Scalar-torsion theories of gravity I:
general formalism and conformal transformations
We discuss the most general class of teleparallel scalar-torsion theories of gravity in their covariant formulation. The only restrictions we impose are the invariance of the action under diffeomorphisms and local Lorentz transformations, as well as vanishing direct coupling of the matter fields to the teleparallel spin connection. In this general setting we discuss the implications of local Lorentz invariance and diffeomorphism invariance and derive the general structure of the field equations. Further, we show how different theories of this class are related to each other by conformal transformations of the tetrad and redefinitions of the scalar field. We finally show how the formalism can be generalized to an arbitrary number of scalar fields, and provide a few examples.
The most successful theory of gravity is general relativity (GR), as it has been proven to be consistent with a wide range of observations on different scales. It attributes the gravitational interaction to the curvature of the Levi-Civita connection of a (pseudo-)Riemannian geometry. Nevertheless, GR is challenged by a number of observations in cosmology, such as the accelerating phases in the early and late universe Ade et al. (2016a, b, c). Further, it is challenged by its tension with quantum theory, as so far no fully conclusive theory of quantum gravity has been derived. These aspects have motivated the study of numerous alternative gravity theories. A large class of such theories is based on the idea that gravity is not attributed to the curvature of the torsion-free Levi-Civita connection, but to the torsion of a flat, i.e., curvature-free connection. This attempt is known as teleparallelism, since the flat connection allows for a path-independent parallel transport of tangent vectors Møller (1961); Aldrovandi and Pereira (2013); Maluf (2013); Golovnev (2018). An interesting feature of teleparallel gravity is its possible interpretation as a gauge theory of the translation group, which brings it closer to other fundamental interactions which are modeled as gauge theories Cho (1976a, b); Hayashi (1977).
In its original formulation, teleparallel theories of gravity have been based on the Weitzenböck connection defined by a tetrad field such that the corresponding spin connection coefficients vanish. However, this formulation has been shown to be problematic due to the breaking of local Lorentz invariance Li et al. (2011a); Sotiriou et al. (2011), as spurious degrees of freedom may appear Li et al. (2011b); Ong et al. (2013); Izumi et al. (2014); Chen et al. (2015a). In order to solve these problems a covariant formulation has been adopted, in which a flat, but in general non-vanishing spin connection is introduced Krššák and Saridakis (2016). It has been shown that a naive variation of the action with respect to the spin connection is too restrictive, as it does not yield the desired field equations, and so either a different method of variation Golovnev et al. (2017) or different physical principles Krssak (2017) must be employed. Further, it has been shown for different classes of teleparallel theories that a constrained variation of the spin connection yields a set of field equations, which is identical to the antisymmetric part of the tetrad field equations Golovnev et al. (2017); Hohmann et al. (2017).
While the simplest and most well-studied teleparallel gravity theory is equivalent to general relativity, and hence known as the teleparallel equivalent of general relativity (TEGR), it provides an alternative starting point for modifications, in order to address the mentioned challenges to GR. A possible modification of TEGR is the introduction of a scalar field non-minimally coupled to torsion Geng et al. (2011); this is similar to introducing a scalar field non-minimally coupled to curvature as a modification of GR. Numerous models of this type have been studied Izumi et al. (2014); Chakrabarti et al. (2017); Otalora (2013); Jamil et al. (2012); Chen et al. (2015b). However, it is common to employ the Weitzenböck connection in order to derive the torsion, which leads to the aforementioned potential issues arising from breaking local Lorentz invariance. These issues are addressed in a covariant formulation of scalar-torsion theories, and it has been shown that the relation between the field equations for the tetrad and the spin connection also holds in this case Hohmann et al. (2018a).
The aim of this article is to generalize the aforementioned results to a larger class of teleparallel scalar-torsion theories of gravity, where we allow the scalar field to couple in an arbitrary way to all other (gravitational and matter) fields, and impose as only restrictions the invariance of the action under diffeomorphisms and local Lorentz transformations, as well as a vanishing coupling between the matter fields and the teleparallel spin connection. For this class of theories we derive the field equations and study the implications of the imposed restriction. In particular, we discuss the question whether the local Lorentz invariance relates the antisymmetric part of the tetrad field equations to the connection field equations, as has been proven for narrower classes of theories. We also discuss how different theories within this class can be related to each other by performing conformal transformations of the tetrad and scalar field redefinitions.
