DOCSLIB.ORG

Math 4140/5530: Differential Geometry How Do We find Geodesics?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 4140/5530: Differential Geometry How Do We find Geodesics? guess and check in Maple or computer software parallel transport—that a tangent vector stays parallel geodesic equations with Christoffel symbols (also useful for Einstein’s field equations) How do we find geodesics? http://i.stack.imgur.com/La4Hj.png https://i1.creativecow.net/u/1027/geodesicvoxel_binding.jpg covering symmetry γ 00 proof on the sphere used the surface normal U = R in γ along with Frenet equations how about more generally? Dr. Sarah Math 4140/5530: Differential Geometry How do we find geodesics? http://i.stack.imgur.com/La4Hj.png https://i1.creativecow.net/u/1027/geodesicvoxel_binding.jpg covering symmetry γ 00 proof on the sphere used the surface normal U = R in γ along with Frenet equations how about more generally? guess and check in Maple or computer software parallel transport—that a tangent vector stays parallel geodesic equations with Christoffel symbols (also useful for Einstein’s field equations) Dr. Sarah Math 4140/5530: Differential Geometry 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt Dr. Sarah Math 4140/5530: Differential Geometry product rule d ~ _ ~ ¨ d ~ _ ~ ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ α00 =α ¨ Dr. Sarah Math 4140/5530: Differential Geometry chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v Dr. Sarah Math 4140/5530: Differential Geometry (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + Dr. Sarah Math 4140/5530: Differential Geometry 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ Dr. Sarah Math 4140/5530: Differential Geometry Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. 0 chain rule ~ du ~ dv α (t) = xu dt + xv dt 0 0 0 α = ~xuu + ~xv v or equivalently α_ = ~xuu_ + ~xv v_ product rule 00 d ~ _ ~ ¨ d ~ _ ~ ¨ α =α ¨ = dt xuu + xuu + dt xv v + xv v chain rule = (~xuuu_ + ~xuv v_ )u_ + ~xuu¨ + (~xvuu_ + ~xvv v_ )v_ + ~xv v¨ 2 2 = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ Dr. Sarah Math 4140/5530: Differential Geometry u v = ~xu + ~xv + lU = Γuu~xu + Γuu~xv + lU c ~ u ~ v ~ ~ Γab called Christoffel symbols. xuv = Γuv xu + Γuv xv + mU = xvu. u v ~xvv = Γvv~xu + Γvv~xv + nU Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. α_ = ~xuu_ + ~xv v_ 2 2 α¨ = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ ~xuu = ~xu + ~xv + U Dr. Sarah Math 4140/5530: Differential Geometry u v = Γuu~xu + Γuu~xv + lU c ~ u ~ v ~ ~ Γab called Christoffel symbols. xuv = Γuv xu + Γuv xv + mU = xvu. u v ~xvv = Γvv~xu + Γvv~xv + nU Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. α_ = ~xuu_ + ~xv v_ 2 2 α¨ = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ ~xuu = ~xu + ~xv + U = ~xu + ~xv + lU Dr. Sarah Math 4140/5530: Differential Geometry u v ~xuv = Γuv~xu + Γuv~xv + mU = ~xvu. u v ~xvv = Γvv~xu + Γvv~xv + nU Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. α_ = ~xuu_ + ~xv v_ 2 2 α¨ = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ u v ~xuu = ~xu + ~xv + U = ~xu + ~xv + lU = Γuu~xu + Γuu~xv + lU c Γab called Christoffel symbols. Dr. Sarah Math 4140/5530: Differential Geometry Geodesic Equations on Surfaces in R3 with F = 0 3 f~xu; x~v ; Ug is a basis for R α00 has to be only in the normal direction, so (eventually) write it in the basis and set ~xu; x~v components = 0. α_ = ~xuu_ + ~xv v_ 2 2 α¨ = ~xuuu_ + 2~xuv u_ v_ + ~xuu¨ + ~xvv v_ + ~xv v¨ u v ~xuu = ~xu + ~xv + U = ~xu + ~xv + lU = Γuu~xu + Γuu~xv + lU c ~ u ~ v ~ ~ Γab called Christoffel symbols. xuv = Γuv xu + Γuv xv + mU = xvu. u v ~xvv = Γvv~xu + Γvv~xv + nU Dr. Sarah Math 4140/5530: Differential