WIMT in Gullstr\"and-Painlev\'e and Reissner-Nordstr\"om metrics: induced Stable Gravito-magnetic Monopoles

The aim of this work is to apply WIMT to Gullstr\'and-Painlev\'e and Reissner-Nordstr\"om metrics in the framework of Weitzeb\"ock Induced Matter Theory (WIMT). This is a newly developed method that extends Induced Matter Theory from a curved 5D manifold using the Weitzeb\"ock's geometry, using the fact that the Riemann-Weitzeb\"ock curvature tensor is always null. We obtain the presence of currents whose interpretation can lead to the presence of stable gravito-magnetic monopoles.


Introduction
In a previous paper [1] we incorporated the Weitzeböck's geometry (using its characteristic connections) into the treatment of extended Induced Matter Theory (IMT) [2,3]. This theory is based on the assumption that ordinary matter and physical fields, which we can observe in our 4D universe, can be geometrically obtained from a 5D space-time which is at least Ricci-flat (in the sense of Levi-Civita connections). We are interested in the cases where the extra dimension is non-compact, 1 and we define a physical vacuum supported by the Ricci-flatness condition [4]. This theory is founded in the Campbell-Magaard embedding theorem [5][6][7][8] as a par- 1 In contrast with the Kaluza-Klein Theory (KK) in which it is assumed that the extra dimension is compact, cyclic, and having a small radius. 2 In our convention the indices "a, b, c, . . . , h" and "A, B, C, . . . , H " are related to 5D space-time and run from 1 to 5. Greek indices "α, β, γ, . . ." are related to 4D space-time and run from 1 to 4, and indices "i, j, k, . . ." are only spatial and run from 2 to 4. We choose our first index related to time coordinate and the last index in 5D space-time is due to the extra coordinate associated with the extra dimension. a e-mail: jesusromero@conicet.gov.ar b e-mail: mbellini@mdp.edu.ar ticular case in which (5D) R ab = λ (5D) g ab , 2 with λ = 0. We must remark that the λ = 0 condition implies Ricciflatness and this makes it possible to define a 5D apparent vacuum that determines the equations of motion for the fields of the theory. Such equations are given by (5D) R ab = 0. The aim of WIMT is to use an alternative description, based on the Weitzenböck geometry, to apply a IMT-like formalism for any 5D space-time, even if this is not flat in the Riemannian sense with Levy-Civita (LC) connections. The central idea of WIMT is to convert the geometrical description of the problem from a non-flat (but torsion-less) Riemannian with LC connections, into a Weitzenböck's geometrical description with a Riemann-Weitzenböck null tensor, 3 but with non-zero torsion. In this case we can use the IMT tools to induce 4D effective space-time dynamics making a foliation over the Riemann-Weitzenböck flat 5D spacetime (this is the vacuum in the Weitzenböck sense). 4 Once this cjoice has been made, we can recover the Riemannian description by doing a transformation over the induced tensors (now obtained in terms of Weitzenböck connections) according to (LC) a bc = (W) a bc + K a bc . Here (LC) a bc is the symbol denoting the LC connection (the usual Christoffel 3 What we refer to as the Riemann-Weitzenböck tensor is the Riemann curvature tensor expressed in terms of Weitzenböck connections. In a coordinate (holonomic) basis the Riemann-Weitzenböck curvature takes the form The label (W) identifies the geometric objects expressed in the Weitzenböck representation and the (W) a dc,b terms are the ordinary partial derivatives of the Weitzenböck connection (W) a dc . We assume the Einstein summation convention for repeated indices. 4 We set the Weitzenböck vacuum as a 5D space-time with null Riemann curvature tensor in Weitzenböck geometry, symbols of the second kind), (W) a bc denotes the Weitzenböck connections and K a bc is the contortion, which depends on the non-metricity and torsion. The shape of the contortion is presented in the next section.
Part of the motivation for the present work comes from two principal ideas: (i) WIMT provides a successfully description for the formation of a spherically symmetric body of finite size, and (ii) the Weitzenböck torsion present in WIMT is a possible geometrical source for the existence of magnetic monopoles.
With respect to the first assertion we can say that already the WIMT was used to address the problem of the formation of a massive compact object [1] as a remanent of a collapsing space-time. This object can be thought of as a black hole. The second one has been worked out using WIMT in the framework of the gravito-magnetic formalism in a variety of cosmological scenarios [9,10], with the result that currents of gravito-magnetic monopoles are strongly linked to the Weitzenböck torsion. With the support of these results, it is generally expected that the density of gravito-magnetic monopoles and charges decay in an accelerated universe due to the strong increasing of the scale factor. Hence, the study of a stationary black hole appears to be an excellent issue to complete our approach to a spherically symmetric object and to try to appreciate the existence of stable gravito-magnetic charges using WIMT. We choose to work in the 5D extension of the Gullstränd-Painlevé (GP) metric, because with this metric it is very simple to get the associated basis of pentads and to operate with them. Additionally, but no less important, the GP metric is widely used in the study of BH and their evolution [11][12][13][14], so that it is an interesting topic that deserves further study.

