. A spacetime ( �M; �g) with �M = M � (a; b)and a warped product metric �g = �dt2 + w2(t)g, satisfying the per-fect uid Einstein equations is called a (local) Friedmann{Robertson{Walker (FRW) space{time. Specifying the equation of state for thematter in the Einstein equations leads to an ODE for w. The FRWspacetimes play a central role in the standard model of cosmology. Inthe vacuum case, only � = �1; 0 are possible, and in this case, the spa-tially compact local FRW spacetimes are for � = 0, the at space{times

16 L. ANDERSSON

Table 2. Bianchi and Thurston geometries

Bianchi type class Thurston geometry commentsI A E3

II A Nil

III = VI�1 B H2 �E1fSL(2;R) cf. Remark 3.1 (iii).

IV | | no compact quotientV B H3 cf. Remark 3.1 (iv)VI0 A Sol

VIh, h 6= 0;�1 | | no compact quotientVII0 A E3

VIIh, h 6= 0 B H3 cf. Remark 3.1 (iv)

VIII A fSL(2;R)IX A S3

covered by T 3 � E1, a special case of Bianchi I, and for � = �1, thelocal FRW space{times discussed in example 2.5, which are Bianchi V.

(ii). The Thurston geometry S2 � R admits no 3{dimensional group ofisometries, and hence it does not correspond to a Bianchi geometry,but to a Kantowski{Sachs geometry [86].

(iii). The type of geometry depends on the subgroup � of the isometry groupused to construct the compacti�cation. The isometry group in turn

depends on the Bianchi data. Note that fSL(2;R) is both maximal andminimal, whereas the isometry group of H2 �R has dimension 4 andis therefore not a minimal (Bianchi) group.

(iv). The compacti�cations of Bianchi V and VIIh, h 6= 0 are both of thetype discussed in Example 2.5, thus no anisotropy is allowed in thecompacti�cation of Bianchi type V and VIIh, h 6= 0.

(v). The compacti�cation of a Bianchi geometry, introduces new (moduli)degrees of freedom, in addition to the local (dynamical) degrees of free-dom, see [94] and references therein for discussion. The resulting pic-ture is complicated and does not appear to have been given a de�nitetreatment in the literature.

In the rest of this section, we will consider only class A Bianchi space{times. We will also refrain from considering the moduli degrees of freedomintroduced by the compacti�cation, as it can be argued that these are notdynamical.Let the expansion tensor �ij be given by

�ij = rjui;


(i.e. �ij = �kij where kij is the second fundamental form). Decompose

�ij as �ij = �ij +�3�ij where � = gij�ij . We are assuming (3.1) and in the

vacuum class A case it follows that �ij is diagonal [126, p. 41].Under these assumptions, cab = cab(t) and we may describe the geom-

etry of ( �M; �g) completely in terms of cab or equivalently in terms of the

3{dimensional commutators kij and the expansion tensor given in terms of�ij ; �.Introduce the dimensionless variables (following Hsu andWainwright [127])

�ij = �ij=�;

Nij = nij=�:

�ij ; Nij can be assumed to be diagonal, �ij = diag(�1;�2;�3), Nij =diag(N1; N2; N3).The dimensionless curvature Bij = Ricg(ei; ej)=�

2 can be written in termsof Ni as Bij = 2Nk

i Nkj � NkkNij . We decompose Bij into trace{free and

trace parts, Sij ; K, where

Sij = Bij � 1

3Bkk �ij ;

K =3

4Bkk :

In addition to these choices we also de�ne a new time � by e� = `, ` thelength scale factor, or dt=d� = 3=�.As � is traceless and diagonal it can be described in terms of ��, given

by

�+ =3

2(�22 +�33);

�� =

p3

2(�22 � �33):

Similarly we can represent Sij in terms of two variables S+; S�. Clearly theequations are be invariant under permutations (�i)! P (�i), (Ni)! P (Ni).Cyclic permutations of (Ni); (�i) correspond to rotations through 2�=3 inthe �+;�� plane.We now specialize to the vacuum case. Then (for � 6= 0) the Einstein equa-

tions are equivalent to the following system of ODE's for �+;��; N1; N2; N3

(where the 0 denotes derivative w.r.t. the time coordinate �):

N 01 = (q � 4�+)N1; (3.3a)

N 02 = (q + 2�+ + 2

p3��)N2; (3.3b)

N 03 = (q + 2�+ � 2

p3��)N3; (3.3c)

�0+ = �(2� q)�+ � 3S+; (3.3d)

�0� = �(2� q)�� 3S�; (3.3e)

18 L. ANDERSSON

where

q = 2(�2+ + �2

�);

S+ =1

2[(N2 �N3)

2 �N1(2N1�N2 �N3)];

S� =

p3

2(N3 �N2)(N1�N2 �N3):

The Hamiltonian constraint (2.2a) is in terms of these variables

�2+ + �2

� +3

4[N2

1 +N22 +N2

3 � 2(N1N2 +N2N3 +N3N1)] = 1: (3.4)

With our conventions, if (��; �+) is the maximal time interval of existencefor the solution to (3.3), then � ! �� corresponds to the direction of thesingularity. For all non{ at vacuum Bianchi models except Bianchi IX, thespace{time undergoes an in�nite expansion as � ! �+ and is geodesicallycomplete in the expanding direction, cf. [108, Theorem 2.1] which covers thevacuum case as a special case. For Bianchi IX, on the other hand, � ! �+corresponds to � ! 0, as follows from the proof of the closed universerecollapse conjecture for Bianchi IX by Lin and Wald [97], and hence to thedimensionless variables becoming ill de�ned.We will now review the basic facts for Bianchi types I, II and IX.

I: Kasner. The Hamiltonian (3.4) constraint reads

�2+ +�2

� = 1; (3.5)

and the induced metric on each time slice is at. These space{times can begiven metrics of the form

ds2 = �dt2 +Xi

t2pidxi dxi; (3.6)

whereP

i pi = 1,P

i p2i = 1, the Kasner relations, which correspond to

the equation (3.5). The �� and pi are related by �+ = 32(p2 + p3) � 1,

�� =p32 (p2� p3). Clearly, the Kasner circle given by (3.5) consists of �xed

points to the system (3.3). The points T1; T2; T3, with coordinates (�1; 0),(1=2;�p3=2) correspond to at spacetimes of type I or VII0 (quotients ofMinkowski space).

II: By using the permutation symmetry of the equations, we may assumeN2 = N3 = 0. The solution curve which is a subset of the cylinder �2

++�2� <

1 has a past endpoint on the longer arc connecting T2 and T3 and futureendpoint on the shorter arc connecting these points see �gure 2. This curverealizes the so called Kasner map, cf. [126, x6.4.1].IX:Mixmaster, characterized by Ni > 0; i = 1; 2; 3. The heuristic picture isthe following. The projection in the (�+;��){plane of a generic orbit in thedirection � & �� moves into the Kasner circle and stays there, undergoing anin�nite sequence of bounces, which are approximately given by the Kasner


−2

0

2

−2

0

2−0.5

0

0.5

1

1.5

−1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 2. A type II solution, the Kasner circle and the tri-angle for the Kasner billiard in the (�+;��){plane areshown.

billiard, cf. �gure 4. This picture is supported by numerical studies of thefull Bianchi IX system, see eg. [27].The Kasner billiard is the dynamical system given by mapping a (non{

at) point p on the Kasner circle to the point on the Kasner circle which isthe end point of the type II orbit starting at p. This map can be describedas follows. Let B be the nearest corner to p of the triangle shown in �gure4. The ray starting at B through p intersects the Kasner circle in a pointq, which is the image of p under the Kasner map, see also [126, Fig. 6.13].Iterating this construction gives a sequence of points fpig on the Kasnercircle, which we may call the Kasner billiard.The exceptional orbits which do not exhibit this in�nite sequence of

bounces are the Taub type IX solutions, cf. �gure 3. Up to a permuta-tion these are given by the conditions

N2 = N3; �� = 0: (3.7)

We call a Bianchi space{time, satisfying (3.7) up to a permutation, a Taubspace{time.The past limit of the Taub type IX solution is the at point (�1; 0).

