DOCSLIB.ORG

Coincident General Relativity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Coincident General Relativity NORDITA-2017-100 IFT-UAM/CSIC-17-093 Coincident General Relativity Jose Beltr´an Jim´eneza,b,∗ Lavinia Heisenbergc,† and Tomi Koivistod‡ aInstituto de F´ısica Te´orica UAM-CSIC, Universidad Aut´onoma de Madrid, Cantoblanco, Madrid, 28049 Spain bDepartamento de F´ısica Fundamental, Universidad de Salamanca, E-37008 Salamanca, Spain. cInstitute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland and dNordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden (Dated: September 17, 2018) The metric-affine variational principle is applied to generate teleparallel and symmetric teleparallel theories of gravity. From the latter is discovered an exceptional class which is consistent with a vanishing affine connection. Based on this remarkable property, this work proposes a simpler geometrical formulation of General Relativity that is oblivious to the affine spacetime structure, thus fundamentally depriving gravity of any inertial character. The resulting theory is described by the Hilbert action purged from the boundary term and is more robustly underpinned by the spin-2 field theory, where an extra symmetry is now manifest, possibly related to the double copy structure of the gravity amplitudes. This construction also provides a novel starting point for modified gravity theories, and the paper presents new and simple generalisations where analytical self-accelerating cosmological solutions arise naturally in the early and late time universe. In the conventional geometrical interpretation of Gen- We start by recalling that the general affine connection eral Relativity (GR), gravitation is described by the cur- admits the decomposition [8] vature α α α α Γ µν = µν + K µν + L µν , (3) α α α λ R βµν 2∂[µΓ ν β + 2Γ µ λ Γ ν β , (1) which includes the contortion ≡ ] [ | | ] α 1 α α α α K µν T µν + T , (4) where the affine connection Γ µν = is the metric- (µ ν) {βγ} ≡ 2 compatible and torsionless Levi-Civita connection that is α α due to torsion T µν 2Γ [µν], and the disformation computed from the metric gµν (with mostly plus Loren- ≡ 1 tizan signature) as Lα Qα Q α , (5) µν ≡ 2 µν − (µ ν) 1 α αλ due to non-metricity Qαµν αgµν . The various ten- βγ g (gβλ,γ + gλγ,β gβγ,λ) . (2) ≡ 2 − sor fields we have already introduced≡ ∇ satisfy important Despite its undeniable observational success, this de- identities, an example being the Bianchi identity [8] µ µ ν µ scription comes hand in hand with the inherent concep- R [αβγ] [αT βγ] + T [αβT γ]ν =0 , (6) tual and technical difficulties of working in a pseudo- − ∇ which we shall need later. In the metric-affine formalism, Riemannian spacetime so that it is desirable to attain a theory can be now defined by a scalar action of the form a simpler formulation of the theory. A step forward in this direction is taken in the teleparallel reformula- n µν α α µν = d x√ gf(g , R ,T ,Q )+ M , (7) tion [1, 2], where the geometry is simplified by the con- S − βµν µν α S α Z straint R βµν = 0, which reduces the connection to the where the independent variables are the metric and the arXiv:1710.03116v2 [gr-qc] 14 Sep 2018 Weizenb¨ock form. In this work we will pursue further affine connection, plus the matter fields contained in M . improvements in the simplifying sequence of geometri- The sources introduced by the latter are the energy-S cal frameworks, and formulate GR in a flat and torsion- momentum tensor and the hypermomentum tensor den- free spacetime. Furthermore, we will introduce a class sity defined as of its generalisations in the trivially connected geometry, α 2 δ M µν 1 δ M i.e with Γ = 0. The standard vierbein formalism of Tµν = S , Hλ = S , (8) µν −√ g δgµν −2 δΓλ GR has been reinterpreted in terms of nonlinear realisa- − µν tions of the group GL(4, R) [3], and teleparallelised in the respectively. metric-affine gauge theory ([4, 5], 5.9 in [6]), but it can still be clarified whether it is natural to stipulate a triv- ial geometry since there, as we shall discover, the inertial connection is a translation1. of translations were presented e.g. in Ref. [7], wherein also the field theoretical approach as an alternative to the conventional geometrical approach to gravitation was emphasised. Note that in the Weitzenb¨ock teleparallelism [1, 2], gravitation is still ge- ometry, and despite the motivation as a translation gauge theory, 1 Compelling arguments to regard gravitation as the gauge theory the gauge connection is a (pseudo-)rotation. 