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.
The dual Komar mass generalizes the concept of the NUT parameter and is akin to the magnetic charge in electrodynamics. In asymptotically flat spacetimes it coincides with the dual supertranslation charge. The dual mass vanishes identically on Riemannian manifolds in General Relativity unless conical singularities corresponding to Misner strings are introduced. In this paper we propose an alternative way to source the dual mass locally. We show that this can be done by enlarging the phase space of the theory to allow for a violation of the algebraic Bianchi identity using local fields. A minim…
We discuss the general structure of metric geometries, and how metricity implies the complete antisymmetry of Cartan tensor; an application in the frame of Lie group theory is given. Interpretations of the completely antisymmetric torsion in physical models are reviewed.
We review how the large $c$ expansion of General Relativity leads to an effective theory in the form of Twistless Torsional Newton-Cartan gravity. We show how this is a strong field expansion around the static sector of General Relativity and illustrate this through two examples.
We re-formulate the notion of a Newton-Cartan manifold and clarify the compatibility conditions of a connection with torsion with the Newton-Cartan structure.
Using the recently advanced Galilean gauge theory (GGT) we give a comprehensive construction of torsional Newton Cartan geometry. The coupling of a Galilean symmetric model with background NC geometry following GGT is illustrated by a free nonrelativistic scalar field theory. The issue of spatial diffeomorphisn \cite{SW, BMM3} is focussed from a new angle. The expression of the torsionful connection is worked out which is in complete parallel with the relativistic theory. Also smooth transition of the connection to its well known torsionless expression is demonstrated. A complete (implicit) ex…
A sequence of generalizations of Cartan's conservation of torsion theorem is given for n-dimensional differentiable manifolds having a general linear connection.
A relativistic theory constructed on Riemann-Cartan manifold with a derived totally antisymmetric torsion is proposed. It follows the coincidence of the autoparallel curve and metric geodesic. The totally antisymmetric torsion naturally appears in the theory without any ad hoc imposed on.
We derive a torsionfull version of three-dimensional N=2 Newton-Cartan supergravity using a non-relativistic notion of the superconformal tensor calculus. The "superconformal" theory that we start with is Schrödinger supergravity which we obtain by gauging the Schrödinger superalgebra. We present two non-relativistic N=2 matter multiplets that can be used as compensators in the superconformal calculus. They lead to two different off-shell formulations which, in analogy with the relativistic case, we call "old minimal" and "new minimal" Newton-Cartan supergravity. We find similarities but also …
In this paper, we formulate the Mimetic theory of gravity in first-order formalism for differential forms, i.e., the mimetic version of Einstein-Cartan-Kibble-Sciama (ECKS) gravity. We consider different possibilities on how torsion is affected by conformal transformations and discuss how this translates into the interpolation between two different conformal transformations of the spin connection, parameterized with a zero-form parameter $λ$. We prove that regardless of the type of transformation one chooses, in this setting torsion remains as a non-propagating field. We also discuss the conse…
This article provides us with a unifying classification of the conformal infinitesimal symmetries of non-relativistic Newton-Cartan spacetime. The Lie algebras of non-relativistic conformal transformations are introduced via the Galilei structure. They form a family of infinite-dimensional Lie algebras labeled by a rational "dynamical exponent", $z$. The Schrödinger-Virasoro algebra of Henkel et al. corresponds to $z=2$. Viewed as projective Newton-Cartan symmetries, they yield, for timelike geodesics, the usual Schrödinger Lie algebra, for which z=2. For lightlike geodesics, they yield, in tu…
This paper reviews the Einstein Cartan theory (ECT), the famous extension of general relativity (GR) in presence of spacetime torsion. The vacuum equations are derived step by step. Vielbein formulation is discussed for determining the field equations in presence of matter. This review would be easily comprehensible for any student familiar with general relativity. Further, ECT is used to describe superstrings with intrinsic torsion, assuming a $D_p$-brane in presence of a curved background of the NS-NS Kalb-Ramond field. D-brane worldvolume is a flat spacetime governed by the Dirac-Born-Infel…
Gravitational stability of torsion and inflaton field in a four-dimensional spacetime de Sitter solution in scalar-tensor cosmology where Cartan torsion propagates is investigated in detail. Inflaton and torsion evolution equations are derived by making use of a Lagrangean method. Stable and unstable modes for torsion and inflatons are found to be dependent of the background torsion and inflaton fields. Present astrophysical observations favour a stable mode for torsion since this would explain why no relic torsion imprint has been found on the Cosmic Background Radiation in the universe.
