DOCSLIB.ORG

Lecture 37. Riemannian Geometry and the General Relativity

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 37. Riemannian Geometry and the General Relativity Lecture 37. Riemannian Geometry and the General Relativity In the 19th century, mathematicians, scientists and philosophers experienced an extraor- dinary shock wave. By the emergence of non-Euclidean geometry, the old belief that math- ematics offers external and immutable truths was collapse. The magnitude of the revolution in thought on such change has been compared to the Darwinian theory of evolution. Birth of differential geometry In Newton's time, the study of curves begins with In- finitesimal Analysis, and he began to study the curvature of plane curves. By the definition, for a curve, the curvature at a point measures the deviation from its tangent. Figure 37.1 Curves and surfaces The theory of surfaces in the euclidean space was developed mainly in the 18th and 19th centuries, and the first half of the 20th century. In the early 19th century, Young and 257 Laplace proved that, for a spherical surface, the inner pressure is always higher than the outer one, and that the difference increases when the radius decreases. The laws of Physics dictate that liquids tend to minimize their surfaces. In the interior of a drop or a bubble in equilibrium, the inner pressure is bigger than the outer one. This difference of pressure is due to the curvature of the boundary surface. Intuitively, it can be concluded that the curvature of a surface at a point measures its deviation from its tangent plane. Although the contributions of Euler, Monge and Dupin were of great importance, the essential part to establish the concept of space is due to Gauss's famous paper \Disquisitiones generales circa superficies curvas," in which the concept of \curvature" of a surface at a point was introduced. Surprisingly the definition of curvature is intrinsic. To do this, Gauss studies the intrinsic properties of the geometry of a surface, by using the first fundamental form as a starting point. \Gauss's Theorema Egregium" can be stated: \At a point of the surface, the curvature is an isometric invariant." As an application of Gauss's \Theorema Egregium," one of the most profound and difficult formulas of Differential Geometry and Algebraic Topology: \The Gauss-Bonnet theorem for surfaces" is proved. Gauss proves it for geodesic polygons and Bonnet extended it for polygons with edges of non vanishing geodesic curvature. This theorem was generalized to an arbitrary dimension nearly a century later, by Allendoerfer, Weyl and Chern. Figure 37.2 Local coordinates and global geometry Birth of Riemannian geometry In 1854, Riemann generalizes Gauss's studies to spaces of arbitrary dimension, which was in a not very rigorous way. As a result, a geometry on a manifold would be a positive-definite quadratic form (i.e., metric form, or the first fundamental form) on each of its tangent spaces. This definition of Riemann allows to generalize much of Gauss's work. 258 As we have mentioned in the previous lecture, Gauss himself had suggested this topic for Riemann's habilitation thesis. On June 10, 1854, in his brilliant lecture entitled \On the Hypotheses That Lie at the Foundations of Geometry," Riemann started by saying that \geometry presupposes the concepts of space, as well as assuming the basic principles constructions in space. It gives only nominal definitions of these things, while their essential specifications appear in the form of axioms. ...... The relationship between these presuppositions is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible." Riemann's spaces of variable curvature include, as particular cases, the space forms, which historically gave rise to the non-euclidean geometries, that are as consistent as the euclidean one. Figure 37.3 Immanuel Kant Impact hitting philosophy The German philosopher Immanuel Kant, who is a central figure of modern philosophy, regarded Euclidean geometry as a status of absolute certainty and unquestionable validity. According to Kant, \if we perceive an object, then necessary this object is spatial and Euclidean." Kant also asserted that information from our sense is organized exclusively along Euclidean templates before it is recorded in our consciousness. 1 In the new horizons that the 19th century opened for geometry. Kant's ideas of space did not survive much longer. Geometers of the 19th century quickly developed intuition in 1Is God a mathematician ? Mario Livio, Simon & Schuster Paperbacks, New York-London-Toronto- Sydney, 2010, p. 152. 259 the non-Euclidean geometries and learned to experience the world along those lines. The Euclidean perception of space turned out to be learned after all, rather than intuitive. All of these dramatic developments let the great French mathematician Henri Poinca´re(1854- 1912) to conclude that the exaioms of geometry are \neither synthetic a priori intuitions nor experimental facts., but it remains free." Newton's omission In Newton universal, mathematical law of motion and gravitation, there was one major question that Newton left completely unanswered: How does gravity really work? How does the Earth, a quarter million miles away from the Moon, affect the Moon's motion? Being aware of this deficiency in his theory, Newton admitted it in the Principia: \Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the Sun and planets...... and propagates its virtue on all sides to immense distances, decreasing always as the inverse square of the distances. But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses." Figure 37.4 The Sun and the Earth Paradox of disappearance of the Sun Even though with certain unanswered questions, Newton's laws were so successful that it took two hundred years for science to take the next fateful step. The next fateful step was made by Albert Einstein (1879-1955) who decided to meet the challenge posed by Newton's omission. 