Teleparallel Theories of Gravity: Illuminating a Fully Invariant Approach

Teleparallel gravity and its popular generalization $f(T)$ gravity can be formulated as fully invariant (under both coordinate transformations and local Lorentz transformations) theories of gravity. Several misconceptions about teleparallel gravity and its generalizations can be found in the literature, especially regarding their local Lorentz invariance. We describe how these misunderstandings may have arisen and attempt to clarify the situation. In particular, the central point of confusion in the literature appears to be related to the inertial spin connection in teleparallel gravity models. While inertial spin connections are commonplace in special relativity, and not something inherent to teleparallel gravity, the role of the inertial spin connection in removing the spurious inertial effects within a given frame of reference is emphasized here. The careful consideration of the inertial spin connection leads to the construction of a fully invariant theory of teleparallel gravity and its generalizations. Indeed, it is the nature of the spin connection that differentiates the relationship between what have been called good tetrads and bad tetrads and clearly shows that, in principle, any tetrad can be utilized. The field equations for the fully invariant formulation of teleparallel gravity and its generalizations are presented and a number of examples using different assumptions on the frame and spin connection are displayed to illustrate the covariant procedure. Various modified teleparallel gravity models are also briefly reviewed.

Although Einstein's General Theory of Relativity (GR) is well studied and tested in many settings [1], alternative theories of gravity continue to be of considerable interest [2,3].
Potential explanations for dark energy and dark matter within the current cosmological paradigm based on general relativity may be investigated using an alternative theory of gravity, rather than changing the matter content of the theory. Furthermore, it is possible that the problems of finding a quantum theory of gravity may be resolved within a theory of gravity that is not general relativity. Therefore, it becomes necessary to challenge our assumptions and assess whether an alternative theory of gravity will lead to different results.
One class of alternative theories of gravity assumes that the motion in the gravitational field is no longer geometrized, as in general relativity, but is encoded in a dynamic gravitational force, as in teleparallel gravity. More specifically, in general relativity the gravitational interaction is realized via the curvature of a zero torsion Lorentz connection, which is used to geometrize the interaction; this means that the motion of a free-falling particle in the gravitational field can be viewed as an inertial motion in the curved spacetime and hence gravity can be viewed as a purely geometric effect. On the other hand, in teleparallel gravity the gravitational interaction is an effect of the torsion of a zero curvature Lorentz connection, which acts as a force and not as geometry. Interestingly, teleparallel gravity and general relativity are found to be completely equivalent theories. For this reason one generally refers to it as the Teleparallel Equivalent of General Relativity (TEGR). Although equivalent, however, they are conceptually quite different. For example, in contrast to general relativity, teleparallel gravity is nicely motivated within a gauge theory context and can be beautifully framed as the gauge theory for the translation group [4]. In fact, like all other gauge theories, its Lagrangian density is quadratic in the torsion tensor, the field strength of the theory. The notions of frame and inertial spin connection are presented in Section II. The fundamentals of teleparallel gravity are described in Sections III and IV.
The geometrical setting of any gravitational theory is the tangent bundle, in which spacetime is the base space and the tangent space at each point of the base space (also known as internal space) is the fiber of the bundle. Spacetime is assumed to be a metric spacetime with a general metric g µν . The tangent space, on the other hand, is by definition a Minkowski spacetime with tangent space metric η ab . Since spacetime and the fibers are both four-dimensional spacetimes, the bundle is said to be soldered. This means that the metrics g µν and η ab are related by with h a µ being the tetrad field, the components of the solder 1-form. It should be noted that this geometrical structure is always present, independent of any prior assumptions.
Teleparallel gravity, a gauge theory for the translation group, is built on this geometrical structure. Gauge transformations are defined as local translations in the tangent Minkowski spacetime, the fiber of the bundle. Of course, like any other relativistic theory, it must also be invariant under both general coordinate transformations and local Lorentz transformations.
Whereas the former is performed in spacetime, the latter is performed in the tangent space.
Local Lorentz transformations define different classes of frames, each one characterized by different inertial effects. Within each class, the infinitely many equivalent frames are related by a global Lorentz transformation. Now, as is well-known, inertial effects are represented by a purely inertial connection (which we define later). In the class of frames in which no inertial effects are present, the inertial Lorentz connection is naturally zero. 1 In all other classes of frames, however, their inertial spin connection will be non-vanishing.
Although they produce physical effects and have energy and momentum, inertial effects cannot be interpreted as a field in the usual sense of classical field theory. For example, in TEGR there are no field equations whose solutions could yield the inertial Lorentz connection. 2 (Indeed, neither the field equations of teleparallel gravity nor the field equations of general relativity are able to determine this.) Since the use of the correct inertial Lorentz connection is crucial for the Lorentz symmetry of any relativistic symmetry, it is then necessary to resort to a different method for retrieving the inertial Lorentz connection associated to a general frame. Such a method is presented in detail in Section V, and some concrete examples are discussed in Section VI.
It is important to remark that in the usual metric formulation of general relativity no frame needs to be specified. In the tetrad formulation of general relativity, we do not face the problem of specifying the Lorentz connection because the Levi-Civita spin connection of general relativity can be fully expressed in terms of the dynamical tetrad and hence 1 In the presence of gravitation, these frames are called "proper frames". In the absence of gravitation they reduce to the class of inertial frames of special relativity. 2 The situation is more subtle in modified theories such as f (T ), and we will discuss this later (e.g., see the comments in the Final Remarks).
can be eliminated from the theory. Furthermore, the Levi-Civita connection includes both gravitational and inertial effects, unlike teleparallel gravity where gravitation is represented by a translational gauge potential and inertial effects are represented by an inertial spin connection. In contrast to general relativity, therefore, the question of specifying the inertial spin connection is part of the process of finding solutions of the teleparallel field equations.
Influenced perhaps by general relativity, in which one does not need to carefully consider the inertial spin connection, many authors have never scrutinized it when working in the context of teleparallel gravity. A consequence of this mindset was the emergence of an approach called "pure tetrad teleparallel gravity". This name stems from the fact that it involves no spin connection, the only variable being the frame field, or tetrad. Of course, owing to the disregard of the inertial spin connection, this theory is not invariant under local Lorentz transformations. In spite of this lack of Lorentz invariance, since the solutions of the field equations of any gravitational theory are independent of the inertial spin connection, the solutions provided by the "pure tetrad teleparallel gravity" coincide with the solutions provided by teleparallel gravity itself. Apart from this peculiarity, all other conclusions obtained from this theory, including the Lorentz non-invariance, are different from those obtained from the standard teleparallel gravity theory. A discussion on "pure tetrad teleparallel gravity", as well as on its differences in relation to teleparallel gravity itself, is presented in Section VII.
Recently there have been many proposals to generalize teleparallel gravity. This was motivated by the example of f ( • R ) gravity, where the Lagrangian density is generalized R is the Ricci scalar of the Levi-Civita connection (see [5] and references within for an overview of f (R) gravity). 3 Similarly, it was suggested to generalize the Lagrangian density of teleparallel gravity from T is the so-called torsion scalar with respect to the teleparallel connection that we will define later (see [6] and references within for an overview of f (T ) teleparallel gravity). Unlike the generalization of general relativity, in which the resulting field equations are no longer second order in derivatives, in f (T ) teleparallel gravity the field equations continue to be second order.
Unfortunately, there appears to be some confusion in the literature with regards to the viability of f (T ) teleparallel gravity and, in particular, to its invariance under local Lorentz transformations [7,8]. In the generalization of the type , since the Ricci scalar 3 The notation used in this paper is summarized in Table I below. is built from the metric, which is Lorentz invariant by definition, the Ricci scalar • R is naturally Lorentz invariant too, and therefore f (R) gravity is also Lorentz invariant. In generalizations of the type T is a combination of scalar invariants of the torsion, and since the torsion tensor is a Lorentz covariant object, the scalar • T is Lorentz invariant, and consequently so is f (T ). It appears that the source of confusion is in the use of the "pure tetrad teleparallel gravity", which is not invariant under local Lorentz transformations. In this case, the non-invariance of the "pure tetrad teleparallel gravity" will of course propagate to the modified f (T ) models.
Fortunately, provided the inertial spin connection is appropriately taken into account, teleparallel gravity can be seen to be fully invariant under local Lorentz transformations. In this case, provided the same care is used, a fully covariant f (T ) theory can be obtained [9].
Details of this construction are presented in Sec. VIII. Using an analogous procedure, it is possible to extend teleparallel gravity to other modified gravity models. In Sec. IX we illustrate this possibility with a number of modified teleparallel theories of gravity, including new general relativity [10], conformal teleparallel gravity [11] and f (T, B) gravity [12].

