DOCSLIB.ORG

Modified Teleparallelism

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Modified Teleparallelism Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Modified teleparallelism: an alternative approach of modified gravity Cecilia Bejarano Instituto de Astronom´ıa y F´ısica del Espacio (CONICET-UBA) Buenos Aires, Argentina Instituto de F´ısica Corpuscular (CSIC-UV) Valencia, Espa˜na Windows on Quantum Gravity 28-30 October, 2015, Madrid Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary The so-called f(T) theories of gravity are a generalization of the well-known teleparallel equivalent of General Relativity (TEGR) in which: ◮ the dynamical relevant object is the tetrad instead of the metric tensor and ◮ the degrees of freedom are encoded in the torsion rather than in the curvature. Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Outline Basic definitions ⋆ Tetrad ⋆ Torsion ⋆ Weitzenb¨ock connection Motivation Basic equations ⋆ TEGR ⋆ f (T ) Lorentz invariance Examples ⋆ Schwarzschild Kerr ֒ Null tetrads ⋆ → Summary Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Motivation Modified gravity Several models of modified gravity have been proposed in order to tackle the shortcomings of General Relativity... Dark matter and Dark energy (deformations of GR at large scales) there exist any geometrical explanantion for the phenomena associated to →֒ the dark sector? Singularities (deformations of GR at small scales) ?it is possible to smooth or even to avoid them →֒ Quantum gravity... Teleparallel gravity tetrad metric torsion (Weitzenb¨ock connection) ↔ curvature (Levi-Civita connection) z ↔}| { Teleparallel Equivalent of General Relativity (TEGR) A f (T ) theory is a deformation of TEGR (likewise f (R) theory is a deformation of GR) Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Tetrad The field of tetrad is a set of four orthonormal vectors defining a local frame at every µ point ea(x ) which constitutes a basis of the tangent space e , e , e , e { 1 2 3 4} µ whose components are represented as ea and are related by the metric tensor through the orthonormal condition Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Torsion It could be represented as the exterior derivative of the tetrad: Ta = dea It is related to the antisymmetric part of any connection The so-called torsion scalar (or Weitzenb¨ock invariant) (like the curvature escalar R) is obtained by the contraction µν ρ T = Sρ T µν ρ where T µν are the components of the torsion tensor µ µ a a ⋆ T νρ = ea (∂ν eρ − ∂ρeν ) µν and Sρ are the components of the so-called superpotential S µν K µν T λµ ν − T λν µ ⋆ 2 ρ = ρ + λ δρ λ δρ W LC λ λ λ contorsion K µν = Γ µν − Γ µν is the difference between →֒ Weitzenb¨ock and Levi-Civita connection Weitzenb¨ock connection →֒ Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Weitzenb¨ock connection W µ µ a a µ Γ ρν = e ∂ν e = e ∂ν e a ρ − ρ a It is curvaturaless connection with torsion The Weitzenb¨ock spacetime is provided by this connection The covariant derivative is obtained through the connection W µ µ µ ρ a µ a µ µ a ν V = ∂ν V + Γ ρν V = ∂ν (V e ) e ∂ν e = e ∂ν V ∇ a − ρ a a Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Weitzenb¨ock connection (TEGR - f (T )) it has torsion but not curvature ֒ → W W W ρ ρ ρ T µν = Γ µν Γ νµ = 0 − 6 W W W W W W W λρ ρ ρ ρ η ρ η R νµ = ∂ν Γ λµ ∂µΓ λν + Γ ην Γ λµ Γ ηµΓ λν = 0 − − Levi-Civita connection (GR - f (R)) it has curvature but not torsion ֒ → LC LC LC ρ ρ ρ T µν = Γ µν Γ νµ = 0 − LC LC LC LC LC LC LC λρ ρ ρ ρ η ρ η R νµ = ∂ν Γ λµ ∂µ Γ λν + Γ ην Γ λµ Γ ηµ Γ λν = 0 − − 6 Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Torsion and curvature GR ⇒ curvature ⇒ angular deficit TG ⇒ torsion ⇒ length deficit well-known in condensed matter ֒ → EC ⇒ curvature + torsion Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary TEGR [Einstein, Sitzungsber.Preuss.Akad.Wiss.Phys.Math. Kl. 217 (1928); 401 (1930); Math.Ann. 102, 685 (1930)] Reformulation of GR Tetrad or vierbein ◮ ea(x) basis of tangent space ◮ {ea(x)} → co-basis of