Charged Wormhole Solutions in Einstein-Cartan gravity

Mohammad Reza Mehdizadeh1 ∗ and Amir Hadi Ziaie2†

1Department of Physics, Shahid Bahonar University,P. O. Box 76175,Kerman, Iran2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM),P.O. Box 55134− 441, Maragha, Iran

November 9, 2018

Abstract

Static solutions representing wormhole configurations in Einstein-Cartan theory (ECT) in thepresence of electric charge are obtained. The solutions are described by a constant redshift func-tion with matter content consists of a Weyssenhoff fluid along with an anisotropic matter andenergy momentum tensor (EMT) of electric field which together generalize the anisotropic energymomentum tensor in Einstein-Maxwell theory in order to include the effects of intrinsic angularmomentum (spin) of particles. Assuming equation of state (EoS) as pr = w1ρ and pt = w2ρ wederive exact wormhole solutions satisfying weak and null energy conditions. Depending on thevalue of spin square density at the wormhole throat these solutions can be asymptotically flat, de-sitter or anti de-sitter. Moreover, the electric field of the wormhole configuration could act as anaccelerator for charged particles and it is the value of charge at wormhole throat that determinesthe intensity of attraction or repulsion.

1 Introduction

Wormholes are topological handles which connect two spacetimes of the same universe (as a bridgeor tunnel) or of different universes altogether by a minimal surface called the throat of the wormhole.This surface respects the flare-out condition [1] through which a traveler can freely pass in bothdirections. Such a configuration was first investigated by Flamm [2] and then led Einstein and Rosen[3] to contribute further advances for constructing wormhole geometries. The concept of wormholewas born in the seminal works of Misner and Wheeler [4] and Wheeler [5], in order to present amechanism for having electric or magnetic “charge without charge” by letting the lines of force threadfrom one spatial asymptotic to another (see also [6] for more details). The most amazing featureof a wormhole configuration is its two-way traversability which happens when the throat remainsopen. Therefore, it is important to recognize the possibility of traveling across the wormhole as ashort-cut in spacetime. Unfortunately, a Schwarzschild wormhole does not possess this property andit is non-traversable, even by a photon [7]. This issue was investigated in a pioneering work byMorris and Thorne [8] and subsequently Morris, Thorne and Yurtsever [9] where they introduced astatic spherically symmetric metric and discussed the required conditions for physically meaningfulLorentzian traversable wormholes. However, the possibility for a wormhole to be traversable leadsinevitably to violation of null energy condition (NEC), in other words, the matter yielding this geometryis known as exotic, i.e. its energy density becomes negative which results in violation of NEC [10].The quest for finding the promising candidates of exotic matter is not an easy task and only ina small area the footprints of exotic matter have been recognized, such as the quantum Casimir

∗mehdizadeh.mr@uk.ac.ir†ah.ziaie@riaam.ac.ir

1

arX

iv:1

811.

0336

4v1

[gr

-qc]

8 N

ov 2

018

effect and semiclassical Hawking radiation. However, all classical matter fields respect the standardenergy conditions. By increasing the accuracy of measurements in observational cosmology and thediscovery of cosmic acceleration of the Universe, different cosmological models of dark energy withexotic EoSs have been put forward [11]. These models among which possible traversable wormholegeometries can be built by matter fields with exotic energy momentum [12], those supported by thecosmological constant [13], phantom or quintom-type energy [14], modified Chaplygin gas [15] andinteracting dark sectors [16]. One of the most important challenges in construction wormhole geometriesis the fulfillment of standard energy conditions and instead of considering non-standard fluids, manyattempts have been done toward modifying GR in order to overcome the issue of energy conditionswithin wormhole settings. In this regard, the study of wormhole solutions has recently attracted manypeople in modified theories of gravity e.g., the presence of higher order terms in curvature would allowfor building thin-shell wormholes supported by ordinary matter [17]. A good deal of work along thisline has been done in order to build and study wormhole solutions, without resorting to exotic matter,within the framework of modified gravity theories among which we can quote: wormhole solutions inhigher dimensional Lovelock theories [18], Rastall gravity [19], Brans-Dicke theory [20], f(R) gravity[21], Born-Infeld theory [22], brane scenarios such as the Randall-Sundram brane [23], Einstein-Gauss-Bonnet theory [24], Kaluza-Klein gravity [25], scalar-tensor [26] and f(R,T) [27] gravity theories andwormhole solutions in noncommutative geometry [28].