Geometry ¨a a _ b _ c Or even more Einstein summation notation! x + Γbcx x = 0 http://scienceblogs.com/startswithabang/files/2013/07/einstein.jpg Geodesic Equations set ~xu; x~v components = 0. u 2 u u 2 1 1 b c u¨ + Γuuu_ + 2Γuv v_ u_ + Γvv v_ = 0 or x¨ + Γbcx_ x_ = 0 v 2 v v 2 2 2 b c v¨ + Γuuu_ + 2Γuv v_ u_ + Γvv v_ = 0 or x¨ + Γbcx_ x_ = 0 Dr. Sarah Math 4140/5530: Differential Geometry Geodesic Equations set ~xu; x~v components = 0. u 2 u u 2 1 1 b c u¨ + Γuuu_ + 2Γuv v_ u_ + Γvv v_ = 0 or x¨ + Γbcx_ x_ = 0 v 2 v v 2 2 2 b c v¨ + Γuuu_ + 2Γuv v_ u_ + Γvv v_ = 0 or x¨ + Γbcx_ x_ = 0 ¨a a _ b _ c Or even more Einstein summation notation! x + Γbcx x = 0 http://scienceblogs.com/startswithabang/files/2013/07/einstein.jpg Dr. Sarah Math 4140/5530: Differential Geometry ~xuu · ~xu + ~xu · ~xuu = 2~xuu · ~xu u v Also from a few slides up, ~xuu = Γuu~xu + Γuu~xv + lU, so u u ~xuu · ~xu = Γuu~xu · ~xu = ΓuuE Eu ~ ~ u u Eu Thus 2 = xuu · xu = ΓuuE so Γuu = 2E v Ev u Ev v Gu u Gu v Gv Similarly Γuu = − 2G ; Γuv = 2E ; Γuv = 2G ; Γvv = − 2E ; Γvv = 2G How do we find the Christoffel symbols? Elwin Bruno Christoffel and Albert Einstein http://www.ethbib.ethz.ch/aktuell/galerie/christoffel/Portr_gross.jpg http://scienceblogs.com/startswithabang/files/2013/07/einstein.jpg Rewrite ~xuu by taking uth partial of E = ~xu · ~xu Eu = Dr. Sarah Math 4140/5530: Differential Geometry u v Also from a few slides up, ~xuu = Γuu~xu + Γuu~xv + lU, so u u ~xuu · ~xu = Γuu~xu · ~xu = ΓuuE Eu ~ ~ u u Eu Thus 2 = xuu · xu = ΓuuE so Γuu = 2E v Ev u Ev v Gu u Gu v Gv Similarly Γuu = − 2G ; Γuv = 2E ; Γuv = 2G ; Γvv = − 2E ; Γvv = 2G How do we find the Christoffel symbols? Elwin Bruno Christoffel and Albert Einstein http://www.ethbib.ethz.ch/aktuell/galerie/christoffel/Portr_gross.jpg http://scienceblogs.com/startswithabang/files/2013/07/einstein.jpg Rewrite ~xuu by taking uth partial of E = ~xu · ~xu Eu = ~xuu · ~xu + ~xu · ~xuu = 2~xuu · ~xu Dr.
Load more

Recommended publications
  • Simply-Riemann-1588263529. Print
    Simply Riemann Simply Riemann JEREMY GRAY SIMPLY CHARLY NEW YORK Copyright © 2020 by Jeremy Gray Cover Illustration by José Ramos Cover Design by Scarlett Rugers All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher, except in the case of brief quotations embodied in critical reviews and certain other noncommercial uses permitted by copyright law. For permission requests, write to the publisher at the address below. [email protected] ISBN: 978-1-943657-21-6 Brought to you by http://simplycharly.com Contents Praise for Simply Riemann vii Other Great Lives x Series Editor's Foreword xi Preface xii Introduction 1 1. Riemann's life and times 7 2. Geometry 41 3. Complex functions 64 4. Primes and the zeta function 87 5. Minimal surfaces 97 6. Real functions 108 7. And another thing . 124 8. Riemann's Legacy 126 References 143 Suggested Reading 150 About the Author 152 A Word from the Publisher 153 Praise for Simply Riemann “Jeremy Gray is one of the world’s leading historians of mathematics, and an accomplished author of popular science. In Simply Riemann he combines both talents to give us clear and accessible insights into the astonishing discoveries of Bernhard Riemann—a brilliant but enigmatic mathematician who laid the foundations for several major areas of today’s mathematics, and for Albert Einstein’s General Theory of Relativity.Readable, organized—and simple. Highly recommended.” —Ian Stewart, Emeritus Professor of Mathematics at Warwick University and author of Significant Figures “Very few mathematicians have exercised an influence on the later development of their science comparable to Riemann’s whose work reshaped whole fields and created new ones.
