Spherically Symmetric Geometries in 𝑓(𝑇) and 𝑓(𝑅) Gravitational Theories

Using the well know relation between Ricci scalar, 𝑅 , and torsion scalar, 𝑇 , that is, 𝑅 = −𝑇−2∇ 𝛼 𝑇 𝛼 , we show that, for any spherically symmetric spacetime whose (i) scalar torsion vanishing, that is, 𝑇 = 𝑇 𝛼𝜇 ] 𝑆 𝜇 ] 𝛼 = 0 or (ii) total derivative term, that is, ∇ 𝛼 𝑇 𝛼 with 𝑇 𝛼 is the contraction of the torsion, vanishing, or (iii) the combination of scalar torsion and total derivative term vanishing, could be solution for 𝑓(𝑇) and 𝑓(𝑅) gravitational theories.


Introduction
In the last few decades many theories of modified gravity have been constructed concerning different physical goals.In the 60s, Brans and Dicke coupled a scalar field  to the metric  ] to obatin a variable effective gravitational constant [1,2].In Brans-Dicke theory, the scalar field is a new degree of freedom of the gravitational field, which is not directly coupled to the matter, but it brings influence by getting in the dynamical equations that govern the spacetime geometry.
Also in the 60s, Einstein-Hilbert Lagrangian has been added with quadratic terms in the curvature to overcome the renormalization of the theory [3].In 1971, Lovelock [4] examined terms of higher order in the curvature as well; but he was motivated by another motivation.While this kind of Lagrangians leads, generally, to fourth-order dynamical equations because they involve second-order derivatives, Lovelock derived a general Lagrangian polynomic in curvature that leads to conserved second-order equations for metric spacetime.Lovelock has shown that the bigger the spacetime dimension is, the bigger the number of terms these Lagrangians can contain is.
In the 70s, Buchdahl and Astron aimed at replacing the Einstein-Hilbert scalar Lagrangian with a function of the scalar curvature and studied its cosmological consequences [5].This kind of amended gravity is for time being called () theory.In 1983, Milgrom [6] thought that the galactic rotation curves were a guidance of the fail of Newtonian gravity to describe gravitation in the weak field regime.According to Milgrom, no dark matter was able to explain the data but a theory of amended gravity.In this regard, Milgrom built amended newtonian dynamics theory.In the last decade, Bekenstein [7] developed a relativistic theory of gravity named tensor-vector-scalar gravity (TeVeS) because it mixes the metric tensor, a vector field, and a scalar field.TeVeS involves Milgrom's weakening of Newtonian gravity in the weak field regime and has also effects for lensing phenomena, cosmology, and so forth.
String theory has been a source of detection for theories of amended gravity.For example, Dvali, Gabadaze, and Poratti (DGP) gravity [8] describes the 4-dimensional universe as being immersed in a 5-dimensional manifold.Therefore, a "normal" 5D gravity can effect large scale effects in 4D as an accelerated expansion with no need of dark energy.These 4D consequences are driven by a scalar field named galileon due to its symmetries it obeys [9,10].
For the present accelerating era of our universe, shown by the recent observations of type Ia supernovae [11][12][13], general relativity (GR) can only describe the dynamics of the universe through an exotic fluid, which is called dark energy (DE), that 2 Advances in High Energy Physics has a negative pressure.Therefore, this phenomena divide researchers into two groups.The first group has been trying to explain the late-time acceleration in the frame work of standard cosmology, supplying the existence of DE.However, till now, the nature of DE is still unknown and is not an easy task in modern physics [14][15][16][17].The various cosmological observational data support the Λ cold dark matter (ΛCDM) model.At the present epoch, the ΛCDM model seems to be a standard cosmological model.However, the theoretical root of the cosmological constant Λ has not been fully catched.A number of models for DE to explain the late-time cosmic acceleration without the cosmological constant have been proposed (for more details of this topic readers are advised to return to the review [18] and references therein).
() theories have been proposed in the last few years as an alternative to () modified gravity [19,20] and they constitute a very promising research area as is witnessed by the increasing interest in this field [21][22][23][24][25][26][27][28][29][30].The dynamical object in () theories is the field of frames (vierbeins or tetrads), which involve the basis   () of the tangent space and its respective cobasis   ().The Lagrangian density of this theory is built with the product of the determinant  of the covariant form of the tetrad    times a function of the scalar , which is quadratic in the set of 2-form   ≡   , where   measures the basis   () departing from a coordinate basis.() actions are preserved only under global Lorentz transformations but not preserved under local Lorentz transformation.Therefore, () theories are dynamical theories not only for the metric but also for the entire tetrad.As will be mentioned later, only in the special case of the so-called teleparallel equivalent of general relativity (TEGR) does the theory acquire invariance under local Lorentz transformations, thus becoming a dynamical theory just for the metric.In general, it could be said that () theory provides the spacetime not only with a metric but also with an absolute parallelization [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45].It is shown that, based on a special frame, some static solutions can be derived with spherical symmetry in () gravity theories [46].It is proved that Schwarzschild spacetime remains as solution to ultraviolet deformations of GR in the framework of () gravitational theories [47].The existence of circularly symmetric solutions was examined in three dimensions [48].For a special form of () = + 2 , a spherically symmetric solution which depends on the parameter  is derived [49].
Within () gravitational theory there are many solutions, spherically symmetric [50,51], spherically symmetric charged [52], homogenous anisotropic [53], and stability of Einstein universe [54].Some cosmological features of ΛCDM model in the framework of the () are investigated [35].It is the aim of this study to show that any spherically symmetric geometries are solutions to the modified gravitational theories () and ().
This paper is organized as follows.In Section 2, basic equations of () gravity theory are presented.In Section 3, a tetrad field with spherical symmetry is given and application to the field equation of () is provided.New analytic, vacuum, and nonvacuum solutions are derived in Section 3. Final section is devoted to the main results.