The Einstein-Cartan theory of spacetime is motivated by the desire to give a simple descriptionof the influence of intrinsic angular momentum of microscopic matter (spin of fermionic particles)within gravitational phenomena. This objective can be achieved by taking the sapcetime as a four-dimensional differential manifold endowed with a metric tensor and a linear connection which in generalis asymmetric. The spacetime torsion tensor is defined as the antisymmetric part of the connectionand is physically generated through the presence of spin of fermionic matter fields. The field equationsof ECT relate certain combinations of the curvature and torsion tensors to the energy-momentumand spin density tensors, respectively. Thus, in ECT, both mass and spin, which are intrinsic andfundamental properties of matter fields would affect the spacetime structure. The essential idea behindECT was advanced by Cartan in early 1923 [29] and further developed by Sciama [30] and Kibble [31]whereas Weyl considered the special case of the Dirac equation in a curved spacetime with torsion [32],see [33] for a beautiful and comprehensive review. Since the advent of ECT various cosmological aswell as astrophysical models have been proposed with the aim of explaining the observed acceleratedexpansion of the Universe and also curing the problem of spacetime singularities which are unavoidablein general relativity GR [34]. Cosmological settings in ECT have been investigated and it is shown thatspacetime torsion may provide a framework by virtue of which the initial singularity of the Universeis replaced by a non-singular bounce [35]. Moreover, within the framework of ECT, the effects of spinhave been investigated in the early Universe [36],[37], inflationary cosmological models [38], emergentUniverse scenario [39], gravitational collapse [40], higher dimensional gravity theories [41] and blackhole physics [42], see also [43] for a careful collection of articles on different aspects of ECT. Recently,static solutions representing traversable wormholes in the context of ECT have been obtained in [44],where the matter sources are taken as two noninteracting scalar fields (one is minimally and theother is non-minimally coupled to gravity) with nonzero potentials, exact static, spherically symmetricwormhole solutions with flat or AdS asymptotic behavior have been obtained. The wormhole solutionssatisfy the NEC and weak energy condition (WEC) with arbitrary throat radius without resortingto exotic matter sources. Moreover, exact wormhole spacetimes with sources in the form of a non-minimally coupled non-phantom scalar field and an electromagnetic field have been found in [45].Work along this line has been also performed in [46] where exact asymptotically flat and anti-de-Sitter spacetimes were obtained which admit traversable wormholes and respect energy conditions.The inclusion of charge within wormhole configurations was considered as a possibility to meet somestability conditions for wormhole geometry [47], though the authors were not concerned with thefulfillment of energy conditions. In the present paper we are interested in finding charged wormhole

2

solutions in ECT for which the supporting matter fields include the a charged spinning fluid togetherwith an anisotropic ordinary matter distribution. We then proceed, in section 2, with introducing thefield equations of ECT with additional part in material components, i.e., the EMT of Maxwell field. Insection 3, assuming that the radial and tangential pressures linearly depend to energy density via twoEoS parameters, we find exact wormhole solutions with zero tidal force satisfying WEC and NEC. Ourconclusion is drawn in section 4.

