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We investigate the Cartan formalism in $F(R)$ gravity. $F(R)$ gravity has been introduced as a theory to explain cosmological accelerated expansion by replacing the Ricci scalar $R$ in the Einstein-Hilbert action with a function of $R$. As is well-known, $F(R)$ gravity is rewritten as a scalar-tensor theory by using the conformal transformation. Cartan $F(R)$ gravity is described based on the Riemann-Cartan geometry formulated by the vierbein. In the Cartan formalism, the Ricci scalar $R$ is divided into two parts, one derived from the Levi-Civita connection and the other from the torsion. Assuming the spin connection independent matter action, we have successfully rewritten the action of Cartan $F(R)$ gravity into the Einstein-Hilbert action and a scalar field with canonical kinetic and potential terms without any conformal transformations. The resulting scalar-tensor theory is useful in applying the usual slow-roll scenario. As a simple case, we employ the Starobinsky model and evaluate fluctuations in the cosmological microwave background radiation., Comment: 8 pages

Theory founded on SO(2,3)* gauge symmetry. One significant feature of thisapproach is that gravitational field, given by the vierbein, becomesmanifest only after a suitable gauge fixing and it is formally united withother gauge fields. Starting from a model of pure noncommutative gravity, weextend it by introducing fermions and Yang-Mills gauge field. Using theenveloping algebra approach and the Seiberg-Witten map we constructcorresponding actions and expand them perturbatively in powers of thecanonical noncommutativity parameter θαβ. Unlike in the case of purenoncommutative gravity, first non-vanishing noncommutative corrections arelinear in the noncommutativity parameter and they describe the coupling ofmatter and gauge fields with gravity due to spacetime noncommutativity. Thisis augmented by the fact that some of these corrections pertain even in flatspacetime where they induce potentially observable noncommutativedeformations. We discuss the effects of noncommutativity on electron’sdispersion relation in the presence of constant background magnetic field -Landau levels. Our results could be useful for further investigation ofphenothe menological consequences of spacetime noncommutativity. [Project ofthe Serbian Ministry of Education, Science and Technological Development,Grant no. ON171031]

An alternative approach to the usual Kaluza-Klein way to field unification is presented which seems conceptually more satisfactory and elegant. The main idea is that of associating each fundamental interaction and matter field with a vector potential which is an eigenvector of the metric tensor of a multidimensional space-time manifold (n-dimensional “vierbein”). We deduce a system of field equations involving both Einstein and Maxwell-like equations for the fundamental fields. Confinement of the fields within the observable 4-dimensional space-time and non-vanishing particles’ rest mass problem are shown to be related to the choice of a scalar boson field (Higgs boson) appearing in the theory as a gauge function. Physical interpretation of the results, in order that all the known fundamental interactions may be included within the metric and connection, requires that the extended space-time is 16-dimensional. Fermions are shown to be included within the additional components of the vector potentials arising because of the increased dimensionality of space-time. A cosmological solution is also presented providing a possible explanation both to space-time flatness and to dark matter and dark energy as arising from the field components hidden within the extra space dimensions. Suggestions for gravity quantization are also examined.

In this paper, we exploit the formal equivalence of the Solution Manifold of two distinct physical systems to create enough symmetries so as to characterize them by Noether Invariants, thus favoring their future quantization. In so doing, we somehow generalize the Arnold Transformation for non-necessarily linear systems. Very particularly, this algorithm applies to the case of the motion on the de Sitter space-time providing a finite-dimensional algebra generalizing the Heisenberg-Weyl algebra globally. In this case, the basic (contact) symmetry is imported from the motion of a (non-relativistic) particle on the sphere S3. © 2021 World Scientific Publishing Company., We thank the Spanish Ministerio de Ciencia e Innovacin (MICINN) for financial support (FIS2017-84440-C2-2-P) and V. A. acknowledges financial support from the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award for the Instituto de Astrofsica de Andaluca (SEV-2017-0709). J. G. acknowledges financial support from the Spanish MICINN (PGC2018-097831B-I00). Discussions with P. Horvathy are also acknowledged.

