The Theory of the Relativity of Motion

Richard Chace Tolman

Informations

• Publié par: enno
• Publié le: 08 décembre 2010
• Nombre de lectures: 21
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

The Project Gutenberg EBook of The Theory of the Relativity of Motion, by Richard Chace Tolman

This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or reuse it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org

Title: The Theory of the Relativity of Motion

Author: Richard Chace Tolman

Release Date: June 17, 2010 [EBook #32857]

Language: English

Character set encoding: ISO88591

*** START OF THIS PROJECT

GUTENBERG EBOOK THE THEORY OF THE RELATIVITY ***

Produced by Andrew D. Hwang, Berj Zamanian, Joshua Hutchinson and the Online Distributed Proofreading Team at http://www.pgdp.net (This file was produced from images from the Cornell University Library: Historical Mathematics Monographs collection.)

transcriber’s note Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for screen viewing, but may be easily A formatted for printing. Please consult the preamble of the LT X E source file for instructions.

THE THEORY OF

THE RELATIVITY OF MOTION

BY

RICHARD C. TOLMAN

UNIVERSITY OF CALIFORNIA PRESS BERKELEY 1917

Press of The New Era Printing Company Lancaster, Pa

TO H. E.

THE THEORY OF THE RELATIVITY OF MOTION.

RICHARD TABLE

BY C. TOLMAN, PH.D. OF CONTENTS.

Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I.Historical Development of Ideas as to the Nature of Space and Time.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I. The Space and Time of Galileo and Newton.. . . . . . . . Newtonian Time.. . . . . . . . . . . . . . . . . . . . . . Newtonian Space.. . . . . . . . . . . . . . . . . . . . . . The Galileo Transformation Equations.. . . . . . . . . . Part II. The Space and Time of the Ether Theory.. . . . . . . . . Rise of the Ether Theory.. . . . . . . . . . . . . . . . . . Idea of a Stationary Ether.. . . . . . . . . . . . . . . . . Ether in the Neighborhood of Moving Bodies.. . . . . . Ether Entrained in Dielectrics.. . . . . . . . . . . . . . . The Lorentz Theory of a Stationary Ether.. . . . . . . . Part III. Rise of the Einstein Theory of Relativity.. . . . . . . . The MichelsonMorley Experiment.. . . . . . . . . . . . The Postulates of Einstein.. . . . . . . . . . . . . . . . . Chapter II.The Two Postulates of the Einstein Theory of Relativity. The First Postulate of Relativity.. . . . . . . . . . . . . . . . The Second Postulate of the Einstein Theory of Relativity.. Suggested Alternative to the Postulate of the Independence of the Velocity of Light and the Velocity of the Source.

iv

1 5 5 7 7 9 11 11 12 12 13 14 17 18 19 21 21 22 24

Evidence Against Emission Theories of Light.. . . . . . Different Forms of Emission Theory.. . . . . . . . . . . Further Postulates of the Theory of Relativity.. . . . . . . . Chapter III.Some Elementary Deductions.. . . . . . . . . . . . . . Measurements of Time in a Moving System.. . . . . . . . . . Measurements of Length in a Moving System.. . . . . . . . . The Setting of Clocks in a Moving System.. . . . . . . . . . The Composition of Velocities.. . . . . . . . . . . . . . . . . The Mass of a Moving Body.. . . . . . . . . . . . . . . . . . The Relation Between Mass and Energy.. . . . . . . . . . . . Chapter IV.The Einstein Transformation Equations for Space and Time.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. The Lorentz Transformation.. . . . . . . . . . . . . . . . . . Deduction of the Fundamental Transformation Equations.. . Three Conditions to be Fulfilled.. . . . . . . . . . . . . The Transformation Equations.. . . . . . . . . . . . . . Further Transformation Equations.. . . . . . . . . . . . . . . Transformation Equations for Velocity.. . . . . . . . . . 1 Transformation Equations for the Function r .. . 2 u 1− 2 c Transformation Equations for Acceleration.. . . . . . . . Chapter V.Kinematical Applications.. . . . . . . . . . . . . . . . . The Kinematical Shape of a Rigid Body.. . . . . . . . . . . . The Kinematical Rate of a Clock.. . . . . . . . . . . . . . . The Idea of Simultaneity.. . . . . . . . . . . . . . . . . . . . The Composition of Velocities.. . . . . . . . . . . . . . . . . The Case of Parallel Velocities.. . . . . . . . . . . . . . Composition of Velocities in General.. . . . . . . . . . . Velocities Greater than that of Light.. . . . . . . . . . . . . Application of the Principles of Kinematics to Certain Optical Problems.. . . . . . . . . . . . . . . . . . . . . . .. The Doppler Effect.. . . . . . . . . . . . . . . . . . . . . The Aberration of Light.. . . . . . . . . . . . . . . . . . Velocity of Light in Moving Media.. . . . . . . . . . . .