2 THE NEW GR where we have referred to the familiar Levi-Civita covari- ant derivative α that comes with the Christoffel symbols D Before proceeding to the central result of this work, it (2). Note that we have removed the symmetrisation from will be interesting to rederive the so-called New GR [9] (16) as a more compact manner of writing all the equa- covariantly in the metric-affine formalism. If we intro- tions: the 10 symmetric components correspond to the duce the superpotential metric field equations and, in general, the equation has also 6 antisymmetric components which are nothing but µν µν [µ ν] [µ ν] Sα aTα + bT α + cδα T , (9) the equations (15). In the standard prescription for stan- ≡ dard matter fields, the hypermomentum decouples from then the general even-parity quadratic theory can be the symmetric equations. The teleparallel equivalent of defined in terms of the three-parameter quadratic form GR (TEGR) is reproduced by the choice of parameters T 1 S µν T α . Thus, instead of imposing a priori the ≡ 2 α µν a = 1 , b = 1 , c = 1 and n = 4. flatness and metricity conditions on the connection, we 4 2 − We have now seen that this theory and its generali- set up the gravitational action as sations can be formulated without introducing the addi- n 1 βµν α α µν tional structure of the frame bundle and its correspond- = d x √ gT + λ R + λ αg , S 2 − α βµν µν ∇ ing extra set of indices. The connection can then be Z (10) reduced to its contortion component entirely determined where the connection is left completely arbitrary and the by the torsion, which is propagating due to the restriction Lagrange multipliers that impose the teleparallelism con- to a teleparallel geometry2. straints are introduced as tensor densities with the obvi- µν (µν) νµρ ν[µρ] ous symmetries λα = λα and λα = λα . We first compute the field equations by varying the action A NEWER GR (10) w.r.t. the metric, yielding We then move to the much less explored case of sym- 1 1 T Sαβ T S αβ g T metric teleparallelisms [11], see also [12–15]. The physi- αβ(µ ν) 2 (µ|αβ| ν) 2 µν − − cal interpretation we suggest extrapolates the successful 2 α = Tµν + ( α + Tα) λ , (11) argument of teleparallelism to its logical conclusion. As √ g ∇ µν − it is well-known, GR cannot distinguish between gravita- while the variations w.r.t. the connection result in tion and inertial effects, but by resorting to frame fields, the gravitational energy can be defined covariantly in the νµρ 1 µ νρσ µν ρ + Tρ λα + T ρσλα = ∆α , (12) teleparallel approach [2, 16, 17]. The canonical frame is ∇ 2 now identified by the absence of curvature and torsion, where we have defined the source term and the canonical coordinates are now identified by the µν µν 1 νµ µν absence of inertial effects. This is a physical rationale ∆α Hα + √ gSα λ α , (13) ≡ 2 − − how to extract quantities of interest such as the gravita- tional energy and the gravitational entropy, and how to and used the symmetries of λ νµρ and λ µν . Now the α α proceed with the quantisation and with the unification. field equations (11) seem to have no dynamics for the tor- The aim is to establish the frame and the coordinate sion. However, we still need to solve for the constraints system wherein the canonical commutation relations can imposed by the Lagrange multipliers. Using the con- be recovered for the operators corresponding to physical straint Rα = 0 and the antisymmetry of the corre- βµν observables. sponding Lagrange multiplier, we can derive the diver- The non-metricity tensor has two independent traces, gence of the source term as α ˜µ µα which we denote as Qµ = Qµ α and Q = Qα . We 1 µ νρσ νρµ νρµ µν can then define the quadratic non-metricity scalar as µT ρσλα µTρλα Tρ µλα = µ∆α . 2∇ − ∇ − ∇ ∇ (14) 1 αβµ 1 βµα 1 α 1 ˜α νρµ = QαβµQ + QαβµQ + QαQ QαQ Then we can use (12) to replace µλα , and yet take Q −4 2 4 − 2 advantage of the Bianchi identity∇ (6) to verify that (17) µν µ + Tµ ∆α =0 . (15) ∇ 2 We can then plug this result in (11) to rewrite them as Due to the teleparallelity constraint, the connection includes the first derivative of a general linear transformation, and the second derivatives resulted in the field equation (16) from solving multi- 2 α α Tµν + ( α + Tα)H = αSµν plier that imposes the metricity constraint. The gauge freedom √ g ∇ µ ν −D − of the Lagrange multipliers and the number of effective compo- αβ 1 nents of the multipliers and equations have been investigated S ν (Tαµβ + Kαβµ) gµν T , (16) − − 2 previously in the gauge formalism [4, 5, 10]. 3 that is special among the general quadratic combination and, thus, the ξλ’s make their appearance as the St¨uck- because, in addition to being invariant under local gen- elberg fields restoring this gauge symmetry.