We consider the role of coordinate dependent tetrads ("Fermi velocities"), momentum space geometry, and torsional Landau levels (LLs) in condensed matter systems with low-energy Weyl quasiparticles. In contrast to their relativistic counterparts, they arise at finite momenta and an explicit cutoff to the linear spectrum. Via the universal coupling of tetrads to momentum, they experience geometric chiral and axial anomalies with gravitational character. More precisely, at low-energy, the fermions experience background fields corresponding to emergent anisotropic Riemann-Cartan and Newton-Cartan spacetimes, depending on the form of the low-energy dispersion. On these backgrounds, we show how torsion and the Nieh-Yan (NY) anomaly appear in condensed matter Weyl systems with a ultraviolet (UV) parameter with dimensions of momentum. The torsional NY anomaly arises in simplest terms from the spectral flow of torsional LLs coupled to the nodes at finite momenta and the linear approximation with a cutoff. We carefully review the torsional anomaly and spectral flow for relativistic fermions at zero momentum and contrast this with the spectral flow, non-zero torsional anomaly and the appearance the dimensionful UV-cutoff parameter in condensed matter systems at finite momentum. We apply this to chiral transport anomalies sensitive to the emergent tetrads in non-homogenous chiral superconductors, superfluids and Weyl semimetals under elastic strain. This leads to the suppression of anomalous density at nodes from geometry, as compared to (pseudo)gauge fields. We also briefly discuss the role torsion in anomalous thermal transport for non-relativistic Weyl fermions, which arises via Luttinger's fictitious gravitational field corresponding to thermal gradients., Comment: 21 pages, several figures, minor changes and updated refs

We propose in this paper a quantization scheme for real Klein-Gordon field in de Sitter spacetime. Our scheme is generally covariant with the help of vierbein, which is necessary usually for spinor field in curved spacetime. We first present a Hamiltonian structure, then quantize the field following the standard approach. For the free field, the time-dependent quantized Hamiltonian is diagonalized by Bogliubov transformation and the eigen-states at each instant are interpreted as the observed particle states at that instant. The interpretation is supported by the known cosmological red-shift formula and the on-shell condition of 4-momentum for a free field. Though the mathematics is carried out in term of conformal coordinates for the sake of convenience, the whole theory can be transformed into any other coordinates based on general covariance. It is concluded that particle states, such as vacuum states in particular are time-dependent and vacuum states at one time evolves into non-vacuum states at later times. Formalism of perturbational is provided with an extended Dirac picture., 15 pages

First-order general relativity in $n$ dimensions ($n \geq 3$) has an internal gauge symmetry that is the higher-dimensional generalization of three-dimensional local translations. We report the extension of this symmetry for $n$-dimensional f(R) gravity with torsion in the Cartan formalism. The new symmetry arises from the direct application of the converse of Noether's second theorem to the action principle of f(R) gravity with torsion. We show that infinitesimal diffeomorphisms can be written as a linear combination of the new internal gauge symmetry, local Lorentz transformations, and terms proportional to the variational derivatives of the f(R) action. It means that the new internal symmetry together with local Lorentz transformations can be used to describe the full gauge symmetry of f(R) gravity with torsion, and thus diffeomorphisms become a derived symmetry in this setting., It contains a detailed derivation of the generalization of 3-dimensional "local translations" for 4-dimensional first-order general relativity

Basis tensor gauge theory (BTGT) is a vierbein analog reformulation of ordinary gauge theories in which the vierbein field describes the Wilson line. After a brief review of the BTGT, we clarify the Lorentz group representation properties associated with the variables used for its quantization. In particular, we show that starting from an SO(1,3) representation satisfying the Lorentz-invariant U(1,3) matrix constraints, BTGT introduces a Lorentz frame choice to pick the Abelian group manifold generated by the Cartan subalgebra of U(1,3) for the convenience of quantization even though the theory is frame independent. This freedom to choose a frame can be viewed as an additional symmetry of BTGT that was not emphasized before. We then show how an [Formula: see text] permutation symmetry and a parity symmetry of frame fields natural in BTGT can be used to construct renormalizable gauge theories that introduce frame-dependent fields but remain frame independent perturbatively without any explicit reference to the usual gauge field.

We consider a 5-dimensional Chern-Simons gauge theory for the isometry group of Anti-de-Sitter spacetime, $\operatorname{AdS}_{4+1}\simeq\operatorname{SO}(4,2)$, and invoke different dimensional reduction schemes in order to relate it to 4-dimensional spin-2 theories. The AdS gauge algebra is isomorphic to a parametrized 4-dimensional conformal algebra, and the gauge fields corresponding to the generators of non-Abelian translations and special conformal transformations reduce to two vierbein fields in $D=4$. Besides these two vierbeine, our reduction schemes leave only the Lorentz spin connection as an additional dynamical field in the 4-dimensional theories. We identify the corresponding actions as particular generalizations of Einstein-Cartan theory, conformal gravity and ghost-free bimetric gravity in first-order form., Comment: 31 pages, no figures, added references

Elie Cartan's “espaces generalizes” are, intuitively, curved geometries where the geometrical structure is that of a flat Klein geometry (a homogeneous space of a group) being rolled around a curved manifold without slipping or twisting. In modern terminology we may think of such a Cartan geometry as a fibre bundle with a means of lifting curves in the base manifold to curves in the Lie groupoid of structure-preserving fibre maps. The infinitesimal geometry will then be the Lie algebroid of certain projectable vector field on the fibre bundle, together with a horizontal lift to represent the connection. This paper considers symmetries of these structures, and explains why any vertical symmetry (projecting to the identity on the base manifold of the bundle) or any vertical infinitesimal symmetry (projecting to zero) must necessarily be trivial.

