Ellipsoidal, Cylindrical, Bipolar and Toroidal Wormholes in 5D Gravity

In this paper we construct and analyze new classes of wormhole and flux tube-like solutions for the 5D vacuum Einstein equations. These 5D solutions possess generic local anisotropy which gives rise to a gravitational running or scaling of the Kaluza-Klein ``electric'' and ``magnetic'' charges of these solutions. It is also shown that it is possible to self-consistently construct these anisotropic solutions with various rotational 3D hypersurface geometries (i.e. ellipsoidal, cylindrical, bipolar and toroidal). The local anisotropy of these solutions is handled using the technique of anholonomic frames with their associated nonlinear connection structures [vst]. Through the use of the anholonomic frames the metrics are diagonalized, in contrast to holonomic coordinate frames where the metrics would have off-diagonal components. In the local isotropic limit these solutions are shown to be equivalent to spherically symmetric 5D wormhole and flux tube solutions.


I. INTRODUCTION
The first solutions describing black holes and wormholes in 4D and higher dimensional gravity were spherical symmetric solutions with diagonal metrics [2]. Later Salam, Strathee and Perracci [3] showed that including off-diagonal components in higher dimensional metrics is equivalent to including gauge fields. They concluded that geometrical gauge fields could act as sources of exotic matter necessary for the construction of a wormhole. Refs. [4,5] examined locally isotropic solutions with off-diagonal metric components for 5D vacuum Einstein equations. These solutions were similar to spherically symmetric 4D wormhole or flux tube metrics with "electric" and/or "magnetic" fields running along the throat of the wormhole. These "electromagnetic" fields arose as a consequence of the off-diagonal elements of the metric. By varying certain free parameters of the off-diagonal elements of the 5D metrics it was possible to change the relative strengths of the fields in the wormhole's throat, and to change the longitudinal and transverse size of the wormhole's throat.
In [6] the we constructed anisotropic wormhole and flux tube solutions, which reduced to the solutions of [4,5] in the isotropic limit. The anisotropy of these metrics was handled using the method of anholonomic frames with associated nonlinear connections, which has been developed by one of the authors (SV) in Refs. [1]. It was shown that these anisotropic solutions exhibited a variation or running of the "electromagnetic" parameters as a result of the angular anisotropies and/or through variations of the extra spatial dimension.
In this paper we extend the investigation of [6] by applying the anholonomic frames method to construct anisotropic wormhole and flux tube solutions to 5D Kaluza-Klein theory which possess a range of different symmetries (elliptic, cylindrical, bipolar, toroidal).
We will discuss the physical consequences of these solutions, in particular the variation of the "electromagnetic" parameters (e.g. the "electric" and "magnetic" charges associated with the solutions). This variation of the "electromagnetic" charges, which here occurs in the context of a higher dimensional gravity theory, can be likened to the variation or running of electric charge that occurs when a real electric charge is placed into some dielectric medium or in a quantum vacuum where quantum fluctuations produce a scale dependent electric charge.
We will sometimes loosely refer to this gravitational variation of the "electromagnetic" parameters of the solutions as the gravitational running, scaling or renormalization of the charges of the solutions.

II. ANHOLONOMIC FRAMES AND 5D VACUUM EINSTEIN EQUATIONS
In this section we outline the basic formulas for 5D Einstein gravity, and introduce the method of anholonomic frames. We construct locally anisotropic metrics which are generalizations of those considered in Ref. [6]. These 5D metrics have a mixture of holonomic and anholonomic variables, and are most naturally dealt with using anholonomic frames.
Finally we analyze the physical and mathematical properties of these 5D, locally anisotropic vacuum solutions.