The influence of space-time torsion on gravitational interaction at cosmological and astrophysical scales is discussed within the framework of gauge gravitation theory in Riemann-Cartan space-time. It is shown that the interaction of the vacuum torsion with proper angular momentums of gravitating objects leads to appearance of additional gravitational force which can be manifested at astrophysical scale.
This article starts with the mathematical definition, concrete description, and physical meaning of Cartan's torsion. I proceed with the argumentation that torsion is required for the description of intrinsic spin. Moreover I argue that the duality between curvature and torsion is analogous to the duality between electricity and magnetism. I conclude this article by pointing out that the aligned rotation axes of the galaxies of the Perseus-Pisces supercluster may be interpreted as a topological defect generated by torsion.
We revisit the manifestly covariant large $c$ expansion of General Relativity, $c$ being the speed of light. Assuming the relativistic connection has no pole in $c^{-2}$, this expansion is known to reproduce Newton-Cartan gravity and a covariant version of Post-Newtonian corrections to it. We show that relaxing this assumption leads to the inclusion of twistless torsion in the effective non-relativistic theory. We argue that the resulting TTNC theory is an effective description of a non-relativistic regime of General Relativity that extends Newtonian physics by including strong gravitational t…
On the basis of an algebraic relation between torsion and a classical spinor field a new interpretation of Einstein-Cartan gravity interacting with classical spinor field is proposed. In this approach the spinor field becomes an auxiliary field and the dynamical equation for this field (the Heisenberg equation) is a dynamical, gravitational equation for torsion. The simplest version of this theory is examined where the metric degrees of freedom are frozen and only torsion plays a role. A spherically symmetric solution of this theory is examined. This solution can be interpreted, in the spirit …
We review the application of torsion in field theory. First we show how the notion of torsion emerges in differential geometry. In the context of a Cartan circuit, torsion is related to translations similar as curvature to rotations. Cartan's investigations started by analyzing Einsteins general relativity theory and by taking recourse to the theory of Cosserat continua. In these continua, the points of which carry independent translational and rotational degrees of freedom, there occur, besides ordinary (force) stresses, additionally spin moment stresses. In a 3-dimensional continuized crysta…
Inflation in the framework of Einstein-Cartan theory is revisited. Einstein-Cartan theory is a natural extension of the General Relativity, with non-vanishing torsion. The connection on Riemann-Cartan spacetime is only compatible with the cosmological principal for a particular form of torsion. We show this form to also be compatible with gauge invariance principle for a non-Abelian and Abelian gauge fields under a certain deviced minimal coupling procedure. We adopt an Abelian gauge field in the form of "cosmic triad". The dynamical field equations are obtained and shown to sustain cosmic inf…
The present paper analyses the Einstein-Cartan theory of gravitation with Elko spinors as sources of curvature and torsion. After minimally coupling the Elko spinors to torsion, the spin angular momentum tensor is derived and its structure is discussed. It shows a much richer structure than the Dirac analogue and hence it is demonstrated that spin one half particles do not necessarily yield only an axial vector torsion component. Moreover, it is argued that the presence of Elko spinors partially solves the problem of minimally coupling Maxwell fields to Einstein-Cartan theory.
Locally Rotationally Symmetric Spacetimes in Einstein-Cartan Theory and Their Classification
Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院理论物理研究所）；DIME Sez. Metodi e Modelli Matematici, Università di Genova（热那亚大学DIME数学方法与模型系）；INFN Sezione di Genova（意大利国家核物理研究所热那亚分部）
We introduce a definition of symmetry generating vector fields on manifolds which are equipped with a first-order reductive Cartan geometry. We apply this definition to a number of physically motivated examples and show that our newly introduced notion of symmetry agrees with the usual notions of symmetry of affine, Riemann-Cartan, Riemannian and Weizenböck geometries, which are conventionally used as spacetime models. Further, we discuss the case of Cartan geometries which can be used to model observer space instead of spacetime. We show which vector fields on an observer space can be interpr…
A modification of Kaluza-Klein theory is proposed in which, as a result of a symmetry breaking, five-dimensional space-time is partially parallelized implying the appearance of torsion fields. A naturally chosen action functional leads to the Einstein-Cartan-Maxwell theory where the electromagnetic field strength is represented by the fifth component of the torsion 2-form. Incorporation of a scalar field reveals that four-dimensional space-time torsion is not induced by the scalar field.
