aa r X i v : . [ m a t h - ph ] A p r Local extendability of Einstein vacuum manifolds
David ParlongueNovember 6, 2018
Abstract
We revisit in this article results of Klainerman and Rodnianski on ageometric breakdown criterion for Einstein vacuum spacetimes. We takeadvantage of the use of a time-harmonic transversal gauge to give a localizedversion (in space and time) of this result.
The Cauchy problem for Einstein vacuum equations has been largely studied inthe last fifty years since the first results on local well-posedness to the non-linearstability of the Minkowski spacetime and beyond low-regularity solutions, relationsbetween general relativity and geometrization ... The question of formation of sin-gularities is however quite open and remains one of the main challenge of classicalgeneral relativity.From a mathematical point of view a geometric breakdown criterion involvingonly control on the L ∞ norm of the second fundamental form and the horizontalderivative of the lapse has been proved for Einstein vacuum equations (EVE) in theconstant mean curvature gauge (CMC) as well as in the asymptotically flat case(see [9]). These results have been extended to criteria with weaker conditions andalso in the direction of Einstein equations coupled with a scalar field or Maxwellequations.These results are however global in space and do not give information on localextendability of vacuum spacetimes. The goal of this article is to localize in spacethese criteria and apply these local theorems to prove a result on formation ofsingularities and local extendability properties of vacuum spacetimes. We firstreview some continuation results for EVE. Global breakdown criteria
We consider in this paper a dimensional Lorentzian manifold ( M , g ) satisfying the EVE : R αβ ( g ) = (1)We suppose that a part of the space-time denoted by M ⊂ M is globallyhyperbolic with respect to a spatial hypersurface Σ and foliated by the levelhypersurfaces of a regular time function denoted by t . We suppose moreover thatthis time function is monotonically increasing towards future, with lapse function n and second fundamental form k . We will use the following sign convention : k ( X, Y ) = − g ( D X T , Y ) , (2) n = ( − g ( Dt , Dt )) − / (3)Throughout this paper T , will denote the future unit normal to the leaf Σ t of the time foliation, ( T ) π its deformation tensor and D the covariant derivativeassociated with g . Let Σ be a slice of the time foliation which will be referredas the initial slice. The EVE can be seen as an initial value problem once aninitial data set is given. An initial data set consists in a 3 dimensional Riemanianhypersurface Σ together with a 3 dimensional Riemannian metric g and a secondfundamental form k . (Σ , g, k ) has moreover to satisfy the following constraint equations which takethe following form in the maximal gauge regime : trk = 0 , (4) ∇ j k i,j = 0 , (5) R = | k | (6)where here R stands for the Riemann curvature of the metric g and ∇ the covariantderivative associated to g . The other equations coming from the reformulation ofEVE are the following evolution equations : ∂ t g ij = − nk ij , (7) ∂ t k ij = −∇ i ∇ j n + n ( R ij − k ia k aj ) (8)together with the lapse equation : ∆ n = | k | n (9)with condition n → at spatial infinity on Σ .2he continuation result of [9] holds true for maximal foliations as well as forconstant mean curvature foliations (CMC). In this case the leaves are compact andthe mean curvature can be taken as being the time function. We will thus considerhere the case of maximal foliation, the one with CMC foliation being simpler dueto the compacity of the leaves. Σ is said to be asymptotically flat if the complement of a compact set K ⊂ Σ is diffeomorphic to the complement of a -sphere and that there exists a system ofcoordinate in which : g ij = (1 − Mr ) δ ij + o ( r − / ) (10) k ij = o ( r − / ) (11) [Geometric assumption (G)] The initial surface Σ is said to satisfy (G) if and only if there exists a converging of Σ by a finite number of charts U suchthat for any fixed chart, the induced metric g verifies : C − | ξ | g ij ( x ) ξ i ξ j C | ξ | , ∀ x ∈ U with C a fixed number.The system of equations (4) to (9) is a determined system of equation. Weare naturally led to ask the question of well-posedness for this system. The firstresult of this theory is the local well-posedness result of Choquet-Bruhat [2], hereas stated in [3]: Theorem 2.0.1. (Local existence theorem) Let (Σ , g , k ) be an initial data setverifying the following conditions :1) (Σ , g ) is a complete Riemannian manifold diffeormorphic to R .2)The ispoperimetric constant of (Σ , g ) is finite.3) Ric ( g ) ∈ H , (Σ , g ) and k ∈ H , (Σ , g ) .4) The initial slice satisfies the constraint equations.Then there exists a unique, local-in-time smooth development, foliated by anormal, maximal time foliation t with range in some [0 , T ] and with t = 0 corre-sponding to the initial slice Σ . Moreover :a) g ( t ) − g ∈ C ([0 , T ] , H , ) b) k ( t ) ∈ C ([0 , T ] , H , ) c) Ric ( g )( t ) ∈ C ([0 , T ] , H , ) where for a given tensorfield h on Σ , k h k H n,s denotes the norm : k h k H n,s = ( n X i =0 Z σ (1 + d ) s + i |∇ i h | ) d stands for the geodesic distance to a basepoint O .The natural question for such a local result is the minimum regularity of theinitial data required for having well-posedness. The scaling property of Einsteinvacuum equation could lead to think that the critical Sobolev exponent for g should be / . In wave coordinates, the local existence regularity can be extendedto / ǫ Sobolev regularity. An important improvement led to the proof for ǫ (see [5]). The so-called L -conjecture asserts that the optimal regularity for theinitial data set is Ric ( g ) ∈ L (Σ ) and ∇ k ∈ L (Σ ) .We are now ready to state the main result of [9] : Theorem 2.0.2.
