In Symmetric Teleparallel General Relativity, gravity is attributed to the non-metricity. The so-called "coincident gauge" is usually taken in this theory so that the affine connection vanishes and the metric is the only fundamental variable. This gauge choice was kept in many studies on the extensions of Symmetric Teleparallel General Relativity, such as the so-called f(Q) theory. In this paper, we point out that sometimes this will make the f(Q) theory inconsistent. To circumvent this problem, we reformulate the f(Q) theory in a covariant way with which we can find suitable non-vanishing affine connection for a given metric. We also apply this method to two important cases: the spherically symmetric spacetime and the homogeneous and isotropic expanding universe.
The covariant Hamilton-Jacobi equation for the Teleparallel Equivalent of General Relativity is derived based on the analysis of the second-class constraints within the covariant Hamiltonian theory of De Donder-Weyl according to the constraints algorithm developed by Kanatchikov.
A pseudo-Riemannian manifold is introduced, with light-cone coordinates in (4+1) dimensional space-time, to describe a Galilei covariant gravity. The notion of 5-bein and torsion are developed and a galilean version of teleparallelism is constructed in this manifold. The formalism is applied to two spherically symmetric configurations. The first one is an ansatz which is inferred by following the Schwarzschild solution in general relativity. The second one is a solution of galilean covariant equations. In addition, this Galilei teleparallel approach provides a prescription to couple the 5-bein…
The transformation properties of the gravitational energy-momentum in the teleparallel gravity are analyzed. It is proved that the gravitational energy-momentum in the teleparallel gravity can be expressed in terms of the Lorentz gauge potential, and therefore is not covariant under local Lorentz transformations. On the other hand, it can also be expressed in terms of the translation gauge field strength, and therefore is covariant under general coordinate transformations. A simplified Hamiltonian formulation of the teleparallel gravity is given. Its constraint algebra has the same structure a…
The observer's frame is the more elementary description of the gravitational field than the metric. The most general covariant, even-parity quadratic form for the frame field in arbitrary dimension generalises the New General Relativity by nine functions of the d'Alembertian operator. The degrees of freedom are clarified by a covariant derivation of the propagator. The consistent and viable models can incorporate an ultra-violet completion of the gravity theory, an additional polarisation of the gravitational wave, and the dynamics of a magnetic scalar potential.
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.
In this article we explore local Lorentz transformations in theories of gravity based on the teleparallel formalism. For the teleparallel equivalent of general relativity (TEGR), the spin connection plays no role in the equations of motion, and therefore it is possible to simply put it equal to zero with no change in physical quantities, and then the theory is formulated purely in terms of the tetrad field which can be freely chosen in any way. In nonlinear modifications of TEGR, this is a more intricate issue, and vanishing spin connection is then the Weitzenböck gauge choice which imposes re…
We present Hamilton's equations for the teleparallel equivalent of general relativity (TEGR), which is a reformulation of general relativity based on a curvatureless, metric compatible, and torsionful connection. For this, we consider the Hamiltonian for TEGR obtained through the vector, antisymmetric, symmetric and trace-free, and trace irreducible decomposition of the phase space variables. We present the Hamiltonian for TEGR in the covariant formalism for the first time in the literature, by considering a spin connection depending on Lorentz matrices. We introduce the mathematical formalism…
We derive a new constraint algebra for a Hamiltonian formulation of the Teleparallel Equivalent of General Relativity treated as a theory of cotetrad fields on a spacetime. The algebra turns out to be closed.
The properties of the gravitational energy-momentum 3-form and of the superpotential 2-form are discussed in the covariant teleparallel framework, where the Weitzenböck connection represents inertial effects related to the choice of the frame. Due to its odd asymptotic behavior, the contribution of the inertial effects often yields unphysical (divergent or trivial) results for the total energy of the system. However, in the covariant teleparallel approach, the energy is always finite and nontrivial. The teleparallel connection plays a role of a regularizing tool which subtracts the inertial ef…
A new expression for the spin connection of teleparallel gravity is proposed, given by minus the contorsion tensor plus a zero connection. The corresponding minimal coupling is covariant under local Lorentz transformation, and equivalent to the minimal coupling prescription of general relativity. With this coupling prescription, therefore, teleparallel gravity turns out to be fully equivalent to general relativity, even in the presence of spinor fields.