Recently, gravitational gauge theories with torsion have been discussed by an increasing number of authors from a classical as well as from a quantum field theoretical point of view. The Einstein-Cartan(-Sciama-Kibble) Lagrangian has been enriched by the parity odd pseudoscalar curvature (Hojman, Mukku, and Sayed) and by torsion square and curvature square pieces, likewise of even and odd parity. (i) We show that the inverse of the so-called Barbero-Immirzi parameter multiplying the pseudoscalar curvature, because of the topological Nieh-Yan form, can only be appropriately discussed if torsion…
In the present paper we consider a theory of gravity in which not only curvature but also torsion is explicitly present in the Lagrangian, both with their own coupling constant. In particular, we discuss the couplings to Dirac fields and spin fluids: in the case of Dirac fields, we discuss how in our approach, the Dirac self-interactions depend on the coupling constant as a parameter that may even make these non-linearities manifest at subatomic scales, showing different applications according to the value of the parameter we have assigned; in the case of spin fluids, we discuss FLRW cosmologi…
The thermodynamics is extented to spacetimes with spin-torsion density.Impplications to Einstein-Cartan-de Sitter inflationary phases are discussed.A relation between the spin-torsion density,entropy and temperature is presented.A lower limit for the radius of the Universe may be obtained from the spin-torsion density and the Planck lenght.
In this work a static solution of Einstein-Cartan (EC) equations in 2+1 dimensional space-time is given by considering classical spin-1/2 field as external source for torsion of the space-time. Here, the torsion tensor is obtained from metricity condition for the connection and the static spinor field is determined as the solution of Dirac equation in 2+1 spacetime with non-zero cosmological constant and torsion. The torsion itself is considered as a non-dynamical field.
In this work, we investigate the effects of the torsion-fermionic interaction on the energy levels of fermions within a Riemann-Cartan geometry using a model-independent approach. We consider the case of fermions minimally coupled to the background torsion as well as non-minimal extensions via additional couplings with the vector and axial fermionic currents which include parity-breaking interactions. In the limit of zero-curvature, and for the cases of constant and spherically symmetric torsion, we find a Zeeman-like effect on the energy levels of fermions and anti-fermions depending on wheth…
Metric perturbations the stability of solution of Einstein-Cartan cosmology (ECC) are given. The first addresses the stability of solutions of Einstein-Cartan (EC) cosmological model against Einstein static universe background. In this solution we show that the metric is stable against first-order perturbations and correspond to acoustic oscillations. The second example deals with the stability of de Sitter metric also against first-order perturbations. Torsion and shear are also computed in these cases. The resultant perturbed anisotropic spacetime with torsion is only de Sitter along one dir…
We address the implementation of the cosmological principle, that is, the assumption of homogeneity and isotropy in the spatial distribution of matter in the Universe, within the context of Einstein-Cartan theory including minimal couplings of both Dirac and Maxwell fields to torsion. This theory gives rise to new physical effects in environments of high spin densities while leaving the vacuum dynamics unaffected. Our approach is to impose the cosmological principle from the onset to the geometrical degrees of freedom (metric and torsion functions), which constrains the torsion components and …
The Einstein-Cartan equations in first-order action of torsion are considered. From Belinfante-Rosenfeld equation special consistence conditions are derived for the torsion parameters relating them to the metric. Inside matter the torsion is given by the spin which leads to an extended Oppenhaimer-Volkov equation. Outside matter a second solution is found besides the torsion-free Schwarzschild one with the torsion completely determined by the metric and vice-versa. This solution is shown to be of non-spherical origin and its uniqueness with respect to the consistence is demonstrated. Unusual p…
Regularization of Gauss-Bonnet Gravity in Riemann-Cartan Geometry
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（基础物理与数学科学学院，杭州高等研究院，UCAS）；University of Chinese Academy of Sciences（中国科学院大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University（湖南师范大学物理与量子效应及应用协同创新中心）
The gravitational weak field of a global monopole in the Einstein-Cartan theory of gravity is investigated.To obtain this solution we assume that Cartan torsion takes the form of the Newtonian gravitational potential.From the geodesics it is possible to show that the torsionic monopole produces a repulsive gravitational force.
