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“Geodesic Principle” in General Relativity∗
A Remark About the “Geodesic Principle” in General Relativity∗ Version 3.0 David B. Malament Department of Logic and Philosophy of Science 3151 Social Science Plaza University of California, Irvine Irvine, CA 92697-5100 [email protected] 1 Introduction General relativity incorporates a number of basic principles that correlate space- time structure with physical objects and processes. Among them is the Geodesic Principle: Free massive point particles traverse timelike geodesics. One can think of it as a relativistic version of Newton’s first law of motion. It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein’s ∗I am grateful to Robert Geroch for giving me the basic idea for the counterexample (proposition 3.2) that is the principal point of interest in this note. Thanks also to Harvey Brown, Erik Curiel, John Earman, David Garfinkle, John Manchak, Wayne Myrvold, John Norton, and Jim Weatherall for comments on an earlier draft. 1 ab equation (or of the conservation principle ∇aT = 0 that is, itself, a conse- quence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic prin- ciple can be recovered as theorem in general relativity, it is not a consequence of Einstein’s equation (or the conservation principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed). -
HÁLÓZATELMÉLET ÉS MŰVÉSZET a Lineáris Információ Nincs Központban
MOHOLY-NAGY MŰVÉSZETI EGYETEM DOKTORI ISKOLA KALLÓ ANGÉLA HÁLÓZATELMÉLET ÉS MŰVÉSZET A Lineáris Információ Nincs Központban. Jöjjön a LINK! DLA ÉRTEKEZÉS TÉMAVEZETŐ: Dr. TILLMANN JÓZSEF BUDAPEST-KOLOZSVÁR 2009 TARTALOMJEGYZÉK DLA ÉRTEKEZÉS – TÉZISEK A DLA értekezés tézisei magyar nyelven – Bevezető – Tézisek Thesis of DLA dissertation (A DLA értekezés tézisei angol nyelven) – Introduction – Thesis DLA ÉRTEKEZÉS Bevezető 1. Hálózatelmélet – Előzmények 1.1. Kiindulópont 1.2. Hálózatelmélet – három név, három cím, három megközelítés 1.2.1. Barabási 1.2.2. Buchanan 1.2.3. Csermely 2. A háló ki van vetve 2.1. A hálózatelmélet hajnala 2.1.1. Sajátos gráfok 2.2. Lánc, lánc… 2.2.1. Az ismeretségi hálózat és a köztéri művészet 2.2.2. Az ismeretségi hálózat és a multimédia művészet 2.2.3. Az ismeretségi hálózat és a net art, avagy ma van a tegnap holnapja 1 2.3. Kis világ 2.3.1. Hidak 2.3.2. Mitől erős egy gyenge kapcsolat? 2.3.3. Centrum és periféria 2.4. Digitális hálózatok 2.4.1. A hálózatok hálózata 2.4.2. A világháló 2.4.3. A Lineáris Információ Nincs Központban. Jöjjön a LINK! 2.4.4. Kicsi világ @ világháló 2.5. Művészet a hálón 2.5.1. A net mint art 2.5.2. Interaktív művészet a hálón és azon túl 2.5.3. A szavak hálózata mint művészet 2.6. Szemléletváltás 2.6.1. A nexus néhány lehetséges módja 2.6.2. Egy lépésnyire a fraktáloktól 2.6.3. Fraktálok – természet, tudomány, művészet 2.6.4. Térképek 3. Összegzés magyar nyelven 4. Summary of DLA dissertation (Összegzés angol nyelven) Bibliográfia Curriculum Vitae 2 HÁLÓZATELMÉLET ÉS MŰVÉSZET A Lineáris Információ Nincs Központban. -
Geodesic Spheres in Grassmann Manifolds
GEODESIC SPHERES IN GRASSMANN MANIFOLDS BY JOSEPH A. WOLF 1. Introduction Let G,(F) denote the Grassmann manifold consisting of all n-dimensional subspaces of a left /c-dimensional hermitian vectorspce F, where F is the real number field, the complex number field, or the algebra of real quater- nions. We view Cn, (1') tS t Riemnnian symmetric space in the usual way, and study the connected totally geodesic submanifolds B in which any two distinct elements have zero intersection as subspaces of F*. Our main result (Theorem 4 in 8) states that the submanifold B is a compact Riemannian symmetric spce of rank one, and gives the conditions under which it is a sphere. The rest of the paper is devoted to the classification (up to a global isometry of G,(F)) of those submanifolds B which ure isometric to spheres (Theorem 8 in 13). If B is not a sphere, then it is a real, complex, or quater- nionic projective space, or the Cyley projective plane; these submanifolds will be studied in a later paper [11]. The key to this study is the observation thut ny two elements of B, viewed as subspaces of F, are at a constant angle (isoclinic in the sense of Y.-C. Wong [12]). Chapter I is concerned with sets of pairwise isoclinic n-dimen- sional subspces of F, and we are able to extend Wong's structure theorem for such sets [12, Theorem 3.2, p. 25] to the complex numbers nd the qua- ternions, giving a unified and basis-free treatment (Theorem 1 in 4). -