Let ( M , g ) be a globally hyperbolic development of an asymptot-ically flat initial data set (Σ , g , k ) satisfying the assumptions of the local in timetheorem and globally foliated by a normal, maximal foliation given by the level setsof a smooth time function t such that Σ corresponds to t = 0 . Suppose moreoverthat Σ satisfies the geometric condition (G) . We suppose that : a ) Z Σ | R | ( x ) dµ ( x ) ∆ (12) b ) (cid:13)(cid:13) n − (cid:13)(cid:13) L ∞ ([0 ,t [ ,L ∞ (Σ t )) ∆ (13) c ) k k k L ∞ ([0 ,t [ ,L ∞ (Σ t )) + k∇ ( log ( n )) k L ∞ ([0 ,t [ ,L ∞ (Σ t )) ∆ (14) than the space-time together with the maximal foliation can be extended beyondtime t . This theorem implies the following breakdown criterion :
Theorem 2.0.3 (Breakdown Criterion) . With the notations of the previous theo-rem, the first time T ∗ with respect to the t -foliation of a breakdown is characterizedby the condition : lim sup t → T ∗ (cid:0) k k k L ∞ (Σ t ) + k∇ ( log ( n )) k L ∞ (Σ t ) (cid:1) = + ∞ (15)The pointwise condition can be relaxed to obtain the following theorem (see[10]) Theorem 2.0.4. [Integral Breakdown Criterion]
We keep here the notations f the previous theorem. We suppose that there exists ∆ , ∆ and ∆ such that : a ) Z Σ | R | ( x ) dµ ( x ) ∆ (16) b ) (cid:13)(cid:13) n − (cid:13)(cid:13) L ∞ ([0 ,t [ ,L ∞ (Σ t )) ∆ (17) c ) Z t ( k k k L ∞ (Σ t ) + k∇ ( log ( n )) k L ∞ (Σ t ) ) ndt ∆ (18) than the space-time together with the maximal foliation can be extended beyondtime t . This theorem implies of course the corresponding breakdown criterion. In bothcases, the breakdown criteria are however (spatially) global. However, the questionof physical relevance related to continuation of solutions of Einstein equations isthe question of local extendability. Obtaining information on local extendabilityof spacetimes requires to localize in space these breakdown criteria.Let us recall some definitions relevant to cosmic censorship conjectures.
Definition 2.0.1.
A spacetime (M,g) is said locally inextendible if there existsno open subset
U ⊂ M with non-compact closure in M such that there exists anisometric imbedding φ : ( U , g |U ) → ( U ′ , g ′ ) where φ ( U ) has compact closure in U ′ . Conjecture 2.0.1 (Strong Censorship Conjecture) . The maximal globally hyper-bolic vacuum extension (MGHVE) of a "generic" initial data set (Σ , g, k ) where Σ is either compact or asymptotically flat is locally inextendible as a C , Lorentzianmanifold.
The difficulty throughout this paper will be that we will have to deal withslices S t which are non-complete Riemmanian manifold. Among the consequences,the solution constructed from data ( S t , g, k ) will typically have a temporal extenttending to zero near the boundary of S t . From another hand, some techniquesbased on elliptic estimates on the slices used in the proof of global criteria with Σ compact or asymptotically flat have to replaced by purely hyperbolic estimates.This will motivate the use of a transversal time-harmonic gauge. The main goal of this paper will be to localize the breakdown criteria of part 2.Among the different choices of gauge possible, some of them have been largely usedbecause of their physical significance (asympotically flat spacetimes and maximal5auge, CMC gauge) or because of their mathematical properties (the wave gaugeenables for instance to reduce the principal symbol of the Ricci tensor to − (cid:3) g g αβ ).We will however use here another choice of gauge well-suited for spatially localizedtheorems, a transversal time-harmonic gauge. The reason why this type of gaugecondition is well-suited for local-in-space applications is that the reduced systemthat we will obtain is hyperbolic instead of being a mixed hyperbolic-elliptic systemas in the case of the CMC or AF gauge where we have to solve an elliptic systemfor the lapse. In comparison with the harmonic gauge, we will also take advantageof the transversality.Fixing a gauge requires to fix four quantities while working on Σ × R withtransported coordinates, where Σ stands for a 3-dimensional Riemannian manifold.The choice made here is having a vanishing shift or equivalently a choice of time-lines orthogonal to the space-sections and an harmonic time index. That is ifdenote by ( x , x , x , x ) by a local set of coordinates : (cid:3) g t = − g αβ Γ αβ = 12 n ( ∂∂x ( log ( g )) − ∂∂x ( log ( n )) = 0 (19)where n stands for the lapse and g for the the determinant of the three-dimensional Riemannian metric on the spatial-sections. We see immediately thatthe harmonicity condition on the time-index leads to the following equality on thewhole four-dimensional manifold : n = √ gf (20)where f > stands for a scalar density of order one given on Σ . Equivalently,we can give ourself a Riemannian metric e on Σ and set f = ( det ( e )) − / .We set moreover : P ij = k ij − trkg ij (21)The reduction of the Cauchy problem for Einstein equations in this particulargauge was performed in [2] : ∂∂x g ij = n (2 P ij − g ij P ) (22) (cid:3) g P ij = nM ij (23) ∂∂x ( log ( n )) = 12 ∂∂x ( log ( g )) = − ntr ( k ) (24)with a set of data S = ( g , P , ∂∂x P , e ) . Where M stands for lower-order terms(of order at most 2 in g or n and one in P ). This is a quasi-diagonal hyperbolicsystem denoted by S for which standard techniques apply.6ecalling that : ∆ n = | k | n − ∂∂x tr ( k ) (25)and differentiating (24), we get : ∂ ∂x n = − n ( ∂∂x n ) trk − n ∂∂x trk (26) = 2 n ( trk ) − n | k | + n ∆ n (27)or (cid:3) g n = n ( | k | − trk ) ) + ( (3) Γ i − (4) Γ i ) ∇ i n = n ( | k | − trk ) + ∇ i ( logn ) ∇ i ( logn )) (28)Note that for a general transversal gauge, we have : (cid:3) g n = n ( | k | − trk ) + ∇ i ( logn ) ∇ i ( logn )) − ∂∂x ( n Γ ) − n (Γ ) + 4 n trk Γ The time-harmonic gauge plays an import role in the simplification leading toa fully hyperbolical structure for ( T ) π µν .Now if we denote by P αβ = g + T α T β the projection operator on Σ t and if wedenote without change of notation k αβ = − P να P µβ ( T ) π µν It has been proved in [2] that : ( 1 n ∂ ∂x − ∆ g )( nk ) ij = − nk h ( i R hj ) + 2 R h mi j nk hm + 2 ∇ ( i log ( n ) ∇ j ) ( ntrk ) + 4 nk ∇ i ∇ j log ( n )+ nk h ( i ∇ j ) ∇ h log ( n ) + ∇ h log ( n ) ∇ h nk ij + 3 nk ∇ i log ( n ) ∇ j log ( n )+4 nk ij Schematically, we can write : (cid:3) g k ij ≈ ( T ) π + R · ( T ) π + ( T ) π · D ( T ) π (29)where we have used that R ≈ R + k . In fact, we remark that ( T ) π satisfies awave equation. Indeed using (28), T = n D t and : (cid:3) g D α D β t = − R µ να β D µ D ν t (30)we get : (cid:3) g ( T ) π ≈ ( T ) π + R · ( T ) π + ( T ) π · D ( T ) π (31)From now on, in this part we will suppose that the two following propertiesare satisfied : 7 the Riemannian manifolds Σ t satisfy uniformly (G) • there exists C > , s.t. C − ≤ n ≤ C These properties will be satisfied as a consequence of (24) in our applications.The idea underlying the rest of this part is that the cubic wave equation islocally well-posed in dimension 3+1 for initial data in H ( R ) . Let us now considera generic equation of the form : (cid:3) m φ = P ( φ, D φ ) (32)where φ stands for a scalar function φ : R → R , and P ( X, Y ) = a + a X + a Y + a X + a , X + a XY where the a ij are supposed to be given in C ∞ ( R ) , we suppose also that a set ofinitial data ( φ t =0 , ∂ t φ t =0 ) = ( φ , φ ) is given in C ∞ ( R ) × C ∞ ( R ) . Then bystandard techniques, the Cauchy problem (32) with initial data ( φ , φ ) is locallywell-posed, and the unique solution φ can be extended to any slab [0 , T ] × R aslong as : Z T k φ k L ∞ dt ′ < ∞ (33)Note that it is crucial that we do not allow terms of the form a Y , a continu-ation condition for such an equation would require a L t L ∞ x control on ∂φ and notonly on φ . This is typically the case of the toy-model for Einstein equation givenby (cid:3) φ = ∂φ · ∂φ .Having in mind this model equation, we can hope use to prove a continuationprinciple for the wave equation satisfied by ( T ) π , provided that we are able to dealwith a curved spacetime with rough metric instead of a Minkowskian one.We consider from now on a regular subset of Σ t , Ω t and denote by I its past and Ω s = Σ s ∩ I . Let H denotes its lateral boundary which is supposed to be Lipschitzand L a null generator. This type of energy estimates for Einstein equationslocalized in a domain has been used in [1]. Theorem 3.1.1. ( T ) π ∈ L t L ∞ (Ω t ) , ( T ) π ∈ L (Ω ) and D ( T ) π ∈ L (Ω ) implies ( T ) π ∈ L ∞ t L (Ω t ) and D ( T ) π ∈ L ∞ t L (Ω t ) .Proof. We introduce the energy-momentum tensor O αβ associated to the previouswave equation : 8 αβ = h δδ ′ h γγ ′ ... D ( T ) α π δγ... D ( T ) β π δ ′ γ ′ ... − g αβ g µν h δδ ′ h γγ ′ ... D ( T ) µ π δγ... D ( T ) ν π δ ′ γ ′ ... O satisfies the following relations : O ( T , T ) = 12 | ( T ) π | as well as the following positivity property, forall pair of fututre oriented timelikeor null vectorfield ( S , S ) , O ( S , S ) ≥ .Moreover, we have : | D α ( O α ) | . ( | ( T ) π | | D ( T ) π | + | R || ( T ) π || D ( T ) π | + | ( T ) π || D ( T ) π | ) Denote by A t = R Ω t | D ( T ) π | − R Ω | D ( T ) π | . As a consequence of Stokes formula : A t + Z H O ( T , L ) dµ H ≤ Z t Z Ω s ( | ( T ) π | | D ( T ) π | + | R || ( T ) π || D ( T ) π | + | ( T ) π || D ( T ) π | ) dµ s nds ≤ Z t (cid:0) n (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ∞ (Ω s ) Z Ω s ( | ( T ) π | | D ( T ) π | + | R || D ( T ) π | + | D ( T ) π | ) (cid:1) dµ s ds . Z t (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ∞ (Ω s ) ( k R k L ∞ t L x + (cid:13)(cid:13) D ( T ) π (cid:13)(cid:13) L (Ω s ) + (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) ) nds using Gronwall lemma and the positivity of the flux R H O ( T , L ) dµ H : (cid:13)(cid:13) D ( T ) π (cid:13)(cid:13) L ∞ t L x (Ω t ) . C + (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ∞ t L x (Ω t ) (34)but remarking that if ≤ u ≤ s ≤ t , F us (Ω s ) ⊂ Ω u , we have for ≤ s ≤ t (where F us : Σ s → Σ u is obtained by following the integral lines of − ∂ t ). (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L x (Ω s ) ≤ (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L x (Ω ) + Z s (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ∞ ( F us (Ω s )) (cid:0) (cid:13)(cid:13) D ( T ) π (cid:13)(cid:13) L ( F us (Ω s )) + (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ( F us (Ω s )) (cid:1) which implies : (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ∞ u L x ( F us (Ω s )) ≤ C + D Z s (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ∞ ( F us (Ω s )) (cid:13)(cid:13) D ( T ) π (cid:13)(cid:13) L ( F us (Ω s )) ndu ≤ C + D Z s (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ∞ (Ω u ) (cid:13)(cid:13) D ( T ) π (cid:13)(cid:13) L (Ω u ) ndu (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L x (Ω s ) ≤ C + D Z s (cid:13)(cid:13) ( T ) π (cid:13)(cid:13) L ∞ Ω s ) (cid:13)(cid:13) D ( T ) π (cid:13)(cid:13) L (Ω s ) ndu plugging in (34) gives ( T ) π ∈ L ∞ t L (Ω t ) and D ( T ) π ∈ L ∞ t L (Ω t ) .Deriving wave equations for D k ( T ) π , | k | ≥ and using the same techniques,we deduce : Theorem 3.1.2.
