Spherically Symmetric Solutions on a Non-Trivial Frame in f(T) Theories of Gravity
SSpherically symmetric solutions on non trivial frame in f ( T ) theories of gravityGamal G.L. Nashed Mathematics Department, Faculty of Science, King Faisal University, P.O. Box.380 Al-Ahsaa 31982, the Kingdom of Saudi Arabia ∗ e-mail:[email protected] New solution with constant torsion is derived using the field equations of f ( T ) . Asymptotic form of energydensity, radial and transversal pressures are shown to met the standard energy conditions. Other solutionsare obtained for vanishing radial pressure and physics relevant to the resulting models are discussed. PACS: 04.50.Kd, 04.70.Bw, 04.20. Jb
DOI:Recent observational data suggest that our universe is accelerating. [1 , This acceleration is explained interms of the so called dark energy (DE). DE could also result from a cosmological constant, from an idealfluid with a different shape of equation of state and negative pressure [3] , etc. It is not clear what type ofDE is more seemingly to explain the current era of the universe. A very attractive possibility is the alreadymentioned as “modification of General Relativity” (GR). Amendments to the Hilbert-Einstein action throughthe introduction of different functions of the Ricci scalar have been systematically explored by the so-called f ( R ) gravity models, which reconstruction has been developed. [4 − In recent times, a new attractive modified gravity to account for the accelerating expansion of the universe,i.e., f ( T ) theory, is suggested by extending the action of teleparallel gravity [8 , − similar to the f ( R ) theory,where T is the torsion scalar. It has been demonstrated that the f ( T ) theory can not only explain the presentcosmic acceleration with no need to dark energy [14] , but also provide an alternative to inflation without aninflation [15 , . Also it is shown that f ( T ) theories are not dynamically equivalent to teleparallel action plusa scalar field under conformal transformation. [17] It therefore has attracted some attention recently. In thisregard, Linder [18] proposed two new f ( T ) models to explain the present accelerating expansion and foundthat the f ( T ) theory can unify a number of interesting extensions of gravity beyond general relativity. [19] The objective of this work is to find spherically symmetric solutions, under the framework of f ( T ),using anisotropic spacetime. In §
2, a brief review of f ( T ) theory is presented. In §
3, non trivial sphericallysymmetric spacetime is provided and application to the field equation of f ( T ) is done. Several new sphericallysymmetric anisotropic solutions are derived in §
3. Several figures to demonstrate the asymptotic behavior ofenergy density and transversal pressure are also given in §
3. Final section is devoted to the key results.In a spacetime with absolute parallelism parallel vector fields h aµ [20] identify the nonsymmetric connectionΓ λµν def . = h aλ h aµ,ν , (1)where h aµ, ν = ∂ ν h aµ . The metric tensor g µν is defined by g µν def . = η ab h aµ h bν , (2)with η ab = ( − , +1 , +1 , +1) is the Minkowski spacetime. The torsion and the contorsion are defined as T αµν def . = Γ ανµ − Γ αµν = h aα ( ∂ µ h aν − ∂ ν h aµ ) , K µν α def . = −
12 ( T µν α − T νµα − T αµν ) = Γ λµν − (cid:8) λµν (cid:9) . (3) ∗ a r X i v : . [ phy s i c s . g e n - ph ] O c t he tensor S αµν and the scalar tensor, T , are defined as S αµν def . = 12 (cid:0) K µν α + δ µα T βν β − δ να T βµβ (cid:1) , T def . = T αµν S αµν . (4)Similar to the f ( R ) theory, one can define the action of f ( T ) theory as L ( h aµ , Φ A ) = (cid:90) d xh (cid:20) π f ( T ) + L Matter (Φ A ) (cid:21) , (5)where h = √− g and Φ A are the matter fields. Assuming the action (5) as a functional of the fields h aµ , Φ A .The vanishing of the variation with respect to the field h aµ gives the following equation of motion [14] S µρν T ,ρ f ( T ) T T + (cid:2) h − h aµ ∂ ρ ( hh aα S αρν ) − T αλµ S ανλ (cid:3) f ( T ) T − δ νµ f ( T ) = 4 π T νµ , (6)where T ,ρ = ∂T∂x ρ , f ( T ) T = ∂f ( T ) ∂T , f ( T ) T T = ∂ f ( T ) ∂T and T νµ is the energy momentum tensor. In this study wewill consider the matter content to have anisotropic form, i.e., given by T µν = diag ( ρ, − p r , − p t , − p t ) , (7)where, ρ , p r and p t are the energy density, the radial and tangential pressures respectively. In the next sectionwe are going to apply the field Eq. (6) to a spherically symmetric spacetime and try to find new solutions.Assuming that the non trivial manifold possesses stationary and spherical symmetry has the form [21] ( h αi ) = e A ( r )2 e B ( r )2 sin θ cos φ r cos θ cos φ − r sin θ sin φ e B ( r )2 sin θ sin φ r cos θ sin φ r sin θ cos φ e B ( r )2 cos θ − r sin θ , (8)where A ( r ) and B ( r ) are two unknown functions of r . The metric associated with (8) takes the form ds = − e A ( r ) dt + e B ( r ) dr + r ( dθ + sin θdφ ) . It is important to note that for the same metric and the same coordinate basis, different frames result indifferent forms of equations of motion. [22]
Using (8), one can obtain h = det ( h aµ ) = e ( A + B )2 r sin θ . With theuse of Eqs. (3) and (4), one can obtain the torsion scalar and its derivatives in terms of r in the form T ( r ) = − (cid:16) − e − B − rA (cid:48) e − B [1 − e − B ] + e − B (cid:17) r , where A (cid:48) = ∂A ( r ) ∂r ,T (cid:48) ( r ) = ∂T∂r = e − B (cid:16) B (cid:48) (cid:104) r A (cid:48) (2 e − B −
1) + 2 r ( e − B − (cid:105) + 2 r [ rA (cid:48)(cid:48) − A (cid:48) ] (cid:104) − e − B (cid:105) − (cid:104) − e − B − e B (cid:105)(cid:17) r . (9)The field equations (6) for an anisotropic fluid have the form4 πρ = − e − B f T T r (cid:18) { [ rA (cid:48) − r A (cid:48)(cid:48) ] + rB (cid:48) + 3 } + 3 r A (cid:48) B (cid:48) + 4 e B + e B (cid:0) r [2 A (cid:48)(cid:48) − A (cid:48) B (cid:48) ] − r [ B (cid:48) + A (cid:48) ] − (cid:1) The energy density, radial and transversal pressures when B ( r ) = r , T = − f ( T ) = 9 and f T ( T ) = 3. + e − B (cid:8) r A (cid:48)(cid:48) − rA (cid:48) − rB (cid:48) − r A (cid:48) B (cid:48) − (cid:9)(cid:19) − e − B f T r (cid:16) − rA (cid:48) + e − B [ rA (cid:48) + rB (cid:48) − (cid:17) + f , πp r = f T r (cid:16) e − B [6 + 3 rA (cid:48) − e − B ] − e − B [ rA (cid:48) + 1] (cid:17) + f , πp t = − e − B f T T r (cid:18) { [8 r A (cid:48) B (cid:48) (1 + rA (cid:48) rA (cid:48) + 8 rB (cid:48) − r A (cid:48) A (cid:48)(cid:48) − r A (cid:48)(cid:48) + 2 r A (cid:48) + 24] } + e B (cid:18) r [ rA (cid:48)(cid:48) − B (cid:48) − A (cid:48) ] − r B (cid:48) A (cid:48) − (cid:19) + 8 e B + e − B (cid:26) r A (cid:48)(cid:48) − r A (cid:48) − rA (cid:48) − rB (cid:48) − r A (cid:48) B (cid:48) − r B (cid:48) A (cid:48) +2 r A (cid:48) A (cid:48)(cid:48) − (cid:27)(cid:19) + f T e − B r (cid:16) − e B + e − B (cid:2) rA (cid:48) + 2 r A (cid:48)(cid:48) + r A (cid:48) − r A (cid:48) B (cid:48) − rB (cid:48) − (cid:3)(cid:17) + f . (10)It is of interest to note that tetrad (8) is used in ([23], Eq. (4 · B ( r ). In the next section we go to find several solutionsto Eq. (10) assuming some conditions on f ( T ).First assumption: T = constnt = T From Eq. (9), it can be shown that A ( r ) which satisfies T = T and T (cid:48) = 0 has the form A ( r ) = 12 (cid:90) (cid:104) − e − B ( r )2 + 2 e − B ( r ) + r T (cid:105) e B ( r )2 r (cid:16) − e − B ( r )2 (cid:17) dr + c , (11)where c is a constant of integration. The weak and null energy conditions have the form ρ ≥ , ρ + p r ≥ , ρ + p t ≥ , ρ + p r ≥ , ρ + p t ≥ . (12)The asymptotic form of energy density, radial and transversal pressures when T = T are shown in figure 1.From figure 1, one can show that the relevant physics met to the standard energy conditions, given by (12).We put restrictions on the values of T , f ( T ), f T ( T ) and B ( r ) such that Eq. (12) is satisfied.In the case of vanishing radial pressure the second of equations (10) reads f ( T ) = 2 f T e − B r (cid:16) rA (cid:48) − e − B [ rA (cid:48) + 