260 Figure 37.5 Albert Einstein Graduated in 1900 with a bachelor's degree from the Polytechnic Institute in Zurich, he found hard time to find a job. In a letter, he confessed that he even considered to ending his life: \The misfortune of my poor parents, who for so many years have not had a happy moment, weights most heavily on me ....... I am noting but a burden to my relatives .... It would surely be better if I did not live at all." 2 Late with a recommendation of a classmates, Einstein was able to find a job as a clerk at the Swiss Patent Office in Bern in 1902. In 1905, Einstein proposed his special relativity, which determines the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum was independent of the motion of all observers. In particular, any speed cannot be faster than the speed of light. Let us go back the Newton's omission about how does gravity really work, from which Einstein's new theory of special relativity appeared to be in direct conflict with Newton's law of gravitation. According to Newton, it was assumed that gravity's action was instantaneous, i.e., it took no time at all for planets to fell the Earth's attraction. On the other hand, by Einstein's special relativity, it should be that no object, energy, or information could travel faster than the speed of light. So how could gravity work instantaneously? Einstein had the following logical thought, which was so-called the paradox of disappearing the Sun. Let us image that the Sun would suddenly disappear. Then without the force holding it to its orbit, the Earth would (according to Newton's theory) immediately start moving along a straight line. However, by the limitation of the light speed, the Sun would actually disappeared from view to an observer at the Earth only about eight minutes later, which was required for light to travel from the Sun to the Earth. Therefore, gravity's action should not be instantaneous, and the change in the Earth's motion would precede the Sun's disappearance. 2Michio Kaku, Parallel Worlds, Anchor books, New York, 2005, p.30. 261 Einstein was seeking a new theory, which should answer Newton's unanswered question, should overcome the above contradiction, should preserve all the remarkable successes of Newton's theory, and should be compatible with his newly discovered special relativity. It is a formidable task. In 1915, Einstein finally reached his goal to propose his theory of general relativity, which is regarded as one of the most beautiful theories ever formulated. Figure 37.6 Gravity was more like a fabric. Einstein's general relativity What is Einstein's general relativity? Let us quote interesting explanation by physicist Michio Kaku as follows. 3 Think of a bowling ball placed on a bed, gently sinking into the mattress. Now shoot a marble along the warped surface of the mattress. It will travel in a curved path, orbiting around the bowling ball. A Newtonian, witnessing the marble circling the bowling ball from a distance, might conclude that there was a mysterious force that the bowling ball exerted on the marble. A Newtonian might say that the bowling ball exerted an instantaneous pull which forced the marble toward the center. To a relativist, who can watch the motion of the motion of the marble on the bed from close up, it obvious that there is no force at all.
Load more

Recommended publications
  • 26-2 Spacetime and the Spacetime Interval We Usually Think of Time and Space As Being Quite Different from One Another
    Answer to Essential Question 26.1: (a) The obvious answer is that you are at rest. However, the question really only makes sense when we ask what the speed is measured with respect to. Typically, we measure our speed with respect to the Earth’s surface. If you answer this question while traveling on a plane, for instance, you might say that your speed is 500 km/h. Even then, however, you would be justified in saying that your speed is zero, because you are probably at rest with respect to the plane. (b) Your speed with respect to a point on the Earth’s axis depends on your latitude. At the latitude of New York City (40.8° north) , for instance, you travel in a circular path of radius equal to the radius of the Earth (6380 km) multiplied by the cosine of the latitude, which is 4830 km. You travel once around this circle in 24 hours, for a speed of 350 m/s (at a latitude of 40.8° north, at least). (c) The radius of the Earth’s orbit is 150 million km. The Earth travels once around this orbit in a year, corresponding to an orbital speed of 3 ! 104 m/s. This sounds like a high speed, but it is too small to see an appreciable effect from relativity. 26-2 Spacetime and the Spacetime Interval We usually think of time and space as being quite different from one another. In relativity, however, we link time and space by giving them the same units, drawing what are called spacetime diagrams, and plotting trajectories of objects through spacetime.