The MVCD of Taub type IX data has a smooth Cauchy horizon, and isextendible, the extension being given by the so{called Taub{NUT space{times. As shown by Chru�sciel and Rendall [44, Theorem 1.2], these are theonly Bianchi IX space{times with a smooth Cauchy horizon, which givesa version of SCC for this class. See [44] for further details on the statusof SCC in the locally homogeneous case. Chru�sciel and Isenberg provedthat the MVCD of Taub type IX data has non{isometric maximal vacuumextensions, further emphasizing its pathological nature, cf. [42].From the point of view of the cosmic censorship conjecture, the follow-

ing theorem appears fundamental. A point which is a past limit point of(�+(�);��(�); N1(�); N2(�); N3(�)) is called an � limit point. We say that

20 L. ANDERSSON

−10

0

10

−2

0

2−0.5

0

0.5

1

1.5

−2 0 2 4 6−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 3. A Taub type IX solution, the Kasner circle andthe triangle for the Kasner billiard in the (�+;��){plane areshown.

the approach to the singularity is oscillatory, if the set of � limit pointscontains at least two points on the Kasner circle, at least one of which isdisctinct from the special points T1; T2; T3, cf. [111].

Theorem 3.1 ([111, 113]). A vacuum Bianchi space{time of class A hasexactly one of the properties

1. The Kretschmann scalar � = �R�� R�� satis�es lim sup�&�� j�j =1

2. The MVCD has a smooth Cauchy horizon and the space{time is a Taubspace{time.

For non{Taub vacuum Bianchi VIII and IX space{times, the approach tothe singularity is oscillatory.

Remark 3.2. (i). It is likely that this result can be extended to class Band to Bianchi space{times with matter.

(ii). Rendall [111] proved the dichotomy in Theorem 3.1 for all Bianchi classA except VIII and IX. These cases as well as the oscillatory behaviorfor type VIII and IX were proved by Ringstrom [113].

(iii). Theorem 3.1 shows that SCC holds in the class of vacuum Bianchiclass A space{times, also with respect to C2 extensions.

(iv). See also Weaver [131] for a related result for Bianchi VI0 with a mag-netic �eld. Weaver proved that the singularity in magnetic BianchiVI0 is oscillatory and that curvature blows up as one approaches thesingularity. It should be noted that vacuum Bianchi VI0, on the otherhand, is non-oscillatory.

The dynamics of the Bianchi space{times has been studied for a long time,from the point of view of dynamical systems. In particular, it is believed thatthe Bianchi IX (mixmaster) solution is chaotic in some appropriate sense, seefor example the paper by Hobill in [126] or the collection [70] as well as therecent work of Cornish and Levin [46], for various points of view. However,


−2

0

2

−2

0

2−0.5

0

0.5

1

1.5

−1 0 1 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Figure 4. Bianchi IX orbit showing a few bounces. Thevertical axis is N1. The Kasner circle and the triangle forthe Kasner billiard in the (�+;��){plane are shown.

it does not yet appear to be clear which is the appropriate de�nition of chaosto be used, and no rigorous analysis exists for the full Bianchi IX system.In the course of the above mentioned work, approximations to the Bianchidynamics have been described and studied, such as the Kasner billiard (cf.�g. 4, and the discussion above) and the BKL map, cf. [126, x11.2.3].

4. U(1)�U(1)

In this section we consider the case when ( �M; �g) is a 3+1{dimensional,spatially compact, globally homogeneous, vacuum space{time, with a 2{dimensional local isometry group G with the action ofG generated by space{like Killing �elds. By passing to the universal cover, we see that the non{degenerate orbits of the G{action are 2{dimensional homogeneous spacesand hence the induced metric on the orbits must have constant curvature.The isometry group of the sphere S2 has no two dimensional subgroup andthus the orbits have geometry E2 or H2.The special case when the group U(1)� U(1) itself acts on �M was con-

sidered by Gowdy [64, 65]. In this case, it follows that orbits are compact(unless there is an extra Killing �eld), andM is covered by T 3; S3 or S1�S2.As SO(3) does not have a 2{dimensional subgroup, any degenerate orbit ofG must be a closed curve.Let �M be a bundle over S1�R with compact 2{dimensional �ber F and

suppose that the orbits of the G{action on the universal cover of �M cover the�bers F . If F has geometry E2 it follows that the Killing �elds generatingthe G{action commute, and hence it is natural, following Rendall [110], touse the term local U(1)�U(1) symmetry for this situation.Space{times with local U(1)�U(1) symmetry have also been considered

by Tanimoto [123], who discussed the question of in which case a space{timewith local U(1)� U(1) symmetry can be considered as a dehomogenizationof a Bianchi space{time. It can be seen from the structure of the Bianchi

22 L. ANDERSSON

groups that the only U(1)�U(1) symmetric space{times which have Bianchi(or Kantowski{Sachs) limits are covered by T 3 or S3, and in the case of S3

(Bianchi IX), it is only the Taub metrics that admits a U(1)� U(1) actionby isometries. If we consider the case with local U(1) � U(1){symmetriccase on the other hand, then all Bianchi models except Bianchi VIII andXI, and in case of Bianchi VIII and IX the Taub metrics, can be viewed aslimits of locally U(1)� U(1){symmetric models, and therefore these serveas dehomogenization of the Bianchi models.For simplicity we consider in the rest of this section, only the case of

U(1) � U(1) symmetric space{times and assume that the twist constantsvanish, i.e. i.e. �1 ^ �2 ^ d�1 = �1 ^ �2 ^ d�2 = 0, where �1; �2 are one formsdual to the generators of the G action. This is a nontrivial restriction only incaseM �= T 3, see [65, p. 211]. Such space{times are known in the literatureas Gowdy spacetimes. We further specialize to the case with Cauchy surfaceM �= T 3

Let t; � be coordinates on the 1+1 dimensional Lorentzian orbit space�M=G and let A(t; �) be the area of the orbit. Gowdy showed that in thenon{twisted case, with M �= T 3, there are no degenerate orbits, i.e. A 6= 0,and further, the level sets of A in �M=G are space{like. We may thereforechoose coordinates so that A = 4�2t.The space{time metric may then be written in the form

�g = e�=2t�1=2(�dt2 + dx2) + thABdyAdyB: (4.1)

Here t; � are coordinates on the orbit space, yA; A = 1; 2 are coordinateson the orbit, 0 � yA � 2�. The G invariance implies that we can assumeall metric components depend on t; � only. Let e = dx2 + (dy1)2 + (dy2)2.With the logarithmic time � = � ln t, which is used in the numerical stud-ies of Gowdy space-times, cf. [23], the Lapse function N satis�es N =pdet(g)= det(e) and hence the time function � is space{time harmonic, cf.

subsection 2.3.By construction, h = h(t; x) is a unit determinant metric which is constant

on each orbit. It therefore represents an element of the Teichm�uller spaceT (T 2). The space T (T 2) with the Weil{Peterson metric is isometric to thehyperbolic plane H2. The identi�cation of T (T 2) with H2 gives a mapu : Gn �M ! H2. This can be realized concretely for example by using themodel for H2 with metric

dP 2 + e2P dQ2 (4.2)

and letting u = (P;Q) with

P = ln(h11);

Q = e�P h12;

(which is the parametrization that is used in the numerical work). Thus theU(1)� U(1) symmetric Einstein equations on M �= T 3 � R can be viewedas equations for the evolution of a loop in H2.