    [Show full text]
  • Fundamental Geometrodynamic Justification of Gravitomagnetism
    Issue 2 (October) PROGRESS IN PHYSICS Volume 16 (2020) Fundamental Geometrodynamic Justification of Gravitomagnetism (I) G. G. Nyambuya National University of Science and Technology, Faculty of Applied Sciences – Department of Applied Physics, Fundamental Theoretical and Astrophysics Group, P.O. Box 939, Ascot, Bulawayo, Republic of Zimbabwe. E-mail: [email protected] At a most fundamental level, gravitomagnetism is generally assumed to emerge from the General Theory of Relativity (GTR) as a first order approximation and not as an exact physical phenomenon. This is despite the fact that one can justify its existence from the Law of Conservation of Mass-Energy-Momentum in much the same manner one can justify Maxwell’s Theory of Electrodynamics. The major reason for this is that in the widely accepted GTR, Einstein cast gravitation as a geometric phenomenon to be understood from the vantage point of the dynamics of the metric of spacetime. In the literature, nowhere has it been demonstrated that one can harness the Maxwell Equa- tions applicable to the case of gravitation – i.e. equations that describe the gravitational phenomenon as having a magnetic-like component just as happens in Maxwellian Elec- trodynamics. Herein, we show that – under certain acceptable conditions where Weyl’s conformal scalar [1] is assumed to be a new kind of pseudo-scalar and the metric of spacetime is decomposed as gµν = AµAν so that it is a direct product of the components of a four-vector Aµ – gravitomagnetism can be given an exact description from within Weyl’s beautiful but supposedly failed geometry. My work always tried to unite the Truth with the Beautiful, consistent mathematical framework that has a direct corre- but when I had to choose one or the other, I usually chose the spondence with physical and natural reality as we know it.
    [Show full text]
  • General Relativity
    Chapter 3 Differential Geometry II 3.1 Connections and covariant derivatives 3.1.1 Linear Connections The curvature of the two-dimensional sphere S 2 can be described by Caution Whitney’s (strong) embedding the sphere into a Euclidean space of the next-higher dimen- embedding theorem states that sion, R3. However, (as far as we know) there is no natural embedding any smooth n-dimensional mani- of our four-dimensional curved spacetime into R5, and thus we need a fold (n > 0) can be smoothly description of curvature which is intrinsic to the manifold. embedded in the 2n-dimensional Euclidean space R2n. Embed- There is a close correspondence between the curvature of a manifold and dings into lower-dimensional Eu- the transport of vectors along curves. clidean spaces may exist, but not As we have seen before, the structure of a manifold does not trivially necessarily so. An embedding allow to compare vectors which are elements of tangent spaces at two f : M → N of a manifold M into different points. We will thus have to introduce an additional structure a manifold N is an injective map which allows us to meaningfully shift vectors from one point to another such that f (M) is a submanifold on the manifold. of N and M → f (M)isdifferen- tiable. Even before we do so, it is intuitively clear how vectors can be trans- ported along closed paths in flat Euclidean space, say R3. There, the vector arriving at the starting point after the transport will be identical to the vector before the transport.
    [Show full text]
  • 6A. Mathematics
    19-th Century ROMANTIC AGE Mathematics Collected and edited by Prof. Zvi Kam, Weizmann Institute, Israel The 19th century is called “Romantic” because of the romantic trend in literature, music and arts opposing the rationalism of the 18th century. The romanticism adored individualism, folklore and nationalism and distanced itself from the universality of humanism and human spirit. Coming back to nature replaced the superiority of logics and reasoning human brain. In Literature: England-Lord byron, Percy Bysshe Shelly Germany –Johann Wolfgang von Goethe, Johann Christoph Friedrich von Schiller, Immanuel Kant. France – Jean-Jacques Rousseau, Alexandre Dumas (The Hunchback from Notre Dam), Victor Hugo (Les Miserable). Russia – Alexander Pushkin. Poland – Adam Mickiewicz (Pan Thaddeus) America – Fennimore Cooper (The last Mohican), Herman Melville (Moby Dick) In Music: Germany – Schumann, Mendelsohn, Brahms, Wagner. France – Berlioz, Offenbach, Meyerbeer, Massenet, Lalo, Ravel. Italy – Bellini, Donizetti, Rossini, Puccini, Verdi, Paganini. Hungary – List. Czech – Dvorak, Smetana. Poland – Chopin, Wieniawski. Russia – Mussorgsky. Finland – Sibelius. America – Gershwin. Painters: England – Turner, Constable. France – Delacroix. Spain – Goya. Economics: 1846 - The American Elias Howe Jr. builds the general purpose sawing machine, launching the clothing industry. 1848 – The communist manifest by Karl Marks & Friedrich Engels is published. Describes struggle between classes and replacement of Capitalism by Communism. But in the sciences, the Romantic era was very “practical”, and established in all fields the infrastructure for the modern sciences. In Mathematics – Differential and Integral Calculus, Logarithms. Theory of functions, defined over Euclidian spaces, developed the field of differential equations, the quantitative basis of physics. Matrix Algebra developed formalism for transformations in space and time, both orthonormal and distortive, preparing the way to Einstein’s relativity.
