Lorentz Distributed Noncommutative Wormhole Solutions in Extended Teleparallel Gravity

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provided by Springer - Publisher Connector Eur. Phys. J. C (2015) 75:173 DOI 10.1140/epjc/s10052-015-3386-9

Lorentz distributed noncommutative wormhole solutions in extended teleparallel gravity

Abdul Jawada, Shamaila Ranib Department of Mathematics, COMSATS Institute of Information Technology, Lahore, Pakistan

Received: 19 January 2015 / Accepted: 2 April 2015 / Published online: 28 April 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com

Abstract In this paper, we study static spherically sym- HDE models [1Ð5]. After GR, Einstein attempted to unify metric wormhole solutions in extended teleparallel grav- gravitation and electromagnetism based on the mathemati- ity with the inclusion of noncommutative geometry under cal structure of absolute or distant parallelism, also referred a Lorentzian distribution. We obtain expressions of matter to as teleparallelism, which led to teleparallel (TP) gravity. components for a non-diagonal tetrad. The effective energy- In this gravity, the gravitational field is established through momentum tensor leads to the violation of energy conditions torsion using the Witzenböck connection instead of curvature which impose a condition on the normal matter to satisfy via the Levi-Civita connection as in GR. these conditions. We explore the noncommutative wormhole The extended TP theory of gravity [or f (T ) gravity where solutions by assuming a viable power-law f (T ) and shape T represents the torsion scalar] is the generalization of TP function models. For the first model, we discuss two cases gravity in the same fashion as f (R) gravity generalizes GR. in which one leads to teleparallel gravity and the other is for This is attained by replacing the torsion scalar in the action of f (T ) gravity. The normal matter violates the weak energy TP gravity by a general differentiable function f (T ). Ferraro condition for the first case, while there exists a possibility for and Fiorini [6,7] firstly introduced this theory by applying a micro physically acceptable wormhole solution. There exists BornÐInfeld strategy and solved the particle horizon problem a physically acceptable wormhole solution for the power-law as well as obtained singularity-free solutions with a positive b(r) model. Also, we check the equilibrium condition for cosmological constant. Afterwards, this theory has been stud- these solutions, which is only satisfied for the teleparallel ied extensively as regards many phenomena, like the accel- case, while for the f (T ) case, these solutions are less stable. erated expansion of the universe, solar system constraints, a discussion of Birkhoff’s theorem, thermodynamics, a recon- struction via dynamical models, static and dynamical worm- 1 Introduction hole solutions, the viability of models through a cosmo- graphic technique, instability ranges of collapsing stars, and General theory of relativity laid down the foundation of mod- many more. A vast area of research in f (T ) gravity uses a ern cosmology. The CDM model is the simplest model spherically symmetric scenario [8Ð22]. compatible with all cosmological observations but suffers A wormhole is a hypothetical path to connect different from some shortcomings, like the fine-tuning and cosmic regions of the universe, which can be regarded as a tunnel or coincidence problems. The modified theories of gravity are bridge from which an observer may traverse easily. Worm- generalized models that came into being by modifying the hole solutions are described by static as well as dynamical gravitational part in the general relativity (GR) action, while configurations. In GR, the exotic matter (i.e., violating the the matter part remains unchanged. At large distances, these energy condition) constitutes a basic ingredient in the devel- theories modify the dynamics of the universe. In another opment of the mathematical structure of the wormhole. The scenario, the modified matter part with unchanged gravita- violation of NEC is the necessary tool to form wormhole solu- tional part results in dynamical models, such as cosmolog- tions which also allow for two-way travel. Also, the inclusion ical constant, quintessence, k-essence, Chaplygin gas, and of an electromagnetic field, a scalar field, noncommutative (NC) geometry, a NC Lorentzian (NCL) distribution, [23Ð26] ae-mails: [email protected]; [email protected] etc. establish more interesting and useful results. The search be-mails: [email protected]; for a realistic source which provides the violation (while [email protected] 123 173 Page 2 of 12 Eur. Phys. J. C (2015) 75 :173 normal matter satisfies the energy conditions) has recently 4, we construct the field equations and matter content for gained a lot of interest. The modified theory of gravity is one the wormhole solutions. Section 5 contains the construction of the directions to establish realistic wormhole solutions. of wormhole solutions for particular f (T ) and b(r) func- In f (T ) gravity, the effective energy-momentum tensor tions. In Sect. 6, we check the equilibrium condition for the is the cause for the corresponding violation, while