co-tangent (dual) space { } → a µ a a ν ν eµeb = δb eµea = δµ ֒ → a a µ µ µ e = e dx ea = e ∂x ֒ → µ a Greek (latin) indices correspond to spacetime (tangent space) There is a link between the tetrad and the metric through the orthonormal condition µ ν ˆabˆ µν ˆa bˆ ηˆabˆ = gµν eˆa ebˆ , η = g eµeν ˆa bˆ µν ˆabˆ µ ν gµν = ηˆabˆeµeν , g = η eˆa ebˆ Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary TEGR [Hayashi & Shirafuji, PRD 19, 3524 (1979)] 1 1 1 S = d 4x eS µν T ρ = d 4x e S T = d 4x e T 2κ ρ µν 2κ · 2κ Z Z Z ~ ˆa κ = 8πG; c = = 1; e = det[eµ]= √ g T is the torsion scalar (or Weitzenb¨ock scalar)− Equivalence −1 µρ R = T + 2e ∂ρ(eT ) − µ the Lagrangians are equal up to a 4-divergence! ⇒ Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Modified teleparallel gravity: f (T ) theories [Ferraro & Fiorini, PRD 75, 084031 (2007), Ferraro, AIP Conf. Proc., 1471, 103 (2012)] A generalization of TEGR... 1 4 S = 2κ d xe f (T ) f f T differents produceR differents theories ( ) ◮ fUV(T ) UV deformations (small scales) ◮ f (T ) → IR deformations (large scales) IR → Equations of motion 2κeλT ν = eν f (T )+ − a λ − a −1 µν λ ρ µν ′ + 4 e ∂µ(e Sa )+ ea Tµλ Sρ f (T )+ h µν ′′ i + 4 Sa ∂µ(T )f (T ) ν ⋆ Tλ is the usual energy-momentum tensor ⋆ 2nd order eqns for the tetrad (the torsion is built from 1st derivatives of the tetrad) f (T )= T TEGR → ⇒ ⋆ 16 independent components extra dof [Li, Miao & Miao, JHEP 07, 108 (2011)] ⇒ Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary A lot of work was done in the context of f (T ) gravity (just for mention the first references...) alternative explanation for the acceleration of the universe [ Bengochea & Ferraro, PRD 79, 124019 (2009); Linder, PRD 81, 127301 (2010); Wu & Yu, PLB 692, 176 (2010); 693, 415 (2010); Bamba, Geng, Lee & Luo, JCAP. 03 (2011); Dent, Dutta & Saridakis (2010), PRD 83, 023508 (2011); Wu & Yu, EPJC 71, 1552 (2011)] successful attempts in order to smooth or even to avoid the singularities present in LFRW cosmologies [Ferraro & Fiorini, PRD 75, 084031 (2007); Ferraro & Fiorini PLB 702, 75 (2011); Tamanini & B¨ohmer PRD 86, 044009 (2012)] and black holes geometries like Schwarzschild [Ferraro & Fiorini PRD, 84, 083518 (2011)] and Kerr spacetimes [Bejarano, Ferraro & Guzm´an, EPJC, 75, 77 (2015)] Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary f (R) vs f (T )... modified GR modified TERG Gravity is encoded in the curvature Gravity is encoded in the torsion LC W λ λ through the LC connection Γ µν through the W connection Γ µν (without torsion) (without curvature) GR TEGR 1 4 1 4 SGR = d x √ g R STEGR = d xeT 2κ Z − 2κ Z f (R) f (T ) 1 4 1 4 Sf R = d x √ g f (R) Sf T = d x e f (T ) ( ) 2κ Z − ( ) 2κ Z R is the curvature scalar T is the torsion scalar (Ricci scalar) (Weitzenb¨ock scalar) 4th order eqns: R contains 2nd 2nd order eqns: T contains 1st derivatives of the metric (metric f.) derivatives of the tetrad extra dof: 1 extra dof: 3 scalar field ֒ massive vector field ֒ no-massive vector field + scalar →֒ → field→ [Buchdahl, MNRAS 150, 1 (1970)] [Ferraro & Fiorini, PRD 75, 084031 (2007)] Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Local Lorentz invariance [Li, Sotiriou & Barrow, PRD 83, 064035 (2011)] ◮ f (T ) gravity is global invariant under Lorentz transformations −1 σν −1 σν T = R + 2e ∂ν (eT ) f ( R + 2e ∂ν (eT ) ) − σ −→ − σ The surface term remains encapsulate insde the function f ֒ breaking→ the local Lorentz invariance. ◮ Of course, if f (T )= T (i.e. TEGR), the local Lorentz invariance is preserved Modified teleparallelism: an alternative approach of modified gravity Introduction TEGR f (T ) Lorentz Invariance Null tetrads Kerr Summary Torsion in teleparallel gravity The torsion Ta could be descompose as a linear combination of three irreducible parts (tensorial(1), vectorial(2), axial(3)) a (1) a (2) a (3) a T = a1 T + a2 T + a3 T (1) 1 1 (2) (2) 1 (2) Tλµν = (Tλµν + Tµλν )+ (gνλ + gνµTλ Tν ) gλµ Tν 2 6 − 3 (2) λ Tµ = Tλµ 1 (3)T = ε T νρσ µ 6 µνρσ which is in general non-local invariant under Lorentz transformation.