Load more

Recommended publications
  • Metric–Affine Gauge Theory of Gravity II. Exact Solutions Abstract
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Metric–affine gauge theory of gravity II. Exact solutions Friedrich W. Hehl∗ and Alfredo Mac´ıas† Departamento de F´ısica Universidad Aut´onoma Metropolitana–Iztapalapa Apartado Postal 55–534, C.P. 09340, M´exico, D.F., Mexico Abstract In continuing our series on metric-affine gravity (see Gronwald, IJMP D6 (1997) 263 for Part I), we review the exact solutions of this theory. file magexac7.tex, 1999-04-09 Typeset using REVTEX ∗E-mail: hehl@thp.uni-koeln.de. Permanent address: Institute for Theoretical Physics, University of Cologne, D–50923 K¨oln, Germany †E-mail: amac@xanum.uam.mx 1 I. METRIC–AFFINE GRAVITY (MAG) In 1976, a new metric–affine theory of gravitation was published [16]. In this model, k the metric gij and the linear (sometimes also called affine) connection Γij were considered to be independent gravitational field variables. The metric carries 10 and the connection 64 independent components. Although nowadays more general Lagrangians are considered, like the one in Eq.(10), the original Lagrangian density of metric–affine gravity reads √ g = − gij Ric (Γ,∂Γ) + βQQ . (1) VGR0 2κ ij i j h i The Ricci tensor Ric ij depends only on the connection but not on the metric, whereas the Weyl covector Q := gkl g /4 depends on both. Here represents the covariant i − ∇i kl ∇i k derivative with respect to the connection Γij , furthermore g =detgkl, κ is Einstein’s gravitational constant, and β a dimensionless coupling constant.
    [Show full text]
  • Arxiv:Cond-Mat/0006280V1 19 Jun 2000 H Ulda Group Euclidean the Fdsoain.Frhroe H Oino Ilcto Shgl Dissip Highly Is Dislocation Point of the Motion Damping)
    Dislocation theory as a 3-dimensional translation gauge theory∗ Markus Lazar† Max-Planck-Institut f¨ur Metallforschung Institut f¨ur Physik D–70569 Stuttgart, Germany (November 1, 2018) We consider the static elastoplastic theory of dislocations in an elastoplastic ma- terial. We use a Yang-Mills type Lagrangian (the teleparallel equivalent of Hilbert- Einstein Lagrangian) and some Lagrangians with anisotropic constitutive laws. The translational part of the generalized affine connection is utilized to describe the the- ory of elastoplasticity in the framework of a translation gauge theory. We obtain a system of Yang-Mills field equations which express the balance of force and moment. Keywords: Dislocation theory; force stress; moment stress to appear in Annalen der Physik (2000) I. INTRODUCTION The field theory of defects in crystals has an old history. First, Kondo [1] and Bilby et al. [2] described independently a dislocation theory in the language of differential geometry. They proved the equivalence of dislocation density to Cartan’s torsion. Kr¨oner and Seeger [3,4] completed this theory to a non-linear theory of elasticity with dislocations and internal stress. Recent develop- ments of this theory are given in [5,6]. The first step towards a gauge theory of defects had been taken by Turski [7]. He derived the equilibrium equations by means of a variational principle. Edelen et al. [8,9] developed a theory of dislocations and disclinations as a gauge theory of the Euclidean group SO(3)×⊃ T (3) and a pure dislocation theory as a translational or T (3)-gauge theory. Unfortunately, they did not distinguish between disclinations in solid and in liquid crystals.