A. Metric ansatz
Let us consider a 5D pseudo-Riemannian spacetime of signature (+, −, −, −, −) and denote the local coordinates u α = (x i , y a ) = (x 1 , x 2 , x 3 , y 4 = s, y 5 = p), -or more compactly u = (x, y) -where the Greek indices are split into two subsets x i (holonomic coordinates) and y a (anholonomic coordinates) labeled respectively by Latin indices i, j, k, ... = 1, 2, 3 and a, b, ... = 4, 5. The local coordinate bases, ∂ α = (∂ i , ∂ a ), and their duals, are written respectively as and We can treat an arbitrary coordinate, x i or y a , as space-like (x, y, z), time-like (t) or as the 5 th spatial coordinate (χ). The aim is then to study anisotropies and anholonomic constraints for various coordinates.
With respect to the coordinate frame base (2) the 5D pseudo-Riemannian metric with its metric coefficients g αβ parameterized as The ansatz functions of this metric are smooth function of the form: The ansatz functions of the metric are taken to depend on two isotropic variables (x 2 , x 3 ) and on one anisotropic variable, y 4 = s.
Metric (3) can be greatly simplified into the form with diagonal coefficients if instead of coordinate bases (1) and (2) one used anholonomic frames (anisotropic bases) and where the N-coefficients are parametrized as They define an associated nonlinear connection (N-connection) structure. (see Refs [1,7]).
Here, we shall not emphasize the N-connection formalism. The anisotropic frames (9) and (10) are anholonomic because, in general, they satisfy some anholonomic relations, with nontrivial anholonomy coefficients where Conventionally, the N-coefficients decompose spacetime objects (e.g. tensors, spinors and connections) into objects with mixed holonomic-anholonomic characteristics. The holonomic parts of an object are indicated with indices of type i, j, k, ..., while the anholonomic parts have indices of type a, b, c, .... Tensors, metrics and linear connections with coefficients defined with respect to anholonomic frames (9) and (10)

B. Einstein equations in holonomic-anholonomic variables
The main "trick" of the anholonomic frames method for integrating Einstein's equations in general relativity and various (super)string and higher / lower dimension gravitational theories, is to find the coefficients N a j such that the block matrices g ij and h ab are diagonalized [1,7]. This greatly simplifies computations. With respect to such anholonomic frames the partial derivatives are N-elongated (locally anisotropic).
Metric (3) with coefficients (4) (or equivalently, the d-metric (7) with coefficients (8)) is assumed to solve the 5D Einstein equations where κ and Υ αβ are respectively the coupling constant and the energy-momentum tensor.
For most of the paper we will consider vacuum solutions, Υ αβ = 0. The nontrivial components of the Ricci tensor (details of the computations are given in Refs. [6,7]), for the ansatz, are where The partial derivatives are denoted as h • = ∂h/∂x 2 , f ′ = ∂f /∂x 3 and f * = ∂f /∂s. We have given the formulas both in terms of h 4,5 and f 4,5 since we will need this later.
Formulas (13)- (16) were obtained with respect to anholonomic frames for a fixed linear connection adapted to the N-connection structure, called the canonical distinguished connection [8]. (Miron and Anastasiei introduced this connection on vector bundles, but it can be used in a similar fashion on (pseudo) Riemannian spaces if the N-connection is considered). The coefficients of a distinguished connection Γ α βγ = L i jk , L a bk , C i jc , C a bc , are computed from the formulas The coefficients in (21) reduce to the Christoffel symbols if the metric components g ij depend only on x-variables, the h ab depend only on y-variables, and the N-connection vanishes.
We emphasize that if the anholonomic frames are introduced into consideration, there is a certain class of linear connections which satisfy the metricity condition for a given metric, or inversely, there is a certain class of metrics which satisfy the metricity conditions for a given linear connection (this result was originally obtained by A. Kawaguchi [9] in 1937. Details can be found in [8]; see Theorems 5.4 and 5.5 in Chapter III). So, we need to state explicitly what type of linear connection is used for the definition of the curvature and Ricci tensor if the space-time is provided with an anholonomic frame structure. In this work and in Refs. [6,7] the linear connection is considered to be of the form (21). The off-diagonal metrics studied in this paper will be compatible with the canonical linear connection, but may not have a trivial limit to a diagonal holonomic metric.
The scalar curvature is using this along with the components of the Ricci tensor in Eqs. (13)-(16) one can show that for the metric ansatz (4) the coefficients of the energy-momentum d-tensor satisfy with respect to anholonomic bases (9) and (10). Thus the Einstein equations can be written as where i = 2, 3.
With this setup it is possible to construct very general classes of solutions to these equations [6,7] which describe locally anisotropic solitons, black holes, black tori and wormhole solutions.

