Wormhole Solutions in the Presence of Nonlinear Maxwell Field

Abstract

In generalizing the Maxwell field to nonlinear electrodynamics, we look for the magnetic solutions. We consider a suitable real metric with a lower bound on the radial coordinate and investigate the properties of the solutions. We find that in order to have a finite electromagnetic field near the lower bound, we should replace the Born-Infeld theory with another nonlinear electrodynamics theory. Also, we use the cut-and-paste method to construct wormhole structure. We generalize the static solutions to rotating spacetime and obtain conserved quantities.

1. Introduction

A wormhole can be defined as a tunnel which can join two universes [1–3]. Since General Relativity does not preclude the existence of (traversable) wormholes, a large number of papers have been written which clarify, support, or contradict much of the research about wormholes.

Morris et al. [1–3] have shown that in order to construct a traversable wormhole, one needs to have extraordinary material, denoted as exotic matter. Exotic matter can guarantee the flare-out condition of the wormhole at its throat. Unlike the classical form of matter [4], it is believed that the exotic matter violates the well-known energy conditions such as the null energy conditions (NEC), weak energy conditions (WEC), strong energy conditions (SEC), and dominant energy conditions (DEC). One of the open questions about the exotic matter is that if it can be formed in macroscopic quantities or not. We should note that these energy conditions are violated by certain states of quantum fields, amongst which one may refer to the Casimir energy, Hawking evaporation, and vacuum polarization [5–12]. Furthermore, it has been shown that one of the effective causes of the (late time) cosmic acceleration is an exotic fluid [13–16]. Hence, one is motivated to study wormhole solutions, at least geometrically.

Many authors have extensively considered the nonlinear electrodynamics and used their results to explain some physical phenomena [17–55]. A charged system whose performance cannot be described by the linear equations may be characterized with nonlinear electrodynamics. From mathematical point of view, since Maxwell equations originated from the empirical nature, we can consider a general nonlinear theory of electrodynamics and state that the Maxwell fields are only approximations of nonlinear electrodynamics, where these approximations break down for the small distances. From physical viewpoint, generalizations of Maxwell theory to nonlinear electrodynamics were introduced to eliminate infinite quantities in theoretical analysis of the electrodynamics [29–40]. In addition, one may find various limitations of the linear electrodynamics in [56, 57].

Recently, we have taken into account new classes of nonlinear electrodynamics, such as Born-Infeld- (BI-) like [53–55] and power-Maxwell invariant (PMI) [41–52] nonlinear electrodynamics, in order to obtain new analytical solutions in Einstein and higher derivative gravity. Traversable wormholes in the Dvali-Gabadadze-Porrati theory with cylindrical symmetry have been studied in [58]. Higher dimensional Lorentzian wormholes have been analyzed by several authors [59–61]. Moreover, wormhole solutions of higher derivative gravity with linear and nonlinear Maxwell fields have been considered in [62–66]. For other kinds of wormhole solutions, we refer the reader to [62–83] and references therein.

Motivated by the above considerations, in this paper we look for the analytical magnetic horizonless solutions of Einstein gravity with nonlinear Maxwell source. Properties of the solution will be investigated.

2. Field Equations and Wormhole Solutions

The field equations of Einstein gravity with an arbitrary 𝑈(1) gauge field as a source may be written as 𝐺_𝜇𝜈+Λ𝑔_𝜇𝜈=12𝑔_𝜇𝜈𝐿(ℱ)−2𝐿_ℱ𝐹_𝜇𝜆𝐹^𝜆_𝜈,𝜕_𝜇√−𝑔𝐿_ℱ𝐹^𝜇𝜈=0, (1)(2) where 𝐺_𝜇𝜈 is the Einstein tensor, Λ=−3/2𝑙² denotes the four-dimensional negative cosmological constant, and 𝐿(ℱ) is an arbitrary function of the closed 2-form Maxwell invariant ℱ=𝐹_𝜇𝜈𝐹^𝜇𝜈 and 𝐿_ℱ=𝑑𝐿(ℱ)/𝑑ℱ.

