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On the Status of the Geodesic Principle in Newtonian and Relativistic Physics1
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PhilSci Archive On the Status of the Geodesic Principle in Newtonian and Relativistic Physics1 James Owen Weatherall2 Logic and Philosophy of Science University of California, Irvine Abstract A theorem due to Bob Geroch and Pong Soo Jang [\Motion of a Body in General Relativ- ity." Journal of Mathematical Physics 16(1), (1975)] provides a sense in which the geodesic principle has the status of a theorem in General Relativity (GR). I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation (Newton- Cartan theory) [Weatherall, J. O. \The Motion of a Body in Newtonian Theories." Journal of Mathematical Physics 52(3), (2011)]. Here I compare the interpretations of these two the- orems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case. 1 Introduction The geodesic principle is the central principle of General Relativity (GR) that describes the inertial motion of test particles. It states that free massive test point particles traverse timelike geodesics. There is a long-standing view, originally due to Einstein, that the geodesic principle has a special status in GR that arises because it can be understood as a theorem, rather than a postulate, of the theory. (It turns out that capturing the geodesic principle as a theorem in GR is non-trivial, but a result due to Bob Geroch and Pong Soo Jang (1975) 1Thank you to David Malament and Jeff Barrett for helpful comments on a previous version of this paper and for many stimulating conversations on this topic. -
The Geodesic Spacetime Equation for Body Motion in Gravitational Fields
The Geodesic Spacetime Equation for Body Motion in Gravitational Fields "The eternal mystery of the world is its comprehensibility" - Albert Einstein ( 1879 - 1955 ) § Einstein's "Law of Equivalence of Acceleration and Gravity" or "The Principle of Equivalence" : Fields of acceleration and fields of gravity are equivalent physical phenomenon. That is, if there's no "upward" acceleration, then there's no "downward" gravity. Preliminary Understanding of the following Mathematical Symbols Naïve realism easily allows ordinary understanding of the mathematical symbols of ; however, for deeper understanding of physical reality it becomes necessary to invent newer and more powerful mathematical tools such as the Christoffel symbol by which a deeper probing of external reality is made entirely possible. Hence, a certain amount of patient indulgence is asked of the serious reader of this mathematical essay since it is strongly suggested that by a thorough reading of these general relativity equations and the tightly woven logic by which they are presented, that with final patience the reader of Relativity Calculator will have a fairly good grasp of the underlying mathematical physics of Einstein's ultimate gravitational equation! A Quick Introductory Meaning of Geodesic Motion Sir Arthur Eddington ( 1882 - 1944 ) on May 29, 1919 tentatively, but empirically, confirmed Einstein's mathematical general relativity conjecture that light passing near the gravity field of the sun was slightly, but convincingly, bent in the spacetime proximate vicinity of the sun, thus showing that on grand cosmic scales there is no such reality for Euclid's straight geometry lines. Likewise, gyroscopic motion will equivalently follow the curvilinear contours of spacetime fabric as influenced by other gravitational fields. -
TWAS Fellowships Worldwide
