Charged Black Holes in Generalized Teleparallel Gravity
M. E. Rodrigues, M. J. S. Houndjo, J. Tossa, D. Momeni, R. Myrzakulov
aa r X i v : . [ g r- q c ] O c t Charged Black Holes in Generalized TeleparallelGravity
M. E. Rodrigues ( a,b )1 , M. J. S. Houndjo ( c,d )2 , J. Tossa ( c )3 , D. Momeni ( e )4 and R. Myrzakulov ( e )5 (a) Faculdade de F´ısica, Universidade Federal do Par´a, 66075-110, Bel´em, Par´a, Brazil(b) Faculdade de Ciˆencias Exatas e Tecnologia, Universidade Federal do Par´a - Campus Universit´ariode Abaetetuba, CEP 68440-000, Abaetetuba, Par´a, Brazil(c) Institut de Math´ematiques et de Sciences Physiques (IMSP) - 01 BP 613 Porto-Novo, B´enin(d) Facult´e des Sciences et Techniques de Natitingou - Universit´e de Parakou - Natitingou - B´enin(e) Eurasian International Center for Theoretical Physics - Eurasian National University, Astana010008, KazakhstanAbstractIn this paper we investigate charged static black holes in 4 D for generalized teleparallel models ofgravity, based on torsion as the geometric object for describing gravity according to the equivalenceprinciple. As a motivated idea, we introduce a set of non-diagonal tetrads and derive the full system ofnon linear differential equations. We prove that the common Schwarzschild gauge is applicable only whenwe study linear f ( T ) case. We reobtain the Reissner-Nordstrom-de Sitter (or RN-AdS) solution for thelinear case of f ( T ) and perform a parametric cosmological reconstruction for two nonlinear models. Wealso study in detail a type of the no-go theorem in the framework of this modified teleparallel gravity.Pacs numbers: 04.50. Kd, 04.70.Bw, 04.20. Jb Gravity as the old fundamental force in world is described by a gauge theory according to the equiv-alence principle. According to Mach’s principle, gravity is view as a deformation of the geometry from e-mail: [email protected] e-mail:[email protected] e-mail: [email protected] e-mail: [email protected] e-mail:[email protected] ∼ = GeometryThe right hand side of the above equation denotes the geometry and it is not clear which kind ofgeometry (Riemannian or non Riemannian) has to be used. Also, the geometry quantities are scalars andcan be constructed from different tensor objects. For example, by using Riemannian geometry, the scalarcan be written in the form of any of the following expressions g ij R ij , R ijkl R ijkl , C ijkl C ijkl and more.The most simple modification is the replacement of scalar curvature “R” by a generic function “f(R)”,originally proposed by [1] and extended in literatures for cosmology and gravity, and also as weak limitof quantum gravity [2].In recent years, more attention is attached to gravity as effect of the torsion of spacetime, originallyintroduced in parallel to curvature description [3], and developed in several years as teleparallel equiva-lence of gravity (TEGR) [4]. Special forms as torsion models have been proposed and applied diversely[5]. Black hole solutions are more compact objects which store informations about entropy on their horizon(originally discovered by Hawking [6]). The horizon of a black hole has a definite topology and tempera-ture, and consequently thermodynamics. In modified TEGR, f ( T ) has some strange and also interestingfeatures. The main important is what happens at the cosmological level, due to the field equations, andalso the fact that the black hole depends on the frame under consideration [7]. This means that, if wechange the tetrad frame from a basic diagonal to a non-diagonal one, performing some kinds of Lorentztransformations, the results are different. This feature of f ( T ) theory, and consequently of its relatedfield equations, leads to the non-invariance under Lorentz transformations [8], absence of evaporation forNariai black hole in diagonal frame [9] and also existence (non existence) of relativistic stars [10, 11].Recent observations from solar system orbital motions in order to constrain f ( T ) gravity have been madeand interesting results have been found [12].Besides neutral black holes there are also charged Maxwell field minimally coupled to gravity. Someearly works have studied charged black holes in f ( T ) theory [13, 14, 15]. Previously, in [14], threedimensional solutions have been investigated in detail and generalized later in D-dimensional frame [15].Our main goal in this paper is to study black hole solutions with Maxwell fields in a general nondiagonal frame in 4 D . We fundamentally undertake this problem from the point of view of analyticsolutions. We choose a non-diagonal frame and study TEGR, recovering the Reissner-Nordstrom solution.The main and interesting feature here is a theorem according to what, in f ( T ), we can use the gaugeof Schwarzschild where g tt g rr = −
