On the Global Embedding of Spacetime into Singular E Σ Einstein Manifolds: Wormholes

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1 arxiv:math/ v1 [math.dg] 7 Mar 2005 On the Global Embedding of Spacetime into Singular E Σ Einstein Manifolds: Wormholes Nikolaos I. Katzourakis University of Athens, Department of Mathematics nkatzourakis@math.uoa.gr Abstract In a recent work [23] it has been proved that any n-dimensional analytic semi-riemannian manifold M can be embedded globally and isometrically into an Einstein manifold E of dimension n + d, d 1 (Λ R). The possible topologies of such an E have been studied, calculating π m (E) and H m (E; G) & both proved to split in every dimension. In this sequel we continue proving that given any (analytic) spacetime manifold M, there exists an Einstein embedding manifold E Σ, admitting unsmoothable differential-topological singularities (0-D points or extended (d k)-d spatial anomalies ) living into E Σ M, presenting also a direct sum Homology splitting for E Σ with 1-point singularity. Certain consequences on MD Relativity and Brane Cosmology are briefly discussed, where these pathological anomalies of the bulk are expounded as wormhole passages allowing distant spacetime travels. Keywords: Semi-Riemannian / Einstein / Ricci-Flat manifold, Campbell- Magaard local embedding, Homology, Singularities; Gravity, General Relativity, Brane Cosmology, Wormholes. MSC Subject Classification 2000: Primary 53C20, 53C25, 32C09, 53Cxx, Secondary 83C75, 83C55, 83E15, 83F05. Address: Troias 6, 18541, Kaminia, Pireaus, Greece 1

2 Global Embedding of Spacetime into Singular Einstein Manifolds 2 Introduction The quest for embedding spaces of given Riemannian manifolds once used to be a pure mathematical problem. Notwithstanding, modern mathematical & theoretical physics has drastically changed this situation, bringing into light new ideas concerning the dimensionality of spacetime and the specific problem has become common in both mathematics & physics. According to Einstein s General Relativity (G.T.R.), the model we adopt to describe the large scale structure of the Universe is a Lorentzian 4-manifold (3 spatial & 1 time dimensions), where the metric is the solution of the field equations, correlating matter, energy, momentum and the curvature of spacetime (e.g. [7], [11]). The idea to add 1 spatial dimension goes back to the 1920 s, where Kaluza & Klein unified relativity & electromagnetism to a 5D vacuum relativity. The 5th dimension was considered topologically compact & infinitesimal, in order to explain its non-visibility. In more modern perspectives ([14]) it may be considered non-compact & then the universe of perception is a hypersurface embedded into a 5D Eintein manifold. Even more recent ideas come from Cosmology, where acc. to the so-called Braneworld scenario, the visible Universe is a possibly moving or rotating (mem-)brane living into an MD Bulk (e.g. [20], [21], [24], [27]) & there may exist more than 1 (colliding or not) branes (e.g. [31]). Mathematically, these theories are modeled upon a local theorem due to Campbell (who outlined a proof in [2]) and Magaard (who proved it in his thesis [8]) (C.-M. Theorem) which claims that local embedding of any analytic semi- Riemannian manifold is always possible in codimension only 1, if the embedding space is Ricci-flat, instead of flat ([29]). This means that 4D-Relativity with matter can be embedded (at least locally) into a vacuum 5D-Relativity. It was conjectured that the condition of Ricci-flatness was not essencial: indeed, it has been proved that it can be replaced by the Einstein space curvature requirement ([18]), which means non-vanishing Cosmological Constant Λ 0. Several other attempts of generalization of C.-M. theorem have appeared, though all in a local framework ([15], [16], [17], [18], [19]) and with more physical than geometrical significance ([26]).