In addition to PMI and BI theories, in this paper, we take into account the recently proposed BI-like models [53–55], which we called as exponential form of nonlinear electrodynamics theory (ENE) and logarithmic form of nonlinear electrodynamics theory (LNE), whose Lagrangians are 𝐿(ℱ)=−ℱ^𝑠,PMI4𝛽²1−ℱ1+2𝛽²𝛽,BI₂−ℱexp𝛽²−1,ENE−8𝛽²ℱln1+8𝛽²,LNE, (3) where 𝑠 and 𝛽 are two nonlinearity parameters. Expanding the mentioned Lagrangians near the linear Maxwell case (𝑠→1 and 𝛽→∞), one can obtain 𝐿(ℱ)⟶𝐿_Max+−ℱln(ℱ)(𝑠−1)+𝑂(𝑠−1)²+,PMI𝜒ℱ²16𝛽²ℱ+𝑂³𝛽⁴,others, (4) where Maxwell Lagrangian 𝐿_Max=−ℱ and 𝜒=1, 2, and 8 for LNE, BI, and ENE branches, respectively.

Investigation of the effects of the higher derivative corrections to the gauge field seems to be an interesting phenomenon. These nonlinear electrodynamics sources have different effects on the physical properties of the solutions. For example, in black hole framework, these nonlinearities may change the electric potential, temperature, horizon geometry, energy density distribution, and also asymptotic behavior of the solutions. In what follows, we study the effects of nonlinearity on the magnetic solutions.

Motivated by the fact that we are looking for the horizonless magnetic solution (instead of electric one), one can start with the following 4-dimensional spacetime: 𝑑𝑠²𝑟=−²𝑙²𝑑𝑡²+𝑑𝑟²𝑓(𝑟)+Υ²𝑙²𝑓(𝑟)𝑑𝜃²+𝑟²𝑑𝜙², (5) where Υ is a constant and will be fixed later. We should note that because of the periodic nature of 𝜃, one can obtain the presented metric (5) with local transformations 𝑡→𝑖𝑙Υ𝜃 and 𝜃→𝑖𝑡/𝑙 in the Schwarzschild metric with zero curvature boundary, 𝑑𝑠²=−𝑓(𝑟)𝑑𝑡²+𝑑𝑟²/𝑓(𝑟)+𝑟²(𝑑𝜃²+𝑑𝜙²). In other words, metric (5) may be locally mapped to Schwarzschild spacetime but not globally. Considering the mentioned local transformation, one can find that the nonzero component of the gauge potential is 𝐴_𝜃: 𝐴_𝜇=ℎ(𝑟)𝛿^𝜃_𝜇, (6) where ℎ(𝑟) is an arbitrary function of 𝑟. Using (2) with the metric (5), we find ℎ(𝑟)=∫𝐸(𝑟)𝑑𝑟 in which 𝐸(𝑟)=2𝑞𝑙²Υ²𝑟^{2/(2𝑠−1)},PMI2𝑞𝑙²Υ²Γ𝑟²,BI2𝑞𝑙²Υ²𝑟²−𝐿exp^𝑊2𝛽,ENE₂𝑟²(1−Γ)𝑞,LNE, (7) where 𝐿_𝑊=Lambert𝑊(𝑋),  𝑋=−16𝑞²𝑙²Υ²/𝛽²𝑟⁴, and Γ=1+𝑋/4 and therefore the nonzero component of electromagnetic field tensor is 𝐹_𝑟𝜃=𝐸(𝑟). (8) We should note that the physical gauge potential should vanish for large values of 𝑟. This condition is satisfied for 1/2<𝑠<3/2 and arbitrary 𝛽 (the mentioned constraint for 𝑠 is used throughout the rest of the paper). In addition, in order to have real solutions, one should restrict the radial coordinate of BI and LNE branches to 𝑟≥𝑟₀ in which 𝑟₀=0,PMI2𝑞𝑙Υ𝛽2,BI𝑞𝑙Υ𝛽exp(1),ENE2𝑞𝑙Υ𝛽,LNE. (9) Now, one can expand (7) to obtain the leading nonlinearity correction of Maxwell field as follows: 𝐸(𝑟)|_{nearthelinearcase}=2𝑞𝑙²Υ²𝑟²+8𝑞𝑙²Υ²ln(𝑟)𝑟²(𝑠−1)+𝑂(𝑠−1)²,PMI2𝜒𝑞³𝑙⁴Υ⁴𝛽²𝑟⁶1+𝑂𝛽⁴.others. (10) In addition, we can investigate the behavior of electromagnetic fields near the lower bound 𝑟₀ as lim_{𝑟→𝑟+0}𝐸(𝑟)=+∞,PMI+∞,BI𝛽Υ𝑙2,ENE2𝛽Υ𝑙,LNE. (11)