Basic Equations of 𝑓(𝑇) Gravity
The mathematical concept of the () gravity theory is based on the Weitzenböck geometry.Our convention and nomenclature are the following: the Latin indices describe the components of the tangent space to the manifold (spacetime), while the Greek ones describe the components of the spacetime.For a general spacetime metric, we write the line element as where   = (−1, +1, +1, +1) is the Minkowski metric.   is the covariant vector fields and its inverse    is the contravariant vector fields satisfying the orthogonality and unitary conditions In a spacetime with absolute parallelism, the parallel vector fields    define the nonsymmetric affine connection (cf., [55]) , where   ,] =  ]    .The curvature tensor defined by the Weitzenböck connection, , is identically vanishing.The torsion and the contortion components are defined as and the torsion scalar is given by .The action of () theory is given by where  = √− = det(   ), L Matter is the Lagrangian of matter field, Λ is the cosmological constant and Φ  are matter fields, and  Pl is the reduced Planck mass, which is related to the gravitational constant  by  Pl = √ℏ/8.Similar to the () theory, one can define the action of () theory as a function of the tetrad fields    and by putting the variation of the function with respect to the tetrad fields    to be vanishing one can obtain the following equations of motion: where and T ]  is the energy-momentum tensor.Now we are going to rewrite the field equations ( 2) in another form.The field equations (2) are written in terms of the tetrad and its partial derivatives.These equations appear to be different from Einstein's field equations.Following [19,20,56], one can obtain an equation relating the scalar torsion  with the Ricci scalar .These will make the equivalence between teleparallel gravity and GR clear.On the other hand, the tetrad cannot be eliminated completely in favor of the metric in (2) because of the lack of local Lorentz symmetry but one can show that the latter can be brought in a form that closely resembles Einstein's equation.This form is more suitable for constructing analytic solutions in the () theory.To start writing the field equations in a covariant version, we must replace partial derivatives in the tensors by covariant derivatives compatible with the metric  ] , that is, ∇  , where ∇   ] = 0. Thus, the definition of the torsion tensor can be written as Using ( 4) in the definition of contorsion and the skew symmetric tensor  ]  , one can obtain On the other hand, from the relation between Weitzenböck connection and the Levi-Civita connection, one can write the Riemann tensor for the Levi-Civita connection, {  ] }, in terms of the nonsymmetric connection, Γ  ] , in the form The associated Ricci tensor can then be written as Now, by using the definition of the contorsion along with the relations  (]) =  (]) =  (]) = 0 and considering that    = 2   = −   , one has [24, 57-65] Equation (8) implies that the torsion scalar, , and Ricci scalar, , differ only by a covariant divergence of a spacetime vector.Therefore, the Einstein-Hilbert action and the teleparallel action (i.e., L TEGR = ∫  4 ||) will both lead to the same field equations and are dynamically equivalent theories.However, the divergence term is the main reason that makes the field equations of () not invariant under local Lorentz transformation (LLT).Let us explain this for some specific form of ().Consider The last term in the R.H.S. of ( 9) is not a total derivative term and therefore it is responsible to make () =  +  2 when written in terms of  and  2 not invariant under LLT in contrast to the linear case, that is, the form of ( 8).Same discussion can be applied to the general form of () and () which shows in general a difference between () and () gravitational theories.It is known that the field equations of () are of fourth-order and invariant under LLT, while the field equations of () are of second-order and not invariant under LLT.Therefore, if the divergence term is vanishing then  = − and hence one can show that any solution in TEGR can be a solution to GR and for a general case one can also show that any solution of () can be a solution to ().The aim of this study is to show that in the spherically symmetric case any solution of () whose divergence term is vanishing will also be a solution to () theory.By using the equations listed above and after some algebraic manipulations, one can get where  ] =  ] − (1/2) ]  is the Einstein tensor.Finally, by using (10), the field equations of () gravity, (2), can be rewritten in the form Equation ( 11) can be taken as the starting point of the () modified gravity model and it has a structure similar to the field equations of () gravity.Note that in the more general case with () ̸ =  the field equations are in covariant form.Nevertheless, the theory is not local Lorentz invariant.In case of () =  and constant torsion, ( 0 ), GR is recovered and field equations are covariant and the theory is Lorentz invariant.