3 Global Embedding of Spacetime into Singular Einstein Manifolds 3 In a recent work ([23]) we proved that global embedding of analytic semi- Riemannian manifolds into Einstein manifolds is always possible in codimension d 1. This result, apart from the pure geometrical importance (interconnected with the Einstein spaces classification), provides the requested existential mathematical framework for all physical theories based on the embedding of relativity to a MD manifold: this is the 1st positive answer to the question whether there exists a consistent mathematical global embedding of any general-relativistic spacetime to an MD vacuum Einstein Bulk with arbitrary cosmological constant Λ R. It was known that local embedding of spacetime was possible, though it was unknown if this was the case to the global level. In addition, the topology of such an Einstein space turns to be extremely simple in every codimension d 1, due to the splitting Homotopy & Homology in every dimension ([23]). Taking into account the natural question: Does there exist any result providing a global framework of embedding General Relativity into (more realistic) singular MD Einstein spacetime manifolds? in this sequel we deal with spacetime manifolds admitting singularities living in the complement of the Bulk that extra (than the 4 of Einstein Relativity) spatial dimensions generate. The subject has a vast bibliography and these singularities are usually expounded as multi-dimensional Black Holes or other exotic anomalies. The basic results Th. 3.1, 3.2 (modifying Th. 2.1) claim that given any spacetime manifold M with a solution g M of the G.T.R. equations 1.1, there exists a manifold admitting 1 singular point in codimension d = 1 and countable many in codimension d > 1, equipped with a solution of equations 1.1 in the respective dimension with vanishing Stress-Energy-Momentum tensor: T 0. In addition there exists no resolution of these differential-topological singularities, provided that the singular spacetime E Σ is not even a manifold! (unless the singularities are extracted.) We also show that the restriction the singularities of E Σ M to be points, is not essential: if d = 1 there can exist topological anomalies of positive 1-measure (lying on the image of a path) and for d > 0 of positive d k-measure (e.g. the diffeomorphic image of a closed ball), for 0 k d.

4 Global Embedding of Spacetime into Singular Einstein Manifolds 4 As a corollary, we obtain a splitting theorem for the (reduced) Homology of the Einstein space admitting a single singularity, with any coefficient group G. In the last section we give an attempt of physical interpretation of these singularities, where they can be expounded as wormholes in spacetime connecting distant point-events of some MD vacuum Relativity lying into the complement E Σ M of the Bulk. Notations: The term manifold denotes a paracompact, Hausdorff, connected smooth manifold of finite dimensions, equipped with a semi-riemannian (indefinite of arbitrary signature) metric structure. 1. Preliminaries In this section we collect for the readers convenience some fundamental notions retrieved from the references that will be employed in the rest of the paper. Definition 1.1. The G.T.R. field equations on a semi-riemannian manifold (M (n) ;, g) in presence of Cosmological Constant Λ R are: where Ric, g, T G Ric (g) 1 2 R(n) g = kt Λg Γ(T M T M), k R, G is the Einstein tensor of Curvature and R (n) is the scalar Curvature, defined as the double contraction with the metric. T is called the Stress-Energy-Momentum tensor. Definition 1.2. A semi-riemannian manifold (M (n) ; M, g M ) is said to be an Einstein space if there exists a Λ R such that: Ric M (g M ) = 2Λ n 2 g M That is, its metric is a solution of the previous equations with T = 0. If Λ = 0 the manifold is said to be Ricci-flat. Theorem 1.3. (Magaard s criterion of local isometrical embeddings) Let (M n, g) be a semi-riemannian manifold and (U; x j = π j φ, 1 j n) is a local chart at p U M and

5 Global Embedding of Spacetime into Singular Einstein Manifolds 5 g U = g ij dx i dx j Γ(T U T U) a local representation of its metric at p U. Then, the following statements are equivalent: (I) There exists a local analytic isometrical embedding of U into some (N, g): ε : U ε(u) N (II) There exist analytic functions g ij, ϕ C α (W R), W φ(u) R for some open set W containing x j e j + 0 e n+1 R n+1 and satisfying the conditions: g ij ι A = g ij, g ij = g ij, g 0, ϕ 0, (ι A is the natural embedding of A R n R n+1 ) such that the metric of (N, g) has the following local representation at ε(p): g V = g ij dx i dx j + ǫ ϕ 2 dx n+1 dx n+1 with ǫ 2 = 1 in some coordinated domain V ε(u) N. Theorem 1.4. (Campbell-Magaard-Dahia-Romero local embedding into Einstein Spaces) Let (M (n), g M ) be a semi-riemannian analytic manifold and g U = g ij dx i dx j Γ α0 (T M T M) a local analytic at 0 φ(p) R n presentation of its metric, with respect to (U, φ) A M. Then, there exists a local isometric embedding (analytic at φ(p)) of U into some (coordinated) semi-riemannian manifold (N (n+1) ; g N, N ): ε : U ε(u) N