In order to examine the effect of nonlinearity on the electromagnetic field, we plot Figures 1 and 2. Figure 1 shows that when we reduce the nonlinearity 𝑠, the electromagnetic field of the PMI branch diverges for 𝑟→0 more rapidly and for large distances it goes to zero more quickly. Figure 2 shows that for all BI-like branches, the electromagnetic field (the same behavior as in Maxwell case) vanishes for large 𝑟. Moreover, in addition to (11), Figure 2 shows that near the lower bound (𝑟₀), unlike BI branch, the electromagnetic fields of LNE and ENE branches have finite values. In other words, considering the LNE and ENE branches, we could remove the lower bound divergency of electromagnetic field.

Figure 1 

𝐸(𝑟) versus 𝑟 for 𝑞=1, Υ=1, 𝑙=1, and 𝑠=1.4 (solid line), 𝑠=1.2 (dashed line), 𝑠=1 “Maxwell field” (bold line), 𝑠=0.8 (dotted line), and 𝑠=0.6 (dash-dotted line).

Figure 2 

𝐸(𝑟) versus 𝑟 for 𝑞=1, Υ=1, 𝑙=1, and 𝛽=1. BI (solid line), ENE (dashed line), LNE (dotted line), and Maxwell field (bold line).

Taking into account the electromagnetic field tensor, we are in a position to find the function 𝑓(𝑟). In order to obtain it, one may use any components of (1). We simplify the components of (1) and find that the nonzero independent components of (1) are 𝑓^′′(𝑟)+2𝑓^′(𝑟)𝑟+2Λ+Δ₁(𝑟)𝑓=0,_′(𝑟)+𝑓(𝑟)𝑟+Λ𝑟+Δ₂(𝑟)=0, (12) withΔ₁(𝑟)=−2ℎ^′2(𝑟)Υ²𝑙^2𝑠,PMI4𝛽²ℎ1+^′2(𝑟)𝛽²Υ²𝑙²𝛽−1,BI₂1−exp−2ℎ^′2(𝑟)𝛽²Υ²𝑙²,ENE8𝛽²ℎln1+^′(𝑟)2𝛽Υ𝑙²Δ,LNE,₂(𝑟)=𝑟2(2𝑠−1)2ℎ^′2(𝑟)Υ²𝑙^2𝑠,PMI2𝑟𝛽²ℎ1+^′2(𝑟)𝛽²Υ²𝑙^2−1/2−1,BI𝑟𝛽²21−1+2ℎ^′(𝑟)𝛽Υ𝑙²exp−2ℎ^′2(𝑟)𝛽²Υ²𝑙²,ENE4𝑟𝛽²ℎln1+^′(𝑟)2𝛽Υ𝑙²−21+2𝛽Υ𝑙/ℎ^′(𝑟)²,LNE, (13)

where prime and double prime are first and second derivatives with respect to 𝑟, respectively. After some cumbersome manipulation, the solutions of (12) can be written as𝑓(𝑟)=−2𝑚𝑟−Λ𝑟²3+𝑟²(2𝑠−1)²2(2𝑠−3)8𝑞²𝑙²Υ²𝑟^{4/(2𝑠−1)𝑠},PMI2𝛽²𝑟²3−2𝛽²𝑟Γ𝑟²−𝛽𝑑𝑟,BI₂𝑟²6−2𝛽𝑞𝑙Υ𝑟1−𝐿_𝑊−−𝐿_𝑊−𝑑𝑟,ENE4𝛽²𝑟𝑟²𝛽ln²𝑟⁴(1−Γ)2𝑞²Υ²𝑙²𝑑𝑟+4𝛽²𝑟𝑟²(1−Γ)𝑑𝑟,LNE, (14)