normal obtained solutions. The last section summarizes the results. matter threads the wormhole solutions. In this regard, the extra terms related to the underlying theory play an effective role in violating the energy conditions, which is necessary 2 Wormhole geometry to keep wormhole throat open to be traversable. Böhmer- break et al. [27] were the first who studied wormhole solu- One of the most fascinating features of GR is the possible tions in this gravity by taking static spherically symmetric existence of spacetimes with a non-trivial topological struc- traversable wormhole solutions and found some constraints ture. Well-known examples of this structure are described by on the wormhole throat. They explored the behavior of the Misner and Wheeler [36] and Wheeler [37] as solutions of energy conditions by taking a particular f (T ), shape, and the Einstein field equations known as wormhole solutions. redshift functions and obtained physically acceptable solu- Basically, the wormhole, having appearance as a tube, a tun- tions. Jamilbreak et al. [28] studied these solutions by tak- nel or a bridge, represents a shortcut way to communicate ing the fluid isotropic and anisotropic, as well as choosing a between two distant regions. If there exist some other paths particular equation of state; they found that the energy con- between these regions, then these are called “intra-universe”, ditions are violated in the anisotropic case, while these are otherwise “inter-universe” wormholes. The simplest exam- satisfied for the remaining two cases. Sharif and Rani [29] ple of such a connection is the EinsteinÐRosen bridge (or explored dynamical wormhole solutions for traceless as well Schwarzschild wormhole), which unfortunately fails to pro- as barotropic equations of state. With the help of analytic vide a way of communication to another region of space to and numerical f (T ) models, they concluded that the weak which it is connected. The reason behind this is that any pho- energy condition (WEC) holds in specific time intervals for ton or particle falling in it reaches the singularity at r = 0 these cases. and thus has no link with the other end of the wormhole. Rahaman et al. [30] examined the NC wormhole solutions The Lorentzian traversable wormholes are the most favor- in GR for a higher dimensional static spherically symmet- able in the sense that a human may traverse from one side of ric spacetime and found their existence in a regular way up the wormhole to the other through these wormholes. Being to four dimensions, and in a very restrictive way for five- the generalized form of the Schwarzschild wormhole (hav- dimensional space. Beyond these dimensions, the possibility ing an event horizon which permits one way travel) with no of wormhole solutions is over. Jamil et al. [31] also explored event horizon, these wormholes lead to two-way travel. Mor- f (R) wormhole solutions in the same background. Recently, ris and Thorne [38] established a static spherically symmetric Sharif and Rani have studied wormhole solutions [32]inthe wormhole spacetime given by background of NC geometry for f (T ) = αT n model as 2 ( ) dr well as the shape function. They concluded that the effec- ds2 = e2 r dt2 − − r 2dθ 2 − r 2 sin2 θdφ2. (1) − b(r) tive energy-momentum tensor serves as the basic ingredient 1 r to thread the wormhole solutions, and normal matter gives ( ) some physically acceptable solutions. They extended [33] Here, r represents the redshift (or potential) function this work to a study of the effects of an electrostatic field. which determines the gravitational redshift of a light parti- The same authors explored these solutions for galactic halo cle (photon). The magnitude of the redshift function must be regions [34] for exponential and logarithmic models but no finite everywhere for the two-way travel through the worm- ( ) physically acceptable solutions are obtained. hole. The function b r denotes the shape function which Recently, Bhar and Rahaman [35] studied the wormhole sets the shape of the wormhole. solutions in Lorentzian distribution as the density function The essential characteristics required for a wormhole in a noncommutativity-inspired spacetime. They obtained a geometry are discussed in [38,39] and given as follows. stable wormhole which is asymptotically flat in the usual four-dimensional spacetime. In order to search for some real- • The no-horizon condition [(r) must be finite everywhere istic wormhole solutions, we extended this work in extended in the spacetime] must be satisfied by the redshift function. teleparallel gravity for specific f (T ) and shape function Usually, it is taken as zero, which gives e2(r) → 1. models. The paper is organized as follows. In the next section, • The shape function must satisfy the flaring out condition we provide wormhole geometry for a static spherically sym- on the throat, i.e., to have the proper shape for a wormhole, metric spacetime and briefly discuss the energy conditions. the ratio of the radial coordinate to the shape function at Section 3 is devoted to a discussion of f (T ) gravity. In Sect. that coordinate must be 1, while this coordinate expresses 123 Eur. Phys. J. C (2015) 75 :173 Page 3 of 12 173

