Variational Principles in Teleparallel Gravity Theories
Manuel Hohmann
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23 pages, no figures, journal version; invited contribution to the Universe Special Issue "Teleparallel Gravity: Foundations and Observational Constraints"
We study the variational principle and derivation of the field equations for different classes of teleparallel gravity theories, using both their metric-affine and covariant tetrad formulations. These theories have in common that in addition to the tetrad or metric, they employ a flat connection as additional field variable, but differ by the presence of absence of torsion and nonmetricity for this independent connection. Besides the different underlying geometric formulation using a tetrad or metric as fundamental field variable, one has different choices to introduce the conditions of vanishing curvature, torsion and nonmetricity, either by imposing them a priori and correspondingly restricting the variation of the action when the field equations are derived, or by using Lagrange multipliers. Special care must be taken, since these conditions form non-holonomic constraints. Here we show explicitly that all of the aforementioned approaches are equivalent, and that the same set of field equations is obtained, independently of the choice of the geometric formulation and variation procedure. We further discuss consequences arising from the diffeomorphism invariance of the gravitational action, and show how they establish relations between the gravitational field equations.
We review the current status of the Lorentz covariance in teleparallel and modified teleparallel theories of gravity, and discuss the controversial features of the different approaches. We also revisit the issue of the remnant Lorentz gauge symmetries in $f(T)$ gravity.
Generalised Proca theories of gravity represent an interesting class of vector-tensor theories where only three propagating degrees of freedom are present. In this work, we propose a new teleparallel gravity analog to Proca theories where the generalised Proca framework is extended due to the lower order nature of torsion based gravity. We develop a new action contribution and explore the example of the Friedmann equations in this regime. We find that teleparallel Proca theories offer the possibility of a much larger class of models in which do have an impact on background cosmology.
The basics of teleparallel gravity and its extensions are reviewed with particular emphasis on the problem of Lorentz-breaking choice of connection in pure-tetrad versions of the theories. Various possible ways to covariantise such models are discussed. A by-product is a new form of $f(T)$ field equations.
Teleparallel theory of gravity and its modifications have been studied extensively in literature. However, gravitational waves has not been studied enough in the framework of teleparallelism. In the present study, we discuss gravitational waves in general theories of teleparallel gravity containing the torsion scalar $T$, the boundary term $B$ and a scalar field $ϕ$. The goal is to classify possible new polarizations generalizing results presented in Ref.[15]. We show that, if the boundary term is minimally coupled to the torsion scalar and the scalar field, gravitational waves have the same p…
In the context of extended Teleparallel gravity theories with a 3+1 dimensions Gauss-Bonnet analog term, we address the possibility of these theories reproducing several well-known cosmological solutions. In particular when applied to a Friedmann-Lemaître-Robertson-Walker geometry in four-dimensional spacetime with standard fluids exclusively. We study different types of gravitational Lagrangians and reconstruct solutions provided by analytical expressions for either the cosmological scale factor or the Hubble parameter. We also show that it is possible to find Lagrangians of this type without…
We consider the variational principle in the covariant formulation of modified teleparallel theories with second order field equations. We vary the action with respect to the spin connection and obtain a consistency condition relating the spin connection with the tetrad. We argue that since the spin connection can be calculated using an additional reference tetrad, modified teleparallel theories can be interpreted as effectively bigravity theories. We conclude with discussion about the relation of our results and those obtained in the usual, non-covariant, formulation of teleparallel theories …
We study teleparallel gravity in five-dimensional spacetime with particular discussions on Kaluza-Klein (KK) and braneworld theories. We directly perform the dimensional reduction by differential forms. In the braneworld theory, the teleparallel gravity formalism in the Friedmann-Lemaître-Robertson-Walker cosmology is equivalent to GR due to the same Friedmann equation, whereas in the KK case the reduction of our formulation does not recover the effect as GR of 4-dimensional spacetime.
