Traversable wormhole solutions in Rastall teleparallel gravity

Journal Pre-proof Traversable wormhole solutions in Rastall teleparallel gravity Kh. Saaidi, N. Nazavari

PII: DOI: Reference:

S2212-6864(19)30353-X https://doi.org/10.1016/j.dark.2020.100464 DARK 100464

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Physics of the Dark Universe

Received date : 30 November 2019 Revised date : 25 December 2019 Accepted date : 6 January 2020 Please cite this article as: Kh. Saaidi and N. Nazavari, Traversable wormhole solutions in Rastall teleparallel gravity, Physics of the Dark Universe (2020), doi: https://doi.org/10.1016/j.dark.2020.100464. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier B.V.

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Traversable wormhole solutions in Rastall teleparallel gravity Kh. Saaidia,∗, N.Nazavaria a

Department of Physics, Faculty of Science, University of Kurdistan, Sanandaj, Iran.

Abstract

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In this manuscript we consider the traversable wormhole solutions in teleparallel gravity formalism with and without Rastall constraint on perfect fluid stress-energy tensor. We have studied the isotropic and non-isotropic form of stress-energy tensor of ideal fluid which the equation of state(EoS) between density and the radial components of pressure is used. This work is done for a diagonal tetrad and we have found some geometries which could describe a traversable wormhole, but the null energy and weak energy conditions were not violated for some sets of parameters. Keywords: traversable wormhole, teleparallel, energy condition. PACS: 98.80.Cq

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1. Introduction

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Wormhole is known as an assumptive object that behave as a tunnel which makes a connection between two different parts of a universe or two different universes. The main challenges about the existence of wormhole are the traversable wormhole concept and the energy conditions of it. With an idea of utilizing a wormhole for time travel or interstellar travel, the concept of traversable wormhole was suggested by Morris and Thorne [1]. It is notable that the concept of traversable wormhole is completely different from the Einstein-Rosen bridge concept, [2] and the work which is done by ∗

Corresponding author Email addresses: [email protected] ( Kh. Saaidi), [email protected] ( N.Nazavari)

Preprint submitted to Physics of The Dark Universe

December 24, 2019

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Wheeler [3] about the charge-carrying microscopic wormhole. For example, authors [4] have investigated a traversable wormhole which is created due to a magnetic monopole. Where as Riemannian geometry can not explain the existence of magnetic monopoles, Dymnikoa metric which has a torsion geometry structure has been used in [4]. Also, by using a foliation θ = π/2 on an extended 5D non-vacuum space-time, a traversable wormhole over an effective Schwarzschild space-time has studied in [5]. In fact, by extending the electrodynamics theory, the authors of [5] have obtained the gravito-electrodynamics theory. Many works about different features of wormhole have been done in [6, 7, 8, 9, 10, 11, 12, 13, 14].

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A geometry with Einstein field equation which describe a wormhole yields a matter distribution that violate the null and weak energy conditions. These types of matter or energy which violate the ordinary energy conditions are called exotic matter. Although, ordinary matter in classical regime obey the energy condition but there are some quantum phenomena such as Casimir effect and Hawking evaporation which the ECs are violated. For studying the other features of wormhole, some outcomes have been performed by scientists. For example, the geometry of wormhole has been studied in f(R) model of gravity and for a certain shape-function some exact solutions have been obtained and for a barotropic matter the static wormhole solution were found in [15, 16]. Behind above subjects, in two past decades, many authors concentrate on matter (energy) that obeys the equation of state(EoS) p = ρω. Where, p and ρ are pressure and density of matter respectively and ω is called the equation of state parameter. There are several values for ω. For example, ω = −1 (ΛCDM model), −1/3 < ω < 1 ( quintessence), ω < −1(Phantom phase of energy).