6 Global Embedding of Spacetime into Singular Einstein Manifolds 6 which is an Einstein space in the local embedding domain: that is Ric ε(u) (g N ε(u) ) = 2Λ n 1 g N ε(u) Γ(T ε(u) T ε(u)) R AB dx A dx B = 2Λ n 1 g AB dx A dx B 2. Global C.M.-type Embedding into Einstein Manifolds The manifolds we employ are considered real analytic. Notwithstanding, it follows from results of Whitney [33] and Grauert [22] that this is not restrictive, provided that any C k or smooth (C ) manifold has a C k (resp. C ) diffeomorphism with an analytic one, that is, it carries a unique compatible real analytic structure. See also [9]. This Section is a shortened version of the respective of [23] without proofs of the intermediate results in the basic proof. It is presented here for complementary only reasons. For details, see [23]. Theorem 2.1. (Global Isometric Embedding into Einstein manifolds in codimension 1) Any n-dimensional real analytic semi-riemannian manifold (M; M, g M ) has a global isometric embedding into an (n + 1)-dimensional Einstein manifold (E; E, g E ) satisfying: for any Λ R. Ric E (g E ) = 2Λ n 1 g E Remark 2.2. It is obvious that if a manifold can be embedded into an Einstein space as a hypersurface, repeated applications of Theorem 2.1 imply the weaker condition to be embedable in any codimension greater than 1. Proof. (Outline) Step 1: Construction of the Bulk E that contains M and the topological cover U of E.

7 Global Embedding of Spacetime into Singular Einstein Manifolds 7 Let F be an abstract analytic curve (1-dimensional manifold). We define the fibre bundle: π := pr 1 : E := M F M Equivalently, we can consider E as the disjoint union of manifolds {p} F parameterized by the points of M: E := {p} F (2.1) p M This construction gives a Bulk (the total space) E that contains as a submanifold the given M (the inclusion map is an embedding). The local (coordinated) trivializations have the form: π 1 = (U) V U (F V ) W A E id where (U; {x a } 1 a n ) A M, (V ; y) A F. Topologically E is paracompact & Hausdorff. This implies the existence of a natural locally finite cover constituted of products charts: U := {(W i ; {x A } 1 a n+1 )/i I} {([U V ] i ; {x a } 1 a n y) / i I} Step 2: Local extensions of each g M U to g W on W U of E. If g M Γ(T M T M) is the semi-riemannian metric of M, (U; {x a } 1 a n ) is a local chart, g has the representation g M U := g M ( a, b ) dx a dx b = g ab dx a dx b, a, b {1,..., n} g ab C α ( {x a (U)} 1 a n R ) This U can be considered as embedded into π 1 (U) = U V. We wish to define a local metric on W U V. Choose an arbitrary function ψ g (n+1)(n+1) C (W R) (analytic at 1 point of W) that will be determined later. If (V ; y) A F, define a g W in

8 Global Embedding of Spacetime into Singular Einstein Manifolds 8 Γ(T W T W) as g W := g ab dx a dx b + ψ dy dy g ab dx a dx b + ψ dx n+1 dx n+1 setting any possible diagonal terms g a(n+1) equal to zero, appending one only one g (n+1)(n+1) ψ term. Step 3: The refinement U B of the cover U, constituted of inversed images of balls. Within every open χ i (W i ) = W i lives an open ball B(a, R i ). Choose an r i < R i. Then B(a, r i ) χ i (W i ) R n+1 transferring χ i (W i ) into R n+1 so that every a i to coincide with the origin 0 R n+1, we may assume that W i χ 1 (B(0, r i )) and we can prove the following: Lemma 2.3. There exists a (locally finite open) refinement U B of the cover U constituted of inversed images of balls U B := {χ 1 i (B(0, r i )) / i I}, with the same indexing set I. i I χ 1 i (B(0, r i )) = E Step 4: A global (smooth) metric structure g E on E, its (local) analyticity and the Levi-Civita linear connection. We introduce a global metric structure on E, extending the given g M on M sewing together the previous local extensions g Wi of the W i s. Deviating from the standard approach, we do not introduce a partition of unity with respect to a locally finite cover, but instead we present a family of functions with similar properties (except for the normalization f i = 1). We can prove that: Statement 2.4. There exists a family {f i } i I of C non-negative functions on E, with properties: f i C (E R [0, + )) & supp(f i ) W i, W i U