Let � = �@2t + @2x. Now the Einstein equations take the form

�uA � t�1@tuA + �ABC(u)@�uB@�u

C�� = 0; (4.3)

where �; � = 0; 1, x0 = t; x1 = x, ��dx�dx� = �dt2 + dx2, the 1+1 dimen-

sional Minkowski metric and �ABC are the Christo�el symbols on H2 withthe metric (4.2). The system (4.3) is supplemented by a pair of equationsfor �, which are implied by the Einstein constraint equations, and which areused to reconstruct the 3+1 metric �g.Equation (4.3) is a semilinear hyperbolic system, which resembles the

wave{map equation, (2.8). Energy estimates or light cone estimates proveglobal existence on (0;1)� S1. The \wave operator" �� t�1@t is singularat t = 0, which corresponds to a singularity in the 3+1 space{time ( �M; �g).We now brie y discuss some results and open problems for the Gowdy

space{times. The following result proves Conjecture 3 for Gowdy space{times with vanishing twist.

Theorem 4.1 (Isenberg and Moncrief [80]). Spatially compact space{timeswith U(1)�U(1) symmetry and vanishing twist, are globally foliated by CMChypersurfaces.

The behavior of CMC foliations for space{times with local U(1) � U(1)symmetry and with non{vanishing twist has been considered by Rendall[110], who proved global existence in CMC time in the direction towardsthe singularity for twisted Gowdy space{times with Vlasov or wave mapmatter, assuming the existence of a CMC Cauchy surface. The existence ofone CMC hypersurface is an open problem in this case. Global existencein the area time coordinate has been proved for U(1) � U(1) symmetricspace{times with non{vanishing twist in the vacuum case [24]. This resulthas recently been generalized by Andr�easson [11] to the case with Vlasovmatter.It was conjectured by Belinskii, Khalatnikov and Lifschitz [96] that spa-

tial points will decouple as one approaches the singularity. This idea hasbeen modi�ed and reformulated by among others Eardley, Liang and Sachs[48] and Isenberg and Moncrief [82], into the notion of asymptotically ve-locity term dominated (AVTD) singularities. Roughly, an AVTD solutionapproaches asymptotically, at generic spatial points, the solution to an ODE,the parameters of which depend on the spatial point. In particular, locallynear a �xed spatial point the space{time approaches a Kasner limit, withparameters depending on the spatial point.One gets the AVTD equations by cancelling the x-derivative terms in

(4.3). Let D = t@t. Then the AVTD equations for untwisted T 3 Gowdy canbe written

D2uA + �ABC(u)DuBDuC = 0; (4.4)

24 L. ANDERSSON

Figure 5. Geodesic motion of a loop shown in the ballmodel of H2. The trajectories of a few individual pointsare shown as dotted lines.

which is just the geodesic equation inH2 with the time � = � ln t. Thereforesolutions to (4.4) are loops in H2, with each point of the loop moving alonga geodesic, cf. �gure 5.Gowdy space{times which are such that the component h12 of the metric

on the orbit vanishes, are called polarized. Equation (4.3) then becomeslinear. It was proved by Isenberg and Moncrief [82] that polarized Gowdyspace{times are AVTD.Numerical studies by Berger et. al., see [23] and references therein, see

also [69, 26], support the idea that general Gowdy space{times are AVTD.Let GAB be the metric on H2. Then for a solution of (4.4), the velocity

v =pGABDuADuB

is time independent.The numerical studies referred to above indicate that as one moves to-

wards the singularity at t = 0, the velocity v is eventually forced to satisfy0 � v � 1, except at isolated x{values, even if v > 1 in some subsets ofS1 initially and that v has a limiting value for each x as one moves towardthe singularity, i.e. as t & 0 and further v < 1 asymptotically except atisolated points. Where v < 1 one expects AVTD behavior, i.e. the solutiontracks a solution to (4.3). The numerical solutions exhibit \spikes" at thosex{values, where v � 1 asymptotically, cf. �gure 6. It should be noted thatthere are polarized Gowdy space{times with v > 1 up to the singularity, butthe above work indicates that this behavior is non{generic.The equation (4.3) may be written as a Fuchsian system

(t@t+ E)U = F [U ];(essentially under the restriction v < 1) and hence using a singular versionof the Cauchy{Kowalewskaya theorem, cf. Kichenassamy and Rendall [89]and references therein, see also [17], AVTD solutions may be constructed


Figure 6. Example of numerical solution of Gowdy �eldequation (4.3). The plot shows the evolution of P;Q w.r.t.the time � = � ln t. The arrow is in the direction of increasing� . Detail of [23, Fig. 5]

.

given real analytic \data on the singularity". This work makes earlier work[66] on formal expansions for the Gowdy �eld equations rigorous. So far,there does not exist a Fuchsian formulation of the Gowdy \spikes".Recent work by Anguige [12] on equations of Fuchsian type in Sobolev

spaces may allow one to generalise this and prove that an open subset ofdata at t 6= 0 lead to AVTD behavior.In view of the above, it is reasonable to make the following conjecture.

Conjecture 4. Generic vacuum, spatially compact U(1)�U(1){symmetricspace{times with vanishing twist are AVTD at the singularity.

Remark 4.1. This is implicit in Grubi�si�c and Moncrief [66]. Here AVTDshould be understood also generically in space in the above sense.

A class of AVTD polarized Gowdy space{times with non{vanishing twisthas been constructed by Isenberg and Kichenassamy [78] using the Fuchsianalgorithm. On the other hand for general non{polarised twisted U(1)�U(1)symmetric space{times, work of Weaver et al. supports the conjecture thatthese space{times show oscillating behavior as one approaches the singu-larity. It is also relevant to mention here that the AVTD behavior forU(1)� U(1) symmetric space{times may be broken by the introduction ofsuitable matter, cf. [130], where numerical evidence for an oscillatory ap-proach to the singularity is presented, for a locally U(1)� U(1) symmetricspace{time with magnetic �eld.Finally we discuss the status of the cosmic censorship conjecture for the

Gowdy space{times. For the class of polarized Gowdy space{times, thiswas proved by Chru�schiel et. al. [43]. It was proved by Kichenassamyand Rendall [89] that for generic AVTD space{times constructed using theFuchsian algorithm, the Kretschmann scalar � blows up at the singularityand hence generically, these space{times do not admit extensions, see also

26 L. ANDERSSON

the discussion in [66, x3{4]. Therefore a reasonable approach to the SCC inthe class of Gowdy space{times, is via conjecture 4.