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Astrophysical observations of supernova of type Ia (SNe Ia) [17, 18] and cosmic microwave background (CMB) data [19, 20] confirms that the universe contain some types of energy whose the equation of state parameter, ω, is negative, namely the pressure of energy is negative. This sort of energy is known as dark energy. Furthermore, observational data shows that the universe is filled of more than seventy percent of dark energy [19] and for phantom phase of dark energy, ω < −1, the null energy condition (NEC) and weak energy condition (WEC) are violated. So this fact suggest that wormhole could be formed and introduced in a dark energy dominated universe [21]. Some 2

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consideration were done in modified model of general relativity for obtaining wormhole solutions that do not require a violation of energy condition [22, 23, 24, 25, 26, 27, 28, 29]. Also, many authors have studied traversable wormhole supported by phantom energy [30, 31, 32, 33, 34, 35, 36, 37]. In this paper we will study static wormhole solutions which supported by dark energy in teleparallel gravity. We do this investigation with and without the Rastall condition on stress- energy tensor. We show that for some range of parameters, the obtained geometry can describe a traversable wormhole which the dominate dark energy is λCDM and the NEC and WEC are not violated. This manuscript is outlined as follows, In Sec.2, we have considered a review on teleparallel gravity. In Sec.3, we have done a review on wormhole concepts and we have found some typical solutions for the case which stress-energy tensor is conserved. In Sec.4, we have investigated our work for the case which the Rastall like condition applied and several solutions are obtained for this case. Sec.5 is about the conclusion of our work. 2. Teleparalle gravity: a review

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We utilize the Greek alphabet (µ, ν, ... = 0, 1, 2, 3) for denoting space-time indices and Latin alphabet (i, j, ... = 0, 1, 2, 3) to denote the tangent space indices. As usual, we assume the tangent space is Minkowski space with the metric ηij = diag(−1, 1, 1, 1). It is remarkable that, the tangent space indices raised and lowered with the Minkowski metric ηij , and for raising and lowering the space-time indices, we use the Riemannian metric which is defined as (1) gµν = ηij eiµ ej ν ,

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here, ei (xµ ) = eiµ ∂ µ is called tetrad field. These nontrivial tetrad fields form an orthonormal basis for the tangent space in which (ei .ej ) = ηij . In fact, nontrivial tetrad fields induce on the spacetime a teleparallel structure and they related to the gravitational field as (1). One can define the Weitzenbock connection based on nontrivial tetrad fields as Γαµν = eiα ∂ν eiµ = −eiµ ∂ν eiα .

(2)

The Weitzenbock covariant derivative of the tetrad field vanishes i.e ∇µ eiν = ∂µ eiν − Γσµν eiσ = 0. 3

(3)

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Eq.(3) is the basic parallelism condition. This connection leads to null curvature and a non-zero torsion, which is defined as
(4) T σµν ≡ Γσνµ − Γσµν = eiσ ∂µ eiν − ∂ν eiµ . The Levi-Civita and Weitzenbock connection are related by ¯ σ = Γσ − K σ , Γ µν µν µν

(5)

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¯ σ is the Levi-Civita connection and K σ is the contorsion tensor where Γ µν µν defined as
1 K σµν = Tµ σ ν + Tν σ µ − T σµν . (6) 2 Finally, we define the torsion scalar as T = S σµν Tσµν ,

(7)

here S σµν is called super-potential and is given by

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1 S σµν = −S σνµ = (K µνσ − g σν T αµα + g σµ T ανα ). 2

(8)

The gravitational field Lagrangian in teleparallel gravity is given by LG =

e T, 16π

(9)

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where, we assume c = G = 1 and e = det(eiµ ). For the case which the space-time is not empty, one can define the action in teleparallel gravity as  Z  T S=− e + Lm d4 x, (10) 16π where Lm is the matter Lagrangian. By Variating this action with respect to tetrad field, we have

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1 e−1 eiµ ∂ρ (eeiα Sα νρ ) + T αλµ Sα νλ + δµν T = 4πΘνµ , 4

(11)

where Θνµ is the stress energy tensor of the perfect fluid. It is shown that   1 ν −1 i α νρ α νλ Dν e e µ ∂ρ (eei Sα ) + T λµ Sα + δµ T = 0, (12) 4 4

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here Dν V µ is the teleparallel version of covariant derivative and it is defined as Dν V µ = ∂ν V µ + (Γµλν − K µλν )V λ , (13) therefore, according to Eq.(11), one can see that Dν Θµν = 0.

