Disformal Transformations in Modified Teleparallel Gravity

In this work, we explore disformal transformations in the context of the teleparallel equivalent of general relativity and modified teleparallel gravity. We present explicit formulas in components for disformal transformations of the main geometric objects in these theories such as torsion tensor, torsion vector and contortion. Most importantly, we consider the boundary term which distinguishes the torsion scalar from the Ricci scalar. With that we show for f (T) gravity that disformal transformations from the Jordan frame representation are unable to straightforwardly remove local Lorentz breaking terms that characterize it. However, we have shown that disformal transformations have interesting properties, which can be useful for future applications in scalar-torsion gravity models, among others.


Introduction
Disformal transformations have been considered by Bekenstein as a deformation of the metric tensor that defines a gravitational theory supplied by two geometries, one for the gravity sector and one for the matter sector. Assuming Finslerian geometry for the matter sector, he finds that in order to preserve both the weak equivalence principle and the causal structure it has to reduce to a Riemannian geometry, whose metricg µν is related to the gravitational one g µν by [1,2]g µν " α 2 g µν`β B µ φB ν φ. (1) This is a non-trivial generalization of a conformal transformation, in the sense that the original space-time g µν is not only isotropically stretched but also considers anisotropic variation in the directions in which the scalar field varies. The causal structure of the transformed metric depends on the coefficients α, β which have to be restricted at least by requiring that the metric is non-degenerate. One of the interesting applications of disformal transformations to gravity is their use for mapping gravitational theories with second order field equations to higher derivative theories that still avoid Ostrogradsky's instability [3]. This sort of transformations has been scarcely discussed in the context of the tetrad formalism [4,5]. We are interested in developing disformal transformations in the teleparallel equivalent of general relativity framework, and apply it to modified teleparallel gravity in order to better understand the issue of the extra degrees of freedom that these theories exhibit. For this, we start with a brief introduction to teleparallel gravity in Sec. 2. We introduce modifications of gravity based on the teleparallel formalism in Sec. 3, together with an introduction to the issue of the degrees of freedom. The basics of disformal transformations in tetrad formalism and transformations of the main geometrical objects are presented in Sec. 4. The main results of this paper are presented in Sec. 5, where we look for an interpretation of the extra d.o.f. of f pTq gravity in terms of a disformal transformation parameterized by a scalar field. Finally, we devote Sec. 6 to the conclusions.

Teleparallel gravity
The spacetime manifold in which gravitational phenomena take place is commonly described by two geometrical structures: a base of orthonormal vectors (tetrad or vierbein) and a spin connection. The first one leads to the definition of metric, while the second one provides the rule for parallel transport. The basis of tetrads e a is defined in the tangent space T p pM q of a 4-dimensional manifold M , and the corresponding co-tetrad basis E a lives in the cotangent space T ‹ p pM q. The Latin indices a, b, . . . " 0, . . . , 3 will denote (co-)tangent space indices, and the Greek indices µ, ν, . . . " 0, . . . , 3 are spacetime ones. When expanded in a coordinate basis as e a " e µ a B µ and E a " E a µ dx µ , the components satisfy completeness relations E a µ e µ b " δ a b and E a µ e ν a " δ ν µ . We recover the usual metric of general relativity by declaring the norm of tetrads in terms of the Minkowski metric, that is where in the second relation we show that the metric can be recovered through the co-tetrad components. We will consider a spacetime endowed with the Weitzenböck connection, defined as This is equivalent to the choice of the spin connection ω a µb " 0, and it renders the Riemann tensor vanish, but leaves non-zero torsion. In such a spacetime, the basis vectors are parallel transported trivially throughout the manifold, coining the term tele-parallel (parallel at a distance).
The Weitzenböck connection is just a subcase of a more general family of flat connections. In general the condition of flatness is imposed as vanishing of the curvature 2-form. This condition can be solved by the so-called inertial connection [6,7], which can also be viewed as covariantization of the Weitzenböck case since it can be obtained by arbitrary Lorentz transformations of the latter.
A gravitational theory that gives dynamics to the tetrad field and has equations of motion equivalent to the Einstein's equations is the teleparallel equivalent of general relativity (TEGR), whose action is written as (we adopt the p`,´,´,´q signature convention, so that the Lagrangian density of the Einstein-Hilbert action reads´?gR, with the minus sign) [6,8] S " 1 2κ where κ " 8πG, E " detpE a µ q, and The TEGR action is equivalent to general relativity (GR) due to the mathematical identity between the curvatures of a metric-compatible torsionful connection Γ and the Levi-Civita connection p0q Γ . With the zero spin connection prescription, the curvature RpΓq is equal to zero, and the TEGR action given in terms of the torsion scalar is equivalent to the Einstein-Hilbert one modulo the surface term of the torsion vector divergence, this iś Given the Weitzenböck choice of the connection, the surface term is not invariant under the local Lorentz transformations (LLT) E a µ Ñ E a1 µ " Λ a1 a pxqE a µ . This fact is harmless once we plug this expression into the TEGR action, since it is integrated out and it gives Lorentz invariant equations of motion fully equivalent to GR. However, it will be a crucial point when taking non-linear extensions of this model. 3 Modified teleparallel gravity and degrees of freedom 3