WIMT
We shall introduce some basic concepts of the Weitzenböck geometry and we expose some results that we developed in previous works, because these will be tools for obtaining some important results of this article.

Weitzenböck geometry and vielbeins
In order to revisit the WIMT we shall use some elements of Weitzenböck geometry [15,16]. The Weitzenböck connection is we assume the transformation which represents a transformation from an element − → E A (belonging to { − → E A }, an orthonormal Lorentzian basis of the 5D tangent space), into − → e a (belonging to { − → e a }, also a basis of 5D tangent space but in general not orthonormal). The elements e A a that perform the transformation are known as vielbeins [17,18] and fulfill the relations e A aē b The vielbeins provide us with a prescription to transform bases and therefore any tensorial element. For example, where η AB is a 5D Minkowski-like metric with signature (+, −, −, −, −), { − → E A } is an orthonormal Lorentzian basis and g ab is the metric tensor expressed in terms of the basis { − → e a }. This clarifies that e A a e B b is an object with five "legs" (corresponding to the five components of the space-time), in the { − → E A } representation for tensorial elements (upper case indices) and the other five "legs" in the { − → e a } representation (lower case indices). For a general tensor T A 1 ...A n B 1 ...B m , which is n times tangent (or contravariant) and m times cotangent (or covariant), we obtain An important property of the Weitzenböck connection This means that the covariant Weitzenböck derivative of the vielbein is null. The expression (1) characterizes the Weitzenböck connection with lower case indices as a transformation of a trivial null connection with upper case indices. About the trivial upper case index connection we must be more formal and construct the following expression: − → E C are linearly independent so that (W) C AB = 0. In this way the Weitzenböck connection represents the extension of the derivative operator with trivial zero connections in the orthonormal basis to the new basis. In the most general case the Riemann curvature tensor in the Weitzenböck geometry takes the following shape in terms of the upper case index connection: due to each upper case index connection being null: = 0, as we can see in (6). The structure coefficients in the expression (7) are defined by Therefore, the following expression is trivial: This relationship is the key of WIMT. It provides us with a tool to define the 5D Weitzenböck vacuum and make use of the usual philosophy of IMT, 5 even in the absence of the 5D Riemannian (LC) vacuum. 6 Finally, the relationships between the LC and Weitzenböck connections are where the Weitzenböck contortion (W) K a bc is In the last expression g ab ; c = (W) Q abc is the Weitzenböck non-metricity. In the absence of such a non-metricity we recover the usual shape for contortion, which is totally given in terms of the torsion tensor [20,21]. In the present case we make a start with an orthonormal basis We must remember that each η AB is constant and then The last expression is general but in the next section we shall study some cases in which this is specially easy to see. We shall work with a holonomic coordinate basis { − → e a }, so that 5 The central ideas of IMT are very didactically developed in [19]. 6 We want to clarify what we mean by "Riemannian (LC) vacuum": it means that the Einstein tensor written in terms of LC connections is zero. This is trivial when the Riemann curvature tensor is null.
because of η AB , c = 0. In the last expression η AB , c = which is the usual well-known contortion tensor [20,21]. This tensor transforms according to (W)

Some results in WIMT
We consider a 5D space-time with coordinates φ( p) = (t, x, y, z, ψ) p dotted by an inner product described by Diag(η AB ) = (+1, −1, −1, −1, −1), expressed in a nonholonomic basis. The action on the 5D apparent vacuum is given by the gravito-electromagnetic fields where the penta-vector potential is is the vector potential, and is the gravitational scalar potential. If we assume the Lorenz gauge A B ; B = 0, the equations of motion will be reduced to In general, we find that the 5D Faraday tensors expressed in terms of the LC and Weitzenböck versions are given, respectively, by The 5D-current of gravito-magnetic monopoles is obtained This is motivated by the symmetry of Maxwell equations. 7 Thus, * must be LC or W according to the corresponding derivative operator. Furthermore, F ABC is the dual tensor of 7 In a 4D space-time, this symmetry is manifest in the expressions F AB . 8 The resulting currents, 2-forms in the 5D space-time, are The Riemannian (LC) current of the gravito-magnetic monopole originates in the Weitzenböck torsion (W) T A BC . 9 It must be noticed that once we choose the Lorenz gauge condition (LC) A N ; N = 0, the same condition is preserved in the Weitzenböck representation: (W) A n ; n = 0. Once we have established the 5D-currents it is easy to obtain the 4D-induced currents by applying a constant foliation on the fifth extra dimension: ψ = ψ 0 . The effective resulting 4D gravito-magnetic current is the 1-form given by so that * must be LC or W and − → n is a vector pointing to the extra direction. The vector − → n is normalized and orthogonal to the 4D tangent sub-space associated with the effective 4D 8 The dual tensor of a 2-form (characterized in this case by ( * ) F AB ) in 5D manifold is a 3-form defined by Here, * must be LC or W and ε ABC DE is the antisymmetric Levi-Civita symbol. We have made use of the fact that √ | η | = 1, η = Det([η] AB ) being the determinant of the metric tensor in the upper case index representation. 9 We focus our attention on the Riemannian (LC) current, but it is formally easy to relate the Riemannian (LC) and Weitzenböck currents of the gravito-magnetic monopole, with the expressions where sub-manifold. Such orthogonality is realized with respect to the complete 5D metric. All expressions must be evaluated in ψ = ψ 0 , so that dx 5 | ψ=ψ 0 = dψ | ψ=ψ 0 = 0 and hence ( * )4D J − → must be a cotangent four-vector (i.e., it is a 1-form in the cotangent 4D space-time).