Load more

Recommended publications
  • Selected Papers on Teleparallelism Ii
    SELECTED PAPERS ON TELEPARALLELISM Edited and translated by D. H. Delphenich Table of contents Page Introduction ……………………………………………………………………… 1 1. The unification of gravitation and electromagnetism 1 2. The geometry of parallelizable manifold 7 3. The field equations 20 4. The topology of parallelizability 24 5. Teleparallelism and the Dirac equation 28 6. Singular teleparallelism 29 References ……………………………………………………………………….. 33 Translations and time line 1928: A. Einstein, “Riemannian geometry, while maintaining the notion of teleparallelism ,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 217- 221………………………………………………………………………………. 35 (Received on June 7) A. Einstein, “A new possibility for a unified field theory of gravitation and electromagnetism” Sitzber. Preuss. Akad. Wiss. 17 (1928), 224-227………… 42 (Received on June 14) R. Weitzenböck, “Differential invariants in EINSTEIN’s theory of teleparallelism,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 466-474……………… 46 (Received on Oct 18) 1929: E. Bortolotti , “ Stars of congruences and absolute parallelism: Geometric basis for a recent theory of Einstein ,” Rend. Reale Acc. dei Lincei 9 (1929), 530- 538...…………………………………………………………………………….. 56 R. Zaycoff, “On the foundations of a new field theory of A. Einstein,” Zeit. Phys. 53 (1929), 719-728…………………………………………………............ 64 (Received on January 13) Hans Reichenbach, “On the classification of the new Einstein Ansatz on gravitation and electricity,” Zeit. Phys. 53 (1929), 683-689…………………….. 76 (Received on January 22) Selected papers on teleparallelism ii A. Einstein, “On unified field theory,” Sitzber. Preuss. Akad. Wiss. 18 (1929), 2-7……………………………………………………………………………….. 82 (Received on Jan 30) R. Zaycoff, “On the foundations of a new field theory of A. Einstein; (Second part),” Zeit. Phys. 54 (1929), 590-593…………………………………………… 89 (Received on March 4) R.