(14)

In this work, we assume the material content is an non-isotropic fluid, that the stress-energy tensor of it is defined as Θνµ = (ρ + pt )uµ uν − pt δµν + (pr − pt )vµ v ν ,

(15)

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where uµ is the space-like velocity vector in which u0 u0 = 1, and v µ is the unitary space like vector in the radial direction in which v1 v 1 = −1. pr and pt are the radial and transverse components of the pressure respectively. It is remarkable that pr − pt 6= 0 shows that non-isotropy is caused due to a surface tension inside the star.

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3. Traversable wormhole solution for conserved stress-energy tensor One of the most common space-time metric that representing a static and spherically wormhole is written as

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ds2 = ea(r) dt2 − eb(r) dr2 − r2 (dθ2 + sin2 θdϕ2 )

(16)

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here a(r) and b(r) are arbitrary functions of radial coordinate r. a(r) is related to gravitational red-shift, so it is denoted as red-shift function. In wormhole, the radial coordinate r has specified geometry and range of it covers [r0 , ∞), where r0 corresponding to the throat of wormhole. Using embedding diagrams it is shown that β(r) = r(1 − e−b(r) ) determines the shape of the wormhole. So β(r) is called the shape function. It is well known that, for getting a traversable wormhole geometry some constraints should be imposed on the functions which have come in the metric. The red-shift function a(r) should be finite for all values of r to avoid an event horizon. The shape function, β(r), should satisfy β(r0 ) = r0 and the flare-out condition at the throat; β 0 (r0 ) ≤ 1 and for r > r0 , β(r) < r has to be satisfied. Using the

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Einstein field equation Gµν = Θµν and Eq.(16), we obtain the stress-energy tensor components as

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T 1 e−b 0 (a + b0 ) − + 2 , 2r 4 2r 1 T − = , 4 2r2  e−b a00 a0 1 0 0 = + ( + )(a − b ) . 2 2 4 2r

4πρ = 4πpr 4πpt

(17) (18) (19)

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Also based on Eq.(16) and using Eqs.(4)-(8), one can find the scalar torsion as   2e−b 1 0 T (r) = a + , (20) r r where the prime denotes derivative with respect to the radial coordinate r. According to Eq.(14), we can obtain the conservation of energy equation as Dν Θµν = 0.

(21)

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In this stage we have known the wormhole geometry and for solving the Einstein field equations we need to specify the form of perfect fluid. Due to this fact, we choose the equation of state (EoS) for perfect fluid as pr = ωρ.

(22)

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Using Eqs.(22), (17) and (18), we have B 0 (r) +

a0 (r) α α B(r) + B(r) = , ω r r

(23)

where e−b = B(r) and α = (1 + ω)/ω. In the following, we have to solve Eq.(23) for different cases.

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3.1. Solutions for a real constant redshift function In this case we assume the redshift function is a real constant, then a0 (r) = 0. So we have 1 (24) T = 2 e−b(r) , 2r and by solving Eq.(23), we have e−b(r) = B(r) = (1 + 6

c ), rα

(25)

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1.0

0.5

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0.0

0.0

ρ

-0.5

-0.5

ρ pr γ=ρ+pt

-1.0 0

2

4

6

8

ρ

ρ pr

γ=ρ+pt

-1.0 0

10

r

2

4

6

8

10

r

(a)

(b)