5. U(1)

The U(1) symmetric vacuum 3+1 Einstein equations is an important casewhich is of intermediate di�culty between the full 3+1 Einstein equationsand the highly symmetric Gowdy equations. In the presence of a hyper-surface orthogonal space{like Killing �eld, the Einstein equations reduce to2 + 1 gravity coupled to wave map matter, the �eld equations and their re-ductions have been derived in [101, 100, 102]. The problem of local existencefor the reduced evolution equations has been studied by Choquet{Bruhatand Moncrief [36, 35], where local existence in Hs spaces was proved.In the above quoted papers, the space{time is assumed to be a U(1)

bundle over a spatially compact 2 + 1 space{time. The case of local U(1)symmetry does not appear to have been studied in connection with theEinstein equations, see however [105] for a study of 3{manifolds with localU(1) action.It is also possible to study the case when the reduced space is asymp-

totically at. This case has been consider in work by Ashtekar and others,[15, 13], see also [14]. In these papers an analogue of the ADM mass at spa-tial in�nity is introduced. It is proved that it is nonnegative and boundedfrom above. It is interesting to study the consequences of the presence of thisconserved quantity for the 2+1 dimensional Einstein{matter system givenby the U(1) problem, as one expects that it gives a stronger bound on the�elds than in the 3+1 case. This appears to be a natural setting for a smalldata version of the U(1) problem.In the following, we will consider the spatially compact case. Let ( �M; �g)

be a 3+1 dimensional space{time, assume �M �= B � R, with � : B ! � aprincipal U(1) bundle, � a compact surface. Further assume the group U(1)acts by isometries on ( �M; �g) with the action generated by the Killing �eldJ , which we assume to be space{like, hJ; Ji > 0.Let the function � on � � R be de�ned by �� = 1

2 log(hJ; Ji), and let

� = e�2��J . Then we can write the space{time metric �g in the form

�g = ��(e�2�g) + e2�� ;

where g is a Lorentzian metric on R� �.Introduce a frame4 e�, � = 0; 1; 2; 3 with e3 = e��J , and let the indices

a; b; c; � � �= 0; 1; 2. We may without loss of generality assume that [J; ea] = 0:We have

dJ�� = �� + 2(e�(�)J� � e�(�)J�);

where ��J� = 0. It follows that �� = ��(e2�F��) where F�� is a 2{form

on R� �. If F�� = 0, then J is hypersurface orthogonal.

4In [36], a frame dxa; �3 is used, where �3� = e�2�J� = e��e3� . Furthermore, theirF�� = d�� = e�2��.


To avoid cluttering up the notation we will in the following make nodistinction between �elds on the orbit space R� B and their pullbacks by�.The components of the 3+1 Ricci tensor are

�Rab = Rab � 2ra�rb�+rcrc�gab � 1

2e4�FacF

ca ;

�Ra3 =1

2e��rc(e

4�F ca );

�R33 = e2�[�gabrarb�+1

4e4�FabF

ab] (5.1)

Let the one{form E on R� � be given by E = � ?g (e4�F ). One of theEinstein equations (5.1) implies that dE = 0.Now the Einstein vacuum equations �R�� = 0 imply the system

Rab =1

2(4ra�rb�+ e�4�EaEb); (5.2a)

rara�+1

2e�4�EaEbg

ab = 0; (5.2b)

rara! � 4ra�Ea = 0: (5.2c)

When Ea = ra!, we recognise (5.2) as the 2+1 dimensional Einsteinequations with wave map matter, for the wave map with components (�; !)with target hyperbolic space H2 with the constant curvature metric

2d�2 +1

2e�4�d!2:

See subsection 2.4 for some discussion of wave map equations.We now specialise even further to the polarized case Ea = 0. This corre-

sponds to assuming that the bundle � : B ! � is trivial, and that the vector�eld J is hypersurface orthogonal. Then ! is constant and the equations(5.2) become

Rab = ra�rb� (5.3a)

rara� = 0 (5.3b)

which is precisely the 2+1 Einstein equations coupled to a massless scalar�eld.In the 2{dimensional case, the operator k ! divk is elliptic on symmetric

2{tensors with vanishing trace. Further, by the uniformization theorem, acompact 2{dimensional Riemannian manifold (�; h) is conformal to (�; [h])where [h] is a representative of the conformal class of g, i.e. a constant cur-vature metric. Therefore working in spatial harmonic gauge with respect to(�; [h]) (i.e. conformal spatial harmonic gauge) and CMC time gauge, theconstraint equations form an elliptic system for (hij ; kij). We get a represen-tation of (hij ; kij) in terms of ([h]; �; k

TT; Y ) where [h] is the conformal classof h, corresponding to a point in Teichm�uller space, � a conformal factor de-termined by the Hamiltonian constraint equation (a nonlinear elliptic system

28 L. ANDERSSON

for �), kTT a trace{free, divergence free 2{tensor on (�; [h]), correspondingto a quadratic di�erential, and �nally, Y is a vector �eld determined fromthe momentum constraint equation. Note that ([h]; kTT) represents a pointin T �T (�), the cotangent bundle of the Teichm�uller space of �.Due to the ellipticity of the constraint equations in the 2+1 dimensional

case, in the gauge as described above, it is possible to eliminate the Ein-stein equation (5.2a) from the system (5.2) and instead solve the elliptic{hyperbolic system consisting of the hyperbolic system (5.2b-5.2c) coupledto the constraint and gauge �xing equations.In the special case of � = S2, the Teichm�uller space is a point, so that the

degrees of freedom in (h; k) are completely represented by (�; Y ). Further,H1(S2) = 0, which means that E = d!. Therefore in case � = S2, the �eldequations are exactly the Einstein{wave map equations in 2+1 dimensions.In the polarized case, the equations take the form

�g� = 0;

g = F [�; @�];

where �g denotes the wave operator on (R�S2; g), and F is de�ned by solv-ing for the components of g using the elliptic constraint and gauge equations.A special case of the polarized U(1) equations is given by setting � �

constant. Then the �eld equations (5.3) are just

Rab = 0;

the 2+1 dimensional vacuum equations. In this case, the space{time is a3-dimensional Lorentzian space{form. The dynamics of 2+1 dimensionalvacuum gravity has been studied by Andersson, Moncrief and Tromba [9]who proved global existence in CMC time for 2 + 1 dimensional vacuumspace{times, with cosmological constant, containing at least one CMC hy-persurface. This proves Conjecture 3 for the class of 2+1 dimensional vac-uum space{times. For cosmological constant � � 0 existence of a CMChypersurface can be proved [8].Recent work by Isenberg and Moncrief [79], shows that the polarized

U(1) equations may be formulated as a Fuchsian system, and therefore andtherefore AVTD solutions may be constructed using a singular version of theCauchy-Kowalewskaya theorem, as was done by Kichenassamy and Rendall[89] for the Gowdy case. This supports the following conjecture.

Conjecture 5. Generic polarized U(1) space{times are AVTD at the sin-gularity.

Remark 5.1. Conjecture 5 was essentially stated by Grubi�si�c and Moncrief[67]. This is supported by numerical work of Berger and Moncrief [29].

It seems reasonable to expect that polarized U(1) space{times which areAVTD at the singularity have a strong curvature singularity generically, andtherefore that proving the AVTD conjecture for polarized U(1) would be abig step towards proving SCC for this class.


