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Advances in High Energy Physics
Volume 2014, Article ID 697914, 10 pages
http://dx.doi.org/10.1155/2014/697914
Research Article

Magnetic String with a Nonlinear Source

1Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
2Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran

Received 21 January 2014; Revised 28 February 2014; Accepted 6 March 2014; Published 2 April 2014

Academic Editor: Christian Corda

Copyright © 2014 S. H. Hendi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3.

Abstract

Considering the Einstein gravity in the presence of Born-Infeld type electromagnetic fields, we introduce a class of 4-dimensional static horizonless solutions which produce longitudinal magnetic fields. Although these solutions do not have any curvature singularity and horizon, there exists a conic singularity. We investigate the effects of nonlinear electromagnetic fields on the properties of the solutions and find that the asymptotic behavior of the solutions is adS. Next, we generalize the static metric to the case of rotating solutions and find that the value of the electric charge depends on the rotation parameter. Furthermore, conserved quantities will be calculated through the use of the counterterm method. Finally, we extend four-dimensional magnetic solutions to higher dimensional solutions. We present higher dimensional rotating magnetic branes with maximum rotation parameters and obtain their conserved quantities.

1. Introduction

One of the interesting topological defects is cosmic string which may be originated during the early universe phase transitions [1] (see Kibble mechanism for more details [2]). Furthermore, considering the inflationary models [3, 4], it has been proposed that cosmic strings can form at the end of inflation. Moreover, one of the predictions of supersymmetric hybrid inflation [5] (and also grand unified models of inflation [6]) is the cosmic string. Interesting properties and interaction of the superconducting cosmic string with astrophysical magnetic fields have been found in [79]. Besides, magnetic strings have been studied in Brans-Dicke theory as well as dilaton gravity [1013]. From cosmological point of view, one can find the properties of the magnetic (cosmic) string in various literatures [1416].

In addition to cosmic strings, other kinds of strings may be considered in QCD and also gravity. Properties of the QCD static strings have been investigated extensively in [1720] and it has been shown that QCD magnetic string can contribute to hadron dynamics [21]. Applications of magnetic string in quantum theories have been presented in [2224]. Magnetic strings in antiferromagnetic crystals have been investigated in [25]. Application of the (chromo)magnetic string model to some experimental data on the inclusive pion asymmetries has been studied in [26, 27]. Some arguments about the magnetic strings in the Yang-Mills plasma have been found in [28].

On gravitational aspect, the horizonless solutions and spacetime with conical singularity have been investigated in gravitating electromagnetic field background (see [7, 8, 2949] and references therein). Interesting properties of the magnetic string in branes, M-theory, and string theory have been investigated [50, 51]. Calculations of the vacuum energy of two different fields in the background of a magnetic string have been analyzed in [52, 53].

One of the generalizations of the Einstein-Maxwell field equations is gravitating nonlinear electrodynamics (NLED), whose most popular theory is Born-Infeld [5459]. In addition to Lorentz and gauge invariances, we know that the Lagrangian of the Maxwell electrodynamics contains only quadratic forms of gauge potential and its first derivative. One can consider both invariances and leave out the third condition to obtain NLED [60]. From historical point of view, NLED were introduced to eliminate infinite quantities in theoretical analysis of charged point-like particles [5459]. Recently, we have more motivations for considering NLED theories, for example, various limitations of the linear electrodynamics [61, 62], clarification of the self-interaction of virtual electron-positron pairs [6365], explanation of electrodynamics on D-branes [6668], and description of radiation propagation inside specific materials [6972]. In addition, from astrophysical viewpoint, we know that the effects of NLED become indeed quite important in superstrongly magnetized compact objects, such as pulsars, and particular neutron stars (some examples include the so-called magnetars and strange quark magnetars) [7375]. Moreover, NLED modifies in a fundamental basis the concept of gravitational redshift and its dependency on any background magnetic field as compared to the well-established method introduced by standard general relativity. Furthermore, it has been recently shown that NLED objects can remove both of the big bang and black hole singularities [7678].

Amongst the nonlinear generalization of Maxwell electrodynamics, the so-called BI type NLED, whose first nonlinear correction is quadratic function of Maxwell invariant, is completely special. It has been shown that BI type NLED may be arisen as a low energy limit of heterotic string theory [66, 68, 7983], which led to an increased interest for BI type NLED theories. In addition, BI type theories have some interesting properties; for example, these theories enjoy the birefringence phenomena, free of the shock waves [84, 85] and electric-magnetic duality [86]. Furthermore, considering the relation between AdS/CFT correspondence and superconductivity phenomenon, it was shown that the BI type theories make a crucial effect on the condensation, the critical temperature, and energy gap of the superconductors [87].