We are concerned with the global weak rigidity of the Gauss-Codazzi-Ricci (GCR) equations on Riemannian manifolds and the corresponding isometric immersions of Riemannian manifolds into the Euclidean spaces. We develop a unified intrinsic approach to establish the global weak rigidity of both the GCR equations and isometric immersions of the Riemannian manifolds, independent of the local coordinates, and provide further insights of the previous local results and arguments. The critical case has also been analyzed. To achieve this, we first reformulate the GCR equations with div-curl structure intrinsically on Riemannian manifolds and develop a global, intrinsic version of the div-curl lemma and other nonlinear techniques to tackle the global weak rigidity on manifolds. In particular, a general functional-analytic compensated compactness theorem on Banach spaces has been established, which includes the intrinsic div-curl lemma on Riemannian manifolds as a special case. The equivalence of global isometric immersions, the Cartan formalism, and the GCR equations on the Riemannian manifolds with lower regularity is established. We also prove a new weak rigidity result along the way, pertaining to the Cartan formalism, for Riemannian manifolds with lower regularity, and extend the weak rigidity results for Riemannian manifolds with unfixed metrics., 42 pages, Journal of Geometric Analysis, 2017

We show that the four-dimensional Lovelock-Cartan action can be derived from a massless gauge theory for the $SO(1,3)$ group with an additional BRST trivial part. The model is originally composed by a topological sector and a BRST exact piece and has no explicit dependence on the metric, the vierbein or a mass parameter. The vierbein is introduced together with a mass parameter through some BRST trivial constraints. The effect of the constraints is to identify the vierbein with some of the additional fields, transforming the original action into the Lovelock-Cartan one. In this scenario, the mass parameter is identified with Newton's constant while the gauge field is identified with the spin-connection. The symmetries of the model are also explored. Moreover, the extension of the model to a quantum version is qualitatively discussed., Comment: 17 pages. No figures. Final version accepted for publication at the EPJC

We study the gravity action built from two gauge fields corresponding to the generators of the conformal group. Starting with the action from which one can obtain Einstein gravity and conformal gravity upon imposing suitable constraints, we keep two independent gauge fields and integrate out the field corresponding to the generator of Lorentz transformations. We identify the two gauge fields with two vierbeins and perturb them around anti--de Sitter space. This gives the linearized equations that differ from both Einstein gravity and conformal gravity linearized equations. We also study the linearized equations for one gauge field perturbed around the flat space and one around zero, and the case in which the gauge fields are proportional to each other.

For internal gauge forces, the result of locally gauging, i.e., of performing the substitution $\partial \rightarrow D$, is physically the same whether performed on the action or on the corresponding Euler-Lagrange equations of motion. Rather unsettling, though, such commutativity fails for the standard way of coupling a Dirac fermion to the gravitational field in the setting of a local Lorentz gauge theory of general relativity in the vierbein formalism, the equivalence principle thus seemingly being here violated. This paper will present a formalism in which commutativity holds for the gravitational force as well, the action for the gravitational field itself being still the Einstein-Hilbert one. Notably, in this formalism, the spinor field will carry a world/coordinate index, rather than a Lorentz spinor index as it does standardly. More generally, no Lorentz indices will figure, neither vector indices nor spinor indices, which from a parsimonious point of view seems quite satisfactory., v3: published version

Lorentz-violating neutrino parameters have been severely constrained on the basis of astrophysical considerations. In the high-energy limit, one generally assumes a superluminal dispersion relation of an incoming neutrino of the form E ~ |p|v, where E is the energy, p is the momentum and $v = sqrt(1 + delta) > 1. Lepton-pair creation due to a Cerenkov-radiation-like process (nu -> nu + e^- + e^+) becomes possible above a certain energy threshold, and bounds on the Lorentz-violating parameter delta can be derived. Here, we investigate a related process, nu_i -> nu_i + nu_f + bar_nu_f, where nu_i is an incoming neutrino mass eigenstate, while nu_f is the final neutrino mass eigenstate, with a superluminal velocity that is slightly slower than that of the initial state. This process is kinematically allowed if the Lorentz-violating parameters at high energy differ for the different neutrino mass eigenstates. Neutrino splitting is not subject to any significant energy threshold condition and could yield quite a substantial contribution to decay and energy loss processes at high energy, even if the differential Lorentz violation among neutrino flavors is severely constrained by other experiments. We also discuss the SU(2)-gauge invariance of the superluminal models and briefly discuss the use of a generalized vierbein formalism in the formulation of the Lorentz-violating Dirac equation., 17 pages; RevTeX; to appear in Physical Review D

