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Concepts of Mass in Contemporary Physics and Philosophy
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Concepts of Mass in Contemporary Physics
and Philosophy
p
Max Jammer
princeton university press princeton, new jersey

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

99-24113

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

1 3 5 7 9 10 8 6 4 2

p Contents p

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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p Preface p
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 calculations 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., Gravitational 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 predominantly 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 = mc2, 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 material. 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 Wissenschaftliche 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.
viii

P R E FA C E
not in the order of chapters or of sections. The references can therefore also be used by the reader of any of these various versions. The present monograph does not presume to resolve the problem of mass. Its purpose is rather to show that the notion of mass, although fundamental to physics, is still shrouded in mystery.
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p Acknowledgments p
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 University 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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Introduction
The concept of mass is one of the most fundamental notions in
physics, comparable in importance only to the concepts of space and time. Isaac Newton, who was the ﬁrst to make systematic use of the concept of mass, was already aware of its importance in physics. It was probably not a matter of fortuity that the very ﬁrst statement in his Principia, the most inﬂuential work in classical physics, presents his deﬁnition of mass or of “quantitas materiae,” as he still used to call it.1 However, his deﬁnition of mass as the measure of the quantity of matter, “arising from its density and bulk conjointly,” was for several reasons soon regarded as inadequate. Since then, the quest for an adequate deﬁnition of mass, combined with the search for a more profound understanding of its meaning, its nature, and its role in the physical sciences, has never ceased to engage the attention of physicists and philosophers alike.
That still today “mass is a mess,” as a contemporary physicist punningly phrased it,2 should not come as a surprise. For “in the world of human thought generally, and in physical science particularly, the most important and most fruitful concepts are those to which it is impossible to attach a well-established meaning.”3
Yet, the remarkable progress in experimental and theoretical physics made during the past few decades has considerably deepened our knowledge concerning the nature of mass. In particular, recent advances in the general theory of relativity and in the theory of elementary particles have opened new vistas that promise to lead us to a more profound understanding of the nature of mass. It is the intention of the present study to review these developments in a rigorous and yet concise fashion.
1 I. Newton, Philosophiae Naturalis Principia Mathematica (London: J. Streater, 1687, 1713, 1726), p. 1; Isaac Newton’s Mathematical Principles of Natural Philosophy and His System of the World (Berkeley: University of California Press, 1934), p. 1.
2 W. T. Padgett, “Problems with the Current Deﬁnitions of Mass,” Physics Essays 3, 178–182 (1990).
3 H. A. Kramers, statement at the Princeton Bicentennial Conference on the Future of Nuclear Energy, 1946, in K. K. Darrow, ed., Physical Science and Human Values (Princeton: Princeton University Press, 1947), p. 196.
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p CHAPTER ONE p
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 Andre´-Marie Ampe`re,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 explain 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 kinematics 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 nidbojli( )uf( doi*, derived from ni( dpς (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-meˆmes . . . j’ai donné le nom de cinématique, de lj(oinb, mouvement.” A.-A. Ampe`re, 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. Barbe`ra Editore, 1967), pp. 1–12.
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CHAPTER ONE

dimensions L, T, and M with their units the meter, the second, and the kilogram. As in the last analysis all measurements in physics are kinematic in nature, to deﬁne the concept of mass and to understand the nature of mass are serious problems. These difﬁculties are further exacerbated by the fact that physicists generally distinguish among three types of masses, which they call inertial mass, active gravitational mass, and passive gravitational mass. For the sake of brevity we shall often denote them by mi, ma, and mp, respectively.
As a perusal of modern textbooks shows, contemporary deﬁnitions of these concepts are no less problematic than those published almost a century ago.5 Today, as then, most authors deﬁne the inertial mass mi of a particle as the ratio between the forced F acting on the particle and the acceleration a of the particle, produced by that force, or brieﬂy as “the proportionality factor between a force and the acceleration produced by it.” Some authors even add the condition that F has to be “mass-independent” (nongravitational), thereby committing the error of circularity.
The deﬁciency of this deﬁnition, based as it is on Newton’s second law of motion

F = mia

(1.1)

is of course its use of the notion of force. For if “force” is regarded as a primitive, that is, as an undeﬁned term, then this deﬁnition deﬁnes an ignotum per ignotius; and if “force” is deﬁned, as it generally is, as the product of acceleration and mass, then the deﬁnition is obviously circular.
The active gravitational mass ma of a body, roughly deﬁned, measures the strength of the gravitational ﬁeld produced by the body, whereas its passive gravitational mass mp measures the body’s susceptibility or response to a given gravitational ﬁeld. More precise deﬁnitions of the gravitational masses will be given later on.
Not all physicists differentiate between ma and mp. Hans C. Ohanian, for example, calls such a distinction “nonsense” because, as he says, “the equality between active and passive mass is required by the equality of action and reaction; an inequality would imply a violation of momentum conservation.”6

5 E. V. Huntington, “Bibliographical Note on the Use of the Word Mass in Current Textbooks,” The American Mathematical Monthly 25, 1–15 (1918).
6 H. C. Ohanian, Gravitation and Spacetime (New York: Norton, 1973), p. 17.

6

INERTIAL MASS
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 electromagnetic, the weak, and the strong interactions. Why then are noninertial 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 conceptually 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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CHAPTER ONE
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 definition 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 velocita` 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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INERTIAL MASS

wrote in 1927: “According to Galileo the same inert mass is attributed to two bodies if neither overruns the other when driven with equal velocities (they may be imagined to stick to each other upon colliding).”12 This statement, which constitutes the ﬁrst step of what we shall call “Weyl’s deﬁnition of inertial mass,” can be rephrased in more detail as follows: If, relative to an inertial reference frame S, two particles A and B of oppositely directed but equal velocities uA and uB = −uA collide inelastically and coalesce into a compound particle A+B, whose velocity uA+B is zero, then the masses mA and mB, respectively, of these particles are equal. In fact, if mA+B denotes the mass of the compound particle, application of the conservation principles of mass and momentum, as used in classical physics, i.e.,

mAuA + mBuB = mA+BuA+B = (mA + mB)uA+B

(1.2)

shows that uB = −uA and uA+B = 0 imply mA = mB. This test is an example of what is often called a “classiﬁcational measurement”: Provided that it has been experimentally conﬁrmed that the result of the test does not depend on the magnitude of the velocities uA and uB and that for any three particles A, B, and C, if mA = mB and mB = mC then the experiment also yields mA = mC (i.e., the “equality” is an equivalence relation), it is possible to classify all particles into equivalence classes such that all members of such a class are equal in mass.
For a “comparative measurement,” which establishes an order among these classes or their members, Weyl’s criterion says: “That body has the larger mass which, at equal speeds, overruns the other.”13 In other words, mA is larger than mB, or mA > mB, if uA = −uB but uA+B = 0 and sign uA = sign uA+B. To ensure that the relation “larger” thus deﬁned is an order relation it has to be experimentally conﬁrmed that it is an asymmetric and transitive relation, i.e., if mA > mB then mB > mA does not hold, and if mA > mB and mB > mC have been obtained then mA > mC will also be obtained for any three particles A, B, and C. Since for uA = −uB equation (1.2) can be written

mA − mB = (uA+B/uA)mA+B

(1.3)

the condition sign uA = sign uA+B shows that the coefﬁcient of mA+B is

12 H. Weyl, “Philosophie der Mathematik und Naturwissenschaft,” in R. Oldenbourg, ed., Handbuch der Philosophie (Munich: Oldenbourg, 1927). Philosophy of Mathematics and Natural Science (Princeton: Princeton University Press, 1949), p. 139.
13 Weyl, Philosophy of Mathematics and Natural Science, p. 139.
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CHAPTER ONE

a positive number and, hence, mA > mB, it being assumed, of course, that all mass values are positive numbers. The experimentally deﬁned relation “>” therefore coincides with the algebraic relation denoted by the same symbol. Finally, to obtain a “metrical measurement” the shortest method is to impose only the condition uA+B = 0 so that equation (1.2) reduces to

mA/mB = −uB/uA.

(1.4)

Hence, purely kinematic measurements of uA and uB determine the mass-ratio mA/mB. Choosing, say, mB as the standard unit of mass (mB = 1) determines the mass mA of any particle A unambiguously.
Weyl called this quantitative determination of mass “a deﬁnition by abstraction” and referred to it as “a typical example of the formation of physical concepts.” For such a deﬁnition, he pointed out, conforms to the characteristic trait of modern science, in contrast to Aristotelian science, to reduce qualitative determinations to quantitative ones, and he quoted Galileo’s dictum that the task of physics is “to measure what is measurable and to try to render measurable what is not so yet.”
Weyl’s deﬁnition of mass raises a number of questions, among them the philosophical question of whether it is really a deﬁnition of inertial mass and not only a prescription of how to measure the magnitude of this mass. It may also be asked whether it does not involve a circularity; for the assumption that the reference frame S is an inertial frame is a necessary condition for its applicability, but for the deﬁnition of an inertial system the notion of force and, therefore, by implication, that of mass may well be indispensable.
Not surprisingly, Weyl’s deﬁnition seems never to have been criticized in the literature on this subject, for the same questions have been discussed in connection with the much better-known deﬁnition of mass that Ernst Mach proposed about sixty years earlier. In fact, these two deﬁnitions have much in common. The difference is essentially only that Weyl’s deﬁnition is based, as we have seen, on the principle of the conservation of momentum while Mach’s rests on the principle of the equality between action and reaction or Newton’s third law. But, as is well known, both principles have the same physical content because the former is only a time-integrated form of the latter.
Although Mach’s deﬁnition of inertial mass is widely known,14 we shall review it brieﬂy for the convenience of the reader. For Mach, just as

14 See, e.g., chapter 8 of COM.
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INERTIAL MASS
for Weyl six decades later, the task of physics is “the abstract quantitative expression of facts.” Physics does not have to “explain” phenomena in terms of purposes or hidden causes, but has only to give a simple but comprehensive account of the relations of dependence among phenomena. Thus he vigorously opposed the use of metaphysical notions in physics and criticized, in particular, Newton’s conceptions of space and time as presented in the Principia.15
Concering Newton’s deﬁnition of mass Mach declared: “With regard to the concept of ‘mass,’ it is to be observed that the formulation of Newton, which deﬁnes mass to be the quantity of matter of a body as measured by the product of its volume and density, is unfortunate. As we can only deﬁne density as the mass of a unit of volume, the circle is manifest.”16
In order to avoid such circularity and any metaphysical obscurities Mach proposed to deﬁne mass with an operational deﬁnition. It applies the dynamical interaction between two bodies, called A and B, that induce in each other opposite accelerations in the direction of their line of junction. If aA/B denotes the acceleration of A owing to B, and aB/A the acceleration of B owing to A, then, as Mach points out, the ratio −aB/A/aA/B is a positive numerical constant independent of the positions or motions of the bodies and deﬁnes what he calls the massratio mA/B = −aB/A/aA/B. By introducing a third body C, interacting with A and B, he shows that the mass-ratios satisfy the transitive relation mA/B = mA/CmC/B and concludes that each mass-ratio is the ratio of two positive numbers, i.e., mA/B = mA/mB, mA/C = mA/mC, and mC/B = mC/mB. Finally, if one of the bodies, say A, is chosen as the standard unit of mass (mA = 1), the masses of the other bodies are uniquely determined.17
Mach’s identiﬁcation of the ratio of the masses of two interacting bodies as the negative inverse ratio of their mutually induced accelerations is essentially only an elimination of the notion of force by combining Newton’s third law of the equality between action and reaction with his second law of motion. In fact, if FAB is the force exerted on A by B and FBA
15 See, e.g., chapter 5 in M. Jammer, Concepts of Space (Cambridge: Harvard University Press, 1954, 1969; enlarged edition, New York: Dover, 1993).
16 E. Mach, Die Mechanik in ihrer Entwicklung (Leipzig: Brockhaus, 1883, 1888, 1897, 1901, 1904, 1908, 1912, 1921, 1933); The Science of Mechanics (La Salle, Ill.: Open Court, 1893, 1902, 1919, 1942, 1960), chapter 2, section 3, paragraph 7. In his Die Principien der Wa¨rmelehre (Leipzig: Barth, 1896, 1900, 1919) Mach called Newton’s deﬁnition of mass “scholastisch.”
17 For further details see chapter 8 in COM.
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CHAPTER ONE
the force exerted on B by A, then according to the third law FAB = −FBA. But according to the second law FAB = mAaA/B and FBA = mBaB/A. Hence, mAaA/B = −mBaB/A or mA/B = mA/mB = −aB/A/aA/B, as stated by Mach, and the mass-ratio mA/B is the ratio between two inertial masses. Thus we see that Mach’s operational deﬁnition is a deﬁnition of inertial masses.
We have brieﬂy reviewed Mach’s deﬁnition not only because it is still restated in one form or another in modern physics texts, but also, and more importantly, because it is still a subject on which philosophers of science disagree just as they did in the early years of the century. In fact, as we shall see, recent arguments pro or contra Mach’s approach were ﬁrst put forth a long time ago, though in different terms. For example, in 1910 the philosopher Paul Volkmann declared that Mach’s “phenomenological deﬁnition of mass,” as he called it, contradicts Mach’s own statement that the notion of mass, since it is a fundamental concept (“Grundbegriff”), does not properly admit any deﬁnition because we deprive it of a great deal of its rich content if we conﬁne its meaning solely to the principle of reaction.18 On the other hand, the epistemologist and historian of philosophy Rudolf Thiele declared that “one can hardly overestimate the merit that is due to Mach for having derived the concept of mass without any recourse to metaphysics. His work is also important for the theory of knowledge, since it provides for the ﬁrst time, an immanent determination of this notion without the necessity of transcending the realm of possible experience.”19
As noted above, many textbooks deﬁne inertial mass mi as the ratio between the force F and the acceleration a in accordance with Newton’s second law of motion, which in Euler’s formulation reads F = mia. Further, they often suppose that the notion of force is immediately known to us by our muscular sensation when overcoming the resistance in moving a heavy body. But there are also quite a few texts on mechanics that follow Mach, even though they do not refer to him explicitly, and introduce mi in terms of an operational deﬁnition based either on Newton’s third law, expressing the equality of action and reaction, or on the principle of the conservation of linear momentum. It is therefore strange that the prominent physicist and philosopher of physics, Percy Williams
18 P. Volkmann, Erkenntnistheoretische Grundzu¨ ge der Naturwissenschaften (Leipzig: Teubner, 1910), p. 138.
19 R. Thiele, “Zur Charakteristik von Mach’s Erkenntnislehre,” in Abhandlungen zur Philosophie und ihrer Geschichte, vol. 45 (Halle: Niemeyer, 1914), p. 101.
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INERTIAL MASS
Bridgman, a staunch proponent of operationalism and probably the ﬁrst to use the term “operational deﬁnition,” never even mentioned Mach’s operational deﬁnition of mass in his inﬂuential book The Logic of Modern Physics, although his comments on Mach’s cosmological ideas clearly show that he had read Mach’s writings.20
Instead, like many physicists and philosophers of the late nineteenth century, among them James Clerk Maxwell and Alois Ho¨ ﬂer,21 Bridgman introduced “mass” essentially in accordance with Newton’s second law, but put, as he phrased it, “the crude concept [of force] on a quantitative basis by substituting a spring balance for our muscles, or instead of the spring balance . . . any elastic body, and [we] measure the force exerted by it in terms of its deformation.” After commenting on the role of force in the case of static systems Bridgman continued:
We next extend the force concept to systems not in equilibrium, in which there are accelerations, and we must conceive that at ﬁrst all our experiments are made in an isolated laboratory far out in empty space, where there is no gravitational ﬁeld. We here encounter a new concept, that of mass, which as it is originally met is entangled with the force concept, but may later be disentangled by a process of successive approximations. The details of the various steps in the process of approximation are very instructive as typical of all methods in physics, but need not be elaborated here. Sufﬁce it to say that we are eventually able to give to each rigid material body a numerical tag characteristic of the body such that the product of this number and the acceleration it receives under the action of any given force applied to it by a spring balance is numerically equal to the force, the force being deﬁned, except for a correction, in terms of the deformation of the balance, exactly as it was in the static case. In particularly, the relation found between mass, force, and acceleration applies to the spring balance itself by which the force is applied, so that a correction has to be applied for a diminution of the force exerted by the balance arising from its own acceleration.22
We have purposely quoted almost all of what Bridgman had to say about the deﬁnition of mass in order to show that the deﬁnition of mass via an operational deﬁnition of force meets with not inconsiderable
20 P. W. Bridgman, The Logic of Modern Physics (New York: Macmillan, 1927, 1961), p. 25. 21 See chapter 8 of COM. 22 Bridgman, Logic of Modern Physics, pp. 102–103.
13