    [Show full text]
  • Book of Abstracts Ii Contents
    Spring workshop on gravity and cosmology Monday, 25 May 2020 - Friday, 29 May 2020 Jagiellonian University Book of Abstracts ii Contents Stochastic gravitational waves from inflaton decays ..................... 1 Approximate Killing symmetries in non-perturbative quantum gravity .......... 1 Disformal transformations in modified teleparallelism .................... 1 Quantum fate of the BKL scenario ............................... 2 Ongoing Efforts to Constrain Lorentz Symmetry Violation in Gravity ........... 2 Proof of the quantum null energy condition for free fermionic field theories . 2 Cosmic Fibers and the parametrization of time in CDT quantum gravity . 3 Stable Cosmology in Generalised Massive Gravity ...................... 3 Constraining the inflationary field content .......................... 3 The (glorious) past, (exciting) present and (foreseeable) future of gravitational wave detec- tors. ............................................. 4 reception ............................................. 4 Testing Inflation with Primordial Messengers ........................ 4 Gravitational waves from inflation ............................... 4 CDT, a theory of quantum geometry. ............................. 4 Cosmic Fibers and the parametrization of time in CDT quantum gravity . 5 Loop-based observables in 4D CDT .............................. 5 Spectral Analysis of Causal Dynamical Triangulations via Finite Element Methods . 5 A consistent theory of D -> 4 Einstein-Gauss-Bonnet gravity ................ 5 Quantum Ostrogradsky theorem
    [Show full text]
  • Alternative Gravitational Theories in Four Dimensions
    Alternative Gravitational Theories in Four Dimensionsa Friedrich W. Hehl Institute for Theoretical Physics, University of Cologne, D-50923 K¨oln, Germany email: [email protected] We argue that from the point of view of gauge theory and of an appropriate interpretation of the interferometer experiments with matter waves in a gravitational field, the Einstein- Cartan theory is the best theory of gravity available. Alternative viable theories are general relativity and a certain teleparallelism model. Objections of Ohanian & Ruffini against the Einstein-Cartan theory are discussed. Subsequently we list the papers which were read at the ‘Alternative 4D Session’ and try to order them, at least partially, in the light of the structures discussed. 1 The best alternative theory? I would call general relativity theory8 GR the best available alternative gravitational 21,14 theory and the next best one its teleparallel equivalent GR||. Because of these two theories, at least, it is good to have this alternative session during the Marcel Grossmann Meeting. Let me try to explain why I grant to GR the distinction of being the best alternative theory. After finally having set up special relativity theory in 1905, Einstein subse- quently addressed the question of how to generalize Newton’s gravitational theory as to make it consistent with special relativity, that is, how to reformulate it in a Poincar´ecovariant way. Newton’s theory was a battle tested theory in the realm of our planetary system and under normal laboratory conditions. It has predictive power as it had been shown by the prediction of the existence of the planet Neptune in the last century.
    [Show full text]
  • Teleparallelism Arxiv:1506.03654V1 [Physics.Pop-Ph] 11 Jun 2015
    Teleparallelism A New Way to Think the Gravitational Interactiony At the time it celebrates one century of existence, general relativity | Einstein's theory for gravitation | is given a companion theory: the so-called teleparallel gravity, or teleparallelism for short. This new theory is fully equivalent to general relativity in what concerns physical results, but is deeply different from the con- ceptual point of view. Its characteristics make of teleparallel gravity an appealing theory, which provides an entirely new way to think the gravitational interaction. R. Aldrovandi and J. G. Pereira Instituto de F´ısica Te´orica Universidade Estadual Paulista S~aoPaulo, Brazil arXiv:1506.03654v1 [physics.pop-ph] 11 Jun 2015 y English translation of the Portuguese version published in Ci^enciaHoje 55 (326), 32 (2015). 1 Gravitation is universal One of the most intriguing properties of the gravitational interaction is its universality. This hallmark states that all particles of nature feel gravity the same, independently of their masses and of the matter they are constituted. If the initial conditions of a motion are the same, all particles will follow the same trajectory when submitted to a gravitational field. The origin of universality is related to the concept of mass. In principle, there should exist two different kinds of mass: the inertial mass mi and the gravitational mass mg. The inertial mass would describe the resistance a particle shows whenever one attempts to change its state of motion, whereas the gravitational mass would describe how a particle reacts to the presence of a gravitational field. Particles with different relations mg=mi, therefore, should feel gravity differently when submitted to a given gravitational field.
    [Show full text]
  • A Teleparallel Effective Geometry for Einstein's Unified Field Theory
    A teleparallel effective geometry for Einstein’s unified field theory L.C. Garcia de Andrade 1 Abstract Riemannian and teleparallel geometrical approaches to the investi- gation of Maxwell electrodynamics shown that a unified field theory of gravitation and electromagnetism a la Einstein can be obtained from a stationary metric. This idea contrasts with the recently proposed pre- metric electrodynamics by Hehl and Obukhov. In the teleparallel case the definition of the electric field is obtained straightforward from the spacetime metric and the orthonormal basis frame of teleparallelism. In this case the only nonvanishing component of Cartan torsion is defined as the effective electric field. In this approach the gravita- tional potentials or metric coefficients are expressed in terms of the effective or analogous electric and magnetic potentials. Thefore the arXiv:gr-qc/0501050v1 16 Jan 2005 Maxwell equations in vacuum can be obtained by derivation of this electric field definition as usual. In the Riemannian case we consider an electrostatic spacetime where the Einstein equations in vacuum in the approximation of linear fields. The constraint of Einstein equa- tions in vacuum are shown to lead or to the Coulomb equation or to a singular behaviour on the metric which would represent a kind of effective electrodynamic black hole event horizon. 1Departamento de F´ısica Te´orica - Instituto de F´ısica Rua S˜ao Fco. Xavier 524, Rio de Janeiro, RJ Maracan˜a, CEP:20550-003 , Brasil. E-Mail.: [email protected] 1 Introduction Recently Hehl and Obukhov (HO) [1] have proposed a pre-metric electrody- namics where the metric would be oobtained from constitutive relations of electrodynamics.