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WEC

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0.0

ρ

-0.5

ρ pr

γ=ρ+pt

2

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6

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r

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Figure 1: WEC versus radial coordinates for c = −1 with different values: (a) ω = −1.5(phantom), (b) ω = −1(ΛCDM), (c) ω = 1.5(exotic matter).

where c is the integration constant. According to Eq.(25) and Eq.(16), one can obtain the metric of this model as  c −1 2 ds2 = ea dt2 − 1 + α dr − r2 dΩ2 . (26) r

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It is clearly seen that at large distance r → ∞, Eq.(26) is asymptotically flat. For dl2 = (1 + c/rα )dr2 and for having a traversable wormhole the condition dl2 /dr2 > 0 given a minimum at r0 for r. For this aim one can obtain r0 = (−c)1/α . r0 must be positive, so that c < 0. Inaddition, we have β(r0 ) = r0 and β 0 (r0 ) = −1/ω 6 1 is satisfied for all value of ω in the range (−∞, −1] ∪ [0, ∞). We are to find the components of stress-energy tensor, which are obtained by Eqs.(17)-(19)as c , 2ωr(2+α) c pr (r) = , (2+α) 2r αc pt (r) = − (2+α) . 4r ρ(r) =

7

(27) (28) (29)

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Energy condition is a concept that we have to check them in order to have a comprehension wormhole. As a first condition on components of stress energy tensor, the density of energy, ρ, should be finite and positive at infinity. This fact is done for the first part of the interval which the traversable wormhole conditions are satisfied, namely, ω ∈ (−∞, −1]. The most common energy conditions are null energy condition (NEC) ρ + pi ≥ 0 and weak energy condition (WEC), ρ ≥ 0 and ρ + pi ≥ 0. Therefore, based on Eqs.(28) - (29) NEC and WEC are valid in ΛCDM model but in phantom phase of dark energy, N EC and W EC are violated. Therefore, one can see that, this geometry can describe a traversable wormhole for ω ∈ (−∞, −1], in which in ΛCDM model the N EC and W EC are not violated and for other values of ω the N EC and W EC are violate. We plot W EC for different value of ω in Fig.1. 3.2. Solution for a specific shape function In this case we use an anzats for b(r) as −b(r)

=1−

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e



r r0

γ−1

,

(30)

where r0 is a constant with length dimension. Note that for γ < 1, grr coefficient of metric is asymptotically flat. Substituting Eq.(30) into Eq.(23), provides the following result for ea(r)

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  1 + γω r γ−1 ea = c 1 − ( )γ−1 . r0

(31)

Eq.(31) states that the gtt coefficient of metric is asymptotically flat for γ < 1. Therefore the metric is asymptotically flat totally and it is given as

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 1 + γω  r γ−1 2 dt − ds2 = c 1 − ( )γ−1 r0

1−



r r0

γ−1 !−1

dr2 − r2 dΩ2 . (32)

The wormhole throat obtain from setting e−b equal to zero, then the throat’s radius of wormhole is r0 . Also β(r) |r=r0 = r0 and the condition β 0 ≤ 1 is satisfied for γ < 1 and all value of ω. One can see that for ω = −1/γ the redshift function has a finite value. It is notable that for 0 < γ < 1 the values 8

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of ω billings to y (−∞, −1) interval. Therefore this wormhole is traversable that connects two asymptotically flat regions for ω ∈ (−∞, −1). Hence, according to Eq.(22), it is clear that in the phantom phase of energy NEC and WEC are violated. 3.3. Solution for specific redshift function In this case we choose the redshift function as a(r) = ln(1 −

r0 ). r

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So using Eq.(33) and Eq.(23), one can obtain e−b as    r0  c −b(r) e = 1− 1+ . r (r − r0 )α

(33)

(34)

By inserting Eq.(33) and Eq.(34) into ds2 we have 

1   dr2 − r2 dΩ2 .  r0 c 1− 1+ r (r − r0 )α

(35)

