On the C^k-embedding of Lorentzian manifolds in Ricci-flat spaces
aa r X i v : . [ g r- q c ] M a y On the C k -embedding of Lorentzian manifolds in Ricci-flat spaces R. Avalos , F. Dahia , C. Romero Departamento de F´ısica, Universidade Federal da Para´ıba,Caixa Postal 5008, 58059-970 Jo˜ao Pessoa, PB, Brazil. Departamento de Matematica - UFC,Bloco 914 Campus do Pici, 60455-760 Fortaleza, Cear´a, Brazil.E-mail: rodrigo.avalos@fisica.ufpb.br; fdahia@fisica.ufpb.br; cromero@fisica.ufpb.br
Abstract
In this paper we investigate the problem of non-analytic embeddings of Lorentzian manifoldsin Ricci-flat semi-Riemannian spaces. In order to do this, we first review some relevant resultsin the area, and then motivate both the mathematical and physical interest in this problem. Weshow that any n -dimensional compact Lorentzian manifold ( M n , g ), with g in the Sobolev space H s +3 , s > n , admits an isometric embedding in an (2 n +2)-dimensional Ricci-flat semi-Riemannianmanifold. The sharpest result available for this type of embeddings, in the general setting, comesas a corollary of Greene’s remarkable embedding theorems [R. Greene, Mem. Am. Math. Soc. 97,1 (1970)], which guarantee the embedding of a compact n -dimensional semi-Riemannian manifoldinto an n ( n + 5)-dimensional semi-Euclidean space, thereby guaranteeing the embedding into aRicci-flat space with the same dimension. The theorem presented here improves this corollary in n + 3 n − n -dimensional globally hyperbolic space-time can be embedded in a (2 n + 2)-dimensionalRicci-flat semi-Riemannian manifold. . INTRODUCTION The problem of embedding general n -dimensional manifolds in higher-dimensional spaceswith particular properties has been an active object of study for more than a century.It is a well known fact that, as mathematicians began to study abstract manifolds, thequestion of whether these structures were actually more general than the submanifoldsof Euclidean spaces naturally arose. In this direction, several very interesting theoremswere proved, such as Whitney’s embedding theorem [2], stating that any n -dimensionalmanifold can be embedded in R n ; Jannet-Cartan’s theorem, which shows the existence oflocal isometric embeddings for n -dimensional Riemannian manifolds in n ( n +1)2 -dimensionalEuclidean space in the case of analytic metrics [3]-[4]; Nash’s famous embedding theorem,which shows the non-analytic and global version of Janet-Cartan’s result [5] and Greene’sgeneralization of Nash’s theorem to semi-Riemannian geometry [1], where, for instance, itis shown that a compact n -dimensional semi-Riemannian manifold can be embedded in an n ( n + 5)-dimensional semi-Euclidean space.Directly related with our present object of study, it is also worth mentioning the resultsobtained by C. J. S. Clarke in [6], where it is shown that a semi-Riemannian manifoldwith metric of signature “ s ” can be embedded in a semi-Euclidean space of dimension d = n (3 n + 12) + 1 − s , for the compact case, and n (2 n + 40) + n + 2 − s for thenon-compact case, and, also, that for the case of globally hyperbolic Lorentzian manifolds,Clarke suggests that the dimension of the Lorentzian embedding space can be shown to be d = n (2 n + 37) + n + 2. This last claim suffered of problem related with the so called folk problems on smoothability , which is addressed and explained, for instance, in anotherrelevant recent result by O. M¨uller and M. S´anchez, where it is shown that any globallyhyperbolic n -dimensional space-time can be isometrically embedded in a Minkowski space of d = N ( n )+1 dimensions, where N ( n ) is the optimal dimension for the case of an embeddingof an n -dimensional Riemannian manifold into an N -dimensional Euclidean space [7] (for adiscussion on the value of N ( n ) see [1], [6], [8]).This type of problem has also become an interesting object of study in the context of mod-ern physical theories. For instance, the Cauchy problem in general relativity (GR) is nothingmore than an embedding problem, in which we look for an embedding of a 3-dimensionalRiemannian manifold in a 4-dimensional Lorentzian manifold, satisfying the Einstein field2quations [9]. This very same problem is of interest in the context of higher-dimensionaltheories of gravity [10],[11],[12],[13],[14]. In this context, in which space-time is supposedto have more than 4 dimensions, and the ordinary space-time of GR is considered as asubmanifold embedded