One of the formalisms that introduces torsion conveniently in gravity is the vierbein-Einstein-Palatini (VEP) formalism. The independent variables are the vierbein (tetrads) and the components of the spin connection. The latter can be eliminated in favor of the tetrads using the equations of motion in the absence of fermions; otherwise there is an effect of torsion on the dynamics of fields. We find that the conformal transformation of off-shell spin connection is not uniquely determined unless additional assumptions are made. One possibility gives rise to Nieh-Yan theory, another one to conformally invariant torsion; a one-parameter family of conformal transformations interpolates between the two. We also find that for dynamically generated torsion the spin connection does not have well defined conformal properties. In particular, it affects fermions and the non-minimally coupled conformal scalar field., Comment: The final publication is available at Springer via https://doi.org/10.1140/epjp/i2018-12070-6

Basis tensor gauge theory is a vierbein analog reformulation of ordinary gauge theories in which the difference of local field degrees of freedom has the interpretation of an object similar to a Wilson line. Here we present a non-Abelian basis tensor gauge theory formalism. Unlike in the Abelian case, the map between the ordinary gauge field and the basis tensor gauge field is nonlinear. To test the formalism, we compute the beta function and the two-point function at the one-loop level in non-Abelian basis tensor gauge theory and show that it reproduces the well-known results from the usual formulation of non-Abelian gauge theory.

We present a purely tensor-spinor theory of gravity in arbitrary even D=2n space-time dimensions. This is a generalization of the purely vector-spinor theory of gravitation by Bars and MacDowell (BM) in 4D to general even dimensions with the signature (2n−1,1). In the original BM-theory in D=(3,1), the conventional Einstein equation emerges from a theory based on the vector-spinor field ψμ from a lagrangian free of both the fundamental metric gμν and the vierbein eμm. We first improve the original BM-formulation by introducing a compensator χ, so that the resulting theory has manifest invariance under the nilpotent local fermionic symmetry: δϵψ=Dμϵ and δϵχ=−ϵ. We next generalize it to D=(2n−1,1), following the same principle based on a lagrangian free of fundamental metric or vielbein now with the field content (ψμ1⋯μn−1,ωμrs,χμ1⋯μn−2), where ψμ1⋯μn−1 (or χμ1⋯μn−2) is a (n−1) (or (n−2)) rank tensor-spinor. Our action is shown to produce the Ricci-flat Einstein equation in arbitrary D=(2n−1,1) space-time dimensions.

Gravity can be regarded as a consequence of local Lorentz (LL) symmetry, which is essential in defining a spinor field in curved spacetime. The gravitational action may admit a zero-field limit of the metric and vierbein at a certain ultraviolet cutoff scale such that the action becomes a linear realization of the LL symmetry. Consequently, only three types of term are allowed in the four-dimensional gravitational action at the cutoff scale: a cosmological constant, a linear term of the LL field strength, and spinor kinetic terms, whose coefficients are in general arbitrary functions of LL and diffeomorphism invariants. In particular, all the kinetic terms are prohibited except for spinor fields, and hence the other fields are auxiliary. Their kinetic terms, including those of the LL gauge field and the vierbein, are induced by spinor loops simultaneously with the LL gauge field mass. The LL symmetry is necessarily broken spontaneously and hence is nothing but a hidden local symmetry, from which gravity is emergent., Comment: 7 pages; version accepted for publication in Phys. Lett. B; a table, footnotes, references, and comments added

Let (M,F) be a Finsler manifold, and let TM0 be the slit tangent bundle of M with a generalized Riemannian metric G, which is induced by F. In this paper, we extract many natural foliations of (TM0,G) and study their geometric properties. Next, we use this approach to obtain new characterizations of Finsler manifolds with positive constant flag curvature. We also investigate the relations between Levi-Civita connection, Cartan connection, Vaisman connection, vertical foliation, and Reinhart spaces.

The Webster scalar curvature is computed for the sphere bundle T1S of a Finsler surface (S, F) subject to the Chern-Hamilton notion of adapted metrics. As an application, it is derived that in this setting (T1S, gSasaki) is a Sasakian manifold homothetic with a generalized Berger sphere, and that a natural Cartan structure is arising from the horizontal 1-forms and the author associates a non-Einstein pseudo-Hermitian structure. Also, one studies when the Sasaki type metric of T1S is generally adapted to the natural co-frame provided by the Finsler structure.

