DOCSLIB.ORG

Redalyc.Aspects of Nonmetricity in Gravity Theories

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Redalyc.Aspects of Nonmetricity in Gravity Theories Brazilian Journal of Physics ISSN: 0103-9733 [email protected] Sociedade Brasileira de Física Brasil Sobreiro, Rodrigo F.; Vasquez Otoya, Victor J. Aspects of nonmetricity in gravity theories Brazilian Journal of Physics, vol. 40, núm. 4, diciembre, 2010, pp. 370-374 Sociedade Brasileira de Física Sâo Paulo, Brasil Available in: http://www.redalyc.org/articulo.oa?id=46415799002 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative 370 Rodrigo F. Sobreiro and Victor J. Vasquez Otoya Aspects of nonmetricity in gravity theories Rodrigo F. Sobreiro and Victor J. Vasquez Otoya UFF − Universidade Federal Fluminense, Instituto de F´ısica, Campus da Praia Vermelha, Avenida Litoraneaˆ s/n, 24210-346 Boa Viagem, Niteroi,´ RJ, Brasil. (Received on 10 February, 2009) In this work, we show that a class of metric-affine gravities can be reduced to a Riemann-Cartan one. The reduction is based on the cancelation of the nonmetricity against the symmetric components of the spin connec- tion. A heuristic proof, in the Einstein-Cartan formalism, is performed in the special case of diagonal unitary tangent metric tensor. The result is that the nonmetric degrees of freedom decouple from the geometry. Thus, from the point of view of isometries on the tangent manifold, the equivalence might be viewed as an isome- try transition from the affine group to the Lorentz group, A(d;R) 7−! SO(d). Furthermore, in this transition, depending on the form of the starting action, the nonmetricity degrees might present a dynamical matter field character, with no geometric interpretation in the Riemann-Cartan geometry. Keywords: Differential geometry, Nonmetricity, Gauge theories of gravity, Metric-affine gravity 1. INTRODUCTION do so, we consider the full covariant derivative of the viel- bein, De = M, where M is the deviation tensor. This equa- tion is taken here as the fundamental equation of the MAG. The renowned article by T. W. B. Kibble [1] describ- When M = 0, we are dealing with the RCG [1]. Further, we ISO(d) ∼ ing gravity in a gauge theoretical approach, with = consider only Euclidean metrics on the tangent space. This SO(d) d nR local symmetry group (with global translations), restriction enforces a particular case of the MAG. We then has inspired many consequent works on the field. In par- show that the nonmetricity eliminates, in a natural way, the ticular, the generalization of the theory to the affine group symmetric degrees of freedom of the spin connection in the A(d; ) ∼ GL(d; ) d R = R n R , with global translations, has led full covariant derivative of the vielbein. To be more specific, to a new class of theories, the Metric-Affine gravities [2, 3]. the symmetric part of the deviation tensor and the symmetric The former [1] puts gravity in a Riemann-Cartan geometry part of the spin connection cancel, eliminating all the sym- T a (RCG), where torsion, , is allowed and spinorial matter metric degrees of freedom of the referred equation. Thus, can couple to gravity. The gauge fields of the theory are we have on this equation just Lorentz algebra valued quan- 1 ea the vielbein , , associated with global translations, and the tities. The antisymm! etric part of the deviation tensor is ab SO(d) 2 spin connection, w ,! associated with the group and then absorbed into the antisymmetric part of the spin connec- is directly related to the affine connection, G. The second case [2, 3], describes a metric-affine geometry (MAG) where tion, resulting in the usual constraint for the RCG, De e = 0, not only torsion, but also nonmetricity, Qab, is allowed. In where De , carries just the Lorentz affine and spin connec- this case, the relation between the spin and affine connec- tions, Ge and we. Thus, as a consequence one identify a kind tions entails also the deviation tensor, Mab. As it will become of geometric reduction in the form A(d;R) 7−! SO(d), on evident in this work, the deviation tensor is directly related the tangent manifold. Obviously, the restriction to Euclidean to the nonmetricity. tangent metrics plays a fundamental hole on the cancelation of the symmetric degrees of freedom. A more general and Basically, the difference between RCG and MAG is the formal proof is left for a future work [5]. nonmetricity tensor, which appears as a non-vanishing quan- tity in the MAG. However, the nonmetricity cannot be de- rived from the algebra of the