in this bulk , natural additional embedding problems arise. For ex-ample, in some of these higher-dimensional models, the bulk is supposed to be characterizedby some geometric property, such as being Ricci-flat or, more generally, an Einstein space[10],[14]. Thus, it becomes a straightforward question whether or not any solution of the4-dimensional Einstein field equations can be embedded in this type of higher-dimensionalstructures. This issue triggered some research and it was shown that, for the local andanalytic case, the embeddings do exist. These theorems are the content of the Campbell-Magaard theorem and its extensions [15]-[16],[17],[18],[19],[20],[21]. Applications of some ofthese theorems to physical situations can be reviewed in [22]. In this direction, it is alsoworth mentioning the approach taken in [23], where it is shown how to construct local iso-metric embeddings of vacuum solutions of the Einstein equations, into either Einstein spacesor solutions of the higher-dimensional Einstein equations sourced by a scalar field.Even though some of the above results have been around for a while, there is no referencein the literature, as far as we know, of non-analytic extensions of the Campbell-Magaardtheorems. These type of results would be of interest not just as mathematically interest-ing problems, but also because some properties of vital importance in relativistic theories,namely causality and stability, may demand non-analytic versions of the Campbell-Magaardtheorems in order for them to be applicable to realistic physical theories (for a discussionon this matter, see [24]-[25]). In particular, regarding stability of the embedding, since theCampbell-Magaard theorem, together with all of its extensions, are proved by means of theCauchy-Kovalesvkya theorem, there is no guarantee that the embedding is stable againstsmall perturbations on the space-time metric, even near the brane representing the originalspace-time. Regarding causality, what should be made clear, is that we have to admit non-analytic solutions to our field equations if we intend to get interactions propagating frominitial data with finite speed. These two very important remarks coming from physics, giveus a strong motivation for the study of non-analytic embeddings for space-time.Another important point to take into account, besides the regularity considerations, isthat if we want to rigorously support the statement that ordinary GR solutions can beembedded in these higher-dimensional space-times, then global theorems are required.3ith all the above motivations in mind, the aim of this paper is to present a first re-sult in this direction, proving a new global embedding theorem for compact n -dimensionalLorentzian manifolds, with metric in the Sobolev space H s +3 , s > n (which requires metricsat least C , and allows in particular smooth metrics, and in general C k metrics with k de-pending on n ), in Ricci-flat semi-Riemannian spaces, where the dimensionality needed forthe embedding manifold is 2 n + 2. It should be noted that the aforementioned embeddingtheorems of semi-Riemannian manifolds in semi-Euclidean spaces, set an upper-bound fornovel results concerning embeddings of Lorenzian manifolds in Ricci-flat semi-Riemannianspaces. In particular, the sharpest result previously known for the compact case, which willbe our object of study, is the one presented by Robert E. Greene in [1], which guaranteesthe much stronger statement that an n -dimensional compact semi-Riemannian manifoldcan be embedded in an n ( n + 5)-dimensional Riemann-flat space. Thus, the existence ofan embedding in an n ( n + 5)-dimensional Ricci-flat space is guaranteed. This means that,using the best result known so far, ( n + 3 n −
2) additional dimensions are needed withrespect to the results we will present in this paper. Furthermore, we will show how the maintheorem presented here, which, as explained above, applies to the compact (without bound-ary) case, can be used to prove the existence of embeddings for any arbitrary finite strip ina globally hyperbolic space-time, with compact Cauchy surfaces, in a Ricci-flat manifold.Moreover, such embeddings can be constructed so as to be stable under small perturba-tions with respect to the “space-time” metric. It is, to say the least, a curious fact that forthe usual 4-dimensional space-time of general relativity the embedding is guaranteed in a10-dimensional Ricci-flat space.
