Gravitomagnetic currents in the inflationary universe from WIMT

Using the Weitzenb\"ock representation of a Riemann-flat 5D spacetime, we study the possible existence of primordial gravito-magnetic currents from Gravito-electromagnetic Inflation (GEMI). We found that these currents decrease exponentially in the Weitzenb\"ock representation, but they are null in a Levi-Civita representation because we are dealing with a 5D Riemann-flat spacetime without structure or torsion.


I. INTRODUCTION AND MOTIVATION
It is well known that magnetic monopoles have been elusive to be detected, despite the efforts. Such monopoles arise as a theoretical possibility from the dual formulation of an electrodynamic theory [1]. The fate of primordial monopoles is very closely linked to the history of the very early universe. Preskill [2] realized that this possible monopole production could create a crisis for cosmology, implying far more monopoles than observational limits allow. Because the expected energy scale of grand unification is quite high, the geometrical size of a monopole core must be quite small. Linde and Vilenkin independently pointed out that such monopoles could expand exponentially in the context of inflationary cosmology [3].
However, there is an even more interesting possibility, which arises from a theory that extends and unify conceptually the electrodynamics with a theory of gravity; the gravitoelectrodynamics. This theory was first outlined in 2006 [4] in a cosmological context and later studied in greater detail [5,6], but the dual formalization has not yet been addressed.
In this paper we shall study the dual formalism from a 5D vacuum and we will try and get formalize their dual streams, that, in a gravito-electrodynamic context are related gravitomagnetic currents. This formalism is inspired in the Induced Matter Theory (IMT), which is based on the assumption that ordinary matter and physical fields that we can observe in our 4D universe can be geometrically induced from a 5D Ricci-flat metric with a space-like noncompact extra dimension on which we define a physical vacuum [7]. The Campbell-Magaard theorem [8][9][10][11] serves as a ladder to go between manifolds whose dimensionality differs by one. This theorem, which is valid in any number of dimensions, implies that every solution of the 4D Einstein equations with arbitrary energy momentum tensor can be embedded, at least locally, in a solution of the 5D Einstein field equations in vacuum.
Because of this, the stress-energy may be a 4D manifestation of the embedding geometry. An extension of the IMT was realized recently using the Weitzenböck Induced Matter Theory (WIMT) [12]. This approach makes possible a geometrical representation of a 5D vacuum (with a zero curvature in the Weitzenböck representation), on nonzero curvature tensor (in the sense of the Levi-Civita representation).