covariant derivatives, and thus, from the gauge procedure standpoint, it seems unnatural to We also discuss the consequences of this geometric tran- consider it as a field strength. On the other hand, the non- sition from the curvature and torsion points of view, as well metricity is commonly used as a field strength or even as a as a from a general gravity action, not explicitly depending propagating spin-3 physical field; see for instance [2–4]. In on the nonmetricity. It is shown that the nonmetricity is not that context, the nonmetricity appears explicitly in the action completely eliminated, but it appears as a matter field de- of a gravity theory. coupled from the geometry. The same result is obtained by Therefore, to highlight the issue of the physical role of considering as starting point an action with an explicit de- the nonmetricity, we discuss in this work if the nonmetric- pendence on the nonmetricity. ity may or may not be eliminated from the geometry. To This work is organized as follows: In Sect.II, we provide 1 The vielbein is identified to global translations. In fact, the gauge field a brief review of the properties, notation and conventions of of translations can be associated to a part of the vielbein. See [3] and the MAG, used in this work. In Sect.III, we establish a few references therein. 2 For mathematical purposes, always that is possible, we avoid noncompact statements in order to show the reduction of MAG to RCG. groups in this work. Also, we call the SO(d) group by Lorentz group After that, the physical consequences of the reduction are since we are considering Euclidean tangent spaces. discussed. Finally, in Sect.IV, we display our Conclusions. Brazilian Journal of Physics, vol. 40, no. 4, December, 2010 371 2. GENERALITIES OF METRIC-AFFINE GEOMETRY where K is the spin torsion K c = ecD eµ : (9) To describe the MAG, we can think of a d-dimensional ab µ [a b] manifold M which is characterized by a metric tensor, g, and an affine connection, G, independent from each other. Also, nonmetricity appears as given by This is the so called metric-affine formalism. The geometry ab ab Qµ = Dµh ; (10) is associated with a A(d;R) diffeomorphism invariance. We define the covariant derivative, ∇, by its action on a tensor where h is the flat metric tensor of T . In general, h = h(x). field v according to However, we restrict ourselves to the case that h is strictly Euclidean. This restriction enforces a particular case of the ∇ vn = ¶ vn + G nva ; µ µ µa MAG. As said before, the results on this work are restricted a ∇µvn = ¶µvn − Gµn va ; (1) to this particular class of the MAG and, therefore, is taken as a heuristic proof of the geometric reduction from the MAG where Greek indices stand for the coordinates in the mani- to the RCG. fold M . The curvature and torsion can be identified from In order to characterize the MAG by a single geometric b b equation, we adopt to work with the full covariant derivative, [∇µ;∇n]va = −Rµna vb − Tµn ∇bva ; (2) D, acting on a M -T mixed object. Here, for the sake of convenience, we take the vielbein itself, where R is the Riemann-Christoffel curvature and T the tor- sion tensor a a a a a a a a b Dµen = Dµen − Gµn ea = ¶µen − Gµn ea + wµ ben : (11) b b g b Rµna (G) = ¶[µGn]a − G[µa Gn]g ; ab ab ab Notice that, with this definition, Qµ = Dµh = Dµh . a a Now, by defining the deviation tensor M as Tµn = G[µn] : (3) M a = ea ; (12) The effect of the independence between g and G is a non- µ n Dµ n metric geometry characterized by a non-trivial nonmetricity, we rewrite expression (11) as a constraint characterizing the Q, MAG Qµna = ∇µgna : (4) a a a a b a ¶µen − Gµn ea + wµ ben = Mµ n ; (13) One may also study the MAG through the isometries of from which we can easily write the relation between the the affine group in the tangent space, T , the well-known affine and spin connections Einstein-Cartan formalism. The gauge fields associated with translations on and GL(d; ) rotations are, respectively, a a a a a b a T R Gµn = ea ¶µen + wµ bea en − Mµ n : (14) the vielbein, e, and spin connection, w. The vielbein maps a a µ M quantities in T quantities, v = eµv . The gauge covariant This constraint fixes G as a function of w, e and M. Thus, G derivative D acts on the tangent space according to is completely determined from the properties of the tangent M a a a b manifold T and . We remark that the RCG is obtained Dµv = ¶µv + wµ bv ; from M = 0. As a consequence we can interpret the deviation b tensor as a measure of how the MAG differs from the RCG.