If for a k ≥ , ( T ) π ∈ L t L ∞ (Ω t ) , D l ( T ) π ( t = 0) ∈ L (Ω ) for l ≤ k and D l R ∈ L ∞ t L (Ω t ) for l ≤ ( k − then : D l ( T ) π ∈ L ∞ t L (Ω t ) for l ≤ k . We also introduce the reduced flux of ( T ) π as : F ( π, p, δ ) = Z N − ( p,δ ) |6 ∇ ( T ) π | + |6 ∇ ( T ) L π | (35)where N − ( p, δ ) stands for the past null cone with vertex p and δ the nullpast radius of injectivity at p . We denote by L the (normalized) null geodesicgenerator of N − ( p, δ ) that is the vectorfield satisfying < L , L > = 0 , D L L = 0 and < T , L > ( p ) = 1 . We denote by s the corresponding affine parameter and S s thelevel spheres. denotes the restriction of D to S s and L the projection to S s of D L .We also have that : Theorem 3.1.3.
For all p ∈ M , F ( π, p, δ ) . C ( k ( T ) π k L t L ∞ (Ω t ) , k ( T ) π k L (Ω ) , k D ( T ) π k L (Ω ) , k R k L (Ω ) ) (36) Once the hyperbolic reduction has been performed, by a simple domain of de-pendance argument it’s sufficient to consider that the initial data is supported in afixed coordinate patch. Gauge conditions are propagated (see [2]) and consideringan initial data set (Σ , g , k ) satisfying the constraint equations and such that ( g, P ) satisfy the previous quasi-linear hyperbolic system S on Σ × R , then themetric : g = − n dt + g ij dx i dx j (37)10atisfies Einstein vacuum equations on this set. Moreover, let Ω be a givensubset of Σ and D (Ω) its domain of dependance for g , if ( g, P ) satisfies S on D (Ω) , the metric g satisfies Einstein vacuum equations on D (Ω) . We will now givea version of the local in time and existence which is gauge invariant. Moreover wetrack the dependance of the temporal extent of the MGHVE on the geometry ofthe initial slice. Theorem 3.2.1. [Local existence and uniqueness result]
Consider an ab-stract initial data set ( S, g, k ) for Einstein Vacuum Equation with ( g, P ) such that k R k L ( S ) + k∇ R k L ( S ) + k∇ R k L ( S ) ≤ α k P k L ( S ) + k∇ P k L ( S ) + k∇ P k L ( S ) + k∇ P k L ( S ) ≤ α then there exists a MGHVE ( M,g ) with Cauchy data ( S, g, k ) unique up toa diffeomorphism such that the future temporal extent of any p ∈ S denoted by T + ( p ) , that is the maximal length of future causal curves initiating at p in M satisfies : T + ( p ) ≥ T ( α, r , h ( p )) = min ( T ⋆ ( α ) , r , h ( p ) T ⋆ ( α )) > (38) where r , h ( p ) stands for the L , harmonic radius at p on S . (see below for thedefinition). Theorem 3.2.2.
Suppose moreover that a scalar density f ∈ H ( S ) is given on S satisfying ν − ≤ f ≤ ν for a ν > , then M can be foliated by a time function t satisfying (cid:3) g t = 0 , t = 0 and ( − g ( D t, D t )) − = n = f p det ( g ) on S in aneigborhood of S in M .Remarks : • contrary to standard local-in-time theorems, this version give an explicitbound on "how big" the MGHVE without compacity or asymptotic flatnesshypothesis . As there is no natural time in general relativity, the most naturalgauge-invariant way to measure the "temporal size" is the one chosen here.• the gauge hypothesis of the second theorem could have been replaced byother gauge choices. It says roughly that given some datas, we know that aneighborhood of S in M can be foliated.• the important aspect of relation (38) is that it enables us not only to comparethe "time of definition" of two solutions of Einstein vacuum equations cor-responding to data sets given on the same manifold but also correspondingto two general abstract initial data sets ( S, g, k ) and ( S ′ , g ′ , k ′ ) . This "timeof definition" depends on analytic and geometric quantities. Globally Hyperbolic Vaccum Extension orollary 3.2.1. If we suppose moreover that S is complete and has radius of in-jectivity bounded by below by i > then for all p ∈ S , r , h ( p ) ≥ r ( k R k H l ( S ) , i ) > ,which implies a uniform lower bound on the future temporal extent for the corre-sponding MGVHE. Instead of a lower bound on the radius of injectivity, we canrequire that the volume radius at scales less than one is uniformly bounded by belowon S . Corollary 3.2.2.
Suppose that S is open and that there exists α > , D > , d > and η > and p ∈ M such that :• d g ( p, ∂M ) ≥ d • vol ( B ( p, s )) ≥ ηs for all s ≤ d/ • k∇ j R k L ( M ) ≤ Q for ≤ | j | ≤ l then r l, h ( p ) ≥ r ( α, d, η ) > , thus : T + ( p ) ≥ T ( Q, d, η ) > (39)The proof will require some results of Cheeger-Gromov theory. Let us begin by the defining the harmonic L l, radius of a point p of a Riemannianmanifold ( M, g ) denoted by r lh ( p ) as the largest radius of a geodesic ball about p on which there exists an harmonic chart in which : δ ij ≤ g ij ≤ δ ij r h ( p ) | j |− / k ∂ j g s k L ( B ( p,r h ( p )) ≤ for every multiindex j, | j | ≤ l. the first relation will be written below : Spectrum ( g ) ∈ [1 / , .Let us now state the theorem : Theorem 3.3.1.
Let
Q > , d > and η > given, then there exists r ( Q, d, η ) > such that for any 3D-Riemannian manifold ( S, g ) and p ∈ M such that :• d g ( p, ∂M ) ≥ d • vol ( B ( p, s )) ≥ ηs for all s ≤ d/ • k∇ j R k L ( M ) ≤ Q for ≤ | j | ≤ l hen r l, h ( p ) ≥ r ( Q, d, η ) > Proof.
