Non-minimal Coupling of Torsion-matter Satisfying Null Energy Condition for Wormhole Solutions
aa r X i v : . [ g r- q c ] D ec Non-minimal Coupling ofTorsion-matter Satisfying Null EnergyCondition for Wormhole Solutions
Abdul Jawad ∗ and Shamaila Rani † Department of Mathematics, COMSATS Institute ofInformation Technology, Lahore-54000, Pakistan.
Abstract
We explore wormhole solutions in a non-minimal torsion-mattercoupled gravity by taking an explicit non-minimal coupling betweenthe matter Lagrangian density and an arbitrary function of torsionscalar. This coupling depicts the transfer of energy and momentumbetween matter and torsion scalar terms. The violation of null en-ergy condition occurred through effective energy-momentum tensorincorporating the torsion-matter non-minimal coupling while normalmatter is responsible for supporting the respective wormhole geome-tries. We consider energy density in the form of non-monotonicallydecreasing function along with two types of models. First model isanalogous to curvature-matter coupling scenario, that is, torsion scalarwith T -matter coupling while the second one involves a quadratic tor-sion term. In both cases, we obtain wormhole solutions satisfying nullenergy condition. Also, we find that the increasing value of couplingconstant minimizes or vanishes the violation of null energy conditionthrough matter. Keywords:
Wormhole; Non-minimal coupling; Torsion; f ( T ) gravity; En-ergy conditions. PACS: ∗ [email protected], [email protected] † [email protected], [email protected] Introduction
The topological handles which connect distant regions of the universe as abridge or tunnel is named as wormhole. The most amazing thing is thetwo-way travel through wormhole tunnel which happened when throat re-mains open. That is, to prevent the wormhole from collapse at non-zerominimum value of radial coordinate. In order to keep the throat open, theexotic matter is used which violate the null energy condition and elaboratedthe wormhole trajectories as hypothetical paths. The violation of null energycondition is the basic ingredient to integrate wormhole solutions. This con-figuration is firstly studied by Flamm [1] and then leads Einstein and Rosen[2] to contribute into the successive steps for the construction of wormholesolutions. The work of Morris and Thorne [3] evoked the wormhole scenarioand leads to the new directions. The usual matter are considered to satisfythe energy conditions, therefore, some exotic type matter is employed forthese solutions. There exists some wormhole solutions in semi-classical grav-ity through quantum effects such as Hawking evaporation and Casimir effects[3, 4] where energy conditions are violated. One may take some such typesof matter which acted as exotic matter, for instance, phantom energy [5],tachyon matter [6], generalized Chaplygin gas [7], some non-minimal kineticcoupling, etc.In order to find some realistic sources that support the wormhole ge-ometry or minimize the usage of exotic matter, different variety of worm-hole solutions are explored. This includes thin-shell, dynamical and rotatingwormholes [8]. However, the concentration goes towards modified theories ofgravity where the effective scenario gives the violation of null energy condi-tion and matter source supports the wormholes. Bronnikov and Starobinsky[9] proved a general no-go theorem which yields no wormholes (static or dy-namic) can be constituted in any ghost-free scalar-tensor theory of gravity(which also includes f ( R ) gravity as a particular case), under some conditionson the non-minimal coupling function. In f ( R ) gravity, Lobo and Oliveira[10] found that the higher curvature terms in effective energy-momentum ten-sor are responsible for the necessary violation for the wormhole solutions inghost-containing scenario. They assumed a particular shape