2 Field equations in Einstein-Cartan theory with a chargedsource

The field equations in ECT are given by [33, 48, 49]

Gµβ ()− Λgµν = κ2 (Tµβ + θµβ) , Qαµβ = −κ2

2

[Σ αµβ +

1

2δαµΣ ρ

βρ −1

2δαβΣ ρ

µρ

], (1)

where κ2 = 8πG/c4 and Λ are the gravitational coupling and cosmological constants, Σµαβ is definedas the spin tensor of matter [33], Qαµβ is the spacetime torsion tensor and Tµβ being the dynamicalEMT [50] which can be obtained in terms of torsion as

θµν =1

κ2

[4QηµηQ

βνβ −

(Qρµε + 2Q ρ

(µε)

)(Qενρ + 2Q ε

(νρ)

)+

1

2gµν

(Qρσε + 2Q(σε)ρ

) (Qεσρ + 2Q(σρ)ε

)− 2gµνQ

ρσρQ

σεσ

]

=1

2κ2[Σ αµα Σ γ

βγ − Σ αγµ Σβγα − Σ αγ

µ Σβαγ

+1

2ΣαγµΣαγβ +

1

4gµβ

(2ΣαγεΣ

αεγ − 2Σ γα γΣαε ε + ΣαγεΣαγε

) ], (2)

where () denotes symmetrization and we have used the second part of (1) in order to omit torsiontensor in favor of spin tensor.

We note that the equation governing the torsion tensor is pure algebraic, thus, the torsion does notpropagate outside the matter distribution as a torsion wave or through any interaction of non-vanishingrange [33] and therefore is only nonzero inside the matter source. Next we proceed to find a suitabledescription for the EMT given in (2) in terms of a spin fluid. Such a fluid can be described by the socalled Weyssenhoff fluid considered as a continuous macroscopic medium whose microscopic elementsare composed of fermionic particles with intrinsic angular momentum. This model which generalizesthe EMT of ordinary matter in GR to include non-vanishing spin was first studied by Weyssenhoff andRaabe [51] and extended by Obukhov and Korotky in order to construct cosmological models basedon the EC theory [52]. In order to consider wormhole solutions in the framework of ECT, we use aclassical description of spin as postulated by Weyssenhoff given by [51],[52],

Σ αµν = sµνuα, sµνuµ = 0, (3)

where uα is the four-velocity of the fluid element and sµν = −sνµ is a second-rank antisymmetric tensordefined as the spin density tensor. The spatial components of spin density tensor include the 3-vector1

(s23, s13, s12) which coincides in the rest frame with the spatial spin density of the matter element. Therest of spacetime components (s01, s02, s03) are assumed to be zero in the rest frame of fluid element,

1We use the convention (t, r, θ, φ) = (0, 1, 2, 3) for labeling the coordinates.

3

which can be covariantly formulated as a constraint given in the second part of (3). This constrainton the spin density tensor is usually called the Frenkel condition which requires the intrinsic spin ofmatter to be spacelike in the rest frame of the fluid2. As we are concerned with a charged spinningfluid, we consider a dynamical EMT which includes three parts i.e., the usual perfect fluid part, TPf

µβ ,

an intrinsic spin part Tsµβ and contributions due to electromagnetic tensors, TEM

µβ . We therefore have[36, 51, 52]

Tαβ = TPfαβ + Ts

αβ + TEMαβ = (ρ+ pt)uαuβ + ptgαβ + (pr − pt)vαvβ

+ u(αsµβ)uνKρµνuρ + uρKµρσuσu(αsβ)µ −

1

2u(αQβ)µνs

µν +1

2Qνµ(αs

µβ)u

ν ,

+1

4π

[F νµ Fβν −

gµβ4

FανFαν], (4)

where the quantities ρ, pr and pt are the usual energy density, radial and tangential pressures of thefluid respectively, vµ is a unit spacelike vector field in radial direction and the quantity Kµνα is thecontorsion tensor defined as

Kµαβ = Qµαβ + Q µαβ + Q µ

βα . (5)

The electromagnetic field tensorFµν = ∂µAν − ∂νAµ, (6)

with Aµ being the electromagnetic potential is an antisymmetric tensor field satisfying the Maxwellfield equations3

∂µFαν + ∂νFµα + ∂αFνµ = 0, ∂µ

[(−g)

