# Variational Problem and Bigravity Nature of Modified Teleparallel Theories

###### Abstract

We consider the variational principle in the covariant formulation of modified teleparallel theories with second order field equations. We show that the variational problem is consistent and leads to non-trivial modifications of teleparallel gravity only if the spin connection is chosen in a such way that the action is finite. Since this is achieved by introducing a second tetrad into the theory, modified teleparallel theories effectively become bigravity theories with two tetrads, where the second tetrad corresponds to the non-dynamical, “reference”, metric tensor and generates the connection. We then discuss the relation of our results and those obtained in the usual, non-covariant, formulation of teleparallel theories.

## 1 Introduction

Teleparallel gravity is an alternative formulation of general relativity that can be traced back to Einstein’s attempt to formulate the unified field theory [1, 2, 3, 4, 5, 6, 7, 8, 9]. Over the last decade, various modifications of teleparallel gravity became a popular tool to address the problem of the accelerated expansion of the Universe without invoking the dark sector [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. The attractiveness of modifying teleparallel gravity–rather than the usual general relativity–lies in the fact that we obtain an entirely new class of modified gravity theories, which have second order field equations. See [39] for the extensive review.

A well-known shortcoming of the original formulation of teleparallel theories is the problem of local Lorentz symmetry violation [40, 41]. This is particularly serious in the modified case, where we often encounter situations that out of two tetrads related by a local Lorentz transformation, only one solves the field equations. Since both tetrads correspond to the same metric, it is not the metric tensor but a specific tetrad that solves a problem in modified teleparallel theories.

This introduces a new complication of how to choose the tetrad. As it turns out, the tetrad must be chosen in a specific way or otherwise the field equations lead us to the condition , which restricts the theory to its general relativity limit. Tetrads that lead to this restriction were nicknamed bad tetrads, while those that avoid it and lead to some interesting new dynamics as good tetrads [42, 43]. An intriguing sub-class of good tetrads are those that lead to , which can be used to show that the solutions of the ordinary GR are the universal solutions of an arbitrary gravity [44, 45].

Recently [46], it was shown that local Lorentz invariance can be restored in modified teleparallel theories, if the starting point is a more general formulation of teleparallel gravity [47, 48, 49, 50, 51, 52, 53, 54], in which teleparallelism is defined by a condition of vanishing curvature. This introduces the purely inertial spin connection to the theory that can be calculated consistently in the ordinary teleparallel gravity [52, 53]. However, in the modified case it was argued to be possible only to “guess” the connection by making some reasonable assumptions about the asymptotic properties of the ansatz tetrad [46]. While it was shown to work, and even to be rather easily achievable, in many physically interesting cases [46, 55, 56, 57], it is obviously not a satisfactory situation and better understanding of covariant modified teleparallel theories is needed.

In this paper, we consider the variational principle in modified teleparallel theories and show that its consistency requires the spin connection to not be treated as an independent variable from the tetrad. We find that it have to be chosen in a such way that the action is finite, what corresponds to the “guess” made in [46]. All other choices of the connection, including those that lead to , reduce the theory to the ordinary teleparallel gravity. We show that these results can be always interpreted in the original, non-covariant, formulation and explain the physical origin of why some tetrads are “good”, and some other “bad”.

Moreover, we argue that we can view modified teleparallel gravity theories as effectively bigravity theories with two tetrads, where the first tetrad determines the spacetime metric, while the second tetrad generates the spin connection and corresponds to the non-dynamical, “reference”, metric.

Notation: We follow notation, where the Latin indices run over the tangent space, while the Greek indices run over spacetime coordinates. The spacetime indices are raised/lowered using the spacetime metric , while the tangent indices using the Minkowski metric of the tangent space.

## 2 Covariant formulation of teleparallel theories

Teleparallel theories are formulated in the framework of tetrad formalism, where the fundamental variable is the tetrad, , related to the spacetime metric through the relation

(1) |

The parallel transport is determined by the spin connection, which is fully characterized by its curvature and torsion in the metric compatible case. In general relativity, we use the connection with vanishing torsion known as the Levi-Civita connection. In teleparallel gravity, we follow a complimentary approach and define the teleparallel connection, , by the condition of zero curvature

(2) |

The most general connection satisfying this constraint is the pure gauge-like connection [47]

(3) |

where .