25 27 29 30 30 32 35 38 40 42

45 45 46 47 49 50 51 51

52 53 53 54 55 56 56 57 59

60 63 64 65

Group Velocity.. . . . . . . . . . . . . . . . . . Chapter VI.The Dynamics of a Particle.. . . . . . . . . . The Laws of Motion.. . . . . . . . . . . . . . . . . .

. . .

. . .

. . . Difference between Newtonian and Relativity Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Mass of a Moving Particle.. . . . . . . . . . . . Transverse Collision.. . . . . . . . . . . . . . . Mass the Same in All Directions.. . . . . . . . Longitudinal Collision.. . . . . . . . . . . . . . Collision of Any Type.. . . . . . . . . . . . . . Transformation Equations for Mass.. . . . . . . . . Equation for the Force Acting on a Moving Particle. Transformation Equations for Force.. . . . . . . . . The Relation between Force and Acceleration.. . . Transverse and Longitudinal Acceleration.. . . . . . The Force Exerted by a Moving Charge.. . . . . . . The Field around a Moving Charge.. . . . . . . Application to a Specific Problem.. . . . . . . . Work.. . . . . . . . . . . . . . . . . . . . . . . . . . Kinetic Energy.. . . . . . . . . . . . . . . . . . . . . Potential Energy.. . . . . . . . . . . . . . . . . . . . The Relation between Mass and Energy.. . . . . . . Application to a Specific Problem.. . . . . . . . Chapter VII.The Dynamics of a System of Particles.. . . On the Nature of a System of Particles.. . . . . . . The Conservation of Momentum.. . . . . . . . . . . The Equation of Angular Momentum.. . . . . . . . The FunctionT.. . . . . . . . . . . . . . . . . . . . The Modified Lagrangian Function.. . . . . . . . . The Principle of Least Action.. . . . . . . . . . . . Lagrange’s Equations.. . . . . . . . . . . . . . . . . Equations of Motion in the Hamiltonian Form.. . . ′ Value of the FunctionT.. . . . . . . . . . . . . The Principle of the Conservation of Energy.. . . . On the Location of Energy in Space.. . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter VIII.The Chaotic Motion of a System of Particles.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 67 67 67 68 69 72 73 74 78 79 80 80 82 84 87 87 89 89 91 91 93 96 96 97 99 101 102 102 104 105 107 109 110 113

The Equations of Motion.. . . . . . . . . . . . . . Representation in Generalized Space.. . . . . . . . Liouville’s Theorem.. . . . . . . . . . . . . . . . . A System of Particles.. . . . . . . . . . . . . . . . Probability of a Given Statistical State.. . . . . . . Equilibrium Relations.. . . . . . . . . . . . . . . . The Energy as a Function of the Momentum.. . . The Distribution Law.. . . . . . . . . . . . . . . . Polar Coördinates.. . . . . . . . . . . . . . . . . . The Law of Equipartition.. . . . . . . . . . . . . . Criterion for Equality of Temperature.. . . . . . . Pressure Exerted by a System of Particles.. . . . . The Relativity Expression for Temperature.. . . . The Partition of Energy.. . . . . . . . . . . . . . . Partition of Energy for Zero Mass.. . . . . . . . .

sired Mass.. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. .

. . . . . . . . . . . . . . . Approximate Partition of Energy for Particles of any De . Chapter IX.The Principle of Relativity and the Principle of Least . . . . . . . . . . . . . . . . .

Action.. . . . . . . . . . . . . . . . . . . . . . . . . . . . The Principle of Least Action.. . . . . . . . . . . . The Equations of Motion in the Lagrangian Form.. Introduction of the Principle of Relativity.. . . . . R R ′ ′ Relation betweenW dtandW dt.. . . . . . . ′ Relation betweenHandH.. . . . . . . . . . . . . Chapter X.The Dynamics of Elastic Bodies.. . . . . . . . . . On the Impossibility of Absolutely Rigid Bodies.. Part I. Stress and Strain.. . . . . . . . . . . . . . . . . . . . Definition of Strain.. . . . . . . . . . . . . . . . . . Definition of Stress.. . . . . . . . . . . . . . . . . . Transformation Equations for Strain.. . . . . . . . Variation in the Strain.. . . . . . . . . . . . . . . . Part II. Introduction of the Principle of Least Action.. . . . The Kinetic Potential for an Elastic Body.. . . . . Lagrange’s Equations.. . . . . . . . . . . . . . . . Transformation Equations for Stress.. . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 114 114 116 116 118 119 121 122 123 124 126 128 130 131

132

135 135 137 138 139 142 145 145 145 146 148 148 149 152 152 153 155

Part IV.

