Torsion Tensor Research Articles

Левоинвариантная парасасакиева структура на групповой модели вещественного расширения плоскости де-Ситтера

In this paper, a group model for a real extension of the de Sitter plane is pro-posed. This group contains a group of special matrices, which is a subgroup of the general linear group. It is established that there exists a left-invariant contact metric structure on this group, which is normal and, therefore, para-Sasakian. The basis vector fields of the Lie algebra of infinitesimal automorphisms are found. The Lie group of automorphisms has the maximum dimension and, in addition to the Levi-Civita connection, it also retains a contact metric connection with skew-symmetric torsion. In this connection, all structural tensors of the para-Sasakian structure, as well as the torsion and curvature tensors, are covariantly constant. Using a nonholonomic field of orthonormal frames adapted to the contact distribution, an orthogonal projection of the Levi-Civita connection onto the contact distribution is found, which is a truncated connection. Passing to natural coordinates, differential equations of geodesics of the truncated connection and Levi-Civita connection are found. Thus, the Levi-Civita contact geodesic connections coincide with the truncated connection geodesics. This means that through each point in each contact direction there is a unique Levi-Civita geodesic connection tangent to the contact distribution. The Levi-Civita connection, like the contact metric connection, is consistent with the contact distribution.

Torsional birefringence in metric-affine Chern-Simons gravity: gravitational waves in late-time cosmology

In the context of the metric-affine Chern-Simons gravity endowed with projective invariance, we derive analytical solutions for torsion and nonmetricity in the homogeneous and isotropic cosmological case, described by a flat Friedmann-Robertson-Walker metric. We discuss in some details the general properties of the cosmological solutions in the presence of a perfect fluid, such as the dynamical stability and the emergence of big bounce points, and we examine the structure of some specific solutions reproducing de Sitter and power law behaviours for the scale factor. Then, we focus on first-order perturbations in the de Sitter scenario, and we study the propagation of gravitational waves in the adiabatic limit, looking at tensor and scalar polarizations. In particular, we find that metric tensor modes couple to torsion tensor components, leading to the appearance, as in the metric version of Chern-Simons gravity, of birefringence, characterized by different dispersion relations for the left and right circularized polarization states. As a result, the purely tensor part of torsion propagates like a wave, while nonmetricity decouples and behaves like a harmonic oscillator. Finally, we discuss scalar modes, outlining as they decay exponentially in time and do not propagate.

Higher-derivative couplings and torsional Riemann curvature

Using the most general higher-derivative field redefinitions for the closed spacetime manifolds, we show that the tree-level couplings of the metric, B-field and dilaton at orders α′2 and α′3 that have been recently found by the T-duality, can be written in a particular scheme in terms of the torsional Riemann curvature mathcal{R} and the torsion tensor H. The couplings at order α′2 have structures mathcal{R} 3, H2 mathcal{R} 2, H6, and the couplings at order α′3 have only structures mathcal{R} 4, H2 mathcal{R} 3. Replacing mathcal{R} with the ordinary Riemann curvature, the couplings in the structure H2 mathcal{R} 3 reproduce the couplings found in the literature by the S-matrix method.

Nearly para-Kähler geometry on Lie groups

In this paper, we introduce left-invariant (strict) nearly para-Kaähler structures on Lie groups (nearly para-Kähler Lie algebras) and present some properties of them. We prove the existence of a unique connection in terms of the characteristic connection on these Lie groups with totally skew-symmetric torsion tensor and we show that the Nijenhuis tensor is parallel with respect to the characteristic connection. We determine conditions that allow the curvature tensor of the characteristic connection satisfies the first Bianchi identity. Finally, we focus on anti-abelian almost para-complex structures on Lie groups and we study nearly para-Kähler conditions for these structures. In this study we encounter bi-invariant metrics as pseudo-Riemannian metrics that accept nearly para-Kähler structures. Moreover, we present some examples of nearly para-Kähler Lie algebras.

Finsler metrics and semi-symmetric compatible linear connections

Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds Wagner (Dokl Acad Sci USSR (N.S.) 39:3–5, 1943). Compatible linear connections are the solutions of the so-called compatibility equations containing the components of the torsion tensor as unknown quantities. Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics Bartelmeß and Matveev (J Diff Geom Appl 58:264–271, 2018), extremal compatible linear connections and algorithmic solutions Vincze (Aequat Math 96:53–70, 2022)), it is very hard to solve them in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved Vincze (Publ Math Debrecen 83(4):741–755, 2013 (see also Vincze (Euro J Math 3:1098–1171, 2017))) that such a compatible linear connection must be uniquely determined. The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the uniqueness by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well: necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities.