The torsion tensor of this connection

(4) |

is in general non-vanishing, and under the local Lorentz transformations

(5) |

transforms covariantly. Therefore, this approach to teleparallel theories is called the covariant approach [52, 46]; in contrast with the more common non-covariant approach that will be discussed in section 6.

The Lagrangian of teleparallel gravity is given by [8]

(6) |

where , is the gravitational constant (in units), and we have defined the torsion scalar

(7) |

where is the superpotential

(8) |

A straightforward calculation shows that the teleparallel Lagrangian (6) and the Einstein-Hilbert one are equivalent up to a total derivative term [8]

(9) |

from where follows that the field equations of general relativity and teleparallel gravity are equivalent.

Since the Lagrangian (6) contains only the first derivatives of the tetrad, we can straightforwardly construct modified gravity theories with second order field equations. The most popular of these theories is the so-called gravity [10, 11, 12, 13] defined by the Lagrangian

(10) |

where is an arbitrary function of the torsion scalar (7).

The variation with respect to the tetrad yields the field equations [46]

(11) |

where

(12) |

and and denote first and second order derivatives of with respect to the torsion scalar . The field equations of the ordinary teleparallel gravity are recovered in the case .

## 3 Variational problem in teleparallel gravity

The difficulty of the variational problem in general relativity is related to the presence of the second derivatives of the metric tensor (or tetrads in tetrad formalism) in the Einstein-Hilbert action. The straightforward variation of the action leads to the field equations that are satisfied only if we require vanishing variations of both the metric and their normal derivatives on the boundary. To satisfy both these requirements simultaneously is too strong condition that leads to the variation problem that is not well-posed, what is usually solved by adding the boundary term to the action [58, 59, 60].

This problem is absent in teleparallel gravity as the teleparallel action contains just the first derivatives of the tetrad. However, we face a new difficulty since the action depends–beside the tetrad–on the spin connection, which needs to be determined.

The straightforward variation of the teleparallel action with respect to the spin connection as independent variable does not take into account the teleparallel constraint (2), and can be shown to be inconsistent [61, 53, 54]. A viable solution is to enforce the teleparallel constraint (2) through the Lagrange multiplier [62, 61, 50, 54], but this introduces additional gauge symmetries associated with the multiplier and the theory becomes significantly more complicated and hard to handle [63, 64].

We follow here another approach that was introduced recently in [53], which is based on writing the torsion scalar (7) in a form that guarantees that the teleparallel constraint (2) is enforced. As it turns out, specifically in the case of the torsion scalar (7) and the teleparallel connection (3), we can write the torsion scalar as [53]

(13) |

where .

We can then easily find that the variation of the action with respect to the spin connection vanishes identically and the spin connection does not affect the field equations of teleparallel gravity (12) (with ); both on the account of variation of a total derivative term [53]. In teleparallel gravity, we can then write

(14) |

what allows us to solve the field equations using an arbitrary spin connection of the form (3).

Even though the spin connection does not affect the variations of the action, it does play an important role in the theory. Beside ensuring the correct tensorial behavior under local Lorentz transformations (5), it is essential for the correct definition of the conserved charges [48, 49, 52], and to obtain the physically relevant finite action [52, 53]. To illustrate the latter problem, let us consider the teleparallel Lagrangian (6) for some tetrad that solves the field equations and some arbitrary teleparallel spin connection of the form (3). We can observe that typically the Lagrangian does not vanish at infinity and we obtain the IR-divergent action [52].

The physical origin of this divergence can be understood if we recall that the tetrad has 16 degrees of freedom and only 10 of them are related to gravity and the metric tensor (1). The remaining 6 represent the freedom to choose the frame, i.e. the inertial effects, which are not related to any actual physical fields and hence do not necessarily vanish at infinity. The torsion tensor that includes these spurious inertial effects does not vanish at infinity and leads to the IR-divergent action.