1 Dalí Museum, Saint Petersburg, Florida
Dalí Museum, Saint Petersburg, Florida Integrated Curriculum Tour Form Education Department, 2015 TITLE: “Salvador Dalí: Elementary School Dalí Museum Collection, Paintings ” SUBJECT AREA: (VISUAL ART, LANGUAGE ARTS, SCIENCE, MATHEMATICS, SOCIAL STUDIES) Visual Art (Next Generation Sunshine State Standards listed at the end of this document) GRADE LEVEL(S): Grades: K-5 DURATION: (NUMBER OF SESSIONS, LENGTH OF SESSION) One session (30 to 45 minutes) Resources: (Books, Links, Films and Information) Books: • The Dalí Museum Collection: Oil Paintings, Objects and Works on Paper. • The Dalí Museum: Museum Guide. • The Dalí Museum: Building + Gardens Guide. • Ades, dawn, Dalí (World of Art), London, Thames and Hudson, 1995. • Dalí’s Optical Illusions, New Heaven and London, Wadsworth Atheneum Museum of Art in association with Yale University Press, 2000. • Dalí, Philadelphia Museum of Art, Rizzoli, 2005. • Anderson, Robert, Salvador Dalí, (Artists in Their Time), New York, Franklin Watts, Inc. Scholastic, (Ages 9-12). • Cook, Theodore Andrea, The Curves of Life, New York, Dover Publications, 1979. • D’Agnese, Joseph, Blockhead, the Life of Fibonacci, New York, henry Holt and Company, 2010. • Dalí, Salvador, The Secret life of Salvador Dalí, New York, Dover publications, 1993. 1 • Diary of a Genius, New York, Creation Publishing Group, 1998. • Fifty Secrets of Magic Craftsmanship, New York, Dover Publications, 1992. • Dalí, Salvador , and Phillipe Halsman, Dalí’s Moustache, New York, Flammarion, 1994. • Elsohn Ross, Michael, Salvador Dalí and the Surrealists: Their Lives and Ideas, 21 Activities, Chicago review Press, 2003 (Ages 9-12) • Ghyka, Matila, The Geometry of Art and Life, New York, Dover Publications, 1977. • Gibson, Ian, The Shameful Life of Salvador Dalí, New York, W.W. -
General Relativity Fall 2019 Lecture 13: Geodesic Deviation; Einstein field Equations
General Relativity Fall 2019 Lecture 13: Geodesic deviation; Einstein field equations Yacine Ali-Ha¨ımoud October 11th, 2019 GEODESIC DEVIATION The principle of equivalence states that one cannot distinguish a uniform gravitational field from being in an accelerated frame. However, tidal fields, i.e. gradients of gravitational fields, are indeed measurable. Here we will show that the Riemann tensor encodes tidal fields. Consider a fiducial free-falling observer, thus moving along a geodesic G. We set up Fermi normal coordinates in µ the vicinity of this geodesic, i.e. coordinates in which gµν = ηµν jG and ΓνσjG = 0. Events along the geodesic have coordinates (x0; xi) = (t; 0), where we denote by t the proper time of the fiducial observer. Now consider another free-falling observer, close enough from the fiducial observer that we can describe its position with the Fermi normal coordinates. We denote by τ the proper time of that second observer. In the Fermi normal coordinates, the spatial components of the geodesic equation for the second observer can be written as d2xi d dxi d2xi dxi d2t dxi dxµ dxν = (dt/dτ)−1 (dt/dτ)−1 = (dt/dτ)−2 − (dt/dτ)−3 = − Γi − Γ0 : (1) dt2 dτ dτ dτ 2 dτ dτ 2 µν µν dt dt dt The Christoffel symbols have to be evaluated along the geodesic of the second observer. If the second observer is close µ µ λ λ µ enough to the fiducial geodesic, we may Taylor-expand Γνσ around G, where they vanish: Γνσ(x ) ≈ x @λΓνσjG + 2 µ 0 µ O(x ). -
Current Technologies and Trends of Retractable Roofs By