α a non-monotonic behavior away from the throat. Taking for any null vector k . This inequality gives ρ+pn ≥ 0, ∀n. ( ) = ( )< ρeff + eff ≥ the throat radius as r0 yields b r0 r0 andb r0 1. In an effective manner, this becomes pn 0. • The proper radial distance, τ(r) =±r √ dr , r0 1−b(r)/r r ≥ r0, should be finite throughout the space. The signs ± associate the two parts which are joined by the throat for 3 f (T) theory of gravity the wormhole configuration. • At large distances, the asymptotic flatness should be a fea- It is well known that curvature and torsion are properties of b(r) → →∞ ture of the spacetime, i.e., r 0asr . a connection and many connections may be defined on the same spacetime. The RiemannÐCartan (generalized Rieman- Notice that the MorrisÐThorne wormhole is not a particular nian) spacetime yields two interesting models of spacetime wormhole solution like the Schwarzschild wormhole, which such as Riemannian and Weitzenböck spacetimes [40,41]. depends on a single parameter, the mass of the wormhole. The concept of a curved manifold is a crucial characteris- Instead, it is a class of solutions for an arbitrarily large number tic of GR which is described by a Riemannian spacetime of redshift as well as shape functions satisfying the above having the metric tensor as the dynamical variable. It uses constraints. a curvature (Riemann) tensor to arrive at a Levi-Civita con- In order to prevent shrinking of the wormhole throat and nection whose curvature remains non-zero while the torsion to make it traversable, the matter distribution of the energy- vanishes due to its symmetry property. On the other hand, the momentum tensor at the throat must be negative. More pre- non-zero torsion with vanishing curvature corresponds to the cisely, the sum of the energy density and the pressure of mat- Weitzenböck spacetime which parallel transports the tetrad ter is negative, representing a violation of the null energy con- field. The TP theory is defined by a Weitzenböck connec- dition (NEC) and such matter is named “exotic”. It is noted tion with a tetrad field as the basic entity. This theory is an that an ordinary energy-momentum tensor satisfies the NEC. alternative description of gravity having a translation group For viability of the wormhole solutions, it is necessary to which is related to a gauge theory. The existence of non- minimize the amount of exotic matter required to support the trivial tetrad field in gauge theory leads to the teleparallelism wormhole solutions. The modified theories of gravity are one structure. The f (T ) theory of gravity is the generalization of of the sources which provide an effective energy-momentum TP theory. tensor to violate the WEC (contains NEC). In this regard, the The geometry of TP theory is uniquely defined by an usual energy-momentum tensor may satisfy this condition. orthonormal set of four-vector fields (three spacelike and one To discuss NEC and WEC, we assume an energy-momentum timelike) named the tetrad field. The simplest type of tetrad = δμ∂ , j = tensor in an appropriate orthonormal frame as field is the trivial tetrad, which has the form ei i μ e j μ i δμdx , where δμ is the Kronecker delta. However, this type αβ T = diag(ρ, p1, p2, p3), of tetrad field gives a zero torsion and is of less interest. The non-trivial tetrad field provides non-zero torsion and yields where ρ is the energy density and pn(n = 1, 2, 3) denote the the construction of TP theory. It can be written as principal pressures. μ j j ν hi = hi ∂μ, h = h νdx , (2) • WEC The relationship between the Raychaudhuri equation and attractiveness of gravity yields the WEC i μ = δi , i ν = satisfying the following properties: h μh j j h μhi ν δμ. This field establishes the metric tensor as a by-product α β Tαβ V V ≥ 0, given as follows:

α i j for any timelike vector V . In terms of the components of gμν = ηijh μh ν. (3) the energy-momentum tensor, this inequality yields The torsion scalar has the following form: ρ ≥ 0,ρ+ pn ≥ 0, ∀n. μν γ T = Sγ T μν. (4) For modiﬁed theories of gravity, we replace the matter eff content of the universe in an effective manner by Tαβ as μν The superpotential tensor Sγ (antisymmetric in its upper well as the matter components ρeff and peff . n indices) is • NEC By continuity, the WEC implies the NEC, 1 α β μν = ( μν + δμ θν − δν θμ ). Tαβ k k ≥ , Sγ K γ γ T θ γ T θ (5) 0 2 123 173 Page 4 of 12 Eur. Phys. J. C (2015) 75 :173

