The Concept of Mass in Physics, Notas de aula de Física

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Concepts of Mass in Contemporary

Physics and Philosophy

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Copyright © 2000 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex All Rights Reserved

Library of Congress Cataloging-in-Publication Data Jammer, Max. Concepts of mass in contemporary physics and philosophy / Max Jammer. p. cm. Includes bibliographical references and index. ISBN 0-691-01017-X (cl: alk. paper)

1. Mass (Physics). 2. Physics—Philosophy. I. Title. QC106.J355 1999 530.11—dc21 99-

This book has been composed in Palatino

The paper used in this publication meets the minimum requirements of ANSI/NISO Z39.48–1992 (R1997) ( Permanence of Paper )

http://pup.princeton.edu

Printed in the United States of America

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k Contents k

Preface vii

Acknowledgments xi

Introduction 3

Chapter 1 Inertial Mass 5

Chapter 2 Relativistic Mass 41

Chapter 3 The Mass-Energy Relation 62

Chapter 4 Gravitational Mass and the Principle of Equivalence 90

Chapter 5 The Nature of Mass 143

Index 169

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k Preface k

This book intends to provide a comprehensive and self-contained

study of the concept of mass as defined, employed, and interpreted in contemporary theoretical and experimental physics and as critically examined in the modern philosophy of science. It studies in particular how far, if at all, present-day physics contributes to a more profound understanding of the nature of mass. In order to make this book accessible not only to the professional physicist but also to the nonspecialist interested in the foundations of physics, unnecessary technicalities and complicated mathematical cal- culations have been avoided without, however, impairing the accuracy and logical rigor of the presentation. Next to space and time, mass is the most fundamental notion in physics, especially once its so-called equivalence with energy had been established by Albert Einstein. Moreover, it has even been argued repeatedly that “space-time does not exist without mass-energy,” as a prominent astrophysicist has phrased it.^1 Although for the sake of completeness and comprehension the text includes some historical and explanatory comments, it deals mainly with developments that occurred after 1960. In fact, the year 1960 marks the beginning of a new era of experimental and theoretical research on gravitation and general relativity, the two main bases of our modern conception of mass. In 1960 the first laboratory measurement of the gravitational redshift was performed by P. V. Pound and G. A. Rebka, and the first recording of a radar echo from a planet (Venus) was made. In 1960 the spinor approach to general relativity was developed by R. Penrose. In the same year V. W. Hughes and independently R.W.P. Drever confirmed the isotropy of inertial mass by what has been called the most precise null experiment ever performed; and R. H. Dicke, together with P. G. Roll and R. Krokov, planned the construction of their famous “Princeton experiment,” which was soon to confirm the equivalence of inertial and gravitational mass with an unprecedented degree of accuracy. All these events rekindled interest in studying the properties of mass and endowed the study with a vigor that has not abated since.

(^1) D. Lynden-Bell, “Inertia,” in O. Lahav, E. Terlevich, and D. J. Terlevich, eds., Gravita- tional Dynamics (Cambridge, Mass.: Cambridge University Press, 1996), p. 235.

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P R E FA C E As this book deals primarily with developments that occurred during the relatively short interval of only four decades, its presentation is pre- dominantly thematic and not chronological. The first chapter discusses the notion of inertial mass and in particular the still problematic issue of its noncircular definability. Chapter 2 deals with problems related to the concept of relativistic or velocity-dependent mass and to the notion of velocity-independent rest mass. Chapter 3 clarifies certain misconceptions concerning the derivations of the mass-energy relation, usually symbolized by the equation E = mc^2 , and comments on various interpretations of this relation. Chapter 4 analyzes the trichotomy of mass into the categories of inertial, active gravitational, and passive gravitational mass and studies the validity of the equivalence principle for test particles and for massive bodies. The final chapter, probably the most controversial one, discusses recently proposed global and local theories of the nature of mass. In order to make the presentation self-contained I found it appropriate to recapitulate very briefly some antecedent developments with which the reader should be familiar in order to understand the new mate- rial. I have also included historical items, irrespective of their dates, whenever their inclusion seemed useful for the comprehension of an important issue of the discussion. The text is fully documented and contains bibliographical references that will enable readers to pursue the study of a particular issue in which they happen to be interested. Some of these bibliographical notes refer to the 1961 Harvard edition of Concepts of Mass in Classical and Modern Physics , abbreviated henceforth as COM.^2 These notes are quoted with reference to the relevant chapter or its section in COM and not to its pagination for the following reason. Later editions of COM in English—such as the 1964 paperback edition in the Torchbook Series of Harper and Row, New York, or translations into other languages (such as the Russian translation by academician N. F. Ovchinnikov, issued in 1967 by Progress Publishers, Moscow; the 1974 German translation by Prof. H. Hartmann, published by Wis- senschaftliche Buchgesellschaft, Darmstadt; the Italian translation by Dr. M. Plassa and Dr. I. Prinetti of the Istituto di Metrologia in Torino, published by G. Feltrinelli Editore, Milan; and the Japanese translation by professors Y. Otsuki, Y. Hatano, and T. Saito, which appeared under the imprint of Kodansha Publishers, Tokyo)—differ in pagination but

