^{1, 2}

^{1}

^{2}

^{3}.

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.

A wormhole can be defined as a tunnel which can join two universes [

Morris et al. [

Many authors have extensively considered the nonlinear electrodynamics and used their results to explain some physical phenomena [

Recently, we have taken into account new classes of nonlinear electrodynamics, such as Born-Infeld- (BI-) like [

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.

The field equations of Einstein gravity with an arbitrary

In addition to PMI and BI theories, in this paper, we take into account the recently proposed BI-like models [

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:

In order to examine the effect of nonlinearity on the electromagnetic field, we plot Figures

Taking into account the electromagnetic field tensor, we are in a position to find the function

where prime and double prime are first and second derivatives with respect to

where

where

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

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 (

Removing this conical singularity by adjusting

One can use the series expansion for

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 [

(a)

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

where

In addition, we plot the energy density

In this section, we want to add angular momentum to the static spacetime (

As we mentioned before, the periodic nature of

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 [

Finally, we are in a position to discuss the electric charge. In order to compute it, we need a nonzero radial electric field

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

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

Then, we obtained the metric function for all branches and found that they reduce to asymptotically adS Einstein-Maxwell solutions for

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

We also studied the effects of nonlinearity parameter on the energy density. For PMI branch, we found that when we reduce

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.

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

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.