We analyze the construction of conformal theories of gravity in the realm of teleparallel theories. We first present a family of conformal theories which are quadratic in the torsion tensor and are constructed out of the tetrad field and of a scalar field. For a particular value of a coupling constant, and in the gauge where the scalar field is restricted to assume a constant value, the theory reduces to the teleparallel equivalent of general relativity, and the tetrad field satisfies Einstein's equations. A second theory is formulated out of the tetrad field only, and is not equivalent to the…
The covariant formulation of teleparallel gravity theories must include the spin connection, which has 6 degrees of freedom. One can, however, always choose a gauge such that the spin connection is put to zero. In principle this gauge may affect counting of degrees of freedom in the Hamiltonian analysis. We show for general teleparallel theories of gravity, that fixing the gauge such that the spin connection vanishes in fact does not affect the counting of degrees of freedom. This manifests in the fact that the momenta of the Lorentz transformations which generate the spin connection are fully…
In the context of extended Teleparallel gravity theories with a 3+1 dimensions Gauss-Bonnet analog term, we address the possibility of these theories reproducing several well-known cosmological bouncing scenarios in a four-dimensional Friedmann-Lemaître-Robertson-Walker geometry. We shall study which types of gravitational Lagrangians are capable of reconstructing bouncing solutions provided by analytical expressions for symmetric, oscillatory, superbounce, the matter bounce and singular bounce. Some of the Lagrangians discovered are both analytical at the origin having Minkowski and Schwarzsc…
We investigate the Lagrange multiplier formulation of teleparallel theories, including f(T) gravity, in which the connection is not set to zero a priori and compare it with the pure frame theory. We show explicitly that the two formulations are equivalent, in the sense that the dynamical equations have the same content. One consequence is that the manifestly local Lorentz invariant f(T) theory cannot be expected to be free of pathologies, which were previously found to plague f(T) gravity formulated in the usual pure frame approach.
Teleparallel geometry utilizes Weitzenböck connection which has nontrivial torsion but no curvature and does not directly follow from the metric like Levi-Civita connection. In extended teleparallel theories, for instance in $f(T)$ or scalar-torsion gravity, the connection must obey its antisymmetric field equations. So far only a few analytic solutions were known. In this note we solve the $f(T,ϕ)$ gravity antisymmetric vacuum field equations for a generic rotating tetrad ansatz in Weyl canonical coordinates, and find the corresponding spin connection coefficients. By a coordinate transformat…
A 2D symmetric teleparallel gravity model is given by a generic 4-parameter action that is quadratic in the non-metricity tensor. Variational field equations are derived. A class of conformally flat solutions is given. We also discuss static and cosmological solutions.
Teleparallel gravity and its popular generalization $f(T)$ gravity can be formulated as fully invariant (under both coordinate transformations and local Lorentz transformations) theories of gravity. Several misconceptions about teleparallel gravity and its generalizations can be found in the literature, especially regarding their local Lorentz invariance. We describe how these misunderstandings may have arisen and attempt to clarify the situation. In particular, the central point of confusion in the literature appears to be related to the inertial spin connection in teleparallel gravity models…
We analyze the relation between teleparallelism and local Lorentz invariance. We show that generic modifications of the teleparallel equivalent to general relativity will not respect local Lorentz symmetry. We clarify the reasons for this and explain why the situation is different in general relativity. We give a prescription for constructing teleparallel equivalents for known theories. We also explicitly consider a recently proposed class of generalized teleparallel theories, called f(T) theories of gravity, and show why restoring local Lorentz symmetry in such theories cannot lead to sensibl…
Recently Kaniel & Itin proposed a gravitational model with the wave type equation [\square+λ(x)]\vartheta^α=0 as vacuum field equation, where \vartheta^αdenotes the coframe of spacetime. They found that the viable Yilmaz-Rosen metric is an exact solution of the tracefree part of their field equation. This model belongs to the teleparallelism class of gravitational gauge theories. Of decisive importance for the evaluation of the Kaniel-Itin model is the question whether the variation of the coframe commutes with the Hodge star. We find a master formula for this commutator and rectify some corre…
It is well known that one cannot apply a conformal transformation to $f(T)$ gravity to obtain a minimally coupled scalar field model, and thus no Einstein frame exists for $f(T)$ gravity. Furthermore nonminimally coupled "teleparallel dark energy models" are not conformally equivalent to $f(T)$ gravity. However, it can be shown that $f(T)$ gravity is conformally equivalent to a teleparallel phantom scalar field model with a nonminimal coupling to a boundary term only. In this work, we extend this analysis by considering a recently studied extended class of models, known as $f(T,B)$ gravity, wh…
There is a growing interest in modified gravity theories based on torsion, as these theories exhibit interesting cosmological implications. In this work, inspired by the teleparallel formulation of general relativity, we present its extension to Lovelock gravity known as the most natural extension of general relativity in higher-dimensional space-times. First, we review the teleparallel equivalent of general relativity and Gauss-Bonnet gravity, and then we construct the teleparallel equivalent of Lovelock gravity. In order to achieve this goal we use the vielbein and the connection without imp…