(^2) Harvard University Press, Cambridge, Mass., 1961; republished in 1997 by Dover Publications, Mineola, New York.

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k Acknowledgments k

It gives me pleasure to acknowledge my indebtedness to Prof. Clifford

M. Will, the leading specialist on experimental gravitation, and to Prof. Jacob Bekenstein, the well-known expert on the theory of relativity, for reading my entire manuscript and for their invaluable critical remarks. I am also grateful to the two anonymous referees of the draft for their constructive critical comments. I thank my friends and colleagues Profs. Abner Shimony, Yuval Ne’eman, Lawrence Horwitz, Nissan Zeldes, and Jacob Levitan for enlightening discussions. Finally, I express my gratitude to Dr. Trevor Lipscombe, the physics editor of Princeton Uni- versity Press, and to Ms. Evelyn Grossberg, the copyeditor for Princeton Editorial Associates, for their fruitful cooperation.

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Concepts of Mass in Contemporary

Physics and Philosophy

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k C H A P T E R O N E k

Inertial Mass

Mechanics, as understood in post-Aristotelian physics,^1 is gen-

erally regarded as consisting of kinematics and dynamics. Kinematics, a term coined by Andr´e-Marie Amp`ere,^2 is the science that deals with the motions of bodies or particles without any regard to the causes of these motions. Studying the positions of bodies as a function of time, kinematics can be conceived as a space-time geometry of motions, the fundamental notions of which are the concepts of length and time. By contrast, dynamics, a term probably used for the first time by Gottfried Wilhelm Leibniz,^3 is the science that studies the motions of bodies as the result of causative interactions. As it is the task of dynamics to ex- plain the motions described by kinematics, dynamics requires concepts additional to those used in kinematics, for “to explain” goes beyond “to describe.”^4 The history of mechanics has shown that the transition from kinemat- ics to dynamics requires only one additional concept—either the concept of mass or the concept of force. Following Isaac Newton, who began his Principia with a definition of mass, and whose second law of motion, in Euler’s formulation F = ma , defines the force F as the product of the mass m and the acceleration a (acceleration being, of course, a kinematical concept), the concept of mass, or more exactly the concept of inertial mass, is usually chosen. The three fundamental notions of mechanics are therefore length, time, and mass, corresponding to the three physical

(^1) In Aristotelian physics the term “mechanics” or nidbojl (i )u (fdoi*, derived from n (idpς (contrivance), meant the application of an artificial device “to cheat nature,” and was therefore not a branch of “physics,” the science of nature. “When we have to produce an effect contrary to nature... we call it mechanical.” Cf. the pseudo-Aristotelian treatise Mechanical Problems (847 a 10). (^2) “C’est a cette science ou les mouvements sont considérés en eux-mˆemes... j’ai donné le nom de cinématique , de l(joinb, mouvement.” A.-A. Ampere, _Essai sur la philosophie des sciences_ (Paris: Bachelier, 1834), p. 52. (^3) G. W. Leibniz, “Essai de Dynamique sur les loix du mouvement,” in C. I. Gerhardt, ed. _Mathematische Schriften_ (Hildesheim: Georg Olms, 1962), vol. 6, pp. 215–231; “Specimen Dynamicum,” ibid., pp. 234–254. (^4) M. Jammer, “Cinematica e dinamica,” in _Saggi su Galileo Galilei_ (Florence: G. Barbera Editore, 1967), pp. 1–12.