1. Recently, this gauge has been used as a solver key of obtainingblack holes in f ( T ) [16]. We will show that g tt g rr = − T = T or when the geometry preserves only TEGR. However, for a generic form of f ( T ) and with T = T or2 ( T ) = T + C , there is no reason to use g tt g rr = −
1. We classify the equations as a system of couplednon linear differential equations. Moreover, we study the no-go theorem in this theory.The paper is organized as follows. In sec. 2, we present general derivation of field equations of f ( T )gravity and also some discussions about the frames. In sec. 3, we formulate charged black holes in anon-diagonal frame. Sec. 4 is devoted to the cosmological reconstruction of two f ( T ) models. In sec. 5we prove and present the no-go theorem for the generalised f ( T ) model. We conclude and summarize insec. 7. The field equations from f ( T ) theory In this section we will show how to obtain the equations of motion for f ( T ) theory and also put outthe choice of the matter as an energy-momentum tensor of a spin-1 Maxwell field.We start defining the line element as dS = g µν dx µ dx ν = η ab θ a θ b , (1) θ a = e aµ dx µ , dx µ = e µa θ a , (2)where g µν is the metric of the space-time, η ab the Minkowski metric, θ a the tetrads and e aµ and theirinverses e µa the tetrads matrices satisfying the relations e aµ e νa = δ νµ and e aµ e µb = δ ab . The root of thedeterminant of the metric is given by √− g = det [ e aµ ] = e . The Weitzenbok connection is defined byΓ αµν = e αi ∂ ν e i µ = − e i µ ∂ ν e αi . (3)Through the connection, we can define the components of the torsion and contorsion tensors as T αµν = Γ ανµ − Γ αµν = e αi (cid:0) ∂ µ e i ν − ∂ ν e i µ (cid:1) , (4) K µνα = −
12 ( T µνα − T νµα − T µνα ) . (5)For facilitating the description of the Lagrangian and the equations of motion, we can define anothertensor from the components of the torsion and contorsion tensors, as S µνα = 12 (cid:16) K µνα + δ µα T βνβ − δ να T βµβ (cid:17) . (6)Now, defining the torsion scalar T = T αµν S µνα , (7)one can write the Lagrangian of the f ( T ) theory, coupled with the matter, as follows L = ef ( T ) + L Matter . (8)3he principle of least action leads to the Euler-Lagrange equations. In order to use these equations, wefirst write the quantities ∂ L ∂e aµ = f ( T ) ee µa + ef T ( T )4 e αa T σνα S µνσ + ∂ L Matter ∂e aµ , (9) ∂ α (cid:20) ∂ L ∂ ( ∂ α e aµ ) (cid:21) = − f T ( T ) ∂ α ( ee σa S µνσ ) − ee σa S µασ ∂ α T f
T T ( T )+ ∂ α (cid:20) ∂ L Matter ∂ ( ∂ α e aµ ) (cid:21) , (10)where f T ( T ) = df ( T ) /dT and f T T ( T ) = d f ( T ) /dT denote the first and second derivatives of thealgebraic function f ( T ) with respect to the torsion scalar T , respectively. The equations of Euler-Lagrangeare given by ∂ L ∂e aµ − ∂ α (cid:20) ∂ L ∂ ( ∂ α e aµ ) (cid:21) = 0 . (11)By multiplying (11) by e − e aβ /
4, one gets S µαβ ∂ α T f
T T ( T ) + (cid:2) e − e aβ ∂ α ( ee σa S µασ ) + T σνβ S µνσ (cid:3) f T ( T ) + 14 δ µβ f ( T ) = 4 π T µβ , (12)where the energy momentum tensor is given by T µβ = − e − e aβ π (cid:26) ∂ L Matter ∂e aµ − ∂ α (cid:20) ∂ L Matter ∂ ( ∂ α e aµ ) (cid:21)(cid:27) . (13)For the Maxwell field, the energy momentum tensor is given by the expression T µβ = 14 π (cid:20) δ µβ F σγ F σγ − F µσ F βσ (cid:21) , (14)where F µν = ∂ µ A ν − ∂ ν A µ is the Maxwell tensor and A µ the electromagnetic quadri-potential. Charged black hole model
We assume that charged static spherically symmetric black hole is described by the following metric: dS = e a ( r ) dt − e b ( r ) dr − r (cid:2) dθ + sin ( θ ) dφ (cid:3) , (15)where the metric parameters { a ( r ) , b ( r ) } are assumed to be functions of radial coordinate r and arenot time dependent. This is a consequence of Birkhoff theorem which governs the generalized gravitywith arbitrary choice of tetrads [17]. In general, this statement shows how static distributions of matter,behaving as a static spacetime manifold (Riemannian or Weitzenbock), is a central key of any theory ofgravity. In absence of this theorem, solar system tests fail and there is no direct way to measure themetric parameters of that theory by using observable parameters.4rom (15) we have different choices of tetrads. The diagonal tetrads restrict the algebraic expressionof f ( T ) to the teleparallel linear form [7]. For constructing a good set of non-diagonal tetrads, it is justnecessary to follow the idea developed in the references [18, 19]. The degree of freedom can be fixed bychoosing the component e µ = u µ , where u µ is a quadri-velocity