C. General properties of the anisotropic vacuum solutions
In the vacuum case Eqs. (22) reduce to: We now discuss general features for the d-metric coefficients, (g 2 , g 3 ) , (h 4 , h 5 ) , and the N-connection coefficients w 2,3 and n 2,3 which solve this system of equations: 1. Eq. (23) relates two functions g 2 (x 2 , x 3 ) and g 3 (x 2 , x 3 ) and their partial derivatives in the isotropic coordinates x 2 and x 3 . If one of the functions is fixed, by some symmetry and boundary conditions the second function is found by solving a second order partial differential equation. For example by redefinition of the coordinates or a conformal transformation one can transform g 3 , (or conversely, g 2 ) into a constant. Using this technique one of the authors (SV) was able to construct various 2D soliton -dilaton and black hole-like configurations [7] 2. Eq. (24) contains partial derivatives of only the anisotropic coordinate s, and relates the two functions h 4 (x 2 , x 3 , s) and h 5 (x 2 , x 3 , s). By fixing one of these functions the second one is found by solving a second or first order differential equation in s (the x-variables being treated as parameters). These equations reduce to the Bernoulli equations [10], and are satisfied by two arbitrary functions h 4,5 (x 2 , x 3 ) for which h * 4,5 = 0. Thus there are three classes of solutions: If the condition h * 5 = 0 is satisfied, we can write (24), in f -variables (see (19)), as which is solved by arbitrary functions f 5 (x 2 , x 3 , s) and Bracketed subscripts indicate "constants" of integration with respect to the s variable.
The general solution of (24) expressing h 5 via h 4 is the integration "constants", 3. If the functions h 4 (x 2 , x 3 , s) and h 5 (x 2 , x 3 , s) are known, then Eqs. (25) become linearly independent algebraic equations for w 2,3 If in the case of vacuum Einstein equations h * 5 = 0, we have α i = β = 0 and as a consequence, Eq. (25) becomes trivial, allowing arbitrary values of the functions w 2,3 (x 2 , x 3 , s). For h * 5 = 0 we must impose the condition α 2,3 = 0, or identify these values with the corresponding non-diagonal components of the energy-momentum tensor. We also note that ansatz (4) admits an arbitrary function w 1 (x 2 ) which is not contained in the vacuum Einstein equations. This function can be fixed by requiring that it be compatible with some locally isotropic solutions.  (20) ) are known, The "renormalization" of h 4,5 into h 4,5 results in the "renormalization" of n 2,3 : n 2,3 → n 2,3 from formula (30) with h 4,5 → h 4,5 and γ → γ.

RUNNING OF CONSTANTS
We give a brief review of the locally isotropic wormhole and flux tube solutions (DSsolutions) constructed in Refs. [5,11], and their anisotropic generalization proposed in Ref. [6]. The isotropic DS-solutions represent 5D gravitational field configurations which carry "electric" and/or "magnetic" charges. Various authors have studied related 5D solutions: Liu and Wesson investigated 5D solitonic solutions [12]; they also considered 5D charged black holes [13]; 5D wormhole configurations with electromagnetic charges were studied in Ref. [14]; Ref. [15] looks at 5D solutions with magnetic charge; a general reference for 5D Kaluza-Klein theory and solutions is [16]. The anisotropic constructions considered in this paper are slightly different from Ref. [6]. Here we use a fixed conformal factor, in order to construct anisotropic solutions with background geometries more general than spherical.
The study of these more general background geometries will be carried out in section V.
The coefficient ω(r) in (32) is treated as the t -component of the electromagnetic potential and n cos θ as the θ-component. These electromagnetic potentials lead to the metric having radial Kaluza-Klein "electrical" and "magnetic" fields. The 5D Kaluza-Klein "electric" field is the "electric" charge q 0 = r 0 ω ′ (0) can be parametrized as The corresponding dual, "magnetic" field is with "magnetic" charge Q 0 = nr 0 parametrized as The following "circle" relation relates the "electric" and "magnetic" charges. As the free parameters of the metric are varied there are five classes of solutions with the properties: 1. Q 0 = 0 or H KK = 0, a wormhole-like "electric" object; 2. q 0 = 0 or E KK = 0, a finite "magnetic" flux tube; 3. q 0 = Q 0 or H KK = E KK , an infinite "electromagnetic" flux tube; 4. Q 0 < q 0 or H KK < E KK , a wormhole-like "electromagnetic" object; 5. Q 0 > q 0 or H KK > E KK , a finite, "magnetic-electric" flux tube.
The 5D gravitational vacuum polarization induced by variation of "constant" r 0 (χ) renormalizes the electromagnetic charge as q(χ) = r 0 (χ)ω ′ (r = 0). In terms of the angular parametrization the "electric" charge becomes q(χ) = 2 a(0) sin α(χ), The "electric" field from (33) becomes The renormalization of the magnetic charge, Q 0 → Q(χ), can be obtained using the renormalized "electric" charge in relationship (35) and solving for Q(χ). The form of (35) implies that the running of Q(χ) will be the opposite that of q(χ). For example, if q(χ) increases with χ then Q(χ) will decrease. The locally anisotropic polarizations α(χ) are either defined from experimental data or computed from a quantum model of 5D gravity. With the coordinates taken as x 1 = t, x 2 = r, x 3 = θ, y 4 = s = χ, y 5 = p = ϕ, one can construct a locally anisotropic solution of the vacuum Einstein equations (23)-(26) by making the following identifications for the ansatz functions from (32) This generalizes the DS-solution (32) by allowing the Kaluza-Klein electric and magnetic charges to be dependent on (i.e. scale with) the 5 th coordinate s = χ. We will call these the χ -solutions).