The crucial observation is that the teleparallel spin connection is related to the inertial effects as well (3), and it can be chosen in a such way that it compensates the inertial effects associated with the tetrad. This results in the torsion tensor that does not include these spurious inertial effects and have correct asymptotic behavior [52]. To this end, we consider the reference tetrad, , representing the same inertial effects as the dynamical tetrad . This is achieved by “switching off” gravity by setting some parameter that controls the strength of gravity to zero. In the case of the asymptotically Minkowski spacetimes–which are of our primary interest here–we can consider the limit

(15) |

since gravity as a physical field is expected to vanish at infinity.

We then define the spin connection by the requirement that the torsion tensor vanishes for the reference tetrad

(16) |

which has an unique solution

(17) |

where is the Levi-Civita connection for the reference tetrad.

The torsion tensor is then guaranteed to be well-behaved in the asymptotic limit by this construction, and the corresponding teleparallel action is finite and free of IR divergences [52, 53]. This argument can be turned around, and spin connection can be defined by the requirement of the finiteness of the action^{1}^{1}1This is in fact a weaker requirement than (16) since there exists a 1-parameter group of local Lorentz transformations that leaves the total derivative term in (13) invariant [65].. Therefore, if we require the action to be finite, the spin connection cannot be treated independently from the tetrad and needs to be chosen according to the procedure discussed above.

## 4 Variational principle for modified teleparallel theories

Let us now move to the modified case and consider the example of the covariant gravity given by the Lagrangian (10), where we can use the relation (13) and most of the results from the ordinary teleparallel gravity.

The crucial observation in teleparallel gravity was that the IR divergences can be always canceled by choosing the spin connection correspondingly to the tetrad. Since the spin connection enters the torsion scalar through the total derivative term only (13), these divergences enter the action through the surface term as well and hence do not affect the field equations.

If we consider a Lagrangian to be a non-linear function of the torsion scalar, as we do in gravity, it implies that the divergences appear not in the surface term but in the “bulk”. This is a far more serious type of divergence that leads to the failure of the variational principle and inconsistent field equations. There are only two possible solutions in the framework of gravity.

The first solution is to force the divergence to appear in the surface term to avoid spoiling the field equations, what lead us trivially back to the ordinary teleparallel gravity. Indeed, if we naively vary the divergent action (10) with respect to the tetrad, we obtain a condition

(18) |

which typically appears as one of the non-symmetric field equations [46]. It can be understood as a consistency condition of the variational principle that allows us to vary the divergent action only in the limit of the ordinary teleparallel gravity.

A second option that lead us to non-trivial modifications of teleparallel gravity, is to require the action to be finite. To this end, based on our discussion from the ordinary teleparallel gravity, we must enforce the constraint (16) and define the spin connection in terms of the reference tetrad (17). The problem in gravity is that the reference tetrad cannot be calculated from the dynamical tetrad (15) since the dynamical tetrad is obtained as a solution of the field equations, which cannot be solved without the knowledge of the spin connection. In [46] it was proposed that this circular problem can be avoided if the reference tetrad is “guessed”. We would like to clarify this point now and show that such a construction naturally follows from the consistent variational principle.

The starting point of any calculation in gravity theories is the ansatz metric that suits the symmetry of the problem under consideration. In tetrad formalism, there is an additional step where we choose the ansatz tetrad in a such way that it corresponds to the ansatz metric according to the relation (1). This fixes 10 components of the tetrad and leave us with 6 inertial degrees of freedom undetermined. Different tetrads leading to the same metric are then in the same equivalence class of tetrads and differ from each other by the inertial effects only. When we pick some definite ansatz tetrad, we choose one tetrad from the equivalence class, and typically we do not know what inertial effects are associated with our chosen tetrad.

The role of the reference tetrad is to remove the inertial effects coming from our choice of the ansatz for the dynamical tetrad. The reference tetrad is not determined by any field equations as it is related to the inertial effects only, which do not represent any actual physical fields and hence should not have dynamics on their own. Instead the reference tetrad is introduced along the dynamical tetrad, what makes modified teleparallel theories effectively bigravity theories as we discuss in section 5. For practical calculations, the reference tetrad is fully determined by the requirement that dynamical tetrad reduces to the reference tetrad in the absence of gravity. In the case of asymptotically Minkowski spacetimes this is given be the condition (15).