5D Gullstränd-Painlevé (GP) metric
The GP coordinates are commonly presented as a set of coordinates obtained by transformation of the Schwarzschild, or Reissner-Nordström (RN) ones, in which the static time is replaced by a GP time t →t(t, r ), preserving the spherical symmetry of the problem [24,25]. We shall use a slightly different method by using the GP cartesian metric and using the vielbein related to an orthonormal basis. In this case we obtain the GP cartesian metric from a normalized one, using a certain vielbein, prescribed in this section. The GP cartesian metric must be obtained from the Schwarzschild or RN BH by doing a space-time transformation. We choose to start with an orthonormal basis where { − → e a } is some coordinate basis. The vielbein is constructed according to and its inverse is in which β i = βx i , such that x i is the ith coordinate asso- is a function, M is a mass parameter, and Q is an electric charge parameter. If Q = 0 we deal with a Schwarzschild BH, but in the case Q = 0, we are in presence of a Reissner-Nördstrom BH. We must remark that M − Q 2 2r is the gravito-electric charge of the BH, and so that, by the definition of β, we have Hence The charge of the BH is dominant at small distances, but the mass dominates at long distances. The metric is In the last expression E N = E N n dx n is an element belonging to a basis for the cotangent space, which fulfills the duality Furthermore, g mn is the corresponding GP metric in cartesian 5D GP coordinates, such that x 1 = t, x 2 = x, x 3 = y, x 4 = z, and x 5 = ψ. 10 Now we must calculate the structure coefficients for the basis (20). According to the definition [27], we obtain The non-zero elements are C 1 r sin(θ) , expressed in GP, or in the usual BH coordinates. Here, f (r ) = 1 − 2M r + Q 2 r 2 for a Reissner-Nordström BH. In the case with Q = 0 one obtains a Schwarzschild BH. On the other hand, if we take the expression (24), will be easily shown that C c ab = 0, which means that { − → e a } is a coordinate or holonomic basis for 5D tangent space. It is very interesting to notice that in the upper case index representation, the torsion tensor is numerically equal to the structure coefficients, due to the fact that (W) Since the Weitzenböck connection extends the trivial null connection in the upper case index basis, we obtain (W) A C B = 0, and hence which is surprising because (W) T A BC is an intrinsic geometrical property of the manifold and C A BC is a coefficient depending on a circumstantially selected basis. In fact, Eq. (25) is not strange and is due to the selection of the Weitzenböck connection and particularly to the geometrical properties to 10 For a most extensive treatment the reader could see the notes of Andrew Hamilton (developed in 4D) [26]. extend a derivative operator linked to a trivial null connection in the upper case index, to a nontrivial connection in the lower case index basis.