Spherically Symmetric Solution in 𝑓(𝑇) Gravity Theory
The spherically symmetric tetrad has the form where  1 () and  2 () are two unknown functions of the radial coordinate, .Tetrad field ( 12) is a solution to the field equations (11) when () = , that is, TEGR, and Reissner Nordström-dS black hole [66].However, our aim is to find a solution to () gravitational theory.Tetrad field (12) gives a nonvanishing value of the scalar torsion; that is, , which means that the extra term appears in the field equation (2); that is,  ]   , ()  will have a nonvanishing value.Therefore, tetrad (12) does not fit to be a solution for () gravitational theories.For this purpose we use the following tetrad: where A 1 () and A 2 () are two unknown functions of the radial coordinate, , and Φ = (,,).Tetrad (13) 1 is a generalization of tetrad (12); that is, when Φ =  we return to tetrad (12).Also we exclude the case in which Φ = 1/ to preserve the symmetry we assume, spherical symmetry.Using tetrad (13) we get the nonvanishing components of the torsion tensor and the skew symmetric tensor   ] in the form where A  1 = A 1 /, A  2 = A 2 /,   = (, , )/,   = (, , )/, and   = (, , )/.
Using (14) we get the scalar torsion in the form Putting the torsion scalar  = 0 we get a solution for  in the form with (, ) being an arbitrary function of  and .Solution ( 16) satisfied (11) under some constraints on the function (0) and   (0) that leads to the vanishing of torsion scalar.We put the constraint  = 0 to ensure that any solution of () will be a solution to ().Other constraints, in the spherically symmetric case, will not satisfy the equality of the two theories.
The other solution comes from the calculations of the total derivatives term, surface term.The nonvanishing components of the vector density of the torsion, that is, √−  ] =  ] , have the form where   ,   , and   are defined after (14).Using (17) we get the divergence term in the form where  , =  2 /,  , =  2 /, and  , =  2 /.Putting ∇    = 0 we get a solution for  in the form where  1 (, ) is an arbitrary function of  and  2 .Using (19) in the relation  = − − 2∇     , gives  = −; that is, any solution in GR will also be a solution to TEGR and vice versa.In general, we can say that any spherically symmetric solution of () theories will also be a solution to () theories.
Finally, we are going to calculate the term − − 2∇    which has the form Equation (20) does not show any dependence on the unknown (, , ) which is consistent with the fact that − − 2∇    =  and Ricci scalar does not have any dependence on the unknown .Therefore, this constraint, −−2∇    , tells us that we work on GR theory which means that any GR solution will be a solution to TEGR or () gravitational theories.

Main Results and Discussion
In this study, we have addressed the problem of finding spherically symmetric solution in the framework of () and () gravitational theories.To achieve this goal, we begin with the relation between Ricci scalar and torsion scalar tensors.From this relation we have studied three different cases.
Case 1.In this case we have applied a special 3 spherically symmetric tetrad field having three unknown functions; one of them is a function of , , and , (, , ).The other two functions are responsible for reproducing all the known spherically symmetric solutions.We have calculated the scalar torsion of this tetrad and got a differential equation.By solving this differential equation with respect to (, , ) we derived a solution that made the scalar torsion vanishing.Using this solution on the identity  ≡ − − 2∇    we get  ≡ −2∇    .Case 2. In this case we have calculated the total derivative term, ∇    , and solved the resulting differential equation with respect to (, , ).Using this solution on the identity  ≡ − − 2∇    we get  ≡ −.
Case 3. In this case we have evaluated the equation − − 2∇    and showed that the resulting differential equation did not show any dependence on (, , ).This is acceptable because we have  = 0 which has no dependence on (, , ); more precisely we work within the domain of GR.
Case 2 above shows that any spherically symmetric solution of GR will also be a solution to TEGR and vise versa.This means that, for example, for Schwarzschild spacetime whose Ricci scalar is vanishing, this solution will also be a solution to () gravity.Let us explain this.The field equation of () gravity has the form [67] where   () = ()/ and   ] are the energy momentum tensor of matter.For Schwarzschild solution we have  = 0 and the field equation of () for this solution takes the form so if we take the constraints that   (0) = (0) = 0 we get a solution within () gravitational theory.At the same time for () gravitational theories we have the problem of local Lorentz transformation which comes from the total derivative term.Therefore, Case 2 above means that if one succeeds to find the condition which makes the total derivatives term vanishing then () and () will be equivalent for the spherically symmetric case only, more precisely for the case where the Ricci scalar torsion has a zero or constant value.

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All the above cases are related to the spherically symmetric geometry in which Ricci scalar either has a vanishing value or constant one like de Sitter.For other symmetries these studies can not be applied science the identity  ≡ −−2∇    will not be in general satisfied.For example, in FRW models in which their symmetries are homogenous and isotropic, we can not apply the above procedure.This is due to the fact that Ricci scalar in those models has a nonvanishing value.Therefore, if one applies the procedure of Case 1 we get  ≡ −2∇    which is a contradiction.This contradiction comes from the fact that Ricci scalar is invariant under local Lorentz transformation while the total derivative term is not.3. Tetrad ( 13) is not the general spherically symmetric one.Same procedure done for tetrad field (13) can be followed for the general spherically symmetric tetrad field [66].