This article is the first in a series of three articles, and develops the general framework for the aforementioned class of scalar-torsion theories. Special subclasses will be discussed in two subsequent articles: a class we denote and whose gravitational action depends on four scalar quantities derived from the gravitational variables is developed in the second article Hohmann and Pfeifer (2018); this class will be further restricted to a class which has properties similar to scalar-tensor gravity in the third and last article Hohmann (2018).
The outline of this article is as follows. In section II we list the fundamental fields present in the class of theories we consider and introduce the notation we will use. These are then used in section III to define the action, and set the notation for its variation. This variation is exploited in the following three sections: to derive the implications of local Lorentz invariance in section IV and diffeomorphism invariance in section V, and to derive the field equations in section VI. Conformal transformations are discussed in section VII. We generalize our discussion to multiple scalar fields in section VIII. A few examples are given in section IX. We end with a conclusion in section X.
Ii Fields in scalar-torsion gravity and their relations
The most important difference between teleparallel theories of gravity and theories based on Riemannian geometry is that the fundamental field defining the geometry is not a metric, but a coframe field , which can be expressed as a set of four one-forms
labeled with a Lorentz index, which constitute a basis of the cotangent space for all spacetime points . The corresponding dual bases of the tangent spaces constitute a frame field , which can be expressed as a set of four vector fields
Since there is a one-to-one correspondence between frame and coframe fields, both a conventionally denoted by the term tetrad. Further, we consider a flat Lorentz spin connection , which is likewise given by one-forms
All quantities related to this connection, which is also called the teleparallel connection, will be denoted by a bullet (). Note that being Lorentzian implies antisymmetry, , where indices are raised and lowered with the Minkowski metric , while flatness implies vanishing curvature
In addition to the tetrad and the spin connection, we consider a scalar field . Finally, we consider a set of matter fields , which we label by an index . For simplicity of notation, we will assume that the matter fields are also given by differential forms of rank . However, this is not essential for our derivation, and more general choices are possible.
For our calculations we will make use of the fact that, given a tetrad, spin connection is uniquely determined by its torsion
where denotes the covariant exterior derivative. In order to invert the relation (5) and determine the spin connection from the torsion, it is most convenient to introduce the contortion
as well as the Levi-Civita connection
whose associated quantities we denote with an open circle (), and which is uniquely defined by having vanishing torsion, . It allows us to write the spin connection as
These are the geometric objects we will need in order to define the action for the class of theories we consider here. This will be done in the next section.
Iii Action and variation
The general form of the action for the dynamical fields listed in the previous section, for the class of theories we consider in this article, is given by
where denotes the gravitational part of the action, while denotes its matter part. Note in particular that we allow for a non-minimal coupling between the scalar field and the matter fields , but no such couplings for the teleparallel spin connection.
Using the formalism of differential forms, the variation of the action with respect to the dynamical fields can be written in a very compact form. For the matter part of the action we write a general variation as
where we introduced the energy-momentum three-forms , a four-form and -forms 111More precisely, the introduced objects are twisted forms, which means that their sign changes under changing the orientation of the spacetime manifold. However, this distinction will not be relevant for the purpose of this article, and so we omit it for simplicity.. Writing the variation in this form implies that any integration by parts, which is necessary in order to eliminate derivatives of the variations of the dynamical fields, has already been performed. Note that we wrote , even though is a scalar; this is done simply for consistency of the notation.
For the variation of the gravitational part of the action, we introduce a similar notation
with three-forms and , as well as a four-form . We choose to be antisymmetric, , since any symmetric part would cancel when contracted with the variation of the (also antisymmetric) Lorentz spin connection.
Using the one-to-one correspondence between spin connections and their torsion discussed in section II, one may also substitute the teleparallel spin connection in the gravitational action with the teleparallel torsion, and write its variation in the form
with three-forms , two-forms and the same four-form as above. Both forms of the variation can easily be related using the definition (5) of the torsion, from which one derives the variation
Substituting this expression in the variation (12) then yields
By comparison with the variation (11) one finds the relations
Conversely, one can make use of the relation (8) to express the variation of the spin connection through the variation of the torsion. Using a similar procedure as given above one obtains
In the following sections we will make use of these formulas, and consider the particular variations induced by local Lorentz transformations and diffeomorphisms.
