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The Geometrical Trinity of Gravity

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universe Article The Geometrical Trinity of Gravity Jose Beltrán Jiménez 1,*, Lavinia Heisenberg 2,* and Tomi S. Koivisto 3,4,* 1 Departamento de Física Fundamental and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain 2 Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland 3 Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-10691 Stockholm, Sweden 4 Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia * Correspondence: [email protected] (J.B.J.); [email protected] (L.H.); [email protected] (T.S.K.) Received: 5 June 2019; Accepted: 5 July 2019; Published: 15 July 2019 Abstract: The geometrical nature of gravity emerges from the universality dictated by the equivalence principle. In the usual formulation of General Relativity, the geometrisation of the gravitational interaction is performed in terms of the spacetime curvature, which is now the standard interpretation of gravity. However, this is not the only possibility. In these notes, we discuss two alternative, though equivalent, formulations of General Relativity in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity, thus putting forward the existence of three seemingly unrelated representations of the same underlying theory. Based on these three alternative formulations of General Relativity, we then discuss some extensions. Keywords: general relativity; modified gravity; torsion; non-metricity 1. Introduction Gravity and geometry have accompanied each other from the very conception of General Relativity (GR) brilliantly formulated by Einstein in terms of the spacetime curvature. This inception of identifying gravity with the curvature has since grown so efficiently that it is now a common practice to recognise the gravitational phenomena as a manifestation of having a curved spacetime. As Einstein ingeniously envisioned, the existence of a geometrical formulation of gravity is granted by the equivalence principle that renders the gravitational interaction oblivious to the specific type of matter and hints towards an intriguing relation of gravity with inertia. Thus, the motion of particles can be naturally associated with the geometrical properties of spacetime. If we embrace the geometrical character of gravity advocated by the equivalence principle, it is pertinent to explore in which equivalent manners gravity can be geometrised. It is then convenient to recall at this point that a spacetime can be endowed with a metric and an affine structure [1–4] determined by a metric a tensor gmn and a connection G mn, respectively. These two structures, although completely independent, enable the definition of geometrical objects that allow for conveniently classify geometries. The failure of the connection to be metric is encoded in the non-metricity Q g , (1) amn ≡ ra mn while its antisymmetric part defines the torsion Ta 2Ga . (2) mn ≡ [mn] Universe 2019, 5, 173; doi:10.3390/universe5070173 www.mdpi.com/journal/universe Universe 2019, 5, 173 2 of 17 Among all possible connections that can be defined on a spacetime, the Levi–Civita connection is the unique connection that is symmetric and metric-compatible. These two conditions fix the Levi–Civita connection to be given by the Christoffel symbols of the metric n o 1 a = galg + g g . (3) mn 2 ln,m ml,n − mn,l The corresponding covariant derivative will be denoted by so that we will have agmn = 0.A a D D general connection G mn then admits the following convenient decomposition: a n a o a a G mn = mn + K mn + L mn (4) with 1 a 1 a Ka = Ta + T La = Qa Q (5) mn 2 mn (m n) mn 2 mn − (m n) the contortion and the disformation pieces, respectively. Notice that, while the Levi–Civita part is non-tensorial, the contortion and the disformation have tensorial transformation properties under changes of coordinates. The curvature is determined by the usual Riemann tensor Ra (G) = ¶ Ga ¶ Ga + Ga Gl Ga Gl . (6) bmn m nb − n mb ml nb − nl mb In Figure1 we give a geometrical intuition for the different objects associated to the affine structure. A relation that will be useful later on is how the Riemann tensor transforms under a shift of the ˆ a a a a connection G mn = G mn + W mn, with W mn an arbitrary tensor. Under such a shift, the Riemann tensor becomes ˆ a a l a a a l R bmn = R bmn + T mnW lb + 2 [mW n]b + 2W [m l W n]b , (7) r j j where Rˆ a and Ra are the Riemann tensors of Gˆ and G, respectively, and is the covariant bmn bmn r derivative associated with G. ↵ R βµ⌫ <latexit 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{ The rotation of a vector transported along a closed curve is given by the curvature: General Relativity. { <latexit sha1_base64="/Qc/70FUMVg/W6zRp2zmxqTMW5U=">AAAB6XicbZC7SgNBFIbPxlsSb1EbwWYwCFZh18aUQRvLKOYCSQizk9lkyOzsMnNWiEvAB7CxUMTWN/BR7AQfxsml0MQfBj7+cw5zzu/HUhh03S8ns7K6tr6RzeU3t7Z3dgt7+3UTJZrxGotkpJs+NVwKxWsoUPJmrDkNfckb/vByUm/ccW1EpG5xFPNOSPtKBIJRtNZNO+0Wim7JnYosgzeHYuXw/jv38HFR7RY+272IJSFXyCQ1puW5MXZSqlEwycf5dmJ4TNmQ9nnLoqIhN510uumYnFinR4JI26eQTN3fEykNjRmFvu0MKQ7MYm1i/ldrJRiUO6lQcYJcsdlHQSIJRmRyNukJzRnKkQXKtLC7EjagmjK04eRtCN7iyctQPyt5lq9tGmWYKQtHcAyn4ME5VOAKqlADBgE8wjO8OEPnyXl13matGWc+cwB/5Lz/ANm2kII=</latexit> <latexit 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    III IIII II I I I I I I I I III IIII II I I I I I I I I Editor Jill Singer, Ph.D. Director, Office of Undergraduate Research Cover Design and Layout: Carol Alex The following individuals and offices are acknowledged for their many contributions: Marc Bayer, Interim Library Director, E.H. Butler Library, Department and Program Coordinators (identified below) and the library staff Donald Schmitter, Hospitality and Tourism, and students and very special thanks to: in HTR 400: Catering Management Kaylene Waite, Graphic Design, Instructional Resources Bruce Fox, Photographer Carol Alex, Center for Development of Human Services Andrew Chambers, Information Management Officer Sean Fox, Ellofex, Inc. Bernadette Gilliam and Mary Beth Wojtaszek, Events Management Department and Program Coordinators for the Eighteenth Annual Student Research and Creativity Celebration Lisa Marie Anselmi, Anthropology Dan MacIsaac, Physics Sarbani Banerjee, Computer Information Systems Candace Masters, Art Education Saziye Bayram, Mathematics Carmen McCallum, Higher Education Administration Carol Beckley, Theater Michaelene Meger, Exceptional Education Lynn Boorady, Fashion and Textile Technology Andrew Nicholls, History and Social Studies Education Louis Colca, Social Work Jill Norvilitis, Psychology Sandra Washington-Copeland, McNair Scholars Program Kathleen O’Brien, Hospitality and Tourism Carol DeNysschen, Health, Nutrition, and Dietetics Lorna Perez, English Eric Dolph, Interior Design Rebecca Ploeger, Art Conservation Reva Fish, Social & Psychological Foundations
  • Teleparallelism Arxiv:1506.03654V1 [Physics.Pop-Ph] 11 Jun 2015

