# Modified teleparallel theories of gravity: Gauss–Bonnet and trace extensions

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## Abstract

We investigate modified theories of gravity in the context of teleparallel geometries with possible Gauss–Bonnet contributions. The possible coupling of gravity with the trace of the energy-momentum tensor is also taken into account. This is motivated by the various different theories formulated in the teleparallel approach and the metric approach without discussing the exact relationship between them. Our formulation clarifies the connections between different well-known theories. For instance, we are able to formulate the correct teleparallel equivalent of Gauss–Bonnet modified general relativity, amongst other results. Finally, we are able to identify modified gravity models which have not been studied in the past. These appear naturally within our setup and would make a interesting starting point for further studies.

## Keywords

Bonnet Torsion Tensor Teleparallel Gravity Torsion Scalar Tetrad Field## 1 Introduction

One possible approach to motivating a geometrical theory of gravity is to compare the geodesic equation of differential geometry with Newton’s force law. This suggests the identification of the gravitational forces with the components of the Christoffel symbols which in turn yields the identification of the gravitational potentials with the metric. Assuming a geometrical framework with a metric compatible covariant derivative without torsion gives the building block of Einstein’s theory of general relativity, one can speak of the metric approach.

Soon after the original formulation of this geometrical theory of gravity, it was noted that there exists an alternative geometrical formulation which is based on a globally flat geometry with torsion. The key mathematical result to this approach goes back to Weitzenböck who noted that it is indeed possible to choose a connection such that the curvature vanishes everywhere. This formulation gives equivalent field equations to those of general relativity and we refer to this as the teleparallel formulation. This naming convention stems from the fact that the notion of parallelism is global instead of local on flat manifolds; see for instance [1, 2] and the references therein.

One of the basic equations of general relativity and its teleparallel equivalent is \(R = -T + B\) where *R* is the Ricci scalar, *T* is the torsion scalar and *B* is a total derivative term which only depends on torsion. Clearly, the Einstein–Hilbert action can now be represented in two distinct ways, either using the Ricci scalar or the torsion scalar, and consequently giving identical equations of motion. A popular modification of general relativity is based on the Lagrangian *f*(*R*) and can be viewed as a natural non-linear extension that results in fourth order field equations [3, 4]. On the other and, one could consider the Lagrangian *f*(*T*) which gives second order field equations [5]. However, this theory is no longer invariant under local Lorentz transformations because the torsion scalar *T* itself is not invariant [6, 7]. Neither is the total derivative term *B* but the particular combination \(-T + B\) is the unique locally Lorentz invariant choice, see also [8]. Hence, *f*(*R*) gravity and *f*(*T*) gravity and not equivalent and correspond to physically different theories. Now, considering the more general family of theories based on *f*(*T*, *B*) one can establish the precise relationship between these theories and it turns out that *f*(*R*) gravity is the unique locally Lorentz invariant theory while *f*(*T*) gravity is the unique second order theory.

The principal aim of this paper is to extend these results to take into account the Gauss–Bonnet term and its teleparallel equivalent. The Gauss–Bonnet scalar is one of the so-called Lovelock scalars [9] which only yields second order field equations in the metric, hence in more than four dimensions the study of the Gauss–Bonnet term is quite natural. In four dimensions, on the other hand, the Gauss–Bonnet term can be written as total derivative and its integral over the manifold is related to the topological Euler number. However, it should be emphasised that topological issues in teleparallel theories are not well understood yet. The only known torsional topological invariant is the Nieh–Yan term; see [10, 11], and in particular [12] in the context of teleparallel theories.

The teleparallel equivalent of the Gauss–Bonnet term was first considered in [13, 14] who studied a theory based on the function \(f(T,T_G)\). As is somewhat expected, the Gauss–Bonnet term differs from its teleparallel equivalent by a divergence term. Hence, as in modified general relativity, it is possible to formulate modified theories based on the Gauss–Bonnet terms or its teleparallel equivalent in such a way that both theories are physically distinct. The link between these theories comes from the divergence term which needs to be taken into account when establishing the relationship between the different possible theories. We also allow our action functional to depend on the trace of the energy-momentum tensor since this is a popular modification that has been investigated in recent years [15].