Gravito-magnetic monopoles from WIMT in GP and RN metrics
If we take the structure coefficients (24) and we employ the non-zero elements in (25), we obtain the Weitzenböck torsion and we could use it in the expressions (15) and (16), in order to write the 5D LC 2-form currents in upper case index basis. After it, we can express such 2-form in the lower case index representation with the transformation (4) adapted to a 2 times cotangent object [as we were doing in (3) for the metric tensor], with the vielbein (22). Notice that we have a 5D LC 2-form current expressed in terms of a Weitzenböck torsion with lower case indices and vielbeins. Hence, we can make a constant foliation getting the effective 4D current from (19). Finally we obtain the expression for the effective 4D current, which in the usual BH coordinates takes the form which is the induced static gravito-magnetic monopole charge density. For the spatial effective 4D currents of gravito-magnetic monopoles, we obtain which is consistent with the magnetic charge conservation in a scenario in which the monopoles are static. Now we must study some important invariants of the problem. We define The invariants (e) J and (m) J have direct counterparts in the 4D space-time, defined by On the other hand, the 4D counterpart of (m) J A (e) J A is The invariant (m) J A (e) J A = 0 only makes sense because we are supposing that the five-vectors (m) J and (e) J are orthogonal on the 5D Weitzenböck vacuum. All the previously defined invariants have a clear meaning, which can easily be explored in a static or stationary case, as the case of a RN BH, which we shall study. Then we must show that Eqs. (28)(29)(30)(31)(32)(33)(34) reduce to In a static 5D space-time in which the observer is moving with arbitrary penta-velocity − → U = U A − → E A , their components U A can be U 1 , U i , U 5 = 0. Hence, one obtains in which we have made use of the fact that (m) J i = 0, so that In a non-static scenario results that (m) J 5 = 0. On the other hand, using the fact that (e) J i = 0, we obtain where in general (e) J 5 = 0. All makes sense when we apply our extended 5D Maxwell equations for (e) J 5 , so that F 5A ; A = (e) J 5 , which in the extended version is If we apply the 5D Lorenz gauge (A ; A A = 0), the last expression evaluated on the foliation reduces to The expression (38) must be interpreted as an effective equation of the form ρ M = − , in which is the gravitational potential for the 4D BH. In this case (e) J 5 must be ρ M ; the effective density of the gravitational mass. We must remark that in the present case the operator does not have a time related covariant derivative part, due to the static nature of the problem here considered (we maintain the complete notation for clarity). We must set Equation (39) justifies the notation used in Eqs. (35) and (37). We shall study the case with U i = 0 and U 5 = 0. For such a scenario we assume that the observer is moving with a penta-velocity Then we must set U 5 to an arbitrary non-zero value. The penta-velocity is defined on a 5D TM and is a penta-vector. The notation used is: • The 5D M is the 5D manifold.
• The 5D TM is tangent space associated to 5D M.
• 4D M is the 4D manifold induced by foliation ψ = ψ 0 and, of course, a sub-manifold of the 5D M. • The 4D TM is the tangent space associated to 4D M and, of course, a sub-space of the 5D TM.
When we apply a constant foliation ψ = ψ 0 on the extra coordinate, the penta-velocity turns into a four-vector by projecting it onto the 4D TM. Then } is a basis of the 5D TM, and { − → e 1 , − → e i } is a basis of the induced 4D TM. Due to the static nature of the studied scenario the invariants (35-37) are not dependent on u i and we must choose the value of u i preserving the validity of the geodesic equations: du α dS + (W) α βγ u β u γ = 0, with the normalization condition g αβ u α u β = 1. Here, g αβ is the effective induced metric with the foliation ψ = ψ 0 and we must take the RN, or the GP ordinary 4D form, depending on the employed coordinates.
The use of WIMT warrants that the 5D space-time is a vacuum in the sense of the Weitzenböck geometry, even if ρ m , ρ e , ρ M = 0. At this point we choose to establish some values for the 5D invariants, in order to obtain a realistic physical behavior in the effective space-time. Hence, we set On the another hand, from (e) J = 0, we obtain so that, equating (40) and (41), we obtain which provides us with the quantization law between the mass density and the electric density charge. In order to obtain the quantization of charge we must use Eqs. (31) and (42), with (gem) J 2 = −n 2 . The resulting expression, which provides the quantization between the electric and magnetic density charges, is

Appendix A: Connecting ONB, GP, and RN
Here we try to order the different representations of spacetime and the relationship between them: • ONB representation: in 5D the orthonormal basis of 5D TM is { − → E A }, which is non-holonomic with structure coefficients given in (24) and the subsequent expressions. In this case the matrix of the metric tensor is η AB with − → E A · − → E B = η AB = ±1. With a constant foliation we must obtain η αβ = ±1, for the basis { − → E α }. In this context the Weitzenböck torsion is specially easy to obtain according to (25), and we can transform it to other representations using the corresponding vielbein.
• GP representation: The 5D GP cartesian metric (23) is linked to the coordinate basis − → e a = − → ∂ ∂ x a associated to x a , with x 1 = t G P , x 2 = x, x 3 = y, x 4 = z, x 5 = l, in which t GP = t + h(r ), where h(r ) is an arbitrary (C ∞ ) function. Using a constant foliation we must obtain the familiar 4D GP cartesian metric [26].
• Tensorial objects: in the ONB representation must be transformed to the GP representation using the vielbein (21) and (22) • Constant foliation: must be viewed as the application of a vielbein e a α = δ a α , e 5 α = 0 to the tensor objects plus a specialization to a certain value for ψ = ψ 0 . Then, for example, F a b → F α β = e α a e b β F a b | ψ=ψ 0 . This idea must be extended to another more general kind of foliation (dynamic foliation) varying the vielbein as required.
Using the previous concepts and formulas we must relate ONB, GP, and RN representations and the effective 4D space-time, tensor objects, and invariants.