function alongwith various fluids to check the validity of energy conditions. Jamil et al. [11]discussed several static wormhole solutions in this gravity with a noncom-mutative geometry background through Gaussian distribution. They con-sidered power-law solution as first and construct wormhole geometry as well2s analyzed the validity of energy conditions. Secondly, they explored thesesolutions with the help of shape function. Taking into account Lorentziandistribution of energy density, Rahaman et al. [12] derived some new exactsolutions under same manner and gave some viable wormhole solutions.In extended teleparallel gravity being the modification of teleparallel grav-ity [13, 14], static as well as dynamical wormhole solutions are also explored.In this way, B¨ohmer et al. [15] investigated wormhole solutions in this grav-ity by taking some specific f ( T ) forms, shape function as well as redshiftfunctions which are the basic characteristics of these solutions. Assumingdifferent fluids such as baro-tropic, isotropic and anisotropic, Jamil et al.[16] enquired the possibility of some realistic sources for wormhole solutions.Sharif and Rani explored the wormhole solutions in this gravity taking non-commutative background with Gaussian distribution [17], dynamical worm-hole solutions [18], for the traceless fluid [19], with the inclusion of charge[20] and galactic halo scenario [21]. They considered some power-law f ( T )functions for which effective energy-momentum tensor depending on torsioncontributed terms violated the null energy condition. Recently, Jawad andRani [22] constructed wormhole solutions via Lorentzian distribution in non-commutative background. They concluded that their exist the possibility ofsome realistic wormhole solutions satisfying energy conditions and stayed inequilibrium. We also studied some higher dimensional wormhole solutions inEinstein Gauss-Bonnet gravity [23].The modification of theories taking some non-minimal coupling betweenmatter and curvature becomes center of interest now-a-days. There existsuch theories involving these coupling as f ( R ) = f ( R ) + [1 + λf ( R )] L m and f ( R, L m ), [24] etc. Harko et al. [25] introduced the most general conditionsin the framework of modified gravity, that is the matter threading the worm-hole throat satisfies all of the energy conditions while the gravitational fluid(such as higher order curvature terms) support the nonstandard wormholegeometries. They explicitly showed that wormhole geometries can be theo-retically constructed without the presence of exotic matter but are sustainedin the context of modified gravity. Taking into account some specific casesof modified theories of gravity, namely, f ( R ) gravity, the curvature-mattercoupling and the f ( R, L m ) generalization, they showed explicitly that onemay choose the parameters of the theory such that the matter threading thewormhole throat satisfies the energy conditions.Following the same scenario as for f ( R ) theory, Harko et al. [26] pro-posed non-minimal torsion-matter coupling as a extension of f ( T ) gravity.3n this gravity, two arbitrary functions f ( T ) and f ( T ) are introduced suchas f is the extension of geometric part while f is coupled with matter La-grangian part through some coupling constant. They discussed this theoryfor cosmological aspects of evolving universe and deduced that the universemay represent quintessence, phantom or crossing of phantom-divide line, in-flationary era, de Sitter accelerating phase, in short, a unified description.In this gravity, Nashed [27], Feng et al. [28] and Carloni et al. [29] studiedspherically symmetric solutions, cosmological evolutions and compared theirresults with observational data and phase space analysis, respectively. Garciaand Lobo [30] explored wormhole solutions by non-minimal curvature-mattercoupling