In this paper, we investigate the horizonless magnetic strings in the presence of two kinds of the BI type NLED [88, 89]. One of the elemental motivations for analyzing the horizonless string solutions is that they may be interpreted as cosmic strings.

2. Basic Field Equations

Our goal in this work is to construct a class of four-dimensional solutions to the Einstein equations with negative cosmological constant in the presence of nonlinear electromagnetic source, , which describes a magnetic string. The Euler-Lagrange equations of motion for the metric and the gauge potential may be written as [90] where , , denotes the Maxwell invariant, and the energy-momentum tensor is given by It is notable that these field equations can be obtained from variation of the following action: where the bulk action (first term) is supplemented with a Gibbons-Hawking surface term (second term) whose variation will cancel the extra normal derivative term in deriving the equation of motion. The quantities and denote the trace of the extrinsic curvature and the induced metric for the boundary , respectively.

In this work, we take into account the recently proposed BI type models of NLED [88, 89]. They have been nominated the Exponential form of Nonlinear Electromagnetic Field (ENEF) and the Logarithmic form of Nonlinear Electromagnetic Field (LNEF), in which their Lagrangians are

Here, we want to obtain magnetic solutions. It is well known that the electric field comes from the time component of the vector potential (), while the magnetic field is associated with the angular component (). Hence one expects that a magnetic solution may be written in a metric gauge in which the components and interchange their roles relatively to that present in the Schwarzschild gauge used to describe electric solution. Therefore, we start with a class of the four-dimensional metrics which produces longitudinal magnetic fields along the direction [48]: where is an arbitrary function of coordinate . It is notable that this metric may be obtained from the horizon flat Schwarzschild-like metric: with the following local transformation: Since the mentioned transformation is not a global mapping and metric (7) can be locally mapped to metric (6), one can find that both (6) and (7) do not describe a unique spacetime. Using the nonlinear Maxwell equation (2) with the metric (6), one can obtain with the following solutions: where the prime denotes differentiation with respect to , the parameter is an integration constant, which satisfies [91, 92], and . It is worthwhile to note that in order to have a real electromagnetic field, we should consider , where

Here, we use the orthonormal contravariant (hatted) basis vectors to study the effect of nonlinearity on the energy density. Considering the mentioned diagonal metric in this basis, one should apply and therefore the component of the stress-energy tensor is We plot versus in Figure 1 and find that, for a fixed value of , as nonlinearity parameter increases, the energy density of the spacetime decreases and therefore, in order to reduce the concentration volume of the energy density, we should increase the nonlinearity parameter.

fig1
Figure 1: versus for , , and (solid line), (bold line), and (dashed line). “ENEF branch (a) and LNEF branch (b).”

Now, we should obtain the metric function . One can take into account (6) and (10) in the gravitational field equation (1) to obtain its nonzero components as where and After some calculations one can show that these equations have the following solutions: where is the integration constant which is related to mass parameter, is equal to and for ENEF and LNEF branches, respectively, and where one may calculate these integrations. We should note that the obtained solutions are the same as asymptotically anti-de Sitter magnetic solution of Einstein-Maxwell gravity [49], asymptotically (large values of radial coordinate). In addition, one may expect to recover the solution of [49] for .

Taking into account the metric (6), it is clearly desirable to have an examination on the geometric structure of the solutions. The first step is investigation of the spacetime curvature. It is easy to show that the Kretschmann scalar is Numerical calculations show that the Kretschmann is finite for nonzero . Furthermore, we can show that where and are different functions of metric parameters , , , and . Equation (19) confirms that the asymptotic behavior of the solutions is adS. In addition, one may take into account (18) to think about the existence of a curvature singularity located at and therefore conclude that there are magnetically charged black hole solutions. Since , one concludes that, for charged solutions with finite , the spacetime never achieves . In addition, we should obtain the zeroes of the function . Considering the largest positive real root of by (suppose ; for , the metric function is positive definite which we are not interested in), one can find that the function is negative for . We should note that and are related by , and therefore negativity of (which occurs for ) leads to negativity of and hence the signature of the metric changes from to . This indicates that we could not extend the spacetime from to . In order to get rid of this incorrect extension, one may introduce a new radial coordinate in the following form: Considering this suitable coordinate transformation, the electromagnetic field can be written as where and . Moreover, the metric (6) in the new coordinate is with , and is now given as where Numerical calculations show that not only Kretschmann scalar but also other curvature invariants are finite in the range () and therefore the mentioned spacetime has no curvature singularity and no horizon. It is notable that the above-mentioned magnetic solutions differ from the electric solutions and the properties of electric and magnetic solutions are distinct. For example, the electric solutions lead to black objects interpretation, while the magnetic solutions do not.