E-Connections on the ε-Anti-Kähler Manifolds

The paper undertakes certain special forms of the quarter symmetric metric and non-metric connections on an ε-anti-Kähler manifold. Firstly, we deduce the relation between the Riemannian connection and the special forms of the quarter symmetric metric and non-metric connections. Then, we present some results concerning the torsion tensors of these connections. In addition, we find the forms of the curvature tensor, the Ricci curvature tensor and scalar curvature of such connections and we search the conditions for the ε-anti-Kähler manifold to be an Einstein space with respect to these connections. Finally, we study U(Ric)-vector fields with respect to these connections and give some results related to them.

Ghost instability in the teleparallel gravity model with parity violations

In this paper we consider the parity violating gravity model within the framework of teleparallel gravity. The parity violations are caused by the couplings of a scalar field to the scalar invariants which are parity-odd and quadratic in the torsion tensor. Totally there are two such type independent invariants, and one of them is the Nieh-Yan density. Through investigations on the cosmological perturbations of this model, we find that in general it suffers from the difficulties of ghost instability in the scalar and vector perturbations. But in the special case only the coupling to the Nieh-Yan density exists, this model is ghost free and reduces to the Nieh-Yan modified Teleparallel Gravity model. We also analyze the severity of the ghost instability by studying the perturbations around the Minkowski background.

Role of the remnant symmetries in gravitational theories based on absolute parallelism: a 2D standpoint

Abstract By using simplified 2D gravitational, non-Lorentz invariant actions constructed from the torsion tensor, we discuss the physical meaning of the remnant symmetries associated with the near-horizon (Milne) geometry experienced by a radial observer in Schwarzschild spacetime. We then fully characterize the remnant symmetries corresponding to this near-horizon 2D geometry by solving the motion equations adapted to 2D Milne space. This symmetries, which represent special or privileged diads, acquire the form of uniformly accelerated (Rindler) observers whose constant acceleration is proportional to the black hole mass M.

Plebański-Demiański solutions with dynamical torsion and nonmetricity fields

We construct Plebański-Demiański stationary and axisymmetric solutions with two expanding and double principal null directions in the framework of Metric-Affine gauge theory of gravity. Starting from the new improved form of the metric with vanishing cosmological constant recently achieved by Podolský and Vrátný, we extend this form in the presence of a cosmological constant and derive the conditions under which the physical sources of the torsion and nonmetricity tensors provide dynamical contributions preserving it in Weyl-Cartan geometry. The resulting black hole configurations are characterised by the mass, orbital angular momentum, acceleration, NUT parameter, cosmological constant and electromagnetic charges of the Riemannian sector of the theory, as well as by the spin and dilation charges of the torsion and nonmetricity fields. The former is subject to a constraint representing a decoupling limit with the parameters responsible of axial symmetry, beyond which the geometry of the space-time is expected to be corrected.

Scalarized black holes in teleparallel gravity

Black holes play a crucial role in the understanding of the gravitational interaction. Through the direct observation of the shadow of a black hole by the event horizon telescope and the detection of gravitational waves of merging black holes we now start to have direct access to their properties and behaviour, which means the properties and behaviour of gravity. This further raised the demand for models to compare with those observations.In this respect, an important question regarding black holes properties is to know if they can support “hairs”. While this is famously forbidden in general relativity, in particular for scalar fields, by the so-called no-hair theorems, hairy black holes have been shown to exist in several class of scalar-tensor theories of gravity.In this article we investigate the existence of scalarized black holes in scalar-torsion theories of gravity.On one hand, we find exact solutions for certain choices of couplings between a scalar field and the torsion tensor of a teleparallel connection and certain scalar field potentials, and thus proof the existence of scalarized black holes in these theories.On the other hand, we show that it is possible to establish no-scalar-hair theorems similar to what is known in general relativity for other choices of these functions.

Cosmological constraints on bulk viscous f(Q,T) gravity

AbstractIn the General Theory of Relativity (GR) gravitational interactions are described by the scalar Ricci curvature. But there exists another class of modified gravity theories equivalent to GR, described by the torsion tensor or non‐metricity scalar. One of such theories, investigated in this paper is gravity, built from the arbitrary function of non‐metricity scalar non‐minimally coupled to the trace of stress‐energy tensor. Current work is aimed on the linear gravity, where and are free MOG parameters to be varied. As well, we assume that the fluid is bulk‐viscous with pressure and the bulk viscosity parameter . From the modified Einstein Field Equations (EFE's) we obtain the Hubble parameter analytically and constrain free parameters of the model through the MCMC sampling of 57 recent Hubble parameter observations. We found that computed Hubble rate for our model from MCMC sampling converges with the observational data at the confidence level. To examine the current accelerated expansion rate of the universe in details we use the statefinder diagnostics. Besides, to distinguish CDM and our model we use analysis.