A. Notation
The notation used when formulating teleparallel theories of gravity resembles that used in general relativity. However, it is necessary to introduce some additional symbols for the various quantities which naturally arise in this framework. Sometimes this notation is identical to the notation used in general relativity but actually denotes a slightly different object. In order to help the reader navigate this notational quagmire, we present a list of symbols used throughout this work.
The whole set of such bases constitutes the bundle of linear frames. A frame field provides, at each point p of spacetime, a basis for the vectors on the tangent space. Of course, on the common domains they are defined, each member of a given basis can be written in terms of the members of any other. For example, e a = e a µ ∂ µ and e a = e a µ dx µ , and conversely, ∂ µ = e a µ e a and dx µ = e a µ e a .
On account of the orthogonality condition (1), the frame components satisfy e a µ e a ν = δ ν µ and e a µ e b µ = δ a b .
Notice that these frames, and their bundles, are constitutive parts of spacetime: they are present as soon as spacetime is taken to be a differentiable manifold [13].
A general linear basis {e a } satisfies the commutation relation with f c ab the so-called structure coefficients, or coefficients of anholonomy, which are functions of the spacetime points. The dual expression of the commutation relation above is the Cartan structure equation The structure coefficients represent the curls of the basis members: A special class of frames is that of inertial frames, denoted e ′ a , for which Notice that f ′c ab = 0 means that de ′a = 0 which, in turn, implies that e ′a is a closed differential form and, consequently, locally exact: i.e., e ′a = dx ′a for some x ′a . The basis {e ′a } is then said to be integrable, or holonomic. Of course, all coordinate bases are holonomic. This is not a local property in the sense that it is valid everywhere for frames belonging to this inertial class.
Consider now the Minkowski spacetime metric written in a holonomic basis {dx µ }. When {x µ } represents a set of Cartesian coordinates, it has the form In any other coordinates, η µν will be a function of the spacetime coordinates. The linear frame e a = e a µ ∂ µ provides a relation between the tangent-space metric η ab and the spacetime metric η µν , given by with the inverse given by Independent of whether e a is holonomic or not, or equivalently, whether they are inertial or not, they always relate the tangent Minkowski space to a Minkowski spacetime. These are the frames appearing in special relativity, which are usually called trivial frames, or trivial tetrads.

B. Spin connections and inertial effects
In special relativity, Lorentz connections represent inertial effects present in a given frame.
In the class of inertial frames, for example, where these effects are absent, the Lorentz connection vanishes identically. To see how an inertial Lorentz connection shows up, let us consider an inertial frame e ′a µ written in a general coordinate system {x µ }, which has the holonomic form with x ′a a point-dependent Lorentz vector: x ′a = x ′a (x µ ). Under a local Lorentz transformation, the holonomic frame (12) transforms into the new frame As a simple computation shows, it has the explicit form is a Lorentz connection that represents the inertial effects present in the Lorentz-rotated frame e a µ , and • Dµ is the associated covariant derivative. Such a connection is sometimes referred to as the Ricci coefficient of rotation [14]. Recalling that under a local Lorentz transformation Λ a b (x) a general spin connection ω a bµ changes according to [15] the spin connection (16) is seen to be the connection obtained from a Lorentz transformation of a vanishing spin connection Starting from an inertial frame, in which the inertial spin connection vanishes, different classes of non-inertial frames are obtained by performing local (point-dependent) Lorentz transformations Λ a b (x µ ). Within each class, the infinitely many frames are related through global (point-independent) Lorentz transformations Λ a b = constant. Now, due to the orthogonality of the tetrads, transformation (14) can be rewritten in the Using this relation, the coefficient of anholonomy (7) of the frame e a µ is found to be where we have identified Of course, as a purely inertial connection, • ω a bµ has vanishing curvature: For e a µ a trivial tetrad, • ω a bµ has also vanishing torsion: • ω a eµ e e ν = 0.

C. Example: Equation of motion of free particles
In the class of inertial frames e ′a µ , a free particle is described by the equation of motion with u ′a the anholonomic particle four-velocity, and the quadratic Minkowski interval. In this form, equation (24) is not manifestly covariant under local Lorentz transformations. This does not mean, however, that it is not covariant.
In fact, in the anholonomic frame e a µ , related to e ′a µ by the local Lorentz transformation (14), the equation of motion of a free particle assumes the Lorentz covariant form where is the Lorentz transformed four-velocity, with u µ = u a e a µ = dx µ /dσ the spacetime holonomic four-velocity. In terms of this four-velocity, the equation of motion (26) assumes the where is the trivial spacetime-indexed version of the inertial spin connection • ω a bµ . The inverse relation is In special relativity, therefore, Lorentz connections represent inertial effects only, and are crucial for the local Lorentz invariance of relativistic physics.

III. BASICS OF TELEPARALLEL GRAVITY
A. Gauge structure Teleparallel gravity can be interpreted as a gauge theory for the translation group [4,16].
The reason for translations can be understood from the gauge paradigm, of which Noether's theorem is a fundamental piece. Recall that the source of the gravitational field is energy and momentum. According to Noether's theorem, the energy-momentum current is covariantly conserved provided the source Lagrangian is invariant under spacetime translations. If gravitation is to present a gauge formulation with energy-momentum as the source, then it must be a gauge theory for the translation group.
A gauge transformation in teleparallel gravity is defined as a local translation of the tangent space coordinates, with ε a (x µ ) the infinitesimal transformation parameter. Under such a transformation, a general source field Ψ = Ψ(x a (x µ )) transforms according to (see Ref. [4], page 42) with ∂ α the translation generators. For a global translation with parameter ε a = constant, the ordinary derivative ∂ µ Ψ transforms covariantly: For a local transformation with parameter ε a (x), however, it does not transform covariantly: In fact, the last term on the right-hand side is a spurious term, which breaks the translational gauge covariance of the transformation. Similar to all other gauge theories [17], in order to recover gauge covariance it is necessary to introduce a (in this case translational) gauge potential B a µ , a 1-form assuming values in the Lie algebra of the translation group: B µ = B a µ ∂ a . This potential can be used to construct the gauge covariant derivative which holds in the class of Lorentz frames in which there are no inertial effects. In fact, provided the gauge potential transforms according to the derivative h µ Ψ is easily seen to transform covariantly under gauge translations: This is the output of the gauge construction applied to the translation group .
Owing to the soldered property of the frame bundle 4 , on which teleparallel gravity is constructed, the gauge covariant derivative (35) can be rewritten in the form where is a non-trivial tetrad field. By non-trivial we mean a tetrad with B a µ = ∂ µ ε a , otherwise it would be just a translational gauge transformation of the trivial tetrad e a µ = ∂ µ x a . Similar to any relativistic theory, the equivalent expressions valid in a general Lorentz frame can be obtained by performing a local Lorentz transformation Considering that the translational gauge potential B a µ is a Lorentz vector in the algebraic index; that is, it is easy to see that, in a general Lorentz frame, the translational covariant derivative (35) assumes the form with • ω a bµ the purely inertial Lorentz connection (16). The tetrad components (38) of the derivative (42) can then be written as The first two terms on the right-hand side make up the trivial tetrad which allows (43) to be rewritten in the form In a general class of frames, therefore, the gauge transformation of B a µ is In the class of frames in which the inertial spin connection we obtain where is the translational field strength, and ∂ a stands for the translations generators. Adding the vanishing piece to the right-hand side of (49), it becomes Consequently, we see that the field strength of teleparallel gravity is just the torsion tensor.
Note that, by construction, the spin connection appearing within the tetrad is the same as that appearing explicitly in the last two terms of the torsion definition. This is fundamental for the definition of torsion. In fact, if these connections are not the same, • T a µν will no longer be a tensor and cannot be called the torsion. In the class of frames in which • ω a bµ vanishes, torsion assumes the form with h a ν the tetrad (39). In Section VII we will return to discuss this point in connection to the so-called "pure tetrad teleparallel gravity".
Through contraction with a tetrad, the torsion tensor can be written in the form where is the non-trivial spacetime-indexed connection corresponding to the inertial spin connection • ω a bµ , also known as the Weitzenböck connection. Its definition is equivalent to the identity In the class of frames in which the spin connection • ω a bµ vanishes, it reduces to In all other classes of frame, it assumes the general form (54).
C. Gravitational coupling prescription

Translational coupling prescription
As discussed in Section III A, in the absence of gravitation the Lorentz covariant derivative in a general frame is written as with e a µ the trivial tetrad (44). In the presence of gravitation, on the other hand, it is given by with h a µ the non-trivial tetrad (43). The translational coupling prescription in a general class of frames can then be written in the form Such a coupling prescription actually amounts to the tetrad replacement which, in turn, amounts to replacing the spacetime Minkowski metric by a general Riemanian metric: As a consequence, the spacetime intervals change according to It is important to remark that in general relativity such replacement is implicitly assumed whenever applying the gravitational coupling prescription. In teleparallel gravity, on the other hand, it emerges naturally as a consequence of the translational coupling prescription.
Furthermore, in contrast to general relativity, it provides an explicit expression for the tetrad field, as given by Eq. (43).