We establish the Hamiltonian formulation of the teleparallel equivalent of general relativity, without fixing the time gauge condition, by rigorously performing the Legendre transform. The time gauge condition, previously considered, restricts the teleparallel geometry to the three-dimensional spacelike hypersurface. Geometrically, the teleparallel geometry is now extended to the four-dimensional space-time. The resulting Hamiltonian formulation is different from the standard ADM formulation in many aspects, the main one being that the dynamics is now governed by the Hamiltonian constraint H_0…
The Hamiltonian formulation of general relativity on a null surface is established in the teleparallel geometry. No particular gauge conditons on the tetrads are imposed, such as the time gauge condition. By means of a 3+1 decomposition the resulting Hamiltonian arises as a completely constrained system. However, it is structurally different from the the standard Arnowitt-Deser-Misner (ADM) type formulation. In this geometrical framework the basic field quantities are tetrads that transform under the global SO(3,1) and the torsion tensor.
We develop a symmetric teleparallel gravity model in a space-time with only the non-metricity is nonzero, in terms of a Lagrangian quadratic in the non-metricity tensor. We present a detailed discussion of the variations that may be used for any gravitational formulation. We seek Schwarzschild-type solutions because of its observational significance and obtain a class of solutions that includes Schwarzschild-type, Schwarzschild-de Sitter-type and Reissner-Nordström-type solutions for certain values of the parameters. We also discuss the physical relevance of these solutions.
There are at least two ways to encode gravity into geometry: Einstein's general theory of relativity (GR) for the metric tensor, and teleparallel gravity, where torsion as opposed to curvature encodes the dynamics of the gravitational degrees of freedom. The main purpose of the paper is to explore the relation between loop gravity and teleparallel gravity. We argue that these two formulations of gravity are related to two different discretizations of the Einstein--Cartan action, which were studied recently in the literature. The first discretization leads to the \emph{loop gravity} kinematical…
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…
The Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR) is developed from an ordinary second-order Lagrangian, which is written as a quadratic form of the coefficients of anholonomy of the orthonormal frames (vielbeins). We analyze the structure of eigenvalues of the multi-index matrix entering the (linear) relation between canonical velocities and momenta to obtain the set of primary constraints. The canonical Hamiltonian is then built with the Moore-Penrose pseudo-inverse of that matrix. The set of constraints, including the subsequent secondary constraints, c…
We consider the most general class of teleparallel gravitational {}{}theories quadratic in the torsion tensor, in three space-time dimensions, and carry out a detailed investigation of its Hamiltonian formulation in Schwinger's time gauge. This general class is given by a family of three-parameter theories. A consistent implementation of the Legendre transform reduces the original theory to a one-parameter family of theories. By calculating Poisson brackets we show explicitly that the constraints of the theory constitute a first-class set. Therefore the resulting theory is well defined with re…
In order to discuss the well-posed initial value formulation of the teleparallel gravity and apply it to numerical relativity a symmetric hyperbolic system in the self-dual teleparallel gravity which is equivalent to the Ashtekar formulation is posed. This system is different from the ones in other works by that the reality condition of the spatial metric is included in the symmetric hyperbolicity and then is no longer an independent condition. In addition the constraint equations of this system are rather simpler than the ones in other works.
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 …
General relativity can be presented in terms of other geometries besides Riemannian. In particular, teleparallel geometry (i.e., curvature vanishes) has some advantages, especially concerning energy-momentum localization and its ``translational gauge theory'' nature. The standard version is metric compatible, with torsion representing the gravitational ``force''. However there are many other possibilities. Here we focus on an interesting alternate extreme: curvature and torsion vanish but the nonmetricity $\nabla g$ does not---it carries the ``gravitational force''. This {\it symmetric telepar…
We give a correct tensor proof of the positive energy problem for the case including momentum on basis of conditions of existence of the two-to-one correspondence between the Sen-Witten spinor field and the Sen-Witten orthonormal frame. These conditions were obtained in our previous publications, but true significance of our works was not estimated properly by G.Y.Chee, and these were not correct quoted in his publication. On other hand, the main result of our work is key argument in favour of geometrical nature of the Sen-Witten spinor field.