9 Global Embedding of Spacetime into Singular Einstein Manifolds 9 and p E, f i (p) > 0 and the f i s are real analytic within the set they are strictly positive i I {p E / f i (p) > 0} {f i > 0} (which is open & coincides with int(supp(f i )) supp(f i )): f i {fi >0} = f i supp(f i ) C α (E {f i > 0} R) We note only that the family of smooth real functions {f i } i I on E is: f i C (E R), i I : p f i (p) := exp ( 1 χ i (p) 2 r 2 i ), p χ 1 i (B(0, r i )) W i 0, p E χ 1 i (B(0, r i )). (2.2) Since E is real analytic, we have f i {fi >0} = f i χ 1 i (B(0,r i )) C α (χ 1 i (B(0, r i )) R) and the supports of the f i s (& their interior) form a locally finite cover of E: i I supp(f i ) = i I ( supp(f i ) = χ 1 i B(0, ri ) ) W i, {f i > 0} = E & f i > 0 Now we can define the requested global section g E Γ(T E T E) as: i I g E = i I { ğ ab dxa dxb + ψ dx n+1 dx n+1 } The index i denotes the components of the metric in the i -coordinate system (W i ; {x A } 1 A n+1}), i I and we have set (each f i given by (2.2))

10 Global Embedding of Spacetime into Singular Einstein Manifolds 10 ğ ab := f i g ab C (E R) ψ := f i ψ C (E R) We recall the Fundamental Theorem of Riemannian Geometry and equip E with the respective unique Levi-Civita linear connection E : (E ; E, g E ). Step 5: Local representations of g E patch. with respect to any coordinated Choose a patch of E (Q ; z A ) A E (that does not necessarily belong to U). Then, we have ( g E Q = { ğ ab i I ( x a ) ( x b ) z A z B + ψ ( x n+1 ) ( x n+1 ) ) } z A z B Q dz A dz B Step 6: The cover V of geodesical coordinates diagonalizing the local restrictions g E V. Consider a locally finite cover V of coordinated V j s that diagonalize the local metrics g E Vj in the last coordinate (the extra dimension of E). Then ( g E Vj = { ğ cd i I ( x c ) ( x c y a (j) y b (j) ) + ψ ( x n+1 y a (j) ) ( x n+1 ) )} y(j) b Vj dy(j) a dyb (j) + ( + { ğ cd i I ( x c y n+1 (j) where 1 a, b, c, d n. ) ( x c y n+1 (j) ) + ψ ( x n+1 ) 2} Vj) y n+1 dy n+1 (j) dy n+1 (j) (j) We define the components of g E with respect to the coordinates of each V j as [g (j) E ] AB := i I {ğ ab ( x a ) ( x b y A (j) y B (j) ) + ψ ( x n+1 y A (j) ) ( x n+1 ) } y(j) B Vj (2.3) Define now V := {(V j ; {y A } 1 A n+1 ) / j J}

11 Global Embedding of Spacetime into Singular Einstein Manifolds 11 with property the a(n + 1)-diagonal terms of (2.3) to vanish: [g (j) E ] (n+1) a 0, 1 a n Step 7: Fixing the topological covers U, U B & V - the analyticity of the local presentations g E Vj. The following is needed in the sequel: Proposition 2.5. The locally finite covers U, U B and V can be fixed so that the following to hold: g E has n-dimensional metric components [g (j) E ] ab given by (2.3) analytic at (at least) 1 point p j of each V j V, ( {y(j) A } 1 A n+1(p j ) 0 R n+1 ) [g (j) E ] ab C α 0 ( Vj R ), 1 a, b n (ii) V has less overlapping patches covering each p E than U, that is p E, with p w p r=1 (w p, v p N depending on p) we can always have W ir and p v p V js s=1 w p > v p Step 8: The requested Curvature condition on E expressed in terms of local coordinates of the cover V using Magaard s Criterion 1.3 and C.-M.-D.-R. local embedding 1.4. The condition E to be an n + 1-dimensional Einstein manifold, with respect to the semi-riemannian structure ( E, g E ) (Def. 1.2) Ric E (g E ) = 2Λ n 1 g E is equivalent to the condition this equation to hold within all the local coordinated open submanifolds of a cover of E, say V: Ric E Vj(gE Vj ) = 2Λ n 1 g E Vj, j J and in the local coordinates ψ j = {y A (j) } of each V j the previous relation reads