The article communicates gravitational baryogenesis in non-minimal $f(T)$ gravity and $f(T,B)$ teleparallel gravity where $T$ denote the torsion scalar and $B$ a boundary term. These extended teleparallel theories of gravity differ from the usual $f(T)$ gravity and therefore could provide key insights in expounding the baryon asymmetry of the universe. Furthermore, such a study also put constraints on the model parameters of these extended teleparallel theory of gravity. I present different baryogenesis interactions proportional to $\partial_{i}T$, $\partial_{i}f(T)$, $\partial_{i}(T+B)$ and $…
It is well known that the teleparallel equivalent of general relativity yields the same vacuum solutions as general relativity does, which ensures that this particular teleparallel model is in good agreement with experiments. A less known result concerns the existence of a wider class of teleparallel models which also admits these solutions when the spacetime is diagonalizable by means of a coordinate change. However, it is stated in Ref. [Phys. Rev. D 67, 044016 (2003).] that the teleparallel equivalent of general relativity is the only teleparallel model which admits black holes. To show tha…
We study tensor perturbations around the most general cosmologically symmetric backgrounds in a class of teleparallel gravity theories known as New General Relativity. These theories comprise a one-parameter class, which is fully consistent with observations at the post-Newtonian level, and which contains the teleparallel equivalent of General Relativity as a special case. We find that for a particular class of cosmological background geometries these theories exhibit gravitational wave birefringence and dispersion, i.e., the dispersion relation for gravitational waves depends on their polariz…
We consider the problem of IR divergences of the action in the covariant formulation of teleparallel gravity in asymptotically Minkowski spacetimes. We show that divergences are caused by inertial effects and can be removed by adding an appropriate surface term, leading to the renormalized action. This process can be viewed as a teleparallel analogue of holographic renormalization. Moreover, we explore the variational problem in teleparallel gravity and explain how the variation with respect to the spin connection should be performed.
We consider the most general class of teleparallel theories of gravity quadratic in the torsion tensor, and carry out a detailed investigation of its Hamiltonian formulation in the time gauge. Such general class is given by a three-parameter family of theories. A consistent implementation of the Legendre transform reduces the original theory to a one-parameter theory determined in terms of first class constraints. The free parameter is fixed by requiring the Newtonian limit. The resulting theory is the teleparallel equivalent of general relativity.
We investigate modified theories of gravity in the context of teleparallel geometries. It is well known that modified gravity models based on the torsion scalar are not invariant under local Lorentz transformations while modifications based on the Ricci scalar are. This motivates the study of a model depending on the torsion scalar and the divergence of the torsion vector. We derive the teleparallel equivalent of $f(R)$ gravity as a particular subset of these models and also show that this is the unique theory in this class that is invariant under local Lorentz transformation. Furthermore one …
Gauge symmetries in teleparallel gravity, together with the identities among the dynamical equations they provide, are analyzed in relation to the way they condition the coupling between matter and gravity. Particularly, the coupling of fermionic matter seems to be excluded in a wide range of teleparallel theories.
We study conditions on a generic connection written in terms of first-order derivatives of the vielbein in order to obtain (possible) equivalent theories to Einstein Gravity. We derive the equations of motion for these theories which are based on the new connections. We recover the Teleparallel Gravity equations of motion as a particular case. The analysis of this work might be useful to Double Field Theory to find other connections determined in terms of the physical fields.
We determine the possible gravitational wave polarizations in two general classes of teleparallel gravity theories, using the metric and symmetric teleparallel geometries. For this purpose we apply the Newman-Penrose formalism, and find that depending on the choice of parameters, the $\mathrm{E}(2)$ class of the theories is one of $\mathrm{N}_2, \mathrm{N}_3, \mathrm{III}_5, \mathrm{II}_6$, corresponding to two to six polarizations, where all of them include the two tensor polarizations known from general relativity. We also find classes of theories apart from general relativity which yield th…
Symmetric teleparallel gravity theories, in which the gravitational interaction is attributed to the nonmetricity of a flat, symmetric, but not metric-compatible affine connection, have been a topic of growing interest in recent studies. Numerous works study the cosmology of symmetric teleparallel gravity assuming a flat Friedmann-Lemaître-Robertson-Walker metric, while working in the so-called "coincident gauge", further assuming that the connection coefficients vanish. However, little attention has been paid to the fact that both of these assumptions rely on the freedom to choose a particula…
Born-Infeld strategy to smooth theories having divergent solutions is applied to teleparallel equivalent of General Relativity. Differing from other theories of modified gravity, modified teleparallelism leads to second order equations, since teleparallel Lagrangian only contains first derivatives of the vierbein. We show that Born-Infeld-modified teleparallelism solves the particle horizon problem in a spatially flat FRW universe by providing an initial exponential expansion without resorting to an inflaton field.