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r0  2 dt −  ds = 1 − r 2

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Note that this metric is not asymptotically flat for ω ∈ (−1, 0) but for the case ω ∈ (∞, −1)∪(0, ∞) the metric (35) is asymptotically flat. From √ −b Eq.(34), it is obviously seen that, relation e = 0 has two roots as, r1 = r0 and r2 = r0 + (−c)1/α . r2 must be a positive real constant, so c has to be positive. For a traversable wormhole, the redshift function should be limited, namely, a(r > r1 ), (r1 is the minimum value of the throat radius) has to be finite. But we see that the value of a for r = r1 = r0 is infinite. So that this geometry can not describe a traversable wormhole. 4. Traversable wormhole solutions for non-conserved stress-energy tensor: Rastall like Teleparallel gravity

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As we know, one of the essential assumption of Einstein theory of gravity is Θν µ;ν = 0. In fact, there are many reasons for accepting that the covariant divergence of the stress-energy tensor vanishes. Peter Rastall claim that based on some assumptions, relation Θν µ;ν = 0 has been accepted and those assumptions are all questionable. So he assumed Θν µ;ν = 9

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aµ , where the functions aµ should be vanished in flat space-time [38]. As we know, curved space-time and gravitation are equivalent, namely gravitation field which is due to existence of matter, makes curvature and vias versa. Therefore, this fact shows Tµν should depends on the curvature. For example, one may take an elastic sphere of an elementary particle. When the curvature does not vanish tidal gravitational force are present which distort the sphere and so that change the rest mass and energy of it[38]. According to the above argumentations, the conservation of energy relation was introduced by P. Rastall as Θν µ;ν = λR,µ ,

(36)

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where λ is a constant [38]. This equation shows an interacting link between matter and geometry and because of this matter-geometry might be created or annihilated. According to Eq.(11) our stressenergy tensor Dν Θµν is a function of torsion which is vanished in flat space-time, then like the Rastall Eq.(36) we assume λ T,µ . (37) 4 Here T is the scalar torsion and λ is a real constant and interacting link between matter and geometry is caused through the scalar torsion of geometry. Based on this assumption, Eq.(11) is reduced to

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Dν Θµν =

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1 e−1 eiµ ∂ρ (eeiα Sα νρ ) + T αλµ Sα νλ + δµν T = 4πΘνµ . (38) 4 Equations of density energy radial and tangential components of pressure is obtained as   λ−1 1 e−b 0 4πρ = T+ 2+ (a + b0 ), (39) 4 2r 2r   λ−1 1 4πpr = − T − 2, (40) 4 2r   e−b a00 a0 1 λ 0 0 4πpt = + ( + )(a − b ) − T. (41) 2 2 4 2r 4 By accepting equation of state as pr = ωρ, we have B 0 (r) + ξa0 (r)B(r) + (ξ + 1) 10

B(r) α = , r r

(42)

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where ξ = α(1 − λ) − 1. Then solution of Eq.(42), is given by   Z r e−ξa(r) ξa(r) ξ e r dr . B(r) = (ξ+1) c + α r r0

(43)

It is obviously seen that, to obtain an clear solution, such as the previous section, we have to solve Eq.(43) for some typical cases.

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4.1. solution for a real constant redshift function In this case we assume the redshift function is a finite constant so one of the traversable constraint is satisfied spontaneously. Then the Eq.(43) is reduced to 1 e−b(r) = + crα(λ−1) . (44) (1 − λ) where c is the integrating constant and the line element is given as   1−λ 2 a 2 ds = e dt − dr2 − r2 dΩ2 . 1 + (1 − λ)crα(λ−1)

(45)

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From Eq.(45), one can see that for two different sets of parameters such as (α ≥ 0, λ < 1 ) and (α < 0, λ > 1), this metric is asymptotically flat. As we mentioned before, the condition of traversable wormhole, e−b(r) ≥ 0, is satisfied for r0 = [1/c(λ − 1)]1/(α(λ−1) . Due to the condition r0 ≥ 0 , c(λ − 1) must be positive, so if c > 0, λ must be bigger than one, and for c < 0 , λ has to satisfy λ < 1. Therefore we encountered with two (α ≥ 0, λ < 1, c < 0) and (α < 0, λ > 1, c > 0) sets of parameters. Moreover, for these two sets of parameters, β(r) = r(1 − e−b(r) ) |r=r0 = r0 , but the last condition of traversable wormhole, i.e β 0 is given as β 0 = 1 − α.

