Torsion Tensor Research Articles

Characteristic conic connections and torsion-free principal connections

We study the relation between torsion tensors of principal connections on G-structures and characteristic conic connections on associated cone structures. We formulate sufficient conditions under which the existence of a characteristic conic connection implies the existence of a torsion-free principal connection. We verify these conditions for adjoint varieties of simple Lie algebras, excluding those of type Aℓ≠2 or Cℓ. As an application, we give a complete classification of the germs of minimal rational curves whose VMRT at a general point is such an adjoint variety: nontrivial ones come from lines on hyperplane sections of certain Grassmannians or minimal rational curves on wonderful group compactifications.

Quantized Integrated Shift Effect in Multigap Topological Phases

We show that certain three-dimensional multigap topological insulators can host quantized integrated shift photoconductivities due to bulk invariants that are defined under reality conditions imposed by additional symmetries. We recast the quantization in terms of the integrated torsion tensor and the non-Abelian Berry connection constituting Chern-Simons forms. Physically, we recognize that the topological quantization emerges purely from virtual transitions contributing to the optical response. Our findings provide another quantized electromagnetic dc response due to the nontrivial band topology, beyond the quantum anomalous Hall effect of Chern insulators and quantized circular photogalvanic effect found in Weyl semimetals. Published by the American Physical Society 2024

Two geometrical invariants for three‐dimensional systems

The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components.

Symmetries in Riemann-Cartan Geometries

Riemann-Cartan geometries are geometries that admit non-zero curvature and torsion tensors. These geometries have been investigated as geometric frameworks for potential theories in physics including quantum gravity theories and have many important differences when compared to Riemannian geometries. One notable difference, is the number of symmetries for a Riemann-Cartan geometry is potentially smaller than the number of Killing vector fields for the metric. In this paper, we will review the investigation of symmetries in Riemann-Cartan geometries and the mathematical tools used to determine geometries that admit a given group of symmetries. As an illustration, we present new results by determining all static spherically symmetric and all stationary spherically symmetric Riemann-Cartan geometries. Furthermore, we have determined the subclasses of spherically symmetric Riemann-Cartan geometries that admit a seven-dimensional group of symmetries.

Brane mechanics and gapped Lie n-algebroids

We draw a parallel between the BV/BRST formalism for higher-dimensional (≥ 2) Hamiltonian mechanics and higher notions of torsion and basic curvature tensors for generalized connections in specific Lie n-algebroids based on homotopy Poisson structures. The gauge systems we consider include Poisson sigma models in any dimension and “generalised R-flux” deformations thereof, such as models with an (n + 2)-form-twisted R-Poisson target space. Their BV/BRST action includes interaction terms among the fields, ghosts and antifields whose coefficients acquire a geometric meaning by considering twisted Koszul multibrackets that endow the target space with a structure that we call a gapped almost Lie n-algebroid. Studying covariant derivatives along n-forms, we define suitable polytorsion and basic polycurvature tensors and identify them with the interaction coefficients in the gauge theory, thus relating models for topological n-branes to differential geometry on Lie n-algebroids.

Semi-Symmetric Metric Gravity

We will study a geometric extension of general relativity, which is based on a connection with a special type of torsion. This connection satisfies that its torsion tensor is fully determined by a vectorial degree of freedom, and it was first introduced by Friedmann and Schouten. We explore its physical implications by presenting three cosmological models within the considered geometric extension of GR, and compare the predictions of the models with those of ΛCDM and the observational data of the Hubble function. Our results show that the geometry envisioned by Friedmann could explain the observational data for the Hubble function without the need of dark energy.

Locally-homogeneous Riemann-Cartan geometries with the largest symmetry group

The symmetry frame formalism is an effective tool for computing the symmetries of a Riemann-Cartan geometry and, in particular, in metric teleparallel geometries. In the case of non-vanishing torsion in a four dimensional Riemann-Cartan geometry, the Minkowski geometry is the only geometry admitting ten affine frame symmetries. Excluding this geometry, the maximal number of affine frame symmetries is seven. A natural question is to ask what four dimensional geometries admit a seven-dimensional group of affine frame symmetries. Such geometries are locally homogeneous and admit the largest isotropy group permitted, and hence are called maximally isotropic. Using the symmetry frame formalism to compute affine frame symmetries along with the additional structure of the torsion tensor, we employ the Cartan-Karlhede algorithm to determine all possible seven-dimensional symmetry groups for Riemann-Cartan geometries.