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.
Hyperbolicity analysis of the linearised 3+1 formulation in the Teleparallel Equivalent of General Relativity
Center for Physical Sciences and Technology (FTMC)（物理科学和技术中心）；Laboratory of Theoretical Physics, Institute of Physics, University of Tartu（塔尔图大学物理研究所理论物理实验室）
We present an extension to the gauge-invariant formulation of the parameterized post-Newtonian (PPN) formalism, which allows its application to symmetric teleparallel gravity theories. In its original formulation, the gauge-invariant PPN formalism makes use of a gauge-invariant perturbative expansion of tensor fields; however, one of the fundamental gravitational field variables in symmetric teleparallel gravity theories is a flat, torsion-free connection, hence not a tensor field. Since connections transform differently from tensor fields under diffeomorphisms, we introduce here an adapted no…
We apply Dirac's Hamiltonian approach to study the canonical structure of the teleparallel form of general relativity without matter fields. It is shown, without any gauge fixing, that the Hamiltonian has the generalized Dirac-ADM form, and constraints satisfy all the consistency requirements. The set of constraints involves some extra first class constraints, which are used to find additional gauge symmetries and clarify the gauge structure of the theory.
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 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.
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.
Symmetric teleparallel gravity is shown to be integrable in the presence of boundaries, given the consistent implementation of constraints in the covariant phase space formalism.
A connection-independent formulation of general relativity is presented, in which the dynamics does not depend on the choice of connection. The gravity action in this formulation includes one additional scalar term in addition to the Einstein-Hilbert action. No conditions on the connection are imposed. Nevertheless, this formulation yields the Einstein equations, without adding the Gibbons-Hawking-York term even when a manifold has a boundary. Furthermore, this formulation yields a unified description of general relativity, teleparallel gravity, and symmetric teleparallel gravity.
A theory of cotetrad fields on a four-dimensional manifold is considered. Its configuration space coincides with that of the Teleparallel Equivalent of General Relativity but its dynamics is much simpler. We carry out the Legendre transformation and derive a Hamiltonian and a constraint algebra.
In the context of the teleparallel equivalent of general relativity we establish the Hamiltonian formulation of the unimodular theory of gravity. Here we do not carry out the usual $3+1$ decomposition of the field quantities in terms of the lapse and shift functions, as in the ADM formalism. The corresponding Lagrange multiplier is the timelike component of the tetrad field. The dynamics is determined by the Hamiltonian constraint ${\cal H}'_0$ and a set of primary constraints. The constraints are first class and satisfy an algebra that is similar to the algebra of the Poincaré group.
This paper elaborates the problem of energy-momentum in the framework of teleparallel equivalent of General Relativity. For this purpose, we consider energy-momentum prescription derived from the integral form of the constraint equations developed in the Hamiltonian formulation of the teleparallel equivalent of General Relativity. We use this technique to investigate energy-momentum of stationary axisymmetric Einstein-Maxwell solutions and cosmic string spacetimes. The angular momentum, gravitational and matter energy-momentum fluxes of these spacetimes are also evaluated. It is concluded that…
We present a simplified expression for the gravitational angular momentum in the framework of the teleparallel equivalent of general relativity (TEGR). The expression arises from the constraints equations of the Hamiltonian formulation of the TEGR. We apply this expression to the calculation of the angular momentum of plane-fronted gravitational waves in an arbitrary three-dimensional volume $V$ of space and compare the results with those obtained for linearised gravitational waves.
In the context of a gauge theory for the translation group, a conserved energy-momentum gauge current for the gravitational field is obtained. It is a true spacetime and gauge tensor, and transforms covariantly under global Lorentz transformations. By rewriting the gauge gravitational field equation in a purely spacetime form, it becomes the teleparallel equivalent of Einstein's equation, and the gauge current reduces to the Møller's canonical energy-momentum density of the gravitational field.
