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We write the field equations of torsion gravity theories and the N\oe ther identity they obey directly in terms of metric and contorsion tensor components expressed with respect to natural coordinates, i.e. without using vierbien but Lagrange multipliers. Then we obtain explicit solutions of these equations, under specific ans\"atze for the contorsion field, by assuming the metric to be respectively of the Bertotti-Robinson, pp-wave, Friedmann-Lema\^itre-Robertson-Walker or static spherically symmetric type. Among these various solutions we obtain some of them have their contorsion tensor depending on arbitrary functions that didn't influence their geometry. This raises question about the predictability of the theory., Comment: 40 pages, a few more typos corrected

We revisit Nye’s lattice curvature tensor in the light of Cartan’s moving frames. Nye’s definition of lattice curvature is based on the assumption that the dislocated body is stress-free, and therefore, it makes sense only for zero-stress (impotent) dislocation distributions. Motivated by the works of Bilby and others, Nye’s construction is extended to arbitrary dislocation distributions. We provide a material definition of the lattice curvature in the form of a triplet of vectors, that are obtained from the material covariant derivative of the lattice frame along its integral curves. While the dislocation density tensor is related to the torsion tensor associated with the Weitzenbock connection, the lattice curvature is related to the contorsion tensor. We also show that under Nye’s assumption, the material lattice curvature is the pullback of Nye’s curvature tensor via the relaxation map. Moreover, the lattice curvature tensor can be used to express the Riemann curvature of the material manifold in the linearized approximation.

The torsion-free connection is one of the important assumptions in Einstein's General theory of Relativity which has not been verified so far from experimental observation. In this article, the effect of torsion on the on-shell action of radiation fields in curved spacetime is reported in order to show certain consequences and possible ways to probe torsion in curved spacetime. In order to describe non-vanishing torsion, we mostly use contorsion tensor in connection which can be written in the combination of torsion tensor and vice versa. We have discussed how does the field equation coming from the minimization of action with respect to torsion changes the matter part of the on-shell action of radiation through substitution of the spin tensor. We have also studied here the effect of torsion on the scalar and vector radiation field in curved spacetime. It is shown that the presence of torsion leads to extra polarizations in Gravitational waves or radiations other than the usual two polarizations already present in General Relativity. Further, it is also shown that in the presence of torsion, there exists a non-trivial vacuum configuration that is different from the trivial vacuum in the absence of coupling with torsion. It is also found that in a generic theory of vector field, the presence of torsion breaks the U(1) gauge invariance just like the presence of mass shows the violation of gauge invariance. The observational consequences of these results are also discussed.

Working within the recent formalism of Poisson-Riemannian geometry, we completely solve the case of generic spherically symmetric metric and spherically symmetric Poisson-bracket to find a unique answer for the quantum differential calculus, quantum metric and quantum Levi-Civita connection at semiclassical order $O(\lambda)$. Here $\lambda$ is the deformation parameter, plausibly the Planck scale. We find that $r,t,dr,dt$ are all forced to be central, i.e. undeformed at order $\lambda$, while for each value of $r,t$ we are forced to have a fuzzy sphere of radius $r$ with a unique differential calculus which is necessarily nonassociative at order $\lambda^2$. We give the spherically symmetric quantisation of the FLRW cosmology in detail and also recover a previous analysis for the Schwarzschild black hole, now showing that the quantum Ricci tensor for the latter vanishes at order $\lambda$. The quantum Laplace-Beltrami operator for spherically symmetric models turns out to be undeformed at order $\lambda$ while more generally in Poisson-Riemannian geometry we show that it deforms to \[ \square f+{\lambda\over 2}\omega^{\alpha\beta}({\rm Ric}^\gamma{}_\alpha-S^\gamma{}_{;\alpha})(\widehat\nabla_\beta d f)_\gamma + O(\lambda^2)\] in terms of the classical Levi-Civita connection $\widehat\nabla$, the contorsion tensor $S$, the Poisson-bivector $\omega$ and the Ricci curvature of the Poisson-connection that controls the quantum differential structure. The Majid-Ruegg spacetime $[x,t]=\lambda x$ with its standard calculus and unique quantum metric provides an example with nontrivial correction to the Laplacian at order $\lambda$., Comment: 47 pages 1 figure

In a recent paper,1 we missed some components of the contorsion tensor due to the minor error. When we insert the correction and use all the components, the final result becomes more comprehensive. Here we put the corrections and obtain the corresponding results.