    [Show full text]
  • (TEGR) As a Gauge Theory: Translation Or Cartan Connection? M Fontanini, E
    Teleparallel gravity (TEGR) as a gauge theory: Translation or Cartan connection? M Fontanini, E. Huguet, M. Le Delliou To cite this version: M Fontanini, E. Huguet, M. Le Delliou. Teleparallel gravity (TEGR) as a gauge theory: Transla- tion or Cartan connection?. Physical Review D, American Physical Society, 2019, 99, pp.064006. 10.1103/PhysRevD.99.064006. hal-01915045 HAL Id: hal-01915045 https://hal.archives-ouvertes.fr/hal-01915045 Submitted on 7 Nov 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Teleparallel gravity (TEGR) as a gauge theory: Translation or Cartan connection? M. Fontanini1, E. Huguet1, and M. Le Delliou2 1 - Universit´eParis Diderot-Paris 7, APC-Astroparticule et Cosmologie (UMR-CNRS 7164), Batiment Condorcet, 10 rue Alice Domon et L´eonieDuquet, F-75205 Paris Cedex 13, France.∗ and 2 - Institute of Theoretical Physics, Physics Department, Lanzhou University, No.222, South Tianshui Road, Lanzhou, Gansu 730000, P R China y (Dated: November 7, 2018) In this paper we question the status of TEGR, the Teleparallel Equivalent of General Relativity, as a gauge theory of translations. We observe that TEGR (in its usual translation-gauge view) does not seem to realize the generally admitted requirements for a gauge theory for some symmetry group G: namely it does not present a mathematical structure underlying the theory which relates to a principal G-bundle and the choice of a connection on it (the gauge field).
    [Show full text]
  • Relativistic Spacetime Structure
    Relativistic Spacetime Structure Samuel C. Fletcher∗ Department of Philosophy University of Minnesota, Twin Cities & Munich Center for Mathematical Philosophy Ludwig Maximilian University of Munich August 12, 2019 Abstract I survey from a modern perspective what spacetime structure there is according to the general theory of relativity, and what of it determines what else. I describe in some detail both the “standard” and various alternative answers to these questions. Besides bringing many underexplored topics to the attention of philosophers of physics and of science, metaphysicians of science, and foundationally minded physicists, I also aim to cast other, more familiar ones in a new light. 1 Introduction and Scope In the broadest sense, spacetime structure consists in the totality of relations between events and processes described in a spacetime theory, including distance, duration, motion, and (more gener- ally) change. A spacetime theory can attribute more or less such structure, and some parts of that structure may determine other parts. The nature of these structures and their relations of determi- nation bear on the interpretation of the theory—what the world would be like if the theory were true (North, 2009). For example, the structures of spacetime might be taken as its ontological or conceptual posits, and the determination relations might indicate which of these structures is more fundamental (North, 2018). Different perspectives on these questions might also reveal structural similarities with other spacetime theories, providing the resources to articulate how the picture of the world that that theory provides is different (if at all) from what came before, and might be different from what is yet to come.1 ∗Juliusz Doboszewski, Laurenz Hudetz, Eleanor Knox, J.