taking linear functions, f = R, f = R . They concluded that thewormhole solutions in realistic manner depends on higher values of couplingparameter.The paper has following symmetry. In next section 2, we describe the f ( T ) gravity and non-minimal torsion-matter coupling extension. Section 3 isdevoted to the gravitational field equations for wormhole geometry in the cou-pling scenario. We find the general conditions on matter part for the validityof null energy condition. Also, we examined the effective energy-momentumtensor being the responsible for violation of null energy condition. In section4, we explore wormhole solutions taking into account two well-known mod-els. These models involving linear torsion scalar coupled with T -matter andquadratic torsion term with matter represented wormhole geometry. In thelast section, we summarize the results. In this section, we mainly review the torsion based gravitational paradigm.The torsional scenario begins with a spacetime undergoing the absolute par-allelism where the parallel vector field h β a determinesΓ abc = h βa h βb,c , (1)which is a non-symmetric affine connection with vanishing curvature and h βa,b = ∂ b h βa . Here we apply the Latin indices as the notation for tangentspacetime while coordinate spacetime indices are presented with the helpof Greek letters. The parallel vector fields (vierbein or tetrad fields) are thebasic dynamical variables which deduce an orthonormal basis for the tangentspace, i.e., h a · h b = η ab where η = diag(1 , − , − , − h βa , the vierbein fields are defined as h a = h βa ∂ β while theinverse components h aβ meet the following conditions h aβ h bβ = δ ab , h aβ h aα = δ βα . The metric tensor is obtained through the relation g βα = η ab h aβ h bα whichgives the metric determinant as √− g = e = det( h aβ ). Using Eq.(1), theantisymmetric part of Weitzenb¨ock connection yields T βασ = e Γ βσα − e Γ βασ = h aβ ( ∂ σ h aα − ∂ α h aσ ) , (2)where T βαγ = − T βγα , i.e., it is antisymmetric in its lower indices. It is notedthat the under the parallel transportation of vierbein field, the curvature ofthe Weitzenb¨ock connection vanishes. Using this tensor, we obtain contorsiontensor as K αγβ = − ( T αγβ − T γαβ − T αγβ ) and superpotential tensor as S βαγ = [ δ αβ T µγµ − δ γβ T µαµ + K αγβ ]. The torsion scalar takes the form T = S βαγ T βαγ . (3)The action of f ( T ) gravity is given by [13, 14] S = 116 π G Z e { f ( T ) + L m } d x. (4)The G is the gravitational constant and f represents the generic differentiablefunction of torsion scalar describing extension of the teleparallel gravity. Theterm L m describes the matter part of the action such as L m = L m ( ρ, p )where ρ, p are the energy density and pressure of matter while we neglectthe radiation section for the sake of simplicity. By the variation of this actionw.r.t vierbein field, the following field equations come out h aβ S βασ ∂ α T f TT + [ h aβ T λαβ S λσα + 1 e ∂ α ( hh aβ S β ασ )] f T + 14 h aσ f = 4 π G h aβ Θ σβ , (5)where the subscripts involving T and T T represent first and second orderderivatives of f with respect to T respectively. For the sake of simplicity, weassume 8 π G = 1 in the following.The f ( T ) field equations in terms of Einstein tensor gain a remarkableimportance in order to discuss various cosmological and astrophysical sce-narios [13, 14]. This type of field equations take place by replacing partial5erivatives to covariant derivatives along with compatibility of metric ten-sor, i.e., ∇ σ g βα = 0. Using the relations T µ ( νγ ) = K ( µν ) γ = S µ ( νγ ) = 0, thetorsion, contorsion and superpotential tensor become T σβα = h σa ( ∇ β h aα − ∇ α h aβ ) , K σβα = h σa ∇ α h aβ ,S σαβ = η ab h βa ∇ σ h αb + δ ασ η ab h τa ∇ τ h βb − δ βσ η ab h τa ∇ τ h αb . The curvature tensor referred to Weitzenb¨ock connection vanishes, while theRiemann tensor related with the Levi-Civita connection Γ γµν is given by R σβλα = ∇ α K σβλ − ∇ λ K σβα + K στα K τ βλ − K στλ K τ βα . We obtain the Ricci tensor and scalar as follows R βα = −∇ σ S ασβ − g βα ∇ σ T λσλ − S σλβ K λσα , R + T = − ∇ σ T ασα , (6)where we use S ασα = − T ασα = 2 K ασα . Substituting Eq.