On the embedding of Weyl manifolds 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 and E-mail: [email protected]; fdahia@fisica.ufpb.br; cromero@fisica.ufpb.br Abstract 7 1 0 We discuss the possibility of extending different versions of the Campbell-Magaard theorem, 2 which have already been established in the context of semi-Riemannian geometry, to the context n a J of Weyl’s geometry. We show that some of the known results can benaturally extended to the new 7 2 geometric scenario, although new difficulties arise. In pursuitof solvingthe embeddingproblem we ] have obtained some no-go theorems. We also highlight some of the difficulties that appear in the c q embedding problem, which are typical of the Weylian character of the geometry. The establishing - r g of these new results may beviewed as part of a program that highlights the possible significance of [ 1 embeddingtheoremsofincreasingdegreesofgeneralityinthecontextofmodernhigher-dimensional v 4 space-time theories. 2 2 8 0 . 1 0 7 1 : v i X r a 1 I. INTRODUCTION The unification of the fundamental forces of nature is now recognized to be one of the most important tasks in theoretical physics. Unification, in fact, has been a feature of all great theories of physics. It is a well knonw fact that Newton, Maxwell and Einstein, they all succeeded in performing some sort of unification. So, not surprisingly in the last two centuries physicists have recurrently pursued this theme. Broadly speaking one can mention two different paths followed by theoreticians to arrive at a unified field theory. First, there are the early attempts of Einstein, Weyl, Cartan, Eddington, Schro¨dinger and many others, whose aim consisted of unifying gravity and electromagnetism [1]. Their approach consisted basically in resorting to different kinds of non-Riemannian geometries capable of accomo- dating new geometrical structures with a sufficient number of degrees of freedom to describe the electromagnetic field. In this way, different types of geometries have been invented, such as affine geometry, Weyl’s geometry (where the notion of parallel transport generalizes that of Levi-Civita’s), Riemann-Cartan geometry (in which torsion is introduced), to quote only a few. In fact, it is not easy to track all further developments of these geometries, most of which were clearly motivated by the desire of extending general relativity to accomodate in its scope the electromagnetic field. However, as we now see, the main problem with all these attempts was that they completely ignored the other two fundamental interactions and did not take into account quantum mechanics, dealing with unification only at classical level. Of course, an approach to unification today would necessarily take into account quantum field theory. The second approach to unification has to do with the rather old idea that our space-time may have more than four dimensions. This program starts with the work of the Finnish physicist Gunnar Nordstr¨om [2], in 1914. Nordstr¨om realised that by postulating the exis- tence of a fifth dimension he was able (in the context of his scalar theory of gravitation) to unify gravity and electromagnetism. Although the idea was quite original and interesting, it seems the paper did not attract much attention due to the fact that his theory of gravi- tation was not accepted at the time. Then, soon after the completion of general relativity, Th´eodor Kaluza, and later, Oscar Klein, launched again the same idea, now entirely based on Einstein’s theory of gravity. Kaluza-Klein theory starts from five-dimensional vacuum Einstein’s equations and shows that, under certain assumptions, the field equations reduce 2 to a four-dimensional system of coupled Einstein-Maxwell equations. This seminal idea has given rise to several different theoretical developments, all of them exploring the possibility of achieving unification from extra dimensionality of space-time. Indeed, through the old and modern versions of Kaluza-Klein theory [3–5], supergravity [6], superstrings [7], and to the more recent braneworld scenario[8, 9], induced-matter [10, 11] and M-theory [12], there has been a strong belief among some physicists that unification might finally be achieved if one is willing to accept that space-time has more than four dimensions. Among all these higher-dimensional theories, one of them, the induced-matter theory (also referred toas space-time-matter theory[10, 11]) hascalled our attentionsince itvividly recalls Einstein’s belief that matter and radiation (not only the gravitational field) should ultimately be viewed as manifestations of pure geometry[13]. Kaluza-Klein theory was a first step in this direction. But it was Paul Wesson [11], from the University of Waterloo, who pursued the matter further. Wesson and collaborators realized that by embedding the ordinary space-time into a five-dimensional vacuum space, it was possible to describe the macroscopic properties of matter in geometrical terms. In a series of interesting papers Wesson and his group showed how to produce standard cosmological models from five- dimensional vacuum space. It looked like as if any energy-momentum tensor could be generated by an embedding mechanism. At the time these facts were