where 𝑚 is the integration constant which is related to the mass parameter. In order to investigate the effect of nonlinearity on the metric function, simplistically, we expand 𝑓(𝑟) for 𝑠→1 for PMI and 𝛽→∞ for other branches. After some manipulation, we obtain𝑓(𝑟)=𝑓_Max(𝑟)+−4𝑞²𝑙²Υ²6+ln8𝑞²Υ²𝑙²𝑟⁴𝑟²(𝑠−1)+𝑂(𝑠−1)²−,PMI2𝜒𝑞⁴𝑙⁴Υ⁴5𝛽²𝑟⁶1+𝑂𝛽⁴,others, (15)

where 𝑓_Max(𝑟) is the magnetic solution of Einstein-Maxwell gravity: 𝑓_Max(𝑟)=−2𝑚𝑟−Λ𝑟²3−4𝑞²𝑙²Υ²𝑟², (16) and the second term on the right hand side of (15) is the leading nonlinearity correction to the Einstein-Maxwell gravity solution.

2.1. Properties of the Solutions

At first step, we should note that the presented solutions are asymptotically anti-de Sitter (adS) and they reduce to asymptotically adS Einstein-Maxwell solutions for 𝑠→1 (PMI branch) or 𝛽→∞ (other branches).

The second step should be devoted to looking for the singularities and hence we should calculate the curvature invariants. One can show that, for the metric (5), the Kretschmann and Ricci scalars are 𝑅_{𝜇𝜈𝜌𝜎}𝑅^{𝜇𝜈𝜌𝜎}=𝑓^′′2(𝑟)+4𝑓^′2(𝑟)𝑟²+4𝑓²(𝑟)𝑟⁴,𝑅=−𝑓^′′(𝑟)−4𝑓^′(𝑟)𝑟−2𝑓(𝑟)𝑟². (17) Inserting (14) into (17) and using numerical calculations, one can show that the Ricci and Kretschmann scalars diverge at 𝑟=𝑟₀ and are finite for 𝑟>𝑟₀ and for 𝑟→∞ one obtains 𝑅_{𝜇𝜈𝜌𝜎}𝑅^{𝜇𝜈𝜌𝜎}_Large𝑟=8Λ²3+𝑂1𝑟^𝜉𝑂1PMI𝑟⁶𝑅|,others,_Large𝑟𝑂1=4Λ+𝑟^𝜉𝑂1PMI𝑟⁸(),,others,𝜉∈3,∞ (18) which confirms that the asymptotic behavior of the solutions is adS. Considering the divergency of the Ricci and Kretschmann scalars at 𝑟₀, one may think that there is a curvature singularity located at 𝑟=𝑟₀. This singularity will be naked if the function 𝑓(𝑟) has no real root larger than 𝑟₀ (singularity is not covered with a horizon) and we are not interested in it. Therefore, we consider the case in which the function 𝑓(𝑟) has at least a nonextreme positive real root larger than 𝑟₀. It is notable that the function 𝑓(𝑟) is negative for 𝑟=𝑟₊−𝜖 (𝜖 is an infinitesimal number) and positive for 𝑟>𝑟₊ where 𝑟₊ is the largest positive real root of 𝑓(𝑟)=0. Negativity of the function 𝑓(𝑟) leads to an apparent change of metric signature and it forces us to consider 𝑟₊≤𝑟<∞. We should state that although the metric function 𝑓(𝑟) vanishes at 𝑟=𝑟₊, we have 𝑓^′(𝑟=𝑟₊)>0. In addition, there is no curvature singularity in the range 𝑟₊≤𝑟<∞. Following the procedure of [84], one may find that there is a conic singularity at 𝑟=𝑟₊.