.1 Generalized teleparallel gravity
The Lagrangian (4) is a suitable starting point for building modifications to Einstein's gravity. This is because the Lagrangian is quadratic in first order derivatives of the tetrad field, therefore any non-linear modification of it would yield equations of motion with only secondorder derivatives. This outcome is not easy to get once we adopt the task of modifying gravity, as it is quite common to get fourth-order equations of motion when such modifications are implemented. This theoretical motivation led to the introduction of the f pTq gravity theories with the action [9,10] S " 1 2κ The associated equations of motion are indeed purely second order in derivatives of the tetrad field, namely where T refers to the usual energy-momentum tensor of any type of matter which couples to the metric. These equations are obtained by taking the variation with respect to E a ν , and we can easily see their second order character. Besides this, the theory has interesting applications to cosmology, a field in which it has been extensively used for getting early and late accelerated phases without introducing additional fields.
When working in the pure tetrad formalism, the f pTq theory, as well as most other modified teleparallel theories, exhibits violation of the local Lorentz invariance. In particular, for f pTq the reason can be traced to Eq. (7). Since the torsion scalar changes by a surface term under a local Lorentz transformation, the non-linear function of it will transform as Therefore, the former surface term can no longer be integrated out, and it gives rise to a theory sensitive to local Lorentz transformations. A different angle over this issue can be obtained in the Jordan frame representation of f pTq gravity. For this, we rewrite the action with the help of an auxiliary scalar field φ as for which the equations of motion are the same as for the original model, and φ " T. Another option is to choose the scalar field as φ " f 1 pTq (related to the previous one via an obvious field redefinition), so that and it can be viewed as a Legendre transform V pφq " Tφ´f pTq, and T " V 1 pφq.
In this formulation one can relate the symmetry breaking with the scalar field φ changing in space and time, since now the former surface term produces effects via derivatives acting upon f 1 pφq or φ at an attempt of integrating by parts. This resonates well with the fact that the Lorentz-violating terms in f pTq equations of motion are propotional to BT, and the scalar φ is set to be given by T.
A natural next step is to try getting rid of the extra factor in front of T via a conformal transformation. A known fact is that in teleparallel gravity this approach falls short of success. A would-be Einstein frame (where the factor in front of T is the unity) comes out with another Lorentz violating term [11,12]. The action that results after this procedure is where hatted quantities refer to conformally transformed ones.
This frame comes together with the presence of a scalar field with an unusual couplinĝ T µ B µ ψ which spoils local Lorentz invariance. This formulation is very suggestive of the presence of new degree(s) of freedom, and also has the worrisome feature that the kinetic term of the ψ field is of the ghostly sign. However one should not forget that there is an intricate kinetic mixing of the scalar field with the tetrad components via the Lorentz breaking term. So that it is not easy to draw definitive conclusions from this form of the action.
It is of course not surprising that we could not cleanly reveal these properties via a conformal transformation. It should not be possible to get rid of Lorentz violating terms by an isotropic field redefinition. However, we could definitely try to explicitly relate those effects to some different kinds of fields. Like, for example, preferred frame effects can come in a disguise of a vector field in the Einstein-Aether model [13]. And, since in our case the origin of violation seems to be related with the gradient of the scalar, we can try a more complex transformation which contains a preferred direction in it, a disformal transformation. The assertion we would like to test is whether it can be used to parameterize the new dynamical mode of f pTq [14], and if it is possible to remove Lorentz violating terms in a teleparallel Einstein frame. Before doing so, let us briefly summarize the state of the art concerning the issue of the degrees of freedom.