    [Show full text]
  • Mathematical Techniques
    Chapter 8 Mathematical Techniques This chapter contains a review of certain basic mathematical concepts and tech- niques that are central to metric prediction generation. A mastery of these fun- damentals will enable the reader not only to understand the principles underlying the prediction algorithms, but to be able to extend them to meet future require- ments. 8.1. Scalars, Vectors, Matrices, and Tensors While the reader may have encountered the concepts of scalars, vectors, and ma- trices in their college courses and subsequent job experiences, the concept of a tensor is likely an unfamiliar one. Each of these concepts is, in a sense, an exten- sion of the preceding one. Each carries a notion of value and operations by which they are transformed and combined. Each is used for representing increas- ingly more complex structures that seem less complex in the higher-order form. Tensors were introduced in the Spacetime chapter to represent a complexity of perhaps an unfathomable incomprehensibility to the untrained mind. However, to one versed in the theory of general relativity, tensors are mathematically the simplest representation that has so far been found to describe its principles. 8.1.1. The Hierarchy of Algebras As a precursor to the coming material, it is perhaps useful to review some ele- mentary algebraic concepts. The material in this subsection describes the succes- 1 2 Explanatory Supplement to Metric Prediction Generation sion of algebraic structures from sets to fields that may be skipped if the reader is already aware of this hierarchy. The basic theory of algebra begins with set theory .
    [Show full text]
  • Einstein's Field Equations and Non-Locality
    International Journal of Mathematics Trends and Technology (IJMTT) – Volume 66 Issue 6 - June 2020 Einstein’s field equations and non-locality Ilija Barukčić#1 #Ilija Barukčić, Horandstrasse, DE-26441 Jever, Germany. Tel: 49-4466-333 [email protected] Received: June, 16 2020; Accepted: June, 20 2020; Published: June, 25 2020. Abstract Objective. Reconciling locality and non-locality in accordance with Einstein’s general theory of relativity appears to be more than only a hopeless endeavour. Theoretically this seems either not necessary or almost impossible. Methods. The usual tensor calculus rules were used. The proof method modus ponens was used to proof the theorems derived. Results. A tensor of locality and a tensor of non-locality is derived from Riemannian curvature tensor. Einstein’s field equations were reformulated in terms of the tensor of locality and the tensor of non-locality without any contradiction, without changing Einstein’s field equations at all and without adding anything new to Einstein’s field equations. Weyl’s curvature tensor does not model non-locality sufficiently well under any circumstances. Under conditions of the general theory of relativity, it is more appropriate and straightforward to use the non- locality curvature tensor which is part of the Riemannian curvature tensor to describe non-locality completely. Conclusions. The view is justified that there is no contradiction at all between Einstein’s field equations and the concept of locality and non-locality. Keywords — principium identitatis, principium contradictionis, theory of relativity, unified field theory, causality. I. INTRODUCTION What is gravity[1] doing over extremely short distances from the point of view of general theory of relativity? Contemporary elementary particle physics or Quantum Field Theory (QFT) as an extension of quantum mechanics (QM) dealing with particles taken seriously in its implications seems to contradict Einstein's general relativity theory especially on this point.