γ ν The torsion tensor is as follows: R =−T −2∇ T γν, where R is a covariant scalar, while T γ ν as well as ∇ T γν are covariant scalars but not local Lorentz γ γ γ γ i i T μν = νμ − μν = hi (∂ν h μ − ∂μh ν), (6) invariant. However, the latter scalar can be neglected inside the integral for TP theory and it becomes equivalent to GR. γ which is antisymmetric in its lower indices, i.e., T μν = The non-invariant theories might be sensitive in order to γ −T νμ. The contorsion tensor can be defined as choose a reasonable tetrad not uniquely determined by the given metric gμν. In general, one comes across a more com- γ 1 γ γ γ K μν = [Tμ + Tν μ − T μν]. (7) 2 ν plicated connection between the tetrad and metric, partic- ularly for a non-diagonal tetrad with a diagonal metric (or To formulate a suitable form of the field equations which even sometimes a diagonal tetrad). Different field equations establishes the equivalent description (up to equations level) are developed by taking a different tetrad, which successively of these theories, we follow the covariant formalism [42Ð45]. induces distinct solutions. In an appropriate limit, some of Incorporating the above equations and after some algebraic these solutions lead to GR counterparts, while others fail to manipulations, it follows that provide a valid GR counterpart. Thus, the choice of tetrad is f (T ) 1 γ σγ a crucial point in theory and needs particular attention. Gμν − gμν T =−∇ Sνγμ − S μ Kγσν, (8) 2 When we deal with spherical coordinates, the diagonal tetrads become unsuitable as they provide some constraints 1 where Gμν = Rμν − gμν R is the Einstein tensor. Finally, on the T and f (T ) models [46]. We obtain an unwanted 2 ˙ we obtain the following field equations for f (T ) gravity: condition, TfTT = 0(orsimply fTT = 0), which yields a constant torsion scalar, or f (T ) = c1 + c2T represent- 1 2 ing TP theory. Thus, the diagonal tetrads do not represent a fT Gμν + gμν( f − TfT ) + Eμν fTT = κ Tμν, (9) 2 useful choice for spherical symmetry. In order to search for ( ) γ a realistic source toward wormhole solutions in f T grav- Eμν = Sνμ ∇γ T where . It is noted that Eq. (9) possesses an ity, we assume the following non-diagonal tetrad for static f (R) equivalent structure like gravity and reduces to GR for spherically symmetric wormhole spacetime (1): f (T ) = T . The trace equation is used to constrain and sim- plify the field equations. Here, the trace of the above equation i is h μ ⎛ ( ) ⎞ e2 r 000 2 ⎜ 1 θ φ θ φ − θ φ ⎟ EfTT − (R + 2T ) fT + 2 f = κ T , ⎜ 0 sin cos r cos cos r sin sin ⎟ ⎜ 1− b ⎟ ⎜ r ⎟ = ⎜ 1 θ φ θ φ θ φ ⎟ . = ν T = T ν ⎜ 0 sin sin r cos sin r sin cos ⎟ with E E ν and ν. In terms of an effective energy- ⎜ 1− b ⎟ ( ) ⎝ r ⎠ momentum tensor, the f T field equations can be rewritten 0 1 cos θ −r sin θ 0 − b as 1 r

2 eff 2 f T Gμν = κ Tμν = κ (Tμν + Tμν). (10) The torsion scalar turns out to be T m 2 b b f μν T =− − − + The term Tμν = corresponds to the matter ﬂuid, while, 2 1 1 fT r r r using the trace equation, the torsion contribution is given by 1 b b − 2 1 − 1 − − . (12) T 1 1 r r r Tμν = −Eμν fTT − gμν(T − EfTT + RfT ) . κ2 f 4 T We assume an anisotropic energy-momentum tensor as (11) μ μ μ μ T ν = (ρ + pt )U Uν − pt δν + (pr − pt )V Vν, (13) 4 Field equations for wormhole construction where pr and pt are the radial and transverse components μ Gravitational theories [like f (R) theory] developed through of the pressure. The four-velocity of the ﬂuid U and the μ μ μ the metric tensor as well as its dependent quantities are unit spacelike vector V satisfy U Uμ = 1, V Vμ = μ local Lorentz invariant. Being the basic entity in f (T ) grav- −1, U Vμ = 0. The corresponding energy-momentum ity, the torsion tensor (taking a tetrad) which induces the tensor is TP structure on spacetime fails to be invariant under local μ Lorentz transformations [42Ð45]. This can be seen from T ν = diag(ρ, −pr , −pt , −pt ). 123 Eur. Phys. J. C (2015) 75 :173 Page 5 of 12 173