Current Technologies and Trends of Retractable Roofs by Julie K. Smith B.S. Civil and Environmental Engineering University of Washington, 2002 Submitted to the Department of Civil and Environmental Engineering in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Civil and Environmental Engineering at the Massachusetts Institute of Technology June 2003 © 2003 Julie K. Smith. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part. Signature of Author: Julie K. Smith Department of Civil and Environmental Engineering 1) May 9, 2003 Certified by:. Jerome Connor Professor/ of C il and Environmental Engineering Thesis Supervisor Accepted by: Oral Buyukozturk Professor of Civil and Environmental Engineering Chairman, Departmental Committee on Graduate Studies MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUN 0 2 2003 LIBRARIES Current Technologies and Trends of Retractable Roofs by Julie K. Smith B.S. Civil and Environmental Engineering University of Washington, 2002 Submitted to the Department of Civil and Environmental Engineering on May 9, 2003 in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Civil and Environmental Engineering Abstract In recent years, retractable roofs have become a popular feature in sport stadiums. However, they have been used throughout time because they allow a building to become more flexible in its use. This thesis reviews the current technologies of retractable roofs and discusses possible innovations for the future. Most retractable roofs use either a 2-D rigid panel system or a 2-D membrane and I-D cable system. -
My MFA Experience Thesis Presented in Partial Fulfillment of The
My MFA Experience Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Fine Arts in the Graduate School of The Ohio State University By Axel Cuevas Santamaría Graduate Program in Art The Ohio State University 2018 Master’s Examination Committee: Ken Rinaldo, Adviser Amy Youngs Alex Oliszewski Copyright by Axel Cuevas Santamaría 2018 Abstract This MFA thesis explores the threshold of phenomenological perception, audience attention and the mystery of imaginary worlds I perceive between microscopic and macroscopic dimensions. In the BioArt projects and digital immersive environments I present in this thesis, I have found the potential to explore real and imaginary landscapes. This exploration further expands, adding new physical and virtual layers to my work that activate the audience. My work incorporates the synthesis of projection mapping, biological living systems and interactive multimedia. It is the vehicle I use to contemplate the impermanence of time and the illusion of reality. i Dedication To the inspiring artists, dancers, doctors, musicians, philosophers, scientists, and friends I crossed paths with during my MFA at The Ohio State University Ken Rinaldo, Amy Youngs, Alex Oliszewski, Norah Zuniga Shaw, Michael Mercil, Ann Hamilton, Trademark Gunderson, Dr. Sarah Iles Johnston, Dani Opossum Restack, Todd Slaughter, Jason Slot, Roger Beebe, George Rush, Kurt Hentschlager, Rafael Lozano-Hemmer, Theresa Schubert, Andrew Adamatzky, Andrew Frueh, Nate Gorgen, Federico Cuatlacuatl, Florence Gouvrit Montoyo, Tess Elliot, Jessica Ann, Cameron Sharp, Kyle Downs, Sa'dia Rehman, Ph.D. Jose Orlando Combita-Heredia, Pelham Johnston, Lynn Kim, Jacklyn Brickman, Mel Mark, Ashlee Daniels Taylor, James D. MacDonald III, Katie Coughlin, Elaine Buss, Max Fletcher, Alicia Little, Eun Young Cho, Morteza Khakshoor, Catelyn Mailloux, Niko Dimitrijevic, Jeff Hazelden, Natalia Sanchez, Terry Hanlon, Travis Casper, Caitlin Waters, Joey Pigg. -
Unique Properties of the Geodesic Dome High
Printing: This poster is 48” wide by 36” Unique Properties of the Geodesic Dome high. It’s designed to be printed on a large-format printer. Verlaunte Hawkins, Timothy Szeltner || Michael Gallagher Washkewicz College of Engineering, Cleveland State University Customizing the Content: 1 Abstract 3 Benefits 4 Drawbacks The placeholders in this poster are • Among structures, domes carry the distinction of • Consider the dome in comparison to a rectangular • Domes are unable to be partitioned effectively containing a maximum amount of volume with the structure of equal height: into rooms, and the surface of the dome may be formatted for you. Type in the minimum amount of material required. Geodesic covered in windows, limiting privacy