discovered, there was no guarantee that any energy-momentum tensor could be obtained in this way. Putting it in mathematical terms, Wesson’s programm would not always work unless one could prove that any solution of Einstein’s field equations could be isometrically embedded in five-dimensional Ricci-flat space [14]. It turns out, however, that this is exactly the content of a beautilful and powerful theorem of differential geometry now known as the Campbell- Magaard theorem [15]. This theorem, little known until recently, was proposed by English mathematician John Campbell in 1926, and was given a complete proof in 1963 by Lorenz Magaard [16]. Campbell [15], as many geometers of his time, was interested in geometrical aspects of Einstein’s general relativity and his works [17] were published a few years before the classical Janet-Cartan [18, 19] theorem on embeddings was established. Compared to the Janet-Cartan theorem the nice thing about the Campbell-Magaard’s result is that the codimension of the embedding space is drastically reduced: one needs only one extra- dimension, and that perfectly fits the requirements of the induced-matter theory. Finally, let us note that both theorems refer to local and analytical embeddings (the global version 3 of Janet-Cartan theorem was worked out by John Nash [20], in 1956, and adapted for semi- Riemannian geometry by R. Greene [21], in 1970, while a discussion of global aspects of Cambell-Magaard has recently appeared in the literature [22]). II. HIGHER-DIMENSIONAL SPACE-TIMES AND RIEMANNIAN EXTEN- SIONS OF THE CAMPBELL-MAGAARD THEOREM Besides the induced-matter proposal, there appeared at the turn of the XX century some other physical models of the Universe, which soon attracted the attention of theoreticians. These models have put forward the idea that the space-time of our everyday perception may be viewed as a four-dimensional hypersurface embedded not in a Ricci-flat space, but in a five-dimensional Einstein space (referred to as the bulk) [8, 9]. Spurred by this proposal new research on the geometrical structure of the proposed models started. It was conjectured [23] and later proved that the Campbell-Magaard theorem could be immediately generalized for embedding Einstein spaces [24]. This was the first extension of the Campbell-Magaard theorem and other extensions, still in the context of Riemannian geometry, were to come. More general local isometric embeddings were next investigated, and it was proved that any n-dimensional semi-Riemannian analytic manifold can be locally embedded in (n+1)- dimensional analytic manifold with a non-degenerate Ricci-tensor, which is equal, up to a local analytical diffeomorphism, to the Ricci-tensor of an arbitrary specified space [25]. Further motivation in this direction came from studying embeddings in the context of non- linear sigma models, a theory proposed by J. Schwinger in the fifties to describe strongly interacting massive particles [26]. It was then showed that any n-dimensional Lorentzian manifold (n ≥ 3) can be harmonically embedded in an (n+1)-dimensional semi-Riemannian Ricci-flat manifold [27]. At this point we should remark that most theories that regard our spacetime as a hyper- surface embedded in a higher-dimensional manifold [28] make the tacit assumption that this hypersurface has a semi-Riemannian geometrical structure. Surely, this assumption avoids possible conflicts with the well-established theory of general relativity, which operates in a Riemannian geometrical frame. However, recently there has been some attempts to broaden this scenario. For instance, new theoretical schemes have been proposed, where one of the most simple generalizations of non-Riemannian geometry, namely the Weyl geometry [29], 4 has been taken into consideration as a viable possibility to describe the geometry of the bulk [30–32]. In some of these approaches, the induced-matter theory is revisited to show that it is even possible to generate a cosmological constant, or rather, a cosmological function, from the extra dimensions and the Weyl field [33]. In a similar context, it has also been shown how the presence of the Weyl field may affect both the confinement and/or stablity of particles motion, and how a purely geometrical field, such as the Weyl field, may effectively act both as a classical and quantum scalar field, which in some theoretical-field modes is the responsible for the confinement of matter in the brane [34, 35]. There is also another very interesting and compelling argument for considering a Weyl structure as a suitable mathematical model for describing space-time. This is based on the well-known axiomatic approach to space-time theory put forward by Ehlers, Pirani and Schild (EPS), which, through an elegant and powerful theoretical construction, shows that bystartingfromaminimumsetofratherplausibleandgeneralaxiomsconcerningthemotion of light signals and freely falling particles, one is naturally led to a Weyl structure as the proper framework of space-time [36]. In order to reduce this more general framework to that of a semi-Riemannian manifold we need an aditional axiom to be added to this minimum set. It turns out, however, that this added axiom does not seem as natural as the others, as