Equation (9) yields the following field equations: It is noted that Eq. (17) gives the violation of the NEC in terms of the flaring out condition. ρ 1 b b f J b − 1 − − 1 − T TT − = , (14) f r r r f f r 2 T T T p J 1 b b 5 Wormhole solutions r + = 2r 1 − − , (15) f f r 2 r r T T Noncommutative geometry is the intrinsic characteristic of p 1 b 1 b b t + 1 − − 1 − − 1 + spacetime and it plays an effective role in several areas. f 2 r r r r T To formulate the NC form of GR, the coordinate coherent f J 1 b state approach is widely used. The NC geometry is used × T TT + = 2r − b − b + + 2 r 2 f f 2r 2 r to eliminate the divergencies that appear in GR by replac- T T ing the point-like structures with smeared objects. Taking a 2 2 2 − 2 rb + 2 r − 2 rb − rb , (16) Lorentzian distribution, the energy density of the particle- like static spherically symmetric gravitational source having where the prime refers to the derivative with respect to r and mass M takes the following form [47]: J(r) is given by √ M φ ρ = , (22) 1 NCL π 2( 2 + φ)2 J(r) = (T − Ef + Rf ). r 4 TT T where φ is the NC parameter. Taking into account the corre- ρ ρ Taking the effective energy density and the pressure from spondence between and NCL using Eqs. (18) and (22), we Eqs. (14) and (15), the NEC yields obtain the following differential equation: √ b 1 b b M φ b r − b 2 b f + 1 − − 1 − T f = , ρeff + peff = + 1 − . (17) r 2 T r r r TT π 2(r 2 + φ)2 r r 3 r r (23) Due to the flaring out condition, we obtain b > b r, leading which contains two unknown functions, b and f . Thus, we < ρeff + eff ≤ to the violation of the NEC with 0, i.e., pr 0, have to choose one of these functions and carried out steps which implies that the effective energy-momentum tensor is for the other one. We discuss the NCL wormhole solutions responsible for the necessary violation of the energy condi- in f (T ) gravity for a non-diagonal tetrad and investigate the tions to support the wormhole geometry. Thus, it may impose behavior of the energy conditions for specific f (T ) and shape a condition on the usual matter to satisfy the energy condi- functions. tions in this scenario and establish some physically acceptable solutions. 5.1 For power-law f (T ) model To be a traversable wormhole solution, the magnitude of its redshift function must be finite. Also, up to equation level We assume a model in power-law form of the torsion scalar for a constant value of other than zero, we note that only which is the generalization of GR and an analogy to f (R) = is used for which constant gives the same scenario. model like f (R) = R + δR2 taken to discuss the wormhole = For the sake of simplicity, setting 0inEqs.(14)Ð(16), solutions. The f (T ) model is the field equations can be written as 2 f (T ) = T + αT . (24) b 1 b b ρ = f + 1 − − 1 − T f , (18) r 2 T r r r TT This model has contributed as the most viable model due b to its simple form and we may directly compare our results p =− f , r 3 T (19) with GR. We discuss wormhole solutions for the following r two cases. 1 b b 1 b pt = 1 − − 1 + T f − b − f , 2r r r TT 2r 2 r T Case I: α = 0 (20) α = where the torsion scalar becomes We consider 0 in model (24) which leads to the whole scenario in teleparallel gravity. In this case, Eq. (23) becomes √ 2 b b φ T = − − − . b M 2 2 2 1 (21) = , (25) r r r r 2 π 2(r 2 + φ)2 123 173 Page 6 of 12 Eur. Phys. J. C (2015) 75 :173

i ii

1.8 12

10 1.6 8

b 1.4 r b 6

1.2 4

2 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.2 0.4 0.6 0.8 1.0 r r

iii iv 1.0 0.5 0.5 0.4

r 0.0 0.3 dr b db 0.2 0.5 0.1 1.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 r r

b − db Fig. 1 Plots of shape function in the teleparallel case: evolution of i b versus r, ii r versus r, iii b r versus r, iv dr versus r

yielding the solution Equations (18)Ð(20) are given as follows: √ √ M φ M r φ − r ρ = , (27) b(r) = + tan 1 √ + c, (26) π 2(r 2 + φ)2 π 2 2 + φ φ √ 2 r φ 1 M r −1 r pr =− + tan √ + c , (28) r 3 2π 2 r 2 + φ φ where c is an arbitrary constant. √ 1 M φ In order to examine the geometry of the wormhole, we pt =− 2r 2 π 2(r 2 + φ)2 draw the shape function taking different conditions in Fig. 1. √ We choose arbitrarily the values of the model parameters, 1 M r φ − r − + tan 1 √ + c . (29) such as φ = 0.5, M = 15, and c = 1. Figure 1i represents r 2π 2 r 2 + φ φ the evolution of the shape function in an increasing manner ρ, ρ + ρ + versus r. For an asymptomatically flat condition, we draw b The behavior of WEC ( pr , and pt )versusr is r ρ, ρ+ p with respect to increasing r. It can be seen from plot ii that b showninFig.2. The curves of t represent a positively r r ρ+ p approaches 0 as r → 0. This implies that the asymptotically decreasing behavior for increasing , while r indicates a flat condition for the wormhole construction is satisfied. In negative behavior, which shows the violation of WEC. Thus plot iii, we draw b − r versus r to find the throat radius. It is no physically acceptable wormhole solution is obtained in noted that the throat radius is the minimum value for which the teleparallel case. These results are compatible with the b−r cuts the r-axis. In this case, the throat radius is obtained results of [35]. = . − b > as r0 1 6. This plot satisfies the condition 1 r 0for r > r0.InFig.1iv, we plot the first derivative of b(r) with Case II: α = 0 respect to r to check the validity of the condition b (r0)<1, which shows that the corresponding condition is satisfied. The case α = 0 leads to the extended teleparallel gravity. Thus, the shape function satisfies the required structure of Inserting Eq. (24)inEq.(23), we obtain the following dif- the wormhole. ferential equation: 123 Eur. Phys. J. C (2015) 75 :173 Page 7 of 12 173

