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Dynamic Wormhole Spacetimes Coupled to Nonlinear Electrodynamics

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Dynamic Wormhole Spacetimes Coupled to Nonlinear Electrodynamics Aar n V. B. Arellano Facultad de Ciencias, Universidad Aut noma del Estado de M xico, M xico. – PowerPoint PPT presentation

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Title: Dynamic Wormhole Spacetimes Coupled to Nonlinear Electrodynamics


1
Dynamic Wormhole Spacetimes Coupled to Nonlinear
Electrodynamics
  • Aarón V. B. Arellano
  • Facultad de Ciencias, Universidad Autónoma del
    Estado de México, México.
  • Francisco S. N. Lobo
  • Centro de Astronomia e Astrofísica da
    Universidade de Lisboa, Portugal.

2
Abstract
  • We explore the possibility of dynamic wormhole
    geometries, within the context of nonlinear
    electrodynamics. The Einstein field equation,
    imposes a contracting wormhole solution and the
    obedience of the weak energy condition.
    Furthermore, in the presence of an electric
    field, the latter presents a singularity at the
    throat, however, for a pure magnetic field the
    solution is regular. Thus, taking into account
    the principle of finiteness, that a satisfactory
    theory should avoid physical quantities becoming
    infinite, one may rule out evolving wormhole
    solutions, in the presence of an electric field,
    coupled to nonlinear electrodynamics.

3
Nonlinear? Electrodynamics
  • Nonlinear Electrodynamics was supposed to
    represent a model of the classical
    singularity-free theory when the concept of the
    point charge is acceptable.
  • Nowadays, Nonlinear Electrodynamics can be
    considered a branch of research on the
    fundamentals of electrodynamics.
  • Pioneering work on Nonlinear Electrodynamics may
    be traced back to Born and Infeld , where the
    latter outlined a model to remedy the fact that
    the standard picture of a point charged particle
    possesses an infinite self-energy. Thus, the
    born-Infeld model was founded on a principle of
    finiteness, that a satisfactory theory should
    avoid physical quantities becoming infinite.
  • Later, Plebanski extended the examples of
    Nonlinear Electrodynamic Lagrangians , and
    demonstrated that the Born-Infeld theory satisfy
    physically acceptable requirements.
  • Nonlinear Electrodynamics has recently found many
    applications in several branches as effective
    theories at different levels of string/M-theory
    , cosmological models , black holes , and
    in wormhole physics , amongst others.

4
Dynamic Wormhole Geometry
  • Spacetime metric representing a dynamic
    spherically symmetric (31)-dimensional wormhole,
    which is conformally related to the static
    wormhole geometry
  • where ? and b are functions of r, and ??(t) is
    the conformal factor, which is finite and
    positive definite throughout the domain of t. ?
    is the redshift function, and b is denoted the
    form function. We shall also assume that these
    functions satisfy all the conditions required for
    a wormhole solution, namely, ?(r) is finite
    everywhere in order to avoid the presence of
    event horizons b(r)/rlt1, with b(r0)r0 at the
    throat and the flaring out condition
    (b-br)/b20, with b(r0)lt1 at the throat.

5
Setup equations
  • The action of (31)-dimensional general
    relativity coupled to nonlinear electrodynamics
    is
  • where R is the Ricci scalar. L(F) is a
    gauge-invariant electromagnetic Lagrangian,
    depending on a single invariant F given by
    FF??F??/4, where F?? is the electromagnetic
    tensor.
  • The stress-energy tensor
  • where LFdL/dF.
  • Electromagnetic field equations
  • where denotes the Hodge dual.

6
Einstein Field Equations
  • For convenience, we workout the Einstein Field
    Equations in an orthonormal reference frame from
    where we verify that ?0, considering the
    non-trivial case d?/dt?0. So without a
    significant loss of generality, we choose ?0.
    Then the components of the Einstein Tensor are
  • were ?0 and ?0 were used.

7
  • And the components of the stress-energy tensor
    are
  • in an orthonormal reference frame and with ?0
    and ?0.

8
Comments
  • It is also important to point out an interesting
    physical feature of this evolving, and in
    particular, contracting geometry, namely, the
    absence of the energy flux term, Ttr0. One can
    interpret this aspect considering that the
    wormhole material is at rest in the rest frame of
    the wormhole geometry, i.e., an observer at rest
    in this frame is at constant r, ?, ?. The latter
    coordinate system coincides with the rest frame
    of the wormhole material, which can be defined as
    the one in which an observer co-moving with the
    material sees zero energy flux.

9
Results
  • From the stress-energy tensor components and the
    Einstein Field Equations we find
  • Equation that can be solved by separation of
    variables to obtain
  • Where ? is the separation constant, C1 and C2 are
    constants of integration. Note that the form
    function reduces to b(r0)r0 at the throat, and
    b(r0)1-2?2r02lt1 is also verified for ??0.
    Relatively to the conformal function, if C1C2,
    then ? is singular at t0.
  • Defining the dimensionless parameter ??r0 we
    rewrite the form function as

10
Energy Conditions
  • Now we explore the energy conditions, in
    particular, and for its significance, the weak
    energy condition (WEC). Its helpful that the
    stress energy tensor is diagonal then we only
    need to check
  • and using the Einstein Tensor components and the
    solutions for b and ?

11
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12
Electromagnetic Field Equations
  • Taking into account the metric, the
    electromagnetic tensor, compatible with the
    symmetries of the geometry
  • where the nonzero components are the following
    Ftr-FrtE, the electric field, and F??-F??B,
    the magnetic field.
  • Using the Electromagnetic Field Equations we find
    the following set of relations
  • and the following restrictions Ftr-FrtE(t,r),
    F??-F??B(?), LFLF(t,r).

13
Results
  • Thus, one may take CEqeconst., and the magnetic
    field
  • where qe and qm are constants related to the
    electric and magnetic charge, respectively.
  • Considering a nonzero electric field, E?0, we
    obtain

14
Particular Cases
  • B0. If we consider B0 we obtain
  • E0. If we consider E0, we obtain
  • that, together with Bqmsin?, Fqm2/(2?4r4) and
    the solutions for b and ?, give a wormhole
    solution without problems at the throat, with
    finite fields.

15
Conclusions
  • It was found that the Einstein field equation
    imposes a contracting wormhole solution and that
    the weak energy condition be satisfied. It was
    also found that in the presence of an electric
    field, a problematic issue was verified, namely,
    that the latter become singular at the throat.
    However, regular solutions of traversable
    wormholes in the presence of a pure magnetic
    field were found.
  • It is also relevant to emphasize that the
    solutions obtained can be obtained using an
    alternative form of nonlinear electrodynamics,
    denoted the P framework.
  • We remind that we have only considered that the
    gauge-invariant electromagnetic Lagrangian L(F)
    be dependent on a single invariant F.
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