of an observer. The other componentsare chosen to be oriented along the directions of the Cartesian axes { x, y, z } .Therefore, according to the technique previously proposed, we choose the following non-diagonaltetrad basis in which we perform a local Lorentz transformation on the diagonal basis, appropriately [20]: { e aµ } = e a/ e b/ sin θ cos φ r cos θ cos φ − r sin θ sin φ e b/ sin θ sin φ r cos θ sin φ r sin θ cos φ e b/ cos θ − r sin θ , (16)where we define the determinant of the tetrad by e = det [ e aµ ] = e ( a + b ) / r sin θ . The non null componentsof the torsion tensor (4) are T = a ′ , T = T = e b/ − r , (17)while the non-null components of the contorsion tensor read K = a ′ e − b , K = K = e − b ( e b/ − r . (18)The non-null components of the tensor S µνα can be computed, giving S = e − b ( e b/ − r , S = S = e − b (cid:0) a ′ r − e b/ + 2 (cid:1) r . (19)From the definition of the torsion scalar (7), one gets T = 2 r (cid:20) − (cid:18) a ′ + 2 r (cid:19) e − b/ + (cid:18) a ′ + 1 r (cid:19) e − b + 1 r (cid:21) . (20)Note that, here, with a general form of the metric, T is not constant.For the charged black hole configurations we need an additional Maxwell field (static). For thecase of static electric potential A µ = [ A ( r ) , , , F = ∂ r A ( r ). The Maxwell equations are ∇ µ F µν = 0 , (21)whose solution provides A ( r ) = qr e ( a + b ) / , (22)where q is the electric charge. 5rom all the calculations above, we can establish the equations of motion from (12), with (14), as2 e − b r (cid:16) e b/ − (cid:17) T ′ f T T + e − b r h b ′ r + (cid:16) e b/ − (cid:17) ( a ′ r + 2) i f T + f q r , (23) e − b r h(cid:16) e b/ − (cid:17) a ′ r + 2 (cid:16) e b/ − (cid:17)i f T + f q r , (24) e − b r h a ′ r + 2 (cid:16) − e b/ (cid:17)i T ′ f T T + e − b r h (cid:0) a ′ b ′ − a ′′ − a ′ (cid:1) r ++ (cid:16) b ′ + 4 a ′ e b/ − a ′ (cid:17) r − e b + 8 e b/ − i f T + f − q r . (25)The system of field equations (23)-(24) is a closed system for three unknown functions { a, b, f ( T ) } highly coupled in non linear forms, yielding a stiff system. For finding the possible exact solution of thissystem we need to have in hand all the three functions by solving simultaneously the system analytically.In fact, this is a very hard work and as a simple case we are only able to fix the form of f ( T ) and solvethe system analytically. Because we do not know how the boundary conditions change on the metricfunctions, the construction of numerical solutions is also difficult. In the next sections, first, we willexamine the system for f ( T ) = T , which corresponds to the TEGR, and later, we also will show how theRN solution appears.Next, we find a family of exact solutions for a viable model of f ( T ). In order to confirm the consistency of the theory, we propose to search whether the usual Reissner-Nordstrom-de Sitter (or RN-AdS) case may be recovered from the teleparallel theory, i.e. f ( T ) = T − e − b r ( a ′ + b ′ ) = 0 . (26)Without loss for generality, one may set a ( r ) = − b ( r ) . Indeed, we can write a ( r ) = − b ( r ) + a but it isstraightforward to combine a in a redefinition of time coordinate t . So, without loss of generality we put a = 0. The physical meaning of this equality is related to the fact that we can interpret metric function a as the potential of Newtonian form if we go to the non relativistic regime in which we approximate g ∼ − G /c , Φ G being Newtonian potential. If we change a → a + a , nothing changes becausethere is a gauge freedom for scalar Newton’s potential.Therefore, Eq. (25) becomes e a (cid:20) a ′ + a ′′ a ′ r (cid:21) − q r + Λ = 0 , (27) This choice is commonly called quasi-global coordinate, for which, in GR, the equations of motion are independent onthe gauge [21]. a ( r ) = ln (cid:18) q r + C r + C − Λ3 r (cid:19) , (28)where C and C are integration constants. The constant C is found by linearising the metric andcomparing it at the Newtonian limit, yielding C = − M , whereas the constant C is obtained byassuming the Minkowskian limit (for Λ = 0), obtaining C = 1. Therefore, one gets a ( r ) = − b ( r ) = ln (cid:18) − Mr + q r − Λ3 r (cid:19) , (29)where M is the mass of the black hole.Note that Wang [13] has obtained the same solution, however with a choice different from the non-diagonal tetrads one. The strong difference between our analysis and that of Wang is that, the choicemade by Wang leads to a restriction on the functional form of f ( T ), in the case where the torsion scalardepends on the radial coordinate r , within a linear dependence on T . This is shown from f T T T ′ = 0, whichis Eq. (52) of [13], yielding black hole