II. THE EMBEDDING PROBLEM.
As we have already stated, the aim of this paper is to prove an embedding theoremfor Lorentzian manifolds in Ricci-flat semi-Riemannian ones. Surprisingly enough, there isno much reference in the literature of embeddings of general manifolds in Ricci-flat spacesbesides the Cauchy problem in GR and the Campbell-Magaard theorem. When trying toadapt any of these results for the case we want to study, we are faced with serious difficulties.On the one hand, the Campbell-Magaard theorem strongly relies on the Cauchy-Kovalesvkytheorem, which depends crucially on the analyticity assumptions for the quantities involved.4hus, it is not well-suited as a starting point for the non-analytic case, which is our presentobject of study. This type of difficulty is not an odd feature of embedding problems. Ananalogous situation was presented when trying to generalize the Janet-Cartan embeddingtheorem, which culminated with Nash’s theorem. On the other hand, the Cauchy problem inGR strongly depends on the hyperbolic character of the evolution equations and the ellipticone of the constraint equations (see, for example, [9],[26]), which depend on the signaturesof the space-time metric and the induced metric on the space-like slices. Nevertheless, theseresults require weak regularity assumptions, and we will, in fact, use some of the resultsknown for the Cauchy problem in GR to show our main theorem.
A. The Cauchy problem in GR
The Cauchy problem in GR consists in the following. Given an initial data set ( M, ¯ g, K )where M is an n-dimensional smooth Riemannian manifold with metric ¯ g and K is a sym-metric second rank tensor field, a development of this initial data set is a space-time ( V, g ),such that there exists an embedding into V with the following properties:i) The metric ¯ g is the pullback of g by the embedding i : M V , that is i ∗ g = ¯ g ;ii) The image by i of K is the second fundamental form of i ( M ) as a submanifold of ( V, g ).In the Cauchy problem in GR we look for a development of an initial data set suchthat the resulting space-time satisfies the Einstein equations. It is usually assumed that V = M × R . We will adopt this usual setting.At this point, to study the Cauchy problem, it is customary to consider an ( n + 1)-dimensional space-time ( V, g ) and then make an “( n + 1)-splitting” for the metric g . Thismeans that we consider local co-frames where we can write the metric g in a convenient way,such that we have a “space-time splitting”. In order to do this, a vector field β , which isconstructed so as to be tangent to each hypersurface M × { t } , is used to define the followinglocal frame e i = ∂ i ; i = 1 , · · · , ne = ∂ t − β θ i = dx i + β i dt ; i = 1 , · · · , nθ = dt Then we can write the metric g in the form g = − N θ ⊗ θ + g ij θ i ⊗ θ j where N is a positive function referred to as the lapse function, while the vector field β iscalled the shift vector. In this adapted frame, the second fundamental form on each M × { t } takes the form K ij = 12 N ( ∂ t g ij − ( ¯ ∇ i β j + ¯ ∇ j β i )) (1)where ¯ ∇ denotes the induced connection compatible with the induced metric ¯ g .It has been shown that if an initial data set satisfies a particular system of constraintequations, plus some low regularity assumptions, then it admits an Einstenian developmentin a spacetime V satisfying the vacuum Einstein equations [9]. The constraints we arereferring to, are the following: ¯ R − | K | g + ( tr ¯ g K ) = 0 (2)¯ ∇ · K − ¯ ∇ tr ¯ g K = 0 (3)where ¯ R represents the Ricci scalar of ¯ g , | · | ¯ g denotes the pointwise-tensor norm in themetric ¯ g , and ¯ ∇ · K denotes the divergence of K . In coordinates, these equations become:¯ R − K ij K ij + ( K ll ) = 0 (4)¯ g ju ¯ ∇ u K ij − ¯ ∇ i K ll = 0 (5)These equations are considered on a particular initial hypersurface M ∼ = M × { t } , forexample, in the hypersuface defined by t = 0.Equations (4)-(5) are generally posed as a set of equations for ¯ g and K . We will insteaduse their thin-sandwich formulation [27], in order to pose them for N and β [28],[29]. Thisis achieved by using the explicit relation between K and ˙ g . = ∂ t g | t =0 , given by (1). Pluggingthis expression in the constraint equations (4)-(5), if R ( g ) <