There exists a large body of literature dealing with the various modified theories of gravity that have been investigated; see for instance the reviews [3, 4, 16, 17, 18].

*Our conventions* Greek indices denote spacetime coordinates, Latin indices are frame or tangent space indices. \(e_{\mu }^a\) stands for the tetrad (1-form), while \(E^{\mu }_a\) denotes the inverse tetrad (vector field). The Minkowski metric is \(\eta _{ab}\) with signature \((-,+,+,+)\). Where possible we follow the conventions of [2].

## 2 Teleparallel gravity and the Gauss–Bonnet term

### 2.1 Teleparallel geometries

*K*is the contortion tensor. The contortion tensor can be expressed using the torsion tensor as follows:

*R*(

*e*) stands for the metric or Levi-Civita Ricci scalar; see also [2].

### 2.2 Teleparallel gravity

*T*so that Eq. (15) can be written very nicely as

An important consideration is the behaviour of the above quantities under local Lorentz transformations. It is clear from (6) that the local Lorentz transformation \(e^a_\mu \mapsto \Lambda ^a{}_b e^b_\mu \) will change the torsion tensor as the Lorentz transformations \(\Lambda ^a{}_b\) are local and hence functions of space and time so that derivatives of \(\Lambda ^a{}_b\) appear. Therefore the torsion tensor does not transform covariantly under local Lorentz transformations; see also [6, 7]. Note that this is a direct consequence of the teleparallel approach and the combination \(-T+B\) is the only combination of *T* and *B* which is locally Lorentz invariant.

In contrast to the standard teleparallel approach, complete Lorentz invariance is preserved when considering metric-affine theories [19] in which the metric and torsion and are treated independently; see also [20]. Consequently, the torsion scalar *T* and the boundary term *B* are both Lorentz scalars in this approach. The metric-affine framework has inspired the recent covariant formulation of *f*(*T*) gravity [21] which is based on the idea of allowing the spin connection to be a dynamical variable in addition to the tetrad fields. This is an interesting alternative treatment to teleparallel theories of gravity which could in also be applied to investigate Gauss–Bonnet extensions.

### 2.3 Gauss–Bonnet term

*G*is a Lorentz scalar, these second derivative terms must be cancelled by terms coming from \(B_{G}\). Consequently the combination \(-T_{G} + B_{G}\) is the unique Lorentz invariant combination which can be constructed. This fact becomes important when considering modified theories of gravity based on the teleparallel equivalent of the Gauss–Bonnet scalar.

In the following we will show some simple examples of the Gauss–Bonnet term and its teleparallel equivalent.

### 2.4 Example: FLRW spacetime with diagonal tetrad

*G*is independent of the Cartesian coordinates. The unique linear combination \(-T_{G}+B_{G}\) is independent of position. Secondly, we have the case of a spatially flat universe; then these terms are absent and the term \(B_{G}\) identically vanishes. The terms depending on the spatial coordinates can be changed be working with a different tetrad, or in other words, these terms are affected by local Lorentz transformations. Finding a tetrad for which \(T_{G}\) and \(B_{G}\) are both independent of the spatial coordinates is a rather involved tasks, however, following the approach outline in [23, 24] we will show that a tetrad with this property can be constructed. Before doing so, we discuss another example with different symmetry properties.

### 2.5 Example: static spherically symmetric spacetime–isotropic coordinates

*t*,

*x*,

*y*,

*z*) to avoid coordinate issues with the tetrads. Including time dependence is straightforward, however, the resulting equations are too involved. We choose

*R*,

*T*and

*B*are given by

*G*, \(T_{G}\) and \(B_{G}\), which are

The tetrads used in (25) and (29) serve as simple examples which are useful to compute the relevant quantities. However, in the context of extended or modified teleparallel theories of gravity, such tetrads should be avoided. The construction of a suitable static and spherically symmetric tetrad in *f*(*T*) gravity, for instance, is rather involved, see [25]. In general the choice of a suitable parallelisation is a subtle and non-trivial issue, the interested reader is referred to [26].