Removing this conical singularity by adjusting Υ=1/[𝑙𝑓^′(𝑟₊)] [84], we desire to interpret the obtained solutions as wormholes. In order to construct wormholes from the gluing, one is required to use the cut-and-paste prescription [67–83]. In this method, we take into account two geodesically incomplete copies of the solutions (removing from each copy the forbidden region Ω) with two copies of the boundaries 𝜕Ω, where Ω≡𝑡,𝑟,𝜃,𝜙∣𝑟<𝑟₊,𝜕Ω≡𝑡,𝑟,𝜃,𝜙∣𝑟=𝑟₊. (19) Now, we identify two copies of the mentioned boundaries to build a geodesically complete manifold. This cut-and-paste method constructs a wormhole with a throat at 𝑟=𝑟₊. In order to confirm this claim, we should check the so-called flare-out condition at the throat. To do this, one can consider a 2-dimensional submanifold of the metric (5), 𝑑𝑠²_2-dim (with constant 𝑡 and 𝜃), and embed it in a 3-dimensional Euclidean flat space in cylindrical coordinates, 𝑑𝑠²_3-dim, where 𝑑𝑠²_2-dim=𝑑𝑟²𝑓(𝑟)+𝑟²𝑑𝜙²,𝑑𝑠²_3-dim=𝑑𝑟²+𝑟²𝑑𝜙²+𝑑𝑧². (20) Regarding the surface 𝑧=𝑧(𝑟), we obtain 𝑑𝑟𝑑𝑧_𝑟=𝑟+=𝑓(𝑟)1−𝑓(𝑟)_𝑟=𝑟+𝑑=0,₂𝑟𝑑𝑧²_𝑟=𝑟+=𝑓^′21−𝑓²_𝑟=𝑟+=12𝑓^′𝑟=𝑟₊>0, (21) which shows that the flare-out condition may be satisfied for the surface 𝑧=𝑧(𝑟) and therefore 𝑟=𝑟₊ is the radius of the wormhole throat. It is clear to find that the metric (the first fundamental form) is continuous on the boundary 𝜕Ω, while its first derivative may be discontinuous. In order to investigate this discontinuity, one should consider the extrinsic curvature (the second fundamental form). Here, we give a brief remark about it [85–91]. The second fundamental forms associated with the two sides of the throat 𝜕Ω are 𝐾^±_𝑖𝑗=−𝑛^±_𝜇𝜕_𝑖𝜕_𝑗𝑥^𝜇+Γ^𝜇_𝛼𝛽𝜕_𝑖𝑥^𝛼𝜕_𝑗𝑥^𝛽_𝜕Ω, (22) in which 𝑛^±_𝜇 are the unit normals to the two boundaries 𝜕Ω with the following explicit form: 𝑛^±_𝜇=±𝜕_𝛼𝐺𝜕^𝛼𝐺^−1/2𝜕_𝜇𝐺, (23) where 𝑖,𝑗=1,2,3 corresponding to the boundary 𝜕Ω, 𝜇,𝛼,𝛽=1,2,3,4 corresponding to original spacetime, and 𝐺=𝑟−𝑟₊. We emphasize that ± sign comes from the fact that a unit’s normal points outward from one side of the throat 𝜕Ω and points inward on the other side. Generally, we have 𝐾⁺_𝑖𝑗≠𝐾⁻_𝑖𝑗, and one can write the following Einstein (Lanczos) equation on the throat: 𝑘_𝑖𝑗−𝑘𝑔_𝑖𝑗=−8𝜋𝑆_𝑖𝑗, (24) where 𝑘_𝑖𝑗=𝐾⁺_𝑖𝑗−𝐾⁻_𝑖𝑗, 𝑘=𝑘^𝑖_𝑖, and 𝑆_𝑖𝑗=diag(𝜎,𝑝_𝜃,𝑝_𝜙) is the stress-energy tensor localized in 𝜕Ω with 𝜎, the surface energy density, and 𝑝_𝜃 and 𝑝_𝜙, the principal surface pressures (tensions): (𝑟𝜎=−𝑑/𝑑𝑟)²𝑓(𝑟)8𝜋𝑙²𝑓(𝑟),𝑝_𝜃=𝑓³(𝑟)𝑙²Υ²,𝑝2𝜋𝑟_𝜙=(𝑑/𝑑𝑟)𝑟²𝑓(𝑟)8𝜋𝑓(𝑟). (25)

One can use the series expansion for 𝑟→𝑟₊ with the obtained metric function 𝑓(𝑟) to find the signs of surface energy density and tensions. Negative surface energy implies the existence of ghost-like matter at the throat and the negative signs of the tensions mean that they are indeed pressures.