(46)

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This result show that only for (α ≥ 0, λ < 1, c < 0) set of parameters which is equivalent with ω ≤ −1 (ΛCDM and phantom models) and ω > 0, (ordinary matter), all constraints of traversable wormhole which connect two flat space-times, are satisfied. In this situation, we have to consider the NEC and WEC conditions. For

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this goal we have to obtain the components of stress-energy tensor as c(λ − 1) ρ(r) = − , 2ωr2+α(1−λ) c(λ − 1) pr (r) = − 2+α(1−λ) , 2r  1 c (α(λ − 1) − 2λ) 2λ pt (r) = + . 4r2 rα(1−λ) λ−1

(47) (48) (49)

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ρ(r) as energy density of perfect fluid should be positive and this fact is satisfied for a set of parameters such as (ω ≤ −1, c < 0, λ < 1). Also, we can see that ρ(r), is vanished at infinity for α(λ − 1) ≤ 2. As we mentioned before, this model is asymptotically flat for α ≥ 0 and λ < 1, hence these value of α and λ satisfy α(λ − 1) ≤ 2, completely. Then based on this argument the phantom and ΛCDM phase of dark energy model can satisfy the traversable conditions of wormhole which connect two flat parts of universe or two different universes. It is obviously seen that in the phantom phase of dark energy the NEC and WEC are violated. For (α = 0, c < 0, λ < 1) which is known as a ΛCDM model of energy, we have two different intervals. First, Λ < 0. In this case we have |c|(|λ| + 1) > 0, (50) ρ(r) = 2r2 ρ + pr (r) = 0, (51)   1 |λ| ρ + pt (r) = + |c| > 0. (52) 2r2 1 + |λ|

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So from Eqs. (51)-(52) obviously seen that the NEC and WEC are not violate for (α = 0, c < 0, λ < 0) set of parameters. Second, for 0 < λ < 1 . In this case we have |c|(1 − λ) > 0, (53) ρ(r) = 2r2 ρ + pr (r) = 0, (54)   1 λ ρ + pt (r) = |c| − . (55) 2 2r 1−λ It is seen that for |c| 0<λ< , (56) 1 + |c| the NEC and WEC in ΛCDM model of energy are not violate as well. We have plotted our result in fig.2 12

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0.0

0.0

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-0.5

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ρ pr γ=ρ+pt

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ρ pr

γ=ρ+pt

-1.0 0

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r

2

4

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r

(a)

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0.0

ρ

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ρ pr

γ=ρ+pt

-1.0 0

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r

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Figure 2: WEC versus radial coordinates for ω=-1(λCDM model) with different values: (a) λ = −1.5, c = −1 (b) λ = 0.5, c = −1.25 (c) λ = 1.5, c = −0.83.

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4.2. Solution for a specific shape function As we mentioned in the previous section, by choosing the shape function as β = r(r/r0 )γ−1 , the flaring out conditions is satisfied i.e β(r0 ) = r0 and β 0 (r0 ) = γ < 1. It is clearly seen that this shape function denote an asymptotically flat space-time geometry. According to this shape function we can obtain the g00 component of metric as a(r)

e

where

ζ  r γ−1 , =cr 1−( ) r0 0

ε

αλ , ξ ξ = α(1 − λ),   1 α ζ = − +1 , ξ γ−1

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ε =

(57)

(58) (59) (60)

and c0 is an another integrating constant. One can see that for ω = −1 13

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ea (r)

0.8

γ=0.5 γ=0

0.2

γ=-0.5

0.0 0

20

40

60

r

80

100

Figure 3: The coefficient grr versus r, for ω = −1, c1 = 1 and r0 = 2.

