Dislocations and internal length measurement in continuized crystals. II. Closed teleparallelism

The Burgers field responsible for dislocations in a continuized crystal is represented by the torsion tensor of a teleparallel connection, metric with respect to the internal length measurement metric tensor. Lattice lines in a continuized dislocated monocrystal are represented by geodesics of the teleparallel connection, and the internal length measurement along these geodesics is analyzed. The closed teleparallelism responsible for uniformly dense distributions of dislocations is discussed, and equations describing slip surfaces for such distributions of dislocations are formulated. The Galilei-like character of the geometry describing uniformly dense distributions of dislocations is pointed out.     

Scalar torsion and a new symmetry of general relativity

We reformulate the general theory of relativity in the language of Riemann–Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to Riemann–Cartan space-times with scalar torsion and obtain the conservation laws for auto-parallel motions in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.     

Dynamical system analysis for an interacting scalar-torsion theory

In this paper, the dynamical system method has been used to study a new model of teleparallel gravity in which a scalar field non-minimally coupled to a boundary term given by the divergence of the torsion vector. An interaction between dark energy and dark matter has also been considered in the model. It has been explored whether there exist late-time accelerated scaling attractors with the ratio of dark matter and dark energy density parameters of the order one. By assuming a non-local form of interaction, a number of scaling solutions were found in the present model. It can therefore be concluded that the present model can alleviate the coincidence problem in cosmology.     

Parameterized post-Newtonian approximation in a teleparallel model of dark energy with a boundary term

We study the parameterized post-Newtonian approximation in teleparallel model of gravity with a scalar field. The scalar field is non-minimally coupled to the scalar torsion as well as to the boundary term introduced in Bahamonde and Wright (Phys Rev D 92:084034 arXiv:1508.06580v4 [gr-qc], 2015). We show that, in contrast to the case where the scalar field is only coupled to the scalar torsion, the presence of the new coupling affects the parameterized post-Newtonian parameters. These parameters for different situations are obtained and discussed.     

On the Møller Energy Associated with Black Holes

In this paper, we consider both Einstein's theory of general relativity and the teleparallel gravity (the tetrad theory of gravitation) analogs of the energy-momentum definition of Møller in order to explicitly evaluate the energy distribution (due to matter and fields including gravity) associated with a general black hole model which includes several well-known black holes. To calculate the special cases of energy distribution, here we consider eight different types of black hole models such as anti-de Sitter Cmetric with spherical topology, charged regular black hole, conformal scalar dyon black hole, dyadosphere of a charged black hole, regular black hole, charged topological black hole, charged massless black hole with a scalar field, and the Schwarzschild-de Sitter space-time. Our teleparallel gravitational result is also independent of the teleparallel dimensionless coupling constant, which means that it is valid not only in teleparallel equivalent of general relativity but also in any teleparallel model. This paper also sustains (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime and (b) the viewpoint of Lessner that the Møller energy-momentum complex is the powerful concept to calculate energy distribution in a given space-time.     

Inequivalence of Jordan and Einstein Frame: What Is the Low Energy Gravity in String Theory?

It is well-known that any scalar can be promoted to a Jordan-Brans-Dicke type scalar coupling to the Einstein-Hilbert term through a field dependent Weyl transformation of the metric. The Weyl rescaling also transforms mass terms into coupling constants between matter and the scalar. It is pointed out that there exists a distinguished metric where all scalars decouple from an arbitrary fiducial fermion, e.g. the nucleon. If bound states of this fermion are used to define distances and to probe the interior of the forward light cone, it seems reasonable to say that the metric in that particular frame defines the local geometry of space-time at low energies, as probed by experimental gravity and cosmology.     

Dynamical instability of shear-free collapsing star in extended teleparallel gravity

We study the spherically symmetric collapsing star in terms of dynamical instability. We take the framework of extended teleparallel gravity with a non-diagonal tetrad, a power-law form of the model presenting torsion and a matter distribution as a non-dissipative anisotropic fluid. The vanishing shear scalar condition is adopted to gain insight in a collapsing star. We apply a first order linear perturbation scheme to the metric, the matter, and f(T) functions. The dynamical equations are formulated under this perturbation scheme to develop collapsing equation for finding dynamical instability limits in two regimes, such as the Newtonian and the post-Newtonian regime. We obtain a constraint-free solution of a perturbed time dependent part with the help of a vanishing shear scalar. The adiabatic index exhibits the instability ranges through the second dynamical equation which depend on physical quantities such as the density, the pressure components, the perturbed parts of the symmetry of the star, etc. We also develop some constraints on the positivity of these quantities and obtain instability ranges to satisfy the dynamical instability condition.     

Gravitation as Anholonomy

A gravitational field can be seen as the anholonomy of the tetrad fields. This is more explicit in the teleparallel approach, in which the gravitational field-strength is the torsion of the ensuing Weitzenböck connection. In a tetrad frame, that torsion is just the anholonomy of that frame. The infinitely many tetrad fields taking the Lorentz metric into a given Riemannian metric differ by point-dependent Lorentz transformations. Inertial frames constitute a smaller infinity of them, differing by fixed-point Lorentz transformations. Holonomic tetrads take the Lorentz metric into itself, and correspond to Minkowski flat spacetime. An accelerated frame is necessarily anholonomic and sees the electromagnetic field strength with an additional term.     