It is a version a Cheeger-Gromov control on the harmonic radius of a man-ifold (see [11]).We remark that if denote by r ( Q, d, η ) the largest r satisfying the previoustheorem, we have the following monotonicity properties :• for any ( d, η ) , r ( ., d, η ) is decreasing• for any ( Q, d ) , r ( Q, d, . ) is increasingMoreover we have that thee exists µ ( Q, η ) > such that : r ( Q, d, η ) ≥ min ( µ ( Q, η ) , dµ ( Q, η )) Proof.
Fix λ > , consider ( S, λ g ) and p ∈ S , denote by r l +2 , h ( p, λ ) its ( l + 2 , harmonic radius on ( S, λ g ) and r l +2 , h ( p ) its ( l + 2 , harmonic radius on ( S, g ) ,then by scaling property of the harmonic radius, r l +2 , h ( p, λ ) = λr l +2 , h ( p ) . But r l +2 , h ( p, λ ) ≥ r ( Q ( S, λ g ) , λd, η ) ≥ r ( max ( λ − / , λ − / ) Q, λd, η ) . Thus : r l +2 , h ( p ) ≥ dr ( max ( d / , d / ) Q, , η ) denoting µ ( Q, r ) = r ( Q, , η ) , for d ≤ , we have r l +2 , h ( p ) ≥ dµ ( Q, η ) . If d ≥ ,consider ( B ( p, , g ) ⊂ ( S, g ) , and r ′′ ( p ) the ( l + 2 , harmonic radius of p on ( B ( p, , g ) , we have r l +2 , h ( p ) ≥ r ′′ ( p ) ≥ r ( Q ( B ( p, , g ) , , η ) ≥ r ( Q, , η ) .Thus (3.3.2) is a corollary of (3.2.1). We will now prove theorem (3.2.1). We begin by using the harmonic coordinates to identify B g ( p, r , h ( p )) ⊂ M witha B ′ g ( p, r h ( p )) ⊂ R . Here R will be endowed with its standard Sobolev norms.Remark that thanks to the definition of the harmonic radius. k g k H ( B ′ g ( p,r h ( p ))) ≤ C ( r h ( p )) k k k H ( B ′ g ( p,r h ( p ))) ≤ C ( r h ( p ) , α ) We can extend smoothly on R , g and k thus constructing (˜ g, ˜ k ) satisfying k ˜ g − e k H ( R ) ≤ C , k ˜ k k H ( R ) ≤ C , where e stands for the euclidian norm of R .We can also suppose that (˜ g, ˜ k ) = ( e, outside B ′ g ( p, r h ( p )) and Spectrum (˜ g ) ⊂ [2 / , / . We do not require of course (˜ g, ˜ k ) to satisfy the constraint equationsoutside B ′ g ( p, r h ( p )) . 13e then solve the hyperbolic system S with data (˜ g, ˜ k ) for a ν − ≤ n ≤ ν given in H . Using the Cauchy stability theorem for this reduced system, weobtain the existence of a time T ( C, n ) > such that the solution of S withinitial data (˜ g, ˜ k ) exists on R × [ − T , T ] and ( g ( x, t ) − e ( x )) ∈ C ([0 , T ]; H ( R )) ∩C ([0 , T ]; H ( R )) depends continuously on the initial data. The propagation ofthe gauge condition ensures that on the domain of dependance of B ′ g ( p, r h ( p )) thepreviously constructed solution is a solution of the Einstein vacuum equations.Using the Cauchy stability theorem for the reduced system as well as Sobolevembedding theorem, there exists T ( r h ( p ) , α, n ) > , such that for every time t ∈ [ − T , T ] and x ∈ B ′ g ( p, r h ( p )) : n ( t, x ) ∈ [ 12 ν − , ν ] (40)Spectrum ˜ g ( t, x ) ⊂ [1 / , (41)Let us now consider p t := ( p, t ) and prove that there exists T such that for | t | < T ( r h ( p ) , α ) ≤ T ( r h ( p ) , α ) , any causal past-directed curve initiating at p t crosses R × { } in B ′ ( p, r h ) that is p t belongs to the domain of dependance of B ′ ( p, r h ) .To prove this property, let us parametrize a causal curve γ ( s ) = ( x ( s ) , t − s ) with s ∈ [0 , t ] and x (0) = p . Using the fact that γ is causal : ˜ g ij ( x, t ) ˙ x i ˙ x j ≤ ν but :
110 ˜ g ij ( x,
0) ˙ x i ˙ x j ≤
14 Σ i | ˙ x i | ≤ ˜ g ij ( x, t ) ˙ x i ˙ x j ≤ ν integrating in s we deduce γ ( t ) ∈ B ′ ( p, νt ) . Thus taking : T ( r h ( p ) , α, ν ) = min ( T ( r h ( p ) , α, ν ) , r h ( p )30 ν ) (42)we deduce that for | t | < T ( α, r h ( p )) ≤ T ( α, r h ( p )) , any causal curve past-directedinitiating in p t crosses R × { } in B ′ ( p, r h ) that is p t belongs to the domain ofdependance of B ′ ( p, r h ) . We thus have : T + ( p ) ≥ inf ( n ) T ( r h ( p ) , α, ν ) ≥ T ′ ( r h ( p ) , α, ν ) > . (43)Once an abstract initial data set is given ( S, g, k ) , we can fix an arbitrary lapseand (42) gives an explicit lower bound for the temporal extent of every point p ∈ S depending only on the initial data set, the given lapse, and the harmonic radius14t p . We patch the local solutions to obtain a GHVE ( M , g ) of ( S, g, k ) foliatedby an harmonic time function. Now if we consider a initial data set ( S, g, k ) for Einstein vacuum extension satisfying the conditions of theorem (3.2.1), thecorresponding MGVHE is such that a neighborhood of S is diffeomorphic to thepreviously constructed GHVE, theorems (3.2.1) and (3.3.2) follow.Let us now consider T ⋆ ( α, r ) the largest time for which theorem 3.2.1 holds,we have the following monotonicity properties :• T ⋆ ( ., r ) is decreasing• T ⋆ ( α, . ) is increasingmoreover there exists T ⋆ ( α ) > s.t. : T ⋆ ( α, r ) ≥ min ( T ⋆ ( α ) , r , h ( p ) T ⋆ ( α )) Proof.