CHAPTER ONE

difﬁculties. Nor do his statements give us any hint as to why he com-

pletely ignored Mach’s operational deﬁnition of mass.

In the late 1930s Mach’s deﬁnition was challenged as having only a

very limited range of applicability insofar as it fails to determine unique

mass-values for dynamical systems composed of an arbitrary number

of bodies. Indeed, C. G. Pendse claimed in 1937 that Mach’s approach breaks down for any system composed of more than four bodies.23

Let us brieﬂy outline Pendse’s argument. If in a system of n bodies ak denotes, in vector notation, the observable induced acceleration of the kth body and ukj(j = k) the observable unit vector in the direction from the kth to the jth body, then clearly

n
ak = αkjukj
j=1

(k = 1, 2, . . . , n),

(1.5)

where αkj(αkk = 0) are n(n − 1) unknown numerical coefﬁcients in 3n algebraic equations. However, these coefﬁcients, which are required for

the determination of the mass-ratios, are uniquely determined only

if their number does not exceed the number of the equations, i.e.,

n(n − 1) ≤ 3n, or n ≤ 4.

Pendse also looked into the question of how this result is affected if

the dynamical system is observed at r different instants. Again using

simple algebra he arrived at the conclusion that “if there be more than

seven particles in the system the observer will be unable to determine

the ratios of the masses of the particles . . . , however large the num-

ber of instants, the accelerations pertaining to which are considered,

may be.”

Pendse’s conclusions were soon challenged by V. V. Narlikar on the

grounds that the Newtonian inverse-square law of gravitation, if applied

to a system of n interacting massive particles, makes it possible to assign a unique mass-value mk(k = 1, 2, . . . , n) to each individual particle of the system. For according to this law, the acceleration aK of the kth particle satisﬁes the equation

ak =

n
Gmj rjk
j=1 j=k

|rjk |3 ,

(1.6)

23 C. G. Pendse, “A Note on the Deﬁnition and Determination of Mass in Newtonian Mechanics,” Philosophical Magazine 24, 1012–1022 (1937). See also References 27 and 28 in chapter 8 of COM.
14

INERTIAL MASS
where G is the constant of gravitation and rjk is the vector pointing from the position of mk to the position of mj. Since all accelerations ak(k = 1, 2, . . . , n) and all rjk are observable, “all the masses become known in this manner.”24
It should be noted, however, that Narlikar established this result for active gravitational masses, for the mj in the above equations are those kinds of masses, and not for inertial masses, which we have seen were the deﬁnienda in Pendse’s approach. It is tempting to claim that this difﬁculty can be resolved within Mach’s conceptual framework by an appeal to his experimental proposition, which says: “The mass-ratios of bodies are independent of the character of the physical states (of the bodies) that condition the mutual accelerations produced, be those states electrical, magnetic, or what not; and they remain, moreover, the same, whether they are mediately or immediately arrived at.”25 Hence one may say that the interactions relative to which the mass-ratios are invariant also include gravitational interactions although these were not explicitly mentioned by Mach. However, this interpretation may be questioned because of Mach’s separate derivation of the measurability of mass by weight.26 As this derivation illustrates, quite a few problematic issues appertaining to Mach’s treatment of mass would have been avoided had he systematically distinguished between inertial and active or passive gravitational mass.
A serious difﬁculty with Mach’s deﬁnition of mass is its dependence on the reference frame relative to which the mutually induced accelerations are to be measured. Let us brieﬂy recall how the mass-ratio mA/B of two particles A and B depends on the reference frame S. In a reference frame S , which is moving with an acceleration a relative to S, we have by deﬁnition mA/B = −aB/A/aA/B = −(aB/A −a)/(aA/B −a) so that mA/B = mA/B[1−(a/aB/A)]/[1−(a/aA/B)] = mA/B (for a = 0). Thus in order to obtain uniquely determined mass-values, Mach assumed, tacitly at least, that the reference frame to be used for the measurement of the induced accelerations is an inertial system However, such a system is deﬁned by the condition that a “free” particle (i.e., a particle not acted upon by a force) moves relative to it in uniform rectilinear motion. This condition involves, as we see, the notion of force, which Mach deﬁned as
24 V. V. Narlikar, “The Concept and Determination of Mass in Newtonian Mechanics,” Philosophical Magazine 27, 33–36 (1938).
25 Mach, The Science of Mechanics, chapter 2, section 7, paragraph 5. 26 Mach, The Science of Mechanics, chapter 2, section 5, paragraph 6.
15

CHAPTER ONE
“the product of the mass-value of a body times the acceleration induced in that body.”27 Hence, Mach’s deﬁnition involves a logical circle.
Nevertheless, in the early decades of the twentieth century Mach’s deﬁnition of mass, as an example of his opposition to the legitimacy of metaphysics in scientiﬁc thought, enjoyed considerable popularity, especially among the members of the Viennese Circle founded by Moritz Schlick. Repudiating Kantian apriorism, logical positivists and scientiﬁc empiricists stressed the importance of the logical analysis of the fundamental concepts of physical science and often regarded Mach’s deﬁnition of mass as a model for such a program. A drastic change occurred only after the 1950s when the positivistic philosophy of science became a subject of critical attack. One of the most eloquent critics was the philosopher Mario Bunge.
According to Bunge, Mach committed a serious error when he “concluded that he has deﬁned the mass concept in terms of observable (kinematic) properties,” for, “Mach confused ‘measuring’ and ‘computing’ with ‘deﬁning.’ ” In particular, the equation mA/mB = −aB/A/aA/B, which establishes an equality between two expressions that differ in meaning—the left-hand side expressing “the inertia of body A relative to the inertia of body B” and the right-hand side standing for a purely kinematical quantity—cannot be interpreted, as Mach contended, as having the meaning of a deﬁnition. It is a numerical, but not a logical, equality and “does not authorize us to eliminate one of the sides in favor of the other.”28
In a similar vein Renate Wahsner and Horst-Heino von Borzeszkowski rejected Mach’s deﬁnition on the grounds that “the real nature” (“das Wesen”) of mass cannot be obtained by merely quantitative determinations.29 Moreover, they charged Mach, as Ludwig Boltzmann had done earlier,30 with contradicting his own precept that a mechanics that transcends experience fails to perform its proper task. Mach’s deﬁnition, based as it is on the interaction between two mutually attracting bodies, has not been proved to be universally valid for all bodies dealt with
27 Mach, The Science of Mechanics, chapter 2, section 7, paragraph 5. 28 M. Bunge, “Mach’s Critique of Newtonian Mechanics,” American Journal of Physics 34, 585–596 (1966); reprinted in J. Blackmore, Ernst Mach—A Deeper Look (Dordrecht: Kluwer, 1992), pp. 243–261. 29 R. Wahsner and H.-H. von Borzeszkowski, epilogue to their new edition of Mach’s Die Mechanik in ihrer Entwicklung (Berlin: Akademie Verlag, 1988), p. 600. 30 L. Boltzmann, “U¨ ber die Grundprinzipien und Grundgleichungen der Mechanik,” in Popula¨re Schriften (Leipzig: J. A. Barth, 1905), p. 293.
16

INERTIAL MASS
in mechanics and his claim that the “experimental propositions” do not go beyond experience is confuted by the fact that they presuppose all principles of mechanics. Similarly, in a recent essay on operational deﬁnitions Andreas Kamlah rejects the claim that the concept of mass can in all cases be deﬁned in a kinematical language containing only the notions of position, time, and velocity (or acceleration). He also argues that “Mach’s deﬁnition is not a deﬁnition in the proper sense . . . [for] it yields the values of mass only for bodies which just by chance collide with other bodies. All other values of that function remain undetermined.”31
In contrast to the preceding unfavorable criticisms (and many others could have been recounted), Mach’s deﬁnition was defended, at least against two major objections, by Arnold Koslow.32 The two objections referred to concern the restricted applicability of the deﬁnition and its noninvariance relative to different reference frames. Koslow’s main argument against the former objection contends that the third experimental proposition has not been taken into account. For according to this proposition the mass-ratios are independent of whether the mutual accelerations are induced by “electric, magnetic, or what not” interactions. Hence, as Koslow shows in mathematical detail, by performing the deﬁnitional operations with respect to different kinds of interactions, the number of the equations can be sufﬁciently increased to ensure the uniqueness of the mass-ratios for any ﬁnite number of particles. Concerning the latter objection, Koslow justiﬁed Mach’s contention that “the earth usually does well enough as a reference system, and for larger scaled motions, or increased accuracy, one can use the system of the ﬁxed stars.”
An operational deﬁnition of inertial mass, which unlike Mach’s deﬁnition seems to be little known even among experts, is the so-called “tabletop deﬁnition” proposed in 1985 by P. A. Goodinson and B. L. Luffman.33 Unlike Mach’s and Weyl’s deﬁnitions of mi, which are based, as we have seen, on Newton’s third law, the Goodinson-Luffman deﬁnition is based on Newton’s second law, which, in Euler’s formulation, says that force
31 A. Kamlah, “The Problem of Operational Deﬁnitions,” in W. Salmon and G. Wolters, eds., Logic, Language, and the Structure of Scientiﬁc Theories (Konstanz: Universita¨tsverlag Konstanz, 1996), pp. 171–189.
32 A. Koslow, “Mach’s Concept of Mass: Program and Deﬁnition,” Synthese 18, 216–233 (1968).
33 P. A. Goodinson and B. L. Luffman, “On the Deﬁnition of Mass in Classical Physics,” American Journal of Physics 53, 40–42 (1985).
17

CHAPTER ONE

is the product of mass and acceleration. However, as the notion of force (or of weight or of friction) as used in this deﬁnition is made part of the operational procedure, an explicit deﬁnition is not required so that from the purely operational point of view they seem to have avoided a logical circularity.
Goodinson and Luffman call their deﬁnition of mi a “table-top definition” because it involves the measurement of the acceleration aB of a body B that is moving on a horizontal table—on “a real table, not the proverbial ‘inﬁnitely smooth table.’ ” The motion of B is produced by means of a (weightless) string that is attached to B, passes over a (frictionless) pulley ﬁxed at the end of the table, and carries a heavy weight W on its other end. At ﬁrst the acceleration a0 of a standard body B0, connected via the string with an appropriate weight W0, is measured. Measurements of distance and time are of course supposed to have been operationally deﬁned antecedently, just as in the operational deﬁnitions by Mach or by Weyl.
The procedure of measuring the acceleration a is repeated for a body B and also for weights W that differ from W0. A plot of a against a0 shows that

a = ka0 + c,

(1.7)

where k and c are constants. Repetition of the whole series of measurements with a different table again yields a linear relation

a = ka0 + d

(1.8)

with the same slope k but with a constant d that differs from c. This shows that the intercepts c and d are table-dependent whereas the slope k is independent of the roughness or friction caused by the table. A series of such measurements for bodies Bq(q = 1, 2, . . .) yields a series of straight-line plots, one plot for each aq against a0 with slope kq. These slopes are seen to have the following properties: if Bq is “heavier than” Bp then

kq < kp

(1.9)

and

1/kq + 1/kp = 1/kq+p,

(1.10)

where kq+p is the slope obtained when Bq and Bp are combined. The inertial mass mi(Bq) of a body Bq, with respect to the standard body B0, is now deﬁned by

18

INERTIAL MASS

mi(Bq) = l/kq.