Load more

Recommended publications
  • Arxiv:0911.0334V2 [Gr-Qc] 4 Jul 2020
    Classical Physics: Spacetime and Fields Nikodem Poplawski Department of Mathematics and Physics, University of New Haven, CT, USA Preface We present a self-contained introduction to the classical theory of spacetime and fields. This expo- sition is based on the most general principles: the principle of general covariance (relativity) and the principle of least action. The order of the exposition is: 1. Spacetime (principle of general covariance and tensors, affine connection, curvature, metric, tetrad and spin connection, Lorentz group, spinors); 2. Fields (principle of least action, action for gravitational field, matter, symmetries and conservation laws, gravitational field equations, spinor fields, electromagnetic field, action for particles). In this order, a particle is a special case of a field existing in spacetime, and classical mechanics can be derived from field theory. I dedicate this book to my Parents: Bo_zennaPop lawska and Janusz Pop lawski. I am also grateful to Chris Cox for inspiring this book. The Laws of Physics are simple, beautiful, and universal. arXiv:0911.0334v2 [gr-qc] 4 Jul 2020 1 Contents 1 Spacetime 5 1.1 Principle of general covariance and tensors . 5 1.1.1 Vectors . 5 1.1.2 Tensors . 6 1.1.3 Densities . 7 1.1.4 Contraction . 7 1.1.5 Kronecker and Levi-Civita symbols . 8 1.1.6 Dual densities . 8 1.1.7 Covariant integrals . 9 1.1.8 Antisymmetric derivatives . 9 1.2 Affine connection . 10 1.2.1 Covariant differentiation of tensors . 10 1.2.2 Parallel transport . 11 1.2.3 Torsion tensor . 11 1.2.4 Covariant differentiation of densities .
    [Show full text]
  • Research Article Weyl-Invariant Extension of the Metric-Affine Gravity
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Open Access Repository Hindawi Publishing Corporation Advances in High Energy Physics Volume 2015, Article ID 902396, 7 pages http://dx.doi.org/10.1155/2015/902396 Research Article Weyl-Invariant Extension of the Metric-Affine Gravity R. Vazirian,1 M. R. Tanhayi,2 and Z. A. Motahar3 1 Plasma Physics Research Center, Islamic Azad University, Science and Research Branch, Tehran 1477893855, Iran 2Department of Physics, Islamic Azad University, Central Tehran Branch, Tehran 8683114676, Iran 3Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia Correspondence should be addressed to M. R. Tanhayi; [email protected] Received 30 September 2014; Accepted 28 November 2014 Academic Editor: Anastasios Petkou Copyright © 2015 R. Vazirian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP3. Metric-affine geometry provides a nontrivial extension of the general relativity where the metric and connection are treated as the two independent fundamental quantities in constructing the spacetime (with nonvanishing torsion and nonmetricity). In this paper,westudythegenericformofactioninthisformalismandthenconstructtheWeyl-invariantversionofthistheory.Itisshown that, in Weitzenbock¨ space, the obtained Weyl-invariant action can cover the conformally invariant teleparallel action. Finally, the related field equations are obtained in the general case. 1. Introduction with the metric and torsion-free condition is relaxed; thus, in addition to Christoffel symbols, the affine connection would Extended theories of gravity have become a field of interest contain an antisymmetric part and nonmetric terms as well.