solutions. The problem with this analysis is that the constraintequation forces to two restricted possibilities: a − f T T = 0, which leads to the linear case; b − T ′ = 0,leading to the constant torsion. But in our approach to charged solutions, we no longer have restrictionon the functional form of f ( T ), as can be seen in (23)-(25). With our choice of tetrads, we have variouspossibilities for the functional form of f ( T ).For Λ = 0, we get the teleparallel version of the Reissner-Nordstrom solution of the GR, because with[22] R = − T − ∇ µ T νµν , (30)the curvature scalar for the solution given by (29), reproduces very well the RN case with R = 0 in (30).As the Maxwell energy-momentum tensor in (14) has a vanishing trace, in GR, we must get R = 0 forRN solution. Here we have the same result, but the non-null function T ( r ) is combined with the secondterm of the right hand side in (30), cancelling identically.We then re-obtain the RN-dS (RN-AdS) solutions, for Λ = 0, the RN, for q = 0, the Schwarzschild-deSitter (S-dS) or S-AdS solution. The vacuum case is recovered in the limit of a vanishing charge andcosmological constant, q → → f ( T ) gravity has equations of motion depending on the choice of the setof tetrads. We also know that it is a theory for which the invariance under local Lorentz transformationno longer is realized [8]. Therefore, as the relation between the torsion scalar and the curvature scalarhas a term which is not invariant under the Lorentz transformations, it is natural to think that there ismore than one possibility to re-obtain the known GR solutions by analogy. A recent and important mechanism for the modified theories of gravity is the reconstruction scheme ofthe functional form of the related action. There are various examples of modified theories, namely, f ( R )[24], f ( G ) [25], f ( R, T ) [26], where R , G and T are the curvature scalar, the Gauss-Bonnet invariant andthe trace of the energy momentum tensor, respectively, and also f ( T ) under consideration in the presentpaper. In the case of static solutions, there are several solutions obtained by this method [13, 27].As the equations of motion (23)-(25) are highly nonlinear and coupled, the usual methods of resolutiondo not apply here. Due to the great difficulty in obtaining new solutions to the equations of motion, inthis section, we will take into account a simplification for these equations, and therefore, being able tofind suitable interpretation to f ( T ) solutions of Maxwell type. A very typical consideration in obtainingsolutions to the f ( T ) theory is taking the matter content as being directly proportional to the algebraicfunction f ( T ) and it derivative [27]. This feature can be directly seen from the equations (12).Hence, we take the first term in brackets on the left hand side of (24), being identically null. Thisyields f ( T ) = 16 π T = q /r , (31) (cid:16) e b/ − (cid:17) a ′ r + 2 (cid:16) e b/ − (cid:17) = 0 . (32)We then determine b ( r ) in terms of the derivative of a ( r ), using (32), and get b ( r ) = 2 ln (cid:20) ra ′ )(2 + ra ′ ) (cid:21) . (33)One can then find ( f T , f T T ), only in terms of r and a ′ ( r ), in (23) and (25), such that they satisfy theseequations. Doing this, we can infer a general f ( T ) solutions of Maxwell type for a ( r ), such that allequations are satisfied. One can make use of the following ansatz a ( r ) = ln (cid:20) − Mr + q r − Λ3 r + a ( r ) (cid:21) , (34)8hich, for a ( r ) = 0, gives rise toexp[ b ( r )] = 9( q − r + Λ r ) r (3 M − r + 2Λ r ) , (35) T ( r ) = − q − M r + Λ r ) r ( q − r + Λ r )[ − q + r (6 M − r + Λ r )] . (36)We can also add the term a ( r ) = a r which is interpreted as the Rindler acceleration for large scales[28], leading to exp[ b ( r )] = 36[ q + r ( − − a r + Λ r )] r [6 M + r ( − − a r + 4Λ r )] , (37) T ( r ) = − [6 q − M r + r ( − a + 2Λ r )] r [ q + r ( − − a r + Λ r )][ − q + r (6 M − r − a r + Λ r )] . (38)Here, we see that by setting a = 0 in (37) and (38), we regain (35) and (36), meaning that the lastsolution is a generalization of the first one. But they possess quite different properties, mainly, regardingthe functional form of each f ( T ) model.We can even do a parametric plot, where the parameter is the radial coordinate r . From (31) we knowthat f ( r ) = q /r , and also we have the solution (36) for T ( r ). Using r as a parameter, we can representthe algebraic function f ( T ) at Figure 1 . Taking the same algebraic function f ( T ) with our ansatz (31)and the solution (38), we describe parametrically f ( T ) at Figure 2.Figure 1: Parametric plot of f ( T ) × T for a ( r ) = 0 , q = 1 , M = 2 , Λ = − .