(6) along withEinstein tensor G βα = R βα − g βα R , we get G βα − g βα T = −∇ σ S ασβ − S τσβ K στα , (7)Finally, inserting this equation in Eq.(5), it yields f T G βα + 12 g βα ( f − T f T ) + Z βα f T T = Θ βα , (8)where Z βα = S αβ σ ∇ σ T . This equation expresses a similar structure like f ( R )gravity at least up to equation level and representing GR for the limit f ( T ) = T . We take trace of the above equation, i.e., Zf T T − ( R + 2 T ) f T + 2 f = Θ,with Z = Z αα and Θ = Θ αα to simplify the field equations. The f ( T ) fieldequations can be rewritten as G βα = 1 f T (cid:2) Θ mβα − Z βα f TT −
12 ( f − T f T ) (cid:3) , (9)where Θ mβα is the energy-momentum tensor corresponding to matter La-grangian.Taking into account non-minimal coupling between torsion and matter,Harko et al. [26] defined the action as follows S = Z e { f + (1 + ωf ) L m } d x, (10)6here ω is the coupling constant having units of mass − and f , f are ar-bitrary differential functions of torsion scalar. Applying the tetrad variationon this action, we obtain the following set of equations h aβ S βασ ∂ α T ( f ′′ + ωf ′′ L m ) + [ h aβ T λαβ S λσα + 1 e ∂ α ( hh aβ S βασ )]( f ′ + ωf ′ L m )+ 14 h aσ f − ω∂ α T h aβ T σαβ f ′ + ωf ′ h aβ S βσα ∂ α L m = 12 (1 + ωf ) h aβ Θ σβ , (11)where the number of primes denotes the correspondingly order derivativewith respect to torsion scalar and T σαa = ∂ L m ∂∂ α h aσ . We assume here that thematter Lagrangian L m is independent of derivatives of tetrad which resultsthe vanishing of T σαa . Also, the Bianchi identities of teleparallel gravityexpress the following relationship ∇ β Θ βτ = 41 + ωf K λβτ S λβα ∇ α ( f ′ + ωf ′ L m ) − ωf ′ ωf (Θ βτ −L m δ βτ ) ∇ β T. (12)This equation represents the substitute of energy and momentum betweentorsion and matter through defined coupling. The f ( T ) field equations forthe torsion-matter coupling in the form of Einstein tensor are given by G βα = 1 f T + ωf T L m (cid:20) Θ + βα − S αβ λ ∇ λ ( f TT + ωf TT L m ) − J g βα (cid:21) , (13)where J = 12 (cid:26) f − T ( f T + ωf T L m ) (cid:27) , Θ + βα = (1 + ωf )Θ mβα − ωf T S αβλ ∇ λ L m . It is noted that the matter Lagrangian density needs to be properly defined inthe analysis of torsion-matter coupling. In the literature for curvature-mattercoupling, the proposals for this matter Lagrangian density are as follows [30].(i) L m = p which reproduce the equation of state for perfect fluid and provedas not unique.(ii) L m = − ne where e denotes physical free energy defined by e = − T s + ρn , T is the temperature, s being the entropy of one particle and n gives theparticle number density.(iii) L m = − ρ representing a natural choice which gives the energy in a localrest frame for the fluid. 7 Gravitational Field Equations for Worm-hole Geometry
In order to construct the gravitational field equations in the framework oftorsion-matter coupling, we firstly describe the wormhole geometry. Let usassume the wormhole metric as follows [17]-[23] ds = e r ) dt − − br dr − r dφ − r sin φdψ , (14)where redshift function Ψ and shape function b are r dependent functions.In order to set geometry of wormhole scenario, some constraints are requiredon both of these functions. These are describe as follows. • Shape function : The shape of the wormhole consists of two openmouths (two asymptomatically flat regions) in different regions of thespace connected through throat which is minimum non-zero value ofradial coordinate. This shape maintains through the shape function b ( r ) with increasing behavior and having ratio 1 with r . At throat, itmust holds b ( r ) = r as well as 1 − b ( r ) r ≥
