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 formatted for printing. Please consult the preamble of the L A T E X 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. BY RICHARD C. TOLMAN, PH.D. TABLE OF CONTENTS. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter I. Historical Development of Ideas as to the Nature of Space and Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Part I. The Space and Time of Galileo and Newton. . . . . . . . . 5 Newtonian Time. . . . . . . . . . . . . . . . . . . . . . . 7 Newtonian Space. . . . . . . . . . . . . . . . . . . . . . . 7 The Galileo Transformation Equations. . . . . . . . . . . 9 Part II. The Space and Time of the Ether Theory. . . . . . . . . . 11 Rise of the Ether Theory. . . . . . . . . . . . . . . . . . . 11 Idea of a Stationary Ether. . . . . . . . . . . . . . . . . . 12 Ether in the Neighborhood of Moving Bodies. . . . . . . 12 Ether Entrained in Dielectrics. . . . . . . . . . . . . . . . 13 The Lorentz Theory of a Stationary Ether. . . . . . . . . 14 Part III. Rise of the Einstein Theory of Relativity. . . . . . . . . 17 The Michelson-Morley Experiment. . . . . . . . . . . . . 18 The Postulates of Einstein. . . . . . . . . . . . . . . . . . 19 Chapter II. The Two Postulates of the Einstein Theory of Relativity. 21 The First Postulate of Relativity. . . . . . . . . . . . . . . . . 21 The Second Postulate of the Einstein Theory of Relativity. . 22 Suggested Alternative to the Postulate of the Independence of the Velocity of Light and the Velocity of the Source. 24 iv Evidence Against Emission Theories of Light. . . . . . . 25 Different Forms of Emission Theory. . . . . . . . . . . . 27 Further Postulates of the Theory of Relativity. . . . . . . . . 29 Chapter III. Some Elementary Deductions. . . . . . . . . . . . . . . 30 Measurements of Time in a Moving System. . . . . . . . . . . 30 Measurements of Length in a Moving System. . . . . . . . . . 32 The Setting of Clocks in a Moving System. . . . . . . . . . . 35 The Composition of Velocities. . . . . . . . . . . . . . . . . . 38 The Mass of a Moving Body. . . . . . . . . . . . . . . . . . . 40 The Relation Between Mass and Energy. . . . . . . . . . . . . 42 Chapter IV. The Einstein Transformation Equations for Space and Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 The Lorentz Transformation. . . . . . . . . . . . . . . . . . . 45 Deduction of the Fundamental Transformation Equations. . . 46 Three Conditions to be Fulfilled. . . . . . . . . . . . . . 47 The Transformation Equations. . . . . . . . . . . . . . . 49 Further Transformation Equations. . . . . . . . . . . . . . . . 50 Transformation Equations for Velocity. . . . . . . . . . . 51 Transformation Equations for the Function 1  1 − u 2 c 2 . . . 51 Transformation Equations for Acceleration. . . . . . . . . 52 Chapter V. Kinematical Applications. . . . . . . . . . . . . . . . . . 53 The Kinematical Shape of a Rigid Body. . . . . . . . . . . . . 53 The Kinematical Rate of a Clock. . . . . . . . . . . . . . . . 54 The Idea of Simultaneity. . . . . . . . . . . . . . . . . . . . . 55 The Composition of Velocities. . . . . . . . . . . . . . . . . . 56 The Case of Parallel Velocities. . . . . . . . . . . . . . . 56 Composition of Velocities in General. . . . . . . . . . . . 57 Velocities Greater than that of Light. . . . . . . . . . . . . . 59 Application of the Principles of Kinematics to Certain Optical Problems. . . . . . . . . . . . . . . . . . . . . . . . . 60 The Doppler Effect. . . . . . . . . . . . . . . . . . . . . . 63 The Aberration of Light. . . . . . . . . . . . . . . . . . . 64 Velocity of Light in Moving Media. . . . . . . . . . . . . 65 Group Velocity. . . . . . . . . . . . . . . . . . . . . . . . 66 Chapter VI. The Dynamics of a Particle. . . . . . . . . . . . . . . . 67 The Laws of Motion. . . . . . . . . . . . . . . . . . . . . . . . 67 Difference between Newtonian and Relativity Mechanics. . . 67 The Mass of a Moving Particle. . . . . . . . . . . . . . . . . . 68 Transverse Collision. . . . . . . . . . . . . . . . . . . . . 69 Mass the Same in All Directions. . . . . . . . . . . . . . 72 Longitudinal Collision. . . . . . . . . . . . . . . . . . . . 73 Collision of Any Type. . . . . . . . . . . . . . . . . . . . 74 Transformation Equations for Mass. . . . . . . . . . . . . . . 78 Equation for the Force Acting on a Moving Particle. . . . . . 79 Transformation Equations for Force. . . . . . . . . . . . . . . 80 The Relation between Force and Acceleration. . . . . . . . . 80 Transverse and Longitudinal Acceleration. . . . . . . . . . . . 82 The Force Exerted by a Moving Charge. . . . . . . . . . . . . 84 The Field around a Moving Charge. . . . . . . . . . . . . 87 Application to a Specific Problem. . . . . . . . . . . . . . 87 Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Kinetic Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Potential Energy. . . . . . . . . . . . . . . . . . . . . . . . . . 