The MGVHE associated with ( S, λ g, λk ) for a fixed λ > is ( M , λ g ) where ( M , g ) stands for the MGVHE of ( S, g, k ) . Consider p ∈ S , its futuretemporal extent in ( M , λ g ) , T + λ ( p ) = λT + ( p ) , we thus deduce that : ∀ λ > , T ⋆ ( α, r ) ≥ λ − T ⋆ ( max ( λ − / , λ − / ) α, λr ) Now let us define T ⋆ ( α ) = T ⋆ ( α, , if r , h ≤ we have : T ⋆ ( α, r ) ≥ r , h T ⋆ (( r , h ) / α, ≥ r , h T ⋆ ( α ) If r , h ≥ , consider T ′ ( p ) the future temporal extent of p in the MGHVE of ( B ( p, , g, k ) , then T + ( p ) ≥ T ′ ( p ) ≥ T ⋆ ( α ′ , where α ′ < α stands for the corre-sponding norm on B ( p, , using the monotonicity of T ⋆ , T + ( p ) ≥ T ⋆ ( α ) .Now consider an initial data set ( S, g, k ) such that diam ( S ) < + ∞ and ( S, g, k ) satisfies the conditions of corollary (3.2.2), we consider a sequence of points p n → ∂S , we have that there exists T ( Q, ν ) > , T + ( p n ) ≥ dist ( p n , ∂S ) T ( Q, ν ) . Weknow that for non-complete initial data set, the temporal extent of the corre-sponding MGVHE typically tends to zero while approaching the boundary of S ,the previous control shows that rate of decrease is at worst linear in the distanceto the boundary. This phenomenon can be checked for instance in the case of thesimplest example of initial data set for corollary 3.2.2 ( S, e, , where S is the openball of radius in R , e the euclidian metric of R . Its MGVHE is ( C , m ) where C = { ( x, y, z, t ) ∈ R | x + y + z < (1 − t ) } and m the Minkovsky metric. Let p ( x, y, z ) ∈ S , then T + ( p ) = p − x − y − z ≥ dist ( p, ∂S ) .15 .5 Application to the local structure of Einstein vacuumspacetimes : Consider a spacetime (M,g) supposed to be strongly causal and an achronalspacelike hypersurface Ω satisfying the condition of local-in-time theorem and (G) . The existence of such a local slice in the neighborhood of a point of a space-time is not a strong requirement. Given a regular spacetime (M,g) , a point p anda future-directed normalized vector T ∈ T p ( M ) , it’s always possible to constructlocally an achronal spacelike hypersurface Ω p such that p ∈ Ω p and T is the nor-mal to the hypersurface at p . It sufficient to remark that in a local coordinatechart ( t, x , x , x ) , we can construct this hypersurface as the graph of a function t = t ( x , x , x ) . Let us consider a domain D of M globally hyperbolic with respectto Ω , foliated by a harmonic time-function with Ω corresponding to t = 0 and suchthat its boundary consists of three parts Ω , Ω t a spacelike hypersurface level setof t and a null lateral boundary H . We suppose the boundary of D to be Lipschitzregular.For a subset of Ω ′ of Ω , we denote by Ω ′ t the subset of (M,g) obtained byfollowing the integral lines of ∂∂t = n T during time t and by F t : Ω ′ → Ω ′ t thecorresponding map.We suppose in this part that the initial data satisfies : a ) Z Ω | R | ( x ) dµ ( x ) ∆ Theorem 3.5.1. [Local theorem]
Suppose that there exists < ∆ < ∞ s.t. : b ) k ( T ) π k L t ( L ∞ (Ω t ) ∆ c ) there exists a non-empty open set Ω ′ ⊂ Ω s.t. F t (Ω ′ ) ⊂ Ω t , ∀ t ∈ [0 , t [ then the spacetime together with the harmonic time function can be extendedbeyond time t in the neighborhood of any point of Ω t . Let us make some remarks :• c ) is a non-crushing condition.• The other conditions are similar to the global criterion in the form provedby Q.Wang (see [13] and [12]), but localized into D .• the choice of D as an appropriate set to localize the breakdown criteriais natural in the sense that if we consider a point p ∈ D + (Ω) and J − ( p ) its causal past than J − ( p ) is divided into J − ( p ) ∩ D + (Ω) and J − ( p ) ∩J − (Ω) thus in particular arguments on the causal structure of N − ( p ) , energyarguments which are local remain valid as we will see later on.16 using (24), we deduce that there exits ∆ > such that ∆ − < n < ∆ − on D .• condition b ) can be replaced by a uniform bound on the Riemann tensorwhich is a formally stronger assumption. Let us denote by Ω ′ t = F t (Ω ′ ) . The previous theorem will be proved using thefollowing lemmas : Lemma 3.6.1. [Geometric control]
The family Ω ′ t satisfy the following prop-erties : there exists s < , C > and D > such that for all t ∈ [0 , t [ and all x ∈ Ω ′ t such that : a ) the metric g t satisfy the geometric assumption (G) uniformly for t ∈ [0 , t [ b ) vol g i B ( x, s ) > Cs for s s dist ( x, ∂ Ω ′ t ) Lemma 3.6.2. [Energy control]
The Ω ′ t satisfy the following condition for all t ∈ [0 , t [ : ( R t , P t ) is uniformly bounded in the H (Ω ′ t ) × H (Ω ′ t ) topology by a constant C (∆ i , t ) k R k H (Ω) . Proof.