On the extra degree(s) of freedom of f(T) gravity
The issue of the nature and the number of the extra degree(s) of freedom of f pTq gravity and other modified teleparallel theories is still not settled, despite intense debate in the last years [14,15,16,17,18]. Here we will briefly summarize some points that we feel important to consider in a discussion: • Lorentz invariance: It is believed by some authors that the loss of local Lorentz invariance could be restored by dropping the assumption of the Weitzenböck connection and resorting to a more general one, inertial connection [6,19,20,21,22]. However, see the Ref. [23] for another viewpoint, especially when it comes to modified teleparallel gravities.
• Weak gravity: It has been shown that there are no new propagating gravitational modes in the linearized approximation around Minkowski spacetime with the trivial choice of the tetrad [24,25]. This follows from a simple observation that f pTq gravity reduces to TEGR in this limit [26], and in particular the local Lorentz invariance is restored.
• Cosmological perturbations: Extra propagating modes also do not appear in the linear cosmological perturbations around spatially flat FRW universe with the standard choice of the background tetrad [27,28,29,30,31]. When a dynamical scalar mode is seen in the perturbations [29], it actually comes from an additional scalar field added as a matter field to the model.
• Hamiltonian formalism: Probably, the first attempt to count the number of degrees of freedom in the Hamiltonian formalism of f pTq gravity can be found in the Ref. [32], where the authors claim that the theory has n´1 extra degrees of freedom in n spacetime dimensions. Later work finds only one extra degree of freedom in any dimension [14,33]. It is suggested that the extra mode could be related with the proper parallelization of spacetime. For other kind of modified teleparallel gravities, the counting it is still pending [34].
• Conformal transformations: We proved above that a clean interpretation of the extra degree(s) of freedom of f pTq gravity can not be achieved with the use of conformal transformations. This result was first obtained in [11], and some more details and refinements, and also extensions have been given in [12].
• Claims of acausality: Assuming presence of extra modes, the cosmological results indicate strong coupling regime for them, bad news for predictability. Also acausality claims for f pTq gravity can be found in the Ref. [35] through the method of characteristics, however they use the number of degrees of freedom calculated in the Ref. [32] for drawing some of their conclusions, which are currently in doubt.
• Galaxy rotation curves: An indirect indication for extra degree(s) of freedom can be found in [36], where they use a model with f pTq " T`αT n to reproduce galactic rotation curves. When the function approaches GR/TEGR (n " 1), the curves are not what is obtained in GR, indicating the presence of some extra mode.
• Remnant symmetry: Even though the full Lorentz group is not a symmetry of f pTq gravity anymore, there still exist subgroups of remnant symmetries which have been studied in the Ref. [37]. Obviously, good understanding of symmetries is an important key to understanding the degrees of freedom.
• Null tetrads: Last but not least, null tetrads can be used to find solutions with T " 0 (or any other constant instead of zero) [38,39,40], which could be useful for exhibiting the extra d.o.f. [41], even though at the background level these solutions are not different from general relativistic ones, up to rescaling of fundamental constants.
Having reviewed all these approaches to the issue of the degrees of freedom in f pTq gravity, it is natural to ask whether disformal transformations could have something to say about this topic. We would like to contribute to this gap in research by presenting basic calculations in (modified) teleparallel theories of gravity.