placeholders to add text, or click domes are a twentieth century development, in • Geodesic domes are exceedingly strong when which the members of the thin shell forming the considering both vertical and wind load • Numerous seams across the surface of the dome an icon to add a table, chart, dome are equilateral triangles. • 25% greater vertical load capacity present the problem of water and wind leakage; SmartArt graphic, picture or • 34% greater shear load capacity dampness within the dome cannot be removed • This union of the sphere and the triangle produces without some difficulty multimedia file. numerous benefits with regards to strength, • Domes are characterized by their “frequency”, the durability, efficiency, and sustainability of the number of struts between pentagonal sections • Acoustic properties of the dome reflect and To add or remove bullet points structure. However, the original desire for • Increasing the frequency of the dome closer amplify sound inside, further undermining privacy from text, click the Bullets button widespread residential, commercial, and industrial approximates a sphere use was hindered by other practical and aesthetic • Zoning laws may prevent construction in certain on the Home tab. -
Italy Creates. Gio Ponti, America and the Shaping of the Italian Design Image
Politecnico di Torino Porto Institutional Repository [Article] ITALY CREATES. GIO PONTI, AMERICA AND THE SHAPING OF THE ITALIAN DESIGN IMAGE Original Citation: Elena, Dellapiana (2018). ITALY CREATES. GIO PONTI, AMERICA AND THE SHAPING OF THE ITALIAN DESIGN IMAGE. In: RES MOBILIS, vol. 7 n. 8, pp. 20-48. - ISSN 2255-2057 Availability: This version is available at : http://porto.polito.it/2698442/ since: January 2018 Publisher: REUNIDO Terms of use: This article is made available under terms and conditions applicable to Open Access Policy Article ("["licenses_typename_cc_by_nc_nd_30_it" not defined]") , as described at http://porto.polito. it/terms_and_conditions.html Porto, the institutional repository of the Politecnico di Torino, is provided by the University Library and the IT-Services. The aim is to enable open access to all the world. Please share with us how this access benefits you. Your story matters. (Article begins on next page) Res Mobilis Revista internacional de investigación en mobiliario y objetos decorativos Vol. 7, nº. 8, 2018 ITALY CREATES. GIO PONTI, AMERICA AND THE SHAPING OF THE ITALIAN DESIGN IMAGE ITALIA CREA. GIO PONTI, AMÉRICA Y LA CONFIGURACIÓN DE LA IMAGEN DEL DISEÑO ITALIANO Elena Dellapiana* Politecnico di Torino Abstract The paper explores transatlantic dialogues in design during the post-war period and how America looked to Italy as alternative to a mainstream modernity defined by industrial consumer capitalism. The focus begins in 1950, when the American and the Italian curated and financed exhibition Italy at Work. Her Renaissance in Design Today embarked on its three-year tour of US museums, showing objects and environments designed in Italy’s post-war reconstruction by leading architects including Carlo Mollino and Gio Ponti. -
GEOMETRIC INTERPRETATIONS of CURVATURE Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Tran
GEOMETRIC INTERPRETATIONS OF CURVATURE ZHENGQU WAN Abstract. This is an expository paper on geometric meaning of various kinds of curvature on a Riemann manifold. Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Transport 3 4. Geodesics and the Exponential Map 4 5. Riemannian Curvature Tensor 5 6. Taylor Expansion of the Metric in Normal Coordinates and the Geometric Interpretation of Ricci and Scalar Curvature 9 Acknowledgments 13 References 13 1. Notation and Summation Conventions We assume knowledge of the basic theory of smooth manifolds, vector fields and tensors. We will assume all manifolds are smooth, i.e. C1, second countable and Hausdorff. All functions, curves and vector fields will also be smooth unless otherwise stated. Einstein summation convention will be adopted in this paper. In some cases, the index types on either side of an equation will not match and @ so a summation will be needed. The