    [Show full text]
  • General Relativity Conflict and Rivalries
    General Relativity Conflict and Rivalries General Relativity Conflict and Rivalries: Einstein's Polemics with Physicists By Galina Weinstein General Relativity Conflict and Rivalries: Einstein's Polemics with Physicists By Galina Weinstein This book first published 2015 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2015 by Galina Weinstein All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-4438-8362-X ISBN (13): 978-1-4438-8362-7 “When I accompanied him [Einstein] home the first day we met, he told me something that I heard from him many times later: ‘In Princeton they regard me as an old fool.’ (‘Hier in Princeton betrachten sie mich als einen alten Trottel’.) […] Before he was thirty-five, Einstein had made the four great discoveries of his life. In order of increasing importance they are: the theory of Brownian motion; the theory of the photoelectric effect; the special theory of relativity; the general theory of relativity. Very few people in the history of science have done half as much. […] For years he looked for a theory which would embrace gravitational, electromagnetic, and quantum phenomena. […] Einstein pursued it relentlessly through ideas which he changed repeatedly and down avenues that led nowhere. The very distinguished professors in Princeton did not understand that Einstein’s mistakes were more important than their correct results.
    [Show full text]
  • From Black Holes to Wormholes in Higher Spin Gravity 2+1-Dimensional Gravity in a Chern-Simons Formulation
    From Black Holes to Wormholes in Higher Spin Gravity 2+1-dimensional gravity in a Chern-Simons formulation Bachelor’s thesis for Engineering Physics Simon Ekhammar, Daniel Erkensten, Marcus Lassila, Torbjörn Nilsson Department of Physics Chalmers University of Technology Gothenburg, Sweden 2017 Bachelor’s thesis 2017 From Black Holes to Wormholes in Higher Spin Gravity 2+1-dimensional gravity in a Chern-Simons formulation Simon Ekhammar Daniel Erkensten Marcus Lassila Torbjörn Nilsson Department of Physics Division of Mathematical Physics Chalmers University of Technology Gothenburg, Sweden 2017 From Black Holes to Wormholes in Higher Spin Gravity 2+1-dimensional gravity in a Chern-Simons formulation Authors: Simon Ekhammar, Daniel Erkensten, Marcus Lassila, Torbjörn Nilsson Contact: [email protected] [email protected] [email protected] [email protected] © Simon Ekhammar, Daniel Erkensten, Marcus Lassila, Torbjörn Nilsson, 2017. TIFX04- Bachelor’s Thesis at Fundamental Physics Bachelor’s Thesis TIFX04-17-29 Supervisor: Bengt E W Nilsson, Division of Mathematical Physics Examiner: Jan Swenson, Fundamental Physics Department of Fundamental Physics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Embedding diagram of wormhole connecting two distant regions of space. The wormhole provides a shorter path between two points than the route going through the regular spacetime, here represented as a folded sheet. The image mapped on the sheet is the Hubble Ultra Deep Field, courtesy of NASA. Image created by Torbjörn Nilsson. Typeset in LATEX Printed by Chalmers Reproservice Gothenburg, Sweden 2017 Abstract The recent ER=EPR conjecture as well as advances in string theory have spurred the interest in worm- holes and their relation to black holes.
    [Show full text]
  • Long-Term History and Ephemeral Configurations
    LONG-TERM HISTORY AND EPHEMERAL CONFIGURATIONS CATHERINE GOLDSTEIN Abstract. Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathe- matics has tended to focus on local topics, over a short term-scale, and on the study of ephemeral configurations of mathematicians, theorems or practices. The first part of the paper explains why this change has taken place: a renewed interest in the connections between mathematics and society, an increased at- tention to the variety of components and aspects of mathematical work, and a critical outlook on historiography itself. The problems of a long-term history are illustrated and tested using a number of episodes in the nineteenth-century history of Hermitian forms, and finally, some open questions are proposed. “Mathematics is the art of giving the same name to different things,” wrote Henri Poincaré at the very beginning of the twentieth century ((Poincaré, 1908, 31)). The sentence, to be found in a chapter entitled “The future of mathematics” seemed particularly relevant around 1900: a structural point of view and a wish to clarify and to firmly found mathematics were then gaining ground and both contributed to shorten chains of argument and gather together under the same word phenomena which had until then been scattered ((Corry, 2004)). Significantly, Poincaré’s examples included uniform convergence and the concept of group. 1. Long-term histories But the view of mathematics encapsulated by this — that it deals somehow with “sameness” — has also found its way into the history of mathematics.