    [Show full text]
  • Nonlinear Invariance of Flat Connection in Gauge Gravity
    International Journal of Pure and Applied Mathematics Volume 107 No. 3 2016, 529-535 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v107i3.2 ijpam.eu NONLINEAR INVARIANCE OF FLAT CONNECTION IN GAUGE GRAVITY Ibrahim˙ S¸ener Altay Mahallesi Domani¸cCaddesi MESA Blokları Aydın Apartmanı D:26, P.B. 06820 Eryaman, Ankara, TURKEY Abstract: At this paper one studies the spinning gravitational field formed by a torsion free and flat connection. In existing such a connection i.e the Maurer - Cartan 1 - form ω on a tangent bundle, the nonlinear invariance of the term tr[ω ∧ ∗ω] gives the spin current form. Also one sees that the action integral of the gravitational field is bonded such that 1 0 Λ(∼ G ) ≤ RM (LMat+LGr) < ∞ depending of the vacuum energy density Λ and gravitational constant G. AMS Subject Classification: 53Z05, 83C05 Key Words: gravity, flat connection, spin current, vacuum energy density 1. Introduction In the Einstein’ s theory of the gravity the curvature of a space - time manifold is created by the energy momentum. However, there is an algebraic relation between the energy - momentum and the curvature and if the energy - momen- tum vanishes (also the vacuum energy density is zero), the curvature continues propagating, that is the vanishing energy - momentum does not imply zero cur- vature, i.e the Schwarzschild solution and gravitational radiation. The solution to the Einstein equation over asymptotically locally flat space gives the plane gravitational wave and this kind of the plane waves is a quadruple.
    [Show full text]
  • Cosmological Solutions to Polynomial Affine Gravity in the Torsion-Free Sector Oscar Castillo-Felisola, José Perdiguero and Oscar Orellana
    Chapter Cosmological Solutions to Polynomial Affine Gravity in the Torsion-Free Sector Oscar Castillo-Felisola, José Perdiguero and Oscar Orellana Abstract We find possible cosmological models of the polynomial affine gravity described by connections that are either compatible or not with a metric. When possible, we compare them with those of general relativity. We show that the set of cosmological vacuum solutions in general relativity are a subset of the solutions of polynomial affine gravity. In our model, the cosmological constant appears as an integration constant, and, additionally, we show that some forms of matter can be emulated by the affine structure—even in the metric compatible case. In the case of connections not compatible with a metric, we obtain formal families of solutions, which should be constrained by physical arguments. We show that for a certain parametrisation of the connection, the affine Ricci-flat condition yields the cosmological field equa- tions of general relativity coupled with a perfect fluid, pointing towards a geomet- rical emulation of—what is interpreted in general relativity as—matter effects. Keywords: affine gravity, exact solutions, cosmological models 1. Introduction All of the fundamental physics is described by four interactions: electromagnetic, weak, strong and gravitational. The former three are bundled into what is known as standard model of particle physics, which explains very accurately the physics at very short scales. These three interactions share common grounds, for example, they are modelled by connections with values in a Lie algebra, they have been successfully quantised and renormalised, and the simplest of them—quantum electrodynamics—gives the most accurate results when compared with the experiments.
    [Show full text]
  • Arxiv:2008.07750V2 [Gr-Qc] 14 Jul 2021 Effect of the Cartan Torsion That in Principle Can Be Measured `Ala the Aharonov-Bohm Effect
    A unified view of curvature and torsion in metric-affine gauge theory of gravity through affine-vector bundles Bo-Hung Chen1, 2, ∗ and Dah-Wei Chiou3, 4, y 1Department of Physics, National Taiwan University, Taipei 10617, Taiwan 2Center for Theoretical Physics, National Taiwan University, Taipei 10617, Taiwan 3Department of Physics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan 4Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan Abstract One of the most appealing results of metric-affine gauge theory of gravity is a close parallel between the Riemann curvature two-form and the Cartan torsion two-form: While the former is the field strength of the Lorentz-group connection one-form, the latter can be understood as the field strength of the coframe one-form. This parallel, unfortunately, is not fully established until one adopts Trautman's idea of introducing an affine-vector-valued zero-from, the meaning of which has not been satisfactorily clarified. This paper aims to derive this parallel from first principles without any ad hoc prescriptions. We propose a new mathematical framework of an associated affine-vector bundle as a more suitable arena for the affine group than a conventional vector bundle, and rigorously derive the covariant derivative of a local section on the affine-vector bundle in the formal Ehresmann-connection approach. The parallel between the Riemann curvature and the Cartan torsion arises naturally on the affine-vector bundle, and their geometric and physical meanings become transparent. The clear picture also leads to a conjecture about a kinematical arXiv:2008.07750v2 [gr-qc] 14 Jul 2021 effect of the Cartan torsion that in principle can be measured `ala the Aharonov-Bohm effect.