Now, we should discuss the energy conditions for the wormhole solutions. On general grounds, it has been shown that traversable wormhole may exist with exotic matter which violates the null energy condition [1–3]. In order to check the energy conditions, we use the following orthonormal contravariant (hatted) basis to simplify the mathematics and physical interpretations: 𝐞𝑡̂=𝑙𝑟𝜕𝜕𝑡,𝐞𝑟̂=𝑓^1/2𝜕,𝐞𝜃=1𝜕𝑟Υ𝑙𝑓^1/2𝜕𝜕𝜃𝜙,𝐞=𝑟⁻¹𝜕.𝜕𝜙 (26) Using the mentioned basis, we can obtain 𝑇𝑡𝑡̂̂𝜙𝜙=1=−𝑇28Υ²𝑞²𝑙²𝑟^{4/(2𝑠−1)𝑠},PMI2𝛽²Γ⁻¹𝛽−1,BI₂21−𝑋𝐿_𝑊,ENE4𝛽²()ln−81−Γ𝑋𝑇𝑟𝑟̂̂𝜃𝜃=(),LNE,=𝑇2𝑠−128Υ²𝑞²𝑙²𝑟^{4/(2𝑠−1)𝑠},PMI2𝛽²()𝛽1−Γ,BI₂2𝑋𝐿_𝑊+𝑋𝐿_𝑊𝛽−1,ENE₂()8−4ln−81−Γ𝑋+𝑋1−Γ,LNE, (27) and therefore for 𝑟≥𝑟₊>𝑟₀, we have 𝑇𝑡𝑡̂̂>0,𝑇𝑡𝑡̂̂+𝑇𝑟𝑟̂̂>0, (28) which shows that the solutions satisfy the null and weak energy conditions, simultaneously (see Figure 3 for more clarification).

(a)

(b)

Figure 3 

(a) 𝑋/𝐿𝑊 (dashed line) and 𝑋𝐿𝑊 (bold line) versus 𝑋. (b) ln(−8(Γ−1)/𝑋) versus 𝑋.

At the end of this section, we desire to study the effects of the nonlinearity on energy density of the spacetime. At the start, we can expand 𝑇𝑡𝑡̂̂ near the linear case to obtain𝑇𝑡𝑡̂̂=𝑇𝑡𝑡̂̂_Maxwell+4𝑞²𝑙²Υ²ln8𝑞²Υ²𝑙²𝑟⁴𝑟⁴(𝑠−1)+𝑂(𝑠−1)²+,PMI6𝜒𝑞⁴𝑙⁴Υ⁴𝛽²𝑟⁸1+𝑂𝛽⁴,others, (29)

where 𝑇𝑡𝑡̂|̂_Maxwell=4Υ²𝑞²𝑙²/𝑟⁴ and the second term on the right hand side of (29) is the leading nonlinearity correction to the energy density of the Einstein-Maxwell theory. Investigations of the energy densities near the lower bound 𝑟₀ show that lim_{𝑟→𝑟+0}𝑇𝑡𝑡̂̂=𝛽+∞,PMI+∞,BI₂2−11−exp2,ENE4𝛽²ln2,LNE. (30)

In addition, we plot the energy density 𝑇𝑡𝑡̂̂ versus 𝑟 for different values of nonlinearity parameter 𝑠 and also various branches of BI-like fields. Figures 4 and 5 show that for the arbitrary choices of 𝑟 the energy density is positive definite. Furthermore, Figure 4 shows that the nonlinearity parameter, 𝑠, has effects on the behavior of the energy density and when we reduce 𝑠, both divergency of energy density near the origin and its vanishing for large values of distance occur more rapidly. Moreover, considering Figure 5 with (30), one can find that 𝑇𝑡𝑡̂̂ has a finite value for an arbitrary allowed distance in ENE and LNE branches. Like Maxwell and PMI theory, for BI branch the energy density diverges near the lower bound 𝑟₀.

Figure 4 

𝑇𝑡𝑡̂̂ versus 𝑟 for 𝑞=1, Υ=1, 𝑙=1, and 𝑠=1.4 (solid line), 𝑠=1.2 (dashed line), 𝑠=1 “Maxwell field” (bold line), 𝑠=0.8 (dotted line), and 𝑠=0.6 (dash-dotted line) “different scales”.