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and γ < 1, the grr coefficient of the metric is asymptotically flat as well. This fact is shown in Fig.3. But the redshift function is infinite for ω = −1, so although the geometry is asymptotically flat but it is not a traversable wormhole for ω = −1. Using Eq.(57), the redshift function is given as   r γ−1 0 a(r) = c + ε ln(r) + ζ ln 1 − ( ) . (61) r0

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From Eq.(61), it is seen that the last term is infinite at r = r0 . Hence if ζ = 0, namely, α = 1 − γ(ω = −1/γ), this term is zero, therefore the redshift function will be finite at r = r0 . It is remarkable that, we accepted γ < 1, than this fact, restrict the value of ω to be belong to ω < −1 and ω > 0 intervals. Therefore the line element in this part is written as ! 1 0 dr2 − r2 dΩ2 , (62) ds2 = c rε dt2 − 1 − ( rro )(γ−1)

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so, this metric which connect two asymptotically flat parts of universe describe a traversable wormhole. In this situation, considering the energy conditions are interesting. So that we explore the components of stress- energy tensor for this model. Density of energy in this model is given as A1 A2  r0 α+2 4πρ(r) = 2 − 2 , (63) r r0 r

where

A1 =

λ 2ωξ

and 14

A2 =

λ+ξ 2ωξ

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Note that for ω < −1 and ω > 0, the quantity of α is positive and therefore for λ(1 + ω) 6= 1 the density of energy is finite for large value of r. Density of energy should be positive at all range of r. So we check it in the throat radius r0 . Therefore the value of ρ at r = r0 and ω = −1 − ε, (ε > 0) is given by 1 . (64) 4πρ(r0 ) = 2 2r0 (1 + )

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Therefore the phantom density of energy is positive. The radial and tangential components of pressure are given as   (λ − 1)A1  r0 α+2 1 (λ − 1)A1 4πpr (r) = +1 , − 2 (65) λr2 r r λ    r0 α+2 3+ξ A1 ξ − 1 A + ( + 1) 4πpt (r) = 1 2 2r0 λ2 4λ r   1 A1 A1 1 + 2λ − + + (66) 2 2 2r λ λ 4A1

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It is well known that for phantom region which the ω < −1 null energy condition is violate. 4.3. Solution for a specific redshift function In subsection, we take the redshift function as

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a(r) = ln(1 −

r0 ). r

Using Eq.(42) we can obtain e−b as    r0  c 1 −b(r) e = 1− + , r 1 − λ (r − r0 )α(1−λ)

(67)

(68)

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where c is an integration constant. And the line element is given as  r0  2 1 2   dr2 − r2 dΩ2 (69) ds = 1 − dt −   r 1 c r 0 1− + r 1 − λ (r − r0 )α(1−λ) It is obviously seen that for α(λ − 1) < 0, the metric of this solution is asymptotically flat. So, for two different sets of parameters 15

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r2 = r0 + [c(1 − λ)]1/α(1−λ)

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such as (α > 0, λ < 1) and (α < 0, λ > 1) Eq.(69) is asymptotically flat. This result shows that for any kind of fluid such as phantom, quintessence, matter, radiation and so on, one can take an appropriate value for λ to bare an asymptotically flat metric. In order √ to cheek the traversable wormhole conditions, we have to solve e−b = 0. The solution of it has two roots as and

r1 = r0

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Notify, r1 and r2 should be positive real constant, so c(1−λ) must be positive and this fact is satisfaied for two set of c and λ parameters as (c > 0, λ < 1)Rand (c < 0, λ > 1). In this case the proper radial disr tance, l(r) = ± r1 eb(r) dr, has to be finite in space-time. The signs ± relate the two different parts which is connected by wormhole configuration. In fact when l(r) = +∞ reduce to l(r) = [c(1 − λ)]1/α(1−λ) at r = r2 and after that reduce to l(r) = 0 at r = r1 at the throat and then increase to −[c(1 − λ)]1/α(1−λ) at the r2 point as the end of throat and after that to the other part l(r) = −∞. It is remarkable that the redshift function for r = r1 = r0 is infinite and therefore in this case the wormhole is not traversable.