In contrast to polarized U(1), the generic U(1) space{times have su�-ciently many degrees of freedom that one expects them to satisfy the BKLpicture of an oscillatory approach to the singularity, as is also expected in thefully 3+1 dimensional case, cf. the remarks in section 7. This is supportedby the numerical evidence so far, see [28].In the expanding direction, on the other hand, a small data semi{global

existence result holds [34], similar to the one for the full 3+1 dimensionalcase discussed below in section 6.1. In this case, the notion of \small data" istaken to be data close to data for a background space{time with space{likeslice M = � � S1 where � is a Riemann surface of genus > 2 with hyper-bolic metric and M has the product metric. In this case, the backgroundspacetimes are products of at spacetimes as in Example 2.5 with n = 2,with the circle (such space{times are of type Bianchi III).

6. 3+1

In the 3+1 dimensional case with no symmetries, the only known facts onthe global properties of space{times are Lorentzian geometry results such asthe Hawking{Penrose singularity theorems [106] and the Lorentzian splittingtheorem of Galloway [61], see [21] for a survey of Lorentzian geometry.Here we are interested in results relevant to the SCC, Conjecture 1 and

the CMC conjecture, Conjecture 3, i.e. results about the global behaviorof solutions to the Cauchy problem for the evolution Einstein equations,in some suitable gauge. The Cauchy problem for the Einstein evolutionequations has been discussed in section 2.2.With this limitation there are essentially 3 types of results known and

all of these are small data results. The results are those of Friedrich onthe \hyperboloidal Cauchy problem", of Christodoulou and Klainerman onthe nonlinear stability of Minkowski space (recently generalized by Christo-doulou, Klainerman and Nicol�o to exterior domains) and the recent workof Andersson and Moncrief [7], cf. Theorem 6.3 below, on global existenceto the future for data close to the data for certain spatially compact at� = �1 (local) FRW space{times, again a nonlinear stability result. We willbrie y discuss the main features of these results.The causal structure of a Lorentz space is a conformal invariant. This

leads to the notion that the asymptotic behavior of space{times can bestudied using conformal compacti�cations or blowup. The notion of iso-lated system in general relativity has been formalized by Penrose in termsof regularity properties of the boundary of a conformally related space{time ( ~M; ~g), with null boundary I, such that ~M is a completion of �M , and

�g = ��(�2~g), where 2 C1( �M) is a conformal factor, � : �M ! ~M isa di�eomorphism of �M to the interior of ~M . Given assumptions on thegeometry of ( ~M; ~g) at I, Penrose proved using the Bianchi identities thatthe components of the Weyl tensor of ( �M; �g) decay at physically reasonablerates.

30 L. ANDERSSON

Friedrich derived a �rst order symmetric hyperbolic system from the Ein-stein equations, the \regular conformal �eld equations". This system in-cludes among its unknowns, components of the Weyl tensor, the conformalfactor and quantities derived from the conformally rescaled metric ~g. Thissystem has the property that under the Penrose regularity conditions at I,the solution can be extended across I. The fact that the regular confor-mal �eld equations gives a well posed evolution equation in the conformallycompacti�ed picture enabled Friedrich to prove small data global existenceresults by using the stability theorem for quasi{linear hyperbolic equations.In [56, Theorem 3.5] Friedrich proved small data, global existence to the

future for data (M; g; k) close to the standard data on a hyperboloid in Min-kowski space, satisfying asymptotic regularity conditions compatible witha Penrose type compacti�cation (the hyperboloidal initial value problem).This was later generalized to Maxwell and Yang{Mills matter in [57]. Initialdata for the hyperboloidal initial value problem were �rst constructed byAndersson, Chru�sciel and Friedrich [5], see also [3], [4].The result of Friedrich is a semi{global existence result, in the sense that

the maximal vacuum Cauchy development D(M) of the data (M; g; k) isproved to be geodesically complete and therefore inextendible to the future,but not to the past. In fact typically D(M) will be extendible to the pastand (M; g; k) may be thought of as a partial Cauchy surface in a largermaximal globally hyperbolic space{time ( �M; �g), cf. �gure 7. In view of theinextendibility to the future of D(M), the result of Friedrich may be viewedas supporting the cosmic censorship conjecture. In the case of the Einsteinequations with positive cosmological constant, the method of Friedrich yieldsa global existence result for data close to the standard data on M = S3 indeSitter space, cf. [56, Theorem 3.3].

I

M

D�(M)

D+(M)

i+

( �M; �g)

Figure 7. The semi{global existence theorem of H. Friedrich

The �rst true small data global existence result for the vacuum Einsteinequations, was proved by D. Christodoulou and S. Klainerman [39]. They


proved that for data (M; g; k) su�ciently close to standard data on a hy-perplane in Minkowski space, with appropriate decay at spatial in�nity,the MVCD ( �M; �g) is geodesically complete and therefore inextendible, cf.�gure 8. The Christodoulou{Klainerman global existence theorem there-fore supports the cosmic censorship conjecture. Recently, Christodoulou,

outgoing gravitationalwaves

M

inner regionouter region

I

Figure 8. The global existence theorem of D. Christodoulouand S. Klainerman

Klainerman and Nicol�o (work in progress, see [92]) have proved a new globalexistence result, for data (M; g; k) which are close to standard at Minkow-ski space data on an exterior region M n K, where K � M is compact sothat M n K �= R n Ball, and with weaker asymptotic conditions comparedto the Christodoulou{Klainerman theorem. The result of Christodoulou{Klainerman{Nicol�o (CKN) states that for a vacuum data set (M; g; k) whichis su�ciently close to the standard at Minkowski data on M nK, the out-going null geodesics in the causal exterior region D(M n K) are completeand D(M nK) is covered by a double null foliation, with precise control overthe asymptotics. It should be noted that the CKN theorem therefore cov-ers a more general class of space{times than the Christodoulou{Klainermantheorem and is not strictly a small data result. In particular, the maximalCauchy vacuum development of (M; g; k) may be singular for data satisfyingthe assumptions of the CKN theorem, cf. �gure 9. If the smallness assump-tion is extended also to the interior region, the Christodoulou{Klainermantheorem is recovered. The Einstein equations are quadratic in �rst deriva-tives, and therefore in 3+1 dimensions, one needs something like a nullcondition in order to get su�cient decay for a global existence argument.The Einstein equations are not known to satisfy a null condition. However,by a detailed construction of approximate Killing �elds and approximate

32 L. ANDERSSON

( �M; �g)

D(MnK)

MnK K

horizon

singularity

strong �eld

MnK

to I

H(MnK)

Figure 9. The exterior global existence theorem of D.Christodoulou, S. Klainerman and F. Nicol�o. The �gure il-lustrates a situation which is covered by this theorem, with asingularity forming due to a strong gravitational �eld in theinterior region I(K) while the exterior region D(M nK) hascomplete outgoing null geodesics reaching I.

conformal Killing �elds, Christodoulou and Klainerman are able to controlthe behavior of components of Bel{Robinson tensors (cf. subsection 6.2below) constructed from the Weyl tensor and its derivatives, and close abootstrap argument which gives global existence for su�ciently small data.As part of this argument, it is necessary to get detailed control over the as-ymptotic behavior of light cones. Christodoulou and Klainerman also studythe asymptotic behavior of components of the Weyl tensor and are able toprove that some, but not all of these have the decay implied by the Penroseconditions on I.It is an open problem to construct a non{ at vacuum space{time which

satis�es the Penrose conditions. The work in [5] and [3] shows that generichyperboloidal data do not satisfy the required regularity at the conformalboundary, which therefore may be viewed as an indication that Penroseregularity at I is non{generic.In contrast to the results of Friedrich and Christodoulou{Klainerman{

Nicol�o, the semi{global existence result of Andersson and Moncrief [7] dealswith spatially compact vacuum space{times. The theorem states that fora vacuum data set (M; g; k), su�ciently close to the standard data in a


spatially compact � = �1 (local) FRW space{time, the MVCD ( �M; �g) iscausally geodesically complete in the expanding direction. It is a conse-quence of the singularity theorem of Hawking and Penrose that ( �M; �g) issingular, i.e. geodesics in the collapsing direction are incomplete, cf. �g-ure 10. In the rest of this section, we discuss the result of Andersson andMoncrief [7].