0, it is possible equate the lapsefunction N from the Hamiltonian constraint (4), giving the following N = s (tr g γ ) − | γ | g − R g (6)6here the tensor γ has components γ ij = 12 (cid:0) ˙ g ij − ( ∇ i β j + ∇ j β i ) (cid:1) . (7)In this setting the momentum constraint (5) becomes a non-linear second order operatoron the shift vector field, defining an equation of the form Φ( ψ, β ) = 0, where ψ = ( g, ˙ g )is regarded as the freely specifiable part of the initial data set ( g, K ). The idea is that thisprocedure can be reversed, i.e , whenever the set of equations Φ( ψ, β ) = 0 are well-posedas equations for the shift, taking (6) as a definition and using (1), we get an initial dataset ( g, K ) satisfying the vacuum constraint equations (see [29] for further details). In thiscontext we will have the following initial data defined in its corresponding functional spaces:(¯ g, ˙ g ) ∈ E . = H s +3 ( T M ) × H s +1 ( T M ) ; s > n . where, as usual, H s ( T pq M ) represents the Sobolev space of ( p, q )-tensor fields with s -generalized derivatives in L (see [9]), and we are left with the analysis of the non-linearequations Φ( ψ, β ) = 0.Using the setting presented above, it has been recently shown that, on any n -dimensionalsmooth compact manifold M , n ≥
3, we can always find a smooth solution of the vacuumconstraint equations ( ψ = ( g , ˙ g ) , N , β ), such that for any ψ = ( g, ˙ g ) ( g a Riemannianmetric) in a sufficiently small E -neighbourhood of ψ there is also a unique solution of thevacuum constraint equations, constructed as dicussed above by means of the thin-sandwichformulation. This is part of the content of Theorem 3 in [29]. From now on, we will typicallyrefer to such solution ( ψ , N , β ) as a reference solution for the constraint equations. Allthis, in turn, guarantees that there is an embedding of the Riemannian manifold ( M n , g )into a Ricci-flat ( n + 1)-dimensional space-time V , where the space-time metric would beat least C (the more regular the initial data is, the more regular the space-time metric willbe) [9]. This new result will be our main tool in proving a new embedding theorem. Themain idea for this proof goes as follows:Suppose we are given a smooth compact n -dimensional Lorentzian manifold ( M, g ) andwe want to embed it in some Ricci-flat manifold. If we can perturb the metric g in such a waythat the perturbed metric g ′ is now a properly Riemannian metric, and furthermore g ′ is partof an initial data set solving the vacuum constraint equations (4)-(5), then, using standardresults on the Cauchy problem for GR, we could guarantee the existence of an isometric7mbedding of ( M, g ′ ) into an ( n + 1)-dimensional space-time ( V, ¯ g ′ ). If we can arrangethings in such a way that the perturbation g ′ lies in a sufficiently small E -neighbourhoodof the reference solution of the constraints used in the main theorem presented in [29] (seeTheorem 3 therein), which has been explained above, then, we can guarantee the existenceof the previous embedding appealing to this theorem, and we would have embeddings forboth g and g ′ . Roughly speaking, the more or less obvious thing to do at that point, wouldbe to embed ( M, g ) into the product of these two embedding Lorentzian manifolds, and usethe embeddings for g and g ′ to construct the embedding for ( M, g ). B. The Main Results
Having established the framework in which we will be operating, we will now make ourassumptions explicit. We will always work on compact manifolds (without boundary), andwe will assume that we have a smooth reference solution of the constraint equations on suchmanifolds, denoted by ( ψ . = ( g , ˙ g ) , N , β ), such that, in the previously defined topologies,there is a neighbourhood of ψ where, for each element in this neighbourhood, the constraintequations have a unique solution. The existence of such a reference solution is guaranteedby Theorem 3 in [29]. Also, in order to define the Sobolev spaces, we make use of the smoothRiemannian metric g . We can now prove the following proposition: Lemma 1.
Given a compact n -dimensional Lorentzian manifold ( M n , g ) , with g ∈ H s +3 , itis always possible to find a Riemannian metric ˜ g on M such that:i) ˜ g is as close of g as we want.ii) g = λ (˜ g − g ) for some positive constant λ .Proof. On M we have both g and g defined. Now let T U . = ⊔ p ∈ M { v ∈ T p M / g ( v, v ) = 1 } be the unit bundle associated to g , and let F : T U R v = ( x, v x ) g x ( v x , v x ) g x ( v x , v x )8oticing that F defines a continuous function on T U , which, since M is compact, is acompact set, then F attains its minimum. Furthermore, since g is Lorentzian, then thisminimum must be a negative number. Thus we get that ∃ α < α ≤ F ( v ) ∀ v ∈ T U.