CDC Round Table, ICTP April 2016 With science and engineering, countries can address challenges in agriculture, climate, health TWAS’s and energy. guiding principles 2 Food security Challenges Water quality for a Energy security new era Biodiversity loss Infectious diseases Climate change 3 A Globally, 81 nations fall troubling into the category of S&T- gap lagging countries. 48 are classified as Least Developed Countries. 4 The role of TWAS The day-to-day work of TWAS is focused in two critical areas: •Improving research infrastructure •Building a corps of PhD scholars 5 TWAS Research Grants 2,202 grants awarded to individuals and research groups (1986-2015) 6 TWAS’ AIM: to train 1000 PhD students by 2017 Training PhD-level scientists: •Researchers and university-level educators •Future leaders for science policy, business and international cooperation Rapidly growing opportunities P BRAZIL A K I N D I CA I RI A S AF TH T SOU A N M KENYA EX ICO C H I MALAYSIA N A IRAN THAILAND TWAS Fellowships Worldwide NRF, South Africa - newly on board 650+ fellowships per year PhD fellowships +460 Postdoctoral fellowships +150 Visiting researchers/professors + 45 17 Programme Partners BRAZIL: CNPq - National Council MALAYSIA: UPM – Universiti for Scientific and Technological Putra Malaysia WorldwideDevelopment CHINA: CAS - Chinese Academy of KENYA: icipe – International Sciences Centre for Insect Physiology and Ecology INDIA: CSIR - Council of Scientific MEXICO: CONACYT– National & Industrial Research Council on Science and Technology PAKISTAN: CEMB – National INDIA: DBT - Department of Centre of Excellence in Molecular Biotechnology Biology PAKISTAN: ICCBS – International Centre for Chemical and INDIA: IACS - Indian Association Biological Sciences for the Cultivation of Science PAKISTAN: CIIT – COMSATS Institute of Information INDIA: S.N. -
Millennium Prize for the Poincaré
FOR IMMEDIATE RELEASE • March 18, 2010 Press contact: James Carlson: [email protected]; 617-852-7490 See also the Clay Mathematics Institute website: • The Poincaré conjecture and Dr. Perelmanʼs work: http://www.claymath.org/poincare • The Millennium Prizes: http://www.claymath.org/millennium/ • Full text: http://www.claymath.org/poincare/millenniumprize.pdf First Clay Mathematics Institute Millennium Prize Announced Today Prize for Resolution of the Poincaré Conjecture a Awarded to Dr. Grigoriy Perelman The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture. The citation for the award reads: The Clay Mathematics Institute hereby awards the Millennium Prize for resolution of the Poincaré conjecture to Grigoriy Perelman. The Poincaré conjecture is one of the seven Millennium Prize Problems established by CMI in 2000. The Prizes were conceived to record some of the most difficult problems with which mathematicians were grappling at the turn of the second millennium; to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasize the importance of working towards a solution of the deepest, most difficult problems; and to recognize achievement in mathematics of historical magnitude. The award of the Millennium Prize to Dr. Perelman was made in accord with their governing rules: recommendation first by a Special Advisory Committee (Simon Donaldson, David Gabai, Mikhail Gromov, Terence Tao, and Andrew Wiles), then by the CMI Scientific Advisory Board (James Carlson, Simon Donaldson, Gregory Margulis, Richard Melrose, Yum-Tong Siu, and Andrew Wiles), with final decision by the Board of Directors (Landon T. -
Prospects in Topology
Annals of Mathematics Studies Number 138 Prospects in Topology PROCEEDINGS OF A CONFERENCE IN HONOR OF WILLIAM BROWDER edited by Frank Quinn PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1995 Copyright © 1995 by Princeton University Press ALL RIGHTS RESERVED The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton Academic Press 10 987654321 Library of Congress Cataloging-in-Publication Data Prospects in topology : proceedings of a conference in honor of W illiam Browder / Edited by Frank Quinn. p. cm. — (Annals of mathematics studies ; no. 138) Conference held Mar. 1994, at Princeton University. Includes bibliographical references. ISB N 0-691-02729-3 (alk. paper). — ISBN 0-691-02728-5 (pbk. : alk. paper) 1. Topology— Congresses. I. Browder, William. II. Quinn, F. (Frank), 1946- . III. Series. QA611.A1P76 1996 514— dc20 95-25751 The publisher would like to acknowledge the editor of this volume for providing the camera-ready copy from which this book was printed PROSPECTS IN TOPOLOGY F r a n k Q u in n , E d it o r Proceedings of a conference in honor of William Browder Princeton, March 1994 Contents Foreword..........................................................................................................vii Program of the conference ................................................................................ix Mathematical descendants of William Browder...............................................xi A. Adem and R. J. Milgram, The mod 2 cohomology rings of rank 3 simple groups are Cohen-Macaulay........................................................................3 A. -