    [Show full text]
  • Spacetime Geometric Structures and the Search for a Quantum Theory of Gravity
    Spacetime geometric structures and the search for a Quantum Theory of Gravity Mihaela Iftime Abstract. One of the biggest challenges to theoretical physics of our time is to ¯nd a background-independent quantum theory of gravity. Today one encounters a profusion of di®erent attempts at quantization, but no fully accepted - or acceptable, theory of quantum gravity. Any such approach requires a response to a question that lies at the heart of this problem. \How shall we resolve the tension between the background de- pendence of all hitherto-successful quantum theories, both non-relativistic quantum mechanics and special-relativistic quantum ¯eld theory, and the background independence of classical general relativity?" (see [28]) The need for a background-independent quantization procedure forms the starting point of my approach. In this paper I shall present a gauge-natural formulation of general relativity, and provide some insights into the struc- ture of the space of geometries, which plays an important role in the construction of a non-perturbative quantum gravity using a path integral approach, as well as in string theory (see e.g., [2, 18, 31]) M.S.C. 2000: 53-XX, 83-XX. Key words: Di®erential geometry, Relativity and gravitational theory. 1 Historical-Physical Motivation The most succesful among all gravitational theories is Einstein's theory of general relativity. In general relativity the ¯eld equations 1 are described in purely geometric terms: the space-time is a fairly smooth manifold M of dim n = 4 with a normal hyperbolic Riemannian structure { Lorentzian metric tensor g transforms at each point to the flat Minkowski metric ´ = diag(1; 1; 1; ¡1), and its timelike and null geodesics represent the paths of freely falling particles and light rays.
    [Show full text]
  • Gauge Theory of Gravity: Electrically Charged Solutions Within the Metric–Affine Framework ∗ F.W
    Vol. 29 (1998) ACTA PHYSICA POLONICA B No 4 GAUGE THEORY OF GRAVITY: ELECTRICALLY CHARGED SOLUTIONS WITHIN THE METRIC–AFFINE FRAMEWORK ∗ F.W. Hehl and J. Socorro† Institute for Theoretical Physics, University of Cologne D-50923 Köln, Germany (Received March 4, 1998) Dedicated to Andrzej Trautman in honour of his 64th birthday We find a class of electrically charged exact solutions for a toy model of metric-affine gravity. Their metric is of the Plebański-Demiański type and their nonmetricity and torsion are represented by a triplet of covectors with dilation, shear, and spin charges. PACS numbers: 04.50. +h, 04.20. Jb, 03.50. Kk 1. Introduction In analyzing the structure of Minkowski space, Kopczyński & Traut- man [1] stressed its underlying affine structure which expresses the inertial properties of spacetime. Superimposed to it, there exists a flat metric which allows to calculate lengths, times, and angles. It is the combination of both structures, the affine and the metric one, which determines the unique fea- tures of the Minkowski space of special relativity. In special relativity the effects of gravity are neglected. Then the group of motions of spacetime is the Poincaré group with its 4+6 parameters. This group is the semidirect product of the translation and the Lorentz group. The source of gravity is the mass–energy of matter. Because of the Noether theorem, the conservation of mass–energy, in Minkowski space, is related to the translation group. Consequently the gauging of the transla- tions should yield gravity, as foreseen by Feynman [2], amongst others. We ∗ Presented at the Workshop on Gauge Theories of Gravitation, Jadwisin, Poland, September 4–10, 1997.