    [Show full text]
  • Spacetime and Geometry: an Introduction to General Relativity Pdf, Epub, Ebook
    SPACETIME AND GEOMETRY: AN INTRODUCTION TO GENERAL RELATIVITY PDF, EPUB, EBOOK Sean M. Carroll,John E. Neely,Richard R. Kibbe | 513 pages | 28 Sep 2003 | Pearson Education (US) | 9780805387322 | English | New Jersey, United States Spacetime and Geometry: An Introduction to General Relativity PDF Book Likewise an explorer from region IV could have a brief look at region I before perishing. Mathematics Kronecker delta Levi-Civita symbol metric tensor nonmetricity tensor Christoffel symbols Ricci curvature Riemann curvature tensor Weyl tensor torsion tensor. Thorne, John Archibald Wheeler Gravitation. When using coordinate transformations as described above, the new coordinate system will often appear to have oblique axes compared to the old system. Indeed, they find some remarkable new regions of spacetime! In general relativity, gravity can be regarded as not a force but a consequence of a curved spacetime geometry where the source of curvature is the stress—energy tensor representing matter, for instance. Views Read Edit View history. Modern cosmological models are a bit more complicated, but retain those features. I was misled even lied to by JG on math. Bitte versuchen Sie es erneut. Considering the number of things I did not know it was a good idea to postpone reading this section until now. Tuesday, 25 February Einstein's equation. So I decided to test the flatness idea in two dimensions Principles of Cosmology and Gravitation. It was supposed to be impossible to travel between regions I and IV of the Kruskal diagram and here Carroll
    [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]
  • Coframe Geometry, Gravity and Electromagnetism
    IARD2012 IOP Publishing Journal of Physics: Conference Series 437 (2013) 012003 doi:10.1088/1742-6596/437/1/012003 Coframe geometry, gravity and electromagnetism Yakov Itin Institute of Mathematics, Hebrew University of Jerusalem and Jerusalem College of Technology, Jerusalem 91904, Israel E-mail: [email protected] Abstract. The extensions of GR for description of fermions on a curved space, for supergravity, and for the loop quantum gravity ordinary use a set of 16 independent variables instead of 10 components of metric. These variables can be assembled in a coframe field, i.e., a set of four linearly independent 1-forms. In this presentation we review a geometrical structure based on the coframe field. We construct a complete class of the coframe connections which are linear in the first order derivatives of the coframe field on an n dimensional manifolds with and without a metric. The subclasses of the torsion-free, metric-compatible and flat connections are derived. We also study the behavior of the geometrical structures under local transformations of the coframe. The remarkable fact is an existence of a subclass of connections which are invariant when the infinitesimal transformations satisfy the Maxwell-like system of equations. 1. Introduction. Why do we have to go beyond Riemannian geometry? General relativity (GR) is, probably, the best of the known theories of gravity. From mathematical and aesthetic points of view, it can be used as a standard of what a physical theory has to be. Einstein's theory is in a very good agreement with all up-to-date observation data.
    [Show full text]
  • On the Interpretation of the Einstein-Cartan Formalism
    33 On the Interpretation of the Einstein-Cartan Formalism Jobn Staebell 33.1 Introduction Hehl and collaborators [I] have suggested the need to generalize Riemannian ge­ ometry to a metric-affine geometry, by admitting torsion and nonmetricity of the connection field. They assume that this geometry represents the microstructure of space-time, with Riemannian geometry emerging as some sort of macroscopic average over the metric-affine microstructure. They thereby generalize the ear­ lier approach to the Einstein-Cartan formalism of Hehl et al. [2] based on a metric connection with torsion. A particularly clear statement of this point of view is found in Hehl, von der Heyde, and Kerlick [3]: "We claim that the [Einstein-Cartan] field equations ... are, at a classical level, the correct microscopic gravitational field equations. Einstein's field equation ought to be considered a macroscopic phenomenological equation oflimited validity, obtained by averaging [the Einstein-Cartan field equations]" (p. 1067). Adamowicz takes an alternate approach [4], asserting that "the relation between the Einstein-Cartan theory and general relativity is similar to that between the Maxwell theory of continuous media and the classical microscopic electrodynam­ ics" (p. 1203). However, he only develops the idea of treating the spin density that enters the Einstein-Cartan theory as the macroscopic average of microscopic angu­ lar momenta in the linear approximation, and does not make explicit the relation he suggests by developing a formal analogy between quantities in macroscopic electrodynamics and in the Einstein-Cartan theory. In this paper, I shall develop such an analogy with macroscopic electrodynamics in detail for the exact, nonlinear version of the Einstein-Cartan theory.