Preface Chapter 1 Chaos Theory and the Dynamics of Narrative
Notes Preface 1. Tom Stoppard, Arcadia (London: Faber and Faber, 1993) 79. 2. Ivar Ekeland discusses the plight of Johannes Kepler, who attempted to chart the trajectories of planets in the early seventeenth century. Despite having sound theories upon which to draw, “He nevertheless had to perform monstrous computa- tions over a number of years.” Ekeland explains that, even with the digital computers of today, “There are still a great many computations that cannot be performed now or in any foresee- able future.” See Ivar Ekeland, Mathematics and the Unexpected (Chicago: University of Chicago Press, 1988) 5, 31; trans. of Le Calcul, l’imprevu: Les figures de temps du Kepler à Thom (Éditions du Seuil, 1984). 3. The phrase is James Gleick’s. His popular science book Chaos: Making a New Science (New York: Viking, 1987) brought chaos theory into the public imagination. 4. I discuss these works in chapter 1. Chapter 1 Chaos Theory and the Dynamics of Narrative 1. James P. Crutchfield, J. Doyne Farmer, Norman H. Packard, and Robert S. Shaw, “Chaos,” Scientific American December 1986: 49. These four were members of the Dynamical Systems Collective of the University of California at Santa Cruz. “Chaos” was one of the first, if not the first, popular texts on the subject of chaos theory. 2. As Crutchfield et al. point out, “The larger framework that chaos emerges from is the so-called theory of dynamical systems” (49). 3. N. Katherine Hayles discusses the concurrent scientific imprecision and cultural resonance of the terms “chaos theory” and “science of chaos” in Chaos Bound: Orderly Disorder in Contemporary 136 NOTES Literature and Science (Ithaca: Cornell University Press, 1990). -
Contemporary Mathematics 432
CONTEMPORARY MATHEMATICS 432 In the Tradition of Ahlfors-Bers/ IV Ahlfors-Bers Colloquium May 19-22, 2005 University of Michigan Ann Arbor, Michigan Dick Canary Jane Gilman Juha Heinonen Howard Masur Editors http://dx.doi.org/10.1090/conm/432 In the Tradition of Ahlfors-Bers, IV CoNTEMPORARY MATHEMATICS 432 In the Tradition of Ahlfors-Bersl IV Ahlfors-Bers Colloquium May 19-22, 2005 University of Michigan Ann Arbor, Michigan Dick Canary Jane Gilman Juha Heinonen Howard Masur Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Andreas Blass Abel Klein 2000 Mathematics Subject Classification. Primary 14H15, 20H10, 28A75, 30F40, 30C62, 32G15, 54E40, 57M50. Library of Congress Cataloging-in-Publication Data Ahlfors-Bers Colloquium (4th : 2005 : Ann Arbor, Michigan) In the tradition of Ahlfors-Bers, IV : the Ahlfors-Bers Colloquium, May 19-22, 2005, Ann Arbor, Michigan / Dick Canary ... [et a!.], editors. p. em. -(Contemporary mathematics, ISSN 0271-4132 ; 432) Includes bibliographical references. ISBN-13: 978-0-8218-4227-0 (alk. paper) ISBN-10: 0-8218-4227-7 (alk. paper) 1. Functions-Congresses. 2. Riemann surfaces-Congresses. 3. Mappings (Mathematics)- Congresses. I. Canary, Richard Douglas. II. Title. QA331.A462 2005 515--dc22 2007060747 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. -
Office of Solicitor General
THE JOURNAL OF APPELLATE PRACTICE AND PROCESS THE OFFICE OF SOLICITOR GENERAL PREFACE If any legal position warrants the appellation, "the appellate lawyer's lawyer," it is that of Solicitor General. Seth Waxman, himself a former Solicitor General, has pointed out that "the office of the Solicitor General of the United States is a wonderful and unique creation,"' noting that only the holder of that office, among all the officers of the federal government, is required by statute to be "learned in the law." 2 President after president has complied with that instruction: The list of Solicitors General that follows this preface includes