The Teleparallel Equivalent of General Relativity and the Gravitational Centre of Mass
J. W. Maluf
备注：
22 pages, no figures, the title has been changed, references added, published in Universe (100 Years of Chronogeometrodynamics: the Status of the Einstein's Theory of Gravitation in Its Centennial Year)
We present a brief review of the teleparallel equivalent of general relativity and analyse the expression for the centre of mass density of the gravitational field. This expression has not been sufficiently discussed in the literature. One motivation for the present analysis is the investigation of the localization of dark energy in the three-dimensional space, induced by a cosmological constant in a simple Schwarzschild-de Sitter space-time. We also investigate the gravitational centre of mass density in a particular model of dark matter, in the space-time of a point massive particle and in an arbitrary space-time with axial symmetry. The results are plausible, and lead to the notion of gravitational centre of mass (COM) distribution function.
In this paper we review the canonical analysis of constrained systems and apply it in the case of teleparallel equivalent of general relativity and teleparallel gravity. For each of them we find all the Poisson brackets, generators of gauge symmetries, as well as the number of degrees of freedom.
After reminder some facts concerning general relativity ({\bf GR}) we pass to teleparallel gravity. We are confining the special model of the teleparallel gravity, which is popular recently, called {\it the teleparallel equivalent of general relativity} ({\bf TEGR}). We are finishing with conclusion and some general remarks.
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.
We calculated the energy and momentum densities of stiff fluid solutions, using Einstein, Bergmann-Thomson and Landau-Lifshitz energy-momentum complexes, in both general relativity and teleparallel gravity. In our analysis we get different results comparing the aforementioned complexes with each other when calculated in the same gravitational theory, either this is in general relativity and teleparallel gravity. However, interestingly enough, each complex's value is the same either in general relativity or teleparallel gravity. Our results sustain that (i) general relativity or teleparallel gr…
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.
We continue our study in 4-dimension to derive non-charged and charged (Anti)-de Sitter black hole solutions in conformal teleparallel equivalent of general relativity. The non-charged and charged equations of motion are applied to a spherically symmetric tetrad and the non-linear differential equations are derived. It is shown that the output solutions of the two cases are identical to those obtained in teleparallel equivalent theory to general relativity. As a result, it is found that in the conformal teleparallel equivalent theory to general relativity, the scalar field cannot influence on …
We derive two new retarded solutions in the teleparallel theory equivalent to general relativity (TEGR). One of these solutions gives a divergent energy. Therefore, we used the regularized expression of the gravitational energy-momentum tensor, which is a coordinate dependent. A detailed analysis of the loss of the mass of Bondi space-time is carried out using the flux of the gravitational energy-momentum.
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.
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…
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…
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…
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 …
Considering the vacuum as characterized by the presence of only the gravitational field, we show that the vacuum energy density of the de Sitter space, in the realm of the teleparallel equivalent of general relativity, can acquire arbitrarily high values. This feature is expected to hold in the consideration of realistic cosmological models, and may possibly provide a simple explanation to the cosmological constant problem.
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 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…
We construct a notion of teleparallelization for Newton-Cartan theory, and show that the teleparallel equivalent of this theory is Newtonian gravity; furthermore, we show that this result is consistent with teleparallelization in general relativity, and can be obtained by null-reducing the teleparallel equivalent of a five-dimensional gravitational wave solution. This work thus strengthens substantially the connections between four theories: Newton-Cartan theory, Newtonian gravitation theory, general relativity, and teleparallel gravity.
This paper has been removed by arXiv administrators because it plagiarizes gr-qc/0603075, 0706.3245, gr-qc/0403097, gr-qc/0404108, gr-qc/0303034, hep-th/0206052, and others. This paper has excessive overlap with the following papers also written by the authors or their collaborators: 0704.0525 and others.
We apply the energy-momentum tensor to calculate energy, momentum and angular-momentum of two different tetrad fields. This tensor is coordinate independent of the gravitational field established in the Hamiltonian structure of the teleparallel equivalent of general relativity (TEGR). The spacetime of these tetrad fields is the charged dilaton. Our results show that the energy associated with one of these tetrad fields is consistent, while the other one does not show this consistency. Therefore, we use the regularized expression of the gravitational energy-momentum tensor of the TEGR. We inves…
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…
This work generalizes the treatment of flat spin connections in the teleparallel equivalent of general relativity. It is shown that a general flat spin connection form a subspace in the affine space of spin connections which is dynamically decoupled from the tetrad and the matter fields. A translation in the affine subspace introduces a torsion term without changing the tetrad. Instead, the change in the torsion is related to the introduction of a global acceleration field term that introduces Lorentz inertial effects in the reference frame. The dynamics of the gravitationally coupled matter f…
The stationary cosmological model without closed timelike curves of Gödel type is obtained for the ideal dust matter source within the framework of the teleparallel gravity. For a specific choice of the teleparallel gravity parameters, this solution reproduces the causality violating stationary Gödel solution in general relativity, in accordance with the teleparallel equivalent of general relativity. The relation between the axial-vector torsion and the cosmic vorticity is clarified.