0. The flare-out conditionbeing the fundamental property of wormhole geometry is defined as1 b ( b − b ′ r ) > . (15)There is another constraint on the derivative of shape function atthroat, i.e., b ′ ( r ) < • Redshift function : The main purpose of wormholes is to give a way tomove two-way through its tunnel which basically depends on non-zerominimum value of r at throat. For this purpose, i.e., to keep throatopen, the redshift function plays its role by maintaining no-horizonat throat. To hold this condition, Ψ must remains finite throughoutthe spacetime. This function calculates the gravitational redshift of alight particle. When this particle moves from potential well to escapeto infinity, there appears a reduction in its frequency which is calledgravitational redshift. At a particular value of r , its infinitely negativevalue expresses an event horizon at throat. To prevent this situation ofappearing horizon so that wormhole solution may provide traversableway, the magnitude of its redshift function must be finite. This maybe taken as Ψ = 0 which gives e r ) → Exotic matter : The existence of wormhole solutions requires someunusual type of matter, called exotic matter having negative pressureto violate the null energy condition. Thus, it becomes the basic fac-tor for wormhole construction as the known classical forms of mattersatisfy this condition. The search for such a source which providesthe necessary violation with matter content obeying the null energycondition occupy a vast range of study in astrophysics.The energy conditions originates through the Raychaudhuri equation inthe realm of general relativity for the expansion regarding positivity of theterm R βα K β K α where K β denotes the null vector. This positivity guaranteesa finite value of the parameter marking points on the geodesics directedby geodesic congruences. In terms of energy-momentum tensor, the aboveterm under positivity condition become R βα K β K α = Θ βα K β K α ≥ G βα = Θ eff βα , where Θ eff βα = 1 f T + ωf T L m [Θ + βα + Θ TM βα ] , (16)where Θ TM βα = − S αβ λ ∇ λ ( f TT + ωf TT L m ) − J g βα representing contributionof torsion-matter coupling in the extended teleparallel gravity. The corre-sponding energy condition becomes R βα K β K α = Θ eff βα K β K α ≥ βα = ( ρ eff + p eff ) U β U α − p eff g βα , this condition yields ρ eff + p eff ≥ mβα K β K α ≥ eff βα K β K α ≤ mβα in order to form wormholes.Consider the violation of null energy condition and Eq.(16), we get1 f T + ωf T L m (cid:20) (1 + ωf )Θ mβα − ωf T S αβλ ∇ λ L m − S αβλ ∇ λ ( f TT + ωf TT L m ) − J g βα (cid:21) K β K α < . f T + ωf T L m ) >
0, then we obtain thefollowing constraint0 ≤ Θ mβα K β K α <
11 + ωf (cid:20) ωf T S αβλ ∇ λ L m + S αβ λ ∇ λ ( f TT + ωf TT L m ) + J g βα (cid:21) K β K α , where 1 + ωf > f T + ωf T L m ) <
0, the nullenergy condition straightforwardly givesΘ mβα K β K α >
11 + ωf (cid:20) ωf T S αβλ ∇ λ L m + S αβ λ ∇ λ ( f TT + ωf TT L m ) + J g βα (cid:21) K β K α . We consider the case L m = − ρ with no horizon condition, that is Ψ = 0to construct the background geometry for wormhole solutions in the frame-work of f ( T ) gravity having torsion-matter coupling. We take anisotropicdistribution of fluid having energy-momentum tensor asΘ mβα = ( ρ + p r ) U β U α − p r g βα + ( p t − p r ) χ β χ α , (17)where p r is the radial directed pressure component and p t denotes tangentialpressure component with ρ = ρ ( r ) , p = p ( r ) satisfying U β U α = − χ β χ α = 1.Taking into account Eqs.(13) and (14), we obtain a set of field equations asfollows b ′ r = 1 f T − ωf T ρ (cid:20) (1 + ωf ) ρ + ωf T ρ ′ r (cid:18) − br − r − br (cid:19)(cid:21) − T ′ ( f TT − ωf TT ρ ) r ( f T − ωf T ρ ) (cid:18) − br − r − br (cid:19) − J f T − ωf T ρ , (18) − br = (1 + ωf ) p r f T − ωf T ρ + J f T − ωf T ρ , (19) − b ′ r − b r = 1 f T − ωf T ρ (cid:20) (1 + ωf ) p t − ωf T ρ ′ r (cid:18) − br − r − br (cid:19)(cid:21) + T ′ ( f TT − ωf TT ρ )2 r ( f T − ωf T ρ ) (cid:18) − br − r − br (cid:19) + J f T − ωf T ρ , (20)10here prime represents derivative with respect to r and T = 2 r (cid:20) (cid:18) − r − br (cid:19) − br (cid:21) . (21)The violation of null energy condition (Θ eff βα K β K α <