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I N E R T I A L M A S S These comments are of course not intended to fault the authors of textbooks, for although it is easy to employ the concepts of mass it is difficult, as we shall see further on, to give them a logically and scientifically satisfactory definition. Even a genius such as Isaac Newton was not very successful in defining inertial mass! The generally accepted classification of masses into mi , ma , and mp , the last two sometimes denoted collectively by mg for gravitational mass, gives rise to a problem. Modern physics, as is well known, recognizes three fundamental forces of nature apart from gravitation—the elec- tromagnetic, the weak, and the strong interactions. Why then are non- inertial masses associated only with the force of gravitation? True, at the end of the nineteenth century the concept of an “electromagnetic mass” played an important role in physical thought.^7 But after the advent of the special theory of relativity it faded into oblivion. The problem of why only gravitational mass brings us to the forefront of current research in particle physics, for it is of course intimately related to the possibility, suggested by modern gauge theories, that the different forces are ultimately but different manifestations of one and the same force. From the historical point of view, the answer is simple. Gravitation was the first of the forces to become the object of a full-fledged theory which, owing to the scalar character of its potential as compared with the vector or tensor character of the potential of the other forces, proved itself less complicated than the theories of the other forces. Although the notions of gravitational mass ma and mp differ conceptu- ally from the notion of inertial mass mi , their definitions, as we shall see later on,^8 presuppose, implicitly at least, the concept of mi. It is therefore logical to begin our discussion of the concepts of mass with an analysis of the notion of inertial mass. There may be an objection here on the grounds that this is not the chronological order in which the various conceptions of mass emerged in the history of civilization and science. It is certainly true that the notion of “weight,” i.e., mpg , where g is the acceleration of free fall, and hence, by implication mp , is much older than mi. That weights were used in the early history of mankind is shown by the fact that the equal-arm balance can be traced back to the year 5000 b.c. “Weights” are also mentioned

(^7) For the history of the notion of “electromagnetic mass” see chapter 11 in M. Jammer, Concepts of Mass in Classical and Modern Physics (Cambridge, Mass.: Harvard University Press, 1961), referred to henceforth as COM. (^8) See the beginning of chapter 4.

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C H A P T E R O N E

in the Bible. In Deuteronomy, chapter 25, verse 13, we read: “You shall not have in your bag two kinds of weights, a large and a small... a full and just weight you shall have.” Or in Proverbs, chapter 11, verse 1, it is said: “A false balance is an abomination to the Lord, but a just weight is his delight.” But that “weight” is a force, given by mpg , and thus involves the notion of gravitational mass could have been recognized only after Newton laid the foundations of classical dynamics, which he could not have done without introducing the concept of inertial mass. Turning, then, to the concept of inertial mass we do not intend to recapitulate the long history of its gradual development from antiquity through Aegidius Romanus, John Buridan, Johannes Kepler, Christiaan Huygens, and Isaac Newton, which has been given elsewhere.^9 Our intention here is to focus on only those aspects that have not yet been treated anywhere else. One of these aspects is what has been supposed, though erroneously as we shall see, to be the earliest operational def- inition of inertial mass. But before beginning that discussion let us recall that, although Kepler and Huygens came close to anticipating the concept of mi , it is Newton who has to be credited with having been the first to define the notion of inertial mass and to employ it systematically. In particular, Galileo Galilei, as was noted elsewhere,^10 never offered an explicit definition of mass. True, he used the term “massa,” but only in a nontechnical sense of “stuff” or “matter.” For him the fundamental quantities of mechanics were space, time, and momentum. He even proposed a method to compare the momenta (“movimenti e lor velocit`a o impeti”) of different bodies, but he never identified momentum as the product of mass and velocity. Richard S. Westfall, a prominent historian of seventeenth-century physics, wrote in this context: “Galileo does not, of course, clearly define mass. His word momento serves both for our ‘moment’ and for our ‘momentum,’ and he frequently uses impeto for ‘momentum.’ ” One of Galileo’s standard devices to measure the momenti of equal bodies was to compare their impacts, that is, their forze of percussion.”^11 It was therefore an anachronistic interpretation of Galileo’s method of comparing momenta when the eminent mathematician Hermann Weyl

(^9) Chapters 2–6 of COM. (^10) Beginning of chapter 5 of COM. (^11) R. S. Westfall, “The Problem of Force in Galileo’s Physics,” in C. L. Golino, ed., Galileo Reappraised (Berkeley: University of California Press, 1966), pp. 67–95.

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