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.
An exact charged solution with axial symmetry is obtained in the teleparallel equivalent of general relativity (TEGR). The associated metric has the structure function $G(ξ)=1-ξ^2-2mAξ^3-q^2A^2ξ^4$. The fourth order nature of the structure function can make calculations cumbersome. Using a coordinate transformation we get a tetrad whose metric has the structure function in a factorisable form $(1-ξ^2)(1+r_{+}Aξ)(1+r_{-}Aξ)$ with $r_{\pm}$ as the horizons of Reissner-Nordstr$\ddot{o}$m space-time. This new form has the advantage that its roots are now trivial to write down. Then, we study the s…
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.
The fundamentals of the teleparallel equivalent of general relativity are presented, and its main properties described. In particular, the field equations, the definition of an energy--momentum density for the gravitational field, the teleparallel version of the equivalence principle, and the dynamical role played by torsion as compared to the corresponding role played by curvature in general relativity, are discussed in some details.
In general relativity (GR), the metric tensor of spacetime is essential since it represents the gravitational potential. In other gauge theories (such as electromagnetism), the so-called premetric approach succeeds in separating the purely topological field equation from the metric-dependent constitutive law. We show here that GR allows for a premetric formulation, too. For this purpose, we apply the teleparallel approach of gravity, which represents GR as a gauge theory based on the translation group. We formulate the metric-free topological field equation and a general linear constitutive la…
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…
It has been shown recently that within the framework of the teleparallel equivalent of general relativity (TEGR) it is possible to define the energy density of the gravitational field. The TEGR amounts to an alternative formulation of Einstein's general relativity, not to an alternative gravity theory. The localizability of the gravitational energy has been investigated in a number of space-times with distinct topologies, and the outcome of these analises agree with previously known results regarding the exact expression of the gravitational energy, and/or with the specific properties of the s…
The energy-momentum tensor, which is coordinate independent, is used to calculate energy, momentum and angular-momentum of two different tetrad fields. Although, the two tetrad fields reproduce the same space-time their energies are different. Therefore, a regularized expression of the gravitational energy-momentum tensor of the teleparallel equivalent of general relativity, (TEGR), is used to make the energies of the two tetrad fields equal. The definition of the gravitational energy-momentum is used to investigate the energy within the external event horizon. The components of angular-moment…
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 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…
The conserved quantities, that are, gravitational energy-momentum and its relevant quantities are investigated for cylindrical and spherical gravitational waves in the framework of teleparallel equivalent of General Relativity using the Hamiltonian approach. For both cylindrical and spherical gravitational waves, we obtain definite energy and constant momentum. The constant momentum shows consistency with the results available in General Relativity and teleparallel gravity. The angular momentum for cylindrical and spherical gravitational waves also turn out to be constant. Further, we evaluate…
Spaniol and Andrade introduced grvitoelectromagnetism in TEGR by considering superpotentials, times the determinant of tetrads, as the gravitoelectromagnetic fields. However, since this defined gravitoelectromagnetic field strength does not give rise to a complete set of Maxwell-like equations, we propose an alternative definition of the gravitoelectromagnetic field strength: instead of superpotentials, torsions are taken as the gravitoelectromagnetic field strengths. Based on this new proposal we are able to derive a complete set of Maxwell-like equations. We then apply them to obtain explici…
In this Letter we consider a general quadratic parity-preserving theory for a general flat connection. Imposing a local symmetry under the general linear group singles out the general teleparallel equivalent of General Relativity carrying both torsion and non-metricity. We provide a detailed discussion on the teleparallel equivalents of General Relativity and how the two known equivalents, formulated on Weitzenböck and symmetric teleparallel geometries respectively, can be interpreted as two gauge-fixed versions of the general teleparallel equivalent. We then explore the viability of the gener…
We consider Bondi's radiating metric in the context of the teleparallel equivalent of general relativity (TEGR). This metric describes the asymptotic form of a radiating solution of Einstein's equations. The total gravitational energy for this solution can be calculated by means of pseudo-tensors in the static case. In the nonstatic case, Bondi defines the {\it mass aspect} $m(u)$, which describes the mass of an isolated system. In this paper we express Bondi's solution in asymptotically spherical 3+1 coordinates, not in radiation coordinates, and obtain Bondi's energy in the static limit by m…
We study 4-dimensional non-charged and charged spherically symmetric spacetimes in conformal teleparallel equivalent of general relativity. We apply the gravitational field equations in non-charged and charged spacetimes to the diagonal and non-diagonal vierbeins and derive their sets of the non-linear differential equations. It is demonstrated that the Schwarzschild, for the non-charged case, and the Reissner-Nordström, for the charged case, are the only black hole solutions for the spherically symmetric case in the framework of the conformal teleparallel equivalent of general relativity theo…