Adding the Lorentz coupling prescription
In addition to being invariant under local translations, any theory must also be invariant under local Lorentz transformations. This second invariance is related to the fact that physics must be the same, independently of the Lorentz frame used to describe it. It should be emphasised that Lorentz invariance by itself is empty of dynamical content as any relativistic equation can be written in a Lorentz covariant form. Although not a dynamic symmetry, however, the local Lorentz invariance introduces an additional coupling prescription, which is a direct consequence of the strong equivalence principle.
The explicit form of the Lorentz gravitational coupling prescription can be obtained from the so-called general covariance principle (see Ref. [18], Section 4.1). In its frame version [19], this principle states that, by writing a special-relativistic equation in a Lorentz covariant form and then using the strong equivalence principle, it is possible to obtain its form in the presence of gravitation. The general covariance principle can be thought of as an active version of the usual (or passive) strong equivalence principle, which says that, given an equation valid in the presence of gravitation, the corresponding special-relativistic equation is recovered locally (that is, at a point or along a trajectory).
Let us consider the ordinary derivative of a general field Ψ. The first step of the general covariance principle is to perform a local Lorentz transformation, such that all ordinary derivatives ∂ µ Ψ assume the Lorentz covariant form, where f c ab is the coefficient of anholonomy (7), and S c b are Lorentz generators written in the representation to which Ψ belongs. The last term in the right-hand side is an inertial compensating term that enforces the Lorentz covariance of the derivative in the new Lorentz frame.
In the presence of gravitation, according to the translational coupling prescription (59), the trivial tetrad e a µ is replaced by the nontrivial one h a µ , and the coupling prescription (62) assumes the form with the coefficient of anholonomy now given by In the specific case of teleparallel gravity, where torsion is non-vanishing, relation (20) as- where • T c ab is the torsion of the purely inertial connection • ω c ab . Use of this equation for three different combinations of indices gives with the contortion tensor The coupling prescription in the presence of gravitation is then obtained by replacing the inertial compensating term of (63) with that given by (66): which defines the full (translational plus Lorentz) gravitational coupling prescription in teleparallel gravity.
Now, due to the fundamental identity of the theory of Lorentz connections [15], with • ω c ba the Levi-Civita spin connection, the teleparallel coupling prescription (68) is found to be equivalent to the general relativity coupling prescription Since both coupling prescriptions were obtained from the general covariance principle, both are consistent with the strong equivalence principle.

Separating inertial effects from gravitation
As is well-known, the spin connection • ω c bµ of general relativity includes both gravitation and inertial effects. Considering that = 0, that identity reduces to from which we see that, in this local frame, inertial effects • ω c bµ exactly compensates for gravitation • K c bµ . As an illustration, let us consider a free particle in Minkowski spacetime, whose equation of motion has the form Since the four-velocity u a is a Lorentz vector, we use the vector representation of the Lorentz generators, which is given by the matrix [20] S c b a In this case, the general relativity coupling prescription (70) assumes the form When applied to the equation of motion (72) describing a free particle, it yields the geodesic equation The vanishing of the right-hand side means that in general relativity there is no gravitational force. In this theory, gravitation and inertial effects are described by the geometry of spacetime, and are included in the spin connection in the left-hand side of the equation.
In a similar way, applying the teleparallel coupling prescription (68) to the free equation of motion (72) yields the teleparallel force equation which is, of course, equivalent to the geodesic equation (75). In this description, however, the inertial effects, represented by the inertial spin connection • ω a bµ , remain geometrized in the left-hand side of the equation, whereas gravitation, represented by the contortion tensor • K a bµ , plays the role of a gravitational force on the right-hand side [21]. This separation of gravitation and inertial effects, beautifully evinced by identity (69), is one of most prominent properties of teleparallel gravity.

A. Teleparallel gravity Lagrangian
Similar to any gauge theory, the Lagrangian density of teleparallel gravity is quadratic in the torsion tensor, the field strength of the theory (we use units in which c = 1) with h = det(h a µ ). The first term corresponds to the usual Lagrangian of internal gauge theories. The existence of the other two terms is related to the soldered character of the bundle, which allows internal and external indices to be treated on the same footing, and consequently new contractions turn out to be possible. Since torsion is a tensorial quantity, each term of this Lagrangian is invariant under both general coordinate and local Lorentz transformations. As a consequence the whole Lagrangian is also invariant, independently of the numerical value of the coefficients.
Introducing the notation κ = 8πG, we note the crucial property of the teleparallel Lagrangian (77) is its equivalence (up to a divergence) to the standard Einstein-Hilbert La- where • T µ = T νµ ν is the vector torsion. Due to this property teleparallel gravity is often called the teleparallel equivalent of general relativity since the dynamical content of the field equations derived from both Lagrangians must be the same.

B. Teleparallel gravity field equations
To derive the field equations of teleparallel gravity, let us consider the Lagrangian with L s the Lagrangian of a general source field. Variation with respect to the gauge potential B a ρ , or equivalently, with respect to the tetrad field h a ρ , yields the teleparallel gravitational field equations where on the left-hand side we have defined the Euler-Lagrange expression and we have introduced the superpotential and the gauge current which in this case represents the Noether energy-momentum pseudo-current of gravitation [22].
The right-hand side of the field equations (80) is the matter energy-momentum tensor We note that the anti-symmetric part of Θ [µν] = h a [µ g ν]ρ Θ ρ a = 0 due to the invariance of the action under local Lorentz transformations [4].

C. Alternative forms of teleparallel gravity field equations
There are many equally valid ways to write the field equations (80) that can be useful in different situations. One of them is to write the field equations using the teleparallel covariant derivative as where we have defined the gravitational energy-momentum tensor The advantage of this form of the field equations is that • Σa ρ is a proper tensor under both diffeomorphisms and local Lorentz transformations. Moreover, it can be shown to be tracefree: as is appropriate for a massless field. Alternatively, the field equations (80) can be rewritten in terms of spacetime-indexed quantities as where is the energy-momentum pseudotensor. Due to the fact that h • t ρ µ is conserved with ordinary derivative, it straightforwardly leads to spacetime conserved charges [23]. Moreover, the left-hand side E µ ρ of (88) is manifestly symmetric, which allows an easy comparison with the field equations of general relativity in the metric formulation.

D. Variations with respect to the spin connection
From the point of view of the gauge approach to teleparallel gravity discussed in Section III-in particular, from the fact that the tetrad can be written in terms of the translational potential and the spin connection, as can be seen from (43)-it is clear that the spin connection is not an independent variable from the tetrad. However, if we decide to use as a starting point, not the gauge paradigm but the Lagrangian (77) written as a function of the tetrad and the spin connection, we need to address the problem of the variation of the Lagrangian with respect to the spin connection.
As we will show shortly, it turns out that the variation with respect to the spin connection is identically satisfied for an arbitrary teleparallel spin connection, and hence there are no extra field equations that would determine it. This is consistent with our interpretation of the spin connection as representing inertial effects only, and hence should not have their own dynamics governed by extra field equations. The problem of how to determine the spin connection will be discussed in detail in Section V.
To compute the variations of the Lagrangian with respect to the spin connection, we follow here the method developed in [24]. We consider a teleparallel Lagrangian corresponding to a vanishing spin connection and a Lagrangian for an arbitrary spin connection: i.e., • L(h a µ , 0) and • ω a bµ ).
Then we use the equivalence of these Lagrangians with the Einstein-Hilbert Lagrangian, as given by Eq. (78). Since both Lagrangians (90) correspond to the same tetrad, the same Einstein-Hilbert Lagrangian can be associated with both of them. Their equivalence can then be written in the form On the other hand, contracting the torsion tensor (50) with h ν a yields where we have used the notation (91), the two teleparallel Lagrangians are found to be related by [24]: This relation shows that the inertial spin connection • ω a bµ enters the Lagrangian as a total derivative, and hence the variation with respect to the spin connection vanishes identically Moreover, relation (93) implies that the spin connection does not contribute to the field equations.

E. Solving the teleparallel gravity field equations
We discuss now a method for solving the teleparallel field equations [23]. To begin with, let us recall that the gravitational field equations are intricate nonlinear differential equations, for which there is not a general constructive method for obtaining solutions. One has to resort to some ad hoc procedure in which some hand work is necessary. Typically one relies on the symmetries of the solution to propose an ansatz for the metric or tetrad, thereby obtaining simpler differential equations that are easier to solve. The important point is that upon proposing some ansatz, one is most likely choosing a tetrad whose associated inertial connection • ω a bµ is non-vanishing. However, as follows from the relation (93), the spin connection enters the action only through a surface term and hence the field equations obtained from • ω a bµ ) and from • L(h a µ , 0) are the same. This means that the field equations can be solved independently of the spin connection, which is left undetermined in the process. It should be noted that the teleparallel field equations determine only the equivalence class of tetrads with respect to the local Lorentz transformations Λ a b (x). In other words, tetrads related through global Lorentz transformations, that is to say, are indistinguishable as far as the teleparallel field equations are concerned. This is a direct consequence of the fact that the teleparallel spin connection (16) is not determined by the field equations. Therefore, the field equations do not determine Λ a b (x), which means that the tetrad is determined up to a local Lorentz transformation. Namely, the field equations effectively determine only the metric tensor. In the next Section we show that the same situation occurs in the tetrad formulation of general relativity.