12Fµν

]= (−g)

12 Jν , (7)

where g is the metric determinant and Jν is the current four-vector defined via the proper chargedensity of the distribution, as

Jν = σ(r)uν . (8)

3 Wormhole Solutions

We consider the general static and spherically symmetric line element representing a wormhole givenby (we set the units so that κ = 1)

ds2 = −e2φ(r)dt2 +

(1− b(r)

r

)−1dr2 + r2dΩ2, (9)

where dΩ2 = dθ2 + sin2 θdΦ2 is the standard line element on a unit two-sphere, φ(r) being the redshiftfunction and b(r) is the wormhole shape function. The radial coordinate has a range that increasesfrom a minimum value at r0 (wormhole’s throat), to spatial infinity. Conditions on φ(r) and b(r) underwhich wormholes are traversable were discussed completely for the first time in [8]. The shape functionmust satisfy the flare-out condition at the throat, i.e., we must have b′(r0) < 1 and b(r) < r for r > r0in the whole spacetime. Our aim in the present model is to determine the shape function b(r) andthe redshift function φ(r) in order to construct physically reasonable wormhole geometries. Following[56, 57] we suppose that the spins of the individual charged particles are all aligned in the radial

2The Weyssenhoff spin fluid has been also described by means of applying the Papapetrou-Nomura-Shirafuji-Hayashimethod of multiple expansion in the Riemann-Cartan spacetime [53] to the conservation law for the spin density (whichresults from the Bianchi identities in the EC gravity [54, 55]) in the point-particle approximation.

3As it has been already pointed out by [57, 58], the electromagnetic field does not couple with the spacetime torsionthus, the Maxwell field equations (7) are written in their usual way. If the electromagnetic field couples to torsion thegauge invariance is broken.

4

direction. Therefore from (3) we obtain s23 = −s32 = S as the only independent non-zero componentof the spin density tensor. We then find the intrinsic angular momentum tensor of matter as

Σ 023 = −Σ 0

32 = S(g00)−12 . (10)

In the present model the four-vector potential is given as

Aµ = [Ψ(r), 0, 0, 0] , (11)

from which the electromagnetic field tensor is obtained as

F01 = −F10 = Ψ′. (12)

The field equations (1) and (7) then read

ρ(r) =b′

r+

S2

4− E2

8π− Λ, (13)

pr(r) =2φ′

r

[1− b

r

]− b

r3+

E2

8π+

S2

4+ Λ, (14)

pt(r) =

[1− b

r

] (φ′′ + φ′2

)− φ′

2r2[rb′ + b− 2r] +

1

2r2

[b

r− b′

]+

S2

4− E2

8π+ Λ, (15)

σ(r) =1

4πr2

(1− b

r

) 12

(rE′ + 2E), (16)

where the electric field strength (the component F01 of the electromagnetic field tensor Fij) is definedas [59]

E(r) = −Ψ′exp(−φ(r))

[1− b

r

] 12

, (17)

and we have used uµ = [exp(−φ(r)), 0, 0, 0] and vµ =[0,√

1− b(r)/r, 0, 0]

as the timelike and spacelike

vector fields. Equation (16) can also be expressed in the following form as

E(r) =4π

r2

ˆ r

r0

x2σ(x)[1− b(x)

x

] 12

dx =Q(r)

r2, (18)

where Q(r) is the total charge of the sphere of radius r. The conservation equation

− φ′[ρ(r) + pr(r)]− p′r +2

r[pt(r)− pr(r)] = 0, (19)

leaves us with the following relation given as

S

2[S′ + Sφ′] +

1

4π

[EE′ +

2

rE2

]= 0, (20)

or equivalently

S

2[S′ + Sφ′] +

1

4πr4QQ′ = 0. (21)

Equations (13)-(15) along with conservation equation (21) constitute a system of differential equationsby taking into account two EoSs, namely, pr = pr(ρ) and pt = pt(ρ). Let us take the radial and tan-gential components of the fluid pressure to be dependent linearly to energy density by the (barotropic)