(1.11)

In the sequel to their paper Goodinson and Luffman prove that equa-
tions (1.9) and (1.10) are independent of the choice of the standard body B0, and that mi(B1) = mi(B2) and mi(B2) = mi(B3) imply mi(B1) = mi(B3) for any three bodies B1, B2, and B3, independently of the choice of B0. In addition to this transitivity of mass, the additivity of mass is obviously assured because of (1.10). That in spite of the fundamental
differences noted above the table-top deﬁnition converges to Mach’s
deﬁnition under certain conditions can be seen as follows. For two arbitrary bodies B1 and B2 with inertial masses mi(B1) = k1−1 and mi(B2) = k2−1, the plots of their respective accelerations a1 and a2 with respect to B0 are

and Hence

a1 = [mi(B1)]−1 a0 + c1 a2 = [mi(B2)]−1 a0 + c2.

(1.12) (1.13)

where

mi(B1)a1 = mi(B2)a2 + c12,

(1.14)

c12 = mi(B1)c1 − mi(B2)c2.

(1.15)

Experience shows that the quantity |c12| is table-dependent and approaches zero in the case of a perfectly smooth table. In the limit,

mi(B1)/mi(B2) = a2/a1,

(1.16)

which agrees with the Machian deﬁnition of the mass-ratio of two bodies as the inverse ratio of their accelerations (the minus sign being ignored). Yet in spite of this agreement the table-top deﬁnition is proof against the criticism leveled against Mach’s deﬁnition as being dependent on the reference frame. In fact, if an observer at rest in a reference frame S graphs the plot for a body B1 with respect to B0 in the form

a1 = [mi(B1)]−1a0 + c1,

(1.17)

then an observer at rest in a reference frame S that moves with an acceleration a relative to S (in the direction of the accelerations involved) will write

a1 = mi(B1) −1 a0 + c1.

(1.18)

19

CHAPTER ONE

But since a1 = a1 − a and a0 = a0 − a, clearly a1 = mi(B1) −1 a0 + c1 ,

(1.19)

where

c1 = c1 + a 1 − mi(B1) −1

(1.20)

Hence, the plot of a1 against a0 has the slope [mi(B1)]−1, which shows, if compared with (1.17), that mi(B1) = mi(B1) since mi is deﬁned only by the slope. Thus, both observers obtain the same result when measuring the inertial mass of the body B1. Of course, this conclusion is valid only within the framework of classical mechanics and does not hold, for instance, in the theory of relativity.
The range of objects to which an operational deﬁnition of inertial mass, such as the Goodinson-Luffman deﬁnition, can be applied is obviously limited to medium-sized bodies. One objection against operationalism raised by philosophers of the School of Scientiﬁc Empiricists, an outgrowth of the Viennese School of Logical Positivists, is that quite generally no operational deﬁnition of a physical concept, and in particular of the concept of mass, can ever be applied to all the objects to which the concept is attributed. Motivated by the apparently unavoidable circularity in Mach’s operational deﬁnition of mass they preferred to regard the notion of mass as what they called a partially interpreted theoretical concept.
A typical example is Rudolf Carnap’s discussion of the notion of mass. The need to refer to different interactions or different physical theories when speaking, e.g., of the mass of an atom or of the mass of a star, led him to challenge the operational approach. Instead of saying that there are various concepts of mass, each deﬁned by a different operational procedure, Carnap maintained that we have merely one concept of mass. “If we restrict its meaning [the meaning of the concept of mass] to a deﬁnition referring to a balance scale, we can apply the term to only a small intermediate range of values. We cannot speak of the mass of the moon. . . . We should have to distinguish between a number of different magnitudes, each with its own operational deﬁnition. . . . It seems best to adopt the language form used by most physicists and regard length, mass and so on as theoretical concepts rather than observational concepts explicitly deﬁned by certain procedures of measurement.”34

34 R. Carnap, An Introduction to the Philosophy of Science (New York: Basic Books, 1966), pp. 103–104.
20

INERTIAL MASS
Carnap’s proposal to regard “mass” as a theoretical concept refers of course to the dichotomization of scientiﬁc terms into observational and theoretical terms, an issue widely discussed in modern analytic philosophy of science. Since, generally speaking, physicists are not familiar with the issue, some brief comments, specially adapted to our subject, may not be out of place.
It has been claimed by philosophers of science that physics owes much of its progress to the use of theories that transcend the realm of purely empirical or observational data by incorporating into their conceptual structure so-called theoretical terms or theoretical concepts. (We ignore the exact distinction between the linguistic entity “term” and the extralinguistic notion “concept” and use these two words as synonyms.)
In contrast to “observational concepts,” such as “red,” “hot,” or “iron rod,” whose meanings are given ostensively, “theoretical concepts,” such as “potential,” “electron,” or “isospin,” are not explicitly deﬁnable by direct observation. Although the precise nature of a criterion for observability or for theoreticity has been a matter of some debate, it has been generally agreed that terms, obtaining their meaning only through the role they play in the theory as a whole, are theoretical terms. This applies, in particular, to terms, such as “mass,” used in axiomatizations of classical mechanics, such as proposed by H. Hermes, H. A. Simon, J.C.C. McKinsey et al., S. Rubin and P. Suppes,35 or more recently by C. W. Mackey, J. D. Sneed, and W. Stegmu¨ ller.36 In these axiomatizations of mechanics “mass” is a theoretical concept because it derives its meaning from certain rules or postulates of correspondence that associate the purely formally axiomatized term with speciﬁc laboratory procedures. Furthermore, the purely formal axiomatization of the term “mass” is justiﬁed as a result of the conﬁrmation that accrues to the axiomatized and thus interpreted theory as a whole and not to an individual theorem that employs the notion of mass.
It is for this reason that Frank Plumpton Ramsey seems to have been the ﬁrst to conceive “mass” as a theoretical concept when he declared in the late 1920s that to say “ ‘there is such a quality as mass’ is nonsense unless it means merely to afﬁrm the consequences of a mechanical
35 See chapter 9 of COM. 36 G. W. Mackey, Mathematical Foundations of Quantum Mechanics (New York: Benjamin, 1963), chapter 1. J. D. Sneed, The Logical Structure of Mathematical Physics (Dordrecht: Reidel, 1971). W. Stegmu¨ ller, Probleme und Resultate der Wissenschaftstheorie und analytischen Philosophie (Vienna: Springer-Verlag, 1973), vol. 2, part 2.
21

CHAPTER ONE
theory.”37 Ramsey was also the ﬁrst to propose a method to eliminate theoretical terms of a theory by what is now called the “Ramsey sentence” of the theory. Brieﬂy expressed, it involves logically conjoining all the axioms of the theory and the correspondence postulates into a single sentence, replacing therein each theoretical term by a predicate variable and quantifying existentially over all the predicate variables thus introduced.38 This sentence, now containing only observational terms, is supposed to have the same logical consequences as the original theory. The term “mass” has been a favorite example in the literature on the “Ramsey sentence.”39
Carnap proposed regarding “mass” as a theoretical concept, as we noted above, because of the inapplicability of one and the same operational deﬁnition of mass for objects that differ greatly in bulk, such as a molecule and the moon, and since different deﬁnitions assign different meanings to their deﬁnienda, the universality of the concept of mass would be untenable. However, this universality would also be violated if the mass, or rather masses, of one and the same object are being deﬁned by operational deﬁnitions based on different physical principles. This was the case, for instance, when Koslow suggested employing different kinds of interactions in order to rebut Pendse’s criticism of Mach’s deﬁnition as failing to account for the masses of arbitrarily many particles. Even if in accordance with Mach’s “experimental proposition” the numerical values of the thus deﬁned masses are equal, the respective concepts of mass may well be different, as is, in fact, the case with inertial and gravitational mass in classical mechanics, and one would have to distinguish between, say, “mechanical mass” (e.g., “harmonic oscillator mass”), “Coulomb law mass,” “magnetic mass,” and so on.
The possibility of such a differentiation of masses was discussed recently by Andreas Kamlah when he distinguished between “energyprinciple mass” (“Energiesatz-Masse”) and “momentum-principle mass” (“Impulssatz-Masse”), corresponding to whether the conservation principle of energy or of momentum is being used for the deﬁnition.40
37 F. P. Ramsey, The Foundations of Mathematics and Other Logical Essays, edited by R. B. Braithwaite (London: Kegan, Paul, Trench, Turner, 1931), pp. 260–261.
38 For details see, e.g., R. Tuomela, Theoretical Concepts (Vienna: Springer-Verlag, 1973), pp. 57–68.
39 See, e.g., Carnap, An Introduction to the Philosophy of Science, p. 249. Another example, soon to be discussed, is P. Lorenzen’s protophysical deﬁnition of mass.
40 A. Kamlah, “Zur Systematik der Massendeﬁnitionen,” Conceptus 22, 69–82 (1988).
22

INERTIAL MASS

Thus, according to Kamlah, the energy-principle masses mk (k = 1, . . . , n) of n free particles can be determined by the system of equations

n

1 2

mku2k(tj) = c,

k=1

(1.21)

where uk(tj) denotes the velocity of the kth particle at the time tj(j = 1, . . . , r) and c is a constant. In the simple case of an elastic collision
between two particles of velocities u1 and u2 before, and u1 and u2 after, the collision, the equation

1 2

m1u21

+

1 2

m2

u22

=

1 2

m1u

2 1

+

1 2

m2u

2 2

(1.22)

determines the mass ratio

m1/m2

=

(u

2 2

−

u22)/(u21

−

u

22).

(1.23)

The momentum-principle masses µk of the same particles are determined by the equations

n
µkuk(tj) = P,
k=1

(1.24)

where P, the total momentum, is a constant. In the simple case of two particles, the equation

µ1u1 + µ2u2 = µ2u1 + µ2u2 determines the mass-ratio,

(1.25)

µ1/µ2 = (u2 − u2)/(u1 − u1)

(1.26)

The equality between m1/m2 and µ1/µ2 cannot be established without further assumptions, but as shown by Kamlah, it is sufﬁcient to postulate the translational and rotational invariance of the laws of nature.
More speciﬁcally, this equality is established by use of the Hamiltonian principle of least action or, equivalently, the Lagrangian formalism of mechanics, both of which, incidentally, are known to have a wide range of applicability in physics. The variational principle δ L/dt = 0 implies that the Lagrangian function L = L(x1, . . . , xn, u1, . . . , un, t) satisﬁes the Euler-Lagrange equation

j

∂2L ∂ ui ∂ uj

u˙j

+

∂2L ∂ ui ∂ xj

uj

−

∂L ∂ xi

=

0

u˙j = duj/dt. (1.27)

23

CHAPTER ONE

By deﬁning generalized masses mij(u1, . . . , un) by mij = ∂L/∂ui∂uj, and masses mi, assumed to be constant, by mij = miδij, and taking into consideration that the spatial invariance implies i ∂L/∂xi, = 0, Kamlah shows that the Euler-Lagrange equation (1.27) reduces to

∂L/∂ui = P = const.,

(1.28)

i

where ∂L/∂ui = miui. Comparison with equation (1.24) yields mi = µi. The fundamental notions of kinematics, such as the position of a

particle in space or its velocity, are generally regarded as observable

or nontheoretical concepts. A proof that the concept of mass cannot be

deﬁned in terms of kinematical notions would therefore support the

thesis of the theoreticity of the concept of mass. In order to study the

logical relations among the fundamental notions of a theory, such as

their logical independence, on the one hand, or their interdeﬁnability,

on the other, it is expedient, if not imperative, to axiomatize the theory

and preferably to do it in such a way that the basic concepts under

discussion are the primitive (undeﬁned) notions in the axiomatized

theory. As far as the concept of mass is concerned, there is hardly an

axiomatization of classical particle mechanics that does not count this

concept among its primitive notions.41 In fact, as Gustav Kirchhoff’s

Lectures on Mechanics,42 or Heinrich Hertz’s Principles of Mechanics,43 or

more recently the axiomatic framework for classical particle mechanics

proposed by Adonai Schlup Sant’Anna44 clearly show, even axiomati-

zations of mechanics that avoid the notion of force need the concept of

mass as a primitive notion.

Any proof of the undeﬁnability of mass in terms of other primitive

notions can, of course, be given only within the framework of an axi-

omatization of mechanics. Let us choose for this purpose the widely

known axiomatic formulation of classical particle mechanics proposed

in 1953 by John Charles Chenoweth McKinsey and his collaborators,45

41 The only exception known to the present author is the (unpublished) study “Mechanik ohne Masse” (1985) by Rudolf Opelt of the Technische Hochschule in Bremen, Germany.
42 G. Kirchhoff, Vorlesungen u¨ ber Mechanik (Leipzig: J. A. Barth, 1876, 1897). 43 H. Hertz, Die Prinzipien der Mechanik in neuem Zusammenhang dargestellt (Leipzig: J. A. Barth, 1894); The Principles of Mechanics Presented in a New Form (New York: Dover, 1956). 44 A. S. Sant’Anna, “An Axiomatic Framework for Classical Particle Mechanics without Force,” Philosophia Naturalis 33, 187–203 (1996). 45 J.C.C. McKinsey, A. C. Sugar, and P. Suppes, “Axiomatic Foundations of Classical Particle Mechanics,” Journal of Rational Mechanics and Analysis 2, 253–272 (1953).
24

INERTIAL MASS
which is closely related to the axiomatization proposed by Patrick Suppes.46 The axiomatization is based on ﬁve primitive notions: P, T, m, s, and f , where P and T are sets and m, s, and f are unary, binary, and ternary functions, respectively. The intended interpretation of P is a set of particles, denoted by p, that of T is a set of real numbers t measuring elapsed times (measured from some origin of time); the interpretation of the unary function m on P, i.e., m(p), is the numerical value of the mass of particle p, while s(p, t) is interpreted as the position vector of particle p at time t, and f (p, t, i) as the ith force acting on particle p at time t, it being assumed that each particle is subjected to a number of different forces.
A system = P, T, m, s, f is called a “system of particle mechanics” if it satisﬁes the following six axioms:

Kinematical axioms
A-1: P is a nonempty, ﬁnite set. A-2: T is an interval of real numbers. A-3: For p ∈ P and t ∈ T, s(p, t) is a twice-differentiable vector with
respect to t.