Disformal transformations in the teleparallel framework 4.1 Generalities on disformal transformation of the tetrad field
We will consider disformal transformations of the tetrad field in the following general form: where we use the standard formulas for manipulating tangent space indices, that is B a " e µ a B µ and B a " η ab B b . The C and D are scalar functions, and for the most part of calculations we will not need to explicitly specify their arguments, though at the end of the day we assume they depend on φ and pBφq 2 " pB µ φqpB µ φq.
Note that the tetrad transformation is quadratic in the field derivatives. However it is easy to prove that the metric transforms the same way, though with the pBφq 2 correction in the coefficient, see below. That is why it would not be natural in our case to restrict the arguments of C and D to the value of φ only. In the current setting, we even could have allowed dependence on T. However, our main purpose is to see whether it is possible to get rid of the local Lorentz violating terms in the Lagrangian. It is hard to imagine this goal achieved if all coefficients in every term depend on the torsion scalar. We introduce the shorthand notation φ µ " B µ φ, and analogously for any other scalar function such as C or D, and use it whenever convenient. With this, we can writẽ for the tetrad and its inverse. Since one is the inverse of another, the orthonormality relations E a µẽ µ b " δ a b andẼ a µẽ ν a " δ ν µ still hold. One can easily obtain the transformation of the metric (2) and its inverse as A useful consequence of the last relation is that which simply states that the length of φ µ transforms according to the corresponding eigenvalue of the transformation matrix. An important requirement when doing transformations in a given theory is to preserve invertibility. For the disformal transformations the most obvious requirement amounts to which is the condition on the eigenvalues of the transformation matrix. If a transformation should be continuously connected with identity, then we substitute the ‰ signs with the ą ones.
Since the transformations above contain field derivatives, they can fail being invertible even with non-zero eigenvalues, thus producing what is known under the name of mimetic gravity [42,43]. The corresponding general conditions for disformal transformations of the metric can be found in the Ref. [44].
Sometimes also conditions on light cones, time directions and other properties of the metrics are imposed. They can be easily derived, too. But we will not need any of these. Now we have all the basic building blocks for studying transformations of the main geometrical objects in teleparallel gravity.

Disformal transformation of the torsion tensor
We start with the calculation of the disformally transformed connection coefficients, defined in Eq. (3). The calculation amounts to replacing the tetrads by their tilde versions given in formulas (15) and (16). Before that, and considering that E a α Γ α µν " B µ E a ν , we can make a useful observation for the object B µ φ a : Now, with all these prerequisites, we get The disformally transformed torsion tensor is the antisymmetric part of this expression (21). Note that if we want to specify the admissible arguments, for example that C " Cpφ, pBφq 2 q, we just need to use the Leibniz rule, that is Another way to obtainT α µν is to use mixed components (differential forms) representation, this isT a µν " B µẼ a ν´BνẼ a µ . With our definitions, we havẽ This result has also been obtained in [5]. However, the expression we want isT α µν "ẽ α aT a µν , which is easily found to coincide with the antisymmetric part of the formula (21): where we have taken into account that

Contortion and torsion vector
A straightforward but somewhat cumbersome way to proceed would be to find the transformation of the torsion scalar. It amounts to calculating We leave it for future work. However let us give an explicit formula for the contortion tensor.
In order to find the contortion tensor, we need to have the torsion tensor with all indices at the same level. This is easily done by consideringT αµν "ẽ b α η abT a µν . We get and then the contortion tensor K αµν " 1 2 pT αµν`Tναµ`Tµαν q, antisymmetric in the lateral indices α´ν, is given bỹ Another very important quantity is the vector part of the disformally transformed torsion tensor T µ " T ρ µρ , and it can be obtained just by contracting the correct indices in (24):