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 relat…
We apply the Hamiltonian formulation of teleparallel theories of gravity in 2+1 dimensions to a circularly symmetric geometry. We find a family of one-parameter black hole solutions. The BTZ solution fixes the unique free parameter of the theory. The resulting field equations coincide with the teleparallel equivalent of Einstein's three-dimensional equations. We calculate the gravitational energy of the black holes by means of the simple expression that arises in the Hamiltonian formulation and conclude that the resulting value is identical to that calculated by means of the Brown-York method.
Despite the fact that General Relativity (GR) has been very successful, many alternative theories of gravity have attracted the attention of a significant number of theoretical physicists. Among these theories, we have theories with conformal symmetry. Here, the use of Weyl geometry to deal with conformal teleparallel gravity is reviewed in great detail. As an application, a model that can be set to be equivalent to the Teleparallel Equivalent of General Relativity (TEGR) and is invariant under diffeomorphisms, local Lorentz transformations (LLT) and Weyl transformations (WT) is created. Some …
We study the matter stability in modified teleparallel gravity or $f(T)$ theories. We show that there is no Dolgov-Kawasaki instability in these types of modified teleparallel gravity theories. This give the $f(T)$ theories a great advantage over their $f(R)$ counterparts because from the stability point of view there isn't any limit on the form of functions that can be chosen.
A class of theories of gravitation that naturally incorporates preferred frames of reference is presented. The underlying space-time geometry consists of a partial parallelization of space-time and has properties of Riemann-Cartan as well as teleparallel geometry. Within this geometry, the kinematic quantities of preferred frames are associated with torsion fields. Using a variational method, it is shown in which way action functionals for this geometry can be constructed. For a special action the field equations are derived and the coupling to spinor fields is discussed.
We derive the post-Newtonian limit of a general class of teleparallel gravity theories, whose action is given by a free function of three scalar quantities obtained from the torsion of the teleparallel connection. This class of theories is chosen to be sufficiently generic in order to include the $f(T)$ class of theories as well as new general relativity as subclasses. To derive its post-Newtonian limit, we first impose the Weitzenböck gauge, and then introduce a post-Newtonian approximation of the tetrad field around a Minkowski background solution. Our results show that the class of theories…
We study the matter density perturbations in modified teleparallel gravity theories, where extra degrees of freedom arise from the local Lorentz violation in the tangent space. We formulate a vierbein perturbation with variables addressing all the 16 components of the vierbein field. By assuming the perfect fluid matter source, we examine the cosmological implication of the 6 unfamiliar new degrees of freedom in modified $f(T)$ gravity theories. We find that despite the new modes in the vierbein scenario provide no explicit significant effect in the small-scale regime, they exhibit some deviat…
We give a pedagogical introduction into the field of (modified) teleparallel theories of gravity. Our presentation is fairly self-contained. In particular, we carefully explain the basic principles of metric-affine approaches to gravity. This contribution is based on our talk "Teleparallel gravity, its modifications, and the local Lorentz invariance" at the 9th Mathematical Physics Meeting: School and Conference on Modern Mathematical Physics in Belgrade, September 2017.
Conformal symmetries appear in many parts of physics and play a unique role in exploring the Universe. In this work, we consider the possibility of constructing conformal theories of gravity in the Symmetric Teleparallel Gravity framework, where gravitation is expressed through non-metricity rather than curvature or torsion. We demonstrate that it is possible to formulate a family of conformal theories with second-order field equations and have the metric tensor as the fundamental variable. Moreover, we consider the addition of a scalar field to the theory, and compare the results to the Telep…
This paper is devoted to the investigation of the energy-momentum problem in two theories, i.e., General Relativity and teleparallel gravity. We use Einstein, Landau-Lifshitz, Bergmann-Thomson and Möller's prescriptions to evaluate energy-momentum distribution of Bell-Szekeres metric in both the theories. It is shown that these prescriptions give the same energy-momentum density components in both General Relativity and teleparallel theory. Möller's prescription yields constant energy in both the theories.