Dynamical axioms

A-4: For p ∈ P, m(p) is a positive real number.

A-5: For p ∈ P and t ∈ T,

∞ i=1

f

(p,

t,

i)

is

an

absolutely

convergent

series.

A-6: For p ∈ P and t ∈ T, m(p)d2s(p, t)/dt2 =

∞ i=1

f

(p,

t,

i).

Clearly, A-6 is a formulation of Newton’s second law of motion and,

since for

∞ i=1

f (p, t, i)

=

0

obviously

s(p, t)

=

a

+

bt,

A-6

also

implies

Newton’s ﬁrst law of motion. However, the question we are interested

in is this: can it be rigorously demonstrated that the primitive m, which

is intended to be interpreted as “mass,” cannot be deﬁned by means

of the other primitive terms of the axiomatization, or at least not by

means of the primitive notions that are used in the kinematical axioms?

The standard procedure followed to prove that a given primitive of an

axiomatization cannot be deﬁned in terms of the other primitives of

that axiomatization is the Padoa method, so called after the logician

Alessandro Padoa, who invented it in 1900. According to this method

it is sufﬁcient to ﬁnd two interpretations of the axiomatic system that

differ in the interpretation of the given primitive but retain the same

46 P. Suppes, Introduction to Logic (New York: Van Nostrand, 1957), pp. 294–295.
25

CHAPTER ONE
interpretation for all the other primitives of the system. For if the given primitive were to depend on the other primitives, the interpretation of the latter would uniquely determine the interpretation of the given primitive so that it would be impossible to ﬁnd two interpretations as described.47
Padoa’s formulation of his undeﬁnability proof has been criticized for not meeting all the requirements of logical rigor and, in particular, for its lack of a rigorous criterion for the “differentness” of interpretations. It has therefore been reformulated by, among others, John C. C. McKinsey,48 Evert Willem Beth,49 and Alfred Tarski.50
That in the above axiomatization m is independent of the other primitive notions can be shown by the Padoa method as follows: P is interpreted as the set whose only member is 1, T as the set of all real numbers, s(1, t) for all t ∈ T as the vector each component of which is unity, f (1, t, i) as the null vector for all t ∈ T and every positive integer i; ﬁnally, it is agreed that m1(1) = 1 and m2(1) = 2. Thus interpreted,
1 = P, T, m1, s, f and 2 = P, T, m2, s, f are systems of particle mechanics, i.e., both systems satisfy all the axioms A-1 to A-6, and agree in all primitives with the exception of m. Hence, according to Padoa’s method, m is not deﬁnable in terms of the other primitives. A similar argument proves the logical independence of m in the axiomatization proposed by Suppes. These considerations seem to suggest that, quite generally, the concept of mass cannot be deﬁned in terms of kinematical conceptions and, as such conceptions correspond to observational notions, mass is thus a theoretical term.
47 A. Padoa, “Essai d’une théorie algébrique des nombres entiers, précédé d’une introduction logique a` une théorie déductive quelconque,” Bibliothe`que du Congre`s International de Philosophie, Paris, 1900 (Paris, 1901), vol. 3, pp. 309–365. English (partial) translation “Logical Introduction to Any Deductive Theory,” in Jean van Heijenoort, ed., From Frege to Go¨del: A Source Book in Mathematical Logic 1879–1931 (Cambridge, Mass.: Harvard University Press, 1967, 1977), pp. 118–123.
48 J.C.C. McKinsey, “On the Independence of Undeﬁned Ideas,” Bulletin of the American Mathematical Society 41, 291–256 (135).
49 E. W. Beth, “On Padoa’s Method in the Theory of Deﬁnition,” Koninklijke Nederlandse Akademie van Wetenschappen, Proceedings of the Science Section 56, Series A, Mathematical Sciences, 330–339 (1953); Indagationes Mathematicae 15, 330–339 (1953).
50 A. Tarski, “Einige methodologische Untersuchungen u¨ ber die Deﬁnierbarkeit der Begriffe,” Erkenntnis 5, 80–100 (1936); “Some Methodological Investigations on the Deﬁnability of Concepts,” in A. Tarski, Logic, Semantics, Metamathematics (Oxford: Clarendon Press, 1956), pp. 296–319.
26

INERTIAL MASS
In 1977 Jon Dorling challenged the general validity of such a conclusion.51 Recalling that in many branches of mathematical physics theoretical terms, e.g., the vector potentials in classical or in quantum electrodynamics, have been successfully eliminated in favor of observational terms, Dorling claimed that the asserted uneliminability results only from the “idiosyncratic choice” of the observational primitives. Referring to G. W. Mackey’s above axiomatization in which the acceleration of each particle is given as a function of its position and the positions of the other particles and not, as in McKinsey’s or Suppes’s axiomatization, of time only, Dorling declared: “The claim that the usual theoretical primitives of classical particle mechanics cannot be eliminated in favor of observational primitives seems therefore not only not to have been established by Suppes’s results, but to be deﬁnitely controverted in the case of more orthodox axiomatizations such as Mackey’s.” The issue raised by Dorling has been revived, though without any reference to him, by the following relatively recent development.
In 1993 Hans-Ju¨ rgen Schmidt offered a new axiomatization of classical particle mechanics intended to lead to an essentially universal concept of mass.52 He noted that in former axiomatizations the inertial mass mk had usually been introduced as a coefﬁcient connected with the acceleration ak of the kth particle in such a way that the products mkak satisfy a certain condition that is not satisﬁed by the ak alone. “If this condition determines the coefﬁcients mk uniquely—up to a common factor—” he declared, “we have got the clue for the deﬁnition of mass. This deﬁnition often works if the deﬁning condition is taken simply as a special force law, but then one will arrive at different concepts of mass.” In order to avoid this deﬁciency Schmidt chose instead of a force-determining condition one that is equivalent to the existence of a Lagrangian. This choice involves the difﬁcult task of solving the socalled “inverse problem of Lagrangian mechanics” to ﬁnd a variational principle for a given differential equation. This problem was studied as early as 1886 by Hermann von Helmholtz and solved insofar as he found the conditions necessary for the existence of a function L such
51 J. Dorling, “The Eliminability of Masses and Forces in Newtonian Particle Mechanics: Suppes Reconsidered,” British Journal for the Philosophy of Science 28, 55–57 (1977).
52 H.-J. Schmidt, “A Deﬁnition of Mass in Newton–Lagrange Mechanics,” Philosophia Naturalis 30, 189–207 (1993).
27

CHAPTER ONE
that a given set of equations Gj = 0 are the Euler-Lagrange equations of the variational principle δ L dt = 0.53
Assisted by Peter Havas’s 1957 study of the applicability of the Lagrange formalism,54 Schmidt, on the basis of a somewhat simpliﬁed solution of the inverse problem, was able to construct his axiomatization, which deﬁnes inertial mass in terms of accelerations. The ﬁve primitive terms of the axiomatization are the set M of space-time events, the differential structure D of M, the simultaneity relation σ on M, the set P of particles, and the set of possible motions of P, the last being bijective mappings or “charts” of M into the four-dimensional continuum R4. Six axioms are postulated in terms of these primitives, none of which represents an equivalent to a force law. The fact that these kinematical axioms lead to a satisfactory deﬁnition of mass is in striking contrast to the earlier axiomatizations for which it could be shown, for instance, by use of the Padoa method, that the dynamical concept of mass is indeﬁnable in kinematical language.55
This apparent contradiction prompted Kamlah to distinguish between two kinds of axiomatic approaches to particle mechanics, differing in their epistemological positions, which he called factualism and potentialism.56 According to factualist ontology, which, as Kamlah points out, was proclaimed most radically in Ludwig Wittgenstein’s 1922 Tractatus Logico-Philosophicus, “there are certain facts in the world which may be described by a basic language for which the rules of predicate logic hold, especially the substitution rule, which makes this language an extensional one. The basic language has not to be an observational language.” According to the ontology of potentialism “the world is a totality of possible experiences. Not all possible experiences actually happen.” By distinguishing between a factualist and a potentialist axiomatization Kamlah claims to resolve that contradiction as follows: The concept of acceleration ak contained in Schmidt’s potentialist kinematics can be “deﬁned” operationally in the language of factualist kinematics. However, Kamlah adds,
53 H. v. Helmholtz, “U¨ ber die physikalische Deutung des Princips der kleinsten Wirkung,” Journal fu¨ r die reine und angewandte Mathematik 100, 137–166, 213–222 (1886).
54 P. Havas, “The Range of Application of the Lagrange Formalism” Nuovo Cimento (Supp.) 5, 363–388 (1957).
55 See chapter 9 of COM. 56 A. Kamlah, “Two Kinds of Axiomatization of Mechanics,” Philosophia Naturalis 32, 27–46 (1995).
28

INERTIAL MASS
such determinations of the meaning of concepts are not proper deﬁnitions though being indispensable in physics, and therefore the acceleration function ak is a theoretical concept in particle kinematics. This theoretical concept seems to be powerful enough in combinations with [Schmidt’s additional axioms] to supply us with an explicit deﬁnition of mass. This result seems to be surprising but does not contradict the wellestablished theorem that mass is theoretical (not explicitly deﬁnable) in particle kinematics.
The thesis of the theoretical status of the concept of inertial mass— whether based on the argument of the alleged impossibility of deﬁning this concept in a noncircular operational way or on the claim that it is implicitly deﬁned by its presence in the laws of motion or in the axioms of mechanics—has been challenged by the proponents of protophysics. The program of protophysics,57 a doctrine that was developed by the Erlangen School of Constructivism but can be partially traced back to Pierre Duhem and Hugo Dingler, is the reconstruction of physics on prescientiﬁc constructive foundations with due consideration for the technical construction of the measuring instruments to be used in physics. Protophysics insists on a rigorous compliance with what it calls the methodical order of the pragmatic dependence of operational procedures, in the sense that an operation O2 is pragmatically dependent upon an operation O1 if O2 can be carried out successfully only after O1 has previously been carried out successfully. In accordance with the three fundamental notions in physics—space, time, and mass— protophysicists distinguish among (constructive) geometry, chronometry, and hylometry, the last one, the protophysics of mass, having been subject to far less attention that the other two. Protophysicists have dealt with the concept of charge, often called the fourth fundamental notion of physics, to an even more limited degree.
Strictly speaking, the ﬁrst to treat “mass” as a hylometrical conception was Bruno Thu¨ ring, who contended that the classical law of gravitation has to form part of the measure-theoretical a priori of empirical physics.58 However, this notion of mass was, of course, the concept of gravitational mass. As far as inertial mass is concerned, the mathematician and philosopher Paul Lorenzen was probably the ﬁrst to treat “mass”
57 G. Bo¨ hme, Protophysik (Frankfurt a.M.: Suhrkamp Verlag, 1976); P. Janich, ed., “Protophysik heute,” Philosophia Naturalis 22, 3–156 (1985).
58 B.Thu¨ ring, Die Gravitation und die philosophischen Grundlagen der Physik (Berlin: Duncke & Humblot, 1967), chapter 3.
29

CHAPTER ONE
from the protophysical point of view.59 Lorenzen’s starting point, as in Weyl’s deﬁnition of mass, is an inelastic collision of two bodies with initial velocities u1 and u2, respectively, where the common velocity of the collision is u. That it is technically possible (“hinreichend gut”) to eliminate friction can be tested by repeating the process with different u1 and u2 and checking that the ratio r of the velocity changes u1 − u and u2 − u is a constant. However, the absence of friction cannot be deﬁned in terms of this constant, for were it veriﬁed in the reference frame of the earth it would not hold in a reference frame in accelerated motion relative to the earth.
If an inertial system is deﬁned as the frame in which this constancy has been established, it is a technical-practical question whether the earth is an inertial system. Foucault’s pendulum shows that it is not. Lorenzen proposed therefore that the astronomical fundamental coordinate system S, relative to which the averaged rotational motion of the galaxies is zero, serves as the inertial system. Any derivation from a constant r must then be regarded and explained as a “perturbation.” This proposed purely kinematical deﬁnition of an inertial system is equivalent to deﬁning such a system by means of the principle of conservation of momentum. The statement that numbers m1 and m2 can be assigned by this method to bodies as measures of their “mass” is then the Ramsey sentence for applying the momentum principle for collision processes in S.
A protophysical determination of inertial mass without any recourse to an inertial reference frame or to “free motion” has been proposed by Peter Janich.60 Janich employs what he calls a “rope balance” (“Seilwaage”), a wheel encircled by a rope that has a body attached to each end. The whole device can be moved, for instance, on a horizontal (frictionless) plane in accelerated motion relative to an arbitrary reference frame. As Janich points out, the facts that the rope is constant in length and taut and that the two end pieces beyond the wheel are parallel and
59 P. Lorenzen, “Zur Deﬁnition der vier fundamentalen Messgro¨ ssen,” Philosophia Naturalis 16, 1–9 (1976); reprinted in J. Pfarr, Protophysik und Relativita¨tstheorie (Bibliographisches Institut, Mannheim, 1981), pp. 25–33. See also P. Lorenzen, “Geometrie als Messtheoretisches Apriori der Physik,” ibid., pp. 35–53.
60 P. Janich, “Ist Masse ein ‘theoretischer Begriff’?,” Journal for General Philosophy of Science 8, 303–313 (1977); “Newton ab omni naevo vindicatus,” Philosophia Naturalis 18, 243–255 (1981); “Die Eindeutigkeit der Massemessung und die Deﬁnition der Tra¨gheit,” Philosophia Naturalis 22, 87–103 (1985); “The Concept of Mass,” in R. E. Butts and J. R. Brown, eds., Constructivism and Science (Dordrecht: Kluwer, 1989), pp. 145–162.
30