F. Comparison with the tetrad formulation of general relativity
It is useful at this point to make a comparison with general relativity. As is well-known, general relativity has both a metric and a tetrad formulation [18]. In the metric formalism we straightforwardly calculate the Riemannian curvature using Christoffel symbols from the metric tensor. The dynamics of the metric tensor is described by the Einstein field equations, which is essentially a set of ten field equations for the ten components of the metric tensor.
On the other hand, in the tetrad formulation the ten-components of the metric tensor are replaced by the sixteen-components of the tetrad field. The Einstein field equations in this case take the similar form where • R a ν is the Ricci curvature calculated directly from the tetrad, and is related to the spacetime-indexed Ricci curvature by From this we can see that the tetrad form of the Einstein field equations (97) is just a projection of their spacetime form (96) along the tetrad components. Therefore, the dynamical content of both forms of field equations is the same and they determine only the metric tensor. This is also clear from the fact that Einstein's field equations (97) are covariant under local Lorentz transformations: This covariance eliminates six of the sixteen equations (97), which means that the tetrad is determined by the field equations (97) only up to a local Lorentz transformation. This means that we actually determine only the metric tensor. Naturally, this is an expected result since both the metric and the tetrad formulations are just two equivalent formulations of the very same theory.

V. TETRAD AND ITS ASSOCIATED SPIN CONNECTION
To each tetrad h a µ there is an associated inertial spin connection • ω a bµ that describes the inertial effects present in the frame. This is clear from the fundamental form of the tetrad in teleparallel gravity, as given by Eq. (43). The class of inertia-free frames, known as proper frames, is characterized by a vanishing spin connection: {h a µ , 0}. In any other class of frames related to the proper frames by a local Lorentz transformation, the spin connection will be non-vanishing, which means that there are infinitely many pairs {h a µ , • ω a bµ }. Each pair defines a different class of frames, characterized by a different inertial spin connection • ω a bµ . However, in all practical cases, it is not immediately possible to identify the spin connection of a given tetrad h a µ . It is then necessary to provide a method to retrieve such a spin connection from a general tetrad. This corresponds to specifying all inertial effects present in the frame represented by h a µ .

A. Determining the inertial spin connection
We begin by defining a "reference tetrad", h a (r) µ , as a tetrad in which gravity is switchedoff. It is, of course, a trivial tetrad in the sense that it relates two Minkowski metrics written in different coordinate systems. This can be done by setting the gravitational constant G equal to zero: In such a tetrad, the gravitational potential B a µ does not appear and the reference tetrad can be written formally as Furthermore, considering that this tetrad represents a trivial frame (see Sec. II), the torsion tensor of the spin connection • ω a bµ vanishes identically: The coefficients of anholonomy f c ab of the general tetrad h a µ , according to Eq. (7), are given by Using Eq. (65), where the torsion is written in terms of f c ab as we find that the condition (102) for the reference tetrad assumes the form with f a bc (h (r) ) the coefficients of anholonomy of the reference tetrad h a (r) µ . Using (105) for three different combination of indices, we can solve for the spin connection [23]: This is the inertial spin connection naturally associated to the reference tetrad h a (r) µ . Since the reference tetrad h a (r) µ and the original tetrad h a µ differ only by their gravitational content-the inertial content of both tetrads are the same-the spin connection (106) is the inertial spin connection naturally associated to the original tetrad h a µ as well. Notice, in addition, that the expression for the teleparallel spin connection (106) coincides with the Levi-Civita spin connection for the reference tetrad, and hence we can write We would like to stress that the Levi-Civita connection is calculated for the reference tetrad corresponding to the Minkowski spacetime and consequently it is guaranteed to have a vanishing curvature; it is hence in the class of teleparallel connections (16).
The crucial point is that the torsion tensor constructed from the "full" tetrad and the spin connection represents purely gravitational torsion in the sense that the spurious contribution from the inertial effects are removed.

B. The regularizing role of the inertial spin connection
Let us begin by considering an action for the reference tetrad (100), which represents only inertial effects. If we naively associate a vanishing spin connection to the reference tetrad h a (r) µ , the gravitational action assumes the form In general this action does not vanish, and is even typically divergent. The reason for this is that it is an action for inertial effects, which in general do not vanish at infinity [25]. If, instead of a vanishing spin connection, we choose the appropriate inertial spin connection (106), then from Eq. (102) we have We now see that the role of spin connection • ω a bµ is to remove all inertial effects of the action, in such a way that it now vanishes-as it should because it represents only inertial effects.
From the point of view of inertial effects, the full and reference tetrads are equivalent in the sense that their inertial content are the same. This consequently means that the spin connection associated with the full tetrad (or reference tetrad) is able to remove the inertial contributions, not only from the inertial action, but from the full action as well. This yields an action that represents gravitational effects only. Considering that the inertial effects are responsible for causing the divergences, the purely gravitational action with the appropriate spin connection, • ω a bµ ), will always be finite for any solution of the gravitational field equations. It can consequently be viewed as a renormalized action [23].
We note that it is possible to achieve the same results in a simpler way. In fact, relation (93) shows that the divergences are removed from the action by adding an appropriate surface term to the action, which is analogous to the process of holographic renormalization.
However, in teleparallel gravity it can be interpreted as the removal of the spurious inertial effects from the theory. Of course, once the spurious inertial contribution to the Lagrangian is removed, all quantities computed using this Lagrangian, such as for example energy and momentum, will also be finite [26]. Furthermore, there is an important difference in relation to other renormalization methods: the inertial effects in teleparallel gravity are removed locally at each point of spacetime and not from the whole integral, as happens in other formalisms. As a consequence, instead of a quasi-local definition, the energy and momentum densities in teleparallel gravity can be defined locally [24].
where A = A(r) and B = B(r) are arbitrary functions to be determined from the field equations.
As we have already discussed, there are infinitely many tetrads corresponding to the metric ansatz (112). We consider here two tetrads: the diagonal tetrad and the off-diagonal tetrad These two tetrads are related byh Obviously both tetrads represent the same metric (112) because, as is well known, the metric is invariant under local Lorentz transformations.

B. Solving the field equations
We choose to solve the field equations in the spacetime form (88), and we assume a zero spin connection initially. We proceed then to obtain all geometrical objects for both the diagonal and the non-diagonal tetrads.
1. Using the diagonal tetrad h a µ For the case of the diagonal ansatz, the non-vanishing components of the superpotential where we do not explicitly display the antisymmetric components non-vanishing components of the energy-momentum pseudotensor Combining them, the nontrivial components of the field equations (88) are found to be The third equation is easily seen not to be an independent equation. Using then the first two equations, we find the same solution as in general relativity; that is, Matching the solution to the Newtonian limit, the integration constant c 1 , as in general relativity, is found to be c 1 = 2GM.

Using the off-diagonal tetradh a µ
It is an interesting exercise to derive the field equations for the off-diagonal ansatzh a µ , which explicitly illustrates that the field equations determine the tetrad up to a local Lorentz transformation. In this case the non-vanishing components of the superpotential Similarly, the non-vanishing components of the energy-momentum pseudotensor Note that both the superpotential and the energy-momentum pseudotensors for the diagonal tetrad (113) and for the off-diagonal tetrad (114) are completely different, due to the fact that when we assumed a vanishing spin connection for both tetrads both quantities

Determining the associated spin connection
We use the method introduced in Section V to find the components of the spin connection associated with the tetrads (113) and (114). The starting point is to define the reference tetrad, which for the diagonal tetrad (113) is Using (106), we find that the non-vanishing components of the spin connection are These components represent the inertial effects present in the diagonal tetrad (113). We can analogously define the reference tetrad for the off-diagonal tetrad (114) as Using (106), we find that the corresponding connection vanishes This means that the off-diagonal tetrad (114) is indeed the proper tetrad, and as such it represents gravitation only. On the other hand, the diagonal tetrad (113) is not proper since the associated inertial effects, represented by the spin connection (126), do not vanish.
We can now check that both tetrads with their associated spin connections do lead to the same prediction for all geometric quantities, that is, • ω a bµ ).
Such quantities transform covariantly under both diffeomorphisms and local Lorentz transformations. Furthermore, they now represent only gravitation, to the exclusion of the spurious inertial effects.

C. Regularization of the action
To see how the spurious inertial effects come into play and why we need to compute the associated spin connection, we consider the action and the corresponding conserved charges [23]. If we compute the action using the diagonal tetrad (113) and vanishing spin connection, we find that which is obviously a divergent quantity and consequently leads to divergent conserved charges. This is precisely due to the fact that the torsion scalar • L(h a µ , 0)-and hence the action (130)-in addition to gravitation, also includes the spurious inertial effects, which means it does not vanish at infinity. The overall integral then leads to a divergent action.
However, if we remove the inertial effects by taking into account the associated spin connection (126), or equivalently use the proper tetrad (114), then we find the well-behaved renormalized action The conserved charges obtained from this action are finite, and represent the correct physical conserved charges.