In spite of the fact that the obtained magnetic solutions have no essential singularity, one can show that and so, when goes to zero, the limit of the ratio “circumference/radius” is not . This indicates that there is a conic singularity located at . In order to remove the conic singularity, one can identify the angular coordinate with the period where the conical singularity has a deficit angle . Expanding the metric function for , one can show that and therefore is given by It is easy to show that the near origin metric can be written as Following the Vilenkin procedure [93], one can identify the near origin metric (28) with a cosmic string and interpret as the mass per unit length of the string [93].

Here, we are in a position to investigate the effect of nonlinearity parameter on the deficit angle . At first, we should note that is a smooth real function for >, where Second, it is interesting to note that the minimum and maximum values of the deficit angle are which means that increasing the nonlinearity parameter leads to decreasing the deficit angle (see Figure 2 for more clarifications).

fig2
Figure 2: The deficit angle versus for , , and . “ENEF branch (a) and LNEF branch (b).”

3. Spinning Magnetic String

In this section, we apply a local rotation boost to the static metric (22) to obtain rotating spacetime solutions. In -dimensional spacetime the rotation group is , and so one can find that there is only one independent rotation parameter. In order to apply rotation, one may use the following local transformation in the plane: where is a rotation parameter. Taking into account the static metric (20) and applying (31), we can obtain where the metric function is the same as that in (23). According to the mentioned transformation, one can find that, in spite of the static case, does not vanish for rotating solutions. Straightforward calculations show that the nonvanishing components of the electromagnetic fields are

Considering (22), (31), and (32), one may think that there is a one-to-one correspondence between static and rotating spacetimes and so they are the same. But this statement is not correct. It is worthwhile to mention that the coordinate is periodic and therefore (31) is not a proper coordinate transformation on the entire manifold. In other words, the metrics (22) and (32) can be locally mapped into each other but not globally, and so (31) generates a new metric (for some details about this local transformation see, e.g., [94]).

In order to finalize this section, we should discuss the conserved quantities of the magnetic string. Using the counterterm method [9598] and following the procedure of magnetic solutions papers [7, 8, 2949], one can find that the mass and angular momentum per unit length of the string can be written as Equation (35) shows that considering leads to vanishing angular momentum and it confirms that is the rotational parameter of the spacetime. In addition, it is interesting to calculate the electric charge of the solutions. Using Gauss’s law and calculating the flux of the electric field at infinity, we find that the electric charge per unit length can be given by We should note that the electric charge may be originated from the electric field. Since, for rotating solutions, besides the magnetic field along the coordinate, there is also a radial electric field (see (33)), one may expect to obtain an electric charge which is related to the rotating parameter.

4. Magnetic Brane Solutions

Here, we start with a class of the ()-dimensional metrics to obtain magnetic brane solutions with the following ansatz: where is the Euclidean metric on the -dimensional submanifold. Using the nonlinear Maxwell equation (2) with the metric (37), we find that the nonzero components of Maxwell field are where and . It is notable that considering the real electromagnetic field leads to , where

Now, we are in a position to obtain the metric function . Considering (37) with (38), we find that the solution of the gravitational field equation (1) is where It is easy to show that the Kretschmann scalar diverges when and is finite for . Following the same method, we find that one could not extend the spacetime from to in which is the largest positive real root of . Therefore, we can use radial coordinate transformation (20) to obtain a real well-defined spacetime for . Here, we leave details for reasons of economy.

Final step is generalization of static magnetic branes to spinning ones. We know that the rotation group in dimensions is and hence the maximum number of independent rotation parameters is integer part of . Generalization of static solutions to the spinning case with rotation parameters leads to the following metric: where ;   is the Euclidean metric on the -dimensional submanifold with volume . The nonvanishing components of electromagnetic field tensor and the metric function are, respectively, where ,    , and the function is

Following the known counterterm procedure and Gauss’s law, it is easy to calculate the conserved quantities of the magnetic brane solutions. Straightforward calculations show that the mass, angular momentum, and electric charge per unit volume of the magnetic branes may be written as

We should note that the electric charge is proportional to the rotation parameter and is zero for the case of static magnetic branes. This is due the fact that radial electric field vanishes for the static solutions.

5. Conclusions

At the first step, we introduced a class of static magnetic string solutions in Einstein gravity in the presence of negative cosmological constant with two types of NLED. In order to have real solutions, we obtained a lower limit for the radial coordinate, . Furthermore, we nominated the largest real root of the metric function as and, in order to get rid of signature changing, we introduced a new radial coordinate .

Calculations of geometric quantities showed that although these solutions do not have curvature singularity, there is a conical singularity at with a deficit angle , where one can interpret as the mass per unit length of the string. Moreover, we found that, unlike the power Maxwell invariant solutions [4547], the nonlinearity does not have any effect on the asymptotic behavior of the solutions and, in other words, obtained solutions are asymptotically adS.