Iv Local Lorentz invariance
We now further demand that the action (9) is invariant under (infinitesimal) local Lorentz transformations with , which act on the tetrad and spin connection as
It thus follows that the torsion transforms as
We remark that the scalar field and the matter fields we consider are Lorentz scalars and hence transform trivially under local Lorentz transformations, such that and .
We start with the variation of the matter action under Lorentz transformations, which is given by
This vanishes for arbitrary local Lorentz transformations if and only if the energy-momentum 3-forms satisfy the symmetry condition
Note that this holds both on-shell and off-shell, since we have not used any field equations.
We then proceed analogously with the gravitational action. Writing its variation in the form (12), one finds that under local Lorentz transformations it transforms by
It follows that the action is locally Lorentz invariant if and only if
Again we remark that this must hold both on-shell and off-shell. We can equivalently start from the variation (11) and find
Hence, the condition for local Lorentz invariance reads
V Diffeomorphism invariance and energy-momentum conservation
We now come to the discussion of diffeomorphism invariance. Recall that under an infinitesimal diffeomorphism generated by a vector field any tensor field changes by its Lie derivative. For a differential form , the Lie derivative can be expressed as
which is also known as Cartan’s (magic) formula.
We are in particular interested in the variation of the matter action under diffeomorphisms, which is given by
Note that on-shell the Euler-Lagrange equations hold, so that the last term vanishes. We will therefore consider only the first two terms, keeping in mind that the resulting formulas hold only on-shell. Using the Cartan formula we can write
Here we have expressed in the tetrad basis to obtain the scalar functions . The induced variation of the matter action vanishes for arbitrary vector fields if and only if
Note that for the Levi-Civita connection (7) holds
This finally allows us to write the diffeomorphism invariance condition as
This is the covariant conservation of the energy-momentum 3-forms. In the case of a minimal (vanishing) coupling of the scalar field to the matter fields, , one obtains the usual energy-momentum conservation .
Vi Field equations
We finally discuss the field equations of the general action (9). Already in the previous section we have mentioned the matter field equations, which take the simple form
The scalar field equation simply reads
Next, we come to the tetrad field equations. These can most easily be read off from the variation (11), using the fact that the tetrad and the spin connection are varied independently, and take the form
Using the relation (15), they can also be written in the form
Note that the structure of the tetrad field equations is the same as for the most general class of teleparallel gravity theories without a scalar field Hohmann et al. (2017). The only difference lies in the fact that now all terms may carry an implicit dependence on the scalar field . It is helpful to consider the antisymmetric part of these field equations separately from the symmetric part. It can be written in different alternative forms. From the expressions (33) and (34) we find
Finally, we come to the field equation for the spin connection. Recall that the spin connection in teleparallel gravity theories is demanded to be flat, i.e., have vanishing curvature (4). Hence, we may allow only variations of the connection which preserve this condition, and must thus be of the form with . From the variation (11) one then immediately finds the field equations
Note that these are simply the antisymmetric tetrad field equations (36), which follows from the fact that the teleparallel connection is a pure gauge degree of freedom. This is a direct generalization of the similar result for a number of sub-classes of teleparallel gravity theories Golovnev et al. (2017); Hohmann et al. (2017, 2018a) and one of the main results of this article.
Vii Conformal transformations
An interesting feature of the class of scalar-torsion theories discussed in this article is its invariance under conformal transformations of the tetrad and redefinitions of the scalar field. In order to show this, we consider a transformation of the tetrad and the scalar field of the form
where and are functions of the scalar field which determine the transformation. Note that we do not transform the spin connection, as it is a pure gauge degree of freedom, and any allowed transformation could be absorbed by a local Lorentz transformation. It then follows that the torsion transforms as
In order to transform the terms in the field equations detailed in section VI, which have been obtained from the variations of the action displayed in section III, we further need to transform the variations of the dynamical fields. For the variation of the tetrad one finds
while the variation of the scalar field transforms as
We also note that the variation of the torsion transforms as
Using these expressions, it is now straightforward to transform the variation of the action. For the variation (10) of the matter action we find
By comparison with the original expression (10) we can thus read off the transformation rules
We finally invert these relations, in order to obtain the transformed (barred) quantities. These read
The same procedure can be applied to the gravitational part of the action. From the variation (11) we obtain the transformation rules
Applying the same procedure to the equivalent variation (12) yields, after a more lengthy calculation, the corresponding transformation rules
The same set of transformation rules can also be obtained by applying the transformation (46) to the relations (15) and (16). One now easily checks that all dynamical and constraint equations derived in sections IV, V and VI retain their form under these transformations, i.e., under replacing all quantities with their transformed (barred) counterparts. Note that for the energy-momentum conservation equation (30) this includes replacing the Levi-Civita connection with the transformed connection
whose proof requires also the symmetry (20) of the energy-momentum three-forms, and is omitted here for brevity. For the Lorentz invariance conditions displayed in section IV and the field equations shown in section VI the same notion of invariance under conformal transformations and redefinitions of the scalar field follows immediately, and will thus likewise be omitted here. This shows that the class of theories considered in this section, as well as all results derived, are invariant under these transformations, which is another important result we present in this article.