i ii 0 4 2

3 4 6 2

r 8 p 10 1 12

14 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r r iii

t 10 p

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 r

Fig. 2 Plots of WEC in teleparallel case: i ρ versus r, ii ρ + pr versus r, iii ρ + pt versus r √ 2 r M φ 4α b b teleparallel case, the shape function satisfies the conditions = b 1 + 2 − − 2 1 − π 2(r 2 + φ)2 r 2 r r of wormhole geometry. To check the behavior of WEC for power-law model, the 4α b b − 1 − − 1 − expressions of matter content using Eqs. (18)Ð(20)aregiven r 2 r r by ⎡ b 4α b b b 2b ρ = 1 + 2 − − 2 1 − × ⎣4 1 − 1 − − r 2 r 2 r r r r ⎡ ⎛ ⎞⎤ 4α b b b 2b − 1 − − 1 − ⎣4 1 − 1 − − − r 2 r r r r + b r b ⎝ − 1 ⎠⎦ . 1 (30) ⎛ ⎞⎤ r 1 − b r br − b 1 + ⎝1 − ⎠⎦ , (31) r − b We check the behavior of the shape function and the flaring 1 r out conditions numerically through the graphs by taking the b 4α b b same values of parameters along with three different values of p =− + − − − , r 3 1 2 2 2 1 (32) α such as α =−2, −3, −5 and the initial value as f (2.2) = r r r r 1. Figure 3i represents an increasing behavior of the shape 1 b 4α b b b function versus r. In the right graph, ii, versus r shows that pt =− b − 1 + 2 − − 2 1 − r 2r 2 r r 2 r r b ⎡ r approaches zero as we increase r, which represents that asymptomatically flatness is obtained. To locate the throat of 2α b b b 2b − 1 − − 1 − ⎣4 1 − 1 − − the wormhole, we plot b(r) − r versus r asshowninplot r 4 r r r r iii of Fig. 3. In this plot, the throat is located at very small ⎛ ⎞⎤ values of r. The first derivative of the shape function is also − db + b r b ⎝ − 1 ⎠⎦ . plotted versus r as shown in plot iv, which indicates that dr 1 (33) ( )< r − b at r0 satisfies the condition b r0 1. Thus, similar to the 1 r 123 173 Page 8 of 12 Eur. Phys. J. C (2015) 75 :173

i ii

0.5 0.8 0.4 0.6 0.3 b r

0.4 b 0.2 0.2 0.1 0.0 0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 r r

iii iv 0.001

0.6 0.000

0.4 0.001 r 0.2 dr db b 0.002 0.0 0.003 0.2 0.004 0.0000 0.0005 0.0010 0.0015 0.0020 0.0 0.5 1.0 1.5 2.0 r r

Fig. 3 Plots of shape function in f (T ) case: red curve for α =−2, green curve for α =−3, blue curve for α =−5. Evolution of i b versus r, b − db ii r versus r, iii b r versus r, iv dr versus r ⎡ ⎤ Figure 4 represents a graph of the WEC expressions versus r, γ −1 γ −1 1 ⎣ − r − − r ⎦ f which show that ρ and ρ + pt show a decreasing behavior but 1 1 r r0 r0 T remain positive. For α =−2,ρindicates a negative value at r ≤ 0.52. The behavior of ρ + p is negative but there exists γ γ −1 r + r f − 1 some part of the curves in the positive panel of the plot. Thus r 2 r T r ⎡ 0 ⎤ there exists a possibility to have a micro or tiny wormhole. γ −1 γ −1 × ⎣ − r − − r ⎦ T f 1 1 2 5.2 For power-law b(r) model r0 r0 T √ M φ Here, we examine the wormhole solution by considering a = , (35) π 2(r 2 + φ)2 speciﬁc shape function in terms of the power-law form and construct the f (T ) function in the NCL background. We take where the following particular shape function [31,50,51]: ⎡ ⎤ γ − γ − γ 2 r 1 r 1 r T = ⎣2 − − 2 1 − ⎦ , b(r) = r0 , (34) r 2 r r r 0 0 0 ⎡ ⎛ ⎞ ⎢ γ −1 γ −1 where γ is any constant. To meet the wormhole geometry, it 2 r r T =− ⎢ ⎝ − − ⎠ − 3 ⎣4 1 1 2 can be seen that b (r0)<1 implies that b (r0) = γ<1 and r r0 r0 b(r0) = r0 hold. The asymptotically ﬂat spacetime is also ( ) −γ ⎛ ⎞⎤ b r = r 1 r γ −1 → obtained for this shape function, i.e., r 0 0 →∞ ρ ( ) γ −1 ⎜ ⎟⎥ as r . Substituting the values of NCL and b r from Eqs. r ⎜ 1 ⎟⎥ + (γ − 1) ⎝1 − ⎠⎦ , (22) and (34)inEq.(18), we obtain the following differential r0 γ −1 1 − r equation: r0 123 Eur. Phys. J. C (2015) 75 :173 Page 9 of 12 173