M

( �M; �g)

Figure 10. The semi{global existence theorem of L. Ander-sson and V. Moncrief.

6.1. A new small data semi{global existence result for a class of

spatially compact vacuum space{times. First we need some de�ni-tions. Let (M; ) be a compact hyperbolic 3{manifold M with the standardsectional curvature �1 metric . Given a compact hyperbolic 3{manifold(M; ) let ( �M; � 0) be the at � = �1 (local) FRW space{time given by�M = (0;1)�M ,

� = �dt2 + t2 :

Then ( �M; � ) is a at space{time of the type discussed in Example 2.5. Wewill call such space{times standard space{times.A at space{time is a geometric structure with group G = ISO(3; 1) =

SO(3; 1) n R4 (where we take SO(3; 1) to mean the connected componentof the identity in O(3; 1)), the inhomogeneous Lorentz group. A standardspace{time is a at space{time with holonomy � � SO(3; 1). The Mostowrigidity theorem says that the moduli space of hyperbolic structures on acompact hyperbolic manifold of dimension � 3 consists of one point, i.e.the deformation space of the holonomy � in SO(3; 1) consists of one point.However, the deformation space of � in the larger groups ISO(3; 1) andSO(4; 1) may be nontrivial. For the case SO(4; 1) the deformation space isthe space of conformally at structures on M , and for the case ISO(3; 1)the deformation space is the space of at space{times. The formal tangent

34 L. ANDERSSON

space at � to the deformation space of � in ISO(3; 1) is H1(�;Adiso(3;1)),the 1-cohomology of � � SO(3; 1) � ISO(3; 1) w.r.t. the adjoint action ofISO(3; 1) on its Lie algebra iso(3; 1), cf. [85] for background.Due to the in�nitesimal version of Mostow rigidity (the Calabi{Weyl rigid-

ity theorem) and the structure of the Lie algebras iso(3; 1) and so(4; 1),

H1(�; vecR4)�= H1(�;Adso(4;1)) �= H1(�;Adiso(3;1)): (6.1)

where vecR4 is the vector representation of SO(3; 1) on R4 = R3;1, cf. [88,x11] for the �rst part of the above identity. Hence the dimension of thedeformation spaces of at conformal structures and of at space{times arethe same.A symmetric 2{tensor hij is called a Codazzi tensor if

rihjk � rjhik = 0:

Note that a Codazzi tensor with zero trace also has zero divergence.

De�nition 6.1. A hyperbolic 3{manifold (M; ) is called rigid if it admitsno non{zero Codazzi tensors with vanishing trace. A standard space time( �M; � ) is called rigid if (M; ) is rigid.

A computation [95] shows that in�nitesimal rigidity in this sense at (M; )in the space of at conformal structures is equivalent to the non{existenceof trace{free Codazzi tensors on (M; ). In view of the above remarks, orby a direct computation, the space of trace{free Codazzi tensors on (M; )is isomorphic to the formal tangent space of the deformation space of atspace{times at ( �M; �g), and thus the two notions of rigidity in De�nition 6.1are equivalent.Kapovich [87, Theorem 2] proved the existence of compact hyperbolic 3{

manifolds which are rigid w.r.t. in�nitesimal deformations in the space of at conformal structures.The above proves

Proposition 6.2. The class of rigid hyperbolic 3{manifolds (M; ) (andrigid standard space{times ( �M; � )), in the sense of De�nition 6.1, is non{empty.

We are now able to state

Theorem 6.3 (Andersson, Moncrief [7]). Let (M; ) be a hyperbolic mani-fold of dimension 3, and assume (M; ) is rigid. Assume that (M; g0; k0)is a constant mean curvature vacuum data set so that �0 = trg0k

0 < 0. Let

( �M; �g) denote the MVCD of (M; g0; k0).There is an � > 0 so that if

jj j�0j2

9g0 � jjH3 + jj j�0j

3k0 + jjH2 < �: (6.2)

then

1. ( �M; �g) is globally foliated by CMC Cauchy surfaces in the expandingdirection (to the future of M�0 with respect to the CMC time � = trgk).


2. ( �M; �g) is future causally geodesically complete.3. ( �M; �g) is has a singularity in the past of M�0 (in the contracting direc-

tion).

Remark 6.1. (i). A scaling argument shows that there are initial data(g0; k0) corresponding to arbitrarily large space{time curvature, so thatTheorem 6.3 applies, in this sense the initial hypersurface may be \closeto the singularity".

(ii). An argument based on the energy estimates used in the proof of Theo-rem 6.3 shows that the standard space{time is unstable in the collapsingdirection, in the sense that there is a neighborhood U of the standarddata of the form given in (6.2), so that the evolution in the collapsingdirection, of data arbitrarily close to, but not equal to standard data,will leave U .

(iii). In the case of a low density � :3 universe,5 a � = �1 (local) FRWmodel is required to explain observations. This allows for the possibilitythat the spatial topology of the universe is that of a compact hyperbolicmanifold, i.e. that the universe is similar to the space{times discussedin Theorem 6.3. Recently, there has been a great deal of interest in thepossibility that the topology of the universe can be detected, using ob-servations on the cosmic background radiation, cf. [47] and referencestherein.

The main features of the proof of Theorem 6.3 are

� local existence in Hs�Hs�1, s > 3=2+ 1, for the gauge �xed Einsteinevolution equations in spatial harmonic gauge and CMC time, see be-low. These gauge conditions lead to an elliptic{hyperbolic system for(g; k;N;X).

� A continuation principle.� Decay estimates for the Bel{Robinson energy (see subsection 6.3 forde�nition) for small data.

� Non{degeneracy of the Bel{Robinson energy if M is rigid.

Let g be a �xed C1 metric on M , for example g = . Let �kij ; �kij be the

Christo�el symbols w.r.t. g; g respectively and let

V i = gmn(�imn � �imn):

Then V i = 0 if and only if the identity map Id : (M; g)! (M; g) is harmonic.Let

�ij =1

2(riVj +rjVi):

Then

Rij � �ij = �1

2�ggij + Sij [g; @g];

5the value of is subject to intense debate and varies with time, the current (Oct.1999) favored values of and �, in view of recent super{nova data and the in ationparadigm are � = 0 and = 1 with a signi�cant positive cosmological constant makingup part of .

36 L. ANDERSSON

where �g is the scalar Laplace{Beltrami operator on (M; g), is elliptic, sothe system

@tgij = �2Nkij + LXgij ; (6.3a)

@tkij = �rirjN +N(Rij + trkkij � 2kimkmj � �ij) + LXkij ; (6.3b)

is hyperbolic.We now consider the elliptic{hyperbolic system consisting of (6.3) and

the following elliptic system of de�ning equations for (N;X).