In addition, we investigated the effects of nonlinearity parameter on the energy density and deficit angle, separately, and found that when one increases the nonlinearity parameter, the concentration volume of the energy density and the deficit angle reduce.

Using a suitable local transformation, we added an angular momentum to the spacetime and found that for rotating solutions there is an electric field in addition to the magnetic one.

Next, we used the counterterm method and Gauss’s law to obtain conserved quantities and electric charge, respectively. It is interesting to note that these quantities depend on the rotation parameter and the static string has no net electric charge.

At the final step, we studied magnetic solutions in higher dimensions. We generalized static magnetic branes to spinning ones and obtained consistent electromagnetic field as well as metric function. Moreover, we obtained conserved quantities of the magnetic branes and found that the electric charge vanishes for the static magnetic branes. In addition, we found that, for , the conserved quantities of the magnetic branes reduce to those of magnetic string, as we expected.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks the anonymous referees for useful criticism and comments which permitted to improve this paper. It is a pleasure to thank C. Corda for useful discussions on NLED. The author wishes to thank Shiraz University Research Council. This work has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM), Iran.

References

  1. H. B. Nielsen and P. Olesen, “Vortex-line models for dual strings,” Nuclear Physics B, vol. 61, pp. 45–61, 1973. View at Publisher · View at Google Scholar
  2. T. Kibble, “Topology of cosmic domains and strings,” Journal of Physics A: Mathematical and General, vol. 9, p. 1378, 1976. View at Publisher · View at Google Scholar
  3. M. Majumdar and A. C. Davis, “Cosmological creation of D-branes and anti-D-branes,” Journal of High Energy Physics, vol. 2002, p. 56, 2002. View at Publisher · View at Google Scholar
  4. S. Sarangi and S. H. H. Tye, “Cosmic string production towards the end of brane inflation,” Physics Letters B, vol. 536, no. 3-4, pp. 185–192, 2002. View at Publisher · View at Google Scholar
  5. D. H. Lyth and A. Riotto, “Particle physics models of inflation and the cosmological density perturbation,” Physics Reports, vol. 314, no. 1-2, pp. 1–146, 1999. View at Publisher · View at Google Scholar
  6. R. Jeannerot, J. Rocher, and M. Sakellariadou, “How generic is cosmic string formation in supersymmetric grand unified theories,” Physical Review D, vol. 68, Article ID 103514, pp. 103–514, 2003. View at Publisher · View at Google Scholar
  7. E. Witten, “Superconducting strings,” Nuclear Physics B, vol. 249, no. 4, pp. 557–592, 1985. View at Publisher · View at Google Scholar
  8. P. Peter, “Spontaneous current generation in cosmic strings,” Physical Review D, vol. 49, p. 5052, 1994. View at Publisher · View at Google Scholar
  9. I. Moss and S. Poletti, “The gravitational field of a superconducting cosmic string,” Physics Letters B, vol. 199, no. 1, pp. 34–36, 1987. View at Publisher · View at Google Scholar
  10. A. A. Sen, “Superconducting cosmic string in Brans-Dicke theory,” Physical Review D, vol. 60, Article ID 067501, 1999. View at Publisher · View at Google Scholar
  11. C. N. Ferreira, M. E. X. Guimarães, and J. A. Helayël-Neto, “Current-carrying cosmic strings in scalar-tensor gravities,” Nuclear Physics B, vol. 581, no. 1-2, pp. 165–178, 2000. View at Google Scholar · View at Scopus
  12. M. H. Dehghani, “Magnetic strings in dilaton gravity,” Physical Review D, vol. 71, Article ID 064010, 2005. View at Publisher · View at Google Scholar
  13. A. Sheykhi, “Magnetic dilaton strings in anti-de Sitter spaces,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 672, no. 2, pp. 101–105, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. L. V. Zadorozhna, B. I. Hnatyk, and Y. A. Sitenko, “Magnetic field of cosmic strings in the early universe,” Ukrainian Journal of Physics, vol. 58, pp. 398–402, 2013. View at Google Scholar
  15. R. H. Brandenberger, A.-C. Davis, A. M. Matheson, and M. Trodden, “Superconducting cosmic strings and primordial magnetic fields,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 293, no. 3-4, pp. 287–293, 1992. View at Google Scholar · View at Scopus
