Over recent decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f (T) gravity, where f (T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f (T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f (T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, alternative to a cosmological constant, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, for a certain class of f (T) models, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations-or, alternatively, the Big Bang singularity can be avoided at even earlier moments due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f (T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f (T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f (R) gravity, trying to illuminate the subject of which formulation, or combination of formulations, might be more suitable for quantization ventures and cosmological applications.

f(T) teleparallel gravity and cosmology

Over the past decades, the role of torsion in gravity has been extensively investigated along the main direction of bringing gravity closer to its gauge formulation and incorporating spin in a geometric description. Here we review various torsional constructions, from teleparallel, to Einstein-Cartan, and metric-affine gauge theories, resulting in extending torsional gravity in the paradigm of f(T) gravity, where f(T) is an arbitrary function of the torsion scalar. Based on this theory, we further review the corresponding cosmological and astrophysical applications. In particular, we study cosmological solutions arising from f(T) gravity, both at the background and perturbation levels, in different eras along the cosmic expansion. The f(T) gravity construction can provide a theoretical interpretation of the late-time universe acceleration, and it can easily accommodate with the regular thermal expanding history including the radiation and cold dark matter dominated phases. Furthermore, if one traces back to very early times, a sufficiently long period of inflation can be achieved and hence can be investigated by cosmic microwave background observations, or alternatively, the Big Bang singularity can be avoided due to the appearance of non-singular bounces. Various observational constraints, especially the bounds coming from the large-scale structure data in the case of f(T) cosmology, as well as the behavior of gravitational waves, are described in detail. Moreover, the spherically symmetric and black hole solutions of the theory are reviewed. Additionally, we discuss various extensions of the f(T) paradigm. Finally, we consider the relation with other modified gravitational theories, such as those based on curvature, like f(R) gravity, trying to enlighten the subject of which formulation might be more suitable for quantization ventures and cosmological applications.

: The final report covering the research performed under the Air Force Office of Scientific Research by the Dallas relativity group, summarizes information on personnel involved, on the scientific work done, and on the results obtained in the four years of the grant's duration. Topics investigated were in areas of General relativity, cosmology, and electrodynamics.

Exact Solutions of Einstein's Field Equations.

Gravitation and cosmology: Stephen Weinberg. 657 pages, diagrams, 6×9 in. New York, John Wiley & Sons, Inc., 1972. Price $18.95 (approx. £6·20).

In previous work, we undertook to study static and anisotropic content in f(T) theory and obtained new spherically symmetric solutions considering a constant torsion and some particular conditions for the pressure. In this paper, still in the framework of f(T) theory, new spherically symmetric solutions are obtained, first considering the general case of an isotropic fluid and later the anisotropic content case in which the generalized conditions for the matter content are considered such that the energy density, the radial and tangential pressures depend on the algebraic f(T) and its derivative f
 T
 (T). Moreover, we obtain the algebraic function f(T) through the reconstruction method for two cases and also study a polytropic model for the stellar structure.

Static Anisotropic Solutions in f(T) Theory

In this paper we establish the definition of the Lie derivative of a second rank tensor in the context of teleparallel theory of gravity and also extend it for a general tensor of rank p + q. This definition is then used to find Killing vectors of the Einstein universe. It turns out that Killing vectors of the Einstein universe in the teleparallel theory are the same as in general relativity.

Teleparallel killing vectors of the einstein universe

In this paper, we find the energy–momentum distribution of stationary axisymmetric spacetimes in the context of teleparallel theory by using Moller prescription. The metric under consideration is the generalization of the Weyl metrics called the Lewis–Papapetrou metric. The class of stationary axisymmetric solutions of the Einstein field equations has been studied by Galtsov to include the gravitational effect of an external source. Such spacetimes are also astrophysically important as they describe the exterior of a body in equilibrium. The energy density turns out to be nonvanishing and well-defined and the momentum becomes constant except along θ-direction. It is interesting to mention that the results reduce to the already available results for the Weyl metrics when we take ω = 0.

Teleparallel energy momentum distribution of lewis papapetrou spacetimes

This work contains the teleparallel version of the stationary axisymmetric solutions. We obtain the tetrad and the torsion fields representing these solutions. The tensor, vector and axial-vector parts of the torsion tensor are evaluated. It is found that the axial-vector has component only along ρ and z directions. The three possibilities of the axial vector depending on the metric function B are discussed. The vector related with spin has also been evaluated and the corresponding extra Hamiltonian is furnished. Further, we use the teleparallel version of Moller prescription to find the energy–momentum distribution of the solutions. It is interesting to note that (for λ = 1) energy and momentum densities in teleparallel theory are equal to the corresponding quantities in GR plus an additional quantity in each, which may become equal under certain conditions. Finally, we discuss the two special cases of the stationary axisymmetric solutions.

Teleparallel version of the stationary axisymmetric solutions and their energy contents

This paper is devoted to investigating the teleparallel versions of the Friedmann models as well as the Lewis–Papapetrou solution. We obtain the tetrad and the torsion fields for both spacetimes. It is shown that the axial-vector vanishes for the Friedmann models. We discuss the different possibilities for the axial-vector, depending on the arbitrary functions ω and ψ in the Lewis–Papapetrou metric. The vector related to spin has also been evaluated.

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Teleparallel versions of Friedmann and Lewis-Papapetrou spacetimes

This paper is devoted to discuss the energy–momentum for static axially symmetric spacetimes in the framework of teleparallel theory of gravity For this purpose, we use the teleparallel versions of Einstein, Landau–Lifshitz, Bergmann and Moller prescriptions A comparison of the results shows that the energy density is different but the momentum turns out to be constant in each prescription This is exactly similar to the results available in literature using the framework of general relativity It is mentioned here that Moller energy–momentum distribution is independent of the coupling constant λ Finally, we calculate energy–momentum distribution for the Curzon metric, a special case of the above-mentioned spacetime

M. Jamil Amir

Papers

Teleparallel killing vectors of the einstein universe

Teleparallel energy momentum distribution of lewis papapetrou spacetimes

Teleparallel version of the stationary axisymmetric solutions and their energy contents

Teleparallel versions of Friedmann and Lewis-Papapetrou spacetimes

Teleparallel energy–momentum distribution of static axially symmetric spacetimes