Riemann-Cartan Spacetime Research Articles

Boundary terms, branes, and AdS/BCFT duality in first-order gravity

We provide an account of the issue of Gibbons-Hawking-York-like boundary terms for a gravity theory defined on a Riemann-Cartan spacetime. We further discuss different criteria for introducing boundary terms in some familiar first-order gravity theories with both on-shell vanishing and nonvanishing torsion, along with considerations regarding the thermodynamics of black holes and profiles of the end-of-the-world branes. Our analysis confirms the expected geodesic profile of the end-of-the-world brane in the BF formulation of Jackiw-Teitelboim gravity. Finally, we present the first realization of the AdS/BCFT duality for spacetime with torsion. Published by the American Physical Society 2024

Generalized Mathisson–Papapetrou–Tulczyjew–Dixon equations

We derive two generalizations of Mathisson–Papapetrou–Tulczyjew–Dixon equations from Casalbuoni–Brink–Schwarz type pseudoclassical Lagrangians of Majorana spinors on a Riemann–Cartan spacetime. One has an additional gauge freedom, which makes the equations of motion also be a generalization of Wong’s equations. The other is a spinor model coupled with a Rarita–Schwinger field to preserve the supersymmetry. The coupling to the torsion is modified due to the existence of the Rarita–Schwinger field. In both extensions, the Tulczyjew condition is automatically satisfied.

Confined chiral dynamo action in torsional dilatonic DWs from CME discontinuities

In this paper, spin polarized domain walls in Einstein–Cartan (EC) gravity have been recently investigated [L. C. Garcia de Andrade, Einstein–Cartan non-supersymmetric dynamo walls, Ann. Phys. 432 (2021).], where the spin polarizes along the direction orthogonal to planar domain wall. In this case, dynamo effects may appear on the wall. Moreover, dilatonic domain walls (DDWs) have shown to be responsible for inflating DWs in EC gravity, where DWs were non-magnetized [L. Garcia de Andrade, Axial and chiral anomalies in Riemann–Cartan spacetime black holes and electrodynamics, Class. Quantum Grav. 16 (1999) 2097–2103.]. In this paper, our goal is to show that when we consider a magnetized DW in the EC–Maxwell cosmology, the dilaton potential in the form of a DDW potential expressed in terms of torsion and the magnetic energy density may drive dynamo effects. Analytical solution of the dynamo equation shows that depending on the handness of DDW metric, the magnetic fields are amplified by dynamo action. From the coupled system of ECM and dilaton field equations, we find a relation between the magnetic energy density of [Formula: see text] and the DDW torsion of [Formula: see text] which leads to a thin DDW thickness of [Formula: see text]. For a galactic magnetic field, energy density of [Formula: see text] leads to a thickness of [Formula: see text], a much thinner DDW. For galactic dynamo amplification, the ratio r between the magnetic energy density and photon energy density is around [Formula: see text]. From this ratio, a relation between photon mass density and torsion is obtained. Chiral Magnetic effect (CME), torsion and magnetic discontinuities along the wall are investigated. Magnetic field discontinuity across the wall allows us to understand why the magnetic fields at DDW must be located along the wall since the orthogonal magnetic field may vanish.

Limiting energy density and gravity in Riemann-Cartan space-time

The gravitational interaction is discussed within the framework of gauge gravitational theoryin the Riemann-Cartan space-time. In the case of spatially homogeneous isotopic gravitatingsystems the gravitational repulsion at extreme conditions near limiting energy density and thetransition from gravitational repulsion to attraction in dependence on energy density isstudied. The conversion to Friedman regime and the transition to gravitational repulsion atvery small energy densities is analyzed.

Local Conservation Laws in a Nonlinear Electrodynamics

By using a generalization of the Lie–Poisson brackets for the dual Maxwell and Born–Infeld field strength tensors, we construct the gauge invariant axial-vector conserved currents for Born–Infeld and Heisenberg–Euler nonlinear electrodynamics in the 4-dimensional Minkowski space-time. The infinite hierarchies of the currents given by Lie brackets for generally covariant conserved vector and axial vector currents are established. These currents are conserved upon action of the gravitational fields, but the conservation is broken in the Einstein–Cartan theory (over a Riemann–Cartan space-time). The axial-vector currents are conserved only in the (3 + 1)-dimensional space-time.