91 The Relation between Mass and Energy. . . . . . . . . . . . . 91 Application to a Specific Problem. . . . . . . . . . . . . . 93 Chapter VII. The Dynamics of a System of Particles. . . . . . . . . 96 On the Nature of a System of Particles. . . . . . . . . . . . . 96 The Conservation of Momentum. . . . . . . . . . . . . . . . . 97 The Equation of Angular Momentum. . . . . . . . . . . . . . 99 The Function T . . . . . . . . . . . . . . . . . . . . . . . . . . 101 The Modified Lagrangian Function. . . . . . . . . . . . . . . 102 The Principle of Least Action. . . . . . . . . . . . . . . . . . 102 Lagrange’s Equations. . . . . . . . . . . . . . . . . . . . . . . 104 Equations of Motion in the Hamiltonian Form. . . . . . . . . 105 Value of the Function T  . . . . . . . . . . . . . . . . . . . 107 The Principle of the Conservation of Energy. . . . . . . . . . 109 On the Location of Energy in Space. . . . . . . . . . . . . . . 110 Chapter VIII. The Chaotic Motion of a System of Particles. . . . . 113 The Equations of Motion. . . . . . . . . . . . . . . . . . 113 Representation in Generalized Space. . . . . . . . . . . . 114 Liouville’s Theorem. . . . . . . . . . . . . . . . . . . . . 114 A System of Particles. . . . . . . . . . . . . . . . . . . . 116 Probability of a Given Statistical State. . . . . . . . . . . 116 Equilibrium Relations. . . . . . . . . . . . . . . . . . . . 118 The Energy as a Function of the Momentum. . . . . . . 119 The Distribution Law. . . . . . . . . . . . . . . . . . . . 121 Polar Coördinates. . . . . . . . . . . . . . . . . . . . . . 122 The Law of Equipartition. . . . . . . . . . . . . . . . . . 123 Criterion for Equality of Temperature. . . . . . . . . . . 124 Pressure Exerted by a System of Particles. . . . . . . . . 126 The Relativity Expression for Temperature. . . . . . . . 128 The Partition of Energy. . . . . . . . . . . . . . . . . . . 130 Partition of Energy for Zero Mass. . . . . . . . . . . . . 131 Approximate Partition of Energy for Particles of any De- sired Mass. . . . . . . . . . . . . . . . . . . . . . 132 Chapter IX. The Principle of Relativity and the Principle of Least Action. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 The Principle of Least Action. . . . . . . . . . . . . . . . 135 The Equations of Motion in the Lagrangian Form. . . . . 137 Introduction of the Principle of Relativity. . . . . . . . . 138 Relation between  W dt and  W  dt  . . . . . . . . . . . 139 Relation between H  and H. . . . . . . . . . . . . . . . . 142 Chapter X. The Dynamics of Elastic Bodies. . . . . . . . . . . . . . 145 On the Impossibility of Absolutely Rigid Bodies. . . . . 145 Part I. Stress and Strain. . . . . . . . . . . . . . . . . . . . . . . . 145 Definition of Strain. . . . . . . . . . . . . . . . . . . . . . 146 Definition of Stress. . . . . . . . . . . . . . . . . . . . . . 148 Transformation Equations for Strain. . . . . . . . . . . . 148 Variation in the Strain. . . . . . . . . . . . . . . . . . . . 149 Part II. Introduction of the Principle of Least Action. . . . . . . . 152 The Kinetic Potential for an Elastic Body. . . . . . . . . 152 Lagrange’s Equations. . . . . . . . . . . . . . . . . . . . 153 Transformation Equations for Stress. . . . . . . . . . . . 155 Value of E ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . 155 The Equations of Motion in the Lagrangian Form. . . . . 156 Density of Momentum. . . . . . . . . . . . . . . . . . . . 158 Density of Energy. . . . . . . . . . . . . . . . . . . . . . 158 Summary of Results Obtained from the Principle of Least Action. . . . . . . . . . . . . . . . . . . . . . . . 159 Part III. Some Mathematical Relations. . . . . . . . . . . . . . . . 160 The Unsymmetrical Stress Tensor t. . . . . . . . . . . . 160 The Symmetrical Tensor p. . . . . . . . . . . . . . . . . 162 Relation between div t and t n . . . . . . . . . . . . . . . . 163 The Equations of Motion in the Eulerian Form. . . . . . 164 Part IV. Applications of the Results. . . . . . . . . . . . . . . . . 165 Relation between Energy and Momentum. . . . . . . . . 165 The Conservation of Momentum. . . . . . . . . . . . . . 167 The Conservation of Angular Momentum. . . . . . . . . 168 Relation between Angular Momentum and the Unsym- metrical Stress Tensor. . . . . . . . . . . . . . . . 169 The Right-Angled Lever. . . . . . . . . . . . . . . . . . . 170 Isolated Systems in a Steady State. . . . . . . . . . . . . 172 The Dynamics of a Particle. . . . . . . . . . . . . . . . . 172 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . 172 Chapter XI. The Dynamics of a Thermodynamic System. . . . . . . 174 The Generalized Coördinates and Forces. . . . . . . . . . 174 Transformation Equation for Volume. . . . . . . . . . . . 174 Transformation Equation for Entropy. . . . . . . . . . . 175 Introduction of the Principle of Least Action. The Ki- netic Potential. . . . . . . . . . . . . . . . . . . . 175 The Lagrangian Equations. . . . . . . . . . . . . . . . . . 