Kinematics and dynamics of null hypersurfaces in the Einstein-Cartan spacetime and related thermodynamic interpretation

A general geometric construction of a generic null hypersurface in presence of torsion in the spacetime (Riemann-Cartan background), generated by a null vector $l^a$, is being developed here. We then explicitly define and structure various corresponding kinematical quantities. The dynamics of the null surface, particularly given by $\hat{G}_{ab}k^al^b$, is also discussed. The later one is constructed under the {\it geodesic constraint} condition. This yields a relation among the rate of change of expansion scalar corresponding to auxiliary null vector $k^a$ and various kinematical entities on the null surface. Using this relation we show that the Einstein-Cartan-Kibble-Sciama equation (which provides the dynamics of the metric and the torsion tensor) on this null hypersurface acquires a thermodynamic interpretation. The thermodynamic entities like temperature, entropy density, energy and pressure are properly identified. In the whole analysis we adopt the geometrical field interpretation of torsion and all discussions are done in a covariant manner.

F(T,Θ) gravity and generalized gravitational baryogenesis

Where is the anti-matter in our universe? This is the most significant open problem of the modern era. This paper is devoted to the study of cosmological impact of gravitational baryogenesis and its generalized case driven by [Davoudiasl et al., Phys. Rev. Lett. 93, 201301 (2004)]. In this class of baryogenesis, the Ricci scalar [Formula: see text] is derivatively coupled to the baryon current [Formula: see text] as [Formula: see text]. We examine the imbalance of baryon and anti-baryon of the universe in the context of [Formula: see text] gravity, where [Formula: see text] and [Formula: see text] are the torsion scalar and the trace of the energy–momentum tensor, respectively. We assume two [Formula: see text] models in which the first one contains the sum of second-order torsion tensors and the trace of energy–momentum tensor while the second one is the product of these terms. For all cases, we find the ratio of baryon number to entropy by considering the power-law scale factor. Finally, we compare our results with the observational value of baryon-number-density-to-entropy ratio. The baryon-to-entropy ratio against the first model lies in the range [Formula: see text] while for the second model it contains [Formula: see text]. For generalized case of gravitational baryogenesis, we also obtain results that are compatible with the observation limit.

The Truth on Gravity and Terrestrial Global Warming. Part 1: The Quantum Gravity Field

Although during the last century both experimental and computational technology progress has accumulated solid factual information on the deeper knowledge of our actual world, the previous approaches derived in most cases from the unfinished Einstein’s theory of gravity using untrue models for the torsion tensor, have failed to explain the gravitational and precessional aspects, causally connected with the morphogenesis of matter engendered by the bending of light. The torsion/ twisted process during the refrigeration of hot matter (lava) is essential for the understanding of gravitation phenomena as embodied/ gravitized mass occurring at the polarization of light. The Lorentz polarized field concept, (4π/3)P(Ker), associated with a quantum partition kernel, Ker = (1/2, 2/3, 3/4), of the light polarization for radiant energy, mass and gravity, respectively, is generalized for the regenerative matter, including the polarization of time as absolute/orbital time and relative/ precessional time (or the half-life of a regenerative substance). The laws of the Sun like regenerative material systems are given and also the explanations of the quantum origin of gravity along with certain aberration of light in its field are considered.

Torsion braneworlds in a tensor-vector gravity

In this paper, we study the properties of gravity and bulk fields living in a torsion warped braneworld. The torsion is driven by a background vector whose norm provides a source for the bulk cosmological constant. For a vector as the derivative of a scalar field, we find new isotropic thick brane geometry. We analyze the features of bosonic and fermionic fields in this scenario. The background vector provides nonminimal coupling between the fields and the geometry leading to modifications in the Kaluza–Klein (KK) states. For a scalar field, the background vector breaks the [Formula: see text] symmetry, whereas for the vector gauge field, a geometric coupling to the torsion tensor traps the massless mode. The spinor connection is modified by the torsion and a geometric derivative Yukawa-like coupling is proposed. The effects of these new couplings are investigated.