We examine a gauge model with a Lagrangian, quadratic in the Riemann-Cardan curvature, without a separate Hilbert Lagrangian. It is argued that torsions must be massive particles, the torsion field does not act on world fields, and the orthogonal components of the contorsion tensor must not vary with variations of the metric. It is shown that a new torsion-metric interaction arises in this case, which generates the gravitational theories of Einstein and Weyl.

We analyse the Einstein-Cartan gravity in its standard form cal-R = R + cal-K^2, where cal-R and R are the Ricci scalar curvatures in the Einstein-Cartan and Einstein gravity, respectively, and cal-K^2 is the quadratic contribution of torsion in terms of the contorsion tensor cal-K. We treat torsion as an external (or a background) field and show that the contribution of torsion to the Einstein equations can be interpreted in terms of the torsion energy-momentum tensor, local conservation of which in a curved spacetime with an arbitrary metric or an arbitrary gravitational field demands a proportionality of the torsion energy--momentum tensor to a metric tensor, a covariant derivative of which vanishes because of the metricity condition. This allows to claim that torsion can serve as origin for vacuum energy density, given by cosmological constant or dark energy density in the Universe. This is a model-independent result may explain a small value of cosmological constant, which is a long--standing problem of cosmology. We show that the obtained result is valid also in the Poincare' gauge gravitational theory by Kibble (T. W. B. Kibble, J. Math. Phys. 2, 212 (1961)), where the Einstein-Hilbert action can be represented in the same form cal-R = R + cal-K^2., 11 pages, no figures, we have corrected the acknowledgements

The effective action from the heterotic string compactification is studied on the manifolds with absolute parallelism. The cosmic string solutions resulting from the effective action are discussed in analogy with the dislocations in three-dimensional crystalline solid. The cosmic string density is concluded to be given by the torsion tensor of space-time, which gives rise to a compactification at least in one-dimensional direction in space-time. It is shown also the contorsion tensor defines the deficit angle. We have found a stringy solution in a sixdimensional model on M 4 × T 2 which is compatible with that of Greene et al. and a cylindrically symmetric exact solution is obtained, which are different from the exact cosmic string solutions of the Einstein theory ever proposed

The effective action from the heterotic string compactification is studied on the manifolds with absolute parallelism. The cosmic string solutions resulting from the effective action are discussed in analogy with the dislocations in 3-dimensional crystalline solid. The cosmic string density is concluded to be given by the torsion tensor of space-time, which gives rise to a compactification at least in 1-dimensional direction in space-time. It is also shown that the contorsion tensor defines the deficit angle. We have found a stringy solution in a 6-dimensional model on M4 × T2 which coincides with that of Greene et al.,5 but the string energy density depends only on torsion, therefore independent of the space-time metric.

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., Comment: 2 pages, RevTeX, to appear in Phys. Rev D (Brief Report)

The metric and contorsion tensors are constructed which yield a combined Riemann curvature tensor of the form R ± μνστ = ( 1 2 a 2 )(g μσ g ντ − g μτ g νσ ± √g ϵ μνστ ) . The metric with euclidean signature ( + + + +) describes a sphere S 4 with radius a , i.e. admits the isometry group O5. For selfdual (antiselfdual) curvature tensor the contorsion tensor is given by the antiselfdual (selfdual) instanton configuration with respect to the spin gauge group SU2 R (SU2 L ). The selfdual (antiselfdual) Riemann tensor admits two covariantly constant right-handed (left-handed) spin 1 2 fermion zero modes, one J = 1 2 and one J = 3 2 right-handed (left-handed) multiplet corresponding to L = 1, transforming as a pseudoreal representation of O4 (SU2 R(L) ). The hermitean Dirac equation retains only the two constant chiral modes.

In order to get to a geometrically based theory of gravitation and electromagnetism, a gauge covariant bimetric tetrad space-time is introduced. The Weylian connection vector is derived from the tetrads and it is identified with the electromagnetic potential vector. The formalism is simplified by the use of gauge-invariant quantities. The theory contains a contorsion tensor that is connected with spinning properties of matter. The electromagnetic field may be induced by conventional sources and by spinning matter. In absence of spinning matter, the equations are identical with those of the gauge-covariant bimetric theory.(23)