    [Show full text]
  • Riemannian Geometry and the General Theory of Relativity
    University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 1967 Riemannian geometry and the general theory of relativity Marie McBride Vanisko The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Vanisko, Marie McBride, "Riemannian geometry and the general theory of relativity" (1967). Graduate Student Theses, Dissertations, & Professional Papers. 8075. https://scholarworks.umt.edu/etd/8075 This Thesis is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. p m TH3 OmERAl THEORY OF RELATIVITY By Marie McBride Vanisko B.A., Carroll College, 1965 Presented in partial fulfillment of the requirements for the degree of Master of Arts UNIVERSITY OF MOKT/JTA 1967 Approved by: Chairman, Board of Examiners D e a ^ Graduante school V AUG 8 1967, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V W Number: EP38876 All rights reserved INFORMATION TO ALL USERS The quality of this reproduotion is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missir^ pages, these will be noted. Also, if matedal had to be removed, a note will indicate the deletion. UMT Oi*MMtion neiitNna UMi EP38876 Published ProQuest LLQ (2013).
    [Show full text]
  • Time Reversal Symmetry in Cosmology and the Creation of a Universe–Antiuniverse Pair
    universe Review Time Reversal Symmetry in Cosmology and the Creation of a Universe–Antiuniverse Pair Salvador J. Robles-Pérez 1,2 1 Estación Ecológica de Biocosmología, Pedro de Alvarado, 14, 06411 Medellín, Spain; [email protected] 2 Departamento de Matemáticas, IES Miguel Delibes, Miguel Hernández 2, 28991 Torrejón de la Calzada, Spain Received: 14 April 2019; Accepted: 12 June 2019; Published: 13 June 2019 Abstract: The classical evolution of the universe can be seen as a parametrised worldline of the minisuperspace, with the time variable t being the parameter that parametrises the worldline. The time reversal symmetry of the field equations implies that for any positive oriented solution there can be a symmetric negative oriented one that, in terms of the same time variable, respectively represent an expanding and a contracting universe. However, the choice of the time variable induced by the correct value of the Schrödinger equation in the two universes makes it so that their physical time variables can be reversely related. In that case, the two universes would both be expanding universes from the perspective of their internal inhabitants, who identify matter with the particles that move in their spacetimes and antimatter with the particles that move in the time reversely symmetric universe. If the assumptions considered are consistent with a realistic scenario of our universe, the creation of a universe–antiuniverse pair might explain two main and related problems in cosmology: the time asymmetry and the primordial matter–antimatter asymmetry of our universe. Keywords: quantum cosmology; origin of the universe; time reversal symmetry PACS: 98.80.Qc; 98.80.Bp; 11.30.-j 1.
    [Show full text]
  • Riemannian Geometry Learning for Disease Progression Modelling Maxime Louis, Raphäel Couronné, Igor Koval, Benjamin Charlier, Stanley Durrleman
    Riemannian geometry learning for disease progression modelling Maxime Louis, Raphäel Couronné, Igor Koval, Benjamin Charlier, Stanley Durrleman To cite this version: Maxime Louis, Raphäel Couronné, Igor Koval, Benjamin Charlier, Stanley Durrleman. Riemannian geometry learning for disease progression modelling. 2019. hal-02079820v2 HAL Id: hal-02079820 https://hal.archives-ouvertes.fr/hal-02079820v2 Preprint submitted on 17 Apr 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Riemannian geometry learning for disease progression modelling Maxime Louis1;2, Rapha¨elCouronn´e1;2, Igor Koval1;2, Benjamin Charlier1;3, and Stanley Durrleman1;2 1 Sorbonne Universit´es,UPMC Univ Paris 06, Inserm, CNRS, Institut du cerveau et de la moelle (ICM) 2 Inria Paris, Aramis project-team, 75013, Paris, France 3 Institut Montpelli`erainAlexander Grothendieck, CNRS, Univ. Montpellier Abstract. The analysis of longitudinal trajectories is a longstanding problem in medical imaging which is often tackled in the context of Riemannian geometry: the set of observations is assumed to lie on an a priori known Riemannian manifold. When dealing with high-dimensional or complex data, it is in general not possible to design a Riemannian geometry of relevance. In this paper, we perform Riemannian manifold learning in association with the statistical task of longitudinal trajectory analysis.