The meaning of torsion in teleparallel theories

The ambiguity of the Weitzenböck connection is investigated from the perspective of frames. In doing so, the pure-tetrad formalism and the formalism with an affine connection are compared with each other; it is shown that the ambiguity in the affine connection is equivalent to that of the frame. The role played by the regularization procedure is discussed, and it is argued that the frame approach gives better results than any regularization procedure, including those that make use of a suitable choice of the affine connection. It is also argued that, for consistency, the Weitzenböck torsion must be either manifesting only gravitational effects or gravitational effects plus accelerations. Two possible interpretations for the torsion tensor are present, and an application to a frame that accelerates along the same direction as that of a [Formula: see text]-wave is used to show the consistence of these interpretations. Finally, some possible solutions to the well-known problems of the [Formula: see text] theories are proposed.

P -brane Galilean and Carrollian geometries and gravities

We study D-dimensional p-brane Galilean geometries via the intrinsic torsion of the adapted connections of their degenerate metric structure. These non-Lorentzian geometries are examples of G-structures whose characteristic tensors consist of two degenerate ‘metrics’ of ranks (p+1) and (D−p−1) . We carry out the analysis in two different ways. In one way, inspired by Cartan geometry, we analyse in detail the space of intrinsic torsions (technically, the cokernel of a Spencer differential) as a representation of G, exhibiting for generic (p, D) five classes of such geometries, which we then proceed to interpret geometrically. We show how to re-interpret this classification in terms of ( D−p−2 )-brane Carrollian geometries. The same result is recovered by methods inspired by similar results in the physics literature: namely by studying how far an adapted connection can be determined by the characteristic tensors and by studying which components of the torsion tensor do not depend on the connection. As an application, we derive a gravity theory with underlying p-brane Galilean geometry as a non-relativistic limit of Einstein–Hilbert gravity and discuss how it gives a gravitational realisation of some of the intrinsic torsion constraints found in this paper. Our results also have implications for gravity theories with an underlying ( D−p−2 )-brane Carrollian geometry.

Statistical conformal Killing vector fields for FLRW space-time

The classification of conformal Killing vector fields for FLRW space-time from Riemannian point of view was done by Maartens-Maharaj in []. In this paper, we introduce conformal Killing vector fields from a new point of view for the FLRW space-time. In particular, we consider three cases for the conformal factor. Then, it is shown that there exist nine conformal vector fields on FLRW in total, such that six of them are Killing and the rest being non-Killing conformal vector fields. Consequently, by recalling the concept of statistical conformal Killing vector fields introduced in [], we classify statistical structures whith repsect to which these vector fields are conformal Killing. We also obtain the form of affine connections that feature a vanishing Lie derivative with respect to these conformal Killing vector fields. Imposing the torsion-free and the Codazzi conditions on these connections, we study statistical structures on FLRW. Finally, for torsionful connections we study the vanishing of the Lie derivative of the torsion tensor with respect to these conformal Killing vector fields and derive the conditions under which this is valid.

Derivative estimates of pluriclosed flow

We provide a derivative estimate for the pluriclosed flow, controlling higher-order derivatives of the Chern curvature and torsion using the Chern curvature. Moreover, we derive an estimate for the torsion tensor using Chern Ricci curvature in dimension two.

Study on generalized BK-5th recurrent Finsler space

In this paper, we present a novel new class and investigate the connection between the K-projective curvature tensor and other tensors of Finsler space Fn, this space is characterized by the property for Cartan’s 4th curvature tensor satisfies the certain relationship with the given covariant vectors field, we define this space as a generalized BK-5th recurrent space and denote it briefly by GBK-5RFn. This paper aims to derive the fifth-order Berwald covariant derivatives of the torsion tensor and the deviation tensor . Additionally, it demonstrates that the curvature vector Kj, the curvature vector Hk, and the curvature scalar H are all non-vanishing within the considered space. We have identified tensors that exhibit self-similarity under specific conditions. Furthermore, we have established the necessary and sufficient conditions for certain tensors in this space to have equal fifth-order Berwald covariant derivatives with their lower-order counterparts.