Figure 5 

𝑇𝑡𝑡̂̂ versus 𝑟 for 𝑞=1, Υ=1, 𝑙=1, and 𝛽=1. BI (solid line), ENE (dashed line), LNE (dotted line), and Maxwell field (bold line).

2.2. Rotating Solutions

In this section, we want to add angular momentum to the static spacetime (5). To do this, one can use the following rotation boost in the 𝑡−𝜃 plane: 𝑎𝑡⟼Ξ𝑡−𝑎𝜃,𝜃⟼Ξ𝜃−𝑙²𝑡, (31) where Ξ=1+𝑎²/𝑙² and 𝑎 is a rotation parameter. Taking into account (31) and applying it to static metric (5), one obtains 𝑑𝑠²𝑟=−²𝑙²(Ξ𝑑𝑡−𝑎𝑑𝜃)²+𝑑𝑟²𝑓(𝑟)+Υ²𝑙²𝑓(𝑟)𝑎𝑙²𝑑𝑡−Ξ𝑑𝜃²+𝑟²𝑑𝜙², (32) where 𝑓(𝑟) is the same as 𝑓(𝑟) given in (14). It is notable that one can obtain the presented metric (32) with local transformations 𝑡→𝑖𝑙Υ(𝑎𝑡/𝑙²−Ξ𝜃) and 𝜃→𝑖(Ξ𝑡−𝑎𝜃)/𝑙 in the Schwarzschild metric with zero curvature boundary; 𝑑𝑠²=−𝑓(𝑟)𝑑𝑡²+𝑑𝑟²/𝑓(𝑟)+𝑟²𝑑𝜃²+𝑟²𝑑𝜙². Thus, the nonzero components of the gauge potential are 𝐴_𝜃 and 𝐴_𝑡∶𝐴_𝜇=ℎ(𝑟)Ξ𝛿^𝜃_𝜇−𝑎𝑙²𝛿^𝑡_𝜇, (33) where ℎ(𝑟) is the same as in the static case. Furthermore, the nonzero components of electromagnetic field tensor are given by 𝐹_𝑡𝑟=𝑎Ξ𝑙²𝐹_𝑟𝜃=𝑎𝑙²𝐸(𝑟). (34)

As we mentioned before, the periodic nature of 𝜃 helps us to conclude that the transformation (31) is not a proper coordinate transformation on the entire manifold and therefore the metrics (5) and (32) are distinct [92]. In addition, it is desired to note that rotating solutions have no horizon and curvature singularity. Moreover, it is worth noting that besides the magnetic field along the 𝜃 coordinate, there is also a radial electric field (𝐹_𝑡𝑟≠0) and, therefore, unlike the static case, the rotating wormhole has a nonzero electric charge which is proportional to the rotation parameter.

2.3. Conserved Quantities

Here we desire to calculate finite conserved quantities. In order to obtain a finite value for these quantities, we can use the counterterm method inspired by the concept of (AdS/CFT) correspondence [93–95]. It has been shown that for asymptotically AdS solutions the finite energy momentum tensor is 𝑇^𝑎𝑏=1𝐾8𝜋^𝑎𝑏−𝐾𝛾^𝑎𝑏−2𝛾^𝑎𝑏𝑙, (35) where 𝐾 is the trace of the extrinsic curvature 𝐾^𝑎𝑏 and 𝛾^𝑎𝑏 is the induced metric of the boundary. Taking into account the Killing vector field 𝜉, one may obtain the quasilocal conserved quantities in the following form: 𝒬(𝜉)=_ℬ𝑑²𝜑√𝜎𝑇_𝑎𝑏𝑛^𝑎𝜉^𝑏, (36) where 𝜎 is the determinant of the boundary metric in ADM (Arnowitt-Deser-Misner) form 𝜎_𝑖𝑗, and 𝑛^𝑎 is the timelike unit vector normal to the boundary ℬ. Considering two Killing vectors 𝜉=𝜕/𝜕𝑡 and 𝜁=𝜕/𝜕𝜃, we can find their associated conserved charges which are mass and angular momentum as follows: 𝑀=4𝜋²3Ξ²−1+1Υ𝑚,𝐽=12𝜋²Υ𝑚Ξ𝑎, (37) where the former equation confirms that 𝑎 is the rotational parameter.