INERTIAL MASS
of equal length can be veriﬁed geometrically. If these conditions are satisﬁed the two bodies are said to be “tractionally equal,” a relation that can be proved to be an equivalence relation. The transition from this classiﬁcation measurement to a metric measurement is established by a deﬁnition of “homogeneous density”: a body is homogeneously dense if any two parts of it, equal in volume, are tractionally equal, it being assumed, of course, that the equality of volume, as that of length before, has been deﬁned in terms of protophysical geometry. The ability to produce technically homogeneously dense bodies such as pure metals or homogeneous alloys is also assumed. Finally, the mass-ratio mA/mB of two arbitrary bodies A and B is deﬁned by the volume ratio VA/VB of two bodies B and C, provided that C is tractionally equal to A, D is tractionally equal to B, and C and D are parts of a homogeneously dense body. Thus the metrics of mass is reduced to the metrics of volume and length. By assigning logical priority to the notion of density over that of mass Janich, in a sense, “vindicated” Newton’s deﬁnition of mass as the product of volume and density—but of course, unlike Newton, without conceiving density as a primitive concept.61
On the basis of this deﬁnition and measurement of inertial mass, an inertial reference system can be deﬁned as that reference frame relative to which, for example, the conservation of linear momentum in an inelastic collision holds by checking the validity of equation (1.2) all the terms of which are now protophysically deﬁned. Kamlah has shown how Janich’s rope balance, which can also be used for a comparative measurement of masses, is an example of the far-reaching applicability of D’Alembert’s principle.62 This does not mean, however, that Kamlah accepts the doctrine of protophysics. His criticism of the claim that the constructivist measurement-instructions cannot be experimentally invalidated without circularity, though directed primarily against the protophysics of time, applies equally well to the protophysics of mass.63 Friedrich Steinle also criticized Janich’s deﬁnition of mass on the grounds that it yields a new conception of mass and not a purged reconstruction of Newton’s conception because for Newton “mass” and
61 Hence the title “Newton ab omni naevo vidicatus” of Janich’s 1981 essay, in analogy to Gerolamo Saccheri’s 1733 work “Euclides ab omni naevo vindicatus.”
62 A. Kamlah, “Die Bedeutung des d’Alembertschen Prinzips fu¨ r die Deﬁnition des Kraftbegriffes,” in W. Balzer and A. Kamlah, Aspekte der physikalischen Begriffsbildung (Braunschweig: Vieweg, 1979), pp. 191–217.
63 A. Kamlah, “Methode oder Dogma,” Journal for General Philosophy of Science 12, 138– 162 (1981).
31

CHAPTER ONE
“weight,” though proportional to one another, were two independent concepts, whereas, Steinle contends in Janich’s reconstruction this proportionality is part of the deﬁnition.64 It may also be that Janich’s deﬁnition of the homogeneous density of a body can hardly be reconciled with the pragmatic program of protophysics; for to verify that any two parts of the body, equal in volume, are also tractionally equal would demand an inﬁnite number of technical operations.
In all the deﬁnitions of inertial mass discussed so far, whether they have been proposed by protophysicists, by operationalists, or by advocates of any other school of the philosophy of physics, one fact has been completely ignored or at least thought to be negligible. This is the inevitable interaction of a physical object—be it a macroscopic body or a microphysical particle—with its environment. (In what follows we shall sometimes use the term “particle” also in the sense of a body and call the environment the “medium” or the “ﬁeld.”)
Under normal conditions the medium is air. But even if the medium is what is usually called a “vacuum,” physics tells us that it is not empty space. In prerelativistic physics a vacuum was thought to be permeated by the ether; in modern physics and in particular in its quantum ﬁeld theories, this so-called vacuum is said to contain quanta of thermal radiation or “virtual particles” that may even have their origin in the particle itself. Nor should we forget that even in classical physics the notion of an absolute or ideal vacuum was merely an idealization never attainable experimentally.
In general, if a particle is acted upon by a force F, its acceleration a in the medium can be expected to be smaller than the hypothetical acceleration a0 it would experience when moving in free space. However, if a < a0 then the mass m, deﬁned by F/a, is greater than the mass m0, deﬁned by F/a0. This allows us to write m = m0 + δm, where m denotes the experimentally observable or “effective” mass of the particle, m0 its hypothetical or “bare” mass, and δm the increase in inertia owing to the interaction of the particle with the medium.
These observations may have some philosophical importance. Should it turn out that there is no way to determine m0, i.e., the inertial behavior of a physical object when it is not affected by an interaction with a ﬁeld, it would go far toward supporting the thesis that the notion of inertial mass is a theoretical concept. Let us therefore discuss in some detail how
64 F. Steinle, “Was ist Masse? Newton’s Begriff der Materiemenge,” Philosophia Naturalis 29, 94–118 (1992).
32

INERTIAL MASS

such interactions complicate the deﬁnition of inertial mass and lead to

different designations of this notion corresponding to the medium being

considered.

Conceptually and mathematically the least complicated notion of this

kind is the concept of “hydrodynamical mass.” Its history can be traced

back to certain early nineteenth-century theories that treated the ether

as a ﬂuid, and in its more proper sense in the mechanics of ﬂuids to Sir

George Gabriel Stokes’s extensive studies in this ﬁeld.65 However, the

term “hydrodynamical mass” was only given currency in 1953 by Sir

Charles Galton Darwin, the grandson of the famous evolutionist Charles

Robert Darwin.66

In order to understand the deﬁnition of this concept let us consider

the motion of a solid cylinder of radius r moving through an inﬁnite

incompressible ﬂuid, say water or air, of density ρ, with constant velocity

v.

The

kinetic

energy

of

the

ﬂuid

is

Efkin

=

1 2

πρ

r2

v2

and

its

mass

per

unit

thickness is M = πρr2.67 If M denotes the mass of the cylinder per

unit thickness, then the total kinetic energy of the ﬂuid and cylinder is

clearly Ekin

=

1 2

(M

+

M

)v2;

and

if

F

denotes the external force in the

direction of the motion of the cylinder, which sustains the motion, then

the rate at which F does work, being equal to the rate of increase in Ekin,

is given by

Fv = dEkin/dt = (M + M )v dv/dt.

(1.29)

This shows that the cylinder experiences a resistance to its motion equal to M dv/dt per unit thickness owing to the presence of the ﬂuid. Comparison with Newton’s second law suggests that M + M be called the “virtual mass” of the cylinder and the added mass M the “hydrodynamical mass.” It can be shown to be quite generally true that every moving body in a ﬂuid medium is affected by an added mass so that its virtual mass is M + kM , where the coefﬁcient k depends on the shape of the body and the nature of the medium. Clearly the notion of “hydrodynamic mass” poses no special problems because it is formulated entirely within the framework of classical mechanics.

65 G. G. Stokes, “On the Steady Motion of Incompressible Fluids,” Transactions of the Cambridge Philosophical Society 7, 439–455 (1842); Mathematical and Physical Papers, vol. 1 (Cambridge, U.K.: The University Press, 1880), pp. 1–16.
66 C. G. Darwin, “Notes on Hydrodynamics,” Proceedings of the Cambridge Philosophical Society 49, 342–354 (1953).
67 For a rigorous proof see, e.g., L. M. Milne-Thomson, Theoretical Hydrodynamics (London: Macmillan, 1968), pp. 246–247.

33

CHAPTER ONE

Much more problematic is the case in which the medium is not a ﬂuid in the mechanical sense of the term but an electromagnetic ﬁeld whether of external origin or one produced by the particle itself if it is a charged particle such as the electron. Theories about electromagnetic radiative reactions have generally been constructed on the basis of balancing the energy-momentum conservation. But the earliest theory that a moving charged body experiences a retardation owing to its own radiation, so that its inertial mass appears to increase, was proposed by the Scottish physicist Balfour Stewart on qualitative thermodynamical arguments.68 Since a rather detailed historical account of the concept of mass in classical electromagnetic theory has been given elsewhere,69 we shall conﬁne ourselves here to the following very brief discussion.
Joseph John Thomson, who is usually credited with having discovered the electron, seems also to have been the ﬁrst to write on the electromagnetic mass of a charged particle. Working within the framework of James Clerk Maxwell’s theory of the electromagnetic ﬁeld, Thomson calculated the ﬁeld produced by a spherical particle of radius r, which carries a charge e and moves with constant velocity v.70 He found that the kinetic energy of the electromagnetic ﬁeld produced by this charge— this ﬁeld playing the role of the medium as described above—is given by the expression

Eeklimn = ke2v2/2rc2,

(1.30)

where the coefﬁcient k, of the order of unity, depends on how the charge e is distributed in, or on, the particle. Comparing (1.30) with the usual equation for kinetic energy (one-half times mass times velocity squared) Thomson concluded that the charged particle has an electromagnetic mass melm given by

melm = ke2/rc2.

(1.31)

Were the particle uncharged, its kinetic energy would be Ekin = m0/2v2, where m0 is its mechanical inertial mass. Hence, Thomson contended, the total kinetic energy of the charged particle is

68 B. Stewart, “On the Temperature Equilibrium of an Enclosure in Which There Is a Body in Visible Motion,” Reports of the British Association for the Advancement of Science, Edinburgh 187, 45–47 (1871).
69 See chapter 11 in COM. 70 J. J. Thomson, “On the Electric and Magnetic Effects Produced by Motion of Electriﬁed Bodies,” Philosophical Magazine 11, 229–249 (1881).
34

INERTIAL MASS

Etkointal = (m0 + melm)v2/2,

(1.32)

an equation that shows that the experimentally observable mass of the particle is given by

m = m0 + melm.

(1.33)

In agreement with our earlier equation m = m0 + δm, m0 can also be called the bare mass and δm = melm the inertia of the ﬁeld produced and surrounding the charged particle.71
Although Thomson still regarded the increase in inertial mass as a phenomenon analogous to a solid moving through a perfect ﬂuid, subsequent elaborations of the concept of electromagnetic mass, such as those carried out by Oliver Heaviside, George Francis Fitzgerald, and, in particular, by Hendrick Antoon Lorentz, suggested that this notion may well have important philosophical consequences. For, whereas the previous tendency had generally been to interpret electromagnetic processes as manifestations of mechanical interactions, the new conception of electromagnetic mass seemed to clear the way toward a reversal of this logical order, i.e., to deduce mechanics from the laws of electromagnetism. If successful, such a theory would explain all processes in nature in terms of convection currents and their electromagnetic radiation, stripping the “stuff” of the world of its material substantiality.
However, such an electromagnetic world-picture could be established only if it could be proved that m0, the mechanical or bare mass of a charged particle, has no real existence. Walter Kaufmann, whose wellknown experiments on the velocity dependence of inertial mass played an important role in these deliberations, claimed in 1902 that m0, which he called the “real mass” (“wirkliche Masse”)—in contrast to melm, which he called the “apparent mass” (“scheinbare Masse”)—is zero, so that “the total mass of the electron is merely an electromagnetic phenomenon.”72 At the same time, Max Abraham, in a study that can be regarded as the ﬁrst ﬁeld-theoretic treatment of elementary particles, showed that, strictly speaking, the electromagnetic mass is not a scalar

71 For a modern derivation of equation (1.31) see, e.g., W.K.H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Reading, Mass.: Addison-Wesley, 1956), pp. 314–317; or J. Vanderlinde, Classical Electromagnetic Theory (New York: John Wiley and Sons, 1993), pp. 317–319.
72 W. Kaufmann, “Die magnetische und elektrische Ablenkbarkeit der Becquerelstrahlen und die scheinbare Masse der Elektronen,” Go¨ttinger Nachrichten 1902, 143–155; “U¨ ber die elektromagnetische Masse des Elektrons,” ibid., pp. 291–296.
35

CHAPTER ONE
but rather a tensor with the symmetry of an ellipsoid of revolution and proclaimed: “The inertia of the electron originates in the electromagnetic ﬁeld.”73 However, he took issue with Kaufmann’s terminology, for, as he put it, “the often used terms of ‘apparent’ and ‘real’ mass lead to confusion. For the ‘apparent’ mass, in the mechanical sense, is real, and the ‘real’ mass is apparently unreal.”74
Lorentz, the revered authority in this ﬁeld, was more reserved. In a talk “On the Apparent Mass of Ions,” as he used to call charged particles, he declared in 1901: “The question of whether the ion possesses in addition to its apparent mass also a real mass is of extraordinary importance; for it touches upon the problem of the connection between ponderable matter and the ether and electricity; I am far from being able to give a decisive answer.”75 Furthermore, in his lectures at Columbia University in 1906 he even admitted: “After all, by our negation of the existence of material mass, the negative electron has lost much of its substantiality. We must make it preserve just so much of it that we can speak of forces acting on its parts, and that we can consider it as maintaining its form and magnitude. This must be regarded as an inherent property, in virtue of which the parts of the electron cannot be torn asunder by the electric forces acting on them (or by their mutual repulsion, as we may say).”76
It should be recalled that at the same time Henri Poincare´ also insisted on the necessity of ascribing nonelectromagnetic stresses to the electron in order to preserve the internal stability of its ﬁnite charge distribution.77 But clearly, such a stratagem would put an end to the theory of a purely electromagnetic nature of inertial mass. The only way to save it would have been to describe the electron as a structureless point charge, which means to take r = 0. But then, as can be seen from equation (1.30), the energy of the self-interaction and thus the mass of the electron would become inﬁnite. Classical electromagnetic theory has never resolved this problem. As we shall see in what follows, the same problem of
73 M. Abraham, “Die Dynamik des Elektrons,” Go¨ttinger Nachrichten 1902, 20–41. 74 Abraham, “Die Dynamik des Elektrons,” p. 24. 75 H. A. Lorentz, “U¨ ber die scheinbare Masse der Ionen,” Physikalische Zeitschrift 2, 78–79 (1901). 76 H. A. Lorentz, The Theory of Electrons (Leipzig: Teubner, 1909, 1916; New York: Dover, 1952), p. 43. 77 H. Poincare´, “Sur la dynamique de l’e´lectron,” Rendiconti del Circolo Matematico di Palermo 21, 129–176 (1906); Oeuvres de Henri Poincare´, vol. 9 (Paris: Gauthier-Villars, 1954), pp. 494–550.
36