Reference frames and black hole thermodynamics

In the context of the absolute parallelism formulation of General Relativity, and because of the fact that the scalar curvature can be written in purely torsional terms, it was known for a long time that a surface term based solely on the torsion tensor appears in the action. It was subsequently suggested that this term might play the role of the Gibbons-Hawking-York boundary term which, in turn, is associated to the free energy in the path integral approach, and then, to the black hole entropy by standard thermodynamic arguments. We show that the identification of the two boundary terms is rather incomplete, and that it strongly depends on the choice of the tetrad (frame) field used to reproduce a given metric. By considering variations of the tetrad field not necessarily subjected to Dirichlet-like conditions on the boundary surface, we find a class of frames adapted to the Schwarzschild spacetime in which the Gibbons-Hawking-York/torsion link is actually established, and conducing to the right black hole entropy without the need of any background subtraction. Remarkably, these frames are also responsible for the correct value of the gravitational energy as computed from the teleparallel energy-momentum pseudo-current.

Internal symmetry in Poincarè gauge gravity

We find a large internal symmetry within 4-dimensional Poincarè gauge theory.In the Riemann-Cartan geometry of Poincaré gauge theory the field equation and geodesics are invariant under projective transformation, just as in affine geometry. However, in the Riemann-Cartan case the torsion and nonmetricity tensors change. By generalizing the Riemann-Cartan geometry to allow both torsion and nonmetricity while maintaining local Lorentz symmetry the difference of the antisymmetric part of the nonmetricity Q and the torsion T is a projectively invariant linear combination S = T - Q with the same symmetry as torsion. The structure equations may be written entirely in terms of S and the corresponding Riemann-Cartan curvature. The new description of the geometry has manifest projective and Lorentz symmetries, and vanishing nonmetricity.Torsion, S and Q lie in the vector space of vector-valued 2-forms. Within the extended geometry we define rotations with axis in the direction of S. These rotate both torsion and nonmetricity while leaving S invariant. In n dimensions and (p, q) signature this gives a large internal symmetry. The four dimensional case acquires SO(11,9) or Spin(11,9) internal symmetry, sufficient for the Standard Model.The most general action up to linearity in second derivatives of the solder form includes combinations quadratic in torsion and nonmetricity, torsion-nonmetricity couplings, and the Einstein-Hilbert action. Imposing projective invariance reduces this to dependence on S and curvature alone. The new internal symmetry decouples from gravity in agreement with the Coleman-Mandula theorem.

Tetrad Fields, Reference Frames, and the Gravitational Energy–Momentum in the Teleparallel Equivalent of General Relativity

AbstractThe concept and definitions of the energy–momentum and angular momentum of the gravitational field in the teleparallel equivalent of general relativity (TEGR) are reviewed. The importance of these definitions is justified by three major reasons. First, the TEGR is a well established and widely accepted formulation of the gravitational field, whose basic field strength is the torsion tensor of the Weitzenböck connection. Second, in the phase space of the TEGR there exists an algebra of the Poincaré group. Not only the definitions of the gravitational energy–momentum and 4‐angular momentum satisfy this algebra, but also the first class constraints related to these definitions satisfy the algebra. And third, numerous applications of these definitions lead to physically consistent results. These definitions follow from a well established Hamiltonian formulation, and rely on the idea of localization of the gravitational energy. In this review, the concept of localizability of the gravitational energy is revisited, in light of results obtained in recent years. The behavior of free particles is studied in the space–time of plane fronted gravitational waves (pp‐waves). Free particles are here understood as particles that are not subject to external forces other than the gravitational acceleration due to pp‐waves. Since these particles acquire or loose kinetic energy locally, the transfer of energy from or to the gravitational field must also be localized. This theoretical result is considered an important and definite argument in favor of the localization of the gravitational energy–momentum, and by extension, of the gravitational 4‐angular momentum.