5

EoS given as pr(r) = w1ρ(r) and pt(r) = w2ρ(r). Therefore, the anisotropic fluid behaves differentlyin radial and tangential directions depending on EoS parameters. Such an EoS has been widely usedin literature for the study wormhole configurations see, e.g. [60]. We then get

b

r3+

w1

r2b′ − 2

r

[1− b

r

]φ′ − (w1 + 1)

8πr4Q2 +

1

4(w1 − 1)S2 − Λ(w1 + 1) = 0, (22)

−[1− b

r

] (φ′′ + (φ′)2

)+

φ′

2r2(rb′ − 2r + b) +

b′

2r2(2w2 + 1)− b

2r3

+Q2

8πr4(1− w2) +

S2

4(w2 − 1)− Λ

8πr4(w2 + 1) = 0, (23)

S

2[S′ + Sφ′] +

1

4πr4QQ′ = 0. (24)

The above system of differential equations is closed once we specify one of the four unknowns. As weknow, one of the features of a traversable wormhole is that the tidal gravitational forces as experiencedby a passenger must be reasonably small. We then proceed with a constant redshift function, i.e.,φ(r) = φ0 implying zero tidal gravitational force as experienced by a hypothetical traveler. We thenfind that the system admits an exact solution given by

Q(r) =±1

w1 + 3w2

[3w1(w1 + 3w2)

(C2r

43 − C1r

2(w1+w2)w1

)] 12

, (25)

S2(r) =3C2w1

4π(w1 + 3w2)r83

− 3C1w1(w1 + w2)

2π(w1 − w2)(w1 + 3w2)r

2w1

(w2−w1) − 2Λ, (26)

b(r) =1

2Λr3 +

9C2w1

16π(w1 + 3w2)r

13 +

3C1w21(w2 − 1)

4π(w1 + 3w2)(w1 + 2w2 + 1)(w1 − w2)r

w1+2w2w1 , (27)

where C1 and C2 are integration constants. The first one can be determined subject to the conditionthat the shape function has to satisfy at the wormhole throat which is b(r0) = r0. This gives

C1 =2(w1 + 2w2 + 1)(w2 − w1)

[98C2w1r

430 + πr20(w1 + 3w2)(Λr20 − 2)

]3w2

1(w2 − 1)r2(w1+w2)

w10

. (28)

The second constant is set so that the charge function be equal to a real and finite value at wormholethroat, i.e., Q(r0) = Q0. We then get

C2 =8πr20

(2− Λr20

)w21 −

[πr20(w2 + 1)(Λr20 − 2)− 1

2Q20(w2 − 1)

]w1 + 2πr20(Λr20 − 2)

(w2 + 1

2

)w2

3w1(3w1 − 2w2 − 1)r430

.

(29)Substituting for C1 and C2 back into the solution we arrive at the following relations for charge functionand square of spin density

Q(r) =

[4A(w1,w2,Q0,Λ, r0)

(r

r0

) 43

+ B(w1,w2,Q0,Λ, r0)(w1 + 2w2 + 1)(w1 − w2)

(r

r0

) 2(w1+w2)w1

] 12

,

(30)

S2(r) =A(w1,w2,Q0,Λ, r0)

πr40

(r0r

) 83

+(w1 + 2w2 + 1)(w1 + w2)B(w1,w2,Q0,Λ, r0)

2πr40

(r

r0

) 2(w2−w1)w1

− 2Λ,

(31)

6

where

A(w1,w2,Q0,Λ, r0) =Q2

0w1(w2 − 1)− 2πr20(Λr20 − 2) (w1 − w2) (w1 + 2w2 + 1)

(w1 + 3w2)(3w1 − 2w2 − 1), (32)

B(w1,w2,Q0,Λ, r0) =(8πr20(Λr20 − 2) + 3Q2

0)

(3w1 − 2w2 − 1)(w1 + 3w2). (33)