We should remark that there is also the third option how to solve these problems, but it lies outside the framework of gravity. We can require the cancellation of the total derivative term in (13) by adding some compensating term to the torsion scalar that would make the new torsion scalar independent of the spin connection. As it turns out, the unique term with this property is the last term in (9), and hence we obtain a theory equivalent to gravity with higher order field equations [30]^{2}^{2}2This result was obtained in [30] in the framework of the non-covariant formulation, where local Lorentz transformations acts on the tetrad only and the spin connection is kept zero in all frames. The authors found that gravity is an unique theory with local Lorentz invariance in this sense. In our covariant formulation this is equivalent to being independent of the spin connection. See section 6 for more details..

### 4.1 Case

We have argued here that if we carefully examine the conditions under which it is possible to derive the field equations for the tetrad, we are unavoidably led to a conclusion that the spin connection is determined by the condition (16). However, let us consider an opposite situation where one would try first to vary the action with respect to the spin connection, solve the resulting equations, and then move to the field equations for the tetrad. We find that it is indeed possible, but it restricts the theory to its general relativity limit.

Using the relation (13), it can be shown straightforwardly that the variation of the action (10) with respect to the spin connection leads to the equation

(19) |

which has a simple solution

(20) |

where , are some constants. This restricts the -function to the linear function and return us back to the ordinary teleparallel gravity. The field equations for the tetrad can be then naturally solved, but they are the field equations of the ordinary teleparallel gravity and hence do not yield any new results.

Let us notice that it is possible to rewrite the condition (19) as

(21) |

which implies

(22) |

This is exactly the peculiar class of solutions that was used to show that the solutions of general relativity are the universal solutions of gravity [43, 44, 45]. Now, we understand that the condition is in fact equivalent to restricting the theory to the general relativity limit, what explains the universality of the solutions.

## 5 Teleparallel theories as bigravity theories

We have shown that in order to obtain a non-trivial modification of teleparallel gravity, we must introduce the reference tetrad to define the spin connection. We can now adopt an alternative viewpoint, and consider the reference tetrad–instead of the spin connection–as a fundamental variable. We can then straightforwardly re-write the field equations and the Lagrangian as functions of two tetrads

(23) |

The modified teleparallel theories can be then viewed as a new kind of bigravity theories that we shall call bitetrad theories. The action is varied with respect to the dynamical tetrad to obtain the field equations (12), while the reference tetrad is determined by the the auxiliary condition that the dynamical tetrad must approach the reference tetrad in absence of gravity. In the case of asymptotically Minkowski spacetimes, this is equivalent to the condition (15).

Effectively this introduces two metrics into the theory; the dynamical tetrad defines the spacetime metric tensor by (1), while the reference tetrad analogously yields the metric of Minkowski spacetime^{3}^{3}3We remind here that we consider only the asymptotically Minkowski spacetimes here. The asymptotically (A)dS spacetimes require further analysis and we will address this problem in the future.

(24) |

We can observe some analogies with other well-known bimetric theories that can be classified into two broad categories. The first category are bimetric theories with the non-dynamical reference metric, e.g. the original bimetric theory of Rosen^{4}^{4}4The analogy with Rosen’s theory is even more intriguing since a separation of gravity and inertia was the original motivation of Rosen’s bimetric theory, same as in the case of teleparallel gravity [52, 53]. [66, 67, 68, 69], or bimetric massive gravity [70, 71]. In the second category are bimetric theories with two dynamical metrics, where the second metric is used to generate the connection [72, 73, 74].

Modified teleparallel theories have features of both kinds of bimetric theories, what can be understood as a consequence of their bitetrad nature. We can recall that a general tetrad has 16 degrees of freedom. However, the reference tetrad is related to the Minkowski metric (24), which is uniquely determined by the choice of the coordinate system and transforms covariantly under its change. Therefore, the metric degrees of freedom of the reference tetrad are fixed by the choice of coordinates and the metric tensor is indeed the non-dynamical “background” metric. However, 6 inertial degrees of freedom are not fixed by the metric, and it is precisely these degrees of freedom–represented by the Lorentz matrix–that generate the teleparallel spin connection through the relation (3). These inertial degrees of freedom are determined then according to the dynamical tetrad as we have discussed in section 4.