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4.4. Isotropic case of stress-energy tensor Our work was based on the non-isotropic version of stress-energy tensor and studying the isotropic case is interesting as well. The isotropic version of stress-energy tensor is obtained by choosing pr = pt = p. Then, using Eqs.(39)-(41), we obtained two expressions as  0  e−b a 1 a00 a02 a0 b0 b0 1 + − − 2+ + − − = 0, (70) 2r2 2 2r r 2 4 4 2r e−b 0 ρ+p= (a + b0 ). (71) 2r By deriving the Eq.(40) and utilizing Eqs. (70) and (71), one can find

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p0 (r) =

λ a0 (λ − 1) + (ρ + p). r3 2

(72)

Finally, substituting Eq.(22) into Eq.(72), we get a differential equation for pressure as follows λ p0 − $a0 p − 3 = 0, (73) r 16

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Where $ = ((λ − 1)α)/2. The solution is given as   Z r λ −$a(r) $a(r) c+ p(r) = e e dr . 3 1 r

(74)

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It is seen that for solving Eq.(74) we have to determine the form of redshift 1 function. In fact this equation is solved for a = a0 and a = ln(1 − ) redshift r functions. Here a0 is a real constant. By doing some algebraic calculations, the following results are obtained for (ea(r) , eb(r) ).   (1 − λ) a0 , , (75) 2cr2 + (1 − λ)   $   2r(r − 1) τ (r − 1)(1 + r − $) 1−r 1 1 + c+ 2 (76) (1 − ), r 1−λ r2 r 2r

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As we mentioned before, the roots of the shape function show minimum value of radius p of the wormhole throat. According to Eq.(75) the radius of throat is r0 = (λ − 2)/4c and based on this radius β 0 (r0 ) = 3 > 1 , then a geometry which is obtained by metric’s coefficient( 75) could not describe a traversable wormhole. From Eq.(76) it is seen that equation e−b(r) = 0 has more than one root and one of those roots is r0 = 1. It is explicitly seen that the redshift function in this equation has infinite value in r0 = 1. Then a metric which is obtained by metric’s coefficient (76) could not describe a traversable wormhole. 5. Conclusions

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In this work, the traversable wormhole (TW) solutions in teleparalell gravity formalism for a static and splenetically symmetric space-time is studied. Based on observational data, we have assumed the dominant component of universe is dark energy and an equation of state such as pr = ωρ is used. We introduced a Rastall like constraint in teleparalell formalism and we considered the model with and without Rastall constraint on stress-energy tensor of perfect fluid. Where as the general solutions were impossible, we studied the TW solutions for some specific cases. The model is investigated for a real constant redshift function and we found a geometry which describe a TW and we obtained that the NEC and WEC are violated in phantom phase of dark energy and they are not violated in ΛCDM phase of dark energy. 17

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Furthermore, by considering the obtained geometry for a(r) = ln(1 − 1r ) we found that in this case the geometry can not describe a traversable wormhole. Inaddition, we continued our investigation for b(r) = − ln(1−( rr0 )(γ−1) ). In this case we obtained a geometry which satisfy the traversable wormhole conditions and it is supported only by phantom phase of dark energy. Finally, we showed that in teleparalell gravity formalism, the geometry which is obtained in the isotropic version of stress-energy tensor of dark energy can not describe a traversable wormhole. References

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