INERTIAL MASS

a divergence to inﬁnity also had to be faced by the modern ﬁeld theory of quantum electrodynamics.
With the advent of the special theory of relativity in the early years of the twentieth century, physicists and philosophers focused their attention on the concept of relativistic mass. Since this notion will be dealt with in the following chapter we shall turn immediately to the quantummechanical treatment of inertial mass but for the time being only insofar as the medium affecting the mass of a particle consists of other particles arranged in a periodic crystal structure. This is a subject studied in the quantum theory of solids or condensed matter and leads to the notion of effective mass. More speciﬁcally, we consider the case of an electron moving under the inﬂuence of an external force F through a crystal.
Let us recall that in accordance with the wave-particle duality in quantum mechanics the electron has to be treated as a wave packet, so that its velocity is given by the equation for the group velocity

vg = v = dω/dk,

(1.34)

where ω denotes the angular frequency and k the wave number. Since its energy E satisﬁes the Einstein energy-frequency relation E = h¯ ω, where h¯ is Planck’s constant h divided by 2π , the velocity of the electron is

v = h¯ −1dE/dk

(1.35)

and its acceleration is

a = dv/dt = (dv/dk)(dk/dt) = h¯ −1(d2E/dk2)(dk/dt). (1.36)

However, in accordance with the work-energy relation Fvdt = dE = (dE/dk)dk, so that by (1.35) F(h¯ −1dE/dk)dt = (dE/dk)dk. Hence,

F = h¯ (dk/dt).

(1.37)

Deﬁning the mass, now called the effective mass and denoted by m∗, in the usual way as the ratio between force and acceleration (F/a), from equation (1.36) we obtain

m∗ = h¯ 2(d2E/dk2)−1.

(1.38)

In fact, if we recall the de Broglie momentum–wave-number relation p = h¯ k and use m∗ in the energy equation E = p2/2m∗, we get

E = h¯ 2k2/2m∗,

(1.39)

which shows that d2E/dk2 = h¯ 2/m∗, which is consistent with the deﬁnition of effective mass.

37

CHAPTER ONE
Obviously, m∗ has a constant value only for energy bands of the form E = E0 ± const. · k2. But even in this case the effective mass may differ from the value of the inertial mass of a free electron. This difference is, of course, to be expected; for in general the acceleration of an electron moving under a given force in a crystal may well differ from the acceleration of an electron that is moving under the same force in free space. What is more difﬁcult to understand intuitively is the fact that, owing to reﬂections by the crystal lattice, an electron can move in a crystal in the direction opposite to that it would have in free space. In this case the effective mass m∗ is negative.78
We conclude this survey with a brief discussion of the concepts of bare mass and experimental or observed mass as they are used in quantum electrodynamics, which, like every ﬁeld theory, ascribes a ﬁeld aspect to particles and all other physical entities and studies, in particular, the interactions of electrons with the electromagnetic ﬁeld or its quanta, the photons.
Soon after the birth of quantum mechanics it became clear that a consistent treatment of the problems of emission, absorption, and scattering of electromagnetic radiation requires the quantization of the electromagnetic ﬁeld. In fact, Planck’s analysis of the spectral distribution of blackbody radiation, which is generally hailed as having inaugurated quantum theory, is, strictly speaking, a subject of quantum electrodynamics.79
Although no other physical theory has ever achieved such spectacular agreement between theoretical predictions and experimental measurements, some physicists, including Paul A. M. Dirac himself, have viewed it with suspicion because of its use of the so-called “renormalization” procedure, which was designed to cope with the divergences of selfenergy or mass, a problem that, as noted above, was left unresolved by classical electromagnetic theory. It reappeared in quantum electrodynamics for the ﬁrst time in 1930 in J. Robert Oppenheimer’s calculation of the interaction between the quantum electromagnetic ﬁeld and an atomic electron. “It appears improbable,” said Oppenheimer, “that the difﬁculties discussed in this work will be soluble without an adequate
78 For details see, e.g., C. Kittel, Introduction to Solid State Physics (New York: John Wiley and Sons, 1953, 1986), chapter 8.
79 For details see M. Jammer, The Conceptual Development of Quantum Mechanics (New York: McGraw-Hill, 1966; enlarged and revised edition, New York: American Institute of Physics, 1989), chapter 3.
38

INERTIAL MASS

theory of the masses of electron and proton, nor is it certain that such a

theory will be possible on the basis of the special theory of relativity.”80

The “adequate theory” envisaged by Oppenheimer took about twenty

years to reach maturity.

As is well known, in modern ﬁeld theory a particle such as an elec-

tron constantly emits and reabsorbs virtual particles such as photons.

The application of quantum-mechanical perturbation theory to such

a process leads to an inﬁnite result for the self-energy or mass of the

electron. (Technically speaking, such divergences are the consequences

of the pointlike nature of the “vertex” in the Feynman diagram of the

process.) Here it is, of course, this “cloud” of virtual photons that plays

the role of the medium in the sense discussed above.

As early as the ﬁrst years of the 1940s, Hendrik A. Kramers, the long-

time collaborator of Niels Bohr, suggested attacking this problem by

sharply distinguishing between what he called mechanical mass, as

used in the Hamiltonian, and observable mass;81 but it was only in the

wake of the famous four-day Shelter Island Conference of June 1947

that a way was found to resolve—or perhaps only to circumvent—

the divergences of mass in quantum electrodynamics. At this confer-

ence Willis E. Lamb reported on the brilliant experiment that he and

Robert C. Retherford had performed using newly invented microwave

techniques, which demonstrated what became known as the Lamb–

Retherford or Lamb shift, namely that the ﬁrst two excited states of

hydrogen, 2s 1 and 2p 1 , are not degenerate but, contrary to Dirac’s

2

2

theory, differ by about 1000 MHz. Perhaps inspired by Kramers’s re-

marks at the conference, Hans Albrecht Bethe realized immediately—

actually during his train ride back from Shelter Island—that the Lamb

shift can be accounted for by quantum electrodynamics if this theory

is appropriately interpreted. He reasoned that when calculating the

self-energy correction for the emission and reabsorption of a photon

by a bound electron, the divergent part of the energy shift can be

identiﬁed with the self-mass of the electron. Hence, in the calculation

of the energy difference for the bound-state levels, as in the Lamb

shift, the energy shift remains ﬁnite since both levels contain the

same, albeit inﬁnite, self-mass terms that cancel each other out in the

80 J. R. Oppenheimer, “Note on the Theory of the Interaction of Field and Matter,” Physical Review 35, 461–477 (1930).
81 See in this context M. Dresden, H. A. Kramers—Between Tradition and Revolution (New York: Springer-Verlag, 1987), chapter 16.
39

CHAPTER ONE
subtraction.82 It is this kind of elimination of inﬁnities, based on the impossibility of measuring the bare mass m0 by any conceivable experiment, that constitutes the renormalization of mass in quantum electrodynamics. A more detailed exposition of the physics of mass renormalization can be found in standard texts on quantum ﬁeld theory,83 and its mathematical features in John Collin’s treatise.84 The reader interested in the historical aspects of the subject is referred to the works of Olivier Darrigol and Seiya Aramaki,85 and the philosopher of contemporary physics to the essays by Paul Teller.86
82 H. Bethe, “The Electromagnetic Shift of Energy Levels,” Physical Review 72, 329–341 (1947); reprinted in J. Schwinger, ed., Selected Papers on Quantum Electrodynamics (New York: Dover, 1958), pp. 139–141.
83 See, e.g., S. Weinberg, The Quantum Theory of Fields (Cambridge: Cambridge University Press, 1995), chapter 12; or M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Reading, Mass.: Addison-Wesley, 1995).
84 J. Collins, Renormalization (Cambridge: Cambridge University Press, 1985). 85 O. Darrigol, Les De´buts de la The´orie Quantique des Champs (Ph.D. Thesis, Universite´ de Paris I, 1982); S. Aramaki, “Formation of the Normalization Theory in Quantum Electrodynamics,” Historia Scientiarum 32, 1–42 (1987), 36, 97–116 (1989), 37, 91–112 (1989). 86 P. Teller, “Three Problems of Renormalization,” in H. R. Brown and R. Harre´, ed., Philosophical Foundations of Quantum Field Theory (Oxford: Clarendon Press, 1998), pp. 73– 89; “Inﬁnite Renormalization,” Philosophy of Science 56, 238–257 (1989).
40

p CHAPTER TWO p
Relativistic Mass

Having confined our attention thus far to the concept of the inertial
mass of classical physics we turn now to its relativistic analogue, the concept of mass in the special theory of relativity. If we ignore for the time being Mach’s principle, which will be discussed in a different context, we can say that in classical physics inertial mass mi is an inherent characteristic property of a particle and, in particular, is independent of the particle’s motion. In contrast, the relativistic mass, which we denote by mr, is well known to depend on the particle’s motion in accordance with the equation

mr = m0(1 − u2/c2)−1/2,

(2.1)

where m0 is a constant with the dimensionality of mass, u is the velocity of the particle as measured in a given reference frame S, and c is the velocity of light. Since u depends on the choice of S relative to which it is being measured, mr also depends on S and is consequently a relativistic quantity and not an intrinsic property of the particle.
In an inertial reference frame S0, in which the particle is at rest, u = 0 and mr obviously reduces to m0. For this reason m0 is usually called the rest mass (or proper mass) of the particle. From a logical point of view, m0 is just a particular case of the relativistic mass and there is not yet any cogent reason to identify it with the Newtonian mass of classical physics. However, as in the so-called nonrelativistic limit, i.e., for velocities that are small compared with the velocity of light (u c), the mathematical equations of special relativity reduce to the corresponding equations of classical physics, many theoreticians regard this correspondence as a warrant to identify m0 with the Newtonian mass of classical physics. However, as we shall see later on, this inference can be challenged—at least on philosophical grounds.
In order to comprehend fully the importance of modern debates on the status of the concept of relativistic mass and its role in physics it seems worthwhile to retrace the historical origins of this concept. Its history is as old as the theory of relativity itself. In his very ﬁrst paper on relativity, the famous 1905 essay, “On the Electrodynamics of Moving

41

CHAPTER TWO

Bodies,”1 Einstein introduced the notion of relativistic mass, though not

in its later accepted form, when he discussed, in the last section of the

essay, the dynamics of a slowly accelerating charged particle.

True, the notion of a velocity-dependent mass and, in particular,

Max Abraham’s conception of longitudinal and transverse masses of

electrons, corresponding to the components of the external force along

or normal to the electron’s trajectory, had been widely discussed even before the theory of relativity was proposed.2 Even equation (2.1) for

the mass of an electron in motion had appeared in the literature prior to 1905.3 However, all these notions and proposals originated within the

framework of theories that were based on speciﬁc assumptions concern-

ing the shape of the electron or the distribution of its charge and were

part of the electromagnetic world-picture, according to which “mass . . .

is of purely electromagnetic nature” and mechanics essentially but a

subdivision of electromagnetism. Thus it should be emphasized that in

spite of the title of Einstein’s ﬁrst relativity paper and regardless of the

importance he attributed to electromagnetic considerations, throughout

that paper, including the derivation of the relativistic equations of mass,

Einstein never did endorse the electromagnetic world-picture nor did

he ever regard mechanics as a subdivision of electromagnetism.

Let us outline brieﬂy—in modern notation—Einstein’s treatment of

the dynamics of a slowly accelerating charged particle in an electromag-

netic ﬁeld and the derivation of his equations of relativistic masses. Let S with coordinates x , y , z , and t be the reference frame in which the

particle is momentarily at rest and thus satisﬁes the equations of motion,

d2x m0 dt 2

= eEx

d2y m0 dt 2

= eEy

d2z m0 dt 2

= eEz,

(2.2)

where e is the charge of the particle, E = (Ex, Ey, Ez) is the electric ﬁeld, and m0 is the mass of the particle, as long as its motion is slow. Using the Lorentz transformation and the relativistic transformation of the

1 A. Einstein, “Zur Elektrodynamik bewegter Ko¨ rper,” Annalen der Physik 17, 891–921 (1905); The Collected Papers of Albert Einstein (Princeton: Princeton University Press, 1989), vol. 2, pp. 276–306. English translation in the Princeton translation project (Princeton University Press, 1989), pp. 140–171; also in A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity (New York: Dover, 1952), pp. 35–65.
2 See chapter 11 of COM. 3 See, e.g., H. A. Lorentz, “Electromagnetic Phenomena in a System Moving with Any Velocity Smaller Than That of Light,” Proceedings of the Academy of Sciences of Amsterdam 6, 809–832 (1904); reprinted in Einstein et al., The Principle of Relativity, pp. 11–34.