12 Global Embedding of Spacetime into Singular Einstein Manifolds 12 [R (j) ] AB dy A (j) dy B (j) = 2Λ n 1 [g(j) E ] AB dy A (j) dy B (j), j J where [g (j) E ] AB are given by (2.3) and [R (j) ] AB are the components of the Ricci tensor of curvature, calculated by (due to the vanishing Torsion of E ): and [R (j) ] AB = M Γ M (j) AB + Γ M (j) ABΓ N (j) MN B Γ M (j) AM Γ M (j) ANΓ N (j) MB Ric E Vj(gE Vj ) = Ric E Vj( A, B ) dy A dy B [R (j) ] AB dy A dy B The local Campbell-Magaard-Dahia-Romero Th. 1.4 on Einstein spaces and the Criterion 1.3 of local isometric embeddings, imply that, since the functions [g (j) E ] ab are analytic (Step 7, Prop. 2.5), there do exist analytic functions (at 1 point at least, say 0 ψ j (p j ) R n+1 ) [g (j) E ] (n+1)(n+1) C α 0 (V j R), j J such that the (local) embeddings to hold (on every patch). Step 9: Fixing the Einstein space metric g E patches of V separately. on any non-intersecting Choose a patch V j of V and recall the relation (2.3) that gives the (n+1)(n+1)- term [g (j) E ] (n+1) of g E on this patch: [g (j) E ] (n+1) w s=1 ( x a ğ (is) (is) ab y n+1 (j) )( x b (is) y n+1 (j) ) Vj }{{} = w ( x n+1 (i f s) is s=1 y n+1 (j) ) 2 Vj }{{} ψ (is) Thus, we obtain Φ (j) Φ (j) = w s=1 Θ (j) ( Θ (i s) (j) V j ) ψ (i s), j J (2.4) Eq. (2.4) is a linear functional equation with smooth (& analytic at 0 {y A (j) }(p j) R n+1 ) coefficients with respect to the ψ s (restricted) on each V j.

13 Global Embedding of Spacetime into Singular Einstein Manifolds 13 We will show that the underlying ψ s living in the W i s intersecting V j s can indeed be defined through (2.4), so that to get the desired curvature on E. We can prove that: Statement 2.6. There exists a partition {A jk / k K} of V j in disjoint sets, each of the A jk s living in a class V j [{f i > 0}] m (see the proof of Prop. 2.5 at [23]): V j = A jk, A jk [{f ik > 0}] m [W ik ] m k K The partition can be chosen so that the set constituted of the union of any A jk s living in the same element of [W i ] m to be non-connected in the subspace topology induced by the respective of the (m + 1)-elements of [W i ] m. Now, we can solve equation 2.4 on each A jk and determine m + 1 local (smooth) functions living in each element of [W i ] m A jk. Since Θ (t,i) (j) A jk0 C (W (t,i) A jk0 R), W (t,i) [W i ] m (0 t m) and since each A jk0 is chosen to belong to one only class [{f i > 0}] m, we obtain that the (m + 1)-functions of eq. (2.4) Θ (j) V j A jk vanish outside this A jk0 ( (j j k0 ) x n+1 ) (t,i) 2} Vj A jk0 = {f (t,i) y n+1 Ajk0 0, (j j k0 ) Θ (t,i) W (t,i) [W i ] m and now eq. (2.4) can be rewritten on each A jk as Φ (j) Ajk0 = Θ (i0) (j) ψ (i 0) + Θ (j) ψ A jk0 [{f i0 >0}] m V j A jk0 }{{}}{{} (m + 1) terms 0 or Φ (j) Ajk0 = Θ (0, i 0) (j) ψ (0, i 0) Vj A (0) j k Θ (m, i 0) (j) ψ (m, i 0) Vj A (m) j k0