    [Show full text]
  • Dynamics in Interacting Scalar-Torsion Cosmology
    universe Article Dynamics in Interacting Scalar-Torsion Cosmology Andronikos Paliathanasis 1,2 1 Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa; [email protected] 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia 5090000, Chile Abstract: In a spatially flat Friedmann–Lemaître–Robertson–Walker background space, we con- sider a scalar-torsion gravitational model which has similar properties to the dilaton theory. This teleparallel model is invariant under a discrete transformation similar to the Gasperini–Veneziano duality transformation. Moreover, in the gravitational action integral, we introduce the Lagrangian function of a pressureless fluid source which is coupled to the teleparallel dilaton field. This specific gravitational theory with interaction in the dark sector of the universe was investigated by using methods of the dynamical system analysis. We calculate that the theory provides various areas of special interest for the evolution of the cosmological history. Inflationary scaling solutions and the de Sitter universe are recovered. Furthermore, we calculate that there exist an attractor which provides a stable solution where the two fluid components, the scalar field and the pressureless matter, contribute in the cosmological fluid. This solution is of special interest because it can describe the present epoch. Finally, the qualitative evolution of the cosmographic parameters is discussed. Keywords: teleparallel; scalar field; dilaton field; scalar-torsion; interaction Citation: Paliathanasis, A. Dynamics in Interacting Scalar-Torsion 1. Introduction Cosmology. Universe 2021, 7, 244. The detailed analysis of the recent cosmological data indicates that General Relativity https://doi.org/10.3390/ may need to be modified in order to describe the observations; for a recent review, we refer universe7070244 the reader to [1].
    [Show full text]
  • On Nye's Lattice Curvature Tensor 1 Introduction
    On Nye’s Lattice Curvature Tensor∗ Fabio Sozio1 and Arash Yavariy1,2 1School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA 2The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA May 4, 2021 Abstract We revisit Nye’s lattice curvature tensor in the light of Cartan’s moving frames. Nye’s definition of lattice curvature is based on the assumption that the dislocated body is stress-free, and therefore, it makes sense only for zero-stress (impotent) dislocation distributions. Motivated by the works of Bilby and others, Nye’s construction is extended to arbitrary dislocation distributions. We provide a material definition of the lattice curvature in the form of a triplet of vectors, that are obtained from the material covariant derivative of the lattice frame along its integral curves. While the dislocation density tensor is related to the torsion tensor associated with the Weitzenböck connection, the lattice curvature is related to the contorsion tensor. We also show that under Nye’s assumption, the material lattice curvature is the pullback of Nye’s curvature tensor via the relaxation map. Moreover, the lattice curvature tensor can be used to express the Riemann curvature of the material manifold in the linearized approximation. Keywords: Plasticity; Defects; Dislocations; Teleparallelism; Torsion; Contorsion; Curvature. 1 Introduction their curvature. His study was carried out under the assump- tion of negligible elastic deformations: “when real crystals are distorted plastically they do in fact contain large-scale dis- The definitions of the dislocation density tensor and the lat- tributions of residual strains, which contribute to the lattice tice curvature tensor are both due to Nye [1].