    [Show full text]
  • Mathematics Is the Art of Giving the Same Name to Different Things
    LONG-TERM HISTORY AND EPHEMERAL CONFIGURATIONS CATHERINE GOLDSTEIN Abstract. Mathematical concepts and results have often been given a long history, stretching far back in time. Yet recent work in the history of mathe- matics has tended to focus on local topics, over a short term-scale, and on the study of ephemeral configurations of mathematicians, theorems or practices. The first part of the paper explains why this change has taken place: a renewed interest in the connections between mathematics and society, an increased at- tention to the variety of components and aspects of mathematical work, and a critical outlook on historiography itself. The problems of a long-term history are illustrated and tested using a number of episodes in the nineteenth-century history of Hermitian forms, and finally, some open questions are proposed. “Mathematics is the art of giving the same name to different things,” wrote Henri Poincaré at the very beginning of the twentieth century (Poincaré, 1908, 31). The sentence, to be found in a chapter entitled “The future of mathematics,” seemed particularly relevant around 1900: a structural point of view and a wish to clarify and to firmly found mathematics were then gaining ground and both contributed to shorten chains of argument and gather together under the same word phenomena which had until then been scattered (Corry, 2004). Significantly, Poincaré’s examples included uniform convergence and the concept of group. 1. Long-term histories But the view of mathematics encapsulated by this—that it deals somehow with “sameness”—has also found its way into the history of mathematics.
    [Show full text]
  • 2020 Volume 16 Issue 2
    Issue 2 2020 Volume 16 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS “All scientists shall have the right to present their scientific research results, in whole or in part, at relevant scientific conferences, and to publish the same in printed scientific journals, electronic archives, and any other media.” — Declaration of Academic Freedom, Article 8 ISSN 1555-5534 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS A quarterly issue scientific journal, registered with the Library of Congress (DC, USA). This journal is peer reviewed and included in the abstracting and indexing coverage of: Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sweden), Scientific Commons of the University of St. Gallen (Switzerland), Open-J-Gate (India), Referativnyi Zhurnal VINITI (Russia), etc. Electronic version of this journal: http://www.ptep-online.com Advisory Board Dmitri Rabounski, Editor-in-Chief, Founder Florentin Smarandache, Associate Editor, Founder Larissa Borissova, Associate Editor, Founder Editorial Board October 2020 Vol. 16, Issue 2 Pierre Millette [email protected] Andreas Ries [email protected] CONTENTS Gunn Quznetsov [email protected] Nyambuya G. G. Fundamental Geometrodynamic Justification of Gravitomagnetism (I) 73 Ebenezer Chifu McCulloch M. E. Can Nano-Materials Push Off the Vacuum? . 92 [email protected] Czerwinski A. New Approach to Measurement in Quantum Tomography . .94 Postal Address Millette P. A. The Physics of Lithospheric Slip Displacements in Plate Tectonics . 97 Department of Mathematics and Science, University of New Mexico, Bei G., Passaro D.
    [Show full text]
  • The Wave Equation for Level Sets Is Not a Huygens' Equation
    THE WAVE EQUATION FOR LEVEL SETS IS NOT A HUYGENS’ EQUATION WOLFGANG QUAPP Mathematisches Institut, Universit¨at Leipzig, PF 100920, D-04009 Leipzig, Germany, [email protected], corresponding author, tel.: 49-(0)341-97-32162 JOSEP MARIA BOFILL Departament de Qu´ımica Org`anica, Universitat de Barcelona; Institut de Qu´ımica Te`orica i Computacional, Universitat de Barcelona, (IQTCUB), Mart´ıi Franqu`es, 1, 08028 Barcelona, Spain, jmbofi[email protected] September 12, 2013 Abstract: Any surface can be foliated into equipotential hypersurfaces of the level sets. A cur- rent result is that the contours are the progressing wave fronts of a certain hyperbolic partial differential equation, a wave equation. It is connected with the gradient lines, as well as with a corresponding eikonal equation. The level of a surface point, seen as an additional coordinate, plays the central role in this treatment. A wave solution can be a sharp front. Here the validity of the Huygens’ principle (HP) is of interest: there is no wake of the wave solutions in every dimension, if a special Cauchy initial value problem is posed. Additionally, there is no distinction into odd or even dimensions. To compare this with Hadamard’s ’minor premise’ for a strong HP, we calculate differ- ential geometric objects like Christoffel symbols, curvature tensors and geodesic lines, to test the validity of the strong HP. However, for the differential equation for level sets, the main criteria are not fulfilled for the strong HP in the sense of Hadamard’s ’minor premise’. Keywords: Contours; steepest ascent; wave equation; progressing waves; Huygens’ principle.
    [Show full text]