The shape function, its derivative and the radial component of the metric (9) also read

b(r) =1

2Λr3 +

3A(w1,w2,Q0,Λ, r0)

4πr0

(r

r0

) 13

− w1(w2 − 1)B(w1,w2,Q0,Λ, r0)

4πr0

(r

r0

) w1+2w2w1

,

(34)

b′(r) =3

2Λr2 +

A(w1,w2,Q0,Λ, r0)

4πr20

(r0r

) 23 − (w2 − 1)(w1 + 2w2)B(w1,w2,Q0,Λ, r0)

4πr20

(r

r0

) 2w2w1

,

(35)

1− b(r)

r= 1− 1

2Λr2 − 3A(w1,w2,Q0,Λ, r0)

4πr20

(r0r

) 23

+w1(w2 − 1)B(w1,w2,Q0,Λ, r0)

4πr20

(r

r0

) 2w2w1

.

(36)

From (35), the flare-out condition (b′(r0) < 1) gives

πr20[8Λr20(w1 − w2) + 2w1 + 4w2 − 6

]− Q2

0(w2 − 1)

πr20(6w1 − 4w2 − 2)< 1. (37)

We also take the value of square of spin density at the wormhole throat to be S2(r0) = S20 whence wehave

Λ =−πr20

[S20r

20(3w1 − 2w2 − 1) + 4w1 + 8w2 + 4

]+ Q2

0(3w1 + 2w2 + 1)

4πr20(w1 − 2w2 − 1). (38)

The solution (36) is asymptotically flat for Λ = 0 and w1 and w2 are of opposite sign. For this casethe square of spin density at throat is found as

S20 =(3w1 + 2w2 + 1)Q2

0 − 8πr20(w1 + 2w2 + 1)

2πr40(3w1 − 2w2 − 1). (39)

We note that equation (38) for vanishing cosmological constant can be also solved for r0. We thenobserve that the throat radius for a specified EoS depends on the value of square of spin density andcharge at the wormhole throat. Next we proceed to find the energy density, radial and tangentialpressures which are obtained as,

ρ(r) =3Q2

0 + 8πr20(Λr20 − 2)

4πr40(3w1 − 2w2 − 1)

(r

r0

) 2(w2−w1)w1

, (40)

pr(r) =w1

[3Q2

0 + 8πr20(Λr20 − 2)]

4πr40(3w1 − 2w2 − 1)

(r

r0

) 2(w2−w1)w1

, (41)

pt(r) =w2

[3Q2

0 + 8πr20(Λr20 − 2)]

4πr40(3w1 − 2w2 − 1)

(r

r0

) 2(w2−w1)w1

. (42)

In GR, the violation of NEC is the basic requirement for the existence of wormhole solutions. TheNEC arises when one refers back to the Raychaudhuri equation, which is a purely geometric statement.

7

Using the condition of attractive nature of gravity for any hypersurface of orthogonal congruences(i.e. zero rotation associated to the congruence defined by null vector field) in these equations gives,Rµνk

µkν ≥ 0 and Rµν lµlν ≥ 0 for null kν and time-like lν vector fields. If we replace the Ricci tensor

with EMT we arrive at the standard energy conditions. Physical reliability requires that the wormholeconfiguration respects the weak energy condition (WEC) stated through the following expressions

ρ(r) ≥ 0, ρ(r) + pr(r) ≥ 0, ρ(r) + pt(r) ≥ 0. (43)

and NEC

ρ(r) + pr(r) ≥ 0, ρ(r) + pt(r) ≥ 0. (44)

We therefor require that the following conditions hold

1. The shape function satisfies the flare-out condition rb′ − b < 0 which turns into inequality (37)at the throat.

2. In order that the wormhole configuration be traversable, the spacetime must be free of horizons(the surfaces with e2φ(r) → 0); therefore the redshift function must be finite everywhere.

3. For asymptotic flat solutions, i.e., b(r)r → 0 as r →∞, we must have w2/w1 < 0. This condition

also satisfies the condition w2/w1 < 1 which is required for density and pressure profiles toconverge at infinity.