Recently it has been established that torsional Newton-Cartan (TNC) geometry is the appropriate geometrical framework to which non-relativistic field theories couple. We show that when these geometries are made dynamical they give rise to Horava-Lifshitz (HL) gravity. Projectable HL gravity corresponds to dynamical Newton-Cartan (NC) geometry without torsion and non-projectable HL gravity corresponds to dynamical NC geometry with twistless torsion (hypersurface orthogonal foliation). We build a precise dictionary relating all fields (including the scalar khronon), their transformations and oth…
Dynamical Resolution of the Cosmic Coincidence Problem in Non-Interacting Holographic Dark Energy via Einstein-Cartan Torsion
Graduate School Department of Physics, Daejin University（大真大学物理学院研究生院）；School of Liberal Arts and Sciences, Korea National University of Transportation（韩国交通大学本科文理学院）；School of Physics, Korea Institute for Advanced Study（韩国高等研究院物理学院）
We use the conservation law of the stress-energy and spin tensors to study the motion of massive brane-like objects in Riemann-Cartan geometry. The world-sheet equations and boundary conditions are obtained in a manifestly covariant form. In the particle case, the resultant world-line equations turn out to exhibit a novel spin-curvature coupling. In particular, the spin of a zero-size particle does not couple to the background curvature. In the string case, the world-sheet dynamics is studied for some special choices of spin and torsion. As a result, the known coupling to the Kalb-Ramond antis…
We consider a gravitational model in a Weyl-Cartan space-time, in which the Weitzenböck condition of the vanishing of the sum of the curvature and torsion scalar is also imposed. Moreover, a kinetic term for the torsion is also included in the gravitational action. The field equations of the model are obtained from a Hilbert-Einstein type variational principle, and they lead to a complete description of the gravitational field in terms of two fields, the Weyl vector and the torsion, respectively, defined in a curved background. The cosmological applications of the model are investigated for a …
We reformulate the general theory of relativity in the language of Riemann-Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan…
Cosmologies based on General Relativity encompassing an anti-symmetric connection (torsion) can display nice desirable features as the absence of the initial singularity and the possibility of inflation in the early stage of the universe. After briefly reviewing the standard approach to the cosmology with torsion, we generalize it to demonstrate that several theories of torsion gravity are possible using different choices of the diffeomorphic invariants in the Lagrangians. As a result, distinct cosmologies emerge. In all of them it is possible that the universe avoids the initial singularity a…
The dynamics of a gravitational torsion kink as a plane symmetric thick domain wall solution of Einstein-Cartan (EC) field equation is given. The spin-torsion energy has to be as high as the gravitational kink potential otherwise torsion will not contribute as an appreciable effect to domain wall.Cartan torsion also contributes to the orthonal pressure of the domain wall.
A Cartan-geometrical perspective on torsion and non-metricity
Scuola Superiore Meridionale（南方高等学院）；INFN– Sezione di Napoli（那不勒斯意大利国家核物理研究所）；University of Tartu（塔尔图大学）；Institute of Theoretical Physics and Astrophysics, University of Gdańsk（格但斯克大学理论物理与天体物理研究所）
We derive the most general effective low-energy potential to order O(1/m) for slow Dirac fermions with mass m, coupled to gravitational, chameleon and torsion fields in the Einstein-Cartan gravity. The obtained results can be applied to the experimental analysis of gravitational, chameleon and torsion interactions in terrestrial laboratories. We discuss the use of rotating coordinate systems, caused by rotations of devices, for measurements of the torsion vector and tensor components, caused by minimal torsion--fermion couplings (Ivanov and Wellenzohn, Phys. Rev. D92, 065006 (2015)). Using the…
The role of space-time torsion in general relativity is reviewed in accordance with some recent results on the subject. It is shown that, according to the connection compatibility condition, the usual Riemannian volume element is not appropriate in the presence of torsion. A new volume element is proposed and used in the Lagrangian formulation for Einstein-Cartan theory of gravity. The dynamical equations for the space-time geometry and for matter fields are obtained, and some of their new predictions and features are discussed. In particular, one has that torsion propagates and that gauge fie…
Einstein-Cartan theory is an extension of the standard formulation of General Relativity characterized by a non-vanishing torsion. The latter is sourced by the matter fields via the spin tensor, and its effects are expected to be important at very high spin densities. In this work we analyze in detail the physics of Einstein-Cartan theory with Dirac and Maxwell fields minimally coupled to the spacetime torsion. This breaks the $U(1)$ gauge symmetry, which is suggested by the possibility of a torsion-induced phase transition in the early Universe. The resulting Dirac-like and Maxwell-like equat…
A global monopole in dilatonic Einstein-Cartan gravity is presented. A linearized solution representing a global monopole interacting with a massless dilaton is found where Cartan torsion does not interact with the monopole Higgs field.Computation of the geodesic equation shows that the monopole-dilaton system generates a repulsive gravitational field.The solution is shown to break the linear approximation for certain values of torsion.