We study the Teleparallel Equivalent of General Relativity (TEGR) with Lagrangian that includes the flat (inertial) spin connection and that is evidently invariant with respect to local Lorentz rotations. Applying directly the Noether theorem, we construct new expressions for conserved currents and related superpotentials. They are covariant both under coordinate transformations and local Lorentz rotations, and allow us to construct well defined conserved charges, unlike earlier approaches. The advantage is achieved by an explicit presence of a displacement vector in the new expressions that c…
We carry out the Hamiltonian formulation of the three- dimensional gravitational teleparallelism without imposing the time gauge condition, by rigorously performing the Legendre transform. Definition of the gravitational angular momentum arises by suitably interpreting the integral form of the constraint equation Gama^ik=0 as an angular momentum equation. The gravitational angular momentum is evaluated for the gravitational field of a rotating BTZ black hole.
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…
We present a new Hamiltonian formulation of the Teleparallel Equivalent of General Relativity (TEGR) meant to serve as the departure point for canonical quantization of the theory. TEGR is considered here as a theory of a cotetrad field on a spacetime. The Hamiltonian formulation is derived by means of an ADM-like 3+1 decomposition of the field and without any gauge fixing. A complete set of constraints on the phase space and their algebra are presented. The formulation is described in terms of differential forms.
We present the Lagrangian and Hamiltonian formulations of a theory for spin 2 fields. The construction is developed in flat space-time. The construction in curved space-time is conceptually straightforward, although it is not unique. The theory is based on a symmetric tensor $S_{μν}$, contains two degrees of freedom of radiation, is motivated by the teleparallel formulation of general relativity, and displays a certain resemblance with Maxwell's theory for the electromagnetic field.
An axiomatization of the so-called Teleparallel Equivalent to General Relativity is presented. A set of formal and semantic postulates are elaborated from where the physical meaning of various key concepts of the theory are clarified. These concepts include those of inertia, Lorentz and diffeomorphism invariance, and reference frame. It is shown that Teleparallel Gravity admits a wider representation of space-time than General Relativity, allowing to define properties of the gravitational field such as energy and momentum that are usually considered problematic. In this sense, although the dyn…
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…
This paper is devoted to the evaluation of the energy-momentum density components for the Friedmann models. For this purpose, we have used M$ø$ller's pseudotensor prescription in General Relativity and a certain energy-momentum density developed from his teleparallel formulation. It is shown that the energy density of the closed Friedmann universe vanishes on the spherical shell at the radius $ρ=2\sqrt{3}$. This coincides with the earlier results available in the literature. We also discuss the energy of the flat and open models. A comparison shows a partial consistency between the M$ø$ller's …
General Relativity and its higher derivative extensions have symmetric teleparallel reformulations in terms of the non-metricity tensor within a torsion-free and flat geometry. These notes present a derivation of the exact propagator for the most general infinite-derivative, even-parity and generally covariant theory in the symmetric teleparallel spacetime. The action made up of the non-metricity tensor and its contractions is decomposed into terms involving the metric and a gauge vector field and is found to complement the previously known non-local ghost- and singularity-free theories.
We show that the well-known problem of frame dependence and violation of local Lorentz invariance in the usual formulation of $f(T)$ gravity is a consequence of neglecting the role of spin connection. We re-formulate $f(T)$ gravity starting, instead of the "pure-tetrad" teleparallel gravity, from the covariant teleparallel gravity, using both the tetrad and the spin connection as dynamical variables, resulting in the fully covariant, consistent, and frame-independent, version of $f(T)$ gravity, which does not suffer from the notorious problems of the usual, pure-tetrad, $f(T)$ theory. We prese…
Symmetric teleparallel gravity (STG) offers an interesting third geometric interpretation of gravitation besides its formulation in terms of a spacetime metric and Levi-Civita connection or its teleparallel formulation. It describes gravity through a connection which is not metric compatible, however is torsion and curvature free. We investigate the propagation velocity of the gravitational waves around Minkowski spacetime and their potential polarizations in a general class of STG theories, the so-called "newer general relativity" class. It is defined in terms of the most general Lagrangian t…
We calculate energy and momentum of a spherically symmetric dilaton frame using the gravitational energy-momentum 3-form within the tetrad formulation of general relativity (GR). The frame we use is characterized by an arbitrary function $Υ$ with the help of which all the previously found solutions can be reproduced. We show how the effect of inertia {\it (which is mainly reproduced from $Υ$)} makes the total energy and momentum always different from the well known result when we use the Riemannian connection ${{\widetilde Γ}_α}^β$. On the other hand, when use is made of the covariant formulat…