    [Show full text]
  • An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity
    An Introduction to Riemannian Geometry with Applications to Mechanics and Relativity Leonor Godinho and Jos´eNat´ario Lisbon, 2004 Contents Chapter 1. Differentiable Manifolds 3 1. Topological Manifolds 3 2. Differentiable Manifolds 9 3. Differentiable Maps 13 4. Tangent Space 15 5. Immersions and Embeddings 22 6. Vector Fields 26 7. Lie Groups 33 8. Orientability 45 9. Manifolds with Boundary 48 10. Notes on Chapter 1 51 Chapter 2. Differential Forms 57 1. Tensors 57 2. Tensor Fields 64 3. Differential Forms 66 4. Integration on Manifolds 72 5. Stokes Theorem 75 6. Orientation and Volume Forms 78 7. Notes on Chapter 2 80 Chapter 3. Riemannian Manifolds 87 1. Riemannian Manifolds 87 2. Affine Connections 94 3. Levi-Civita Connection 98 4. Minimizing Properties of Geodesics 104 5. Hopf-Rinow Theorem 111 6. Notes on Chapter 3 114 Chapter 4. Curvature 115 1. Curvature 115 2. Cartan’s Structure Equations 122 3. Gauss-Bonnet Theorem 131 4. Manifolds of Constant Curvature 137 5. Isometric Immersions 144 6. Notes on Chapter 4 150 1 2 CONTENTS Chapter 5. Geometric Mechanics 151 1. Mechanical Systems 151 2. Holonomic Constraints 160 3. Rigid Body 164 4. Non-Holonomic Constraints 177 5. Lagrangian Mechanics 186 6. Hamiltonian Mechanics 194 7. Completely Integrable Systems 203 8. Notes on Chapter 5 209 Chapter 6. Relativity 211 1. Galileo Spacetime 211 2. Special Relativity 213 3. The Cartan Connection 223 4. General Relativity 224 5. The Schwarzschild Solution 229 6. Cosmology 240 7. Causality 245 8. Singularity Theorem 253 9. Notes on Chapter 6 263 Bibliography 265 Index 267 CHAPTER 1 Differentiable Manifolds This chapter introduces the basic notions of differential geometry.
    [Show full text]
  • Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md
    International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 157 ISSN 2229-5518 Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md. Abdul Halim Sajal Saha Md Shafiqul Islam Abstract-In the paper some aspects of Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields are focused. The purpose of this paper is to develop the theory of manifolds equipped with Riemannian metric. I have developed some theorems on torsion and Riemannian curvature tensors using affine connection. A Theorem 1.20 named “Fundamental Theorem of Pseudo-Riemannian Geometry” has been established on Riemannian geometry using tensors with metric. The main tools used in the theorem of pseudo Riemannian are tensors fields defined on a Riemannian manifold. Keywords: Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields. —————————— —————————— I. Introduction (c) { } is a family of open sets which covers , that is, 푖 = . Riemannian manifold is a pair ( , g) consisting of smooth 푈 푀 manifold and Riemannian metric g. A manifold may carry a (d) ⋃ is푈 푖푖 a homeomorphism푀 from onto an open subset of 푀 ′ further structure if it is endowed with a metric tensor, which is a 푖 . 푖 푖 휑 푈 푈 natural generation푀 of the inner product between two vectors in 푛 ℝ to an arbitrary manifold. Riemannian metrics, affine (e) Given and such that , the map = connections,푛 parallel transport, curvature tensors, torsion tensors, ( ( ) killingℝ vector fields and conformal killing vector fields play from푖 푗 ) to 푖 푗 is infinitely푖푗 −1 푈 푈 푈 ∩ 푈 ≠ ∅ 휓 important role to develop the theorem of Riemannian manifolds.