the names of some of this country's most distinguished lawyers. There may even be those who think of the Solicitors General as a corps of immortals, for as Waxman discovered, "[s]ome 60 years ago, a letter found its way into the United States mail addressed simply 'The Celestial General, Washington, D.C." 3 The inadequacy of the address notwithstanding, the Post Office "apparently had no trouble discerning to whom it should be delivered. It went to Robert H. 1. Seth P. Waxman, Speech, Presenting the Case of the United States As It Should Be: The Solicitor General in Historical Context (address to the Supreme Court Historical Society, Washington, June 1, 1998) at I (available at <http://www.usdoj.gov/osg/about osg/sgarticle.html>). 2. Id. 3. Id. THE JOURNAL OF APPELLATE PRACTICE AND PROCESS Vol. 3, No. 2 (Fall 2001) THE JOURNAL OF APPELLATE PRACTICE AND PROCESS Jackson, then Solicitor General of the United States." 4 Waxman is quick to point out that neither he nor any of his predecessors had "pretensions of other-worldliness," but he does acknowledge that they "have all been fortunate to have been able to serve in what Thurgood Marshall called 'the best job I've ever had." We in the law can see that it is indeed a special job, for the Solicitor General is the only lawyer who, as Francis Biddle put it, "has no master to serve except his country." 6 The responsibilities of the job are great, but so are the rewards. -
Selected New Aspects of the Calculus of Variations in the Large
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 39, Number 2, Pages 207{265 S 0273-0979(02)00929-1 Article electronically published on January 4, 2002 SELECTED NEW ASPECTS OF THE CALCULUS OF VARIATIONS IN THE LARGE IVAR EKELAND AND NASSIF GHOUSSOUB Abstract. We discuss some of the recent developments in variational meth- ods while emphasizing new applications to nonlinear problems. We touch on several issues: (i) the formulation of variational set-ups which provide more information on the location of critical points and therefore on the qualitative properties of the solutions of corresponding Euler-Lagrange equations; (ii) the relationships between the energy of variationally generated solutions, their Morse indices, and the Hausdorff measure of their nodal sets; (iii) the gluing of several topological obstructions; (iv) the preservation of critical levels after deformation of functionals; (v) and the various ways to recover compactness in certain borderline variational problems. Contents 1. Introduction 208 2. Critical points and non-linear variational problems 209 2.1. Variational proofs of non-existence: Pohozaev phenomenon 210 2.2. The min-max identification of critical levels 211 2.3. Analysis of Palais-Smale sequences in borderline variational problems 214 3. Locating critical points via dual sets 221 3.1. Improved compactness around dual sets 223 3.2. Relaxing boundary conditions in the presence of dual sets 225 4. Duality and the multiplicity of critical points 227 4.1. Group actions and topological indices 227 4.2. Multiplicity resultsa ` la Ljusternik-Schnirelmann 228 4.3. Comparable functions and other types of multiplicity 230 5. Morse indices of variationally generated critical points 231 5.1. -
TME Volume 7, Number 1
The Mathematics Enthusiast Volume 7 Number 1 Article 10 1-2010 TME Volume 7, Number 1 Follow this and additional works at: https://scholarworks.umt.edu/tme Part of the Mathematics Commons Let us know how access to this document benefits ou.y Recommended Citation (2010) "TME Volume 7, Number 1," The Mathematics Enthusiast: Vol. 7 : No. 1 , Article 10. Available at: https://scholarworks.umt.edu/tme/vol7/iss1/10 This Full Volume is brought to you for free and open access by ScholarWorks at University of Montana. It has been accepted for inclusion in The Mathematics Enthusiast by an authorized editor of ScholarWorks at University of Montana. For more information, please contact [email protected]. The Montana Mathematics Enthusiast ISSN 1551-3440 VOL. 7, NO.1, January 2010, pp.1-174 Editor-in-Chief Bharath Sriraman, The University of Montana Associate Editors: Lyn D. English, Queensland University