14 Global Embedding of Spacetime into Singular Einstein Manifolds 14 where A (t) j k0 [{f i0 > 0}] m, 0 t m. This is a linear functional equation that admits infinite (m + 1)-tuples of solutions ψ, one for each element of [W i ] m. Thus, we can choose a solution so that the Einstein space curvature condition to hold on every V j separately. The components of the metric g E are analytic (at 0 = {y A (j) }(p j)) functions ψ (t, i 0) = f i0 ψ (t, i 0) C α 0 (E W (t, i0 ) R), W (t, i0 ) [W i0 ] m (where 0 t m). The above prove that for every V j in U, we can fix g E Vj so that Ric E Vj(gE Vj ) = 2Λ n 1 (g E Vj ), j J Step 10: Fixing the Einstein space metric g E on any finite-many intersecting patches of V. We will now show that on any overlapping V j patches of the fixed cover V, we can determine the ψ s on the W i s of U, so that E to be an Einstein manifold as desired. Having proved that the number of intersecting elements of U covering each p E is greater than the respective of V, we proceed by considering a partition of E in disjoint sets {B y / y Y}, B y Bȳ =, y ȳ, of the form: v B y { V jr } { v r=1 r=1 V jr j r j r V j r } such that non of the B y s intersecting a V j has a non-empty difference B y V j : B y : B y V j, if B y V j We take the equations 2.4 (now on all the V j s simultaneously) and using the partition of E in disjoint B y s, we restrict them on the B y s, y Y: Φ (j) By = i I ( Θ (j) V j B y ) ψ, B y E Provided that V is locally finite, each B y consists of finite intersecting V j s. This partition of E implies a system of linear equations with smooth functional

15 Global Embedding of Spacetime into Singular Einstein Manifolds 15 coefficients on every B y, each of which admits infinite solutions, due to the Proposition 2.5 that claims that the number w p of the intersecting W i s can be fixed greater than the respective v p of the V j s at each p E. Thus, on every B m we obtain: w p ( Φ (j) By = Θ (j) ) B y ψ, i=1 1 j v p w p In order to solve this system and determine the ψ s, we apply the working philosophy of the previous Step 9 and separate each B y in disjoint A yl s: B y = l L The ψ s can be fixed on every A yl B y, since A yl [{f iyl > 0}] m [W iyl ] m by choosing a solution of the v p -equations of w p -variables (w p > v p ) of the system A yl Φ (j1 ) Ayl = (Θ (i 1) (j 1 ) A yl ) ψ (i 1) (Θ (i 1) (j 1 ) A yl ) ψ (i 1) Φ (j2 ) Ayl = (Θ (i 1) (j 2 ) A yl ) ψ (i 1) (Θ (i 1) (j 2 ) A yl ) ψ (i 1)... Φ (jv) Ayl and the components are = (Θ (iw) (j v) A yl ) ψ (iw) (Θ (iw) (j v) A yl ) ψ (iw) ψ (i l) = f iyl ψ (iy l ) C α 0 ( E A yl R) 3. Embedding into Cα -Manifolds EΣ admitting Differential-Topological Singularities The basic embedding result 2.1 of Section 2 states that given any analytic semi- Riemannian manifold, there exists an Einstein manifold having as hypersurface

16 Global Embedding of Spacetime into Singular Einstein Manifolds 16 (or generally submanifold) the given. This manifold is smooth (& analytic) without any singular (in any sense) points. Notwithstanding, this is not necessarily the case: the complement of the Bulk may admit singular points of the differential & topological structure. These singularities are by construction pathological and there does not exist any resolution of them: they are obtained by certain topological identitifications of distinct point and the obtain Bulk is not even a manifold. Theorem 3.1. (Embedding into Einstein manifold with unsmoothable differential-topological singularities of zero measure) Given a semi-riemannian analytic manifold M, there exist: (I) a topological space (E Σ, T E Σ) (II) an analytic manifold (E, A E ), with dim(e) = dim(m) + d, d 1 (III) a countable set of points F Σ E Σ, satisfying: F Σ = { {f σ }, if d = 1 {f σ (1), fσ (2),...} = N, if d 1. such that the following topological homeomorphism holds (under inclusion to a subspace) (E Σ F Σ, T E Σ E Σ F Σ) = (E, T AE ) and E is an Einstein space of global isometric embedding for M. In addition, E Σ is not even a topological manifold and there does not exists any smoothing resolution of the singularities F Σ E Σ. T AE denotes the (induced from the differential structure) topology of E. Proof. Let Ĕ := M F be a Cartesian product of the given semi-riemannian manifold M with an analytic manifold F, dim(f) 1 satisfying the property: deleting any countable many points F Σ of an m-dimensional and a single point {f σ } from an 1-dimensional F, the respective complements F (d) F Σ and F (1) {f σ } are connected ( If dim(f) = 1, we can choose F = S 1 and if dim(f) = d > 1, F = R d. ) Recalling 2.1, we rewrite Ĕ as the disjoint union of manifolds