Finally, we are in a position to discuss the electric charge. In order to compute it, we need a nonzero radial electric field 𝐹_𝑡𝑟 and therefore one expects that vanishing 𝐹_𝑡𝑟 (static case) leads to zero electric charge. Taking into account the Gauss law for the rotating solutions and computing the flux of the electric field at infinity, one can find 2𝑄=^3𝑠+1𝜋²Υ𝑠4𝑙(2𝑠−1)𝑞(3−2𝑠)𝑙^2𝑠−1𝑎,PMI4𝜋²Υ𝑞𝑙²𝑎,others, (38) which confirms that the static wormholes do not have electric charge.

3. Closing Remarks

In this paper, we took into account a class of magnetic Einsteinian solutions in the presence of nonlinear source. The magnetic spacetime which we used in this paper may be obtained from the Schwarzschild metric with zero curvature boundary by the local transformations 𝑡→𝑖𝑙Υ𝜃 and 𝜃→𝑖𝑡/𝑙. It is notable that because of the periodic nature of 𝜃, the mentioned transformations cannot be global.

We considered four forms of nonlinear electrodynamics, namely, PMI, BI, ENE, and LNE theories, whose asymptotic behavior leads to Maxwell theory. Obtaining real solutions forced us to define a lower bound 𝑟₀ for the radial coordinate of spacetime. We investigated the effect of nonlinearity parameter on the electromagnetic field and found that, for PMI branch, if one reduces the nonlinearity parameter 𝑠, then the electromagnetic field diverges near the origin more rapidly and for large distances it goes to zero more quickly. In addition, we found that for all BI-like branches, the behavior of the electromagnetic field is the same as Maxwell case for large values of distance, but near the lower bound, the electromagnetic field of the ENE and LNE branches is finite and it diverges for the BI branch. It is interesting to note that the divergency of the BI branch has less strength in comparison to the the Maxwell field. Consequently, in order to have a finite electromagnetic field near the lower bound, we could not use BI theory for our magnetic spacetime and we may apply LNE and ENE theories for this purpose.

Then, we obtained the metric function for all branches and found that they reduce to asymptotically adS Einstein-Maxwell solutions for 𝑠→1 (PMI branch) or 𝛽→∞ (other branches). We also expanded the metric function near the linear Maxwell field and calculated the curvature scalars for large 𝑟 to find that obtained solutions are asymptotically anti-de Sitter (adS). Taking into account the presented metric, one can find that the function 𝑓(𝑟) cannot be negative since its negativity leads to an apparent change of metric signature. This limitation forced us to consider 𝑟₊≤𝑟<∞. Using numerical calculations, one can find that there is no curvature singularity in the range 𝑟₊≤𝑟<∞, but one may find a conic singularity at 𝑟=𝑟₊.

After that, we removed the mentioned conic singularity and used the cut-and-paste prescription to construct a wormhole from the gluing and then we checked the so-called flare-out condition at the throat 𝑟=𝑟₊. Since it has been stated before that traversable wormhole may or may not exist with exotic matter [1–3], we investigated the energy conditions for the obtained wormhole solutions and found that the null and weak energy conditions are satisfied, which means that there is no exotic matter near the throat.

We also studied the effects of nonlinearity parameter on the energy density. For PMI branch, we found that when we reduce 𝑠, both divergency of energy density near the origin and its vanishing for large values of distance occur more rapidly. Moreover, one can find that energy density has a finite value for an arbitrary distance for LNE and ENE branches and it diverges near the lower bound for BI branch.

We generalized the static solutions to rotating ones and obtained the conserved quantities. We found that, unlike the static case, for the spinning spacetime, the wormhole has a net electric charge density. We also found that in spite of the fact that the mentioned nonlinear theories change the properties of the solutions significantly, they do not have any effect on mass and angular momentum.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks the anonymous referees for the useful criticisms and comments which permitted him to improve this paper. The author is indebted to A. Poostforush and H. Mohammadpour for reading the paper. The author also wishes to thank Shiraz University Research Council. This work has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Iran.