42

RELATIVISTIC MASS

components of the electric ﬁeld E = (Ex, Ey, Ez) and the magnetic ﬁeld B = (Bx, By, Bz), previously established in his paper, Einstein derived the equations of motion in a reference frame S relative to which both the particle and the frame S are moving with velocity u along the positive x-axis:

d2x dt2

=

e m0γu3 Ex

d2y dt2

=

e m0γu [Ey

− (u/c)Bz]

d2z dt2

=

e m0γu [Ez

+ (u/c)By],

(2.3)

where γu = (1 − u2/c2)−1/2, or equivalently,

m0γu3

d2x dt2

=

eEx

=

eEx

m0γu2

d2y dt2

=

eγu[Ey

−

(u/c)Bz]

=

eEy

m0γu2

d2z dt2

=

eγu

Ez + (u/c)By

= eEz.

(2.4)

Einstein now argued as follows: since the force that acts on the particle in the reference frame co-moving with the particle is eE and “might be measured, e.g., by a spring balance at rest in this frame,” the equation mass × acceleration = force implies that the longitudinal mass is

m = m0γu3 = m0(1 − u2/c2)−3/2

(2.5)

and the transverse mass is m = m0γu2 = m0 1 − u2/c2 −1 .

(2.6)

Einstein concludes this derivation with two comments: a generalization based on a continuity argument and a qualiﬁcation concerning the terminology. He generalizes his conclusion by extending its validity to uncharged particles on the grounds that these “can be made into charged particles by the addition of an electric charge, no matter how small”; and he qualiﬁes his result by admitting that “with a different deﬁnition of force and acceleration we should naturally obtain other values for the masses.”
This is precisely what happened when, less than a year later, Max Planck proposed a different deﬁnition of force, which turned out to be

43

CHAPTER TWO

more advantageous because it made it possible to establish a HamiltonLagrange formulation for relativistic mechanics.4 Planck showed that equations (2.4) can be written in the form

d dt

(mu)

=

e

[E

+

(1/c)u

×

B]

,

(2.7)

where

m = m0γu = m0(1 − u2/c2)−1/2.

(2.8)

Unlike Planck, who wrote (2.7) as three scalar equations for the different components, we write it as a vector equation in order to show that as a logical consequence of the special theory of relativity, Einstein’s derivation of his equations for relativistic mass also implied the wellknown equation e(E + (1/c)u × B) for the Lorentz force, which until then had to be postulated as a separate axiom added to the Maxwell equations. Furthermore, if, as Newton did, we deﬁne force as the (time) rate of change of momentum and momentum as the product of mass and velocity, then clearly equation (2.7) implies that the relativistic momentum is given by

p = m0γuu

(2.9)

and the relativistic mass by equation (2.8). A new chapter in the history of the concept of relativistic mass began
in 1909 when Gilbert N. Lewis and Richard C. Tolman took exception to the fact that relativistic mechanics had been based on electrodynamics and that, in particular, the relativistic velocity dependence of mass had always been derived by recourse to the theory of the electromagnetic ﬁeld. Convinced of the conceptual autonomy of mechanics, they insisted that the expression for relativistic mass, the most fundamental notion in mechanics, should “be obtained merely from the conservation laws and the principle of relativity, without any reference to electromagnetics.”5
To prove the feasibility of such a procedure they designed a thought experiment in which two identical bodies are assumed to move toward each other with equal velocities, to collide elastically, and then to re-

4 M. Planck, “Das Prinzip der Relativita¨t und die Grundgleichungen der Mechanik,” Verhandlungen der Deutschen Physikalischen Gesellschaft 4, 136–141 (1906); reprinted in: M. Planck, Physikalische Abhandlungen und Vortra¨ge (Braunschweig: E. Vieweg, 1958), vol. 2, pp. 115–120.
5 G. N. Lewis and R. C. Tolman, “The Principle of Relativity and Non-Newtonian Mechanics,” Philosophical Magazine 18, 510–523 (1909).
44

RELATIVISTIC MASS

bound on their original paths in a direction perpendicular to that of

the relative motion of two inertial observers. Applying the principles of

conservation of mass and conservation of momentum and the relativistic

addition theorem of velocities, they derived equation (2.1).6 Three years

later Tolman generalized this method to the case of a “longitudinal

collision” in which, unlike in the “transverse collision,” the two bodies

move toward each other in the same direction as the relative velocity

of the two observers.7 He also broadened his proof to account for “the

general case of any type of collision between any two bodies—elastic or

otherwise.”

For elastic longitudinal collision Tolman proceeded as follows: He

assumed that two identical bodies moving along the x-axis of an inertial

reference frame S with velocities +u and −u are at rest in S at the moment

they collide and then rebound over their original paths with velocities

−u and +u, respectively. If in the reference frame S of another observer,

who moves with a constant velocity v relative to S along the x-axis of S,

the velocities and masses before the collision are denoted, respectively,

by u1 and u2 and m1 and m2, then according to the addition theorem,

u1

=

1

u−v − uv/c2

and

u2

=

−u − v 1 + uv/c2

.

(2.10)

At the moment of the collision, when both bodies are moving in S with

velocity −v, their momentum is −(m1 +m2)v, which by the conservation

principle is equal to the original momentum before the collision. Hence,

− (m1

+ m2)v

=

m1u1

+ m2u2

=

u−v m1 1 − uv/c2

+

m2

−u − v 1 + uv/c2

,

(2.11)

which means that

m1 m2

=

1 − uv/c2 1 + uv/c2

(2.12)

and after a simple algebraic transformation

m1 m2

=

(1 − u22/c2)1/2 (1 − u21/c2)1/2

=

γu1 . γu2

(2.13)

“Remembering that these were bodies that had the same mass m0 when
at rest, we see that the mass of a body is inversely proportional to (1 − u2/c2)1/2, where u is its velocity, and have thus derived the desired

6 For details see chapter 12 of COM. 7 R. C. Tolman, “Non-Newtonian Mechanics: The Mass of a Moving Body,” Philosophical Magazine 23, 375–380 (1912).
45

CHAPTER TWO
relation m = m0(1−u2/c2)−1/2.” Tolman therefore declared emphatically that “the expression m0(1 − u2/c2)−1/2 is best suited for THE mass [sic] of a moving body,”8 Tolman’s method of introducing relativistic mass has been adopted by many authors of textbooks on relativity, among them P. G. Bergmann, M. Born, C. Møller, W.G.V. Rosser, and M. Schwartz, to mention only a few. In his own treatise on relativity, which he dedicated to G. N. Lewis, Tolman introduced the notion of relativistic mass by means of an elastic longitudinal collision, just as he had done in his 1912 easay.9 It was due, at least in part, to the work of Tolman and Lewis that in 1909 the Fortschritte der Physik, the time-honored German equivalent of Science Abstracts, stopped listing papers on relativity under the heading of “Elektrizita¨t und Magnetismus.”
But did Tolman really establish m = m0γu, and thereby relativistic mechanics or, as he called it “non-Newtonian” mechanics, “without any reference to electromagnetics” as he claimed? Does not the very presence of c, the velocity of light, in γu cast some doubt on this claim. The c appears in Tolman’s’derivation because of his use of the relativistic composition theorem of velocities, which is a consequence of the Lorentz transformation, and the latter is, in turn, a consequence of Einstein’s’postulate of the universal invariance of the velocity of light. But light, after all, is an electromagnetic phenomenon, the propagation of electromagnetic waves with the velocity c = (ε0µ0)−1/2, where ε0 is the electromagnetic permissibility and µ0 the electromagnetic permeability of space.
A conceptually rigorous realization of Tolman’s procedure would require divesting c of its electromagnetic connotations by conceiving it, for instance, as the maximum velocity attainable in mechanics in agreement with the divergence of m0γu to inﬁnity for u = c. However, there is a better alternative, which follows from a remarkable, but little known, study by Basil V. Landau and Sam Sampanthar, who showed that c can be introduced as a constant of integration.10 The assumptions that these mathematicians postulate are these: (1) the mass of a particle depends somehow on its speed; (2) conservation of mass; (3) conservation of momentum; and (4) some very general conditions, such as the isotropy of space, assumptions about velocities of frames of reference S, S , and
8 Tolman, Philosophical Magazine 23, 376 (1912). 9 R. C. Tolman, Relativity, Thermodynamics, and Cosmology (Oxford: Clarendon Press, 1934), pp. 43–45. 10 B. V. Landau and S. Sampanthar, “A New Derivation of the Lorentz Transformation,” American Journal of Physics 40, 599–602 (1972).
46

RELATIVISTIC MASS

S in uniform motion relative to each other, and the assumption that the functions encountered are differentiable.
They ﬁrst introduce a velocity composition operation ⊕, which is so deﬁned that if v is the velocity of S relative to S and u is the velocity of S relative to S , then v ⊕ u is the velocity of S relative to S, and show that these relative velocities form an abelian group under this operation. This enables them to associate with every velocity u a real number, called the pseudovelocity, denoted by the corresponding capital letter U, such that whenever v⊕u = w, then V +U = W , or in terms of a function g, deﬁned by u = g(U), g(V) + g(U) = g(V + U). A simple argument, based on considerations of a particle coalescing at almost the same speed shows that assumption (1) can be expressed in the form

m = m0f (U),

(2.14)

where f (U) is still an unknown function of U but is equal to unity for u = 0. Since for u = 0 the mass m equals m0, m0 is the rest mass of the particle. A thought experiment in which a particle of rest mass M0 at rest in S disintegrates symmetrically into two particles, each of rest mass m0 and pseudovelocity +V or −V, respectively, shows that (1) and (2) imply

M0 = 2m0f (V)

(2.15)

and that f is an even function. In S , where m0 has the pseudovelocity U, the pseudovelocities of the daughter particles are U + V and U − V, respectively, so that (2) results in

M0f (U) = m0f (U + V) + m0f (U − V)

(2.16)

or from equation (2.15)

2f (V)f (U) = f (U + V) + f (U − V).

(2.17)

Differentiating twice with respect to V and putting V = 0 yields the differential equation

f (0)f (U) = f (U)

(2.18)

and its solution

f (U) = cosh u.

(2.19)

Postulate (3) applied to S gives

M0f (U)g(U) = m0f (U + V)g(U + V) + m0f (U − V)g(U − v), (2.20)

47

CHAPTER TWO

which, by virtue of (2.15) and (2.19), becomes

2m0 cosh V cosh Ug(U) = m0 cosh (U + V)g(U + V) + cosh (U − V)g(U − V).

(2.21)

Differentiating twice again with respect to V and putting V = 0 yields the differential equation

2 sinh Ug (U) + cosh Ug (U) = 0

(2.22)

and its solution

c = g(U) = c tanh U,

(2.23)

where c is a constant of integration. Finally, from equations (2.14), (2.19), and (2.23) it follows that

m = m0f (U) = m0 cosh U = m0 cosh (tanh −1u/c) = m0(1 − u2/c2)−1/2

(2.24)

or

m = m0γu.

(2.25)

Equation (2.25) provides the physical interpretation of the constant of integration c. As the mass value m of a particle is a real number if and only if

|u| < |c|,

(2.26)

c signiﬁes the upper limit of possible velocities of massive particles. Within the present context, the fact that this upper limit happens to coincide with the velocity of electromagnetic waves (or light) in vacuo remains a mystery.
Undoubtedly, Lewis and Tolman would have welcomed this result had they been alive in 1972.11 Landau and Sampanthar did not mention the fact that their derivation of m = m0γu closed the gap that had interfered with the complete realization of Tolman’s work. They considered

11 Lewis died in 1946, and Tolman in 1948. Only a few years after their deaths W. Macke showed in a remarkable but little-known paper, “Begru¨ ndung der speziellen Relativita¨tstheorie aus der Hamiltonschen Mechanik,” Zeitschrift fu¨ r Naturforschung 7a, 76–78 (1952), that the Hamiltonian canonical formalism, which includes energy and time, leads to a velocity-dependent mass and, provided that the limiting velocity is identiﬁed with the velocity of light, to the Lorentz transformations in compliance with Tolman’s program.
48

RELATIVISTIC MASS

this derivation only as a prelude to their main objective, which was the derivation of the Lorentz transformations from the equation for relativistic mass and the conservation laws for mass and momentum.
As this work is not germane to our present concern, we will describe the way in which it was carried out only brieﬂy. The analysis of a symmetric disintegration of a particle into two fragments with respect to two different inertial reference frames, combined with the relativistic mass equation, led to the relativistic composition rule of velocities. This rule implied that the Galilean transformation had to be replaced by another transformation, which from the assumption that it transforms a uniform motion along a straight line in one reference frame into the same kind of motion in the other frame, turned out to be the Lorentz transformation.
The fact that the Lorentz transformation and the relativistic mass equation mutually imply one another seems to indicate that the relation between these two is more intimate than commonly thought. Indeed, we shall show that the equation m = m0γu is a direct consequence of the Lorentz transformation without recourse to any collision experiments or other auxiliary devices. Since the Lorentz transformations transform four-vectors, such as the space-time position four-vector, X = (x0 = ct, x1 = x, x2 = y, x3 = z) = (x0, x), of an inertial reference frame S into a four-vector such as X = (x0, x ) of another reference frame S , it is clear that the formalism we have to use is that of four-vectors. We assume, of course, that the mass of a particle, as measured in a reference frame, may depend on the particle’s velocity relative to this frame and that the particle’s rest mass m0 is its mass as measured in a frame in which the particle is at rest. We denote the Lorentz transform of any quantity q by q . Let

P = (cq0 = p0, px = mux, py = muy, pz = muz) = (p0, p) (2.27)
be a four-vector in S, where m is the mass of the particle in S, ux, uy, uz are the components of the velocity u of the particle in S, and q0 is an as yet uninterpreted quantity subject to the condition that P transforms like a four-vector. For a particle moving with velocity u = ux = 0 along the x-axis of S the four-vector P reduces to

P = (cq0, mu, 0, 0).