We present some gauge conditions to eliminate all second time derivative terms in the vierbein forms of the ten Einstein equations of general relativity; at the same time, we present the corresponding Lagrangian in which there is not any quadratic term of first time derivative that can leads to those vierbein forms of the Einstein equations without second time derivative term by the corresponding Euler-Lagrange equations. General relativity thus becomes a fully singular Lagrange system., 25 pages, no figure

This work proposes a new gravitational theory formulated in terms of the vierbein field. The vierbein contains components which can be shifted by local Lorentz transformations and therefore do not show up in the spacetime metric. These components are given dynamics and become physical in our setup. They enter the massless theory in the form of an antisymmetric tensor field which makes the action reminiscent of the bosonic sector of supergravity. We then demonstrate that both the metric and the antisymmetric tensor can be made massive by adding a potential term for the vierbein. The form of this mass potential is inspired by ghost-free massive gravity. We confirm the absence of additional and potentially pathological degrees of freedom in an ADM analysis. However, at the linearized level around maximally symmetric solutions, the fluctuation of the antisymmetric tensor has a tachyonic mass pole., 23 pages, v2: Modified discussion on causality, modified convention in eq. 2.10 and thereafter. References added. v3: Added extended discussion of tachyonic mass pole. v4: matches published version

Starting with a subclass of the four-dimensional spaces possessing two commuting Killing vectors and a non-trivial Killing tensor, we fully integrate Einstein's vacuum equation with a cosmological constant. Although most of the solutions happen to be already known, we have found a solution that, as far as we could search for, has not been attained before. We also characterize the geometric properties of this new solution, it is a Kundt spacetime of Petrov type II possessing a null Killing vector field and an isometry algebra that is three-dimensional and abelian. In particular, such solution becomes a pp-wave spacetime when the cosmological constant is set to zero., Comment: 11 pages

A gauge theory of the Lorentz group with a mass-dimension one gauge field coupling to matter of any spin is developed. As a completely new feature the "Vierbein" assuring local gauge invariance enters not as an independent dynamical field, but emerges as a functional of the Lorentz gauge field. The underlying geometry of the theory turns out to be a $SO(1,3)$ Banach bundle. The most general action which is renormalizable by power-counting is constructed in terms of the gauge field and its first derivatives. It contains no higher derivative terms in the gauge field which destroy unitarity in the usual renormalizable $R^2$-theories of gravitation. Finally equivalence of the Lorentz gauge field theory coupled to spin zero matter with General Relativity is established, 25 pages

We present the Hamiltonian formalism for $f(T)$ gravity, and prove that the theory has $\frac{n(n-3)}{2}+1$ degrees of freedom (d.o.f.) in $n$ dimensions. We start from a scalar-tensor action for the theory, which represents a scalar field minimally coupled with the torsion scalar $T$ that defines the teleparallel equivalent of general relativity (TEGR) Lagrangian. $T$ is written as a quadratic form of the coefficients of anholonomy of the vierbein. We obtain the primary constraints through the analysis of the structure of the eigenvalues of the multi-index matrix involved in the definition of the canonical momenta. The auxiliary scalar field generates one extra primary constraint when compared with the TEGR case. The secondary constraints are the super-Hamiltonian and supermomenta constraints, that are preserved from the Arnowitt-Deser-Misner formulation of GR. There is a set of $\frac{n(n-1)}{2}$ primary constraints that represent the local Lorentz transformations of the theory, which can be combined to form a set of $\frac{n(n-1)}{2}-1$ first-class constraints, while one of them becomes second-class. This result is irrespective of the dimension, due to the structure of the matrix of the brackets between the constraints. The first-class canonical Hamiltonian is modified due to this local Lorentz violation, and the only one local Lorentz transformation that becomes second-class pairs up with the second-class constraint $\pi \approx 0$ to remove one d.o.f. from the $n^2+1$ pairs of canonical variables. The remaining $\frac{n(n-1)}{2}+2n-1$ primary constraints remove the same number of d.o.f., leaving the theory with $\frac{n(n-3)}{2}+1$ d.o.f. This means that $f(T)$ gravity has only one extra d.o.f., which could be interpreted as a scalar d.o.f., Comment: 15 pages, no figures, comments welcome; references added; typos corrected, matches the published version