Gravitational leptogenesis in teleparallel and symmetric teleparallel gravities

In this study, we investigate the possibilities of generating baryon number asymmetry under thermal equilibrium within the frameworks of teleparallel and symmetric teleparallel gravities. Through the derivative couplings of the torsion scalar and the non-metricity scalar to baryons, baryon number asymmetry is produced in the radiation dominated epoch. For gravitational baryogenesis mechanisms in these two frameworks, the produced baryon-to-entropy ratio is too small to be consistent with observations. However, the gravitational leptogenesis models within both frameworks have the potential to explain the observed baryon-antibaryon asymmetry.     

Semi-Teleparallel Theories of Gravitation

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.     

The teleparallel equivalent of general relativity

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 observers in space–time is discussed, and the tensor of inertial accelerations on frames is obtained. It is shown that the Lagrangian and Hamiltonian field equations allow us to define the energy, momentum and angular momentum of the gravitational field, as surface integrals of the field quantities. In the phase space of the theory, these quantities satisfy the algebra of the Poincaré group.     

An axially symmetric scalar field and teleparallelism

An axially symmetric scalar field is considered in teleparallel gravity. We calculate, respectively, the tensor, the vector and the axial-vector parts of torsion and energy, momentum and angular momentum in the ASSF. We find the vector parts are in the radial and \(\hat{e}_{\theta}\) directions, the axial-vector, momentum and angular momentum vanish identically, but the energy distribution is different from zero. The vanishing axial-vector part of torsion gives us the result that there occurs no deviation in the spherical symmetry of the spacetime. Consequently, there exists no inertia field with respect to a Dirac particle, and the spin vector of a Dirac particle becomes constant. The result for the energy is the same as obtained by Radinschi. Next, this work also (a) supports the viewpoint of Lessner that the Møller energy-momentum complex is a powerful concept for the energy-momentum, (b) sustains the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime, and (c) supports the hypothesis by Cooperstock that the energy is confined to the region of non-vanishing energy-momentum tensor of the matter and all non-gravitational fields.     

Charged Noncommutative Wormhole Solutions via Power-Law f(T) Models

In this paper, we explore static spherically symmetric charged wormhole solutions in extended teleparallel gravity taking power-law f(T) models. We consider noncommutative geometry under Lorentzian distribution. In order to obtain matter components, we develop field equations using effective energy-momentum tensor for non-diagonal tetrad. We explore solutions by considering various viable power-law f(T) models, which also include teleparallel gravity case. The violation of energy conditions obtain by exotic matter to form wormhole solutions in teleparallel case while, physical acceptable wormhole solutions exist for charged noncommutative wormhole solutions for some cases of power-law models. The effective energy-momentum tensor and charge are responsible for the violation of the energy conditions. Also, we check the equilibrium condition for these solutions. The equilibrium condition meets for the teleparallel case and some power-law solutions while remaining solutions are either in less equilibrium or in disequilibrium situation.     

Variational principle for general relativity with torsion and non-metricity

The variation of the field action with the scalar Lagrangian density $\text{?g}\text{R}$ is calculated with respect to the metric, torsion and non-metricity. When the non-metricity vanishes, the variation reduces to the U₄ theory of gravitation when a relativistic matter Lagrangian with minimal coupling is included.     

Energy in the Schwarzschild-de Sitter Spacetime

The energy (due to matter and fields including gravitation) of the Schwarzschild-de Sitter spacetime is investigated by using the Møller energy-momentum definition in both general relativity and teleparallel gravity. We found the same energy distribution for a given metric in both of these different gravitation theories. It is also independent of the teleparallel dimensionless coupling constant, which means that it is valid in any teleparallel model. Our results sustain that (a) the importance of the energy-momentum definitions in the evaluation of the energy distribution of a given spacetime and (b) the viewpoint of Lessner that the Møller energy-momentum complex is a powerfifi concept of energy and momentum.     

A comment on the geometry of some scalar-tensor theories

We show that the scalar field in scalar-tensor theories such as the Jordan-Brans-Dicke theory has an interpretation as a potential for the torsion in a Riemannian manifold. The relation is similar to that of the metric to the connection.     

On the Interaction of Fermion and Boson Fields with the Gravitational Field

We point out that the gravitational field taken by itself cannot be considered as a gauge field. Only an affinity and not a metric can serve as a gauge field. Originally, metric and affinity are completely independent of each other. This fact allows in a natural way to formulate a restricted principle of relativity, according to which only fermion fields may show that there exist a priori distinguished frames of reference. Furthermore, we can couple the gravitational field to boson and fermion fields such that the flat metric or tetrads orthonormalized with respect to this flat metric appearing in the special relativistic matter Lagrangian, are replaced by a Riemannian metric and tetrads orthonormalized with respect to this metric (principle of most minimal gravitational coupling). This coupling principle is a strong restriction on the existence of independent boson fields. Only scalar and vector fields and their different pseudoquantities are possible as independent fields. Boson fields of higher rank are to be considered as fusions of these (pseudo)scalar and (pseudo)vector fields. Theire field equations follow from those of the (pseudo)scalar and (pseudo)vector fields.     

Stringy differential geometry for double field theory, beyond Riemann

While the fundamental object in Riemannian geometry is a metric, closed string theories call for us to put a two-form gauge field and a scalar dilaton on an equal footing with the metric. Here we propose a novel differential geometry which treats the three objects in a unified manner, manifests not only diffeomorphism and one-form gauge symmetry but also O(D, D) T-duality, and enables us to rewrite the known low energy effective action of them as a single term. We comment that the notion of cosmological constant naturally changes.