Disformal transformations in f(T) gravity and the search for the Einstein frame
Let us recall that our main goal was to search for an Einstein frame, i.e. a frame in which the action will be written in terms of usual gravity with minimally coupled matter, and in particular without local Lorentz violating terms. Since we assume that the disformal transformation depends only on the scalar field φ, and does not depend on any Lorentzbreaking quantities, then according to the relation (7), instead of transforming the torsion scalar it is enough to look at the transformation of the Levi-Civita divergence of the torsion vector. Let us substitute the Eq. (7) into the Jordan frame action (11): or into the Legendre transform version of it (12): Of course, those two are different by an obvious scalar field redefinition. (This action can also be interpreted as a relevant part of any scalar-torsion theory with a non-minimal coupling to the torsion scalar.) The only Lorentz-violation term is now the one with p0q ▽ µ T µ , and integrating it by parts we reduce the Lorentz-breaking effects to the action term of the form depending on the chosen representation. The question was whether it is possible to get rid of this term by a disformal transformation in terms of the same scalar field which enters the Jordan frame. We have already computed all the necessary ingredients for that. The calculation gives a remarkably simple resultφ ( 32) where we have used that φ α φ µ φ β T βµα " 0 due to antisymmetry of the torsion tensor. Note that the latter expression would not have vanished if an independent scalar field was used for the transformation, leaving us with even more Lorentz-violating terms.
We still need to improve this expression to make it more comprehensible. Indeed, the covariant derivatives in (32) are given in terms of the Weitzenböck connection, and in order to separate the Lorentz violating effects, we must reduce them to the Levi-Civita ones. This is easily done. One expression does not require anything: where we have used antisymmetry of the contortion tensor, and for the d'Alembertian operator we get:˝φ Mixing all these ingredients in a proper way, we get a very nice result for the disformally transformed Lorentz violating term φ¸ff .
(35) If we recall now thatẼ " C 3 pC`DpBφq 2 qE, then we readily see that the coefficient in front of the ET µ B µ φ term changes simply by the factor of C 2 : Therefore, it is not possible to remove the Lorentz-violating term by means of disformal transformations.

Discussion and conclusions
We have presented explicit formulas for disformal transformations of the basic quantities in teleparallel theories of gravity, mostly having in mind the f pTq theory. They are not able to bring the theory to a genuine Einstein frame form. An obvious caveat is that we could also allow for Lorentz-violating arguments in C and D. However as we have discussed above it is highly unlikely that it will help eliminate Lorentz-breaking terms from the action. And, in this case, it would not be enough for our purposes to study only the transformation of the boundary term.
On the other hand, this Lorentz-violating term in the action vanishes whenever the torsion vector is orthogonal to the scalar gradient. One can think of that in terms of the fact that the Lorentz violation in f pTq gravity is related to the direction in which the torsion scalar, or the auxiliary scalar field, changes. Though, of course, vanishing of an action term for a particular combination of fields does not have a very clear meaning since it is not respected by generic variations.
Note however, that after the disformal transformation this vanishing can happen in other, more interesting cases, too. Let us choose a very simple transformation with C " D " 1. For it the final answer is even simpler: This term can be set to zero as a whole if the torsion vector is related to the scalar field in a very particular way, namely: What is a possible physical meanings of this result? We do not know. The formula (37) relates Lorentz-breaking and Lorentz-preserving quantities in it, and therefore is not well in line with the whole procedure we followed. However it looks intriguing. Can it be related somehow to the cases when linear perturbative analyses show no new propagating modes? To the remnant symmetry?
Another related idea would be to use a two-field disformal transformation of the form where ψ is an independent field at our disposal. All the calculations above apply to this case just by setting C " 1 and D " ψ. Then we see that the violating term (35) gets transformed to zero if Of course, the rest of the Lagrangian receives then Lorentz-violation through the ψ field, but formally it looks like a metric theory with two scalar fields, φ and ψ. Can we now treat those just as two independent scalar fields? Whatever the answers are, we think that it shows that disformal transformations have a good potential to shed some new light on f pTq gravity models. And since the latter are currently so important in cosmological model building, it is worth to further pursue this topic.