  16. M. Giovannini, “Magnetic fields, strings and cosmology,” Lecture Notes in Physics, vol. 737, pp. 863–939, 2008. View at Publisher · View at Google Scholar · View at Scopus
  17. L. Del Debbio, A. Di Giacomo, and Y. A. Simonov, “Field-strength correlators in SU(2) gauge theory,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 332, no. 1-2, pp. 111–117, 1994. View at Google Scholar · View at Scopus
  18. M. Rueter and H. Guenter Dosch, “SU(3) flux tubes in a model of the stochastic vacuum,” Zeitschrift für Physik C Particles and Fields, vol. 66, no. 1-2, pp. 245–252, 1995. View at Publisher · View at Google Scholar · View at Scopus
  19. H. G. Dosch, O. Nachtmann, and M. Rueter, arXiv:hep-ph/9503386.
  20. A. Gorsky and V. Zakharov, “Magnetic strings in QCD as non-Abelian vortices,” Physical Review D, Particles, Fields, Gravitation and Cosmology, vol. 77, no. 4, Article ID 045017, 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. Y. A. Simonov and J. A. Tjon, “Magnetic string contribution to hadron dynamics in QCD,” Physical Review D, Particles, Fields, Gravitation and Cosmology, vol. 62, no. 9, Article ID 094511, pp. 1–7, 2000. View at Google Scholar · View at Scopus
  22. M. A. Giorgi, “Recent results on CP violation in B decay,” International Journal of Modern Physics A, vol. 20, no. 22, pp. 5069–5079, 2005. View at Publisher · View at Google Scholar · View at Scopus
  23. Y. A. Sitenko, “The Aharonov-Bohm effect and the inducing of vacuum charge by a singular magnetic string,” Nuclear Physics B, vol. 372, no. 3, pp. 622–634, 1992. View at Google Scholar · View at Scopus
  24. M. Bordag and S. Voropaev, “Bound states of an electron in the field of the magnetic string,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 333, no. 1-2, pp. 238–244, 1994. View at Google Scholar · View at Scopus
  25. E. L. Nagaev, “Large nonzero-moment magnetic strings in antiferromagnetic crystals of the manganite type,” Physical Review B, Condensed Matter and Materials Physics, vol. 66, no. 10, Article ID 104431, pp. 1044311–1044317, 2002. View at Google Scholar · View at Scopus
  26. M. G. Ryskin, “Polarizations in inclusive processes,” Soviet Journal of Nuclear Physics, vol. 48, pp. 708–712, 1988. View at Google Scholar
  27. S. B. Nurushev and M. G. Ryskin, “Experimental data on the single-spin asymmetry and their interpretations by the chromomagnetic string model,” Physics of Atomic Nuclei, vol. 69, no. 1, pp. 133–141, 2006. View at Publisher · View at Google Scholar · View at Scopus
  28. M. N. Chernodub and V. I. Zakharov, arXiv:hep-ph/0702245.
  29. W. B. Bonnor, “Static magnetic fields in general relativity,” Proceedings of the Physical Society A, vol. 67, no. 3, article 305, pp. 225–232, 1954. View at Publisher · View at Google Scholar · View at Scopus
  30. M. A. Melvin, “Pure magnetic and electric geons,” Physics Letters, vol. 8, no. 1, pp. 65–68, 1964. View at Google Scholar · View at Scopus
  31. A. Vilenkin, “Gravitational field of vacuum domain walls and strings,” Physical Review D, vol. 23, no. 4, pp. 852–857, 1981. View at Publisher · View at Google Scholar · View at Scopus
  32. W. A. Hiscock, “Exact gravitational field of a string,” Physical Review D, vol. 31, no. 12, pp. 3288–3290, 1985. View at Publisher · View at Google Scholar · View at Scopus
  33. M. Aryal, L. H. Ford, and A. Vilenkin, “Cosmic strings and black holes,” Physical Review D, vol. 34, no. 8, pp. 2263–2266, 1986. View at Publisher · View at Google Scholar · View at Scopus
  34. B. Linet, “A vortex-line model for infinite straight cosmic strings,” Physics Letters A, vol. 124, no. 4-5, pp. 240–242, 1987. View at Publisher · View at Google Scholar
  35. D. Harari and P. Sikivie, “Gravitational field of a global string,” Physical Review D, vol. 37, no. 12, pp. 3438–3440, 1988. View at Publisher · View at Google Scholar · View at Scopus
  36. A. D. Cohen and D. B. Kaplan, “The exact metric about global cosmic strings,” Physics Letters B, vol. 215, p. 65, 1988. View at Publisher · View at Google Scholar
  37. R. Gregory, “Global string singularities,” Physics Letters B, vol. 215, no. 4, pp. 663–668, 1988. View at Publisher · View at Google Scholar