i ii 2.5 0.02

2.0 0.01 0.00 1.5

r 0.01 p 1.0 0.02 0.03 0.5 0.04

0.0 0.05 0.5 1.0 1.5 2.0 0 2 4 6 8 10 12 14 r r

iii

2 t p 1

1 0.0 0.5 1.0 1.5 2.0 r

Fig. 4 Plots of WEC in f (T ) case: red curve for α =−2, green curve for α =−3, blue curve for α =−5. Evolution of i ρ versus r, ii ρ + pr versus r, iii ρ + pt versus r

⎡ ⎛ ⎞ γ −1 γ −1 γ − 1 r f 1 r ⎢ γ −1 γ −1 ρ + p = + 1 − 12 r r r 2 T = ⎢ ⎝ − − ⎠ − r r0 T r r0 4 ⎣2 1 1 r r0 r0 ⎞ γ − r 1 f T f ⎛ ⎞ − − ⎠ − , 1 2 (37) r0 T T γ −1 ⎜ ⎟ (γ − 1)(γ − 6) r ⎜ 1 ⎟ − ⎝1 − ⎠ γ −1 γ −1 6 r γ −1 γ + 1 r f 1 r 0 r ρ + p = + − 1 − t 2 1 r0 2r r0 T 2r r0 ⎤ ⎞ − 3 2 2(γ −1) γ −1 2 γ − (γ − 1) r r ⎦ r 1 f T f + 1 − . − 1 − ⎠ − . (38) 12 r r 2 0 0 r0 T T

We evaluate the f (T ) function numerically and draw its plot Figure 5(iiÐiv) shows the plots of these expressions versus r. as well as WEC versus T and r, respectively, as shown This shows that ρ, ρ+ pr ,ρ+ pt indicate a positive behavior in Fig. 5. Keeping the same values of NCL parameters of these expressions. Thus, physically acceptable wormhole along with γ = 0.5 for three values of the throat radius, solutions are obtained for the speciﬁc shape function where = . , . , . r0 0 93 0 95 0 99. The plot i indicates the positively the basic role is played by the effective energy-momentum decreasing behavior of f . tensor. The expressions for the WEC become γ γ −1 γ −1 ρ = r f + 1 − r 6 Equilibrium conditions 2 1 r r0 T r r0 ⎞ γ − To discuss the equilibrium conﬁguration of the wormhole r 1 f T f − − ⎠ − , solutions, we consider the generalized TolmanÐOppenheimer 1 2 (36) r0 T T ÐVolkov equation [48,49], 123 173 Page 10 of 12 Eur. Phys. J. C (2015) 75 :173

i ii 0.4 0.8 0.3 0.6

T 0.2

f 0.4

0.2 0.1

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.0 1.5 2.0 2.5 3.0 T r

iii iv 0.8 0.20 0.7 0.15 0.6 r t p p 0.5 0.10

0.4 0.05

0.3 0.00 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 r r

Fig. 5 Plots of f (T ) and WEC: red curve for r0 = 0.93, green curve for r0 = 0.95, blue curve for r0 = 0.99. Evolution of i f (T ) versus T , ii ρ versus r, iii ρ + pr versus r, iv ρ + pt versus r

σ d pr 2 constant . This shows that Fgf becomes zero and we are + (ρ + pr ) + (pr − pt ) = 0, (39) dr 2 r left with hydrostatic and anisotropic forces with the corresponding equilibrium condition Fhf + Faf = 0. For the for the metric ds2 = diag(eσ(r), −eμ(r), −r 2, −r 2 sin2 θ). teleparallel case, speciﬁcally the f (T ) and b(r) cases, Fhf Ponce de León suggested this equation for an anisotropic and Faf take the following form, respectively: mass distribution as follows:

μ−σ e 2 M p √ 2 eff d r φ (pt − pr ) − (ρ + pr ) − = 0. (40) 3 M r −1 r r r 2 dr Fhf =− + tan √ + c r 4 2π 2 r 2 + φ φ σ−μ 1 2 √ Here M = r e 2 σ is the effective gravitational mass 3 eff 2 1 M φ(5r + 3φ) 3c 3M − r which is measured from the throat to some arbitrary radius r. + − tan 1 √ , r 3 2π 2r 3(r 2 + φ)2 r 4 2πr 4 φ This equation indicates the equilibrium conﬁguration for the √ √ φ φ wormhole solutions by taking gravitational, hydrostatic as F =−1 M − 3 M r af 3 π 2( 2 + φ)2 π 2 2 + φ well as anisotropic force due to an anisotropic matter distri- r r r 2 r bution. Using Eq. (40), these forces are deﬁned, respectively, − r + tan 1 √ + c , as φ σ (ρ + pr ) d pr 2(pt − pr ) b 3b 4α b b Fgf =− , Fhf =− , Faf = . F = − 1 + 2 − − 2 1 − 2 dr r hf r 3 r 4 r 2 r r For the wormhole solutions to be in equilibrium, it is required ⎡ that b b b + ⎣ − 8α 2 − − 2 1 − r 6 r r Fgf + Fhf + Faf = 0. (41) ⎛ ⎞⎤ 4α(br − b) 1 It is noted here that σ represents the gravitational redshift, − ⎝1 − ⎠⎦ , σ = σ = r − b which is taken constant, i.e., 2 leads to 0for 1 r 123 Eur. Phys. J. C (2015) 75 :173 Page 11 of 12 173 ⎡ Evolution of Faf and Fhf 1 3b 4α F =− ⎣ b − 1 + 2 1.0 af r 3 r r 2 0.5 b b 4α b 0.0 − − 2 1 − + − 1

r r r r hf 0.5 ⎡ F , af

b b 2b F 1.0 + 1 − ⎣4 1 − 1 − − r r r 1.5 ⎛ ⎞⎤⎤ 2.0 br − b 1 + ⎝1 − ⎠⎦⎦ , 2.5 r − b 0.5 1.0 1.5 2.0 1 r r γ − γ − γ − 3 r 1 f 1 r 1 f T f F = + − , F F ( ) hf 3 2 2 Fig. 7 Plot of hf (dotted)and af (solid) in speciﬁc f T case versus r r0 T r r0 T T ⎛ r. Evolution of Faf and Fhf γ − γ − 3 − γ r 1 f 1 r 1 F = + ⎝ Evolution of F and F forces af 3 2 hf af r r0 T r r0 0.5 ⎞ γ − r 1 f T f − + − ⎠ − . 0.0 1 1 2 r0 T T af F

, 0.5 hf We plot these equations for the obtained wormhole solu- F tions as shown in Figs. 6, 7, and 8, respectively. For the 1.0 teleparallel case, we examine that hydrostatic and anisotropic forces show the same behavior but in opposite directions and 1.5 thus balance each other. This implies that the wormhole solu- 1.5 2.0 2.5 3.0 tion in the teleparallel case satisfies the equilibrium condi- r tion. In the case of a power-law f (T ) function, this condition shows wormhole solutions in an equilibrium state as r Fig. 8 Plot of Fhf (dotted)andFaf (solid) in specific b(r) case versus F F increases. For smaller values of r, these solutions do not sat- r. Evolution of hf and af forces isfy the equilibrium condition properly or we may remark ( ) that these solutions are less stable. For a specific b r func- ity. Therefore the physically acceptable wormhole solutions F F tion, the forces hf and af do not balance each other. Since are not in equilibrium form. all the curves behave in opposite manners there is no similar- 7 Conclusion Evolution of Fhf and Faf 2 A wormhole represents a shortcut distance to connect different regions of the universe. To study these solutions, the 1 violation of NEC plays the key role which is associated with the exotic matter. To minimize the usage of exotic matter is an af F

, 0 important issue, which leads one to explore a realistic model hf

F in favor of the wormhole. In this paper, we have studied static spherically symmetric wormhole solutions in f (T ) gravity 1 by taking a NCL background. We have developed the f (T ) ﬁeld equations in terms of an effective energy-momentum 2 tensor by taking a non-diagonal tetrad and proved that this 1.0 1.5 2.0 2.5 3.0 3.5 4.0 tensor is responsible for the WEC violation. By imposing r the condition on the matter content to thread the wormhole ( ) Fig. 6 Plot of Fhf and Faf in teleparallel case versus r: red curve solutions, we have assumed either the f T or the shape func- represents Fhf and blue curve represents Faf . Evolution of Fhf and tion and constructed the other one. The graphical behavior F af of these solutions is discussed. 123 173 Page 12 of 12 Eur. Phys. J. C (2015) 75 :173

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