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 prop…
We analyse the kinematics of cosmological spacetimes with nonzero torsion, in the framework of the classical Einstein-Cartan gravity. After a brief introduction to the basic features of spaces with non-vanishing torsion, we consider a family of observers moving along timelike worldlines and focus on their kinematic behaviour. In so doing, we isolate the irreducible variables monitoring the observers' motion and derive their evolution formulae and associated constraint equations. Our aim is to identify the effects of spacetime torsion, and the changes they introduce into the kinematics of the s…
In general relativity the affine connection is required to be symmetric so torsion is zero while according to the Einsten- Cartan's theory torsion is connected to the spin tensor as expressed by the Cartan's equations. We consider the theory of spinors in general relativity in the light of the results of Einstein Cartan's theory.In general relativity the affine connection is required to be symmetric so torsion is zero while according to the Einsten- Cartan's theory torsion is connected to the spin tensor as expressed by the Cartan's equations. We consider the theory of spinors in general relat…
We study the Weyl vector fields which can play an important role in quantum gravity. The metric obtains its dynamical content after dynamical symmetry breaking in the phase of the effective Einstein gravity which is induced by quantum Weyl corrections. In low energy regime with scalar field there is a relation between the Weyl vector fields and the torsion fields. If this condition is given to Weyl vector fields and torsions, then the Lagrangian becomes like Maxwell type.
The "metric" structure of nonrelativistic spacetimes consists of a one-form (the absolute clock) whose kernel is endowed with a positive-definite metric. Contrarily to the relativistic case, the metric structure and the torsion do not determine a unique Galilean (i.e. compatible) connection. This subtlety is intimately related to the fact that the timelike part of the torsion is proportional to the exterior derivative of the absolute clock. When the latter is not closed, torsionfreeness and metric-compatibility are thus mutually exclusive. We will explore generalisations of Galilean connection…
We analyse a spin precession of slow neutrons in the Einstein-Cartan gravity with torsion, chameleon and magnetic field. For the derivation of the Heisenberg equation of motion of the neutron spin we use the effective low-energy potential, derived by Ivanov and Wellenzohn (Phys. Rev. D92, 125004 (2015)) for slow neutrons, coupled to gravitational, chameleon and torsion fields to order 1/m, where m is the neutron mass. In addition to this low-energy interactions we switch on the interaction of slow neutrons with a magnetic field. We show that to linear order approximation with respect to gravit…
We analyze the projected effective Einstein equation in a 4-dimensional arbitrary manifold embedded in a 5-dimensional Riemann-Cartan manifold. The Israel-Darmois matching conditions are investigated, in the context where the torsion discontinuity is orthogonal to the brane. Unexpectedly, the presence of torsion terms in the connection does not modify such conditions whatsoever, despite of the modification in the extrinsic curvature and in the connection. Then, by imposing the Z_2-symmetry, the Einstein equation obtained via Gauss-Codazzi formalism is extended, in order to now encompass the to…
Fluctuations on de Sitter solution of Einstein-Cartan field equations are obtained in terms of the matter density primordial density fluctuations and spin-torsion density and matter density fluctuations obtained from COBE data. Einstein-de Sitter solution is shown to be unstable even in the absence of torsion.The spin-torsion density fluctuation is simply computed from the Einstein-Cartan equations and from COBE data.
Distributional sources of cosmic walls crossed by cosmic strings from Riemann-Cartan (RC) Geometry. The matter density of the planar wall is maximum at the point where the cosmic string crosses the cosmic wall. Cartan torsion is has a support on the cosmic string given by the Dirac $ δ$-function. Off the sources are left with a torsionless vacuum.
In (1+2)-dimensional Poincaré gauge gravity, we start from a Lagrangian depending on torsion and curvature which includes additionally {\em translational} and {\em Lorentzian} Chern-Simons terms. Limiting ourselves to to a specific subcase, the Mielke-Baekler (MB) model, we derive the corresponding field equations (of Einstein-Cartan-Chern-Simons type) and find the general vacuum solution. We determine the properties of this solution, in particular its mass and its angular momentum. For vanishing torsion, we recover the BTZ-solution. We also derive the general conformally flat vacuum solution …
We use the Papapetrou method of multipole expansion to show that a Dirac field in the Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity cannot form singular configurations concentrated on one- or two-dimensional surfaces in spacetime. Instead, such a field describes a nonsingular particle whose spatial dimension is at least on the order of its Cartan radius. In particular, torsion modifies Burinskii's model of the Dirac electron as a Kerr-Newman singular ring of the Compton size, by replacing the ring with a toroidal structure with the outer radius of the Compton size and the inner radius…
