Relational Mechanics Andre K.T. Assis Apeiron Montreal Published by Apeiron 4405, rue St-Dominique Montreal, Quebec H2W 2B2 Canada http://redshift.vif.com © Andre K.T. Assis Dr. Andre Koch Torres Assis Institute of Physics State University of Campinas 13083-970 Campinas - SP, Brazil E-mail: [email protected] Homepage: www.ifi.unicamp.br/~assis First Published 1999 Canadian Cataloguing in Publication Data Assis, Andre Koch Torres, 1962- Relational mechanics Translation of Mecanica Relacional Includes bibliographical references and index. ISBN 0-9683689-2-1 1. Mechanics. 2. Gravitation. 3. Inertia (Mechanics). I. Title. QC125.2.A88 1999 531 C99-900382-8 Front cover: Newton’s bucket experiment. The water rises in the second case because of its rotation. But: rotation relative to what? This is the central theme of this book. Back cover: Galileo’s free fall experiment. Why do two bodies of different weight fall in vacuum with the same acceleration toward the surface of the earth? Relational Mechanics offers a simple explanation for this remarkable fact. In memory of Isaac Newton who paved the way for past, present and future generations 2 Contents I Old World 13 1 Newtonian Mechanics 15 1.1 Introduction . 15 1.2 Newtonian Mechanics . 15 1.3 Energy . 26 2 Applications of Newtonian Mechanics 29 2.1 Uniform Rectilinear Motion . 30 2.2 Constant Force . 30 2.2.1 Free Fall . 30 2.2.2 Charge Moving Inside an Ideal Capacitor . 33 2.2.3 Accelerated Train . 36 2.3 Oscillatory Motions . 37 2.3.1 Spring . 37 2.3.2 Simple Pendulum . 38 2.3.3 Electrically Charged Pendulum . 42 2.4 Uniform Circular Motion . 46 2.4.1 Circular Orbit of a Planet . 47 2.4.2 Two Globes . 49 2.4.3 Newton’s Bucket Experiment . 51 3 Non-inertial Frames of Reference 59 3.1 Constant Force . 59 3.1.1 Free Fall . 59 3.1.2 Accelerated Train . 61 3.2 Uniform Circular Motion . 62 3.2.1 Circular Orbit of a Planet . 62 3.2.2 Two Globes . 65 3.2.3 Newton’s Bucket Experiment . 66 3.3 Rotation of the Earth . 67 3 4 CONTENTS 3.3.1 Kinematical Rotation of the Earth . 67 3.3.2 The Figure of the Earth . 71 3.3.3 Foucault’s Pendulum . 75 3.3.4 Comparison of the Kinematical and Dynamical Rotations 81 3.4 General fictitious Force . 82 4 Gravitational Paradox 85 4.1 Newton and the Infinite Universe . 85 4.2 The Force Paradox . 86 4.3 The Paradox based on Potential . 88 4.4 Solutions of the Paradox . 90 4.5 Absorption of Gravity . 94 5 Leibniz and Berkeley 97 5.1 Leibniz and Relative Motion . 97 5.2 Berkeley and Relative Motion . 101 6 Mach and Newton’s Mechanics 107 6.1 Inertial Frame of Reference . 107 6.2 Absolute Time . 109 6.3 The Two Rotations of the Earth . 110 6.4 Inertial Mass . 111 6.5 Mach’s Formulation of Mechanics . 114 6.6 Relational Mechanics . 115 6.7 Mach and the Bucket Experiment . 116 6.8 Mach’s Principle . 119 6.9 What Mach did not Show . 122 7 Einstein’s Theories of Relativity 125 7.1 Introduction . 125 7.2 Einstein’s Special Theory of Relativity . 125 7.2.1 Asymmetry in Electromagnetic Induction . 127 7.2.2 Postulate of Relativity . 131 7.2.3 Twin Paradox . 132 7.2.4 Constancy of the Velocity of Light . 133 7.2.5 Velocity in Lorentz’s Force . 140 7.2.6 Michelson-Morley Experiment . 144 7.3 Einstein’s General Theory of Relativity . 146 7.3.1 Relational Quantities . 146 7.3.2 Invariance in the Form of the Equations . 148 7.3.3 Implementation of Mach’s Ideas . 148 7.3.4 Newton’s Bucket Experiment . 152 CONTENTS 5 7.4 General Comments . 157 II New World 161 8 Relational Mechanics 163 8.1 Basic Concepts and Postulates . 163 8.2 Electromagnetic and Gravitational Forces . 166 8.3 Spherical Shell Interacting with a Particle . 170 8.4 Implementation of Mach’s Principle . 173 8.5 General Consequences . 180 8.6 Cosmology . 188 8.7 Ptolemaic and Copernican World Views . 190 8.8 Implementation of Einstein’s Ideas . 192 9 Applications of Relational Mechanics 197 9.1 Uniform Rectilinear Motion . 197 9.2 Constant Force . 198 9.2.1 Free Fall . 199 9.2.2 Charge Moving Inside an Ideal Capacitor . 205 9.2.3 Accelerated Train . 206 9.3 Harmonic Oscillator . 208 9.3.1 Spring . 208 9.3.2 Simple Pendulum . 209 9.4 Uniform Circular Motion . 210 9.4.1 Circular Orbit of a Planet . 210 9.4.2 Two Globes . 212 9.4.3 Newton’s Bucket Experiment . 214 9.5 Rotation of the Earth . 218 9.5.1 The Figure of the Earth . 218 9.5.2 Foucault’s Pendulum . 219 10 Beyond Newton 227 10.1 Precession of the Perihelion of the Planets . 227 10.2 Anisotropy of Inertial Mass . 231 10.3 High Velocity Particles . 233 10.4 Experimental Tests of Relational Mechanics . 236 11 History of Relational Mechanics 243 11.1 Gravitation . 243 11.2 Electromagnetism . 244 11.3 Weber’s Law Applied to Gravitation . 