VII. SOME REMARKS ON THE PURE TETRAD TELEPARALLEL GRAVITY
As already discussed, to any tetrad h a µ there is associated a specific inertial spin connection • ω a bµ that represents the inertial effects present in that frame: On the other hand, torsion is defined as the covariant derivative of the tetrad: Note that the spin connection is essential for torsion (133) to be a tensorial quantity; that is, an object that transforms covariantly under both local Lorentz and general coordinate transformations. As a consequence, the action of teleparallel gravity, which similarly to the action of any gauge theory is quadratic in the field strength (in this case, torsion), will be invariant under both local Lorentz and general coordinate transformations. Of course, the corresponding field equations will transform covariantly under those transformations.
The crucial observation of teleparallel gravity is that the spin connection associated with the tetrad (132) and the one used in the torsion (133) are the very same spin connection. This is particularly clear within the approach to teleparallel gravity as a gauge theory for the translation group. As can be seen in Section III B, the torsion tensor is explicitly constructed in a such way that the spin connection • ω a bµ appearing in the covariant derivative is the same spin connection associated to the given tetrad h a µ according to (43). Nevertheless, it is important to note that we do not usually follow this gauge approach in practical calculations. Instead, we typically start with some ansatz tetrad that suits the symmetry of the problem under consideration and we do not know a priori its associated spin connection. Due to the peculiar structure of the teleparallel action discussed in section IV D, we can first solve the problem for the tetrad and then determine the spin connection according to the method discussed in Section V. Only then do the physical quantities computed from the torsion tensor, such as for example the action of the gravitational field or the gravitational conserved charges, give the correct, finite, physically relevant results.
Therefore, the requirement of the finiteness of the action and conserved charges motivates the necessity to associate to each tetrad a spin connection according to the method discussed in Section V. Now, there is in the literature a different approach to teleparallel gravity, known as pure tetrad teleparallel gravity (see Ref. [27] for a review). Its name stems from the fact that torsion is defined not as the covariant derivative of the tetrad [15], but instead as an ordinary derivative, with the subscript zero denoting the "torsion" of the pure tetrad theory. In other words, the spin connection appearing explicitly in the torsion definition (133) is always assumed to vanish, although to each tetrad there is associated a (generally) non-vanishing spin connection (132). The pure tetrad formulation ignores the this fact and treats the tetrad and the spin connection as genuinely independent variables. Due to the pure gauge form (16), the teleparallel spin connection can be transformed to zero independently of transformations of the tetrad and hence the vanishing spin connection can be associated with each tetrad.
There are multiple problems with this approach. First, even though (134) is mathematically and physically meaningful, it is not torsion. As a matter of fact, it is minus the coefficient of anholonomy f c ab of the frame h a µ , whose components, according to Eq. (103), are given by Only in the class of frames in which the inertial spin connection • ω a bµ vanishes will the coefficient of anholonomy coincide with torsion. In all other classes of frames, they will not coincide.
In spite of this problem, it is still possible to use the pure tetrad teleparallel gravity for some specific purposes. The Lagrangian of the pure tetrad teleparallel gravity is obtained from the Lagrangian (77) by setting the spin connection to zero and replacing torsion by the coefficient of anholonomy Despite the fact that f a bc is not a tensor, the field equations derived from this Lagrangian are the same as those derived from (77) and it then follows, as discussed in Section IV D, that the spin connection does not contribute to the field equations. Therefore, as far as we are interested in the solutions of the field equations, both approaches, our invariant approach and the pure tetrad one, lead to the same result.
Second, the Lagrangian (136) is not invariant with respect to local Lorentz transformations, but only quasi-invariant; i.e., changes by the surface term [28]. Nevertheless, these contributions through the surface term do play an important role when the total value of the action is considered and in derivation of the conserved charges. It is then found that only for certain preferred class of the frames do we obtain finite, physically relevant results.
Third, local Lorentz symmmetry is a fundamental symmetry of general relativity and it is hard to establish a full equivalence with some theory that violates it in a meaningful way. Therefore, only the invariant formulation of teleparallel gravity presented here can be considered to be a genuine teleparallel equivalent of general relativity.
In the literature, properties of pure tetrad teleparallel gravity are often erroneously attributed to teleparallel gravity. For example, it is quite common to find statements that in teleparallel gravity torsion is not a tensor, or that the theory is not invariant under local Lorentz transformations, or that there are prefered frames. Obviously all of these statements apply to the pure tetrad formulation but not to the invariant formulation presented here. This confusion can be easily understood from a modern perspective where the pure tetrad teleparallel gravity can be viewed as a teleparallel gravity formulated in the specific gauge in which the spin connection vanishes. After fixing the gauge, we are obviously not allowed to perform any local Lorentz transformations and naturally there are only certain tetrads in which this gauge holds. The whole discussion of local Lorentz invariance in pure tetrad formulation can be then viewed as rather misguided and not an indicator of any problem of teleparallel gravity. The analogous situation in electromagnetism would be to discuss gauge invariance after fixing a specific gauge (e.g., the Coulomb gauge), which obviously does not imply any problem of the theory but rather the limitations of our chosen gauge.

VIII. MODIFICATION OF TELEPARALLEL GRAVITY: f (T ) GRAVITY
The discovery of the accelerated expansion of the Universe has motivated the study of various extensions of general relativity. A very popular extension is the so-called f (R) gravity where the Lagrangian is taken to be a function of the Ricci scalar. This relatively simple model has a number of interesting features and rich phenomenology [5].
In a similar fashion to f (R) gravity, Ferraro and Fiorini [29][30][31][32] have proposed the f (T ) gravity model, where the Lagrangian is given by where • T is the so-called torsion scalar representing the same quadratic torsion pieces appearing in the Lagrangian of teleparallel gravity (77); i.e., 5 Following the equivalence between the teleparallel and Einstein-Hilbert Lagrangians (78), we can obtain the relation between the torsion and curvature scalars is the so-called boundary term.
The boundary term does not contribute to the field equations in the case of teleparallel gravity. However, modified gravity models based on the idea of replacing the actions linear in • R or linear in • T with arbitrary non-linear functions, f (R) and f (T ) respectively, are no longer equivalent. This simply follows from the fact that an arbitrary function of a boundary term is, in general, no longer a boundary term. As we discuss later in Section IX F, it is possible to relate f (R) gravity and f (T ) gravity if we consider teleparallel gravity theories with higher derivative terms in torsion.

A. Field equations and variations of the action in f (T ) gravity
The Lagrangian of f (T ) gravity is a function of both the tetrad and the spin connection, and hence we should consider variations with respect to both variables. We remind the reader about the situation in the ordinary teleparallel gravity discussed in Section IV, where the variation with respect to the spin connection turned out to be trivial (94). This led us to the conclusion that the field equations do not determine the spin connection and, instead, it needed to be calculated using the reference spacetime as discussed in Section V.
The situation is rather different when we consider modified teleparallel gravity models such as f (T ) gravity. As we will show shortly, we find that the variations with respect to both the tetrad and the spin connection are non-trivial. However, the two variations are closely related since the variation with respect to the spin connection leads to the identical field equations as the antisymmetric part of the field equations for the tetrad.
Let us consider the f (T ) Lagrangian (137) and a general source field and vary the action with respect to the tetrad. This leads to the field equations for the where the Euler-Lagrange expression on the left-hand side (see [9] for details) is given by where f T and f T T denotes first and second order derivatives of the f -function with respect to the torsion scalar.
In order to analyze the symmetric and antisymmetric part of the field equations, it is useful to define the fully Lorentz-indexed Euler-Lagrange expression which can be explicitly written as where • Gab is the symmetric Einstein tensor of the Levi-Civita connection calculated from the tetrad only. In this form it is straightforward to see that the last two terms are symmetric and hence the antisymmetric part of the Euler-Lagrange expression is given by Given that the energy-momentum tensor has vanishing anti-symmetric part, Θ [ab] = 0, the antisymmetric part of the field equations is We can now consider the variation with respect to the spin connection and show that it leads to the same equations as (147). We follow here the method introduced in [33], but it is possible to derive the same result using alternative methods [34,35].
The relation (93) for the torsion scalar can be re-expressed as We then find that the variation of δ ω • L f with respect to the spin connection is given by Now, it is important to consider only those variations of the spin connection that preserve the local flatness and the teleparallel form of the spin connection (16). Since the teleparallel connection (16) is entirely given by the local Lorentz transformation matrix Λ a b , it is sufficient to consider only its changes under infinitesimal local Lorentz transformations [34], The variation of the spin connection is then given by We can then substitute this expression into Eq. (149), set the variation to vanish, and integrate by parts to obtain δ ω which can be then written as where we have used [36], we find that the field equations for the spin connection (153) coincide with the antisymmetric part of the field equations for the tetrad (147).
We notice that a very special situation occurs when we have known Schwarzschild [37], Kerr [38] and McVittie [39] solutions of general relativity solve f (T ) gravity as well.