II. WEITZENBÖCK INDUCED MATTER THEORY (WIMT).
We consider the basic elements for the extension of the IMT to a geometrical description with the Weitzenböck connections. The connections are constructed from certain 5D vielbein related to the transformation defined by where − → E A is an element of a base { − → E A } that we shall "started base" (SB). In our case we shall work with a 5D Minkowski space. Furthermore − → e a is an element of the "arrival base" (AB), which is obtained trough the transformation (1 If we use the Weitzenböck , can be seen that It can see that the expressions • Lowercase latin letters a, b, c, .., h = 0, 1, 2, 3, 4 run on the 5D "arrival space" (AS).
ST. In general, for any vector field Now we consider a 5D spacetime described by the metric g AB in the SS, and g ab describing the AS. It is obvious that this last space is Weitzenböck-flat in the sense that the Riemann tensor constructed trough this kind of connections is null: (W ) R a bcd = 0. However, it cannot be Riemann-flat with respect to the Levi-Civita connections: (lc) R a bcd = 0. The Riemann tensor written with the Weitzenböck representation for the spacetime characterized by the metric g ab , is given by where the Weitzenböck contortion (W ) K a bc is related to the Weitzenböck torsion (W ) T a bc 2 We call ";" the covariant derivative . This is not only valid for the superscript 0, but so does for other indices, so Here, we have considered the nonzero no-metricity g ab; c = (W ) Q abc . When g ab; c = 0, the tensor (W ) K a bc reduces to the well known contortion tensor in the Weitzenböck representation On the other hand, if the Weitzenböck torsion becomes zero (it holds when the SB has no structure), we have (W ) K a bc = g ma 2 {g bm;c + g mc;b − g bc;m }. By contracting the null-tensor (W ) R a bcd we obtain the following tensors: From (4), (5), (7) and (8) we can obtain the expressions for the corresponding curvature tensors with the Weitzenböck representation by means of those of the Levi-Civita representation (and viceversa). Hence, we have Using the eqs. (7) and (8) in the last expression we found the analogous equations for 3 In this work we shall use this definition with zero nonmetricity. 4 The expressions (W ) Γ n (d|a (W ) Γ a n|c) indicate the symmetrization of the indices d y c, inside the perentheses, but excepting the indices a y n inside the vertical bars "|". Now we shall consider the Einstein equations with the Weitzenböck representation. We shall try to obtain the efective 4D equations after making a constant foliation from a 5D Weitzenböck vacuum. Taking the equation (8), we obtain that In this work we shall deal with canonical metrics [13]. An interesting example is [14] where l is related to the noncompact extra dimension and l 0 is a constant. After making the constant foliation, we obtain βα is a Weitzenböck connection defined on the embedded 4D hypersurface obtained trough the foliation: l = l 0 .
It makes possible to obtain the effective 4D Ricci-Weitzenböck tensor which is symmetric with respect to the indices ζ, δ. The antisymmetric tensor is obtained The Ricci-Weitzenböck scalar curvature can be obtained from a 5D vacuum, as Hence, from eq. (14) we can obtain which means that the scalar Ricci-Weitzenböck curvature has the source and finally the induced Einstein-Cartan-Weitzenböck equations are where we have taken into account in (18) that g βγ (W ) S βγ = 0. Furthermore, the symmetric and antisymmetric parts of the energy momentum tensor in eqs. (17) and (18)  A. Effective 4D dynamics with the Levi-Civita representation.
Now we shall intend to write the curvature and the Ricci tensors (in the Levi-Civita representation) with respect to the Weitzenböck connections and contortions. The Ricci in the Levi-Civita representation is related to the Ricci tensor in the Weitzenböck representation plus additional terms that depends with contortions and structure The scalar curvature is Hence, the Einstein-Cartan equations are given by joined with (W ) R a5 = 0 y (W ) S a5 = 0 that are additional conditions. Here,

III. DUAL ACTION AND EQUATIONS OF MOTION
We shall consider the conditions by which we can induce curvature and currents by means of WIMT [12], on a 5D spacetime represented by cartesian coordinates. The 5D tensor metric can be written as We can construct a AB with interesting cosmological properties if we take the appropriate vielbein. The action for the gravito-electromagnetic fields in a 5D vacuum can be written in the form 5 The equations (24) take into account the Cartan equations which describe spinor contributions.
In order to obtain the dual currents may be interesting rewrite the last action in terms of the dual tensors F ABC (28) It is evident that , so that when k = 1 3! , we obtain that both actions describes the same physical system: In our case, when we use the Lorentz gauge and we deal with a 5D vacuum with R = 0, we have S 1 ∼ S and both actions give us the same equations of motion. In order to describe the dual sources of these equations we shall deal with the action S 1 . The dynamics of the gravitoelectromagnetic fields obtained taking extreme the action (27)), is ; K = 0. On a 5D vacuum (R = 0), and when we take λ = 1 and the

Lorentz gauge A B
; B = 0, they reduce to that are Klein-Gordon equations for massless fields. The gravitomagnetic currents become from the solutions for the fields (29). The last equations are compatible with a current which has its source in where we have used (29) and the Lorentz gauge in absence of nonmetricity. We have used the fact that where clearly we have imposed the absence of currents on the 5D vacuum with respect to the Levi-Civita representation. Hence 6 . Notice that in eq. (31) we use the covariant derivative with respect to the Christoffel symbols, but in the expression (32) we use the derivative covariant with respect to the Weitzenböck connections. Hence, we can adopt both representations in a complementary mode to describe the 5D vacuum.
The Weitzenböck currents are related by one expression analogous to eq. (31), but more Once required the gauve condition (lc) A N ; N = 0, it is preserved in the Weitzenböck representation: (W ) A n ; n = 0 7 . 6 Using the expression (lc) Γ A BC = (W ) Γ A BC + K A BC it is possible to obtain the following expression between both Faraday tensors: (lc) If we make the derivative of this expression and we use (31), we can check the validity of eq. (32). 7 One can show that (lc) e N n ;m = (W ) e N n ;m + e N k K k nm , but since the Weitzenböck connections comply with (W ) e N n ;m = 0, it is possible to express the covariant derivative of the vielbein in the Levi-Civita representation as a function of the contortion of Weitzenböck: (lc) e N n ;m = e N k K k nm . If we use this on the gauge condition and we take into account that A N = e N n A n , we can prove that (lc) A N ; N = 0 ⇒ (lc) A n ; n +K m mn A n = (W ) A n ; n = 0. Hence, one can show the following equalities:

IV. GRAVITOMAGNETIC CURRENTS FROM WIMT
We shall intend to explore the flat-spacetime in order to know under what conditions we can introduce gravitomagnetic currents using WIMT on 5D. We shall consider a orthogonal base without structure and trivial connections.