## 6 Relation to the non-covariant teleparallel theories

The teleparallel connection (3) is a pure gauge-like, and hence there always exists a local Lorentz transformation, , that transforms the connection to the vanishing one. In the original, non-covariant, approach to teleparallel theories [9, 10, 11, 12, 13, 39], it is assumed that the spin connection is independent variable, and hence can be gauged away independently of transformation of the tetrad. This results into the theory where the only variable is the tetrad, which was nicknamed pure tetrad teleparallel gravity [46].

However, as our analysis in this paper shows, this in not correct and modified teleparallel theories are consistent only if the spin connection is determined accordingly to the tetrad. Therefore, a local Lorentz transformation of the spin connection, should be always accompanied by a transformation of the corresponding tetrad

(25) |

what shows that only to a very special subclass of tetrads–called proper tetrads [52, 46]–corresponds the vanishing zero connection. These proper tetrads are exactly those that are considered to be “good” in the non-covariant formulation.

Nevertheless, we would like to show that it is always possible to do the transformation (25) and formulate the theory directly in terms of the proper tetrads , and obtain analogous results to our covariant approach. The crucial observation is that proper tetrads are those that lead to the consistent variational principle, which is possible only if the torsion scalar vanishes in absence of gravity. In the case of the asymptotically Minkowski spacetime, this implies that the torsion tensor vanish at asymptotic infinity. Since in the pure tetrad theory we do not have the spin connection that could fix this asymptotic behavior, we have to enforce a condition

(26) |

while performing the variation of the action. This condition singles out “good” tetrads and leads to a consistent theory.

In this approach we loose local Lorentz invariance since we restrict ourselves to the class of good tetrads, i.e. we fix the particular frame. We can view this analogously to formulating electromagnetism in the particular gauge; we loose gauge invariance, but it is not a sign of any pathology.

## 7 Conclusions

We have analyzed the variational principle in the covariant formulation of teleparallel theories, where the fundamental variables are the tetrad and the spin connection subject to the teleparallel constraint (2). In the ordinary teleparallel gravity the spin connection does not affect the field equations as it enters the action through the surface terms only. However, the correct conserved charges and the physically relevant finite action can be obtained only if the spin connection is chosen according to the tetrad by a method that removes the spurious inertial effects from the theory [52] .

We have shown that in the modified case, the action must be finite in order to derive the field equations consistently and to obtain new non-trivial solutions of the theory. This makes the removal of the inertial effects necessity. The spin connection cannot be treated as an independent variable anymore and always have to be chosen accordingly to the tetrad. The only known viable way how to achieve this is to introduce the reference tetrad representing the same inertial effects as the dynamical tetrad.

We have then argued that the reference tetrad can be considered as a fundamental variable instead of the spin connection. Modified teleparallel theories can be interpreted as bitetrad theories, where the dynamical tetrad generates the spacetime metric, while the reference tetrad introduces the background reference metric and its inertial components generate the spin connection. This is a novel viewpoint on modified teleparallel theories and opens new way to study their underlying physical structure, as well as intriguing analogies with other bimetric theories of gravity.

Our results were demonstrated on the example of gravity, but they can be easily extended to all modified teleparallel theories with second order field equations, e.g. see [75, 76, 77]. If the derivatives of torsion are included in the action, it is possible to show that the teleparallel equivalent of theory does not require the above procedure of fixing the spin connection [30]. However, this leads to the fourth order field equations and introduces another kind of difficulties known from the curvature-based modified theories.

Let us conclude with the statement that modified teleparallel theories are a viable approach to obtain new modified gravity theories with second order field equations, but the covariant formulation of these theories with a well-defined variational problem is achievable only if we introduce the reference tetrad and effectively adopt our bitetrad viewpoint.

## 8 Acknowledgments

The author would like to thank S. Bahamonde, R. Ferraro, A. Golovnev, M.-J. Guzmán, M. Hohmann, L. Järv, K. Koivisto, L.-W. Luo, J. N. Nester, J. W. Maluf, J. G. Pereira, C. Pfeifer, E. N. Saridakis, H.-H. Tseng, Y.-P. Wu for interesting and stimulating discussions, and C.-Q. Geng for hospitality during the stay at NCTS (Hsinchu, Taiwan). This research is funded by the European Regional Development Fund through the Centre of Excellence TK 133 The Dark Side of the Universe.

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