Let us begin with the Lemma 1 : ∂ t g ij = − nk ij (44)Consider a point p of D , following the integral lines of − ∂ t and using theglobal hyperbolicity of D , the integral line intersects Ω at a point p . Denot-ing by C the constant of the hypothesis (G) , we get thanks to the use Gron-wall’s lemma that (G) is satisfied uniformly on the slides Ω ′ t for a constant C ′ = Cexp ( R t k nk k L ∞ (Ω mt ) dt ) that is as matrices : C ′− δ ≤ g t ≤ C ′ δ We thus obtain that F t : Ω ′ → Ω ′ t is a quasi-isometry with constant not dependingon t ∈ [0 , t [ , more precisely, there exists J > such that for every ( p, q ) ∈ Ω ′ andevery t ∈ [0 , t [ : 17 − d g t ( F t ( p ) , F t ( q )) ≤ d g ( p, q ) ≤ J d g t ( F t ( p ) , F t ( q )) To control the volume radius at fixed scales, we first remark that there exists C ′′ such that for all x , for all t , x ∈ Ω ′ t , B g t ( x, a ) ⊃ B e ( x, C ′′ a ) moreover ∂ t log ( detg ) = − ntr ( k ) , thus : V ol ( B g t ( x, a )) ≥ Z B e ( x,C ′′ a ) p det ( g t ) dx ≥ J (∆ , ∆ , C ′′ ) a We only sketch in this part the proof of the energy lemma as its proof isessentially the same as the one given in [9]. We insist only on the differencesrelated to the use of the time-harmonic transversal gauge and the local energyestimates. We refer the reader to this paper for the details of the proof.Let us consider a point p ∈ D , the local geometry near the vertex of the past nullcone N − ( p ) can be controlled exactly as in the global in space case. We can proveexactly as in the global case a bound from below for the null radius of injectivity of N − ( p ) assuming only a bound of a reduced L flux of the curvature along the nullcones. implying a pointwise control on the Riemann curvature tensor as well as a H × H control on ( g, k ) as in [9] and [10]. This requires sophisticated techniqueswhich require to consider from the one hand the Bel-Robinson tensor and from theother hand the non-linear wave equation satisfied by R and its covariant derivativetogether with control on the causal geometry of null cones (see [9], [7], [4], [6]).This control depends only on the constants ∆ i and t . We refer the reader to thepreviously cited article for details. To be more precise : Theorem 3.6.1.
Let R be the Riemann curvature tensor of a solution of the EVEand satisfying the hypothesis a ) , b ) and c ) of the local theorem, then for all t in [0 , t [ and p ∈ D : a ) Z Ω t | R | ( x ) dµ ( x ) C (∆ i , t ) (45) b ) Z N − ( p ) ∩J + (Ω) Q [ R ]( T , T , T , L ) C (∆ i , t ) (46) where N − ( p ) stands for the past null cone initiating at the point p that is theboundary of the causal past of p and the integral is taken in time between and t nd L for the null geodesic generator of N − ( p ) , that is the vectorfield on N − ( p ) satisfying : D L L = 0 < L , L > = 0 (47) and the normalization condition at p : < L , T > ( p ) = 1 Moreover for higher order derivatives, we use the wave equation for R : (cid:3) g R = R ⋆ R where ⋆ is a bilinear symmetric operator on the four-times covariant tensor.Its covariant derivative satisfies : (cid:3) g DR ≈ DR · R using local energy estimates for wave equations : Theorem 3.6.2.
Under the hypothesis of the local theorem, there exist a constant C = C ( t , ∆ i ) such that for all t in the time slab [0 , t [ : k DR k L (Ω t ) C (cid:0) k DR k L (Ω ) + Z t k R k L (Ω s ) nds (cid:1) Theorem 3.6.3.
With the hypothesis of the local theorem, there exist a constant C ′ = C ′ ( t , ∆ i ) such that for all t in the time slab [0 , t [ : (cid:13)(cid:13) D R (cid:13)(cid:13) L (Ω t ) C ′ (cid:0) (cid:13)(cid:13) D R (cid:13)(cid:13) L (Ω ) + Z t k R k L ∞ (Ω s ) k DR k L (Ω s ) nds (cid:1) As in the global case, our problem is thus to control the pointwise norm of R . Such a control is obtained by proving that there exists a control on the pastnull radius of injectivity of p , that is there exists i > such that ∀ p ∈ D , i − ( p ) ≥ min ( d ( p, Ω) , i ) . Such a property as well as the parametrix of [7] whichare local can be used as in [9] to close the estimate and obtain a uniform H (Ω ′ t ) control on R . It remains however to derive control on k and n from the controlon R . The method used previously for a CMC or an AF gauge is from the onehand a three-dimensional Hodge system for k and form the other hand the lapseequation (see [9]) : ∇ i k ij = 0 ∇ ∧ k = Etrk = 0 (AF gauge), trk = t (CMC gauge) ∆ n = n | k | − ǫ ǫ = 0 in the case of the AF gauge, and ǫ = 1 in the case of a CMC gaugeand E stands for the electric part of the electric-magnetic decomposition of R . Ona manifold without boundary we have using integration by part that if R ∈ L (Ω) , ∇ k and ∇ n ∈ L (Ω) . Moreover, the time derivatives can be controlled using theevolution equation for k , obtaining ∂ t k ∈ L (Ω) and ∇ ∂ t n ∈ L (Ω) . This approachis not well-suited for local in space and time application due to the ellipticity ofthe estimates used.Instead of these techniques , we use hyperbolic estimates of theorem (3.1.1) todeduce that : Theorem 3.6.4.