A. Quantization of the fields
Starting from the vacuum action (27) with (lc) R = 0, one obtains the dynamics of the gravitoelectromagnetic fields The differential equations (35) are separable, so that we can propose a solution of the kind The vector field can be written as a Frourier expansion We have expanded the vector field as a function of the components of the tangent base { − → E N }.

These fields comply with
scribe a inner product trough the application on the elements of the tangent space. It is clear that the connections are null. The fields can be expanded with any base with the following requirement for the polarization vectors: • e N n η MC (W ) F N M; C = g mc (W ) F nm; c , In general the choice of the polarization vectors is independent of − → K N , which is the wave vector of the N-component for the field related to the directional propagation of flat waves.

The expression
The adjoint operator is given by so that we see from the expression (37), that We see that both conditions are equivalent.
With respect to the interpretation of − → E * N , we claim that it is is merely symbolic because the bases are given by real vector fields. Hence as second rank tensors, in the following manner: These expressions provide us the algebra in an arbitrary metric obtained from a 5D Minkowski spacetime, which is free of structure. In order to illustrate the formalism, in the following section we shall apply it to an de Sitter expansion, which describes the early inflationary universe.
V. AN EXAMPLE: MONOPOLES FROM A 5D MINKOWSKI SPACETIME

WITH CONTORTION
We shall study an example in which we take as a started spacetime a 5D Minkowski described with the cartesian coordinates φ(p) = (t, x, y, z, l) p , in which the tensor metric is given by the orthonormal matrix η AB (26). We choose a SB which is not coordinated, and the structure coefficients are given by and C 0 04 = −1 l 2 . The vielbein transform asē n N = diag(1/l, e −N /l, e −N /l, e −N /l, 1), so that we obtain the metric with ψ 2 (l) = l 2 /l 2 0 . The arrival spacetime is described by a coordinated base. This implies that the Weitzenböck torsion in the arrival spacetime will be nonzero. This torsion will be a possible geometrical source for the emergence of gravito-magnetic monopoles, once it has been made a constant foliation on the extra noncompact coordinate. The current components in this case are given by and (W )(m) J 0 = (lc)(m) J 0 = 0. Notice that the spatial components of the magnetic currents decay with time in the Weitzenböck representation. The study of the dynamics for the field fluctuations during a de Sitter inflation in the framework of GEMI was studied with detail in [17], and goes beyond the scope of this paper. However, as has been demonstrated in eq.
(42), from the point of view of the Levi-Civita representation there are no currents related to gravito-magnetic sources.

VI. CONCLUSIONS
We have extended the WIMT formalism to GEMI with the aim to show that gravitomagnetic currents may be obtained, at least in a Weitzenböck representation. The WIMT formalism was introduced with the idea to generalize the foliation method in the Induced Matter Theory of gravity, in which foliations which are no static result to be very difficult.
With the WIMT formalism, on can make static foliations from a 5D curved spacetime on which one defines a 5D vacuum from the point of view of a Weitzenböck representation.
Once done the foliation is possible to pass to the representation of Levi-Civita. It open a huge versatility to make static foliations to obtain arbitrary 4D hypersurfaces from 5D curved manifolds, which could be very important for quantum field theories, gravitation, cosmology, etc.
In particular, we have centered our study of the WIMT in the dual formalization of the GEMI applied to the cosmology of the early inflationary universe. We have obtained nonzero gravito-magnetic currents with the representation of Weitzenböck. The currents decrease exponentially with time with the expansion of the universe, so that at the end of inflation becomes negligible. It should be agree with present observations. However, these currents are null in a Levi-Civita representation because in this geometrical representation the base coordinated has not structure or torsion. In a future work we shall study an example where gravito-magnetic sources are nonzero in any representation [18].