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R} \times \mathbb{R}^3$. The Einstein-Hilbert action $S(e,\omega)$ is defined on the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $\omega$, which are the dynamical variables in General Relativity. Define a functional $F_S(\omega)$, by integrating the curvature $d\omega + \omega \wedge \omega$ over the surface $S$, which is $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued. We integrate $F_S(\omega)$ against a holonomy operator of a hyperlink $L$, disjoint from $S$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connections $\omega$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the quantized curvature can be computed from the linking number between $L$ and $S$., Comment: arXiv admin note: text overlap with arXiv:1705.06577, arXiv:1706.01011, arXiv:1705.00396

Unimodular gravity is an interesting approach to address the cosmological constant problem, since the vacuum energy density of quantum fields does not gravitate in this framework, and the cosmological constant appears as an integration constant. These features arise as a consequence of considering a constrained volume element 4-form that breaks the diffeomorphisms invariance down to volume preserving diffeomorphisms. In this work, the first-order formulation of unimodular gravity is presented by considering the spin density of matter fields as a source of spacetime torsion. Even though the most general matter Lagrangian allowed by the symmetries is considered, dynamical restrictions arise on their functional dependence. The field equations are obtained and the conservation laws associated with the symmetries are derived. It is found that, analogous to torsion-free unimodular gravity, the field equation for the vierbein is traceless; nevertheless, torsion is algebraically related to the spin density as in standard Einstein-Cartan theory. The particular example of massless Dirac spinors is studied, and comparisons with standard Einstein-Cartan theory are shown., Comment: 8 pages, 1 Appendix, matches published version

In this paper, the application of the concepts of Cartan connection and covariant derivative to robotics and mechanisms is presented. In particular, how Cartan connection can be used in kinematic calculation for rigid serial manipulators. The ideas were inspired in books of differential geometry, Lie groups and Lie algebras and its applications. The important concept of extended Newton’s law (or covariant formulations of dynamics) is also presented. It is presented first for the case of pure rotations and then for general motion of the robot’s links. Some calculations are presented for the case of a two-link planar robot, which is an example of serial mechanism, and for a planar n-bar mechanism, which is a closed-loop system. Finally, conclusions and future work directions are presented.

In this paper we consider the product of two complex Cartan manifolds, the outcome being a class of product complex Cartan spaces. Then, we study the relationships between the geometric objects of a product complex Cartan space and its components, (e.g. Chern-Cartan complex nonlinear connection, Cartan tensors). By means of these, we establish the necessary and sufficient conditions under which a product complex Cartan space is Landsberg-Cartan or Berwald-Cartan or it has some other properties.

We present an action for chiral $N=(1,0)$ supergravity in $2+2$ dimensions. The fields of the theory are organized into an $OSp(1|4)$ connection supermatrix, and are given by the usual vierbein $V^a$, spin connection $\omega^{ab}$, and Majorana gravitino $\psi$. In analogy with a construction used for $D=10+2$ gauge supergravity, the action is given by $\int STr ({\bf R}^2 {\bf \Gamma})$, where ${\bf R}$ is the $OSp(1|4)$ curvature supermatrix two-form, and ${\bf \Gamma}$ a constant supermatrix containing $\gamma_5$. It is similar, but not identical to the MacDowell-Mansouri action for $D=2+2$ supergravity. The constant supermatrix breaks $OSp(1|4)$ gauge invariance to a subalgebra $OSp(1|2) \oplus Sp(2)$, including a Majorana-Weyl supercharge. Thus half of the $OSp(1|4)$ gauge supersymmetry survives. The gauge fields are the selfdual part of $\omega^{ab}$ and the Weyl projection of $\psi$ for $OSp(1|2)$, and the antiselfdual part of $\omega^{ab}$ for $Sp(2)$. Supersymmetry transformations, being part of a gauge superalgebra, close off-shell. The selfduality condition on the spin connection can be consistently imposed, and the resulting "projected" action is $OSp(1|2)$ gauge invariant., Comment: LaTeX, 10 pages

Einstein's vierbein formulation of general relativity based on the notion of distant parallelism (teleparallelism) naturally introduces a covariant surface term in addition to the Einstein-Hilbert action. We investigate the action principle in teleparallelism with the existence of spacetime boundaries and find that the covariant surface term exactly eliminates all the unwanted surface terms that reside in the metric formulation of general relativity, in the role of a Gibbons-Hawking-York (GHY) term. The identity of such a covariant GHY term is further confirmed by the recovery of the correct black hole entropy from the free energy due to the spacetime boundary. These results indicate that the vierbein formulation of gravity generally exhibits a well-posed action principle and readily admits the path-integral approach to quantization.