    [Show full text]
  • Spacetime Symmetries and the Cpt Theorem
    SPACETIME SYMMETRIES AND THE CPT THEOREM BY HILARY GREAVES A dissertation submitted to the Graduate School|New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Philosophy Written under the direction of Frank Arntzenius and approved by New Brunswick, New Jersey May, 2008 ABSTRACT OF THE DISSERTATION Spacetime symmetries and the CPT theorem by Hilary Greaves Dissertation Director: Frank Arntzenius This dissertation explores several issues related to the CPT theorem. Chapter 2 explores the meaning of spacetime symmetries in general and time reversal in particular. It is proposed that a third conception of time reversal, `geometric time reversal', is more appropriate for certain theoretical purposes than the existing `active' and `passive' conceptions. It is argued that, in the case of classical electromagnetism, a particular nonstandard time reversal operation is at least as defensible as the standard view. This unorthodox time reversal operation is of interest because it is the classical counterpart of a view according to which the so-called `CPT theorem' of quantum field theory is better called `PT theorem'; on this view, a puzzle about how an operation as apparently non- spatio-temporal as charge conjugation can be linked to spacetime symmetries in as intimate a way as a CPT theorem would seem to suggest dissolves. In chapter 3, we turn to the question of whether the CPT theorem is an essentially quantum-theoretic result. We state and prove a classical analogue of the CPT theorem for systems of tensor fields. This classical analogue, however, ii appears not to extend to systems of spinor fields.
    [Show full text]
  • Geometric GSI’19 Science of Information Toulouse, 27Th - 29Th August 2019
    ALEAE GEOMETRIA Geometric GSI’19 Science of Information Toulouse, 27th - 29th August 2019 // Program // GSI’19 Geometric Science of Information On behalf of both the organizing and the scientific committees, it is // Welcome message our great pleasure to welcome all delegates, representatives and participants from around the world to the fourth International SEE from GSI’19 chairmen conference on “Geometric Science of Information” (GSI’19), hosted at ENAC in Toulouse, 27th to 29th August 2019. GSI’19 benefits from scientific sponsor and financial sponsors. The 3-day conference is also organized in the frame of the relations set up between SEE and scientific institutions or academic laboratories: ENAC, Institut Mathématique de Bordeaux, Ecole Polytechnique, Ecole des Mines ParisTech, INRIA, CentraleSupélec, Institut Mathématique de Bordeaux, Sony Computer Science Laboratories. We would like to express all our thanks to the local organizers (ENAC, IMT and CIMI Labex) for hosting this event at the interface between Geometry, Probability and Information Geometry. The GSI conference cycle has been initiated by the Brillouin Seminar Team as soon as 2009. The GSI’19 event has been motivated in the continuity of first initiatives launched in 2013 at Mines PatisTech, consolidated in 2015 at Ecole Polytechnique and opened to new communities in 2017 at Mines ParisTech. We mention that in 2011, we // Frank Nielsen, co-chair Ecole Polytechnique, Palaiseau, France organized an indo-french workshop on “Matrix Information Geometry” Sony Computer Science Laboratories, that yielded an edited book in 2013, and in 2017, collaborate to CIRM Tokyo, Japan seminar in Luminy TGSI’17 “Topoplogical & Geometrical Structures of Information”.
    [Show full text]
  • Riemann's Contribution to Differential Geometry
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematics 9 (1982) l-18 RIEMANN'S CONTRIBUTION TO DIFFERENTIAL GEOMETRY BY ESTHER PORTNOY UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA, IL 61801 SUMMARIES In order to make a reasonable assessment of the significance of Riemann's role in the history of dif- ferential geometry, not unduly influenced by his rep- utation as a great mathematician, we must examine the contents of his geometric writings and consider the response of other mathematicians in the years immedi- ately following their publication. Pour juger adkquatement le role de Riemann dans le developpement de la geometric differentielle sans etre influence outre mesure par sa reputation de trks grand mathematicien, nous devons &udier le contenu de ses travaux en geometric et prendre en consideration les reactions des autres mathematiciens au tours de trois an&es qui suivirent leur publication. Urn Riemann's Einfluss auf die Entwicklung der Differentialgeometrie richtig einzuschZtzen, ohne sich von seinem Ruf als bedeutender Mathematiker iiberm;issig beeindrucken zu lassen, ist es notwendig den Inhalt seiner geometrischen Schriften und die Haltung zeitgen&sischer Mathematiker unmittelbar nach ihrer Verijffentlichung zu untersuchen. On June 10, 1854, Georg Friedrich Bernhard Riemann read his probationary lecture, "iber die Hypothesen welche der Geometrie zu Grunde liegen," before the Philosophical Faculty at Gdttingen ill. His biographer, Dedekind [1892, 5491, reported that Riemann had worked hard to make the lecture understandable to nonmathematicians in the audience, and that the result was a masterpiece of presentation, in which the ideas were set forth clearly without the aid of analytic techniques.