4. The square of spin density must be positive throughout the spacetime and also at the throat.

5. The coefficient of energy density i.e., ρ(r0) be positive so that the first inequality in (43) issatisfied.

6. w1 > −1 and w2 > −1 along with condition (3) so that the second and third inequalities in (43)are fulfilled. This conditions also guarantee that the WEC is satisfied at the wormhole throat,i.e., ρ(r0) + pr(r0) > 0 and ρ(r0) + pt(r0) > 0. We also note that WEC implies the null form.

Figure (1) shows the allowed values of w1 and w2 for which the above four conditions are respected. Infigure (2) we have plotted for the radial component of the metric and the ratio of shape function overr. It is seen that the radial metric component stays positive for r > r0 and thus the metric signaturedoes not change.

In figure (3) we have plotted the electric field for fixed values of EoS parameters but differentvalues of charge at the throat. It is seen that the electric field gets different maximums for differentvalues of Q0. As the electric field reaches a maximum value around the throat, then in this wormholeconfiguration, the charges particles will experience maximum attractive or repulsive forces dependingon their sign. In figure (4) we plotted for energy density, ρ(r) + pr(r) and ρ(r) + pt(r) where it is seenthat the WEC is satisfied for our wormhole solutions.

In order to visualize the wormhole, we consider the slice given by t = constant and θ = π/2 section4

of the wormhole spacetime. The respective line element is then given by

ds2 =dr2

1− b(r)r

+ r2dΦ2. (45)

According to Morris and Thorne [8], for visualizing the above slice, we embed the metric (45) into a3D Euclidean space for which the line element in cylindrical coordinates (Z, r,Φ) is written as

ds2 = dZ2 + dr2 + r2dΦ2. (46)

4Due to the spherically symmetry we can consider an equatorial slice, θ = π/2, without loss of generality.

8

Figure 1: The space parameter for allowed values of equation of state parameters for Q0 = 1.1, Λ = 0and r0 = 1.

Figure 2: The behavior of radial metric component and the ratio of shape function over radius forQ0 = 1.1, Λ = 0 and r0 = 1, w1 = −0.33, w2 = 1 (solid curve), w1 = −1, w2 = 0.33 (dashed curve)and w1 = −1, w2 = 0.63 (dot-dashed curve).

9

We note that in the 3D Euclidean space the embedded surface has equation Z = Z(r), hence the lineelement of the surface can be written as,

ds2 =

[1 +

(dZ

dr

)2]dr2 + r2dΦ2, (47)

whereby matching with the line element (45) we get the following differential equation, and the corre-sponding integral, for the embedding surface of the wormhole

dZ

dr= ±

[r

b(r)− 1

]− 12

, Z(r) = ±ˆ r

r0

dy√yb(y) − 1

. (48)

From the above equation we observe that the geometry has the minimum radius r0 = b(r0) (worm-hole’s throat) at which the embedded surface is vertical, i.e., dZ/dr|r→r0 → ∞. Moreover, the radialcoordinate r is ill-behaved near the throat, however, the proper radial distance defined as

`(r) = ±ˆ r

r0

dy√1− b(y)

y

, (49)

must be well behaved everywhere, i.e., `(r) must be finite at all finite r, throughout the spacetime. The± signs refer to the two asymptotically flat regions (` → ±∞ or equivalently r → ∞ then b/r → 0)which are connected by the wormhole. Next, we proceed to evaluate the above integrals using thesolution (34). Unfortunately the integration can not be carried out analytically but we can performit using numerical methods. The left panel in figure (5) shows the behavior of embedding functionversus the radial coordinate for the same parameters chosen for Fig. (2). The embedding diagramshows that the wormhole extends from the throat located at r = 1 to infinity. In the right panel wehave plotted for proper radial distance of the wormhole. It is observed that as ` increases from −∞ tozero the radial coordinate decreases monotonically to a minimum value at the wormhole’s throat; andas ` tends to +∞, r increases monotonically. Figure (6) represents the full visualization of wormholeobtained by rotating the embedding curve around the Z-axis.