01 and r ∈ [0 . , We can see from the two figures that we are dealing with the cases of nonlinear algebraic f ( T ) function.We cannot reconstruct algebraically this function, since the equations (36) and (38) cannot be invertedto obtain r ( T ) and substituting in (31). Hence, our parametric analysis is necessary. We remove the condition T ( r ) = 0, where the graph are represented by dashed lines in the two figures. - f H T L @ R e d D Figure 2:
Parametric plot of f ( T ) × T for a ( r ) = a r, q = 1 , M = 2 , Λ = − . , a = 1 and r ∈ [0 . , a ′ + b ′ = 0 in the nonTEGR gravity and with T = T ? In the previous sections we derived some exact black hole solutions in the non constant torsion scalarcase. Now, in this section, we prove that very useful and suitable gauge a ′ + b ′ = 0 which simplifiescalculations, is only accessible in the cases of T = T , f T ( T ) = 0, or only when we study TEGRaction. This means that if we try to solve the system of equations (23-25) in another cases, not inthe two mentioned cases (constant torsion of TEGR), we are not able to use the simplification solver“key” a ′ + b ′ = 0. So, there is no Schwarzschild like solution for f ( T ) with variable T or far away fromTEGR. This is a very significant result, because some works studied black holes with special case of gauge a ′ + b ′ = 0 without notice this important fact. Explicitly, we put out the following theorem: Theorem : For a non TEGR case and without constant torsion scalar, i.e. with T = T , the metricfunctions { a, b } cannot satisfy the reducible constraint a ′ + b ′ = 0 . So, it is impossible to solve black holeequations of motion in f ( T ) with the gauge a ( r ) = − b ( r ). Proof.
By subtracting one of the equations (23) and (24) from other, in general, one gets2( f T ) ′ f T + a ′ + b ′ e b/ − . (39)This equation is valid without any additional assumption on the form of f ( T ) or T . From (39), weintegrate and find explicitly f T = α exp {− Z a ′ + b ′ e b/ − dr } . (40)We analyse (40) in the following cases.TEGR case: with TEGR, we have: f T = 1 = ⇒ ( f T ) ′ = 0 ⇐⇒ a + b = constant. For this reason wecan recover the RN spacetime in the TEGR limit using the field equations, successfully.10onstant torsion scalar case, T = T : in this case, T ′ = 0 = ⇒ ( f T ) ′ = 0 ⇐⇒ f T ( T )( a ′ + b ′ ) = 0.Now, we have two possibilities:A - First: if f T ( T ) = 0, then (39) becomes an identity; so our preposition is valid;B - Second: if f T ( T ) = 0, we consequently have a ′ + b ′ = 0, which again proves our theorem.So, in general, for a generic form of f ( T ) and with a non constant torsion T = T , we loss simplificationof gauge fixing a ′ + b ′ = 0, i.e. Schwarzschild like metric.For this reason, it is a very hard task to find exact solutions for T = T . A simple reason to this isthat, if we substitute (40) in the new set of equations (23+25) and (24+25), we find the following systemof coupled differential equations for { a, b, f ( T ) } : b ′ = h ( a, b, a ′ , a ′′ , f ; r ) , (41) a ′′ = k ( a, b, a ′ , b ′ , f ; r ) . (42)The system is highly non-linear and we cannot easily solve it analytically. Note that here f = f ( T ) = f ( a ′ , b ′ , b ; r ) , and satisfies (40). So, we have a system of three unknown functions. It is possible to eliminate f ( T ) fromthe above equations and obtain a pair of differential equations for { a, b } and solve them analytically.For example, considering a very simple ansatz a = kb, k = −
1, from (40), we find f T = α (1 − e − b/ ) − ( k +1) = ⇒ f ′ = dfdr = αT ′ (1 − e − b/ ) − ( k +1) = ⇒ f = α Z r T ′ ( x )(1 − e − b ( x ) / ) − ( k +1) dx. (43)We substitute (43) in (24) for k = 1 , α = 1 and q = 1 /
3, and find (cid:0) − r (cid:1) w + (cid:0) − r (cid:1) w − (cid:0) r (cid:1) w + (cid:0) r + 9 r w ′′ (cid:1) w − r (cid:0) w ′ + w ′′ (cid:1) w − r w ′ = 0 , (44)where we have w ( r ) = exp[ b ( r ) / r → + ∞ we have w → + ∞ . Consequently, the spacetime is non-asymptotically flat (NAF) (see Fig. 3). Also,we have an event horizon in r = r H = 1 ( g ( r H ) = w ( r H ) = 0). Our numerical solution resembles aspecific class of NAF black holes, for { γ = 1 . , r = b = r H = 1 } in (2 .
8) and (2 .