Constants of motion associated with canonoid transformations for non-autonomous systems

We consider non-autonomous systems of ordinary differential equations that can be expressed in Hamiltonian form in terms of two different coordinate systems, not related by a canonical transformation. We show that the relationship between these coordinate systems leads to a, possibly time-dependent, tensor field, $S^{\alpha}_{\beta}$, whose eigenvalues are constants of motion. We prove that if the Nijenhuis torsion tensor of $S^{\alpha}_{\beta}$ is equal to zero then the eigenvalues of $S^{\alpha}_{\beta}$ are in involution, and that these eigenvalues may be in involution even if the Nijenhuis tensor is not zero.

Exploring the geometry of supersymmetric double field theory

The geometry of mathcal{N} = 1 supersymmetric double field theory is revisited in superspace. In order to maintain the constraints on the torsion tensor, the local tangent space group of O(D) × O(D) must be expanded to include a tower of higher dimension generators. These include a generator in the irreducible hook representation of the Lorentz group, which gauges the shift symmetry (or ambiguity) of the spin connection. This gauging is possible even in the purely bosonic theory, where it leads to a Lorentz curvature whose only non-vanishing pieces are the physical ones: the generalized Einstein tensor and the generalized scalar curvature. A relation to the super-Maxwell∞ algebra is proposed. The superspace Bianchi identities are solved up through dimension two, and the component supersymmetry transformations and equations of motion are explicitly (re)derived.

The $\partial$-operator and real holomorphic vector fields

Let $(M,h)$ be a Hermitian manifold and $\psi$ a smooth weight function on $M$. The $\partial$-complex on weighted Bergman spaces $A^2_{(p,0)}(M,h, e^{-\psi})$ of holomorphic $(p,0)$-forms was recently studied in [[10] and [9]. It was shown that if $h$ is K\"ahler and a suitable density condition holds, the $\partial$-complex exhibits an interesting holomorphicity/duality property when $(\bar\partial\psi)^{\sharp}$ is holomorphic (i.e., when the real gradient field $\mathrm{grad}_h\psi$ is a real holomorphic vector field). For general Hermitian metrics this property does not hold without the holomorphicity of the torsion tensor $T_p{}^{rs}$. In this paper, we investigate the existence of real-valued weight functions with real holomorphic gradient fields on K\"ahler and conformally K\"ahler manifolds and their relationship to the $\partial$-complex on weighted Bergman spaces. For K\"ahler metrics with multi-radial potential functions on $\mathbb C^n$ we determine all multi-radial weight functions with real holomorphic gradient fields. For conformally K\"ahler metrics on complex space forms we first identify the metrics having holomorphic torsion leading to several interesting examples such as the Hopf manifold $\mathbb{S}^{2n-1} \times \mathbb{S}^1$, and the "half" hyperbolic metric on the unit ball. For some of these metrics, we further determine weight functions $\psi$ with real holomorphic gradient fields. They provide a wealth of triples $(M,h,e^{-\psi})$ of Hermitian non-K\"ahler manifolds with weights for which the $\partial$-complex exhibits the aforementioned holomorphicity/duality property. Among these examples, we study in detail the $\partial$-complex on the unit ball with the half hyperbolic metric and derive a new estimate for the $\partial$-equation.

Метрики пространства с линейной связностью, не являющейся полусимметрической

It is well-known Levi-Chivita’s construction of object for affine connection (in modern terminology — linear connection) by the field of non-degenerate metric on a smooth manifold. An inverse problem (a construction of metric by given linear connection) is solved ambiguously, besides, the metric may turn out to be degenerate and indefinite. On the one hand, two metrics differing in a sign are obviously build: by curvature tensor contractionwith subsequent symmetrization. Оn the other hand, Vranceanu’s metric is a double contraction of multiplication of a torsion tensor’s components. In this paper Levi-Chivita’s inverse problem is solved in other way using the field of connection object. It is proved that in the general case, when the linear connection is not semi-symmetric, six metrics can be constructed. In the special case, when the linear connection is semi-symmetric (in particular, torsion-free), the constructed metrics vanish. The investigation is done on a semi-holonomic smooth manifold by means of two prolongation its structure equations.

On the extremal compatible linear connection of a generalized Berwald manifold

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). It is known (Vincze in J AMAPN 21:199–204, 2005) that such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is some strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. The paper presents the idea of the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is uniquely determined because the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Using the reference element method, the extremal compatible linear connection can be expressed in terms of the canonical data as well. It is an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.