A hyperlink is a finite set of non-intersecting simple closed curves in $$\mathbb {R} \times \mathbb {R}^3$$. Let R be a compact set inside $$\mathbb {R}^3$$. The dynamical variables in General Relativity are the vierbein e and a $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$-valued connection $$\omega $$. Together with Minkowski metric, e will define a metric g on the manifold. Denote $$V_R(e)$$ as the volume of R, for a given choice of e. The Einstein–Hilbert action $$S(e,\omega )$$ is defined on e and $$\omega $$. We will quantize the volume of R by integrating $$V_R(e)$$ against a holonomy operator of a hyperlink L, disjoint from R, and the exponential of the Einstein–Hilbert action, over the space of vierbein e and $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$-valued connection $$\omega $$. Using our earlier work done on Chern–Simons path integrals in $$\mathbb {R}^3$$, we will write this infinite-dimensional path integral as the limit of a sequence of Chern–Simons integrals. Our main result shows that the volume operator can be computed by counting the number of nodes on the projected hyperlink in $$\mathbb {R}^3$$, which lie inside the interior of R. By assigning an irreducible representation of $$\mathfrak {su}(2)\times \mathfrak {su}(2)$$ to each component of L, the volume operator gives the total kinetic energy, which comes from translational and angular momentum.

We derive a Tolman-Oppenheimer-Volkoff equation in neutron star systems within the modified $f(T, \mathcal{T})$-gravity class of models using a perturbative approach. In our approach $f(T, \mathcal{T})$-gravity is considered to be a static spherically symmetric space-time. In this instance the metric is built from a more fundamental tetrad vierbein which can be used to relate inertial and global coordinates. A linear function $f = T(r) + \mathcal{T}(r) + \chi h(T, \mathcal{T}) + \mathcal{O}(\chi^{2})$ is taken as the Lagrangian density for the gravitational action. Finally we impose the polytropic equation of state of neutron star upon the derived equations in order to derive the mass profile and mass-central density relations of the neutron star in $f(T, \mathcal{T})$-gravity., Comment: arXiv admin note: text overlap with arXiv:1701.04761

A hyperlink is a finite set of non-intersecting simple closed curves in $\mathbb{R} \times \mathbb{R}^3$. Let $S$ be an orientable surface in $\mathbb{R}^3$. The dynamical variables in General Relativity are the vierbein $e$ and a $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connection $\omega$. Together with Minkowski metric, $e$ will define a metric $g$ on the manifold. Denote $A_S(e)$ as the area of $S$, for a given choice of $e$. The Einstein-Hilbert action $S(e,\omega)$ is defined on $e$ and $\omega$. We will quantize the area of the surface $S$ by integrating $A_S(e)$ against a holonomy operator of a hyperlink $L$, disjoint from $S$, and the exponential of the Einstein-Hilbert action, over the space of vierbeins $e$ and $\mathfrak{su}(2)\times\mathfrak{su}(2)$-valued connections $\omega$. Using our earlier work done on Chern-Simons path integrals in $\mathbb{R}^3$, we will write this infinite dimensional path integral as the limit of a sequence of Chern-Simons integrals. Our main result shows that the area operator can be computed from a link-surface diagram between $L$ and $S$. By assigning an irreducible representation of $\mathfrak{su}(2)\times\mathfrak{su}(2)$ to each component of $L$, the area operator gives the total net momentum impact on the surface $S$., Comment: arXiv admin note: text overlap with arXiv:1701.04397, arXiv:1705.00396

In teleparallel gravity, we apply Lorenz type gauge fixing to cope with redundant degrees of freedom in the vierbein field. This condition is mainly to restore the Lorentz symmetry of the teleparallel torsion scalar. In cosmological application, this technique provides standard cosmology, turnaround, bounce or $\Lambda$CDM as separate scenarios. We reconstruct the $f(T)$ gravity which generates these models. We study the stability of the solutions by analyzing the corresponding phase portraits. Also, we investigate Lorenz gauge in the unimodular coordinates, it leads to unify a nonsingular bounce and standard model cosmology in a single model, where crossing the phantom divide line is achievable through a finite-time singularity of Type IV associated with a de Sitter fixed point. We reconstruct the unimodular $f(T)$ gravity which generates the unified cosmic evolution showing the role of the torsion gravity to establish a healthy bounce scenario., Comment: Minor modifications, some references added to match the published version

Basis tensor gauge theory (BTGT) is a reformulation of ordinary gauge theory that is an analog of the vierbein formulation of gravity and is related to the Wilson line formulation. To match ordinary gauge theories coupled to matter, the BTGT formalism requires a continuous symmetry that we call the BTGT symmetry in addition to the ordinary gauge symmetry. After classically interpreting the BTGT symmetry, we construct using the BTGT formalism the Ward identities associated with the BTGT symmetry and the ordinary gauge symmetry. As a way of testing the quantum stability and the consistency of the Ward identities with a known regularization method, we explicitly renormalize the scalar QED at one-loop using dimensional regularization using the BTGT formalism., Comment: 23 pages