9) of [29]. We plot r γ ( r − w ( r ), where our solution is more convex downward. This is thesecond example of a solution of charged black hole with a non-linear f ( T ) (see the first in [13]). Here, wecannot reconstruct the algebraic function f ( T ). We will develop this interesting feature in a future work.11 (r) Figure 3:
Plot of metric function w ( r ) and r γ ( r −
1) for α = 1 , k = 1 , q = 1 / , w (1) = 0 . , w ′ (1) = 2 and γ = 1 .
28 versus the radial coordinate r . The graphs started from horizon ( r H = 1) to infinity and show thatspacetime is non-asymptotically flat. Our solution is more convex downward. We consider a diagonal frame of tetrads in which we minimally coupled a Maxwell field tensor F togravity via torsion. It is easy to show that no-go theorem in diagonal frame is satisfied identically. In [15],a version of no-go theorem stated that there is no possibility to have two non vanishing components ofelectric (magnetic) field simultaneously. For example, only radial of azimuthal component of field (electricor magnetic) exists and satisfies all field equations appropriately, and if we insert both components, aserious inconsistency happens. In the previous section, we proved a simple but very useful theorem onthe non existence of Schwarzschild gauge in a generic model of f ( T ) with T = constant. For electric field,a no-go theorem has been proved and in three dimensional cases an explicit proof has been presented in[14]. But the situation in non-diagonal frame like our case is so complicated and different. Our aim inthis section is to check the validity of a type of no-go theorem for our model in the non-diagonal case.Here, the statement of no-go theorem is given as follows. Theorem : There is no consistent metric in the form of g µν = diag ( e a ( r ) , − e − a ( r ) , − r Ω ) for chargedMaxwell field non minimally coupled to torsion via f ( T ) gravity . The proof was given in the previoussection.So, we also presented a “new” no-go theorem for f ( T ). The charged black hole solutions in generalized teleparallel gravity models in Weitzenbock spacetimeare revisited in a non-diagonal tetrads basis in 4 D . As advantage of this non-diagonal components, weavoid the restriction f T T = 0 which leads to TEGR where the result is well known as Reissner-Nordstrom-dS (RN-AdS) spacetime. 12e derived field equations of a general f ( T ) gravity in the first steps. Then, by assuming a non-diagonal tetrads basis in static coordinates we derived the full system of field equations. We proved animportant theorem which shows why the common Schwarzschild form of metric in new tetrads formalismcannot be used for a general f ( T ) gravity without f T T = 0. According to this theorem we concludedthat charged black holes in a general f ( T ) is a complicated system of coupled differential equations. Asexamples, we separately analyse the cases of TEGR, recovering our field equations, re-obtaining the usualsolutions of GR, as RN-dS (RN-AdS), RN, S-dS (S-AdS) and Schwarzschild.The important point here is that, charged static black hole in TEGR has a non constant torsion scalar T . This holds in GR in spite of the spacetime having a vanishing curvature scalar, R = 0. We show thatthe analogy is still possible because the two terms that result from the curvature scalar can mutuallycancel each other, then, leading to a new interpretation for these solutions in f ( T ) gravity. We testedand proved a no-go theorem based on the inconsistency of charged black holes in non-diagonal case witha Schwarzschild type metric. Our work exposes new features of f ( T ) gravity as an alternative to GR. Acknowledgement:
M. E. Rodrigues wishes to thank PPGF of the UFPA for the hospitality duringthe development of this work and and thanks CNPq for partial financial support.