0) for field equations(18)-(20) is checked through the consideration of radial null vector whichyields ρ eff + p eff r = 1 f T − ωf T ρ (cid:20) (1 + ωf )( ρ + p r ) + 1 r (cid:26) ωf T ρ ′ − T ′ ( f TT − ωf TT ρ ) (cid:27)(cid:18) − br − r − br (cid:19)(cid:21) = b ′ r − br < , ⇒ ρ eff + p eff r < , where the inequality comes through the flaring out condition of shape func-tion. In order to discuss the above scenario at throat, we obtain the followingrelationship ρ eff ( r ) + p eff r ( r ) = 1 f T − ωf T ρ (cid:20) (1 + ωf ( r ))( ρ + p r ) + 1 r (cid:26) ωf T ρ ′ − T ′ ( f T T − ωf T T ρ ) (cid:27)(cid:18) − b r − r − b r (cid:19)(cid:21) , which must satisfy the following constraint in order to meet the above in-equality, that is(1 + ωf ( r ))( ρ + p r ) + 1 r ωf T ρ ′ (cid:18) − b r − r − b r (cid:19)
75 and σ = 3. As a first model, we consider the models f = T and f = T analogous tothe case of curvature-matter coupling scenario where these models are takenas f = R = f [30]. Substituting these values of models along with Eq.(25)in (22), we obtain the following differential equation for the shape function b ′ − ωσρ (cid:0) r r (cid:1) σ − ωρ (cid:0) r r (cid:1) σ (cid:18) br + r − br (cid:19) = − σρ (cid:0) r r (cid:1) σ − ωρ (cid:0) r r (cid:1) σ [ r (1 + ωT ) + ω ] . (26)We plot the shape function by numerically in order to study the wormholegeometry as shown in Figure 1 fixing initial condition as b (1) = 1. Wetake some particular values of constants as r = 1 , ρ = 0 . , ω = 0 . r . The plot of b represents positively increasing behavior withrespect to r . The trajectory of 1 − br depicts the positive behavior for r ≤ b < r . Figure 2 shows the null energy conditiontaking both components of pressure along with chosen energy density. Theblack curve represents ρ + p r and blue curve describes ρ + p t versus r takinginto account Eqs.(23)-(25). We see that the null energy condition holds inthis case. Thus the possibility of wormhole solutions for which the normalmatter satisfying the null energy condition in the background of torsion-matter coupling exists. Figure 3 represents the general relativistic deviation profile of null energycondition for radial pressure component which gives the range of coupling14arameter. At throat, the null energy condition implies( ρ + p r ) | r = r = ρ − − ωρ r + 2 ω . For the chosen values of parameters, we obtain the range of coupling constantas ω ≥ .
11 for which null energy condition holds. This depicts that theincreasing value of coupling constant minimizes or vanishes the violationof null energy condition through matter. For the case when there is nocoupling, i.e., ω = 0, Eq.(22) reduces to b ′ = − σρ r σ r − σ . The solution ofthis equation is b = − σρ r σ − σ r − σ + c , where c is an integration constant andcan be determined by b ( r ) = r . After applying this condition, finally theshape function becomes b ( r ) = r (cid:18) σρ r − σ (cid:19) − σρ r σ − σ r − σ . This shape function shows a asymptomatically flat geometry as br → r →∞ . Taking same values of parameters, this function gives b = − .
25 + . r representing decreasing but positive behavior for r < . − br holdsfor two sets of ranges ( r > − .
25 and r >
1) or ( r < − .
25 and r < b ′ ( r ) < b ′ = − σρ r <
1. The expression ρ + p r = r (2 r − .
25) leads to r > .
125 to meet the null energy conditionwhile ρ + p t = r (0 . r + 4 .
5) which remains positive.