The Noether charge associated to diffeomorphism invariance in teleparallel gravity is derived. It is shown that the latter yields the ADM mass of an asymptotically flat spacetime. The black hole entropy is then investigated based on Wald's prescription that relies on the Noether charge. It is shown that, like in general relativity, the surface gravity can be factored out from such a charge. Consequently, the similarity with the first law of thermodynamics implied by such an approach in general relativity does show up also in teleparallel gravity. It is found that, based on the expression of th…
The recently proposed Nieh-Yan modified teleparallel gravity is a parity-violating gravity model that modifies the general relativity equivalent teleparallel gravity by a Nieh-Yan term. This model is healthy and simple in form. In this paper, we consider the application of this model to the Solar System and investigate its slow-motion and weak-field approximation in terms of the parametrized post-Newtonian formalism. We find that all the post-Newtonian parameters of the model are the same as those of general relativity, which makes the model compatible with the Solar System experiments.
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…
Even though it is not possible to differentiate General Relativity from Teleparallel Gravity using classical experiments, it could be possible to discriminate between them by quantum gravitational effects. These effects have motivated the introduction of nonlocal deformations of General Relativity, and similar effects are also expected to occur in Teleparallel Gravity. Here, we study nonlocal deformations of Teleparallel Gravity along with its cosmological solutions. We observe that Nonlocal Teleparallel Gravity (like nonlocal General Relativity) is consistent with the present cosmological dat…
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…
Teleparallel gravity, a gauge theory for the translation group, turns up as fully equivalent to Einstein's general relativity. In spite of this equivalence, it provides a whole new insight into gravitation. It breaks several paradigms related to the geometric approach of general relativity, and introduces new concepts in the description of the gravitational interaction. The purpose of this chapter is to explore some of these concepts, as well as discuss possible consequences for gravitation, mainly those that could be relevant for the quantization of the gravitational field.
We investigate propagation of gravitational waves in the most general teleparallel gravity model with second order field equations around the Minkowski background. We argue that in this case the most general Lagrangian at the first order of perturbations is given by the linear combination of quadratic invariants and hence coincides with the well-known new general relativity model. We derive the linearized field equations and analyze them using the principal polynomial and the Newman-Penrose formalism. We demonstrate that all gravitational wave modes propagate at the speed of light and there ar…
Using the Dirac procedure to treat constraints dynamical sistems applied to gravitation, as described in the context of Teleparallel Equivalent of General Relativity (TEGR), we investigate, from the first class constraints, the gauge transformations in the fundamental field: the components of tetrads. We have shown that there is no an isotropy in physical space with respect to gauge transformations, i.e., given an arbitrary gravitational field, coming from a gauge transformation in the internal space, physical space reacts differently in the spatial and temporal components. By making an approp…
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…
A review of the teleparallel equivalent of general relativity is presented. It is emphasized that general relativity may be formulated in terms of the tetrad fields and of the torsion tensor, and that this geometrical formulation leads to alternative insights into the theory. The equivalence with the standard formulation in terms of the metric and curvature tensors takes place at the level of field equations. The review starts with a brief account of the history of teleparallel theories of gravity. Then the ordinary interpretation of the tetrad fields as reference frames adapted to arbitrary o…
The main purpose of this paper is to explicitly verify the consistency of the energy-momentum and angular momentum tensor of the gravitational field established in the Hamiltonian structure of the Teleparallel Equivalent of General Relativity (TEGR). In order to reach these objectives, we obtained the total energy and angular momentum (matter plus gravitational field) of the closed universe of the Friedmann-Lemaitre-Robertson-Walker (FLRW). The result is compared with those obtained from the pseudotensors of Einstein and Landau-Lifshitz. We also applied the field equations (TEGR) in an expandi…
The purpose of this paper is to illustrate the problem of energy and momentum distributions of Van Stockum space-time within the framework of two different theories of gravity, general relativity and teleparallel gravity. We have shown that for all homogeneous space-times with metric components $g_{μν}$ being functions of time variable, $t$, alone and independent of space variables the total gravitational energy for any finite volume is identically zero. By working with general relativity, we have calculated the energy-momentum density for Van Stockum space-time using double index complexes an…