References [1] H. A. Buchdahl, Mon. Not. Roy. Astron. Soc.,150,1 (1970).[2] S. Nojiri, S. D. Odintsov, [arXiv: 1008.4275]; E. Elizalde, S. Nojiri, S. D. Odintsov, L. Sebastianiand S. Zerbini [arXiv: 1012.2280]; G. Cognola, et al., Phys. Rev. D77,046009(2008); S. H. Hendi, D.Momeni, arXiv: 1201.0061 [gr-qc].[3] A. Einstein 1928,Sitz. Preuss. Akad. Wiss. p. 217;ibid p.224.[4] J. W. Maluf, Ann. Phys. (Berlin) : 339-357 (2013) [arXiv:1303.3897 [gr-qc]].[5] G. R. Bengochea, R. Ferraro and , Phys. Rev. D , 124019 (2009) [arXiv:0812.1205 [astro-ph]]; E. V. Linder, Phys. Rev. D , 127301 (2010) [Erratum-ibid. D , 109902 (2010)][arXiv:1005.3039 [astro-ph.CO]] ; M. Jamil, D. Momeni and R. Myrzakulov, Eur. Phys. J. C (2012) 2267 [arXiv:1212.6017 [gr-qc]]; R. Myrzakulov, Entropy (2012) 1627 [arXiv:1212.2155 [gr-qc]]; M. R. Setare and N. Mohammadipour, JCAP (2012) 030 [arXiv:1211.1375 [gr-qc]]; M.R. Setare, N. Mohammadipour, JCAP (2013) 015 [arXiv: 1301.4891]; M. Jamil, D. Momeni,R. Myrzakulov and P. Rudra, J. Phys. Soc. Jap. (2012) 114004 [arXiv:1211.0018 [physics.gen-ph]]; M. E. Rodrigues, M. J. S. Houndjo, D. Saez-Gomez and F. Rahaman, Phys. Rev. D (2012) 104059 [arXiv:1209.4859 [gr-qc]]; M. Jamil, D. Momeni and R. Myrzakulov, Eur. Phys.13. C (2012) 2122 [arXiv:1209.1298 [gr-qc]]; R. Myrzakulov, Eur. Phys. J. C (2012) 2203[arXiv:1207.1039 [gr-qc]]; M. J. S. Houndjo, D. Momeni and R. Myrzakulov, Int. J. Mod. Phys.D (2012) 1250093 [arXiv:1206.3938 [physics.gen-ph]]; M. E. Rodrigues, M. H. Daouda andM. J. S. Houndjo, arXiv:1205.0565 [gr-qc]; M. R. Setare and M. J. S. Houndjo, arXiv:1203.1315[gr-qc]; K. Bamba, M. Jamil, D. Momeni and R. Myrzakulov, arXiv:1202.6114 [physics.gen-ph];K. Bamba, R. Myrzakulov, S. ’i. Nojiri and S. D. Odintsov, Phys. Rev. D (2012) 104036[arXiv:1202.4057 [gr-qc]]; M. Jamil, D. Momeni and R. Myrzakulov, Eur. Phys. J. C (2012)2267 [arXiv:1212.6017 [gr-qc]]; M. Jamil, D. Momeni and R. Myrzakulov, Gen. Rel. Grav. (2013)263 [arXiv:1211.3740 [physics.gen-ph]]; M. Jamil, D. Momeni and R. Myrzakulov, Eur. Phys. J. C (2012) 2122 [arXiv:1209.1298 [gr-qc]]; M. Jamil, D. Momeni and R. Myrzakulov, Eur. Phys. J. C (2012) 2075 [arXiv:1208.0025 [gr-qc]]; M. Jamil, K. Yesmakhanova, D. Momeni and R. Myrzakulov,Central Eur. J. Phys. (2012) 1065 [arXiv:1207.2735 [gr-qc]]; M. J. S. Houndjo, D. Momeni andR. Myrzakulov, Int. J. Mod. Phys. D (2012) 1250093 [arXiv:1206.3938 [physics.gen-ph]]; M. Jamil,D. Momeni and R. Myrzakulov, Eur. Phys. J. C (2012) 1959 [arXiv:1202.4926 [physics.gen-ph]];M. Jamil, S. Ali, D. Momeni, R. Myrzakulov and , Eur. Phys. J. C , 1998 (2012) [arXiv:1201.0895[physics.gen-ph]]; M. Jamil, D. Momeni, N. S. Serikbayev, R. Myrzakulov and , Astrophys. Space Sci. , 37 (2012) [arXiv:1112.4472 [physics.gen-ph]]; M. Jamil, D. Momeni, M. A. Rashid and , Eur.Phys. J. C , 1711 (2011) [arXiv:1107.1558 [physics.gen-ph]]; M. Hamani Daouda, M. E. Rodriguesand M. J. S. Houndjo, Eur. Phys. J. C (2012) 1893 [arXiv:1111.6575 [gr-qc]]; R. Myrzakulov,Gen. Rel. Grav. (2012) 3059 [arXiv:1008.4486 [physics.gen-ph]]; K. K. Yerzhanov, S. .R. Myrza-kul, I. I. Kulnazarov and R. Myrzakulov, arXiv:1006.3879 [gr-qc]; R. Myrzakulov, Eur. Phys. J.C (2011) 1752 [arXiv:1006.1120 [gr-qc]]; M. E. Rodrigues, M. J. S. Houndjo, D. Momeni,R. Myrzakulov and , arXiv:1302.4372 [physics.gen-ph]; Ratbay Myrzakulov, Eur.Phys.J.C : 1752(2011); General Relativity and Gravitation, v. , N12, 3059 (2012); P.Y. Tsyba, I.I. Kulnazarov,K.K. Yerzhanov, R. Myrzakulov, IJT Physics. :1876 (2011); K.K.Yerzhanov, Sh.R.Myrzakul,I.I.Kulnazarov, R.Myrzakulov, arXiv:1006.3879 [gr-qc]; J.W. Maluf, F.F. Faria, Annalen Phys. :366-370 (2012) [arXiv:1203.0040 [gr-qc]].