    [Show full text]
  • Autoparallel Vs. Geodesic Trajectories in a Model of Torsion Gravity
    Universe 2015, 1, 422-445; doi:10.3390/universe1030422 OPEN ACCESS universe ISSN 2218-1997 www.mdpi.com/journal/universe Article Autoparallel vs. Geodesic Trajectories in a Model of Torsion Gravity Luis Acedo Instituto Universitario de Matemática Multidisciplinar, Universitat Polit`ecnica de Val`encia, Building 8G, 2o Floor, Camino de Vera 46022, Valencia, Spain; E-Mail: [email protected] Academic Editor: Lorenzo Iorio Received: 14 October 2015 / Accepted: 13 November 2015 / Published: 25 November 2015 Abstract: We consider a parametrized torsion gravity model for Riemann–Cartan geometry around a rotating axisymmetric massive body. In this model, the source of torsion is given by a circulating vector potential following the celestial parallels around the rotating object. Ours is a variant of the Mao, Tegmark, Guth and Cabi (MTGC model) in which the total angular momentum is proposed as a source of torsion. We study the motion of bodies around the rotating object in terms of autoparallel trajectories and determine the leading perturbations of the orbital elements by using standard celestial mechanics techniques. We find that this torsion model implies new gravitational physical consequences in the Solar system and, in particular, secular variations of the semi-major axis of the planetary orbits. Perturbations on the longitude of the ascending node and the perihelion of the planets are already under discussion in the astronomical community, and if confirmed as truly non-zero effects at a statistically significant level, we might be at the dawn of an era of torsion phenomenology in the Solar system. Keywords: Solar system anomalies; Riemann–Cartan spacetime; gravitation models; autoparallel curves; geodesic curves 1.
    [Show full text]
  • On Quantization of a Slowly Rotating Kerr Black Hole in Teleparallel Gravity
    universe Article On Quantization of a Slowly Rotating Kerr Black Hole in Teleparallel Gravity Sérgio Costa Ulhoa 1,2,* , Ednardo Paulo Spaniol 3,4 and Ronni Geraldo Gomes Amorim 5 1 Instituto de Física, Universidade de Brasília, Brasília, DF 70910-900, Brazil 2 International Center of Physics, Instituto de Física, Universidade de Brasília, Brasília, DF 70910-900, Brazil 3 UDF Centro Universitário, Brasília, DF 70390-045, Brazil; [email protected] 4 Faculdade de Tecnologia e Ciências Sociais Aplicadas, Centro Universitário de Brasília (UniCEUB), Brasília, DF 70790-075, Brazil 5 Faculdade Gama, Universidade de Brasília, Brasília, DF 70910-900, Brazil; [email protected] * Correspondence: [email protected] Received: 3 December 2018; Accepted: 11 January 2019; Published: 16 January 2019 Abstract: In this article we calculate the total angular momentum for Kerr space-time for slow rotations in the context of teleparallel gravity. In order to analyze the role of such a quantity, we apply Weyl quantization method to obtain a quantum equation for the z-component of the angular momentum density, and for the squared angular momentum density as well. We present an approximate solution using the Adomian decomposition method (AM), which reveals a discrete characteristic for angular momentum. Keywords: quantum gravity; teleparallelism; angular momentum 1. Introduction In the old quantum theory, the discrete nature of physical systems played an important role. Such a feature evolved, giving rise to the so called quantum mechanics. In this process, the passage from classical theory to its quantum counterpart was developed by quantization techniques. Perhaps one of the first of such rules was the Bohr-Sommerfeld quantization [1–5].
    [Show full text]
  • Gravitational and Torsion Waves in Linearised Teleparallel Gravity
    Gravitational and torsion waves in linearised teleparallel gravity By L.C. Garcia de Andrade1 Abstract Spin-2, spin-1 and spin-0 modes in linearised teleparallelism are obtained where the totally skew-symmetric complex contortion tensor generates scalar torsion waves and the symmetric contortion in the last two indices generates gravitational waves as gravitational pertur- bations of flat spacetime with contortion tensor. A gedanken exper- iment with this gravitational-torsion wave hitting a ring of spinless particles is proposed which allows us to estimate the contortion of the Earth by making use of data from LISA GW detector. This value co- incides with previous value obtained by Nitsch in teleparallelism using another type of experiment. arXiv:gr-qc/0206007v1 3 Jun 2002 1Departamento de F´ısica Te´orica - IF - UERJ - Rua S˜ao Francisco Xavier 524, Rio de Janeiro, RJ, Maracan˜a, CEP:20550.e-mail:[email protected] 1 Introduction Since the early proposals of experiments [1, 2, 3, 4, 5] to detect spacetime torsion or Cartan contortion [6] and to give an experimental basis to al- ternative gravity theories of Einstein-Cartan type [7] theoretical physicists working on the field have noticed that torsion detection by the technique of spin precession have been proved hopeless since all computations are very far beyond all quantum capabilities of laboratory devices and even astrophys- ical probes would be difficult to construct for this task. In this letter we propose a new method to estimate spacetime contortion which is based on the GW detectors in linearised teleparallel space. This space which has been applied to gravity by Einstein in 1932 in Berlin [8] in the attempt to build a unified geometrical theory of gravitation and electrodynamics proved re- cently [9] to be very important in solving theoretical problems in gravitation such as the tetrad complex of gravitational energy.