249 6 CONTENTS 11.4 Relational Mechanics . 250 12 Conclusion 259 Acknowledgments To the undergraduate and graduate students who followed our courses on Rela- tional Mechanics, Mach’s Principle, Weber’s Electrodynamics and Cosmology, for the many constructive remarks they presented. To our undergraduate and Ph.D. students who are developing researches in these areas. To our friends who are giving us support to continue this work. In special we wish to thank those to whom we gave a first version of this book: C. Roy Keys, Thomas E. Phipps Jr., A. Ghosh, D. Roscoe, A. Martin, R. de A. Martins, M. A. Bueno, W. M. Vieira, M. A. de Faria Rosa, A. Zylbersztajn, D. S. L. Soares, J. I. Cisneros, H. C. Velho, D. Gardelli, J. A. Hernandez and J. E. Lamesa. The suggestions and ideas which we received contributed greatly to the improvement of the book. To the following Publishers and individuals for granting permission to quote from their publications: University of California Press (I. Newton, Mathemat- ical Principles of Natural Philosophy, 1934, Cajori edition); Open Court Pub- lishing Company, a division of Carus Publishing Company, Peru, IL (E. Mach, The Science of Mechanics, 1960, translated by T. J. McCormack); Everyman’s Library, David Campbell Publications Ltd. (G. Berkeley, Philosophical Works - including De Motu and A Treatise Concerning the Principles of Human Knowl- edge, 1992); Dover Publications (A. Einstein, H. A. Lorentz, H. Weyl and H. Minkowski, The Principle of Relativity, 1952); Cambridge University Press (J. B. Barbour, Absolute or Relative Motion?, 1989); Albert Einstein Archives, The Hebrew University of Jerusalem, Israel (A. Einstein, The Meaning of Rel- ativity, 1980); Dr. Peter Gray Lucas, in name of Dr. H. G. Alexander (The Leibniz-Clarke Correspondence, edited by H. G. Alexander, 1984); Birkhauser Boston and Dr. J. B. Barbour for his translations of E. Schr¨odinger, H. Reiss- ner, A. Einstein, B. and I. Friedlaender (Mach’s Principle - From Newton’s Bucket to Quantum Gravity, edited by J. B. Barbour and H. Pfister, 1995). To the Center for Electromagnetics Research, Northeastern University (Bos- ton, USA), which received us for one year in which we had the first idea to write this book and discussed its contents with some friends. To the Institutes of Physics and Mathematics and to the Center of Logics, Epistemology and History of Sciences of the State University of Campinas - UNICAMP (Brazil), 7 8 CONTENTS that gave the necessary support to undertake this work. Above all, to my parents, my wife and children for helping me grow as a human being. Preface This book presents Relational Mechanics, a new mechanics which implements the ideas of Leibniz, Berkeley, Mach and many others. Relational mechan- ics is based only on relative quantities, such as the distance between material bodies, their relative radial velocity and relative radial acceleration. In this new mechanics the absolute concepts of space, time and motion do not appear. The same can be said of inertia, inertial mass and inertial frames of reference. When we compare relational mechanics with Newtonian mechanics, we will gain a new and clear understanding of these old concepts. Relational mechanics is a quantitative implementation of Mach’s ideas utilizing a Weber’s force law for gravitation. Many people have contributed to its development, including Erwin Schr¨odinger. This is the first time such a book has been written, bringing together all the features and characteristics of this new world view. This allows it to be seen in its proper light, and a comparison with old worldviews is easily accomplished. Considerable emphasis is placed on Galileo’s free fall experiment and on Newton’s bucket experiment. These are some of the simplest experiments ever performed in physics. Despite this fact, no other experiment has had such far-reaching consequences for the foundations of classical mechanics. An ex- planation of these two experiments without utilizing the concepts of absolute space and inertia is one of the major accomplishments of relational mechanics. In order to show all the power of relational mechanics and put it in perspec- tive, we first present Newtonian mechanics and Einstein’s theories of relativity. We address the criticisms of Newton’s theory made by Leibniz, Berkeley and Mach. Then we present relational mechanics and show how it solves all these problems quantitatively with a clarity and simplicity unsurpassed by any other model. We also discuss the history of relational mechanics in detail, emphasiz- ing the achievements and limitations of all major works along these lines.