  38. A. Banerjee, N. Banerjee, and A. A. Sen, “Static and nonstatic global string,” Physical Review D, Particles, Fields, Gravitation and Cosmology, vol. 53, no. 10, pp. 5508–5512, 1996. View at Google Scholar · View at Scopus
  39. M. H. Dehghani and T. Jalali, “Abelian Higgs hair for a static charged black string,” Physical Review D, vol. 66, no. 12, Article ID 124014, 2002. View at Publisher · View at Google Scholar · View at Scopus
  40. M. H. Dehghani, “Horizonless rotating solutions in (n+1)-dimensional Einstein-Maxwell gravity,” Physical Review D, vol. 69, Article ID 044024, 2004. View at Publisher · View at Google Scholar
  41. M. H. Dehghani and A. Khodam-Mohammadi, “Hairy rotating black string in the Einstein-Maxwell-Higgs system,” Canadian Journal of Physics, vol. 83, no. 3, pp. 229–242, 2005. View at Publisher · View at Google Scholar · View at Scopus
  42. M. H. Dehghani, A. Sheykhi, and S. H. Hendi, “Magnetic strings in Einstein-Born-Infeld-dilaton gravity,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 659, no. 3, pp. 476–482, 2008. View at Publisher · View at Google Scholar · View at Scopus
  43. M. H. Dehghani and S. H. Hendi, “Wormhole solutions in Gauss-Bonnet-Born-Infeld gravity,” General Relativity and Gravitation, vol. 41, no. 8, pp. 1853–1863, 2009. View at Publisher · View at Google Scholar · View at Scopus
  44. S. H. Hendi, “Higher dimensional charged BTZ-like wormhole,” Progress of Theoretical Physics, vol. 127, no. 5, pp. 907–919, 2012. View at Publisher · View at Google Scholar
  45. S. H. Hendi, “Magnetic branes supported by a nonlinear electromagnetic field,” Classical and Quantum Gravity, vol. 26, no. 22, Article ID 225014, 2009. View at Publisher · View at Google Scholar · View at Scopus
  46. S. H. Hendi, “Magnetic string coupled to nonlinear electromagnetic field,” Physics Letters B, vol. 678, no. 5, pp. 438–443, 2009. View at Publisher · View at Google Scholar
  47. S. H. Hendi, S. Kordestani, and S. N. Doosti Motlagh, “The effects of nonlinear Maxwell source on the magnetic solutions in Einstein-Gauss-Bonnet gravity,” Progress of Theoretical Physics, vol. 124, no. 6, pp. 1067–1082, 2010. View at Publisher · View at Google Scholar · View at Scopus
  48. O. J. C. Dias and J. P. S. Lemos, “Rotating magnetic solution in three dimensional Einstein gravity,” Journal of High Energy Physics, vol. 2002, p. 006, 2002. View at Publisher · View at Google Scholar
  49. Ó. J. C. Dias and J. P. S. Lemos, “Magnetic strings in anti-de Sitter general relativity,” Classical and Quantum Gravity, vol. 19, no. 8, pp. 2265–2276, 2002. View at Publisher · View at Google Scholar · View at Scopus
  50. A. H. Chamseddine and W. A. Sabra, “Calabi-Yau black holes and enhancement of supersymmetry in five dimensions,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 460, no. 1-2, pp. 63–70, 1999. View at Google Scholar · View at Scopus
  51. A. H. Chamseddine and W. A. Sabra, “Magnetic strings in five dimensional gauged supergravity theories,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 477, no. 1-3, pp. 329–334, 2000. View at Publisher · View at Google Scholar · View at Scopus
  52. M. Bordag and K. Kirsten, “Ground state energy of a spinor field in the background of a finite radius flux tube,” Physical Review D, Particles, Fields, Gravitation and Cosmology, vol. 60, no. 10, pp. 1–14, 1999. View at Google Scholar · View at Scopus
  53. M. Scandurra, “Vacuum energy in the presence of a magnetic string with a delta function profile,” Physical Review D, Particles, Fields, Gravitation and Cosmology, vol. 62, no. 8, Article ID 085024, pp. 1–10, 2000. View at Google Scholar · View at Scopus
  54. M. Born and L. Infeld, “On the quantum theory of the electromagnetic,” Proceedings of the Royal Society A, vol. 143, no. 849, pp. 410–437, 1934. View at Publisher · View at Google Scholar
  55. M. Born and L. Infeld, “Foundations of the new field theory,” Proceedings of the Royal Society A, vol. 144, no. 852, pp. 425–451, 1934. View at Publisher · View at Google Scholar
  56. M. H. Dehghani and H. R. R. Sedehi, “Thermodynamics of rotating black branes in (n+1)-dimensional Einstein-Born-Infeld gravity,” Physical Review D, Particles, Fields, Gravitation and Cosmology, vol. 74, no. 12, Article ID 124018, 2006. View at Publisher · View at Google Scholar · View at Scopus