Geometric structures of micropolar continuum with elastic and plastic deformations based on generalized Finsler space

Elasto-plastic deformations of micropolar continuum are discussed by a non-Riemannian geometry. The non-locality of micropolar continuum is described in a second-order vector bundle of displacements and microrotations. With a decomposition of total elasto-plastic field, geometric quantities are divided into the elastic and plastic components independently. Especially, when an intrinsic parallelism of displacements and microrotations holds, integrability conditions of the elasto-plastic field are represented by a torsion tensor or the curvature of nonlinear connection. Then, Burgers and Frank vectors and an energy release rate around crack tips are related to the torsion tensor or the curvature of nonlinear connection. Moreover, the non-locality of microrotation is discussed based on a kink band as a disclination. It is found a generalized expression of Burgers vector which can describe the kink interface including the disclination.

Harmonic G2-structures on almost Abelian Lie groups

We consider left-invariant G2-structures on 7-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket A of the corresponding Lie algebra. In those terms, we establish the algebraic condition on A for each of the possible 16-torsion classes of a G2-structure. In particular, we show that four of those torsion classes are not admissible, since τ3=0 implies τ0=0. Finally, we use the above results to provide the algebraic criteria on A, satisfying the harmonic condition divT=0, and this allows to identify which torsion classes are harmonic.

Metric-affine gravity: Nonmetricity of space as dark matter/energy ?

A possibility of using a metric-affine theory to consider dark matter and dark energy through the properties of nonmetricity is discussed. In the framework of proposed metric-affine theory, the gravity field is described by the standard Einstein tensor, which depends only on the metric tensor and its derivatives. The nonmetricity and torsion are considered as property of space–time, which is source gravity that can be interpreted as a specific (maybe dark) form of matter and energy. This form of matter and energy, which are described by nonmetricity and torsion tensors, are sources of ”metric” gravity. For the observers, these non-metric properties are manifested as the specific properties of perfect (or non-perfect) fluid interacting with metric gravity only. The proposed metric-affine theory cannot be considered as standard theory of metric-affine gravity. The suggested theory is the metric theory of gravity and the affine theory of a specific (dark) form of matter and energy. The stress–energy tensor of this specific matter and energy is described by the non-metric part of the generalized (metric-affine) Einstein tensor. The specific matter and energy manifest itself as specific perfect or non-perfect fluid (and its energy), which interacts with standard matter only through gravity. The non-metric part is described as a stress–energy tensor, which is supposed to be considered as a candidate for the role of a stress–energy tensor of dark matter/energy. The article discusses a potential possibility to assume that nonmetricity simulates dark matter/energy by introducing a new source term of gravity into general relativity.

Covariant spin-parity decomposition of the torsion and path integrals

We propose a functional measure over the torsion tensor. We discuss two completely equivalent choices for the Wheeler–DeWitt supermetric for this field, the first one is based on its algebraic decomposition and the other is inspired by teleparallel theories of gravity. The measure is formally defined by requiring the normalization of the Gaußian integral. To achieve such a result we split the torsion tensor into its spin-parity eigenstates by constructing a new, York-like, decomposition. Of course, such a decomposition has a wider range of applicability to any kind of tensor sharing the symmetries of the torsion. As a result of this procedure a functional Jacobian naturally arises, whose formal expression is given exactly in the phenomenologically interesting limit of maximally symmetric spaces. We also discuss the explicit computation of this Jacobian in the case of a four-dimensional sphereS 4 with particular emphasis on its logarithmic divergences.

Parity violating scalar-tensor model in teleparallel gravity and its cosmological application

The parity violating model based on teleparallel gravity is a competitive scheme for parity violating gravity, which has been preliminary studied in the literature. To further investigate the parity violating model in teleparallel gravity, in this paper, we construct all independent parity-odd terms that are quadratic in torsion tensor and coupled to a scalar field in a way without higher-order derivatives. Using these parity-odd terms, we formulate a general parity violating scalar-tensor model in teleparallel gravity and obtain its equations of motion. To explore potentially viable models within the general model, we investigate the cosmological application of a submodel of the general model in which terms above the second power of torsion are eliminated. We focus on analyzing cosmological perturbations and identify the conditions that preserve the parity violating signal of gravitational waves at linear order while avoiding the ghost instability.