DS-solutions with anisotropy via ϕ
In a similar fashion we can consider anisotropic dependencies with respect to s = ϕ.
There is again an elliptical variation in θ, and also an anisotropic dependence in ϕ.
Comparing formulas (41) and (43)  For the χ-solutions this variable n will affect n 3 , while for the ϕ-solutions it will affect w 3 .
The form of the χ and ϕ -solutions from the previous two subsections gets modified by this "elliptic" renormalization of the Kaluza-Klein charges.
• With the coordinates defined as x 1 = t, x 2 = r, x 3 = θ, y 4 = s = ϕ, y 5 = p = χ the ϕ-solutions with κ r = [ η 4 (r, θ, χ)] −1 take the form can be generalized to other rotational hypersurface geometry backgrounds. In this section we will give the explicit forms for these generalized solutions and analyze their basic properties.
The notations and metric relations for the 3D Euclidean rotational hypersurfaces that we use will be those of Ref. [17].

A. Elongated rotation ellipsoid hypersurfaces
An elongated rotation ellipsoid hypersurface (a 3D e-ellipsoid) is given by the formula where σ ≥ 1, and x, y, z here are the usual Cartesian coordinates. a(r) is similar to the radius in the spherical symmetric case. The 3D, ellipsoidal coordinate system is defined x = a sinh u sin v cos s, y = a sinh u sin v sin s, z = a cosh u cos v, where σ = cosh u and 0 ≤ u < ∞, 0 ≤ v ≤ π, 0 ≤ s < 2π. The hypersurface metric is It will be more useful to consider a conformally transformed metric, where the components in Eq.(49) are multiplied by the conformal factor a −2 sinh 2 u + sin 2 v −1 , giving g ss (u, v) = sinh 2 u sin 2 v/(sinh 2 u + sin 2 v).

B. Flattened rotation ellipsoid hypersurfaces
In a similar fashion we consider the hypersurface equation for a flattened rotation ellipsoid (a 3D f-ellipsoid), here σ ≥ 0 and σ = sinh u. In this case the 3D coordinate system is defined as x = a cosh u sin v cos s, y = a cosh u sin v sin s, z = a sinh u cos v, where 0 ≤ u < ∞, 0 ≤ v ≤ π, 0 ≤ s < 2π. The hypersurface metric is Again for later convenience we consider a conformally transformed version of this metric g ss (u, v) = sinh 2 u cos 2 v/(sinh 2 u + cos 2 v).

The χ-solutions
For the five 3D geometries given above, the d-metrics (8) for the χ-solutions from Eqs.

VI. CONCLUSIONS
The construction of wormhole and/or flux tube solutions in modern string theory, extra dimensional gravity and quantum chromodynamics is of fundamental importance in understanding these theories (especially their non-perturbative aspects). Such solutions are difficult to find, and the solutions which are known usually have a high degree of symmetry.
In this paper we have applied the method of anholonomic frames to construct the general form of wormholes and flux tubes in 5D Kaluza-Klein theory. These solutions have local anisotropy which would make their study using holonomic frames difficult. This helps to demonstrate the usefulness of the anholonomic frames method in studying anisotropic solutions. Most physical situations do not possess a high degree of symmetry, and so the anholonomic frames method provides a useful mathematical framework for studying these less symmetric configurations.
The key result of this paper is the demonstration that off-diagonal metrics in 5D Kaluza-Klein theory can be parametrized into forms that define new, interesting classes of solutions of Einstein's vacuum equations. These solutions represent wormhole and flux tube configurations which are locally anisotropic. These anisotropic solutions reduce to previously known spherically symmetric wormhole metrics [4,5,11] in the local isotropic limit. These anisotropic solutions also extend the idea of Salam, Strathee and Perracci [3] that including off-diagonal components in higher dimensional metrics gives rise to gauge fields and charges.
Not only do we find "electric" and"magnetic" charges for our solutions, but the anisotropies in the 5 th coordinate (χ) and/or in the angular coordinate (ϕ) give a gravitational scaling or running of these Kaluza-Klein charges. Such a gravitational scaling of charges could provide an experimental signature for the presence of extra dimensions (i.e. if some charge were observed to exhibit a running which was not in agreement with that given by 4D quantum