    [Show full text]
  • A Note on Parallel Transportation in Symmetric Teleparallel Geometry
    A novel approach to autoparallels for the theories of symmetric teleparallel gravity Caglar Pala1∗ and Muzaffer Adak2† , 1 2Department of Physics, Faculty of Arts and Sciences, Pamukkale University, 20017 Denizli, Turkey 1Department of Physics, Faculty of Science, Erciyes University, 38030 Kayseri, Turkey Abstract Although the autoparallel curves and the geodesics coincide in the Riemannian geometry in which only the curvature is nonzero among the nonmetricity, the torsion and the curvature, they define different curves in the non-Riemannian ones. We give a novel approach to autoparallel curves and geodesics for theories of the symmetric teleparallel gravity written in the coincident gauge. PACS numbers: 04.50.Kd, 11.15.Kc, 02.40.Yy Keywords: Non-Riemannian geometry, Geodesic, Autoparallel curve arXiv:1102.1878v2 [physics.gen-ph] 8 Sep 2021 ∗[email protected] †[email protected] 1 Introduction There have been two main ingredients for the theories of gravity since Isaac Newton; field equations and trajectory equations. The field equations are cast for describing the dynamics of a source and the trajectory ones are the for determining path followed by a spinless massive point test particle. For the Newton’s theory of gravity they are written respectively as ∇2φ =4πGρ (1) d2~r + ∇~ φ =0 (2) dt2 where G is the Newton’s constant of gravity, ~r is the three-dimensional Euclidean position vector of the test particle, t is the time disjoint from the Euclidean space, ρ is the volume density of mass of the source, ∇~ is the gradient operator in the three-dimensional Euclidean space and φ is the gravitational field from a source at the position of ~r.
    [Show full text]
  • Constraining Spacetime Nonmetricity with Neutron Spin Rotation in Liquid
    Physics Letters B 772 (2017) 865–869 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Constraining spacetime nonmetricity with neutron spin rotation 4 in liquid He ∗ Ralf Lehnert a,b, , W.M. Snow a,c,d, Zhi Xiao a,e, Rui Xu c a Indiana University Center for Spacetime Symmetries, Bloomington, IN 47405, USA b Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany c Physics Department, Indiana University, Bloomington, IN 47405, USA d Center for Exploration of Energy and Matter, Indiana University, Bloomington, IN 47408, USA e Department of Mathematics and Physics, North China Electric Power University, Beijing 102206, China a r t i c l e i n f o a b s t r a c t Article history: General spacetime nonmetricity coupled to neutrons is studied. In this context, it is shown that certain Received 3 June 2017 nonmetricity components can generate a rotation of the neutron’s spin. Available data on this effect Received in revised form 19 July 2017 obtained from slow-neutron propagation in liquid helium are used to constrain isotropic nonmetricity Accepted 27 July 2017 − (6) components at the level of 10 22 GeV. These results represent the first limit on the nonmetricity ζ S Available online 1 August 2017 2 000 Editor: A. Ringwald parameter as well as the first measurement of nonmetricity inside matter. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction The specialized situation in which the nonmetricity tensor van- ishes Nαβγ = 0 and only torsion is nonzero represents the widely The idea that spacetime geometry represents a dynamical phys- known Einstein–Cartan theory [3].
    [Show full text]
  • Appendix a May a Torus with Null Riemann Curvature Exist on E3?
    Appendix A May a Torus with Null Riemann Curvature Exist on E3? 1 1 1 We can also give a flat Riemann curvature tensor for T3 S × S T1 T2. First parametrize 1 2 1 2 3 1 2 1 2 T3 by (x ,x ), x ,x ∈ R, such that its coordinates in R are x(x ,x )=x(x +2π, x +2π)and x(x1,x2)=(h(x1)cosx2,h(x1)sinx2,l(x1), (A.1) with h(x1)=R + r cos x1, l(x1)=r sin x1, (A.2) where here R and r are real positive constants and R>r(See Figure A.1). 3 Figure A.1: Some Possible GSS for the Torus T3 living in E where the grid defines the parallelism 1 Recall that T1 and T2 have been defined in Section 1.1. 110 Appendix A. May a Torus with Null Riemann Curvature Exist on E3? 1 2 Now, T T3 has a global coordinate basis {∂/∂x ,∂/∂x }, and the induced metric on T3 is easily found as g = r2dx1 ⊗ dx1 +(R + r cos x1)2dx2 ⊗ dx2 (A.3) Now, let us introduce a connection ∇ on T3 such that j ∇∂/∂xi ∂/∂x =0. (A.4) With respect to this connection, it is immediately verified that its torsion and curvature tensors are null, but the nonmetricity of the connection is non null. Indeed, 1 1 ∇∂/∂xi g = −2(R + r cos x )sinx . (A.5) So, in this case, we have a particular Riemann-Cartan-Weyl GSS (T3, g, ∇)withA =0 , Θ=0, R =0.