As the second choice for the model, we consider the viable model withquadratic torsion term as [26] f = − Λ , f = µT + νT , (27)where Λ > , µ and ν are constants. This model describes a well-depictedresult in cosmological scenario, i.e., a matter-dominated phase followed byphantom phase of the universe. Substituting Eqs.(25) and (27) in (22), weobtain the following differential equation b ′ + rµ + 2 νT (cid:26) σ ( µ + 2 νT ) r − νT ′ (cid:27)(cid:18) br + r − br − (cid:19) = − r ω ( µ + 2 νT )15 - br b H r L = rb H r L Figure 4: Plots of shape function b, − br , b = r versus r for Model 2 using r = 1 , ρ = 0 . , σ = 3 , µ = 0 . , ν = − ω = 0 . Ρ + p t Ρ + p r Figure 5: Plots of null energy condition ρ + p r , ρ + p t versus r for Model 2using r = 1 , ρ = 0 . , σ = 3 , µ = 0 . , ν = − ω = 0 . .0 0.2 0.4 0.6 0.8 1.0 - - Ω Ρ + p r Figure 6: The general relativistic deviation profile versus ω for Model 2 using r = 1 , ρ = 0 . , σ = 3 , µ = 0 . , ν = − × (cid:18) ω ( µ + νT ) T (cid:19) . (28)Figure 4 represents the plots of shape function under different conditionsthrough numerical computations. The trajectory of b describes positivelyincreasing behavior for r ≥ − br respects positive behavior. Thus, for model 2, we obtain wormholegeometry for torsion-matter coupling. The null energy condition for thismodel is plotted in Figure 5 which shows the positivity of the condition. Inorder to check the relativistic deviation profile taking into account null energycondition for radial pressure component at throat, we find the equation asfollows ( ρ + p r ) | r = r = ρ + ω ( µr + 4 ν ) ρ r + 2 ω ( µr + 2 ν ) . Its plot is shown in Figure 6 representing ω > .
15 for the range where nullenergy condition holds. This depicts that the increasing value of couplingconstant minimizes or vanishes the violation of null energy condition throughmatter. Also, we may discuss the case of zero coupling in the similar way asdiscussed for Model 1.
The search for wormhole solutions satisfying energy conditions becomes themost interesting configuration now-a-days. Wormhole is a tube like shapeor tunnel which is assumed to be a source to link distant regions in the17niverse. The most amazing thing is the two-way travel through wormholetunnel which happened when throat remains open. That is, to prevent thewormhole from collapse at non-zero minimum value of radial coordinate. Inorder to keep the throat open, the exotic matter is used which violate the nullenergy condition and elaborated the wormhole trajectories as hypotheticalpaths. In order to find some realistic sources which support the wormholegeometry, the concentration is goes towards modified theories of gravity. Inthese theories, effective scenario gives the violation of null energy conditionand matter source supports the wormholes. In this paper, the wormholegeometries are explored taking a non-minimal coupling between torsion andmatter part in extended teleparallel gravity. This coupling expresses theexchange of energy and momentum between both parts torsion and matter.The extension of f ( T ) gravity appeared in terms of two arbitrary func-tions f ( T ) and f ( T ) where f is the extension of geometric part while f is coupled with matter Lagrangian part through some coupling constant. Atthroat, the general conditions imposed by the null energy condition takingenergy-momentum tensor of matter Lagrangian are presented in terms ofnon-minimal torsion-matter coupling. The field equations appeared in non-linear form which are difficult to solve for analytical solutions. Presentedvarious strategies to solve these equations, we have adopted to assume twoviable models with a non-monotonically decreasing function of energy den-sity. These models involving linear torsion scalar coupled with T -matter andquadratic torsion term with matter represented wormhole geometry. Forthese solutions, the null energy condition is satisfied. It is concluded thatthat the null energy condition is satisfied for increasing values of couplingconstant. This depicted that, the usage of exotic matter can be reducedor vanished with the higher values of coupling constant. Thus through thetorsion-matter coupling, we have obtained some wormhole solutions in realis-tic way such that matter source satisfied the energy conditions while effectivepart having torsion-matter coupled terms provided the necessary violation.Finally, we remark here that this work may be a useful contribution for thepresent theory as well as astrophysical aspects. References [1] Flamm, L.: Phys. Z. (1916)448.182] Einstein, A. and Rosen, N.: Phys. Rev. (1935)73.[3] Morris, M.S. and Thorne, K.S.: Am. J. Phys. (1988)1446.[5] Lobo, F.S.N.: Phys. Rev. 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