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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.
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.
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.
Dynamic Wormhole Geometry
• Spacetime metric representing a dynamic spherically symmetric
(3+1)-dimensional wormhole, which is conformally related to the
static wormhole geometry
2

dr
2
2
2 r 
2
2
2
2
2 
ds   t  e
dt 
 r d  sin d 
1  br  / r




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)/r<1, with b(r0)=r0 at the
throat; and the flaring out condition (b-b´r)/b2≥0, with b´(r0)<1 at the
throat.
“Setup” equations
•
The action of (3+1)-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 F=FF/4,
where F is the electromagnetic
tensor.
The stress-energy tensor
•
where LF=dL/dF.
Electromagnetic field equations
*
where denotes the Hodge dual.
S
 R
 4
g
 LF  d x,
16

T  g  LF   F F LF ,

F

LF

;

 0, * F 

;
0
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.
• And the components of the stress-energy tensor are
in an orthonormal reference frame and with =0 and ´=0.
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, T^t^r=0. 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.
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



br   r 1   2 r 2  r0 , t  
•
•
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)=122r02<1 is also verified for ≠0. Relatively to the
conformal function, if C1=C2, then  is singular at
t=0.
Defining the dimensionless parameter =r0 we
rewrite the form function as
2
2
C1et  C2 e t

 r  2  
br   r 1   2    1 
 r0 
 

Energy Conditions
•
Now we explore the energy conditions, in particular, and for it’s significance, the weak energy condition
(WEC). It’s 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 
•
The middle inequality reduces to the null energy condition (NEC) which is obviously complied for arbitrary t
and r. The last inequality reduces to 2r(1+2)≥0 which is always fulfilled. And for the first inequality, we use the
graphical representation with F(r,t)=[b´/r2+3((d/dt)/)2]
Electromagnetic Field Equations
• Taking into account the metric, the electromagnetic tensor,
compatible with the symmetries of the geometry




F  E x          Bx         ,
t
r
r
t




where the nonzero components are the following: Ftr=-Frt=E, the
electric field, and F=-F=B, the magnetic field.
• Using the Electromagnetic Field Equations we find the following
set of relations
ELF 
CE
4 4


,
BL

C
t
,
r

r sin  ,
F
B
1/ 2
2
r 1  b / r 
and the following
LF=LF(t,r).
restrictions:
Ftr=-Frt=E(t,r),
F=-F=B(),
Results
• Thus, one may take CE=qe=const., and the
magnetic field
B   qm sin  ,
where qe and qm are constants related to the
electric and magnetic charge, respectively.
• Considering a nonzero electric field, E≠0, we
obtain
b´r  3b  2 r  b´r  3b 2  4 r 2  32qe qm 2
E t , r  
,
1/ 2
2
32qe r 1  b / r 
LF 
32qe
2
b´r  3b 2 r  b´r  3b 2  4 r 2  32qe qm 2
Particular Cases
• B=0. If we consider B=0 we 2
2
E LF 
obtain
b´r  3b
2
b´r  3b
,
E

,
16 r 3 1  b / r 
16qe r 1  b / r 1/ 2
16qe
LF  2
 r b´r  3b 
2
• E=0. If we consider E=0, we
obtain
1
1
2


LF 

r
b
´
r

3
b
,
L


2
8 2
16qm
that, together with B=qmsin,
F=qm2/(24r4)
and
the
solutions for b and , give a
wormhole solution without
problems at the throat, with
finite fields.
 b´  d / dt  2 
 
 2  3
r


 

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 gaugeinvariant electromagnetic Lagrangian L(F) be dependent on a
single invariant F.