��N + jkj2N = 1; (6.4a)

�X i +RifX

f = (�2kab + 2raXb)(�iab � �iab)

+ LXV i + 2raNkia � riNk aa : (6.4b)

on (N;X).We prove local existence and uniqueness for the system (6.3), (6.4) in

Hs � Hs�1, s > n=2 + 1, using an adaptation of standard methods. Theproof relies on the fact that the operator

X i 7! LX i = �X i + RifX

f � LXV i;

has no terms containing second order derivatives of gij and therefore isregularizing from Hs�1 to Hs+1.Let the gauge and constraint quantities A; F; V i; Di be de�ned by

A = trk � t; B = R+ (trk)2 � jkj2;�ij =

1

2(riVj +rjVi); � = riV

i;

F = B � �; Di = ritrk � 2rmkmi:

Let now (g0; k0) 2 Hs �Hs�1, s > 3=2 + 1 and assume (g0; k0) are closeto ( ;� ). A stability argument shows that the elliptic system (6.4), hasunique solutions in Hs+1 � Hs+1 for (g; k) 2 Hs � Hs�1, close to ( ;� ).Let (N0; X0) be given by solving (6.4) at the given data (g0; k0). Thisconstructs initial data (g0; k0; N0; X0) at some initial time t0 for the elliptic{hyperbolic system (6.3), (6.4), and by the local existence we get a curve(g; k;N;X) 2 C((T0; T1);Hs �Hs�1 �Hs+1 �Hs+1) solving (6.3), (6.4).Let now (g; k;N;X)2 C((T0; T1);Hs�Hs�1�Hs+1�Hs+1) be a solution

curve to (6.3), (6.4) as above.The system (6.3), (6.4) implies that the quantities (A; F; V i; Di) satisfy a

hyperbolic system, and hence energy estimates show that if A; F; V i; Di arezero initially, they are zero during the interval of existence (T0; T1).This means that a solution to the system (6.3), (6.4) with vacuum initial

data (g0; k0) satisfying the gauge conditions

V i[g0] = 0 spatial harmonic gauge; (6.6a)

trg0k0 = t0 CMC time gauge; (6.6b)


leads to a solution curve (g; k;N;X) with (A; F; V i; Di) � 0 and hencethis solution curve is a curve of solutions to the original Einstein evolutionequation (3.3). We have thus constructed a vacuum space time.For small data g0 close to there is a unique harmonic map �0 : (M; g)!

(M; ) which, by an implicit function theorem argument, is a di�eomor-phism. Replacing (g0; k0) by the pushforward under �, (��g0; ��k0) gives adata set which satis�es the spatial harmonic gauge condition. This showsthat there is no loss of generality for small data to assume in the followingthat the spatial harmonic gauge condition is satis�ed.Let T� be a maximal time of existence for the system (6.3), (6.4) with

initial data satisfying the gauge conditions (6.6) with t0 = trg0k0 < 0. Then

by the above t = trgk for T0 < t < T�. By assumption M is of hyperbolictype and therefore of Yamabe type �1, so by remark 2.1, T� � 0.A continuation principle for the elliptic{hyperbolic system (6.3),(6.4) with

(g; k;N;X) 2 Hs � Hs�1 � Hs+1 � Hs+1, s > n=2 + 1 shows that if T� ismaximal then either T� = 0 or (g; k;N;X) diverges in W 1;1 � W 0;1 �W 1;1 � W 1;1, as t % T�. It follows that it is su�cient to control the

rescaled data (~g; ~k) = ( jtj2

9 g; jtj3 k), t = trgk, in H3 � H2 in order to getT� = 0, i.e. semi{global existence. We use the higher order Bel{Robinsonenergy estimates to get control over the rescaled data.

6.2. Weyl �elds. A trace free tensor �eld Wabcd with the symmetries ofthe Riemann tensor is called a Weyl �eld. Let �abcd be the volume elementof ( �M; �g) and let

�Wabcd =1

2� efab Wefcd;

W �abcd =

1

2Wabef �

efcd:

Given a Weyl �eld Wabcd in ( �M; �g) we de�ne

Jbcd = �raWabcd; J�bcd = �raW �abcd:

Then J = J� = 0 if and only if W satis�es the homogeneous Bianchi iden-tities �r[aWbc]de = 0.

Remark 6.2. If Wabcd = Cabcd, the Weyl tensor of the space{time ( �M; �g),then Wabcd satis�es the homogeneous Bianchi identities if �Rab = 0, i.e. if( �M; �g) is a vacuum space{time.

Let T be the future directed time{like normal to the foliation Mt anddecompose W into electric and magnetic parts w.r.t. T ,

E(W )ab = WaebfTeT f ; B(W )ab = W �

aebfTeTf:

Then E;B are symmetric, satisfy EabTa = BabT

a = 0, and are trace{free.De�ne the �rst order Weyl �eld by

(1)

W abcd = T f �rfWabcd

38 L. ANDERSSON

Then(1)

W abcd and Wabcd are related by the following Maxwell like equations,

E((1)

W)ij = curlBij � 3

2(E � k)ij +

1

2trkEij + J(ij)aT

a; (6.7a)

B((1)

W )ij = �curlEij � 3

2(B � k)ij +

1

2trkBij � J�(ij)aT

a; (6.7b)

where

curlAij =1

2(� mni rnAmj + � mn

j rnAmi)

(A� B)ij = � k`i � mn

j AkmB`n +1

3(A �B)gij ;

for symmetric 2{tensors Aij ; Bij with A trace free. In the right hand side of(6.7), E;B; J; J� are de�ned w.r.t. W .

6.3. The Bel{Robinson tensor and the Bel{Robinson energy. Givena Weyl �eld Wabcd, de�ne a tensor Qabcd by

Qabcd = WaecfWe f

b d +W �aecfW

� e fb d :

Qabcd is called the Bel{Robinson tensor of W . Qabcd is totally symmetricand trace{free. Computations show

QabcdTaT bT cT d = jEj2 + jBj2;

�raQabcdTbT cT d = 2EijJicjT

c + 2BijJ�icjTc:

The Bel-Robinson energy is de�ned by

Q(�;W ) =

ZM�

QabcdTaT bT cT dd�g =

ZM�

(jEj2+ jBj2)d�g;

and satis�es the propagation equation

@�Q(�;W ) = �2ZM�

N(EijJicjTc+BijJ�icjT

c)d�g�3ZM�

NQabcd�abT cT dd�g;

where �ab = �raTb.Let �ab = �ab +

trk3 (�gab + TaTb). Then

J((1)

W )icjTc = �trk

3E(

(1)

W ij + �af �rfWai0j + terms quadratic in W; (6.8)

and similarly for J�((1)

W ). Now note that

� � is small for small data (g; k).

� The term � trk3 E(

(1)

W) in the expression (6.8) for J((1)

W ) gives a negative

term in @�Q(�;(1)

W ).� Only spatial derivatives ofW occur in @�Q and these may be controlledin terms of Q.


If (M; ) is rigid, it can be shown that if ( �M; �g) is vacuum, �Rab = 0,Wabcd = �Rabcd, the Weyl tensor of ( �M; �g), then with � = trgk, there is an� > 0 so that for (g; k) satisfying the smallness condition (6.2), the estimate

jj j� j2

9g � jj2H2 + jj j� j

3k + jj2H1 � CQ(t;W )

holds for some constant C, and for (g; k) satisfying the constraint equa-tions (2.2) and the gauge conditions (6.6). It follows that under the sameconditions, the higher order estimate

jj j� j2

9g � jj2H3 + jj j� j

3k + jj2H2 � C

Q(�;W ) +Q(�;

(1)

W )

!holds.