    [Show full text]
  • Gravitational Theories with Torsion
    Department of Physics National Tsing Hua University Doctoral Dissertation Gravitational Theories with Torsion Huan-Hsin Tseng Advisor Prof. Dr. Chao-Qiang Geng arXiv:1812.00314v1 [gr-qc] 2 Dec 2018 June, 2015 Abstract We give a complete formulation of Poincar´egauge theory, starting from the fibre bundle formulation to the resultant Riemann-Cartan spacetime. We also introduce several diverse gravity theories descendent from the Poincar´egauge theory. Especially, the cosmological effect of the simple scalar-torsion (0+) mode in Poincar´egauge theory of gravity is studied. In the theory, we treat the geometric effect of torsion as an effective quantity, which behaves like dark energy, and study the effective equation of state (EoS) of the model. We concentrate on the two cases of the constant curvature solution and positive kinetic energy. In the former, we find that the torsion EoS has different values corresponding to the stages of the universe. For example, it behaves like the radiation (matter) EoS of wr = 1=3 (wm = 0) in the radiation (matter) dominant epoch, while in the late time the torsion density is supportive for the accelerating universe. In the latter case of positive kinetic energy, we find the (affine) curvature is not constant in general and hence requires numerical solution. Our numerical analysis shows that the EoS in general has an asymptotic behavior in the high redshift regime, while it could cross the phantom divide line in the low redshift regime. By further analysis of the Laurent series expansion, we find that the early evolution of the torsion density ρT has a radiation-like asymptotic behavior of O(a−4) where a(t) denotes the scale factor, along with a stable point of the torsion pressure (PT ) and a density ratio PT /ρT ! 1=3 in the high redshift regime (z 0), this is different from the previous result in the literature.
    [Show full text]
  • Arxiv:2101.00432V2 [Gr-Qc] 1 Feb 2021
    Noether Symmetries and Quantum Cosmology in Extended Teleparallel Gravity Francesco Bajardi1, 2, ∗ and Salvatore Capozziello1, 2, 3, 4, y 1Department of Physics “E. Pancini”, University of Naples “Federico II”, Naples, Italy. 2INFN Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Naples, Italy. 3Scuola Superiore Meridionale, Largo San Marcellino 10, I-80138, Naples, Italy. 4Tomsk State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia. (Dated: February 2, 2021) We apply the Noether Symmetry Approach to point-like teleparallel Lagrangians in view to derive minisuper- spaces suitable for Quantum Cosmology. Adopting the Arnowitt–Deser–Misner formalism, we find out related Wave Functions of the Universe. Specifically, by means of appropriate changes of variables suggested by the existence of Noether symmetries, it is possible to obtain the cosmological Hamiltonians whose solutions are classical trajectories interpretable as observable universes. Keywords: Teleparallel gravity; quantum cosmology; Noether symmetries. I. INTRODUCTION infinite-dimensional superspace constructed on the 3D- spa- tial metrics. Once a Super-Hamiltonian is defined on it, it is General Relativity (GR) is considered the best accepted the- possible to define a geometrodynamics to fix the evolution of ory describing gravity. Despite its successes, related to the re- these 3-metrics. cent observational discovery of gravitational waves and black The problem is extremely difficult to be handled from a holes, after more than one hundred years from its formula- mathematical point of view, nevertheless, in 1983 J.B. Har- tion, the theory presents many theoretical and experimental tle [13], using the Wentzel-Kramers-Brillouin approximation, issues, both at low and high energy scales, which need to be showed that it is possible to restrict the superspace to finite- addressed.
    [Show full text]