Dynamic Wormhole Spacetimes Coupled to Nonlinear

ElectrodynamicsAarón V. B. Arellano

Facultad de Ciencias, Universidad Autónoma del Estado de México, México.

Francisco S. N. LoboCentro 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

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.

2222

22222 sin

/1 ddr

rrbdr

dtetds r

“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=FF/4, where F is the electromagnetic tensor.

• The stress-energy tensor

where LF=dL/dF.• Electromagnetic field equations

where * denotes the Hodge dual.

,16

4

xdFLR

gS

0,0 ;*

;

FLF F

,FLFFFLgT

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

• 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)=1-22r0

2<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

tt eCeCtrrrrb

21

20

22 2,1

11

2

0

2

rr

rrb

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

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

and the following restrictions: Ftr=-Frt=E(t,r), F=-F=B(), LF=LF(t,r).

,sin,,

/144

2/12rrtCBL

rbr

CEL BF

EF

,

xBxEF trrt

Results

• Thus, one may take CE=qe=const., 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

,sin mqB

22422

2

2/12

22422

323´3´

32

,/132

323´3´,

me

eF

e

me

qqrbrbrbrb

qL

rbrq

qqrbrbrbrbrtE

Particular Cases• B=0. If we consider B=0 we

obtain

• E=0. If we consider E=0, we obtain

that, together with B=qmsin, F=qm

2/(24r4) and the solutions for b and , give a wormhole solution without problems at the throat, with finite fields.

brbrq

L

rbr

brbq

Erbrbrb

LE

eF

eF

3´16

,/1

3´16

,/13´

16

2

2

2/1

2

3

22

2

222

2

/3

´81

,3´16

1 dtdrb

Lbrbrq

Lm

F

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.