of Technology, Australia Simon Goodchild, University of Agder, Norway Brian Greer, Portland State University, USA Luis Moreno-Armella, Cinvestav-IPN, México International Editorial Advisory Board Mehdi Alaeiyan, Iran University of Science and Technology, Iran Miriam Amit, Ben-Gurion University of the Negev, Israel Ziya Argun, Gazi University, Turkey Ahmet Arikan, Gazi University, Turkey. Astrid Beckmann, University of Education, Schwäbisch Gmünd, Germany Raymond Bjuland, University of Stavanger, Norway Morten Blomhøj, Roskilde University, Denmark Robert Carson, Montana State University- Bozeman, USA Mohan Chinnappan, University of Wollongong, -
Problems in Low-Dimensional Topology
Problems in Low-Dimensional Topology Edited by Rob Kirby Berkeley - 22 Dec 95 Contents 1 Knot Theory 7 2 Surfaces 85 3 3-Manifolds 97 4 4-Manifolds 179 5 Miscellany 259 Index of Conjectures 282 Index 284 Old Problem Lists 294 Bibliography 301 1 2 CONTENTS Introduction In April, 1977 when my first problem list [38,Kirby,1978] was finished, a good topologist could reasonably hope to understand the main topics in all of low dimensional topology. But at that time Bill Thurston was already starting to greatly influence the study of 2- and 3-manifolds through the introduction of geometry, especially hyperbolic. Four years later in September, 1981, Mike Freedman turned a subject, topological 4-manifolds, in which we expected no progress for years, into a subject in which it seemed we knew everything. A few months later in spring 1982, Simon Donaldson brought gauge theory to 4-manifolds with the first of a remarkable string of theorems showing that smooth 4-manifolds which might not exist or might not be diffeomorphic, in fact, didn’t and weren’t. Exotic R4’s, the strangest of smooth manifolds, followed. And then in late spring 1984, Vaughan Jones brought us the Jones polynomial and later Witten a host of other topological quantum field theories (TQFT’s). Physics has had for at least two decades a remarkable record for guiding mathematicians to remarkable mathematics (Seiberg–Witten gauge theory, new in October, 1994, is the latest example). Lest one think that progress was only made using non-topological techniques, note that Freedman’s work, and other results like knot complements determining knots (Gordon- Luecke) or the Seifert fibered space conjecture (Mess, Scott, Gabai, Casson & Jungreis) were all or mostly classical topology. -
“Selecta” Books Scrapbook Forewords & Excerpts from Reviews
“SELECTA” BOOKS SCRAPBOOK FOREWORDS & EXCERPTS FROM REVIEWS Benoit B. Mandelbrot July 1, 2004 SE Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. New York: Springer, 1997, x+551pp. SFE Fractales, hasard et finance (1959 - 1997). Paris: Flammarion (Collection Champs), 1997, 246pp. SN Multifractals and 1/f Noise : Wild Self-affinity in Physics. New York: Springer. 1999, viii + 442 pp. SH Gaussian Self-Affinity and Fractals: Globality, the Earth, 1/f, and R/S. New York: Springer. 2002, ix + 654 pp. SC Fractals and Chaos: The Mandelbrot Set and Beyond. New York: Springer. 2004, xii + 308 pp. B.B. MANDELBROT ◊ SCRAPBOOK OF REVIEWS OF “SELECTA” BOOKS ◊ JULY 1, 2004 ◊ 2 FOREWORDS “FRACTALS AND SCALING IN FINANCE” ◊ Ralph E. Gomory (President, Sloan Foundation) In 1959-61, while the huge Saarinen-designed research laboratory at Yorktown Heights was being built, much of IBM's Research was housed nearby. My group occupied one of the many little houses on the Lamb Estate complex which had been a sanatorium housing wealthy alcoholics. Even in a Lamb Estate populated exclusively with bright research-oriented people, Benoit always stood out. His thinking was always fresh, and I enjoyed talking with him about any subject, whether technical, political, or historical. He introduced me to the idea that distributions having infinite second moments could be more than a mathematical curiosity an a source of counter-examples. This was a foretaste of the line of thought that eventually led to fractals and to the notion that major pieces of the physical world could be, and in fact could only be, modeled by distributions and sets that had fractional dimensions.