The relativistic generalization of the Newtonian Lagrangian perturbation theory is investigated. In previous works, the perturbation and solution schemes that are generated by the spatially projected gravitoelectric part of the Weyl tensor were given to any order of the perturbations, together with extensions and applications for accessing the nonperturbative regime. We here discuss more in detail the general first-order scheme within the Cartan formalism including and concentrating on the gravitational wave propagation in matter. We provide master equations for all parts of Lagrangian-linearized perturbations propagating in the perturbed spacetime, and we outline the solution procedure that allows one to find general solutions. Particular emphasis is given to global properties of the Lagrangian perturbation fields by employing results of Hodge-de Rham theory. We here discuss how the Hodge decomposition relates to the standard scalar-vector-tensor decomposition. Finally, we demonstrate that we obtain the known linear perturbation solutions of the standard relativistic perturbation scheme by performing two steps: first, by restricting our solutions to perturbations that propagate on a flat unperturbed background spacetime and, second, by transforming to Eulerian background coordinates with truncation of nonlinear terms., Comment: 15 pages, matches published version in PRD

In this work the concept of generally covariant duality is treated with the introduction of generalized Levi-Civita tensor within the framework of vierbein formalism. The equations for the attached Lorentz scalar fields are derived from vierbein postulate. It is shown that the masses of the associated particles including that with negative square mass are completely determined by Einstein’s cosmological constant.

In this brief review we make some remarks on an interesting mathematical method which is, in our opinion, wrongfully undervalued. In many physics textbooks, the Cartan formalism is not explained and students who specialize in another branch of physics or mathematics, do not know the formal elegance of this approach. The aim of this short note is to stimulate curiosity, and it wants to be a starting point to explore this topic

In this work we propose a new gravitational setup formulated in terms of two interacting vierbein fields. The theory is the fully diffeomorphism and local Lorentz invariant extension of a previous construction which involved a fixed reference vierbein. Certain vierbein components can be shifted by local Lorentz transformations and do not enter the associated metric tensors. We parameterize these components by an antisymmetric tensor field and give them a kinetic term in the action, thereby promoting them to dynamical variables. In addition, the action contains two Einstein-Hilbert terms and an interaction potential whose form is inspired by ghost-free massive gravity and bimetric theory. The resulting theory describes the interactions of a massless spin-2, a massive spin-2 and an antisymmetric tensor field. It can be generalized to the case of multiple massive spin-2 fields and multiple antisymmetric tensor fields. The absence of additional and potentially pathological degrees of freedom is verified in an ADM analysis. However, the antisymmetric tensor fluctuation around the maximally symmetric background solution has a tachyonic mass pole., Comment: 17 pages, 4 figures. v2: added extended discussion of tachyonic mass pole

In this article we present the algebraic rearrangement, or matrix inversion of the Dirac equation in a curved Riemann-Cartan spacetime with torsion, the presence of non-vanishing torsion is implied by the intrinsic spin-1/2 of the Dirac field. We then demonstrate how the inversion leads to a reformulation of the fully non-linear and self-interactive Einstein-Cartan-Dirac field equations in terms of Dirac bilinears. It has been known for some decades that the Dirac equation for charged fermions interacting with an electromagnetic field can be algebraically inverted, so as to obtain an explicit rational expression of the four-vector potential of the gauge field in terms of the spinors. Substitution of this expression into Maxwell's equations yields the bilinear form of the self-interactive Maxwell-Dirac equations. In the present (purely gravitational) case, the inversion process yields \emph{two} rational four-vector expressions in terms of Dirac bilinears, which act as gravitational analogues of the electromagnetic vector potential. These "potentials" also appear as irreducible summand components of the connection, along with a traceless residual term of mixed symmetry. When taking the torsion field equation into account, the residual term can be written as a function of the object of anholonomity. Using the local tetrad frame associated with observers co-moving with the Dirac matter, a generic vierbein frame can described in terms of four Dirac bilinear vector fields, normalized by a scalar and pseudoscalar field. A corollary of this is that in regions where the Dirac field is non-vanishing, the self-coupled Einstein-Cartan-Dirac equations can in principle be expressed in terms of Dirac bilinears only., 21 pages, updated version accepted for publication

In this paper, we flnd a relation between the norm of Cartan and mean Cartan torsions of Finsler metrics deflned by a Riemannian metric and a 1-form on a manifold. Then, we flnd a subclass of these metrics which have bounded Cartan torsion. It turns out that every C-reducible Finsler metric has bounded Cartan torsion.