B. Local Lorentz symmetry in f (T ) gravity
The issue that has caused a lot of attention and raised some doubts about the consistency of f (T ) gravity and other modified teleparallel gravity models is the question of local Lorentz invariance [5,8]. We will now explain the origins of this problem and how it is avoided in the covariant formulation of the theory that we use here.
In Section VII we have mentioned the so-called pure tetrad approach to teleparallel gravity, where the only variable is the tetrad. This originates from the fact that the teleparallel connection (16) is a pure gauge connection and hence it is always possible to perform a local Lorentz transformation such that the connection is transformed to zero. This is then equivalent to choosing a specific frame in which the spin connection vanishes and hence formulating the theory in this very specific class of frames. Strictly speaking, the question of local Lorentz invariance is ill-defined in this case since we choose the specific frame and hence we are not allowed to perform local Lorentz transformations.
Nevertheless, in the case of the ordinary teleparallel gravity with Lagrangian density (77) this approach gained some popularity not only because it was originally used in Einstein's teleparallelism [40,41] but also since it can be justified in some cases. Particularly, on account of property (93), the spin connection does not dynamically affect the field equations (80) and hence setting it to zero can be used to obtain solutions. In fact, as we have explained in Section IV E, in our covariant approach to ordinary teleparallel gravity, for the sake of convenience we can also set the spin connection to zero as an intermediate In particular, this leads to the situation where only some frames in the same equivalence class (i.e., corresponding to the same metric), were able to solve the field equations. Such frames were referred to as good tetrads [42], while the so-called bad tetrads-related to good tetrads by a local Lorentz transformation-were solutions only in the limit of the ordinary teleparallel gravity. For example, in the case of spherical symmetry, the diagonal tetrad (113) was considered to be a bad tetrad, while the off-diagonal (114) was a good tetrad.
It is easy to understand this problem of the original f (T ) gravity from the viewpoint of our covariant formulation. As we have argued in the previous Section VIII A, the field equations of f (T ) gravity determine both the tetrad and the spin connection, and hence the solution is always a pair of variables {h a µ , • ω a bµ }. Since these variables are not independent, a transformation of the spin connection must be always accompanied by a transformation of the corresponding tetrad. In particular, if we transform the spin connection to zero, then we must perform a simultaneous transformation on the tetrad; i.e., We can then identify the class of framesh a µ which corresponds to the zero connection within the class of good frames. However, we can see that there is nothing fundamentally special about these frames. All other frames related by local Lorentz transformation are equally viable; we just need to use the corresponding spin connections with them.
At least in principle, it is possible to write down the field equations directly in terms of the good framesh a µ , which are obtained by using the transformation (154) on the field equations (142). As a result, we will obtain 16 field equations that completely determine the tetrad, including the local Lorentz degrees freedom of the tetrad. This is in contrast with the situation in the ordinary teleparallel gravity discussed in Section IV E where the field equations determined only the equivalence class of the tetrads. The tetrad that solves the f (T ) field equations after the transformation (154) is completely fixed.
However, this does not imply that local Lorentz symmetry is violated in f (T ) gravity because we need to keep in mind that we have obtained this solution using the transformation (154) and hence we work only in this class of frames. If we want to discuss local Lorentz symmetry we need to act with local Lorentz transformations on both variables which necessarily generates a new spin connection. We can picture this as havingh a µ that solves the field equations with a zero spin connection, and then we perform an inverse of a local Lorentz transformation (154) with an arbitrary Λ a b . This will then generate for each tetrad h a b a corresponding spin connection We consider two different tetrads representing the Minkowski spacetime. The first one is the diagonal tetrad in the Cartesian coordinate system: which-if we set the spin connection to zero-leads to zero torsion and hence for any function f (T ) with f (0) = 0 then the field equations are trivially satisfied. Therefore, the tetrad (155) is a proper tetrad to which corresponds a vanishing zero connection or, in the terminology of [42], a good tetrad.
On the other hand, if we consider a Minkowski diagonal tetrad in the spherical coordinate system h a µ = diag (1, 1, r, r sin θ) , and use it with the vanishing zero spin connection, we find that the corresponding torsion scalar is non-zero. Additionally, we observe that one of the field equations, has a solution only if we set f T T = 0. This reduces the theory back to the ordinary teleparallel gravity and hence (156) is, in the terminology of [42], a bad tetrad.
The problem of the original formulation of f (T ) gravity was that in order to be able to use the spherical coordinate system in f (T ) gravity to describe Minkowski spacetime we had to use the tetradh a µ =Λ a b h a µ withΛ a b given by (116) and h a µ given by (156), which yields the equivalent of the reference tetrad in Eq. (127). Since gravity is absent in Minkowski spacetime, this demonstrates very well that the nature of the Lorentz invariance problem is not related to the modification of gravity but is just a matter of the consistent formulation of the theory. On the other hand, in the covariant formulation of f (T ) gravity, the problem is absent and we are able to use all tetrads in arbitrary coordinate systems. In the case of Minkowski spacetime, for instance, we are free to use the diagonal tetrad (156) but we need to use it along with its corresponding spin connection (126).

Example: FLRW spacetime
We now move on to the more non-trivial example of the zero-curvature FLRW spacetime describing the evolution of the Universe. Using the Cartesian coordinate system we can choose a diagonal tetrad that represents the zero-curvature FLRW metric as, h a µ = diag(1, a(t), a(t), a(t)), with a(t) being the scale factor. Similar to the case of Minkowski spacetime, if we want to use the spherical coordinate system we need to either transform the corresponding diagonal tetrad by the local Lorentz transformation (116) or accompany the diagonal tetrad with its corresponding spin connection (126).
The above tetrad leads to the torsion scalar where H =ȧ/a is the Hubble parameter, and gives rise to the Friedmann equations where ρ M and p M are the energy density and pressure of the matter fluid, respectively. These are the correct f (T ) modified Friedmann equations capable of explaining the accelerated expansion of the Universe, as shown originally in [29][30][31][32] and later extensively studied in . For a complete list of references see the review [68].
Similar to the case of Minkowski spacetime, if we would like to use instead of the Cartesian diagonal tetrad (158) the diagonal tetrad in the radial coordinate system h a µ = diag(1, a(t), a(t) r, a(t) r sin θ), we have to either use the spin connection corresponding to (162), which is given by (126), or make the tetrad (162) proper by the local Lorentz transformation given by (116). Both methods will lead to the same set of Friedmann equations (160)-(161) as when working with the Cartesian tetrad (158).

Example: Spherically symmetric vacuum spacetime
The spherically symmetric solutions of the field equations in any gravitational theory are of the crucial importance since they describe the gravitational field outside a massive spherical body which is important for understanding the dynamics of the solar system and the dynamics of more exotic objects such as black holes. In the framework of f (T ) gravity theories, spherically symmetric solutions have attracted a lot of attention lately [37,42,[69][70][71][72][73][74][75][76][77][78]. Interestingly, in f (T ) gravity the spherically symmetric vacuum spacetime is necessarily static, as shown in [79].
The metric (see also Eq. (112)) has the form The most natural choice of the tetrad that corresponds to this metric has the simple diagonal form (see also in Eq. (113)), the same choice as in the ordinary teleparallel gravity, h a µ = diag (A, B, r, r sin θ) .
It was argued that only very specific forms of the tetrad, with off-diagonal components, can lead to a physical outcome when using f (T ) gravity, but the physical motivation for this was not fully understood. Having constructed the covariant formulation of f (T ) gravity in the previous sections, we can now show that every tetrad corresponding to the desired metric is equally good and leads to solutions, as long as the correct spin connection is used.
Additionally, we will also provide the explanation for the necessity to use a complicated off-diagonal tetrad in previous non-covariant formulations of f (T ) gravity.
It is straightforward to check that if we use a diagonal tetrad (113) and assume a trivial spin connection ω a bµ = 0, then we obtain the field equation In general • T ′ = 0, and therefore this then necessarily gives us the condition f T T = 0, which restricts the theory to ordinary teleparallel gravity. In the literature this feature is wrongly interpreted as "the diagonal tetrad is not a good tetrad for spherically-symmetric solutions in f (T ) gravity".
Let us now show that the above issue is an artifact of the non-covariant formulation of f (T ) gravity that was used. In particular, using the covariant formulation presented in the previous sections we will calculate the appropriate spin connection which will allow us to use any tetrad giving the metric (163), without restricting the functional dependence of the Lagrangian. Similar to the previous examples, due to the fact that the solution of the field equations is unknown, we have to start with an ansatz for the reference tetrad corresponding to the tetrad (164). It is natural to expect that in the absence of gravity the diagonal tetrad (164) should reduce to the tetrad (156) representing the Minkowski spacetime in spherical coordinates. Therefore, the corresponding spin connection is given by (see also Eq. (126)) Using this spin connection we can now remove the spurious inertial contributions and obtain the gravitational torsion tensor. The torsion scalar constructed from it is then given by where a prime denotes the derivative with respect to the radial coordinate r. Thus, using the tetrad (164) and the non-zero spin connection (166), the field equations (142) become where T ′ is the derivative of the torsion scalar (167) with respect to the coordinate r.
As a brief inspection shows, the field equations (168) is a proper tetrad (i.e., a tetrad in which the inertial spin connection vanishes), and that is why the obtained field equations coincide with the ones of the covariant formulation. In other words, in the usual, non-covariant formulation of f (T ) gravity, one considers specific peculiar non-diagonal tetrads, thus making the theory frame-dependent, as a naive way to be consistent with a vanishing spin connection. However, as we have shown, the correct and general way to acquire consistency is to use the covariant formulation of f (T ) gravity, in which case frame-dependence is absent. In particular, it is permissible to use any form of the tetrad provided that the corresponding spin connection is calculated. The off-diagonal tetrad (164) has no privileged position anymore; it is just a specific tetrad in which the corresponding spin connection happens to be zero.

IX. OTHER MODIFIED TELEPARALLEL MODELS
The most popular modified teleparallel gravity model in the recent decade is f (T ) gravity and much of the attention has been focused on this particular model. However, the teleparallel structure allows us to formulate a plethora of other interesting modified teleparallel gravity models. We briefly review here some of the more popular models and provide a classification scheme based on their essential features and/or the purpose for which they were proposed.