17 Global Embedding of Spacetime into Singular Einstein Manifolds 17 Choose a countable set of points Ĕ = {p} F p M F Σ := {f(1) σ, fσ (2),...} = N of F (if dim(f) > 1) or a single point {f σ } (if dim(f) = 1). We will prove the general case for F Σ reducing obviously to the trivial case when d = 1. The previous choice implies a choice of points for each {q} F of Ĕ: {q} F Σ = {q} {f(m)} σ m N = {(q, f σ ) (m) } m N {fq σ (m)} m N = N denoted as F Σ p := {f σ p (m) } m N {p} F, p M Considering the (disjoint) union of all such points in Ĕ p M F Σ p Ĕ = p M we may get a more subtle expression of them, as: {p} F F Σ (m) := {fp σ (m) } p M Ĕ = M F, m N Define now a topological space E Σ as the quotient space, produced identifying for each m in N and p M, all the distinct points F Σ (m) as one fσ (m) : E Σ := Ĕ / m N F Σ (m) = p M{{p} F} / m N F Σ (m) with the quotient topology, say T E Σ, induced by the topologies T AM and T AF. In the case of 1-dimensional manifold F, this is exactly the wedge sum of the manifolds {p} F with respect to (the 1 term sequence) F Σ (each fp σ is the chosen base point of {p} F): E Σ F Σ {{p} F} (3.5)

18 Global Embedding of Spacetime into Singular Einstein Manifolds 18 Back to the general case, Ĕ = M F has a natural (product) analytic manifold structure. Deleting all the (countable many) distinct points {fp σ (m) }, p M and m N, we obtain an analytic connected manifold structure in the (open in the induced topology) complement: If we set E := Ĕ {fp σ (m) } m N p M then E = Ĕ F Σ p = {p} {F F Σ p } with the obvious C α -atlas p M p M A E := AĔ E Deleting the sequence of (identified) points {F Σ (m) } m N of E Σ, we obtain the differential-topological structure of an analytic manifold, E Σ F Σ without any singularities, since: E Σ m NF Σ (m) = {Ĕ / m N F Σ (m)} F Σ (m) = {p} {F F Σ p } E m N which proves the analytic structure on E Σ p M F Σ (m) m N. Applying now the embedding theorem 2.1 for this analytic manifold, we can make it an Einstein global isometric embedding manifold of M. The remaining task is to show that E Σ is not a manifold. To see this, suppose that taking the wedge sum of 2 distinct fibres ({p} F) f 0 ({q} F), there exists a homeomorphism of a domain V f0 of f 0 to R n+d. If this holds, then there must exist a homeomorphism φ : V {f 0 } = φ(v ) {φ(f 0 )} R n+d choose an open ball B n+d (0, r) of maximum radius living into φ(v f0 ) and containing φ(f 0 ). Then, the following homeomorphism V f0 φ 1 (B n+d (0, r)) {f 0 } = B n+d (0, r) {φ(f 0 )}