    [Show full text]
  • 1 How Could Relativity Be Anything Other Than Physical?
    How Could Relativity be Anything Other Than Physical? Wayne C. Myrvold Department of Philosophy The University of Western Ontario [email protected] Forthcoming in Studies in History and Philosophy of Modern Physics. Special Issue: Physical Relativity, 10 years on Abstract Harvey Brown’s Physical Relativity defends a view, the dynamical perspective, on the nature of spacetime that goes beyond the familiar dichotomy of substantivalist/relationist views. A full defense of this view requires attention to the way that our use of spacetime concepts connect with the physical world. Reflection on such matters, I argue, reveals that the dynamical perspective affords the only possible view about the ontological status of spacetime, in that putative rivals fail to express anything, either true or false. I conclude with remarks aimed at clarifying what is and isn’t in dispute with regards to the explanatory priority of spacetime and dynamics, at countering an objection raised by John Norton to views of this sort, and at clarifying the relation between background and effective spacetime structure. 1. Introduction Harvey Brown’s Physical Relativity is a delightful book, rich in historical details, whose main thrust is to an advance a view of the nature of spacetime structure, which he calls the dynamical perspective, that goes beyond the familiar dichotomy of substantivalism and relationism. The view holds that spacetime structure and dynamics are intrinsically conceptually intertwined and that talk of spacetime symmetries and asymmetries is nothing else than talk of the symmetries and asymmetries of dynamical laws. Brown has precursors in this; I count, for example, Howard Stein (1967) and Robert DiSalle (1995) among them.
    [Show full text]
  • A SPACETIME GEOMETRODYNAMIC MODEL (GDM) of the PHYSICAL REALITY Shlomo Barak
    A SPACETIME GEOMETRODYNAMIC MODEL (GDM) OF THE PHYSICAL REALITY Shlomo Barak To cite this version: Shlomo Barak. A SPACETIME GEOMETRODYNAMIC MODEL (GDM) OF THE PHYSICAL REALITY. 2018. hal-01935260 HAL Id: hal-01935260 https://hal.archives-ouvertes.fr/hal-01935260 Preprint submitted on 14 Jan 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A SPACETIME GEOMETRODYNAMIC MODEL (GDM) Shlomo Barak Shlomo Barak OF THE PHYSICAL REALITY Shlomo Barak II The Book of the GDM A Realization of Einstein’s Vision Dr. Shlomo Barak Editor: Roger M. Kaye A collection of 18 papers published Nov 2016 to Nov 2018. III A. Einstein (1933) …. the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented… Copyright © 2018 Shlomo Barak The right of Shlomo Barak to be identified as the author of this work has been asserted by him. All rights reserved. No part of this book may be reproduced or copied in any form or by any means, graphic, electronic or mechanical, or otherwise, including photocopying, recording, or information retrieval systems - without written permission. ISBN 978-965-90727-1-2 IV This GDM book is a collection of 18 papers published from November 2016 to November 2018.
    [Show full text]
  • SPACETIME & GRAVITY: the GENERAL THEORY of RELATIVITY
    1 SPACETIME & GRAVITY: The GENERAL THEORY of RELATIVITY We now come to one of the most extraordinary developments in the history of science - the picture of gravitation, spacetime, and matter embodied in the General Theory of Relativity (GR). This theory was revolutionary in several di®erent ways when ¯rst proposed, and involved a fundamental change how we understand space, time, and ¯elds. It was also almost entirely the work of one person, viz. Albert Einstein. No other scientist since Newton had wrought such a fundamental change in our understanding of the world; and even more than Newton, the way in which Einstein came to his ideas has had a deep and lasting influence on the way in which physics is done. It took Einstein nearly 8 years to ¯nd the ¯nal and correct form of the theory, after he arrived at his 'Principle of Equivalence'. During this time he changed not only the fundamental ideas of physics, but also how they were expressed; and the whole style of theoretical research, the relationship between theory and experiment, and the role of theory, were transformed by him. Physics has become much more mathematical since then, and it has become commonplace to use purely theoretical arguments, largely divorced from experiment, to advance new ideas. In what follows little attempt is made to discuss the history of this ¯eld. Instead I concentrate on explaining the main ideas in simple language. Part A discusses the new ideas about geometry that were crucial to the theory - ideas about curved space and the way in which spacetime itself became a physical entity, essentially a new kind of ¯eld in its own right.
    [Show full text]