Dynamo seeds from gravitational torsional anomalies and de Sitter magnetized metrics

Recently gravitational and Nieh–Yan (NY) chiral anomalies have been obtained in Riemann–Cartan space–time (L.C. Garcia de Andrade. Class Quantum Grav. 38(6), 065005 (2021). doi: 10.1088/1361-6382/abd25f ), where electrodynamics is encoded in the metric geometry. In this paper we follow the same pathway by obtaining a class of deformed de Sitter metrics in teleparallelism. The existence of the unmagnetized de Sitter metric (DSMM) without axial anomalies is proved. Unified theories à la Einstein, Eddington, and Schroedinger, called modified de Sitter metrics, present some novel features. First, we show that a pure DSMM in T4 does not induce gravitational anomalies. This is a motivation to study modifications of DSMM. NY torsional anomaly in DSMM teleparallel T4 geometry is shown to vanish in all cases. Gravitational non-trivial anomalies are obtained from these metrics. Torsional anomaly, much used in condensed matter physics, does not vanish. From these deformed DSMM, we show that a dynamo equation with torsional gradient sources is valid from class III of the metrics but is torsionless sourced in class II. We show that in the gravitational anomaly of new deformed de Sitter metric one may cancel the gravitational anomaly, by a proper choice of the metric function. The axial anomaly is obtained for some metric deformation as well. A simple deformation leads to the existence of the NY density in the case of DSMM. This would be class IV of DSMM.

Imprints from a Riemann–Cartan space-time on the energy levels of Dirac spinors

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 whether they are aligned/anti-aligned with respect to the axial vector part of the torsion (or to specific combination of torsion quantities), and determine the corresponding fine-structure energy transitions. We also discuss non-minimal couplings between fermionic fields and torsion within the Einstein–Cartan theory and its extension to include the (parity-breaking) Holst term. Finally we elaborate on the detection of torsion effects related to the splitting of energy levels in astrophysics, cosmology and solid state physics using current capabilities.

Gauge-field-induced torsion and cosmic inflation

In this paper, inflation in the framework of Einstein–Cartan theory is revisited. Einstein–Cartan theory is a natural extension of the General Relativity with nonvanishing torsion. The connection on Riemann–Cartan space–time is only compatible with the cosmological principal for a particular form of torsion. We also show this form to be compatible with gauge invariance principle for non-Abelian and Abelian gauge fields under a certain deviced 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 inflation with a large number of e-folds. We emphasize that at the end of inflation, torsion vanishes and the theory of Einstein–Cartan reduces to the General Relativity with the usual FRW geometry.

Unified theories metrics and primordial magnetic fields in Riemann–Cartan spacetime

Several spacetime metrics in teleparallel geometry of Einstein like unified field theory (UFT) are presented. Cosmic magnetic fields are obtained in terms of the early universe torsion and other stages of the universe. For example, in one of the metrics, integration of the 2-form torsion 0-component is written in terms of magnetic flux, from Faraday’s induced equation. These ideas were obtained from a recent paper we published [Class. Quantum Grav. (2015)] on non-stationary teleparallel metrics, where at coherent length of [Formula: see text] a magnetic field of [Formula: see text] is obtained. At early universe, a torsion of [Formula: see text] leads us to a magnetic field of the order of [Formula: see text] which is weaker than the Bierman battery effect magnetic field of the order of [Formula: see text]. Hence this new metric indicates that unifield theory metrics a la Schrödinger may be used to obtain primordial magnetic fields. Other tests of this metric led to the nowadays magnetic field of [Formula: see text] from the torsion at present universe (at the laboratory using dual maser obtained by Kostelecky et al. [Phys. Rev. Lett.]) of [Formula: see text]. Cartan torsion has been frequently associated to topological defects in crystals or in pseudo-magnetic torsional fields. In this paper, we discuss how from teleparallel gravity one may obtain a theory of electromagnetism from metrics in spacetime. Topological defects given by Letelier [Class. Quantum Grav. 12 (1995) 1133] and Tod [Class. Quantum Grav. 11(5) (1994)] metrics can also be associated with pseudo-magnetic fields. Electromagnetism is geometrized via a bimetric theory of gravity where one metric is responsible for Ampere’s law and the other by the Faraday induction equation which gives rise to dynamo equation.