tangent vector field @xi induced by local i coordinates (x ) will be denoted as @i. 2. Affine Connections Riemann curvature is a measure of the noncommutativity of parallel transporta- tion of tangent vectors. To define parallel transport, we need the notion of affine connections. Definition 2.1. Let M be an n-dimensional manifold. An affine connection, or connection, is a map r : X(M) × X(M) ! X(M), where X(M) denotes the space of smooth vector fields, such that for vector fields V1;V2; V; W1;W2 2 X(M) and function f : M! R, (1) r(fV1 + V2;W ) = fr(V1;W ) + r(V2;W ), (2) r(V; aW1 + W2) = ar(V; W1) + r(V; W2), for all a 2 R. -
Differential Geometry Lecture 17: Geodesics and the Exponential
Differential geometry Lecture 17: Geodesics and the exponential map David Lindemann University of Hamburg Department of Mathematics Analysis and Differential Geometry & RTG 1670 3. July 2020 David Lindemann DG lecture 17 3. July 2020 1 / 44 1 Geodesics 2 The exponential map 3 Geodesics as critical points of the energy functional 4 Riemannian normal coordinates 5 Some global Riemannian geometry David Lindemann DG lecture 17 3. July 2020 2 / 44 Recap of lecture 16: constructed covariant derivatives along curves defined parallel transport studied the relation between a given connection in the tangent bundle and its parallel transport maps introduced torsion tensor and metric connections, studied geometric interpretation defined the Levi-Civita connection of a pseudo-Riemannian manifold David Lindemann DG lecture 17 3. July 2020 3 / 44 Geodesics Recall the definition of the acceleration of a smooth curve n 00 n γ : I ! R , that is γ 2 Γγ (T R ). Question: Is there a coordinate-free analogue of this construc- tion involving connections? Answer: Yes, uses covariant derivative along curves. Definition Let M be a smooth manifold, r a connection in TM ! M, 0 and γ : I ! M a smooth curve. Then rγ0 γ 2 Γγ (TM) is called the acceleration of γ (with respect to r). Of particular interest is the case if the acceleration of a curve vanishes, that is if its velocity vector field is parallel: Definition A smooth curve γ : I ! M is called geodesic with respect to a 0 given connection r in TM ! M if rγ0 γ = 0. David Lindemann DG lecture 17 3. -
Topology Optimized Hemispherical Shell Under Asymmetric Loads by Grace Melcher
Topology Optimized Hemispherical Shell under Asymmetric Loads by Grace Melcher B.S. in Civil and Environmental Engineering Massachusetts Institute of Technology, 2019 Submitted to the Department of Civil and Environmental Engineering in Partial Fulfillment of the Requirements for the degree of MASTER OF ENGINEERING IN CIVIL AND ENVIRONMENTAL ENGINEERING at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2020 ©2020 Grace E Melcher. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part in any medium now known or hereafter created. Signature of Author: Grace Melcher Department of Civil and Environmental Engineering May 8, 2020 Certified by: John Ochsendorf Class of 1942 Professor of Civil and Environmental Engineering and Architecture Thesis Co-Supervisor Certified by: Josephine Carstensen Assistant Professor of Civil and Environmental Engineering Thesis Co-Supervisor Accepted by: Colette L. Heald Professor of Civil and Environmental Engineering Chair, Graduate Program Committee 2 Topology Optimized Hemispherical Shell under Asymmetric Loads by Grace Melcher Submitted to the Department of Civil and Environmental Engineering on May 8, 2020 in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Civil and Environment Engineering ABSTRACT Domes are an efficient way to span long distances and resist gravity loads. The two kinds of classical domes prevalent in architecture are continuous shells and grid shells. Continuous shells are monolithic concrete or masonry; grid shells reorient material in lattice members to create depth throughout the shell thickness. This thesis considers the design of a topology optimized hemisphere, a hemispherical continuous shell, and a hemispherical grid shell, and compares the structural performance of these three shells under asymmetric loads.