    [Show full text]
  • On the Dynamical Content of Mags
    Home Search Collections Journals About Contact us My IOPscience On the dynamical content of MAGs This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Conf. Ser. 600 012043 (http://iopscience.iop.org/1742-6596/600/1/012043) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 131.169.4.70 This content was downloaded on 07/03/2016 at 23:01 Please note that terms and conditions apply. Spanish Relativity Meeting (ERE 2014): almost 100 years after Einstein’s revolution IOP Publishing Journal of Physics: Conference Series 600 (2015) 012043 doi:10.1088/1742-6596/600/1/012043 On the dynamical content of MAGs Vincenzo Vitagliano CENTRA, Departamento de F´ısica,Instituto Superior T´ecnico,Universidade de Lisboa - UL, Av. Rovisco Pais 1, 1049 Lisboa, Portugal E-mail: [email protected] Abstract. Adopting a procedure borrowed from the effective field theory prescriptions, we study the dynamics of metric-affine theories of increasing order, that in the complete version include invariants built from curvature, nonmetricity and torsion. We show that even including terms obtained from nonmetricity and torsion to the second order density Lagrangian, the connection lacks dynamics and acts as an auxiliary field that can be algebraically eliminated, resulting in some extra interactions between metric and matter fields. Introduction. The intriguing choice to treat alternative theories of gravity by means of the Palatini approach, namely elevating the affine connection to the role of independent variable, contains the seed of some interesting (usually under-explored) generalizations of General Relativity, the metric-affine theories of gravity.
    [Show full text]
  • On the Ontology of Spacetime in a Frame of Reference
    On the Ontology of Spacetime in a Frame of Reference Alexander Poltorak 1 Abstract The spacetime ontology is considered in General Relativity (GR) in view of the choice of a frame of reference (FR). Various approaches to a description of the FR, such as coordinate systems, monads and tetrads are reviewed. It is shown that any of the existing FR definitions require a preexisting background spacetime, which, if defined independently of the FR, renders the spacetime absolute in violation of the principle of relativity, or, if defined within an inertial FR (IFR), as it is usually done, makes the argument circular. Consequently, defining a FR in a preexisting spacetime is unacceptable. We show that a FR defines a differentiable manifold with, generally, non-Euclidean geometry. In a noninertial FR (NIFR) the observer must chose a coordinative definition either admitting existence of a universal – inertial – force or settling for non-Euclidean spacetime. Following Reichenbach, it is preferable to eliminate all universal forces and opt for a non-Euclidean geometry. It is shown that a metric-affine space ( L4,g) is best suited to describe the geometry of spacetime within a FR. Considering a gravitational field in an arbitrary FR, we show within the framework of Einstein’s GR that the gravity is described by nonmetricity of spacetime. This result may shed new light on the nature of the cosmological constant and dark energy. I. Introduction One of the fundamental problems of spacetime ontology is how matter affects the geometry of spacetime and, vice versa, how spacetime affects the behavior of the matter therein.
    [Show full text]
  • Cosmology in Symmetric Teleparallel Gravity and Its Dynamical System
    Eur. Phys. J. C (2019) 79:530 https://doi.org/10.1140/epjc/s10052-019-7038-3 Regular Article - Theoretical Physics Cosmology in symmetric teleparallel gravity and its dynamical system Jianbo Lua, Xin Zhao, Guoying Cheeb Department of Physics, Liaoning Normal University, Dalian 116029, People’s Republic of China Received: 28 February 2019 / Accepted: 10 June 2019 / Published online: 20 June 2019 © The Author(s) 2019 Abstract We explore an extension of the symmetric telepa- symmetric teleparallel connection that is not metric compati- rallel gravity denoted the f (Q) theory, by considering a ble. This classification highlights that curvature is a property function of the nonmetricity invariant Q as the gravitational of the connection and not of the metric tensor or the mani- Lagrangian. Some interesting properties could be found in fold. It becomes a property of the metric only through the use the f (Q) theory by comparing with the f (R) and f (T ) the- of the Levi–Civita connection. GR can be equivalently for- ories. The field equations are derived in the f (Q) theory. mulated in terms of either of these connections. All of them The cosmological application is investigated. In this theory can be used to define Lagrangians of which Euler-Lagrange the accelerating expansion is an intrinsic property of the uni- equations coincide with the Einstein equations for a particu- verse geometry without need of either exotic dark energy or lar choice of contributing terms. extra fields. And the state equation of the geometrical dark In recent years teleparallel theories have gained more energy can cross over the phantom divide line in the f (Q) attention as alternative theories of gravity.
    [Show full text]