◦ Value ofE.. . . . . . . . . . . . . . . . . . . . . . . . . The Equations of Motion in the Lagrangian Form.. . . . Density of Momentum.. . . . . . . . . . . . . . . . . . . Density of Energy.. . . . . . . . . . . . . . . . . . . . . Summary of Results Obtained from the Principle of Least Action.. . . . . . . . . . . . . . . . . . . . . . . Part III. Some Mathematical Relations.. . . . . . . . . . . . . . . The Unsymmetrical Stress Tensort.. . . . . . . . . . . The Symmetrical Tensorp.. . . . . . . . . . . . . . . . Relation betweendivtandt n .. . . . . . . . . . . . . . . The Equations of Motion in the Eulerian Form.. . . . . Applications of the Results.. . . . . . . . . . . . . . . . Relation between Energy and Momentum.. . . . . . . . The Conservation of Momentum.. . . . . . . . . . . . . The Conservation of Angular Momentum.. . . . . . . . Relation between Angular Momentum and the Unsym metrical Stress Tensor.. . . . . . . . . . . . . . . The RightAngled Lever.. . . . . . . . . . . . . . . . . . Isolated Systems in a Steady State.. . . . . . . . . . . . The Dynamics of a Particle.. . . . . . . . . . . . . . . . Conclusion.. . . . . . . . . . . . . . . . . . . . . . . . . Chapter XI.The Dynamics of a Thermodynamic System.. . . . . . The Generalized Coördinates and Forces.. . . . . . . . . Transformation Equation for Volume.. . . . . . . . . . . Transformation Equation for Entropy.. . . . . . . . . . Introduction of the Principle of Least Action. The Ki netic Potential.. . . . . . . . . . . . . . . . . . . The Lagrangian Equations.. . . . . . . . . . . . . . . . . Transformation Equation for Pressure.. . . . . . . . . . Transformation Equation for Temperature.. . . . . . . . The Equations of Motion for Quasistationary Adiabatic Acceleration.. . . . . . . . . . . . . . . . . . . . The Energy of a Moving Thermodynamic System.. . . . The Momentum of a Moving Thermodynamic System.. The Dynamics of a Hohlraum.. . . . . . . . . . . . . . .

155 156 158 158

159 160 160 162 163 164 165 165 167 168

169 170 172 172 172 174 174 174 175

175 176 177 178

178 179 180 181

Chapter XII.Electromagnetic Theory.. . . . . . . . . . . . . . . . . The Form of the Kinetic Potential.. . . . . . . . . . . . The Principle of Least Action.. . . . . . . . . . . . . . . The Partial Integrations.. . . . . . . . . . . . . . . . . . Derivation of the Fundamental Equations of Electromag netic Theory.. . . . . . . . . . . . . . . . . . . . The Transformation Equations fore,handρ.. . . . . . The Invariance of Electric Charge.. . . . . . . . . . . . . The Relativity of Magnetic and Electric Fields.. . . . . Nature of Electromotive Force.. . . . . . . . . . . . . . Derivation of the Fifth Fundamental Equation.. . . . . . . . Difference between the Ether and the Relativity Theories of Electromagnetism.. . . . . . . . . . . . . . . . . . . Applications to Electromagnetic Theory.. . . . . . . . . . . .

183 183 184 184 185 188 190 191 191 192 193 196 The Electric and Magnetic Fields around a Moving Charge.196 198 201 202 203 204 207 210 210 211 214 214 215 215 217 217 219 219 221

The Energy of a Moving Electromagnetic System.. . . . Relation between Mass and Energy.. . . . . . . . . . . . The Theory of Moving Dielectrics.. . . . . . . . . . . . . . . Relation between Field Equations for Material Media and Electron Theory.. . . . . . . . . . . . . . . . . . Transformation Equations for Moving Media.. . . . . . Theory of the Wilson Experiment.. . . . . . . . . . . . . Chapter XIII.FourDimensional Analysis.. . . . . . . . . . . . . . Idea of a Time Axis.. . . . . . . . . . . . . . . . . . . . NonEuclidean Character of the Space.. . . . . . . . . . Part I. Vector Analysis of the NonEuclidean FourDimensional Manifold.. . . . . . . . . . . . . . . . . . . . . . . . . . . Space, Time and Singular Vectors.. . . . . . . . . . . . 2 2 2 2 2 Invariance ofx+y+z−c t.. . . . . . . . . . . . . Inner Product of OneVectors.. . . . . . . . . . . . . . . NonEuclidean Angle.. . . . . . . . . . . . . . . . . . . . Kinematical Interpretation of Angle in Terms of Velocity. Vectors of Higher Dimensions. . . . . . . . . . . . . . . . . . Outer Products.. . . . . . . . . . . . . . . . . . . . . . . Inner Product of Vectors in General.. . . . . . . . . . .