A. New General Relativity
The so-called New General Relativity, introduced in 1979 by Hayashi and Shirafuji [10,80], is the oldest modified teleparallel gravity model where the teleparallel Lagrangian (77) is straightforwardly generalized by considering arbitrary coefficients of the quadratic scalar torsion terms. We follow here the original approach used in [10,80], which will be also useful later in Section IX C, and decompose the torsion tensor into irreducible parts with respect to the Lorentz group where are known as the vector, axial, and purely tensorial torsions, respectively. We can then construct three parity preserving quadratic invariants T ten = t λµν t λµν = 1 2 T ax = a µ a µ = 1 18 The action of the teleparallel gravity (77) in terms of these quadratic invariants takes the New General Relativity is a straightforward generalization of the teleparallel Lagrangian where the coefficients in front of the quadratic invariants take arbitrary values; i.e., where the four a i are arbitrary constants and a 0 can be interpreted as the cosmological constant. Since, up to a divergence, with • R the scalar curvature of the Levi-Civita connection, Hayashi and Shirafuji rewrote the Lagrangian (178) in the form with the new coefficients given by In this theory, torsion would represent additional degrees of freedom relative to the curvature, which would thus produce deviations in relation to general relativity, or equivalently, in relation to teleparallel gravity. In the original new general relativity by Hayashi and Shirafuji [10,80], only the b 1 parameter was considered to be non zero since solar system experiments put strong constraints on b 2 and b 3 . Further problems and limitations of this model were discussed in [81][82][83].

B. Conformal Teleparallel Gravity
Recently there has been increased interest in gravitational theories with conformal invariance, which is expected to be recovered as a fundamental symmetry at the Planck scale [84][85][86]. In the standard Riemannian framework the Lagrangian is usually assumed to be quadratic in the Weyl tensor and hence leads to a theory with fourth order field equations.
Within the teleparallel framework we can use the fact that the torsion contains only first derivatives of the tetrad and construct a conformally invariant theory of gravity with second order field equations [11]. The most general Lagrangian of the teleparallel conformal gravity is then given by where L 1 is a generalization of the torsion scalar (138) given by where a 1 , a 2 , a 3 are three constants satisfying the relation and L 2 is defined analogously to (183) with generally three different constants a ′ 1 , a ′ 2 , a ′ 3 satisfying the analogous constraint to (184).
Using the quadratic invariants of the irreducible parts of the torsion tensor (174)-(176), it is possible to write the torsion scalar (183) in a simpler form where b 1 , b 2 are arbitrary constants [87]. The second scalar L 2 can be written analogously Since the overall normalization fixes one of the constants, the teleparallel conformal model (182) has 3 free parameters that can be chosen arbitrarily.
C. f (T ax , T ten , T vec ) Gravity A natural generalization combining elements of both f (T ) gravity and new general relativity is f (T ax , T ten , T vec ) gravity [87], where the Lagrangian is taken to be an arbitrary function of quadratic invariants of the irreducible parts of the torsion tensor (174)-(176) This model includes other models discussed in previous sections as special cases and is particularly suitable to study the behavior of teleparallel models under conformal transformations of the metricĝ µν = Ω 2 (x)g µν , or equally of the tetradĥ a µ = Ω(x)h a µ , where Ω(x) is the conformal factor. The quadratic invariants of the irreducible parts of the torsion tensor (174)-(176) transform as T ax = Ω 2T ax , T ten = Ω 2T ten , T vec = Ω 2T vec + 6Ωv µ∂ µ Ω + 9ĝ µν (∂ µ Ω)(∂ ν Ω). (187) Analyzing these transformation properties it can then be concluded, unlike the situation in f (R) gravity [5], that it is not possible to find an "Einstein frame" where the theory reduces to ordinary general relativity and a minimally coupled scalar field [87]. This is a generalization of the result previously obtained in f (T ) gravity [92,93]. Moreover, we can also quickly confirm the observation that (185) transforms properly under conformal transformations and hence the Lagrangian (182) is, indeed, conformally invariant.

D. Gravity Models Inspired by Axiomatic Electrodynamics
A novel approach to teleparallel theories was recently proposed by Itin et al. [94] that utilizes the similarities between electromagnetism and teleparallel gravity. The field equations of electromagnetism can be written as where F is the electromagnetic field strength 2-form, H is the excitation 2-form and J the current 3-form. The specific case of Maxwell electrodynamics is then defined by the relation between the excitation form and the field strength form using the Hodge dual map The crucial observation of the so-called axiomatic electrodynamics is that the field equations (188)-(189) describe a consistent theory of electrodynamics even if the relation (190) is generalized to a more general constitutive relation H = κ(F ). Using this generalization we can then describe various effects in media and non-linear theories of electrodynamics in one unified language [95].
Teleparallel gravity can be cast into a similar form by writing the Lagrangian in the language of differential forms as where • Ha is the gravitational excitation 2-form related to the superpotential (82) through The Bianchi identities and the field equations of teleparallel gravity then take a form that closely resembles the equations of electrodynamics (188)-(189): where D is the teleparallel covariant exterior derivative, and In [94] it was demonstrated that the same generalization to that in the case of axiomatic electrodynamics can be realized in teleparallel gravity by replacing the constitutive relation (192) by a general constitutive relation is a function of the tetrad and the torsion. It was shown that the most general local and linear constitutive relation defines new general relativity discussed in Section IX A [94]. In [35] this approach was generalized to include arbitrary non-linear constitutive relations and it was shown that all previously studied modified teleparallel gravity models with second order field equations can be naturally realized by finding corresponding constitutive relations. For example, f (T ) gravity can be realized through the constitutive The advantage of such an axiomatic approach is that the field equations of all modified teleparallel theories then take the same form (193)- (194), which allows us to analyze them in full generality and better understand their generic features. For instance, in [35] it was shown that the result from f (T ) gravity, discussed in Section VIII A, that the field equations for the spin connection coincide with the antisymmetric part of the field equations for the tetrad, applies to all modified gravity models with second order field equations. Moreover, we obtain also a new framework where modified gravity theories are viewed analogously to various non-linear electrodynamics theories. Using such analogies we can then construct new modified gravity models inspired by electrodynamics theories [35].

E. Teleparallel Dark Energy and Scalar-Torsion Models
Another popular way to modify gravity is by introducing scalar fields. In the standard curvature-based approach, the earliest such model was Brans-Dicke gravity [97], and more recent models include various quintessence and scalar-tensor gravity models [98][99][100][101].
In the teleparallel framework, we can follow the same path and formulate modified gravity theories with scalar fields. The first model introduced by Geng et al. [102] under the name teleparallel dark energy considers the torsion scalar (138) non-minimally coupled to the scalar field with kinetic and potential terms for the scalar field: It was shown that such a model leads to interesting cosmological dynamics consistent with observations yet distinct from f (T ) gravity or curvature-based analogues [47,[103][104][105][106][107]. Note that this model, in the same way as f (T ) gravity discussed in Section VIII, was originally presented as violating local Lorentz invariance, but it can be reformulated in a covariant way using the non-trivial spin connection [108].
Following this example other modified gravity models with a scalar field were proposed, including the possibility of coupling the gradient of the scalar field with the trace of the torsion tensor [109], and a tachyonic scalar field [110]. Recently, the most general extension was proposed as f (T, X, Y, φ), where X is the kinetic term for the scalar field, and Y is the term representing the coupling between the torsion and gradient of the scalar field [111].
A similar yet distinct model is f (T, T ) gravity [112], where the scalar T = Θ µ µ is the trace of the energy-momentum of matter (84).

F. Higher Derivative Models: f (T, B) and Others
All of the previous teleparallel models considered the Lagrangian to be a function of the torsion scalar only and did not include its derivatives; as a result the equations of motion were always second order. However, it is possible to include derivatives of torsion and deal with fourth (or possibly higher) order equations. Among such higher derivative type models, the best motivated one is perhaps f (T, B) gravity, where we include the 'boundary' term • B = (2/e) ∂ µ (ev µ ) (see Eq.(140)).
Interestingly, for a particular form of the arguments, we can obtain the usual f (R) gravity: It is precisely the boundary term is a teleparallel theory that is locally Lorentz invariant even in the pure tetrad formulation discussed in section VII.
Recently, a number of other teleparallel theories with higher order field equations were proposed. These include, for instance, the so-called teleparallel Gauss-Bonnet gravity [113,114] inspired by analogous work in the curvature approach [115], where the following Lagrangian was considered Here • T G is the so-called teleparallel Gauss-Bonnet term related to the usual curvature Gauss-Bonnet by a boundary term in a similar fashion to the relationship of curvature and torsion scalars in Eq.(140). Considering a f -function of such a teleparallel Gauss-Bonnet term in (200) leads then to a distinctive gravity model to curvature Gauss-Bonnet gravity. It is also possible to introduce derivatives of torsion in other ways (see [116]).