176 Transformation Equation for Pressure. . . . . . . . . . . 177 Transformation Equation for Temperature. . . . . . . . . 178 The Equations of Motion for Quasistationary Adiabatic Acceleration. . . . . . . . . . . . . . . . . . . . . 178 The Energy of a Moving Thermodynamic System. . . . . 179 The Momentum of a Moving Thermodynamic System. . 180 The Dynamics of a Hohlraum. . . . . . . . . . . . . . . . 181 [...]... applications of the theory of relativity following quite closely Einstein’s original method of development In particular we may call attention to the ease with which we may handle the optics of moving media by the methods of the theory of relativity as compared with the difficulty of treatment on the basis of the ether theory In Chapters VI, VII and VIII we develop and apply a theory of the dynamics of a particle... performing these functions of explanation and prediction, have been the development of the modern theory of electrons, the application of thermodynamic and statistical reasoning to the phenomena of radiation, and the development of Einstein’s brilliant theory of the relativity of motion It has been the endeavor of the following book to present an introduction to this theory of relativity, which in the decade... now codified as the second postulate of relativity part ii the space and time of the ether theory 7 Rise of the Ether Theory Twelve years before the appearance of the Principia, Römer, a Danish astronomer, observed that an eclipse of one of the satellites of Jupiter occurred some ten minutes later than the time predicted for the event from the known period of the satellite and the time of the preceding... combining the principle of the relativity of motion with the postulate that the velocity of light is independent of the velocity of its source, a number of attempts have been made to develop so-called emission theories of relativity based on the principle of the relativity of motion and the further postulate that the velocity of light and the velocity of its source are additive Before examining the available... but in particular the Einstein theory of relativity takes as the basis for its second postulate a principle that has long been familiar to the ether theory, namely that the velocity of light is independent of the velocity of the source We shall see in following chapters that it is the combination of this principle with the first postulate of relativity that leads to the whole theory of relativity and to... with the facts The theory of Lorentz developed from that of Maxwell by the addition of the idea of the electron, as the atom of electricity, and his treatment is often called the “electron theory. ” This atomistic conception of electricity was foreshadowed by Faraday’s discovery of the quantitative relations between the amount of electricity associated with chemical reactions in electrolytes and the. .. experiment, with the further work of Morley and Miller For details as to the nature of these experiments the reader may refer to the original articles or to an excellent discussion by Laub of the experimental basis of the theory of relativity. ∗ In none of the above investigations was it possible to detect any effect attributable to the earth’s motion through the ether Nevertheless a number of these experiments... to the assumed stationary ether We have devoted this space to the Lorentz theory, since his work marks the culmination of the ether theory of light and electromagnetism, and for us the particularly significant fact is that by this line of attack science was inevitably led to the idea of an absolutely immovable and stationary ether 13 We have thus briefly traced the development of the ether theory of. .. conclusions drawn from the theory are neither self-contradictory nor contradictory of each other, and furthermore that they agree with the facts of the external world, we may again feel that our theory has achieved a measure of success In the present chapter we shall present the two main postulates of the theory of relativity, and indicate the direct experimental evidence in favor of their truth In following... = ∗ in which the letters have their usual significance (See Chapter XII.) Now the whole of the Lorentz theory, including of course his treatment of moving media, is derivable from these five equations, and the fact that the idea of a stationary ether does lie at the basis of his theory is most clearly shown by the first and last of these equations, which contain the velocity u with which the charge in . attention to the ease with which we may handle the optics of moving media by the methods of the theory of relativity as compared with the difficulty of treatment on the basis of the ether theory. In. 19 Chapter II. The Two Postulates of the Einstein Theory of Relativity. 21 The First Postulate of Relativity. . . . . . . . . . . . . . . . . 21 The Second Postulate of the Einstein Theory of Relativity. . 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