### 2.6 Example: FLRW spacetime–good tetrad

*a*(

*t*) is the scale factor of the universe and \(k=\{0,\pm 1\}\) is the spatial curvature which corresponds to flat, close and open cosmologies, respectively.

*f*(

*T*) gravity it implies an off-diagonal field equation which is highly restrictive, namely the condition \(f_{TT}=0\). Such a theory is equivalent to general relativity and hence it is not a modification. In order to avoid this issue, one can follow the procedure outlined [23, 24] which allows for the construction of tetrads which result in more favourable field equations. Consider the tetrad (39) and perform a general 3-dimensional rotation \(\mathcal {R}\) in the tangent space parametrised by three Euler angles \(\alpha \), \(\beta \), \(\gamma \) so that

*t*and

*r*. Doing this means we will work with the rotated tetrad

*T*and the boundary term

*B*becomes

*r*and

*t*, respectively. In order to have

*T*and

*B*position independent we must choose our function \(\gamma \) to satisfy

*T*and

*B*time dependent only. Therefore, the rotated tetrad (42) with \(k=-1\) and the function \(\gamma \) given by (46) is a ‘good’ tetrad in the sense of [24]. Independently of the choice of tetrad we always obtain the usual Ricci scalar

*r*, one can verify that

## 3 Modified theories of gravity and their teleparallel equivalents

We are now ready to discuss the general framework of modified theories of gravity and their teleparallel counterparts. In principle our approach could be applied to any metric theory of gravity whose action is based on objects derived from the Riemann curvature tensor. Any such theory can in principle be rewritten using the torsion tensor thereby allowing for a teleparallel representation of that same theory.

### 3.1 Equations of motion

*f*is a smooth function of the scalar torsion

*T*, the boundary term

*B*, the Gauss–Bonnet scalar torsion \(T_{G}\) and the boundary Gauss–Bonnet term \(B_{G}\).

The field equations are very complicated, however, when considering a homogeneous and isotropic spacetime, they simplify substantially and can be presented in closed form. Comparison of these equations with previous results serves as a good consistency check of our calculations.

### 3.2 FLRW equations with \(k=0\)

*p*. One should make explicit that \(\dot{f}_{B}=f_{BB}\dot{B}+f_{BT}\dot{T}+f_{BT_{G}}\dot{T}_{G}+f_{BB_{G}}\dot{B}_{G}\) using the chain rule, so that dot denotes differentiation with respect to cosmic time. It is clear that by setting \(f(T,B,T_{G},B_{G})=\mathfrak {f}(R,G)\) the equations for the flat FRWL in \(\mathfrak {f}(R,G)\) theory are recovered and are explicitly given by

### 3.3 FRWL equations with \(k=+1\)

### 3.4 Theories with energy-momentum trace

*f*is a function of the trace of the energy-momentum tensor \(\mathcal {T}=E_{a}^{\beta }\mathcal {T}_{\beta }^{a}\). As before \(L_\mathrm{m}\) denotes an arbitrary matter Lagrangian density. We can define the energy-momentum tensor as

## 4 Conclusions

*f*(

*T*,

*B*) one can formulate the teleparallel equivalent of

*f*(

*R*) gravity and identify those parts of the field equations which are part of

*f*(

*T*) gravity, the second order part of the equations which is not locally Lorentz invariant. In analogy to this one can also make the relationships between various modified theories of gravity clear which are based on the Gauss–Bonnet term.

The top left corner of Fig. 1 refers to \(f(T,B,T_{G},B_{G})\) gravity, the most general theory one can formulate based on the four variables. One can think of the top entries as the teleparallel row and the bottom as the metric row. The arrows indicate the specific choices that have to be made in order to move from one theory to the other.