Then, ∀ λ > − λ < α it holds that: − λ ≤ g x ( v x , v x ) g x ( v x , v x ) ∀ v ∈ T U, which gives us the following0 ≤ λ g x ( v x , v x ) + g x ( v x , v x ) ∀ v ∈ T U. (8)Now define ˜ g . = λ g + g . The claim is that this metric is positive-definite. In order to seethis, simply note that for arbitrary m ∈ M and ∀ V ∈ T m M , V = 0, the following holds:˜ g m ( V, V ) = k V k g ˜ g m (cid:16) V k V k g , V k V k g (cid:17) > V k V k g ∈ T m U , where k V k g = ( g ( V, V )) . This last relation shows that ˜ g is a well-defined Riemannian metric on M . Now, in order to show our first statement, we simplynote that λ can in fact be taken as large as we want, thus, given ǫ > ∃ λ > k ˜ g − g k H s +3 = 1 λ k g k H s +3 < ǫ. (9)Having established our first claim, we see that the second one is a trivial consequence of thedefinition of ˜ g , and thus the proposition holds.We now present the main theorem: Theorem 1.
Any n -dimensional compact Lorentzian manifold ( M, g ) , with n ≥ and g ∈ H s +3 , s > n , admits an embedding in a Ricci-flat (2 n +2) -dimensional semi-Riemannianmanifold with index n + 1 (that is, with n + 1 time-like dimensions).Proof. We will start by appealing to Lemma 1 and writing the Lorentzian metric g in thefollowing way g = λ (˜ g − g ) (10)9here both g and ˜ g are Riemannian metrics, and g is part of a reference solution( g , ˙ g , N , β ) of the vacuum constraint equation (4)-(5) on M . As we have already statedabove, the main theorem presented in [29], guarantees that under our hypotheses we canalways pick such a solution of the constraint equations, with g being actually smooth, andguarantee that for initial data ( g, ˙ g ) in a small enough neighbourhood of ( g , ˙ g ), there is aunique solution ( N, β ) of the constraint equations (4)-(5). Thus, picking λ and ˜ g so that˜ g lies in a small enough H s +3 -neighbourhood of g , then, given the initial data (˜ g, ˙ g ), weknow that there is a solution of the constraint equations, and, thus, that there are isomet-ric embeddings φ and φ , of ( M, ˜ g ) and ( M, g ), into the Ricci-flat Lorentzian manifolds( V . = M × [0 , T ) , h ) and ( V, h ) respectively, for some T >
0, with h and h at least C .We now consider the semi-Riemannian manifold ( V × V, h ), where h . = λ ( π ∗ h − σ ∗ h ) (11)and π and σ denote the projections of V × V onto its first and second factors respectively.Thus, the manifold is the product semi-Riemannian manifold resulting from ( V, h ) and( V, − h ). We claim that the following map is an isometric embedding φ : M V × Vm ( φ ( m ) , φ ( m )) (12)The fact that φ is an embedding comes from the fact that both φ and φ are embeddings.To check the isometry condition, given v, w ∈ T m M , we compute the following: φ ∗ ( h ) m ( v, w ) = h ( dφ m ( v ) , dφ m ( w ))= λ ( h ( dπ φ ( m ) ◦ dφ m ( v ) , dπ φ ( m ) ◦ dφ m ( w )) − h ( dσ φ ( m ) ◦ dφ m ( v ) , dπ φ ( m ) ◦ dφ m ( w )))= λ ( h ( dφ m ( v ) , dφ m ( w )) − h ( dφ m ( v ) , dφ m ( w )))= λ ( φ ∗ ( h ) m ( v, w ) − φ ∗ ( h ) m ( v, w ))= λ (˜ g − g ) m ( v, w )= g m ( v, w ) . This last equality shows the isometry condition. As a final step, we have to show that( V × V, h ) is Ricci-flat. But, since the semi-Riemannian product manifold of Ricci-flat spaces,with the usual product structure, is again Ricci-flat, the conclusion follows immediately.10his theorem possesses quite some intrinsic geometric value. It provides a general em-bedding result for compact Lorentzian manifolds with smooth metrics into Ricci-flat spaces,and the codimension needed for the embedding space, even though greater than in theCampbell-Magaard theorem, is much lower than the one needed using the result obtainedby R. Greene for the much stronger condition of Riemann-flatness on the embedding space.Explicitly, ( n + 3 n − Corollary 1.