19 Global Embedding of Spacetime into Singular Einstein Manifolds 19 can not hold, since B n+d (0, r) {φ(f 0 )} has 1 connected component while φ 1 (B n+d (0, r)) {f 0 } has 2 connected components. The mathematical mechanism of topological identitification we used to produce the pathological singularities on the Bulk raises the idea that it is not really necessary these anomalies to be points: there may exist extended problematic domains of the complement E Σ M. We simply identify distinct homeomorphic domains on the fibres F. This more realistic case may correspond to extended object in the Universe with positive (multi-dimensional) volume. Theorem 3.2. (Embedding into Einstein manifold with unsmoothable differential-topological singularities of positive (d k)-measure) Given a semi-riemannian analytic manifold M, there exist: (I) a topological space (E Σ, T E Σ) (II) an analytic manifold (E, A E ), with dim(e) = dim(m) + d, d 1 (III) countable many closed path connected & simply connected sets F Σ E Σ, satisfying: { {f Σ } = [0, 1], if d = 1 F Σ = {f Σ (1), fσ (2),...} = N B d k (0, 1), if d 1. (0 k d) such that the following topological homeomorphism holds (under inclusion to a subspace) (E Σ F Σ, T E Σ E Σ F Σ) = (E, T AE ) and E is an Einstein space of global isometric embedding for M. In addition, E Σ is not a topological manifold. Proof. The proof is a replica of the previous result. The only fact that needs to be noted is that extracting from a closed (abstract) curve a closed (image of) an interval, (e.g. a closed arc from S 1 ) or a closed (image of a) (d k)-ball from F (choosing e.g. R d ) for dim(f) = d), we obtain open submanifolds in the analytic subspace structure.

20 Global Embedding of Spacetime into Singular Einstein Manifolds 20 In addition, the unability of resolution of the singularities, owes to the fact that choosing from each of the F s a (diffeomorphic image of a) closed ball, the quotient space when they are all identified is not a manifold, by the same topological argument as in 3.1, extracting a closed inversed image of ball instead of a point. Countability of these extended anomalies can be easily shown, since choosing for the (diffeomorphic images of) balls, diam(b d k (0, r)) = 2 r < 2, we can embed in R d countable many balls, centered at the points of N d. If k 0, it suffices to reduce embedding of balls to a d k-dimensional linear subspace of R d. If we wish to give a topological description of the singular spacetime E Σ as in Sec. 3 of [23], we can use reduced Homology theory, since the axioms (e.g. [1], [6]) provide a direct sum splitting for the wedge topological sum of spaces. The following gives a Homological description of the E Σ with a single 1-point singularity: Proposition 3.3. (Homology of E Σ with 1-point Singularity) The Einstein manifold E Σ with 1-point singularity, has the splitting (reduced) Homology H m (E Σ ; G) = M c H m (F; G), m N for any coefficient group G. Proof. The wedge sum relation (3.5) for E Σ implies the following Homological splitting, due to the respective property of the singular theory: H m (E Σ ; G) = H m ({q} F; G), m N f q {q} F and provided that the diffeomorphisms {q} F = F, q M imply the group isomorphisms H m ({p} F; G) = H m (F; G), p M

21 Global Embedding of Spacetime into Singular Einstein Manifolds 21 we obtain the direct sum splitting H m (E Σ ; G) = M c H m (F; G), m N as claimed. 4. An Attempt of Physical Interpretation. Summarizing the results obtained in this paper, we have proved that given any spacetime manifold M with an indefinite analytic metric, there exist Einstein spaces E Σ wherein the given is globally & isometrically embedded. If M is a hypersurface in this Bulk, there may exist an 1-point singularity or an extended singular 1-dimensional domain, tearing and deformating the smooth spacetime. In the case of embedding codimension greater than 1, the case changes dramatically, provided that the complement of the Bulk E Σ M may admit infinite-many countable singularities! These pathologically problematic domains can be points, or have positive volume, of dimensionality at most that of the embedding codimension. Suppose now that a test-particle living into the Bulk (not confined on M), an electromagnetic wave or some other physical entity living into E Σ enters the singularity f Σ living into E Σ M. If the entity comes out, nothing guarantees that will appear to the same fibre (say {p} F) it entered: this will in general be some different {q} F. Thus, the entity will have traveled in spacetime through this passage between 2 distant points of the Universe! This may be thought of some kind of wormhole of spacetime allowing spacetime travels. Said differently, if an object enters the anomaly moving on a (not necessarily) geodesical orbit γ : I E Σ lying into one {p} F γ([0, 1]) {p} F with γ(1) = f Σ {p} F, it can come out being now at some different {q} F and in the orbit appears a discontinuity at γ(1) E Σ, since γ([1, 2]) {q} F

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