Now, the left half of the figure corresponds to the metric approach while the right half corresponds to the teleparallel framework. The four main theories of the previous discussions are highlighted by boxes. Many of these theories were considered in isolation in the past and their relationship with other similarly looking theories was only made implicitly. We should also point out that our representation of these theories is only one of the many possibilities and moreover, Fig. 2 is incomplete. There are many more theories one could potentially construct which we have not mentioned so far. The diagram was constructed having in mind those theories which have been studied in the past.

In constructing the diagram we also made the interesting observation that the theory based on the function *f*(*R*, *T*) should be viewed as a special case of the teleparallel gravity theory *f*(*T*, *B*). To see this, simply recall the principal identity \(R=-T+B\) which shows that the special choice \(f(-T+B,T)\) is the teleparallel equivalent of *f*(*R*, *T*) theory and also that the teleparallel framework should be viewed as the slightly more natural choice for this theory.

Short list of previously studied theories covered by the function \(f(T,B,T_{G},B_{G},\mathcal {T})\)

Of the many possible theories one could potentially construct from \(f(T,B,T_{G},B_{G},\mathcal {T})\), we identified some which might of interest for future studies. Clearly, there are many theories which do not have a general relativistic counter part like \(f(B,T_G,B_B,\mathcal {T})\) since no theory in this class can reduce to general relativity. However, it is always possible to consider such a theory in addition to general relativity by considering for instance a theory based on \(-T+f(B,T_{G},B_{G},\mathcal {T})\). For a function linear in its arguments this yields the teleparallel equivalent of general relativity.

It is also useful to make explicit the limitations of the current approach put forward by us. In essence, we are dealing with modified theories of gravity which are based on scalars derived from tensorial quantities of interest, for instance the Ricci scalar or the trace of the energy-momentum tensor. However, theories containing the square of the Ricci tensor or theories containing the term \(R_{\mu \nu }\mathcal {T}^{\mu \nu }\) are not currently covered. In principle, it is straightforward though to extend our formalism to such theories. In case of the quantity \(R_{\mu \nu }\mathcal {T}^{\mu \nu }\), we would have to recall Eq. (12) so that this term can be expressed in the teleparallel setting, something that also has not been done yet. Likewise, we could also address quadratic gravity models [32] which contain squares of the Riemann tensor and again use Eq. (11). Theories depending on higher order derivative terms [33] also require a separate treatment.

The current approach is entirely based on the torsion scalar *T* which is motivated by its close relation to the Ricci tensor. However, in principle one could follow the work of [34] and decompose the torsion tensor into its three irreducible pieces and construct their respective scalars. This would allow us to study a larger class of models based on those three scalars and the boundary term. To the best of our knowledge this has not been considered in the past and would make an interesting further development.

For many years now, an ever increasing number of modifications of general relativity has been considered. In this work we focussed only on theories where the gravitational field can either be modelled using the metric of the tetrad. Hence, we excluded all types of metric-affine theories where the metric and the torsion tensor are treated as two independent dynamical variables. It would be almost impossible to present a visualisation that encompasses all those theories as well. Even this would represent only a fraction of what is referred to as modified gravity. It would still exclude higher dimensional models, Einstein–Aether models, Hořava–Lifshitz theory and many others. It is also interesting to note that *f*(*R*) gravity for instance can be formulated as a theory based on a non-minimally coupled scalar field. Hence, many of the theories in Fig. 2 might also have various other representations which in turn might be connected in different manners.

This discussion motivates the process of classifying the different families of modifications of general relativity and their possible interrelations, followed by a broad investigation of which theories should not be studied further due to incompatibilities with well-established observational bounds. It appears that we are reaching the point where we possibly do not need more theories but rather an improved sense of direction for future developments.

## Notes

### Acknowledgements

We would like to thank Franco Fiorini and Emmanuel Saridakis for their valuable comments on this manuscript. SB is supported by the Comisión Nacional de Investigación Científica y Tecnológica (Becas Chile Grant No. 72150066). This article is partly based upon work from COST Action CA15117 (Cosmology and Astrophysics Network for Theoretical Advances and Training Actions), supported by COST (European Cooperation in Science and Technology).

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