    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.
  • Building a Popular Science Library Collection for High School to Adult Learners: ISSUES and RECOMMENDED RESOURCES

    Building a Popular Science Library Collection for High School to Adult Learners: ISSUES and RECOMMENDED RESOURCES

    Building a Popular Science Library Collection for High School to Adult Learners: ISSUES AND RECOMMENDED RESOURCES Gregg Sapp GREENWOOD PRESS BUILDING A POPULAR SCIENCE LIBRARY COLLECTION FOR HIGH SCHOOL TO ADULT LEARNERS Building a Popular Science Library Collection for High School to Adult Learners ISSUES AND RECOMMENDED RESOURCES Gregg Sapp GREENWOOD PRESS Westport, Connecticut • London Library of Congress Cataloging-in-Publication Data Sapp, Gregg. Building a popular science library collection for high school to adult learners : issues and recommended resources / Gregg Sapp. p. cm. Includes bibliographical references and index. ISBN 0–313–28936–0 1. Libraries—United States—Special collections—Science. I. Title. Z688.S3S27 1995 025.2'75—dc20 94–46939 British Library Cataloguing in Publication Data is available. Copyright ᭧ 1995 by Gregg Sapp All rights reserved. No portion of this book may be reproduced, by any process or technique, without the express written consent of the publisher. Library of Congress Catalog Card Number: 94–46939 ISBN: 0–313–28936–0 First published in 1995 Greenwood Press, 88 Post Road West, Westport, CT 06881 An imprint of Greenwood Publishing Group, Inc. Printed in the United States of America TM The paper used in this book complies with the Permanent Paper Standard issued by the National Information Standards Organization (Z39.48–1984). 10987654321 To Kelsey and Keegan, with love, I hope that you never stop learning. Contents Preface ix Part I: Scientific Information, Popular Science, and Lifelong Learning 1
  • Kinetic Models for the in Situ Reaction Between Cu-Ti Melt and Graphite

    Kinetic Models for the in Situ Reaction Between Cu-Ti Melt and Graphite

    metals Article Kinetic Models for the in Situ Reaction between Cu-Ti Melt and Graphite Lei Guo, Xiaochun Wen and Zhancheng Guo * State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Xueyuan Road No.30, Haidian District, Beijing 100083, China; [email protected] (L.G.); [email protected] (X.W.) * Correspondence: [email protected]; Tel.: +86-135-2244-2020 Received: 16 January 2020; Accepted: 17 February 2020; Published: 18 February 2020 Abstract: The in situ reaction method for preparing metal matrix composites has the advantages of a simple process, good combination of the reinforcing phase and matrix, etc. Based on the mechanism of forming TiCx particles via the dissolution reaction of solid carbon (C) particles in Cu-Ti melt, the kinetic models for C particle dissolution reaction were established. The kinetic models of the dissolution reaction of spherical, cylindrical, and flat C source particles in Cu-Ti melt were deduced, and the expressions of the time for the complete reaction of C source particles of different sizes were obtained. The mathematical relationship between the degree of reaction of C source and the reaction time was deduced by introducing the shape factor. By immersing a cylindrical C rod in a Cu-Ti melt and placing it in a super-gravity field for the dissolution reaction, it was found that the super-gravity field could cause the precipitated TiCx particles to aggregate toward the upper part of the sample under the action of buoyancy. Therefore, the consuming rate of the C rod was significantly accelerated. Based on the flat C source reaction kinetic model, the relationship between the floating speed of TiCx particles in the Cu-Ti melt and the centrifugal velocity (or the coefficient of super-gravity G) was derived.
  • 3. Introducing Riemannian Geometry

    3. Introducing Riemannian Geometry

    3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4).