was pointed out by Perlick [37]. We take Perlick’s point of view as one of the motivations for investigating the geometry of Weyl spaces. In this paper we shall consider the mathematical problem of extending different ver- sions of the Campbell-Magaard theorem from the Riemannian context to Weyl’s geometry. Specifically, we shall first analyze the possibility of locally and analytically embedding an n- dimensional Weyl manifold in an (n+1)-dimensional Weyl space, the latter being Ricci-flat. We then weaken this condition to investigate the problem of embedding manifolds whose symmetric part of the Ricci tensor vanishes. These problems can be regarded as extensions of the Campbell-Magaard theorems, which hold in Riemannian geometry, to a more general geometrical setting, namely that of Weyl’s geometry. We believe that an investigation of these seemingly purely geometrical problems may also shed some light on the physics of higher-dimensional theories in which there are extra degrees of freedom coming from the geometric structure of space-time, in particular, those in which there are mechanisms for generating matter and fields from extra dimensions in the case of theories of gravitation whose geometrical framework is based on the Weyl theory, and other higher-dimensional 5 proposals formulated in Weyl manifolds, such as D-dimensional dilaton gravity [38], higher- dimensional WIST theories [30],[31],[33],[39] and others. Finally, a few words should be said with regard to the Campbell-Maggard theorem and its application to physics. First, let us note that the proof provided by Magaard is based on the Cauchy-Kovalevskaya theorem. Therefore, some properties of relevance to physics, such as the stability of the embedding, cannot be guaranteed to hold [43]. Nevertheless, the problem of embedding space-time into five-dimensional spaces can be considered in the context of the Cauchy problem for general relativity [44]. Specifically, it has recently been shown that the embedded space-time may arise as a result of physical evolution of proper initial data. This new perspective has some advantages in comparison with the original Campbell-Magaard formulation because, by exploring the hyperbolic character of the field equations, it allows to show that the embedding has stability and domain of dependence (causality) properties [45]. III. WEYL GEOMETRY When working in Riemannian geometry we consider a pair (M,g), where M is a differen- tiable manifold and g a (semi)-Riemannian metric defined on M. The fundamental theorem of Riemannian geometry states that there is a unique torsionless linear connection compati- ble with g [46]. By compatibility we mean the following. When we endow any differentiable manifold with a linear connection ∇ we have an associated notion of parallel transport. It is well known that parallel transport defines isomorphisms between tangent spaces. The com- patibility condition is defined as the requirement that this isomorphism is also an isometry. This turns out to be equivalent to the following requirement ∇g = 0. It turns out that in Weyl’s geometry we relax the requirement of ∇ being compatible with g, and this means that parallel transports are not required to define isometries anymore. We first endow M not only with a semi-Riemannian metric, but also with a one-form field ω, so that instead of the pair (M,g) we now consider a triple (M,g,ω). Weyl’s connection is defined by requiring it to be torsionless and that, for any parallel vector field V along any 6 smooth curve γ, the following condition is satisfied: d ′ g(V(t),V(t)) = ω(γ (t))g(V(t),V(t)). (1) dt Before presenting the main existence and uniqueness theorems for such connection, we shall try to get some insight on what this condition means geometrically. First of all, note that because parallel transport is a linear application, if V,W are parallel fields along some curve γ, then V +W also is a parallel field along γ. On the other hand, by polarization we get 1 g(V,W) = (g(V +W,V +W)−g(V,V)−g(W,W)), 2 which together with (1) gives d ′ g(V,W) = ω(γ )g(V,W). dt We thus say that the connection ∇ is Weyl compatible with (M,g,ω) if for any pair of parallel vectors along any smooth curve γ = γ(t) the condition below is satisfied d ′ g(V(t),W(t)) = ω(γ (t))g(V(t),W(t)), dt where γ′ denotes the tangent vector of γ. Integrating the above equation along the curve γ leads to g(V(t),W(t)) = g(V(0),W(0))eR0tω(γ′(s))ds. (2) Inparticular, ifV = W thislastexpressiongivesuspreciselyhowmuchtheparalleltransport fails to be an isometry: g(V(t),V(t)) = g(V(0),V(0))eR0tω(γ′(s))ds. Note that if the vectors V(0) and W(0) are orthogonal, then (2) implies that they remain orthogonal when parallel transported along the curve, although their respective ”norms” may change. Let us now state some results that hold for a Weyl connection which are analogues to those valid for a Riemannian connection. All these results are proven in very much the same way as in Riemannian geometry. Proposition 1. A connection ∇ is compatible with a Weyl structure (M,g,ω) iff for any pair of vector fields V,W along any smooth curve γ in M the following holds: d DV DW ′ g(V,W) = g( ,W)+g(V, )+ω(γ )g(V,W) (3) dt dt dt 7 Corollary 1. A linear connection ∇ is