Consider a spacetime foliated by a time-harmonic function t ,and such that D l R ∈ L ∞ t L x (Ω t ) for | l | ≤ and ( T ) π ∈ L t L ∞ (Ω t ) , then D l ( T ) π ∈ L ∞ t L x (Ω t ) , | l | ≤ D l R ∈ L ∞ t L x (Ω t ) , | l | ≤ We thus avoid completely in this gauge the trace theorems from Σ t → C where C is a null cone and the associated loss of Sobolev regularity of and get a controlon the whole flux of ( T ) π on C and not only on the flux associated to the waveequation for (4) k .As in [9], [10] and [13], we thus have to derive a H (Ω t ) control on R . If insteadof a L t L ∞ x control on ( T ) π , we had only required a L t L ∞ x control, the proof wouldhave been similar to the one in [10], however the stronger assumption made hererequires to prove that the L t L ∞ x bound is sufficient to control the geometry of pastnull cones. This require the use of the main result of [12] whose statement requiressome definitions. Note that in this paper the author gave a proof assuming ageneric time foliation in comparison with [4] in which the authors studied only thecase of geodesic foliation of a (troncated) null cone.Let ( M , g ) be a smooth 3+1 Einstein vacuum spacetime foliated by Σ t , thelevel hypersurfaces of a time function t with lapse function n . Consider an outgoingnull hypersurface H = ∪ The Ricci coefficients relative to an adapted ull frame ( e A , e B , L , L ) are defined as follows : χ AB = h D A L , e B i χ AB = h D A L , e B i ξ A = 12 h D L L , e A i ξ A = 12 h D L L, e A i η A = 12 h D L L , e A i η A = 12 h D L L , e A i ω = 14 h D L L , L i ω = 14 h D L L , L i V A = 12 h D A L , L i We recall the definition of the null components of a Weyl Field W relative to anull pair ( L , L ) : α ( W )( X, Y ) = W ( X, L , Y, L ) β ( W )( X ) = 12 W ( X, L , L , L ) ρ ( W ) = 14 W ( L , L , L , L ) σ ( W ) = 14 ⋆ W ( L , L , L , L ) α ( W )( X, Y ) = W ( X, L , Y, L ) β ( W )( X ) = W ( X, L , L , L ) denote moreover by : a − = − < L , T >µ = − D L trχ + a tr ( χ )) − ωtr ( χ )4 πr ( t ) = Z S t dµ S t moreover µ stands for the mass function, a for the null lapse, r for the radius ofthe level sets { t = cte } foliation of a null cone. For the definition of the appropriatefunctional spaces P and B see [4].More precisely : Theorem 3.6.6. [Control on the null Ricci coefficients of null cones,Q.Wang, [12]] Using the previously introduced notations, assume C − < n < C on H for C > and : ( H ) + F ( π ) ≤ R , on H with R sufficiently small. Then the following estimates hold true : k trχ − s k L ∞ ( H ) . R | a − | ≤ k Z | ˆ χ | nadt k L ∞ ω + k Z | ζ | nadt k L ∞ ω . R k Z | ν | nadt k L ∞ ω + k Z | ζ | nadt k L ∞ ω . R k6 ∇ trχ k P + k µ k P + k6 ∇ trχ k L H + k µ k L H . R N ( ˆ χ ) + N ( ζ ) + N ( trχ − r ) + N ( trχ − ( an ) − antrχ ) . R k trχ − r k L t L ∞ ω + k trχ − ( an ) − antrχ k L t L ∞ ω . R k sup t ≤ | r / trχ |k L ω + k sup t ≤ | r / µ |k L ω + k r / trχ k B + k r / µ k B . R Remark : We have here a slighlty better control on the flux of order one of ( T ) π due to the fact that we have obtained a wave equation on ( T ) π and not onlyon k αβ = P αν P βµ ( T ) π µν like in [13] and [12].The control of the geometry of the null cones require also to use the followingtheorem : Theorem 3.6.7. [Control on the radius of injectivity of null hypersur-faces, Klainerman and Rodnianski, [8]] Let ( M , g ) be a smooth 3+1 Einstein vacuum spacetime foliated by Σ t , the levelhypersurfaces of a time function t with lapse function n . Suppose that Σ satisfies ( G ) . Consider an outgoing null hypersurface H = ∪ Remark : The reader will note that the two theorems come together : somecontrol on the null Ricci coefficients are needed to control the radius of injectivity,in particular trχ (see [8]). From another hand, the bootstrap argument of [4], [10]22r [12] requires to work on a smooth part of a past null cone that is on a N − ( p, δ ) which requires some uniform control on the radius of injectivity of these null cones.Finally if the null cones satisfy uniformly the control on the radius of conjugacyand on the null Ricci coefficients, one can obtain using a parametrix of Kirchoff-Sobolev type (see [7]) following the orginal strategy that R , DR and D R arebounded in L (Ω s ) by a constant depending only on the initial data, the ∆ i and t . Thus the proof of the energy estimates follows the original strategy of [ ? ] toderive uniform L control on R and its covariant derivatives with two slight modi-fications. From the one hand, we use purely hyperbolic estimates to control energyestimates on ( T ) π from the energy estimates on R using the structure of the re-duced system in a time harmonic transversal gauge and from the other hand weuse local energy estimates instead of estimates in a strip. Proof. Consider a point p ∈ Ω ′ , denote by p t = F t ( p ) . Using the equivalence ofthe uniform ellipticity of the metrics g t given by energy lemma we can find r p > and C p > such that the volume radius of p t at scales ≤ r p on Ω ′ t is bounded bybelow by C p independently of t . Using the energy lemma as well as the geometriclemma, we can apply the local in time existence theorem to (Ω t , g t , k t ) and points p t to deduce that there exists a Cauchy development of (Ω t , g t , k t ) for proper time T ( p ) > (not depending of t) to the future of p t ∈ Ω t . From another hand, theconditions on the theorem imply that the maximal proper time between the slices Ω i and Ω j for ≤ t i < t j < t denoted by T ji satisfies : T ji ∆ ( t j − t i ) (49)Indeed using : g = − n dt + g ij dx i dx j (50)a uniform bound on the lapse implies directly (49).We have as a consequence that for t large enough the maximal Cauchy develop-ment of (Ω t , g t , k t ) extends as a globally hyperbolic spacetime to the future of any F t (Ω ′ ) , ≤ t < t ′ < t in the neighborhood of p t , which concludes the proof. Acknowledgement : I would like to thank Professor Planchon for usefuldiscussions. Professor Klainerman suggested the subject to me and the interest ofusing a time-harmonic gauge to localize the criterion. I am very grateful to himfor the discussions we had on the subject and the time he spent in discussions withme during the months i spent in Princeton.23 eferences [1] Choquet-Bruhat Yvonne Anderson Arlen and York James W. 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