4 Concluding Remarks

In the present work we have constructed models of static wormholes within the frame work of ECT byconsidering usual (non-exotic) spinning matter (Weyssenhoff fluid) along with an anisotropic energymomentum tensor (EMT) and a Maxwell field as supporting matters for wormhole geometry. Theradial and tangential components for anisotropic fluid pressures were taken to depend linearly to energydensity via different EoS parameters. These EoS parameters along with the value of charge and spinsquare density at the throat construct a space of parameters. The allowed regions of this parameterspace are subject to fulfillment of physical conditions on wormhole configuration that a subspace ofwhich is sketched in Fig. (1). In general, our solutions include the ranges w1 < 0 and w2 > 0 for EoSparameters, that is the wormhole configurations are supported by positive and negative pressures alonglateral and radial directions, respectively. Thus the anisotropy parameter, which for our model is givenby ∆(r) = pt − pr = (w2 − w1)ρ(r) is always positive, unless the WEC is violated. This implies thatthe geometry is repulsive due to the anisotropy of the system. From the viewpoint of the equilibriumcondition for wormhole configuration we can divide conservation equation (19), which is the well-known Tolman-Oppenheimer-Volkov equation, into three parts [61]: the anisotropic force Fa = 2∆/r,the hydrostatic force Fh = −p′r and force due to gravitation contribution fg = −φ′[ρ(r) + pr(r)],where the last one is absent in our model since the redshift function is constant. Taking the derivative

10

Figure 3: The behavior of electric field for w1 = −0.3, w2 = 0.8, Q0 = 1.1 (solid curve) and Q0 = 3.1(dashed curve). Dotted and dot-dashed curves are plotted for the same values but negative sign of(25). We have set Λ = 0 and r0 = 1.

of radial pressure, it is easy to check that Fh = −Fa and hence we can deduce that the wormholesolutions are in equilibrium as the anisotropic and hydrostatic forces cancel each other [62]. As Fig.(3) shows, the electric field takes a maximum near the throat and this provides a setting to acceleratecharged particles toward wormhole, namely, the more a negative charge approaches the wormholethroat the more attractive force (E(r) > 0) it feels; this scenario also occurs for positive chargesconsidering a negative sign for charge function. Beside the present model, wormhole solutions in thepresence of electric charge have been studied earlier. For example, by adding an electric charge, theauthors studied the possibility of stabilizing a wormhole supported by a ghost scalar field [63]. Chargedwormholes in Einstein-Maxwell theory have been constructed by real feasible matter sources in [64]where the solutions respect the energy conditions throughout the spacetime but the NEC is violatedat the throat.

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Figure 4: The behavior of energy density (upper left), ρ(r)+pr(r) (upper right) and ρ(r)+pt(r) (lowerleft) for Q0 = 1.1, Λ = 0, r0 = 1, w1 = −0.33, w2 = 1 (solid curve), w1 = −1, w2 = 0.33 (dashedcurve), w1 = −1, w2 = 0.63 (dot-dashed curve), w1 = −0.2, w2 = 2 (dotted curve). (Lower right) Thebehavior of square of spin density for Q0 = 1.1, Λ = 0, r0 = 1, w1 = −0.33, w2 = 1 (solid curve),w1 = −1, w2 = 0.33 (dashed curve), w1 = −1, w2 = 0.63 (dot-dashed curve).

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Figure 5: (Left Panel) The embedding diagram of the wormhole (45) obtained from solution (34) forthe parameter values w1 = −0.33, w2 = 1, Λ = 0, r0 = 1 and Q0 = 1.1, where the throat is located.The solid line correspond to + sign and the dashed line to − sign in (48). (Right Panel) Plot of theradial proper distance of the wormhole against radial coordinate for same values taken for the leftpanel.

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