compatible with a Weyl structure (M,g,ω) iff ∀ p ∈ M and for every vector fields X,Y,Z on M the condition below holds X (g(Y,Z)) = g (∇ Y,Z )+g (Y ,∇ Z)+ω (X )g (Y ,Z ) (4) p p Xp p p p Xp p p p p p In the last proposition we can actually drop p and write X(g(Y,Z)) = g(∇ Y,Z)+g(Y,∇ Z)+ω(X)g(Y,Z), X X which, then, can be used to prove the following: Proposition 2. A linear connection ∇ is compatible with a Weyl structure (M,g,ω) iff it satisfies ∇g = ω ⊗g (5) Now the following result is easily established. Proposition 3. There is a unique torsionless connection compatible with the Weyl structure (M,g,ω). In the proof of this proposition it is found that the Weyl connection, in a particular coordinate system, takes the following form: 1 1 Γu = gbu(∂ g +∂ g −∂ g )+ gbu(ω g −ω g −ω g ) (6) ac 2 a bc c ab b ca 2 b ca a bc c ab ItisimportanttonotethataWeylmanifolddefinesanequivalence classofsuchstructures all linked by the following group of transformations: g = e−fg  (7)  ω = ω −df  where f is an arbitrary smooth function defined on M. It is easy to check that these transformations define an equivalence relation between Weyl manifolds, and that if ∇ is compatible with (M,g,ω), then it is also compatible with (M,g,ω). In this way every member of the class is compatible with the same connection, hence has the same geodesics, curvature tensor and any other property that depends only on the connection. This is the reason why it is regarded more natural, when dealing with Weyl manifolds, to consider the whole class of equivalence (M,[g],[ω]) rather than working with a particular element 8 of this class. In this sense, it is argued that only geometrical quantities that are invariant under (7) are of real significance in the case of Weyl geometry. Following the same line of argument it is assumed that only physical theories and physical quantities presenting this kind of invariance should be considered of interest in this context. To conclude this section, we remark that when the one-form field ω is an exact form, then the Weyl structure is called integrable. A. Weyl submanifolds Definition 1. Let (M,g,ω) be a Weyl manifold and M ֒→ M be a submanifold of M. If the pullback i∗(g) is a metric tensor on M then (M,i∗(g),i∗(ω)) is a Weyl submanifold of M. In this case we will use the notation g = i∗(g) and ω = i∗(ω) for the induced metric and 1-form. Using the same conventions as in the previous definition, we denote by ∇ the Weyl- compatible connection associated with (M,g,ω). We define the induced connection ∇ on M following the same reasoning as in the Riemannian case. Thus if X,Y are vector fields . on M, and X,Y are extensions of these vector fields to M, then ∇ Y = (∇ Y)T. It is a X X well-known fact that this definition does not depend on the extensions [46]. It is worth noticing that both the definition of Weyl submanifold and of induced con- nection make sense in the whole class (M,[g],[ω]). We can see that the definition of Weyl submanifold satisfies this condition since every such structure that can be obtained from an element of (M,[g],[ω]) lies in (M,[g],[ω]) and vice versa, every element of (M,[g],[ω]) can be obtained from some element of (M,[g],[ω]). The fact that the definition of induced con- nection is invariant in the whole class (M,[g],[ω]), is because two conformal metrics make the same splitting of the tangent spaces: T M = T M ⊕T M⊥. p p p The following results are obtained in the same way as in Riemannian geometry. Proposition 4. Given a Weyl manifold (M,g,ω) and a Weyl submanifold M ֒→ M, the in- duced connection ∇ on M is the Weyl connection compatible with the induced Weyl structure on (M,g,ω). 9 As usual, we define the second fundamental form α on M as ⊥ α : TM ×TM 7→ TM . ⊥ α(X,Y) = (∇ Y) X One can easily check that this definition does not depend on how we extend X and Y to M. Thus, if X,Y are fields on M we write ∇ Y = ∇ Y +α(X,Y) X X The next proposition is analogous to its Riemannian counterpart: Proposition 5. The second fundamental form α is symmetric and F(M)-linear in both arguments. From now on we shall consider only hypersurfaces. In this case, we can define a unit normal vector field η, which, at least locally, is unique up to a sign. We define the scalar second fundamental form l as given by l : TM ×TM 7→ F(M) (X,Y) 7→ g(α(X,Y),η) We note that although the choice of the unit normal field η depends on a particular element of (M,[g],[ω]), the definition of l does not. Now from the last proposition it follows that l is symmetric and F(M)-linear, i.e., l is a symmetric (0,2)-tensor field on M. Following a procedure entirely analogous to what is done in Riemannian geometry, we obtain the Gauss-Codazzi equations for hypersurfaces. Thus, if X,Y and Z are vector fileds on M, Gauss’ equation takes the form g(R(X,Y)Z,W) = g(R(X,Y)Z,W)+g(α(X,Z),α(Y,W))−g(α(Y,Z),α(X,W)). (8) If ξ is a unit field normal to M, then Codazzi’s equation reads 1 g(R(X,Y)Z,ξ) = ǫ((∇ l)(Y,Z)−(∇ l)(X,Z)+ (ω(Y)l(X,Z)−ω(X)l(Y,Z))). (9) X Y 2 . Where ǫ = g(ξ,ξ) = ±1 and the sign depends on whether the restriction of g to each T M⊥ p is positive or negative definite. Let us now have a look at the Bianchi identities in Weyl geometry, as they will be useful in our investigation of the embedding problem. 10