[6] J. M. Bardeen, B. Carter, S. W. Hawking , Commun. Math. Phys. 31 (1973) 161-170.[7] N. Tamanini and C. G. Boehmer, Phys. Rev. D , 044009 (2012) [arXiv:1204.4593 [gr-qc]].[8] Baojiu Li, T. P. Sotiriou and J. D. Barrow, Phys. Rev. D , 064035 (2011); Phys. Rev. D ,104030 (2011).[9] M. J. S. Houndjo, D. Momeni, R. Myrzakulov and M. E. Rodrigues, arXiv:1304.1147.[10] C. Deliduman and B. Yapiskan, arXiv:1103.2225v3 [gr-qc].1411] M. Hamani Daouda, M. E. Rodrigues and M. J. S. Houndjo, Eur. Phys. J. C (2011) 1817[arXiv:1108.2920 [astro-ph.CO]].[12] L. Iorio and E. N. Saridakis, Mon. Not. Roy. Astron. Soc. : 1555 (2012) [arXiv:1203.5781 [gr-qc]].[13] T. Wang, Phys. Rev. D (2011) 024042 [arXiv:1102.4410 [gr-qc]].[14] P. A. Gonzaleza, E. N. Saridakis, Y. Vasquez, arXiv:1110.4024v2 [gr-qc].[15] S. Capozziello, P. A. Gonzalez, E. N. Saridakis, Y. Vasquez, JHEP 1302(2013)039 [arXiv:1210.1098[hep-th]].[16] K. Atazadeh, M. Mousavi, Eur. Phys. J. C 72 (2012) 2272 [arXiv: 1212.3764].[17] Han Dong, Ying-bin Wang and Xin-he Meng, Eur.Phys.J. C : 2002 (2012); Eur.Phys.J. : 2201(2012).[18] J.W. Maluf and S.C. Ulhoa, Gen.Rel.Grav. : 1233-1247 (2009) [arXiv:0810.1934 [gr-qc]].[19] J.W. Maluf, F.F. Faria and S.C. Ulhoa, Class.Quant.Grav. : 2743-2754 (2007) [arXiv:0704.0986[gr-qc]]; J.W. Maluf, S. C. Ulhoa and J. F. da Rocha-Neto, Phys. Rev. D : 044050 (2012).[20] M. Hamani Daouda, M. E. Rodrigues and M. J. S. Houndjo, Phys. Lett. B (2012) 241[arXiv:1202.1147 [gr-qc]].[21] Dmitri V. Gal’tsov, Jose P. S. Lemos and Gerard Clement, Phys.Rev. D : 024011 (2004).[22] Thomas P. Sotiriou, Baojiu Li and John D. Barrow, Phys.Rev.D :104030 (2011).[23] Rafael Ferraro and Franco Fiorini, Phys. Rev. D : 083518 (2011).[24] S. Nojiri and S. D. Odintsov, ECONF C 0602061 , 06 (2006); Int. J. Geom. Meth. Mod. Phys. , 115-146 (2007) [arXiv:hep-th/0601213]; Phys. Rept. , 59-144 (2011) [arXiv:1011.0544]. Phys.Rept. (2011)024020. [arXiv:1104.2669 [gr-qc]]; M. J. S. Houndjo, Int. J. Mod. Phys. D. , 1250003 (2012). arXiv:1107.3887 [astro-ph.CO]; M. J. S. Houndjo and O. F. Piattella, Int. J. Mod. Phys. D. , 1250024(2012). arXiv: 1111.4275 [gr.qc]; D. Momeni, M. Jamil and R. Myrzakulov, Euro. Phys. J. C ,arXiv: 1107.5807[physics.gen-ph]; M. J. S. Houndjo, C. E. M. Batista, J. P. Campos and O. F.Piattella, [arXiv:1203.6084 [gr-qc]]; F. G. Alvarenga, M. J. S. Houndjo, A. V. Monwanou and Jean.B. Chabi-Orou, J. Mod. Phys , 130-139 (2013), arXiv: 1205.4678 [gr-qc]; F. G. Alvarenga, A. dela Cruz-Dombriz, M. J. S. Houndjo, M. E. Rodrigues, D. S´aez-G´omez, Phys. Rev. D , 103526(2013), arXiv:1302.1866 [gr-qc].[27] M. Hamani Daouda, M. E. Rodrigues and M. J. S. Houndjo, Eur. Phys. J. C (2012) 1890[arXiv:1109.0528 [physics.gen-ph]].[28] Daniel Grumiller, Phys.Rev.Lett. : 211303 (2010); Erratum-ibid. : 039901 (2011)[arXiv:1011.3625 [astro-ph.CO]]; D. Grumiller, W. Kummer, D.V. Vassilevich, Phys.Rept. : 327-430 (2002) [arXiv:hep-th/0204253].[29] Gerard Clement and Cedric Leygnac, Phys.Rev. D70