Any given compact n -dimensional Lorentzian manifold ( M, g ) , with metric inthe Sobolev space H s +3 , s > n , admits an isometric embedding in a n + 1) -dimensionalRicci-flat semi-Riemannian manifold ( ˜ M n +2 , h ) , with index h = n + 1 , where both the em-bedding and h depend continuously on g .Proof. From the proof of the previous theorem we get that the embedding φ : M V × V ,depends on g only through the embedding φ : ( M, ˜ g ) ( V = M × [0 , T ) , h ), T >
0. Now,from basic facts about the Cauchy problem for the vacuum Einstein equations, we knowthat φ is just the inclusion map, and that, under our functional hypotheses, h is continuouswith respect to ˜ g = λ ( g ) g + g . Now, from (9), we see that we any λ ( g ) > || g || Hs +3 ǫ makes theprocedure work. In particular, we can fix a choice which satisfies this condition and makes λ a continuous function on g , for instance λ ( g ) . = || g || Hs +3 ǫ + 1. With this choice, it is clearthat ˜ g . = λ ( g ) g + g depends continuously on g , with respect to the H s +3 topology. Thus,using the fact that the Cauchy development of vacuum initial data for the Einstein fieldequations is continuous with respect to the initial data, then h is continuous with respectto g , which proves our statement.The physical interpretation of this corollary would be that the embedding is stable against small perturbations on the space-time metric g . Thus, if we think of a compact space-time ( M, g ) as an embedded submanifold in ( M n +2 , h ), then, we can guarantee that we cando this embedding in such a way that, near the submanifold representing the 4-dimensionalspace-time, the embedding is stable against small perturbations on the space-time metric g .Now, we will present an important corollary which shows that Theorem 1 can be used toguarantee the embedding of strips of globally hyperbolic space-times with compact space-slices. 11 orollary 2. Suppose we have a strip ( M × [0 , T ) , g ) , T > , of an n -dimensional globallyhyperbolic space-time, where M is an ( n − -dimensional closed manifold, and the Lorentzianmetric g is smooth. Then, for any < T < T < T , the closed strip M × [ T , T ] admits anisometric embedding in a (2 n + 2) -dimensional Ricci-flat manifold ( ˜ M n +2 , h ) . Furthermore,both the embedding and h can be chosen to be continuous with respect to the space-timemetric g .Proof. Given T and T satisfying the hypotheses of the corollary, take T and T such that0 < T < T < T < T < T . Then, consider two bump functions f , f : [0 , T ] R + . Pick f such that f | [0 ,T ] ≡ f | [ T ,T ] ≡
0, and pick f such that f | [0 ,T ] ≡ f | [ T ,T ] ≡ M × [ T , T ) the following (0 , g ( · , t ) . = f ( t ) g ( · , t − ( T − T )) , (13)and then extend it for t ∈ [0 , T ] as zero. In order for the previous expression to be well-defined, take T and T sufficiently close, and T sufficiently close to T . This choice has tobe done such that T + T > T . Thus, on M × ( T , T ) g defines a Lorentzian metric. Now,define the following Lorentzian metric on M × [ T , T ]: G . = f g + g . (14)Notice that the metric G is actually well-defined on the whole strip M × [0 , T ), but we arerestricting it to the closed strip M × [ T , T ]. Also, note that G ( · , t ) = g ( · , t ) ∀ T ≤ t ≤ T ,and G ( · , t ) = g ( · , t − ( T − T ) for T ≤ t ≤ T . In particular, this gives us that: ∂ kt G ( · , T ) = ∂ kt G ( · , T ) = ∂ kt g ( · , T ) ∀ k ∈ N ,∂ ka G ( · , T ) = ∂ ka G ( · , T ) = ∂ ka g ( · , T ) ∀ k ∈ N , where ∂ a denote derivatives on the coordinates on M . The above relations give us thatthe metric G can be glued smoothly when identifying the slices M × { T } with M × { T } .This gluing gives us a quotient manifold (cid:16) M × [ T , T ] (cid:17) / ∼ ∼ = M × S , with an inducedsmooth Lorentzian metric G ′ , such that the embedding ( M × [ T , T ] , g ) ֒ → ( M × S , G ′ ) isisometric. We can thus apply Theorem 1 to ( M × S , G ′ ) and get an isometric embedding( M × S , G ′ ) φ −→ ( ˜ M n +2 , h ). The composition of these two embeddings gives us an isometricembedding of ( M × [ T , T ] , g ) in ( ˜ M n +2 , h ).Finally, the continuous dependence with respect to the metric g is a consequence ofCorollary 1. 12 II. DISCUSSION