  57. D. L. Wiltshire, “Black holes in string-generated gravity models,” Physical Review D, vol. 38, no. 8, pp. 2445–2456, 1988. View at Publisher · View at Google Scholar · View at Scopus
  58. M. Aiello, R. Ferraro, and G. Giribet, “Exact solutions of Lovelock-Born-Infeld black holes,” Physical Review D, Particles, Fields, Gravitation and Cosmology, vol. 70, no. 10, Article ID 104014, 2004. View at Publisher · View at Google Scholar · View at Scopus
  59. S. H. Hendi, “Rotating black branes in Brans-Dicke-Born-Infeld theory,” Journal of Mathematical Physics, vol. 49, no. 8, Article ID 082501, 15 pages, 2008. View at Publisher · View at Google Scholar
  60. J. Plebansky, Lectures on Non-Linear Electrodynamics, Nordita, Copenhagen, Denmark, 1968.
  61. D. H. Delphenich, Nonlinear electrodynamics and QED, arXiv:hep-th/0309108.
  62. D. H. Delphenich, Nonlinear optical analogies in quantum electrodynamics, arXiv:hep-th/0610088.
  63. J. Schwinger, “On gauge invariance and vacuum polarization,” Physical Review, vol. 82, no. 5, pp. 664–679, 1951. View at Publisher · View at Google Scholar · View at Scopus
  64. W. Heisenberg and H. Euler, “Folgerungen aus der diracschen theorie des positrons,” Zeitschrift für Physik, vol. 98, no. 11-12, pp. 714–732, 1936, Translation by: W. Korolevski and H. Kleinert, “Consequences of Dirac's Theory of the Positron”. View at Publisher · View at Google Scholar
  65. H. Yajima and T. Tamaki, “Black hole solutions in Euler-Heisenberg theory,” Physical Review D, vol. 63, no. 6, Article ID 064007, 2001. View at Publisher · View at Google Scholar · View at Scopus
  66. E. S. Fradkin and A. A. Tseytlin, “Non-linear electrodynamics from quantized strings,” Physics Letters B, vol. 163, no. 1–4, pp. 123–130, 1985. View at Google Scholar · View at Scopus
  67. J. Ambjorn, Y. M. Makeenko, J. Nishimura, and R. J. Szabo, “Nonperturbative dynamics of noncommutative gauge theory,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 480, no. 3-4, pp. 399–408, 2000. View at Publisher · View at Google Scholar · View at Scopus
  68. N. Seiberg and E. Witten, “String theory and noncommutative geometry,” Journal of High Energy Physics, vol. 3, no. 9, 1999. View at Google Scholar · View at Scopus
  69. V. A. De Lorenci and M. A. Souza, “Electromagnetic wave propagation inside a material medium: an effective geometry interpretation,” Physics Letters B: Nuclear, Elementary Particle and High-Energy Physics, vol. 512, no. 3-4, pp. 417–422, 2001. View at Publisher · View at Google Scholar · View at Scopus
  70. V. A. De Lorenci and R. Klippert, “Analogue gravity from electrodynamics in nonlinear media,” Physical Review D, vol. 65, no. 6, Article ID 064027, 2002. View at Publisher · View at Google Scholar · View at Scopus
  71. M. Novello and E. Bittencourt, “Gordon metric revisited,” Physical Review D, vol. 86, Article ID 124024, 2012. View at Publisher · View at Google Scholar
  72. M. Novello, S. P. Bergliaffa, J. Salim et al., “Analogue black holes in flowing dielectrics,” Classical and Quantum Gravity, vol. 20, no. 5, p. 859, 2003. View at Publisher · View at Google Scholar
  73. H. J. M. Cuesta and J. M. Salim, “Non-linear electrodynamics and the gravitational redshift of highly magnetized neutron stars,” Monthly Notices of the Royal Astronomical Society, vol. 354, no. 4, pp. L55–L59, 2004. View at Publisher · View at Google Scholar · View at Scopus
  74. H. J. M. Cuesta and J. M. Salim, “Nonlinear electrodynamics and the surface redshift of pulsars,” Astrophysical Journal Letters, vol. 608, no. 2, pp. 925–929, 2004. View at Publisher · View at Google Scholar · View at Scopus
  75. Z. Bialynicka-Birula and I. Bialynicki-Birula, “Nonlinear effects in quantum electrodynamics. Photon propagation and photon splitting in an external field,” Physical Review D, vol. 2, no. 10, pp. 2341–2345, 1970. View at Publisher · View at Google Scholar · View at Scopus
  76. C. Corda and H. J. M. Cuesta, “Inflation from R2 gravity: a new approach using nonlinear electrodynamics,” Astroparticle Physics, vol. 34, no. 7, pp. 587–590, 2011. View at Publisher · View at Google Scholar · View at Scopus
  77. V. A. De Lorenci, R. Klippert, M. Novello, and J. M. Salim, “Nonlinear electrodynamics and FRW cosmology,” Physical Review D, vol. 65, no. 6, Article ID 063501, 2002. View at Publisher · View at Google Scholar · View at Scopus