6.4. The model case. Let ( �M; � ) be the standard at space{time with�M = (0;1)�M ,

� = �d�2 + �2 ;

so that g(�) = �2 . Then T = @�, the mean curvature is � = �3=�. Further

N =3

�2; �ab = ��

3(�gab + TaTb);

g(�) =9

�2 ; k(�) =

3

� :

Now consider linearized gravity on this background. Let s 7! �gs be a curveof vacuum metrics �R[�gs]ab � 0, with �g0 = � . Let

�g0 =@

@s

��s=0

�gs:

Then �R0abcd is a Weyl �eld, which satis�es the homogeneous Bianchi iden-tities. Now apply the propagation law for the Bel{Robinson energy. Usingthe above identities this gives

@�Q(�;W ) = �3ZM�

NQabcd�abT cT dd�g

= 3

ZM�

3

�2Qabcd

�

3(�gab + T aT b)T cT dd�g

=3

�Q(t;W ):

This implies Q(�;W ) = Q(�0)j� j3 ! 0 as � % 0.We now consider the small data case. With � = trgk, and �0 = trg0k

0 <0, as � % 0, the space{time expands, and hence the Sobolev constantsdegenerate. This means that in order to control the smallness condition(6.2), we must consider a rescaled energy.

40 L. ANDERSSON

Let � := j� j3 and let t = trk be a scale{free time parameter. Let Wabcd =

�Rabcd and assume �Rab = 0. Let

E = ��1Q(�;W ) + ��3Q(�;(1)

W ):

Then E is scale independent.By the above, for small data there is a constant C so that for (g; k)

satisfying the constraint equations (2.2) and the gauge conditions (6.6), wehave the estimate

jj�2g � jj2H3 + jj�k+ jj2H2 � CE : (6.9)

A lengthy argument based on the Maxwell type identity (6.7), the propa-gation law for Q, and using the estimate (6.9) to control (N;X) via (6.4)shows that there is a constant C so the energy estimate

@tE � 2� 2CE1=2t

E (6.10)

holds. The model equation for the di�erential inequality (6.10),

y0 =2� 2Cy1=2

ty; y(t0) = y0 > 0; t0 < 0;

has global existence as t % 0 for small data y0 and y � Cjtj2 for some

constant C, as t % 0. Further, y 7! 2�2Cy1=2t y is monotone decreasing for

small data (as t < 0). Now a comparison argument implies decay for therescaled energy.It follows from the above argument that we can choose � > 0 small enough

so that the rescaled data (�2g; �k) stay close to ( ;� ). This means thatthe continuation principle shows that the maximal existence interval for thesystem (6.3), (6.4) with small data satisfying the gauge conditions (6.6) andthe constraints (2.2), is of the form (T0; 0). This proves point 1 of Theorem6.3. Point 2 is proved by estimating the Christo�el symbols, and point 3 isa direct consequence of the the Hawking{Penrose singularity theorems.

7. Concluding remarks

The results and numerical studies so far can be argued to �t with theso called BKL picture of cosmological singularities, which states that thegeneric singularity should be space{like, local and oscillatory, see [25] for arecent review. Roughly, one expects that the \points on the singularity"are causally separated and that locally at the singularity, the dynamicsundergoes a chaotic sequence of curvature driven \bounces" interspersedwith relatively uneventful \coasting" epochs, with Kasner like dynamics(i.e. the metric is, locally in space, approximately of the form given in(3.6)). The locality at the singularity, is easily checked for the nonvacuumFriedman-Robertson-Walker models of the standard model of cosmology,and is the cause of the so{called \horizon problem" in cosmology. Localityat the singularity can be proved for Gowdy and polarized U(1).


The picture with bounces and coasting is to a large extent inspired bythe Mixmaster dynamics (Bianchi IX, cf. section 3) and the above scenariomay therefore be described by the slogan \generic local mixmaster at thesingularity". The mixmaster behavior may be prevented by the presence ofcertain types of matter such as a scalar �eld (sti� uid). In fact recent workby Andersson and Rendall [10], shows that the 3+1 dimensional Einstein-scalar �eld equations may be formulated as a Fuchsian system near thesingularity, and AVTD solutions may be constructed using a singular versionof the Cauchy{Kowalevskaya theorem. This may also be relevant for thepolarized U(1) case, which leads to 2+1 dimensional Einstein-scalar �eldequations, as mentioned in section 5. In view of the above mentioned work,it is likely that AVTD solutions to the polarized U(1) equations can beconstructed using the Fuchsian algorithm.It appears likely that some aspect of the picture sketched above will be

relevant for the �nal analysis of the large data, global behavior of vacuumspace{times.

Appendix A. Basic Causality concepts

Here we introduce the basic causality concepts, see [128, Chapter 8],[21,68, 106] for details.A vector V 2 T �M is called space{like, null or time{like if �g(V; V ) > 0;= 0

or < 0 respectively. V is called causal if it is null or time{like. A C1 curvec � �M is called time{like (causal) if _c is time{like (causal). This extendsnaturally to continuous curves. A hypersurface M � �M is called space{like(acausal) if the normal of M is time{like (causal).Given a closed subset S � �M , the domain of dependence D(S) is the

set of points p 2 �M such that any inextendible causal curve containing pmust intersect S. If S � �M is a space{like hypersurface and D(S) = �M , Sis called a Cauchy surface in �M . If �M has a Cauchy surface �M is calledglobally hyperbolic. A globally hyperbolic space{time has a global timefunction, i.e. a function t on �M so that �rt is time{like, with the level sets oft being Cauchy surfaces. In particular, �M �=M �Rwhere M is any Cauchysurface.

�M is called time oriented if there is a global time{like vector �eld on �M .A globally hyperbolic space{time is time oriented by the above, and henceit makes sense to talk about the future and past domains of dependenceD+(M) and D�(M) of a Cauchy surface M . In a time oriented space time,the chronological future I+(S) of S � �M is the set of all points reached byfuture directed time{like curves, of nonzero length, starting on S. The time{like past I�(S) is de�ned analogously. The causal future and past J+(S) andJ�(S) is de�ned analogously to I�(S) with causal curve replacing time{likecurve (the causal curve is allowed to be trivial).The variational problem for geodesics (w.r.t. Lorentzian length) is well

behaved precisely when ( �M; �g) is globally hyperbolic. Let C(p; q) be the setof continuous causal curves between p; q 2 �M . Then C(p; q) is compact w.r.t.

42 L. ANDERSSON

uniform convergence for all p; q 2 �M , if ( �M; �g) is globally hyperbolic. Thismakes global hyperbolicity a natural assumption in Lorentzian geometry,just as completeness is a natural assumption in Riemannian geometry.Global hyperbolicity may also be characterized by strong causality and

compactness of C(p; q) for all p; q 2 �M , or compactness of J+(p)\J�(q) forall p; q 2 �M , see [128, Chapter 8] for details.Given a space{time ( �M; �g) and a space{like hypersurface M � �M , the fu-

ture Cauchy horizon H+(M) is given by H+(M) = D+(M)nI�[D+(M)],with the past Cauchy horizonH�(M) de�ned analogously. The setH(M) =H+(M)[H�(M) is called the Cauchy horizon of M . It can be proved thatH(M) = @D(M), the boundary of D(M).

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Department of Mathematics, Royal Institute of Technology, 100 44 Stock-

holm, Sweden,

E-mail address: [email protected]

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