Photon mass limits driven by axions, Lorentz violation, chiral and gravitational anomalies in non-Riemannian spaces, and helical dynamo seeds

In this paper two types of applications of Cartan torsion to cosmology and quantum field theory are given. The first is given by massive photon axial torsion mixing through non-minimal coupling. It is shown that chiral anomalies induce upper limits on the photon mass and galactic dynamo seeds. In this example we obtain magnetic field seeds of the order of 10−16 G from a strong chiral battery magnetic field of the order of 1030 G. This cosmic magnetic field is in accordance with previous magnetic dynamo seed limits as 10−22 G ≤ Bseed ≤ 10−12 G given by Barrow et al. (Phys. Rev. D, 86, 023533 (2012). doi: 10.1103/PhysRevD.86.023533 ). In this example, as in the Prokopec et al. (Phys. Rev. Lett. 89, 101301 (2002). doi: 10.1103/PhysRevLett.89.101301 ) torsionless case, which associates Dolgov anomalies to photon mass limits, we show that at early universe scales, 10−32 eV is a very stringent limit. Here we can show that the parameter [Formula: see text], which is well within the lower bound of r needed for the magnetic field to seed galactic dynamos. Helical magnetic fields are obtained. One can also derive massive photon limits from axial chiral anomalies with torsion degrees of freedom (Duncan et al. Nucl. Phys. B, 387, 215 (1992). doi: 10.1016/0550-3213(92)90052-D ) leading to photon mass of the order of 10−11 eV. Dynamo equations are obtained from anomalous Maxwell equations as well as magnetic wave equations with photon mass, which are analytically solved. In the second example one shows that Chern–Simons (CS) current is affected by the presence of minimal coupled torsion. In this case as well one obtains a photon mass limit of the order 10−18 eV. Gravitational anomalies in Riemann–Cartan spacetime with axial torsion are also obtained. Recently, Terrano et al. (Phys. Rev. Lett. 122, 231301 (2019). doi: 10.1103/PhysRevLett.122.231301 ) computed dark matter mass constraints of the order of 10−23 eV from spin-polarised torsion balances, which has given some support for torsion theories of Einstein–Cartan type. Here, however, we deal with torsion with dynamical degrees of freedom. Lorentz violation in non-Riemannian CS currents are investigated.

Emergent Spacetime and Gravitational Nieh-Yan Anomaly in Chiral p+ip Weyl Superfluids and Superconductors.

Momentum transport is anomalous in chiral p+ip superfluids and superconductors in the presence of textures and superflow. Using the gradient expansion of the semiclassical approximation, we show how gauge and Galilean symmetries induce an emergent curved spacetime with torsion and curvature for the quasirelativistic low-energy Majorana-Weyl quasiparticles. We explicitly show the emergence of the spin connection and curvature, in addition to torsion, using the superfluid hydrodynamics. The background constitutes an emergent quasirelativistic Riemann-Cartan spacetime for the Weyl quasiparticles which satisfy the conservation laws associated with local Lorentz symmetry restricted to the plane of uniaxial anisotropy of the superfluid (or superconductor). Moreover, we show that the anomalous Galilean momentum conservation is a consequence of the gravitational Nieh-Yan (NY) chiral anomaly the Weyl fermions experience on the background geometry. Notably, the NY anomaly coefficient features a nonuniversal ultraviolet cutoff scale Λ, with canonical dimensions of momentum. Comparison of the anomaly equation and the hydrodynamic equations suggests that the value of the cutoff parameter Λ is determined by the normal state Fermi liquid and nonrelativistic uniaxial symmetry of the p-wave superfluid or superconductor.

Gravitational interaction in astrophysics in Riemann-Cartan space-time and vacuum torsion

The role of space-time torsion in astrophysics in the frame of minimum gauge gravitational theory in Riemann-Cartan space-time is discussed. The influence of the vacuum torsion on dynamics of astrophysical objects induced by the interaction of their spin momenta with vacuum torsion is studied. Possible manifestations of such interaction are considered.

Exact Solutions in Poincaré Gauge Gravity Theory

In the framework of the gauge theory based on the Poincaré symmetry group, the gravitational field is described in terms of the coframe and the local Lorentz connection. Considered as gauge field potentials, they give rise to the corresponding field strength which are naturally identified with the torsion and the curvature on the Riemann–Cartan spacetime. We study the class of quadratic Poincaré gauge gravity models with the most general Yang–Mills type Lagrangian which contains all possible parity-even and parity-odd invariants built from the torsion and the curvature. Exact vacuum solutions of the gravitational field equations are constructed as a certain deformation of de Sitter geometry. They are black holes with nontrivial torsion.