  78. C. Corda and H. J. M. Cuesta, “Removing black hole singularities with nonlinear electrodynamics,” Modern Physics Letters A, vol. 25, no. 28, pp. 2423–2429, 2010. View at Publisher · View at Google Scholar · View at Scopus
  79. R. R. Metsaev, M. A. Rahmanov, and A. A. Tseytlin, “The born-infeld action as the effective action in the open superstring theory,” Physics Letters B, vol. 193, no. 2-3, pp. 207–212, 1987. View at Google Scholar · View at Scopus
  80. E. Bergshoeff, E. Sezgin, C. N. Pope, and P. K. Townsend, “The Born-Infeld action from conformal invariance of the open superstring,” Physics Letters B, vol. 188, no. 1, pp. 70–74, 1987. View at Google Scholar · View at Scopus
  81. Y. Kats, L. Motl, and M. Padi, “Higher-order corrections to mass-charge relation of extremal black holes,” Journal of High Energy Physics, vol. 2007, no. 12, article 068, 2007. View at Publisher · View at Google Scholar · View at Scopus
  82. R.-G. Cai, Z.-Y. Nie, and Y.-W. Sun, “Shear viscosity from effective couplings of gravitons,” Physical Review D, Particles, Fields, Gravitation and Cosmology, vol. 78, no. 12, Article ID 126007, 2008. View at Publisher · View at Google Scholar · View at Scopus
  83. D. Anninos and G. Pastras, “Thermodynamics of the Maxwell-gauss-bonnet anti-de Sitter black hole with higher derivative gauge corrections,” Journal of High Energy Physics, vol. 2009, no. 7, article 030, 2009. View at Publisher · View at Google Scholar · View at Scopus
  84. G. Boillat, “Nonlinear electrodynamics: lagrangians and equations of motion,” Journal of Mathematical Physics, vol. 11, no. 3, pp. 941–951, 1970. View at Google Scholar · View at Scopus
  85. G. Boillat, “Simple waves in N-dimensional propagation,” Journal of Mathematical Physics, vol. 11, no. 4, pp. 1482–1483, 1970. View at Google Scholar · View at Scopus
  86. G. W. Gibbons and D. A. Rasheed, “Electric-magnetic duality rotations in non-linear electrodynamics,” Nuclear Physics B, vol. 454, no. 1-2, pp. 185–206, 1995. View at Google Scholar · View at Scopus
  87. Z. Zhao, Q. Pan, S. Chen, and J. Jing, “Notes on holographic superconductor models with the nonlinear electrodynamics,” Nuclear Physics B, vol. 871, no. 1, pp. 98–100, 2013. View at Publisher · View at Google Scholar
  88. S. H. Hendi, “Asymptotic charged BTZ black hole solutions,” Journal of High Energy Physics, vol. 2012, p. 065, 2012. View at Publisher · View at Google Scholar
  89. S. H. Hendi, “Asymptotic reissner-nordström black holes,” Annals of Physics, vol. 333, pp. 282–289, 2013. View at Publisher · View at Google Scholar
  90. S. H. Hendi, “Rotating black string with nonlinear source,” Physical Review D, vol. 82, Article ID 064040, 2010. View at Publisher · View at Google Scholar
  91. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 1972.
  92. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the lambert W function,” Advances in Computational Mathematics, vol. 5, no. 4, pp. 329–359, 1996. View at Google Scholar · View at Scopus
  93. A. Vilenkin, “Cosmic strings and domain walls,” Physics Reports, vol. 121, no. 5, pp. 263–315, 1985. View at Google Scholar · View at Scopus
  94. J. Stachel, “Globally stationary but locally static space-times: a gravitational analog of the Aharonov-Bohm effect,” Physical Review D, vol. 26, no. 6, pp. 1281–1290, 1982. View at Publisher · View at Google Scholar · View at Scopus
  95. J. Maldacena, “The large N Limit of superconformal field theories and supergravity,” Advances in Theoretical and Mathematical Physics, vol. 2, no. 2, pp. 231–252, 1998. View at Google Scholar · View at Scopus
  96. E. Witten, “Anti de sitter space and holography,” Advances in Theoretical and Mathematical Physics, vol. 2, no. 2, pp. 253–290, 1998. View at Google Scholar · View at Scopus
  97. P. Kraus, F. Larsen, and R. Siebelink, “The gravitational action in asymptotically AdS and flat space-times,” Nuclear Physics B, vol. 563, no. 1-2, pp. 259–278, 1999. View at Google Scholar · View at Scopus
  98. O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, “Large N field theories, string theory and gravity,” Physics Report, vol. 323, no. 3-4, pp. 183–386, 2000. View at Google Scholar · View at Scopus