In Theorem 1, we have obtained a non-analytic embedding theorem for compactLorentzian manifolds in Ricci-flat spaces, where the regularity assumptions are rather weak.In fact, by means of the Sobolev embedding theorems, we get that the minimal regularityneeded for the Lorentzian mentric g is C , since H s +3 embeds in C for s > n . In general,in order for g ∈ H s +3 , we will need g in some C k space, with k depending on n , and inparticular a smooth g always satisfies this condition. As far as we know, this is a new result,and we regard this theorem as a first step in the search of sharper theorems in this context.Even tough the direct application of this theorem to physics would seem to be lim-ited, since the compactness condition on space-time poses strong consequences, such asthe existence of a closed time-like curve (see, for instance, [30]), we have been able to ex-tract two important corollaries from Theorem 1, which give us that any closed strip in an n -dimensional globally hyperbolic space-time, with a compact (without boundary) Cauchysurface, can be isometrically embedded in a 2( n + 1)-dimensional Ricci-flat semi-Riemannianmanifold. Notice that such strip can be taken to be as large as we like as long as it remainsfinite. Thus, for instance, any cosmological model with compact space-slices, consideredfrom some arbitrary finite time T > T < ∞ can be isometrically embedded in a 2( n + 1)-dimensional Ricci-flat manifold. Nevertheless,it is essential to highlight that the lack of control on the number of time-like dimensions ofthe ambient space represents a serious restriction for physical applications, which leaves theissue of controlling the signature of the ambient space as a physically relevant open problem.We would also like to point out that, in our opinion, the main theorem presented here hasan intrinsic geometric value, offering a substantial improvement in the number of dimensionsneeded to get the embedding. In fact, compared with the sharpest results we are aware offor this kind of embedding we are saving as much as n + 3 n − n = 4, 26 extra dimensions are saved.It is also worth pointing out that results such as the Campbell-Magaard theorems, whichhave been invoked in the context of some higher-dimensional space-time theories, are bothlocal and analytic theorems. By local we mean that they guarantee the existence of theembedding in a neighbourhood of an arbitrary point in space-time, and, in fact, the sizeof such neighbourhood is not controlled. In this sense, we have shown that for globally13yperbolic space-times with compact Cauchy hypersurfaces, we can prove the existence ofembeddings which are global in space and local in time , where by local in time we mean thatas long as we chose a finite time interval, the embedding will exist. Moreover, regradinganalyticity, we have dramatically lowered the regularity needed for the space-time metric,and this has enabled us to prove the stability of the embedding. On the other hand, we wouldlike to remark that it would be desirable to control the number of time-like dimensions of theembedding manifold, even though this has been known to be a difficult task in analogouscontexts (see [1]). Also, we leave as a future research perspective the weakening of thecompactness condition.Finally, we would like to stress that we do not expect the results obtained in this paperto be sharp, not regarding the number of extra dimensions needed for the embedding, nei-ther regarding the signature of the embedding space. Probably, embeddings of even lowerregularity might be attainable, but we do not regard this technical issue as one of the mostimportant points to be pursued in upcoming research. The reason why we expect sharperresults to be attainable, is the strategy adopted to get the embedding, which besides its orig-inality, does not seem to be optimal. Conjecturing which would be the sharpest codimensionfor this type of embeddings does not seem to be sensible at this point. Certainly the mainmotivation behind this program is trying to give a response to this issue. Even though theCampbell-Maagaard theorem may suggest that we should ideally look for codimension one,and some explicit examples support this idea, this could easily turn out to be completelymisleading, since, as previously explained, this theorem is local and depends crucially on theanalyticity assumptions. Since the strategy of the proof has to be substantially modified inorder to get global and smooth embeddings, it could easily be the case that the general resultmay demand more than one codimension. The parallelism with history behind Riemann-flatembeddings, going from Janet-Cartan’s theorem to Nash’s theorem and continuing up topresent days, shows that we should be cautious in conjecturing such optimal codimension.It is certainly a very interesting open problem to establish sufficient conditions which enableus to attain this optimal codimension one global C k -Ricci-flat embedding.14 cknowledgements R. A. and C. R. would like to thank CNPq and CLAF for financial support. R. 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