X. FINAL REMARKS
Teleparallel gravity and its generalizations can be formulated as fully invariant (both coordinate and local Lorentz) theories of gravity. Nevertheless, it is often suggested in the literature that torsion is not a tensor in teleparallel gravity or likewise that the local Lorentz symmetry is violated and teleparallel gravity theories are frame dependent. These notions originated from the fact that the teleparallel spin connection is of a pure-gauge form and hence it is always possible to choose a special gauge in which it vanishes. This is similar to choosing a specific gauge in gauge theories. This non-covariant approach where one restricts the analysis to a vanishing connection is what has been coined pure tetrad teleparallel gravity.
One of the primary goals of this paper is to distinguish between what happens in pure tetrad teleparallel gravity and teleparallel gravity or its generalizations, clearly illuminating the properties of the Lorentz connections and their pivotal role in understanding and determining the equations in Lorentz invariant gravitational theories.
In the invariant framework described here for teleparallel theories of gravity, the torsion tensor is a covariant object under both diffeomorphisms and local Lorentz transformations.
However, unlike the familiar situation in general relativity where the curvature tensor depends only on the metric tensor, the torsion tensor of teleparallel gravity is a function of both the tetrad and the spin connection. The teleparallel spin connection is independent of the metric tensor and represents only the inertial effects associated with the choice of the frame. This is the crucial difference when comparing teleparallel theories with general relativity and other curvature-based theories of gravity, which introduces the very pressing practical problem of how to determine both the tetrad and the teleparallel spin connection.
In the teleparallel equivalent of general relativity the teleparallel spin connection does not enter the field equations for the tetrad, and the field equations for the spin connection turn out to be identically satisfied. Both of these properties can be easily understood as a consequence of the spin connection contributing to the teleparallel action through the surface term only. Therefore, as far as the solutions of the field equations are concerned, the spin connection can be chosen arbitrarily in order to solve the field equations. In particular, this allows us to set the spin connection to zero and effectively obtain the purely tetrad formulation of teleparallel gravity.
However, in revealing the underlying problem of the pure tetrad formulation, the crucial point is that the spin connection can be chosen arbitrarily only when we are interested in solutions of the field equations. The spin connection still plays an important role as it contributes to the action through the surface term which manifests itself in many situations where the total value of the action is of interest; e.g., calculations of the energy-momentum and black hole thermodynamics. As was shown in [23], in order to determine the spin connection corresponding to the tetrad, we can use the fact that the spin connection regularizes the action and hence we can define it by the requirement of the finiteness of the action.
From a physical perspective, this amounts to the removal of spurious inertial contributions causing divergences of the action and obtaining a purely gravitational action. In practice, this can be achieved by introducing a reference tetrad which represents the same inertial effects as the full tetrad. This leads to the procedure described in Section V, which was demonstrated explicitly for the spherically symmetric solution in Section VI.
Note that the problem of how to determine the spin connection corresponding to the tetrad arises only if our starting pointing is the Lagrangian (77) depending on both the tetrad and the spin connection as a priori independent variables. The procedure discussed above can then be viewed as a method of determining their mutual relation. However, if we follow the gauge construction reviewed in Section III A, we naturally avoid this problem.
We start with an inertial frame together with a gauge translational potential and a gauge covariant derivative which can be naturally introduced. The spin connection appears when we pass to the general frame by performing a local Lorentz transformation, in a similar fashion to how it appears in special relativity. We can then see that the general tetrad (43) is given by a combination of the spin connection and the translational gauge potential and that the translational field strength (49) coincides with the torsion tensor (50) for the general tetrad. In this construction, it is obvious that the tetrad is not an independent variable from the spin connection and that, in fact, to each tetrad corresponds a certain teleparallel spin connection.
It is useful to remember that the field equations of teleparallel gravity are non-linear coupled PDEs that can only be solved analytically in certain highly symmetric situations.
Therefore, as in the case in general relativity, a starting point of many explicit calculations is a certain ansatz which respects the assumed symmetry. From this ansatz metric we then choose an ansatz tetrad and solve the field equations. This is the reason why in all practical calculations we start with the tetrad and the spin connection as a priori independent vari-ables instead of constructing the tetrad from the translational gauge potential and the spin connection. This approach of solving the field equations is very much within the spirit of general relativity and it remains an open question as to whether one could fully follow the gauge construction in practice and use translational gauge potentials instead of the tetrad (and whether there would be any advantage to such an approach).
The situation is radically different in the case of modified teleparallel theories of gravity where the teleparallel spin connection contributes to the action in a more intricate way. The variation with respect to the tetrad and the spin connection results in a system of coupled field equations that depend on both variables in a non-trivial way. However, it turns out that the resulting field equations for the spin connection are equivalent to the antisymmetric part of the field equations for the tetrad. This means that, unlike in the case of the ordinary teleparallel gravity, there is no freedom to choose the spin connection when solving the field equations; instead the spin connection is determined by the field equations. Therefore, the solution to the problem in modified teleparallel theories is always the pair h a µ and • ω a bµ . It is interesting to note that in many highly symmetric situations, such as spherically symmetric spacetimes or isotropic cosmologies, the antisymmetric and symmetric parts of the field equations for the tetrad do decouple from each other [9,117]. As a result, it is often possible to solve the antisymmetric part of the field equations, and hence determine the spin connection, independently from obtaining the solution for the symmetric part of the field equations that determines the full tetrad and the metric tensor.
We can now clearly understand the problem of the pure tetrad formulation in the modified case and why these theories can easily be misunderstood regarding their local Lorentz invariance. Since the solution of the field equations is always the pair h a µ and • ω a bµ , the field equations in the pure tetrad formulation are non-trivially satisfied only in the case when the tetrad corresponds to a vanishing spin connection. These tetrads were originally nicknamed good tetrads in the case of f (T ) gravity, since they lead to non-trivial solutions of the field equations; this is in contrast with the so-called bad tetrads, in which case f (T ) gravity reduces trivially to ordinary teleparallel gravity [42]. We now see that this concept of good and bad tetrads is just the result of neglecting the role of the teleparallel spin connection and one cannot draw any conclusions about the preferred frames in teleparallel theories.
Nevertheless, we should mention that despite these conceptual and fundamental flaws, the pure tetrad formulation -if one properly uses good tetrads only -can be utilized to suc-cessfully solve the field equations. Therefore, most of the results found in the literatureobtained using the pure tetrad formulation -are correct.
In Section IX, we reviewed the covariant formulation of other modified teleparallel theories and classified various models based on their essential features. With the exception of higher derivative theories, equivalent to some curvature based models as discussed in Section IX F, much of the previous discussion about the tetrad and the spin connection from f (T ) gravity generally applies to all modified teleparallel models [35].
It is worth noting that alternatively one could set up an action for teleparallel theories and through the use of Lagrange multipliers ensure that the curvature of the spin connection is zero via a metric affine gauge approach [118,119]. In this case the action would be a functional of both the frame field and the spin connection. The result of varying with respect to the frame field will yield a set of equivalent field equations to the covariant version presented here and in [9]. The result of varying with respect to the spin connection could, of course, result in a difference in the matter sector of the theory unless additional assumptions are placed on the nature of the matter Lagrangian (such as, for example, independence from the spin connection). Assuming that there is no hyper-momentum or spin current matter source [119], the zero curvature constraint results in a Lorentz connection as in the covariant representation of the theory.
An interesting and very fundamental open problem is the question of the propagating degrees of freedom in modified teleparallel theories of gravity. In the case of f (T ) gravity it was shown that there are no propagating extra degrees of freedom at the linear level [43,55,120]. However, the Hamiltonian analysis revealed that there are five propagating degrees of freedom [121]. It was then argued that the presence of these extra non-perturbative degrees of freedom poses a serious problem for the causality of f (T ) gravity [122], and these claims were then further discussed in [123][124][125]. Recently, these results were questioned and it was argued that f (T ) gravity has only one extra propagating degree of freedom [126,127].
Most of these results were obtained using the non-covariant pure tetrad teleparallel gravity, and therefore their applicability within the invariant framework presented here is not yet clear. The recent analysis in the covariant formulation of new general relativity, discussed in Section IX A, implies that the presence of the spin connection does not influence the total number of degrees of freedom [128]. This is clearly an important open question for future consideration.
Let us conclude with the statement that teleparallel theories of gravity, which have experienced a renaissance recently, are an intriguing approach to understand gravity. In the case of ordinary teleparallel gravity, we are able to obtain a number of fundamental insights into the nature of gravity, which are not readily available (or are, at least, more hidden) in standard general relativity. Among those which we have reviewed here include the manifest gauge nature of gravity, and new approaches to understand the problems of the definition of the gravitational energy-momentum and regularization of the gravitational action. There are a number of other areas where teleparallel gravity can improve our understanding of gravity. For instance, it has been argued that coarse graining of the gravitational field equations of general relativity might be more naturally achieved within a teleparallel frame formulation of gravity [129,130]. The ability to parallelly transport in a path independent manner facilitates an integration procedure useful for the development an averaged theory of gravity.
We have also discussed a number of modified teleparallel gravity models within the covariant formulation. Since these theories are distinct new models of gravity, it is ultimately up to observations to discriminate between them. Nevertheless, a number of theoretical challenges arise. In particular, we have focused on the question of local Lorentz invariance, and we have clearly demonstrated that the question is resolved due to the existence of the Lorentz connection.