Viii Generalization to multiple scalar fields
Instead of a single scalar field we now consider a scalar field multiplet of scalar fields. The action (9) of the theory now depends on all scalar fields, and thus takes the form
This action must accordingly be varied with respect to all scalar fields, and thus the expressions for the variation generalize correspondingly. For the variation of the matter action (10) we now have the structure
while the variation (11) of the gravitational action in connection variables reads
or, equivalently, the variation (12) in torsion variables,
The dynamical equations derived from these variations receive only minor modifications. Most notably, the energy-momentum conservation (30) for the matter fields now reads
and there are scalar field equations of the form (32),
However, all other field equations retain their structure, although the terms therein depend on a larger number of scalar fields. Also the constraint equations associated to local Lorentz invariance detailed in section IV remain unchanged, since the scalar fields are not affected by local Lorentz transformations.
The difference between a single and multiple scalar fields becomes most apparent in the discussion of invariance under conformal transformations and scalar field redefinitions, which we presented in section VII. In the multi-scalar-torsion case the combined transformation (38) reads
Hence, instead of a single function there are functions defining the scalar field redefinition, and these functions as well as the conformal transformation depend on the multiplet of all scalar fields. Alternatively, we may write for the scalar field redefinition. Also the formulas for the transformation behavior of derived quantities receive correction terms. In particular, we find the transformation (39) of the torsion
as well as the transformation (48) of the Levi-Civita connection
where we introduced the notation
for the derivative with respect to one of the scalar fields. This now allows us to generalize the transformation behavior of the variation of the action. In particular, the formula (40) generalizes to
while the formula (41) generalizes to
Here the factor on the right hand side is simply the Jacobian of the function , which determines the scalar field redefinition. Further, the variation of the torsion transforms as
generalizing the transformation formula (42). By comparison with the variation of the action we then find that the terms in the field equations transform as
while the remaining terms keep their transformation behavior detailed by the relations (45), (46) and (47). One now easily checks that also this larger class of theories, with scalar fields instead of a single field, satisfies the same notion of being closed under conformal transformations of the type (56), in the sense that the action and the field equations retain their form.
In order to show the applicability of the results we obtained in the previous sections, we finally display two examples. Here we restrict ourselves to classes of theories, which are by themselves very general, and contain a wide range of well studied teleparallel scalar-torsion models. In particular, we discuss gravity theories defined by a constitutive relation in section IX.1, and a newly introduced class of theories defined by a free function of four scalar quantities in section IX.2.
ix.1 Constitutive relations and second order field equations
In a recent work Hohmann et al. (2017) we showed how the action of any teleparallel gravity theory with second order field equations can be expressed in the form
where the excitation two-forms are defined by a constitutive relation as an algebraic function of the coframe and the torsion. We argued that also a scalar field may be included in the constitutive relation, , and gave a specific example. This motivates considering a class of theories based on a constitutive relation involving an arbitrary number of scalar fields, whose action reads
where the scalar field action determines the dynamics of the scalar fields through suitable kinetic and potential terms, and is also assumed to be locally Lorentz invariant. It is now straightforward to apply the analysis performed in this article to this class of theories; see Hohmann et al. (2017), where this is already partially done.
We conclude the discussion of this class of theories with a remark on its behavior under conformal transformations. It can be seen from the transformation laws (39) and (57) of the torsion that the appearing factor , which rescales the torsion tensor, can be absorbed into the transformation of the constitutive relation . However, this is not the case for the additive term involving derivatives of the scalar field, which is in general non-minimally coupled to torsion, and thus cannot be decomposed into contributions to the constitutive relation and the scalar field action .