1] (cid:17) . (13)We go to study various cases of Eq. (13): First form of f ( T )In this case let us assume f ( T ) to has the form [24] : f ( T ) = a + a T + a n T n , (14)3igure 2: The energy density and the tangential pressure when a = a = c = 1, B ( r ) ∼ r .Figure 3: The energy density and the tangential pressure when a = a = a = c = 1, B ( r ) ∼ r .where a , a and a n are constants. From Eqs. (13) and (14) one can get for the linear case of f ( T ) A ( r ) = 12 (cid:90) (cid:2) e B ( r ) (2 a − a r ) − a (cid:3) ra dr + c , (15)with c being another constant of integration. The asymptote behavior of energy density and the transversalpressure, can be obtained from Eq. (10) using Eq. (15), are shown in figure 2. From figure 2, one can showthat the standard energy conditions are satisfied which is physically acceptable. This model is based on theasymptote of B ( r ) ∼ r . Other asymptote of B ( r ) violate not only the standard energy conditions but alsocausality.By the same way one can obtain a solution for the non-linear case , i.e., when n = 2, in the form A ( r ) = 14 (cid:90) r (3 + e B ( r ) [1 − e − B ( r )2 ]) (cid:18) e B ( r ) (cid:104) e B ( a r + 24 a e B ) − a (1 + e B ( r ) ) (cid:105) − (cid:26) a e − B ( r ) + 16 a e − B [ a r − a ] − e − B [12 a a r − a a r + a r + 96 a ]+ e − B [32 a a r − a a r − a ] + e − B [4 a a r − a a r ] + 16 a e − B (cid:27) / (cid:19) dr + c , (16)with c being constant of integration. From figure 3, it is clear that this model meets the standard energycondition when a = a = a = c = 1 and B ( r ) ∼ r . Other options are not permitted due to the reasonsdiscussed above. 4igure 4: The energy density and the tangential pressure when a = a = a = 1 and B ( r ) ∼ r .Using Eq. (13) in Eq. (14) we get1 r (cid:26) [ r { a + a T + a n T n } ] − e − B ( r )2 [4 a + 2 ra A (cid:48) + 2 nra n T n − A (cid:48) + 4 na n T n − ]+ e − B ( r ) [4 na n T n − + 4 ra A (cid:48) + 4 a + 4 nra n A (cid:48) T n − ] (cid:27) = 0 . (17)According to the model of f ( R ), also used for the f ( T ) theory, as an alternative to the dark energy [14] bytaking the function f ( T ) presented by Eq. (14) when n = −
1, one can obtain a relation between the unknownfunction A ( r ) in terms of the unknown function B ( r ). The asymptote behavior of energy density and thetangential pressure are plot in figure 4. Using the same above procedure we get physically acceptable modelas shown in figure 4. Main results and discussionAll the solutions obtained in this work can be summarized as follow: • When the scalar torsion is taken to be constant, i.e., T = T a quite general differential equation that gov-erned the two unknown functions has been obtained. By this relation calculations of energy density, radialand transversal pressures are provided. The asymptote behavior of these quantities are drawn in figure 1 forspecific asymptote of B ( r ). From figure 1, it became clear that the asymptote of energy density, radial andtransversal pressures depend on the constants as well as on the asymptote behavior of B ( r ). • The condition of vanishing radial pressure is derived. A quite general assumption on the from of f ( T ) hasbeen employed. Different cases have been studied using this assumption:i) When n = 0, following the procedure done in the case of constant scalar torsion, a differential equation thatlinks the two unknown functions A ( r ) and B ( r ) has been derived. The asymptote behavior of the densityand transversal pressure are given in figure 2. From this figure one can show that both energy density andtransversal pressure are positive.ii) When the condition n = 2 is employed a solution is obtained for a specific form of one of the two unknownfunctions. The density and tangential pressure are given in figure 3. From this figure one can conclude thatthe character of this model is physically acceptable since it satisfied the standard energy conditions.iii) When the condition n = −
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