    [Show full text]
  • Building a Connection Through Obstruction; Relating Gauge
    BUILDING A CONNECTION THROUGH OBSTRUCTION; RELATING GAUGE GRAVITY AND STRING THEORY. by Casey Cartwright c Copyright by Casey Cartwright, 2016 All Rights Reserved A thesis submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Master of Science (Applied Physics). Golden, Colorado Date Signed: Casey Cartwright Signed: Dr. Jeff Squier Thesis Advisor Signed: Dr. Alex Flournoy Thesis Advisor Golden, Colorado Date Signed: Dr. Jeff Squier Professor and Head Department of Physics ii ABSTRACT Gauge theories of internal symmetries, e.g. the strong and electroweak forces of the Standard Model, have a geometric description in terms of standard fiber bundles. It is tempting then to ask if the remaining force, gravitation, has a description as a gauge theory. The answer is yes, however unlike the internal symmetries of the Standard Model, the story is not so simple. There are dozens of renditions of gravitational gauge theory and no standard fiber bundle description. The main issue in the construction of gravitational gauge theory is the inclusion of translational symmetry. While the Lorentz group, like internal symmetries, acts only at each point, the translational symmetry shifts points in spacetime. For this reason a gauge theory of gravity requires a somewhat more sophisticated fiber bundle known as a composite fiber bundle. When constructing gauge theories of internal symmetries it is easy to take certain topological conditions for granted, like orientability or the ability to define spinors. However it is known that there exist spaces which do not have the properties required to define sensible field theories.
    [Show full text]
  • Gr-Qc/9402012
    1 0.1 Introduction: the need to go beyond Rieman- nian manifolds A number of developments in physics in recent years have evoked the possibility that the treatment of spacetime might involve more than just the Riemannian spacetime of Einstein’s general relativity: (1) The vain effort so far to quantize gravity is, perhaps, the strongest piece of evidence for going beyond a geometry which is dominated by the classical distance concept. (2) Intuitively, the generalization of the three–dimensional theory of elastic continua with microstructure to the four–dimensional spacetime of gravity suggests, in a rather convincing manner, physical interpretations for the newly emerging struc- tures in post–Riemannian spacetime geometry. (3) The description of hadron (or nuclear) matter in terms of extended structures. In particular, the quadrupole pulsation rates of that matter and, in a rest frame, their relation to representations of the volume–preserving three–dimensional lin- ear group SL(3, R) — with the rotation group SO(3) as subgroup — have been established experimentally. (4) The study of the early universe — in the light of the various theorems about a sin- gular origin, the ideas about unification of the fundamental interactions (mostly involving additional dimensions, later compactified) and inflationary models with dilaton–induced Weyl covector — ...and each of these developments necessitates the study of dynamical theories involving post–Riemannian geometries, whether in the context of local field theories or within the arXiv:gr-qc/9402012v1 4 Feb 1994 framework of string theories. We explain the interest in continua with microstructure, in extended structures, and the problematics of the early universe – as far as these are relevant as motivations for a relaxation of the Riemannian constraint in gravity – in the rest of this chapter, leaving the rather involved issue of quantum gravity to Chap.2.
    [Show full text]
  • Emergent Planck Mass and Dark Energy from Affine Gravity
    INR-TH-2020-005 Emergent Planck mass and dark energy from affine gravity I. Kharuka,b aMoscow Institute of Physics and Technology, Institutsky lane 9, Dolgoprudny, Moscow region, 141700, Russia bInstitute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect, 7a, Moscow, 117312, Russia E-mail: ivan.kharuk@phystech.edu Abstract: We introduce a novel model of affine gravity, which implements the no–scale scenario. Namely, in our model the Planck mass and Hubble constant emerge dynamically, through the mechanism of spontaneous breaking of scale–invariance. This naturally gives rise to the inflation, thus introducing a new inflationary mechanism. Moreover, the time direction and non–degenerate metric emerge dynamically as well, which allows one to consider the usual General Relativity as an effective theory. We show that our model is phenomenologically viable, both from the perspective of the direct tests of gravity and cosmological evolution. arXiv:2002.12178v2 [gr-qc] 1 Jun 2021 Contents 1 Introduction 1 2 Affine gravity 2 2.1 Overview of affine gravity 2 2.2 Perspectives and problems of affine gravity 5 3 Scalar–affine gravity 6 4 Phenomenological validity 10 4.1 Linearized limit 10 4.2 Cosmology 10 5 Discussion 11 5.1 Possible physical implications 11 5.2 Gravity as a gauge theory of connection 12 A Linearized limit 13 1 Introduction The power of General Relativity (GR) comes from the fact that it combines profound physical ideas with elegant mathematical apparatus. By generalizing the latter, one may hope to gain a deeper insight into gravity. One of the possible routes in this direction is known as the affine gravity [1–9].
    [Show full text]