The non-minimal derivative coupling between the scalar field and the torsion tensor arising from a conformal transformation of the action (65) motivates studying actions in which such terms are included. We therefore propose a new generic class of theories, which is constructed by assuming a Lagrangian that is a free function of four scalar quantities, which we define as follows. First, we write the torsion in the tetrad basis as
We then define the superpotential
where the constitutive tensor is given by
From this we can define the torsion scalar
We further define the scalar field kinetic term
as well as the kinetic coupling term
Finally, the gravitational Lagrangian of the theory if given as a free function of the three aforementioned quantities and the scalar field, such that the gravitational part of the action reads
With the definitions given above it is now straightforward to use the formalism detailed in this article in order to derive the gravitational field equations; see Hohmann and Pfeifer (2018) for a detailed discussion and full derivation of various properties of this class of theories. In particular, we show that this class of theories is algebraically closed under conformal transformation, i.e., a conformal transformation of the action (72) yields another action of the same form, but with a different functional form of the Lagrangian . We finally remark that also in this case the field equations are (at most) of second derivative order, since the action functional contains at most first order derivatives of the fundamental field variables.
In this article we have discussed the most general class of scalar-torsion theories of gravity, in which one or more scalar fields are coupled to the tetrad and flat spin connection of teleparallel geometry. We have restricted the theories only by demanding invariance under local Lorentz transformations and diffeomorphisms, as well as by excluding a direct coupling of matter fields to the teleparallel spin connection. As the main result, we have investigated a number of statements, which have been previously discussed in the context of narrower classes of teleparallel gravitational theories, and proven that they also hold for this very broad class of theories. In particular, we have shown how the condition of local Lorentz invariance causes the field equations for the teleparallel spin connection to be equal to the antisymmetric part of the tetrad field equations. As a consequence, the spin connection becomes a pure gauge degree of freedom. Further, we have shown that conformal transformations of the tetrad and scalar field redefinitions relate different theories within this class.
In order to show the applicability of the results we derived in this work, we listed two classes of example theories which are covered by the formalism we derived, and which by itself are broad enough to contain plenty of interesting theories and subclasses. Both classes have in common that their field equations are at most of second derivative order, since their actions are constructed only from terms that are of first derivative order in the fundamental fields. However, note that we made no restriction on the derivative orders in the theories we considered in this article. Hence, more general examples, involving also higher derivatives, may be conceived.
Various other aspects of the generic class of scalar-tensor theories we discussed here may be studied. For example, it may be discussed how spacetime symmetries can be exploited in order to narrow the possible solutions for the fundamental fields, to simplify the field equations, and potentially derive generic solutions for a larger class of theories Hohmann (2016); Hohmann et al. (2018b). One may also consider using the derivations presented here in order to discuss phenomenological aspects, or construct a general formalism for the phenomenology of scalar-torsion theories of gravity. Finally, one may exploit alternative formulations of teleparallel gravity, for example, as a higher gauge theory Baez and Wise (2015), in order to extend our results to related classes of gravity theories based on more general geometric backgrounds.
The work we presented here is foundational, and allows for various generalizations. Here we considered only scalar fields which are non-minimally coupled to the tetrad and spin connection of teleparallel geometry. One may relax this condition and also allow for more general tensor fields in the gravitational part of the action. Of particular interest could be additional fields carrying Lorentz indices, which would therefore transform non-trivially under local Lorentz transformations. As with the scalar fields in this article, one may also allow these additional fields to couple non-minimally to matter fields. Finally, one may relax the condition that there is no direct coupling between matter and the teleparallel spin connection. This would be interesting in particular in the presence of spinning matter, which is the source of torsion in Einstein-Cartan gravity, where torsion appears as an independent degree of freedom in addition to the metric.
Acknowledgements.The author thanks Martin Krššák and Christian Pfeifer for helpful comments and discussions. He gratefully acknowledges the full financial support of the Estonian Ministry for Education and Science through the Institutional Research Support Project IUT02-27 and Startup Research Grant PUT790, as well as the European Regional Development Fund through the Center of Excellence TK133 “The Dark Side of the Universe”.
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