Teleparallel geometry offers a platform on which to build up theories of gravity where torsion rather than curvature mediates gravitational interaction. The teleparallel analogue of Horndeski gravity is an approach to teleparallel geometry where scalar-tensor theories are considered in this torsional framework. Being teleparallel gravity of lower order in dynamics, this turns out to be more general than metric Horndeski gravity. In other words, the class of teleparallel Horndeski gravity models is much broader than the standard metric one. In this work, we explore constraints on this wide rang…
The teleparallel formulation of gravity theories reveals close structural analogies to electrodynamics, which are more hidden in their usual formulation in terms of the curvature of spacetime. We show how every locally Lorentz invariant teleparallel theory of gravity with second order field equations can be understood as built from a gravitational field strength and excitation tensor which are related to each other by a constitutive relation, analogous to the axiomatic construction of theories of electrodynamics. We demonstrate how the previously studied models of $f(\mathbb{T})$ and $f(T_\tex…
The variation procedure on a teleparallel manifold is studied. The main problem is the non-commutativity of the variation with the Hodge dual map. We establish certain useful formulas for variations and restate the master formula due to Hehl and his collaborates. Our approach is different and sometimes easier for applications. By introducing the technique of the variational matrix we find necessary and sufficient conditions for commutativity (anti-commutativity) of the variation derivative with the Hodge dual operator. A general formula for the variation of the quadratic-type expression is obt…
Teleparallel gravity theories employ a tetrad and a Lorentz spin connection as independent variables in their covariant formulation. In order to solve their field equations, it is helpful to search for solutions which exhibit certain amounts of symmetry, such as spherical or cosmological symmetry. In this article we present how to apply the notion of spacetime symmetries known from Cartan geometry to teleparallel geometries. We explicitly derive the most general tetrads and spin connections which are compatible with axial, spherical, cosmological and maximal symmetry. For homogeneous and isotr…
In this paper, we provide a general framework for the construction of the Einstein frame within non-linear extensions of the teleparallel equivalents of General Relativity. These include the metric teleparallel and the symmetric teleparallel, but also the general teleparallel theories. We write the actions in a form where we separate the Einstein--Hilbert term, the conformal mode due to the non-linear nature of the theories (which is analogous to the extra degree of freedom in $f(R)$ theories), and the sector that manifestly shows the dynamics arising from the breaking of local symmetries. Thi…
A brief discussion is made about the relevance of surface terms in the Lagrangian and Hamiltonian formulations of theories of gravity. These surface terms play an important role in the variation of the action integral and in the definition of field quantities such as the gravitational energy-momentum. Then we point out several inconsistencies of a recently proposed formulation of teleparallel theories of gravity with local Lorentz symmetry.
We investigate the gravitational waves and their properties in various modified teleparallel theories, such as $f(T), f(T,B)$ and $f(T,T_G)$ gravities. We perform the perturbation analysis both around a Minkowski background, as well as in the case where a cosmological constant is present, and for clarity we use both the metric and tetrad language. For $f(T)$ gravity we verify the result that no further polarization modes comparing to general relativity are present at first-order perturbation level, and we show that in order to see extra modes one should look at third-order perturbations. For n…
In the context of extended Teleparallel gravity theories, we address the issue of junction conditions required to guarantee the correct matching of different regions of spacetime. In the absence of shells/branes, these conditions turn out to be more restrictive than their counterparts in General Relativity as in other extended theories of gravity. In fact, the general junction conditions on the matching hypersurfaces depend on the underlying theory and a new condition on the induced tetrads in order to avoid delta-like distributions in the field equations. This result imposes strict consequenc…
New classes of modified teleparallel theories of gravity are introduced. The action of this theory is constructed to be a function of the irreducible parts of torsion $f(T_{\rm ax},T_{\rm ten},T_{\rm vec})$, where $T_{\rm ax},T_{\rm ten}$ and $T_{\rm vec}$ are squares of the axial, tensor and vector components of torsion, respectively. This is the most general (well-motivated) second order teleparallel theory of gravity that can be constructed from the torsion tensor. Different particular second order theories can be recovered from this theory such as new general relativity, conformal telepara…
Different models of universes are considered in the context of teleparallel theories. Assuming that the universe is filled by a fluid with equation of state (EoS) $P=-ρ-f(ρ)$, for different teleparallel theories and different EoS we study its dynamics. Two particular cases are studied in detail: in the first one we consider a function $f$ with two zeros (two de Sitter solutions) that mimics a huge cosmological constant at early times and a pressureless fluid at late times. In the second one, in the context of loop quantum cosmology with a small cosmological constant, we consider a pressureless…
Although the autoparallel curves and the geodesics coincide in the Riemannian geometry in which only the curvature is nonzero among the nonmetricity, the torsion and the curvature, they define different curves in the non-Riemannian ones. We give a novel approach to autoparallel curves and geodesics for theories of the symmetric teleparallel gravity written in the coincident gauge. Then we apply our autoparallel equation to a Schwarzschild-type metric and give remarks about dark matter and orbit equation.