Introdução à Teoria de Gauge da Gravitação

The fundamental interactions of nature, the electroweak and the quantum chromodynamics, are described in the Standard Model by the Gauge Theory under internal symmetries that maintain the invariance of the functional action. The fundamental interaction of gravitation is very well described by Einstein's General Relativity in a Riemannian spacetime metric, but General Relativity has been over time a gravitational field theory apart from the Standard Model. The theory of Gauge allows under symmetries of the group of Poincar\'e to impose invariances in the functional of the action of the spinor field that result in the gravitational interaction with the fermions. In this approach the gravitational field, besides being described by the equation similar to General Relativity, also brings a spin-gravitational interaction in a Riemann-Cartan spacetime.

About gravitational interaction in astrophysics in Riemann–Cartan space-time

The gravitational interaction at astrophysical scale is discussed within the framework of minimum gauge gravitation theory in Riemann–Cartan space-time. It is shown that the interaction of the vacuum torsion with the spin momenta of gravitating objects can lead into appearance of additional gravitational force in non-relativistic approximation, that can be manifested on astrophysical scale (galaxies, galactic clusters).

The self-coupled Einstein–Cartan–Dirac equations in terms of Dirac bilinears

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 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.

Revisiting wormhole energy conditions in Riemann-Cartan spacetimes and under Weyl transformations

We show that the violation of the null energy condition by matter, required by traversable wormholes, can be removed in spacetimes with torsion. In addition, we show that this violation can also be removed in the conformal frame obtained by a Weyl transformation. This comes about because both conformally transformed wormholes and wormholes in spacetimes with torsion become sustained thanks to a combination of 'normal' matter, that satisfies the null energy condition, and an induced 'geometric' energy-momentum tensor that is able to greatly violate the null energy condition.

Deviation equation in Riemann-Cartan spacetime

We derive a generalized deviation equation in Riemann-Cartan spacetime. The equation describes the dynamics of the connecting vector which links events on two general adjacent world lines. Our result is valid for any theory in a Riemann-Cartan background, in particular, it is applicable to a large class of gravitational theories which go beyond the general relativistic framework.

Tetrad-Gauge Theory of Gravity

We present a tetrad–gauge theory of gravity based on the local Lorentz group in a four-dimensional Riemann–Cartan space–time. Using the tetrad formalism allows avoiding problems connected with the noncompactness of the group and includes the possibility of choosing the local inertial reference frame arbitrarily at any point in the space–time. The initial quantities of the theory are the tetrad and gauge fields in terms of which we express the metric, connection, torsion, and curvature tensor. The gauge fields of the theory are coupled only to the gravitational field described by the tetrad fields. The equations in the theory can be solved both for a many-body system like the Solar System and in the general case of a static centrally symmetric field. The metric thus found coincides with the metric obtained in general relativity using the same approximations, but the interpretation of gravity is quite different. Here, the space–time torsion is responsible for gravity, and there is no curvature because the curvature tensor is a linear combination of the gauge field tensors, which are absent in the case of pure gravity. The gauge fields of the theory, which (together with the tetrad fields) define the structure of space–time, are not directly coupled to ordinary matter and can be interpreted as the fields describing dark energy and dark matter.

Torsion in gauge theory

The potential conflict between torsion and gauge symmetry in the Riemann-Cartan curved space-time was noted by Kibble in his 1961 pioneering paper, and has since been discussed by many authors. Kibble suggested that, to preserve gauge symmetry, one should forgo the covariant derivative in favor of the ordinary derivative in the definition of the field strength F_{\mu}{\nu} for massless gauge theories, while for massive vector fields covariant derivatives should be adopted. This view was further emphasized by Hehl and collaborators in their influential 1976 review paper. We address the question of whether this deviation from normal procedure of forgoing covariant derivatives in curved spacetime with torsion could give rise to inconsistencies in the theory, such as the quantum renormalizability of a realistic interacting theory. We demonstrate in this note the one-loop renormalizability of a realistic gauge theory of gauge bosons interacting with Dirac spinors, such as the SU(3) chromodynamics, for the case of a curved Riemann-Cartan spacetime with totally anti-symmetric torsion. This affirmative confirmation is one step towards providing justification for the assertion that the flat-space definition of the gauge field strength should be adopted as the proper definition.