(2.28)

In an inertial frame S , in standard conﬁguration with S and with its origin attached to the particle, the particle’s mass, according to the assumptions we made above, is

49

CHAPTER TWO

and P transforms into

m = m0

(2.29)

P = (cq0 = p0, px, py, pz) = (cq0, 0, 0, 0),

(2.30)

where px is given by

px = γu(px − uq0)

(2.31)

in accordance with the Lorentz transformation x = γu(x − ut). Hence by (2.28) and (2.30)

0 = γu(mu − uq0)

(2.32)

or, since u = 0,

q0 = m

(2.33)

and therefore

Furthermore,

q0 = m = m0.

(2.34)

q0 = γu[q0 − (u/c2)px]

(2.35)

in accordance with the Lorentz transformation t = γu[t − (u/c2)x]. Hence by (2.28), (2.33), and (2.34)

m0 = γu(m − mu2/c2) = mγu−1

(2.36)

or

m = m0γu.

(2.37)

It will have been noted that only the Lorentz transformations have been used in this derivation of (2.37). As the special theory of relativity is characterized by invariance under the Lorentz (or rather Poincare´) group, this derivation of (2.37) seems to support Tolman’s designation of the relativistic mass as the mass of a particle.
Yet, particle physicists generally ignore the notion of relativistic mass and, as a rule, use only the concept of the velocity-independent mass m0, which they measure in units of MeV/c2 in accordance with the mass-energy relation, usually symbolized by the equation E = mc2. This relation will be dealt with in detail only in chapter 3, but we ﬁnd it appropriate to refer to it in the present context insofar as it is relevant to the notion of relativistic mass.

50

RELATIVISTIC MASS
First of all, it will have been noted that the four-vector P as deﬁned in (2.27) is precisely the relativistic momentum four-vector usually deﬁned as the product of m0 and the four-velocity U, with U deﬁned as the derivative of the space-time position four-vector X with respect to the invariant proper time τ , i.e.,

P = (p0, p) = m0U = m0dX/dτ.

(2.38)

In the nonrelativistic limit, where γu −→ 1, expansion of cp0 = c2q0 or, by

(2.33),

expansion

of mc2

gives

mc2

=

m0c2(1−u2/c2)−1/2

=

m0

c2

+

1 2

m0

u2

+

terms of higher order in u.

Since

1 2

mu2

is

the

classical

kinetic

energy

of

the

particle

to

which

the relativistic kinetic energy should reduce in this limit, the relativ-

istic kinetic energy is deﬁned by Ekin = mc2 − m0c2, the rest energy by E0 = m0c2, and the total energy of the particle by E = e0 + Ekin = mc2.

The preceding remarks concerning the mass-energy relation have

been referred to, in anticipation of chapter 3, because of the role they

have played in what has probably been the most vigorous campaign ever

waged against the concept of relativistic mass. In 1989, Lev Borisovich

Okun, a prominent particle physicist known for his work on weak inter-

actions, published some essays in which he emphatically declared that

“in the modern language of relativity there is only one mass, the New-

tonian mass m, which does not vary with velocity,” and “there is only one

mass in physics which does not depend on the reference frame.”12 Okun

blamed all those who, like Tolman or Wolfgang Pauli, distinguished

between “rest mass” and “relativistic velocity-dependent mass” and

caused thereby widespread confusion that has marred even the “most

serious monographs on relativistic physics.” Okun maintained that the

main reason for this confusion was the popular expression of Einstein’s

mass-energy relation given by E = mc2.

In order to illustrate the widespread extent of this confusion even

among professional physicists Okun reports on an opinion poll that he

conducted among his colleagues at the Moscow Institute for Theoretical

and Experimental Physics. In this poll he presented the following four

equations:

12 L. B. Okun, “The Concept of Mass (Mass, Energy, Relativity),” Uspekhi Fisicevskikh Nauk 158, 511–530 (1989). Soviet Physics Uspekhi 32, 629–638 (1989). “The Concept of Mass,” Physics Today 42, 31–36 (June 1989).
51

(I) E0 = mc2 (IV) E = m0c2

CHAPTER TWO
(II) E = mc2 (III) E0 = m0c2

(2.39)

and asked the following two questions:

(Q1) Which of these equations most rationally follows from special relativity and expresses one of its main consequences and predictions?
(Q2) Which of these equations was ﬁrst written by Einstein and was considered by him a consequence of special relativity?

As Okun recounts it, most of his colleagues opted for equations (II) or (III) as the answer to both questions and not for equation (I), which according to Okun is the only correct answer to both. To prove his contention Okun refers to the two fundamental equations of special relativity: to the energy-momentum four-vector equation

E2 − p2m2 = m2c4,

(2.40)

in which each side is a scalar and m is the ordinary mass, “the same as in Newtonian mechanics,” and to the equation for the momentum

p = uE/c2.

(2.41)

Since for u = 0, Okun continues, equation (2.41) yields p = 0 and E becomes the rest energy E0, equation (2.40) reduces to E0 = mc2, i.e., equation (I), where of course, in accordance with Okun’s above quoted declaration, m denotes the ordinary Newtonian mass. For “as soon as you reject the ‘relativistic mass’ there is no need to call the other mass the ‘rest mass’ and to mark it with the index 0.” Okun then asks the following question: if the notation m0 and the term “rest mass” have to be rejected, why should the notation E0 and the term “rest energy” be retained? His answer is: “because mass is a relativistic invariant and is the same in different reference systems, while energy is the fourth [timelike] component of a four-vector (E, p) and is different in different reference systems. The index 0 in E0 indicates the rest system of the body.”
As we shall see in what follows, Okun’s position on this issue can well be defended and is, in fact, very similar to that adopted by Edwin F. Taylor and John Archibald Wheeler in their inﬂuential text Spacetime Physics, which will be referred to in due course. However, the answer he gives to his second question is more problematic. As this question is of an historical nature, it can be interpreted in two different ways. If it asks which of the four equations (I) to (IV) did Einstein write in

52

RELATIVISTIC MASS

his “ﬁrst” (1905) paper on the mass-energy relation, the answer, as we shall see in chapter 3, is “none.” If it asks which of these four equations did Einstein write when he expressed this relation for the “ﬁrst” time in the form of an equation, and not in words as he had done in his early papers on this issue, the answer is equation (I), but written in the notation µv2 = ε0, in a footnote on p. 425 of his 1907 essay “U¨ ber die vom Relativita¨tsprinzip geforderte Tra¨gheit der Energie.” Okun’s answer that “Einstein formulated the famous mass-energy relation in the second of his 1905 papers on relativity in the form E0 = mc2,” though conceptually correct, is not found in that paper in this mathematical formulation. More details and references on Einstein’s treatment of the mass-energy relation will be presented in chapter 3.
It is instructive to compare Okun’s argument in favor of equation (I) with the counterargument offered by the proponents of the notion of relativistic mass and equation (II) with m being the relativistic mass. They start with the above statement that the total energy of a particle is the sum of its rest energy and its kinetic energy, the work done on the particle from its position of rest. They then show that the latter satisﬁes the equation dEkin = d(mγuc2), where m denotes the Newtonian mass and γu stands for (1−u2/c2)−1/2. Since u = 0 implies Ekin = 0, integration yields Ekin = mc2(γu − 1) = mrc2 − mc2 and E0 = mc2, which mr denotes the relativistic mass. Finally, E = E0 + Ekin implies

E = mrc2,

(2.42)

which is, of course, equation (II) with m interpreted as mr. Let us also point out that Tolman’s approach was adopted by many
authors of the earlier textbooks on relativity. Thus, for example, in his inﬂuential treatise on relativity Max Born using conservation of momentum in the case of an inelastic collision concluded that it is impossible to “retain the axiom of classical mechanics that mass is a constant quantity peculiar to each body.” Rather, he wrote, “mass is to have different values according to the system of reference from which it is measured, or, if measured from a deﬁnite system of reference, according to the velocity of the moving body.”13 This point of view is diametrically opposed to that of those who reject the legitimacy of mr on the grounds that it is objectionable that the mass of a particle decreases or increases for no

13 M. Born, Die Relativita¨tstheorie Einsteins und ihre physikalischen Grundlagen (Berlin: J. Springer, 1920, 1922, 1964); Einstein’s Theory of Relativity (New York: Dover, 1962, 1965), p. 269.
53

CHAPTER TWO
physical reason, merely by being observed from different perspectives. Moreover, alluding to the early notions of longitudinal and transverse mass,14 they claim that “no unique dependence of mass on velocity follows from the mechanics of special relativity” and that it would be unreasonable to assume that the mass of a particle, supposed to be an inherent property, should depend on purely geometrical details such as the spatial direction of the force or the acceleration of the moving particle.15
We shall not give a detailed account of the heated debate pro and contra mr that has been going on for the last two or three decades but shall conﬁne our discussion to a few brief comments. First of all, textual evidence shows that the use of four-vectors for the presentation of relativity does not enforce any preference in this matter. Thus Joseph Aharoni, who develops relativistic dynamics in four-vector notation, writes: “the theory of relativity forces us to the conclusion that what is regarded in the classical theory of mass cannot be assumed (as is done in the classical theory) to be independent of velocity.”16 In contrast, Robert W. Brehme17 and Andrew Whitaker,18 who regard the four-vector calculus as the “clearest and simplest” way of thinking, reject mr on the grounds that “it gives the impression that the effects of relativity are due to ‘something happening’ to the particle, whereas they are of course due to the properties of space-time.”
Still, there has been a general tendency in recent years to dispense with mr. Thus, as Carl G. Adler noted,19 a widely used textbook ascribed in its earlier editions (1963) to the concept of relativistic mass “the greatest importance when dealing with atomic and subatomic particles,” but in its later editions (1976, 1980) describes the very same concept as “misleading” and “not necessary” at all.20
14 See equations (15) and (16) in chapter 12 of COM. 15 V. L. Ginzburg, “Who Developed the Theory of Relativity, and How?,” in V. A. Ugarov (ed.), Special Theory of Relativity (Moscow: Mir, 1979), p. 352. 16 J. Aharoni, The Theory of Relativity (Oxford: Clarendon Press, 1959), p. 140. 17 R. W. Brehme, “The Advantage of Teaching Relativity with Four-Vectors,” American Journal of Physics 36, 896–901 (1968). 18 M.A.B. Whitaker, “Deﬁnition of Mass in Special Relativity,” Physics Education 11, 55–57 (January 1976). 19 C. G. Adler, “Does Mass Really Depend on Velocity, Dad?,” American Journal of Physics 55, 739–743 (1987). 20 F. W. Sears and M. W. Zemansky, University Physics (Reading, Mass.: Addison–Wesley, 1963, 1970, 1976, 1980, 1982).
54

RELATIVISTIC MASS
According to Taylor and Wheeler the root of this controversy lies in the fact that the term “mass” is being used in two different connotations— once in the sense of the invariant (scalar) magnitude of the energymomentum four-vector P = (E/c, p) divided by c2, i.e., m = E2 − p2c2 /c2, and once as the time component of this very same four-vector, i.e., mr = E/c2. Taylor and Wheeler discourage the use of mass in the latter sense because it leads to the erroneous belief that the increase in the energy, alias “mass,” of a particle with velocity or momentum results from some change in the internal structure of the particle and not in the geometric properties of space-time itself.21
More recently, Okun’s polemic condemnation of mr gave rise to an animated debate in a series of “Letters” in the May 1990 issue of Physics Today. While Michael A. Vanyck, for example, fully endorses Okun’s rejection of mr and suggests even further revisions in this spirit, Wolfgang Rindler declares: “Okun’s earnest tirade against the use of the concept of relativistic mass” is harmful for the understanding of relativity. Further, he adds, “to me, mr is a useful heuristic concept. It gives me a feeling for the magnitude of the momentum p = mru at various speeds. The formula E = mrc2 reminds me that energy has masslike properties such as inertia and gravity, and it tells me how energy varies with speed.”22 In another article, written in 1991, Thomas R. Sandin defends mr even on aesthetic grounds because “relativistic mass paints a picture of nature that is beautiful in its simplicity” and its elimination would be “a form of unnecessary censorship.”23
Although, as noted above, the general trend, especially in the literature on elementary particle physics, is toward the elimination of mr, there are quite a few exceptions, mainly in the textbook literature. Thus, for instance, Richard A. Mould in his recently published text on relativity argues strongly against the belief that only rest mass should be admitted. Although he acknowledges the importance of rest mass because of its invariance under coordinate transformations, he recommends using relativistic mass as well because “it retains the gravitational and inertial properties long associated with mass, just as energy retains its familiar association with work-related activity.”24 In order to reinforce
21 E. F. Taylor and J. A. Wheeler, Spacetime Physics (San Francisco: Freeman, 1963, 1966); see in particular Table 14: Uses and abuses of the concept of mass, pp. 134–137.
22 “Putting to Rest Mass Misconceptions,” Physics Today 43, 13–15, 115–119 (May 1990). 23 T. R. Sandin, “In Defense of Relativistic Mass,” American Journal of Physics 59, 1032– 1036 (1991). 24 R. A. Mould, Basic Relativity (New York: Springer-Verlag, 1994), p. 119.
55

CHAPTER TWO
his position he illustrates it in terms of a photon gas, which has a rest mass equal to zero but, contrary to what is commonly thought, is not weightless. “Its passive gravitational mass is equal to its relativistic mass (which equals its total energy i hνi/c2), so that when it is placed on a scale in a gravitational ﬁeld g its weight is equal to i hνi/c2 × g. Furthermore, if the gas is accelerated horizontally, it will display inertial properties also equal to i hνi/c2, even at nonrelativistic accelerations.” The use of mr is therefore fully justiﬁed.
That the crux of this controversy is not a matter of aesthetic simplicity, terminological convention, or practical applicability but rather, as Taylor and Wheeler intimated, the result of different mathematical approaches has recently been argued by R. Paul Bickerstaff and George Patsakos.25 As they point out, a quantity that is an invariant in the nonrelativistic limit of the Lorentz transformations can be generalized in the relativistic realm to two quantities with different tensorial characters. The bestknown example, though not mentioned by them, is the concept of time in classical physics: with respect to the nonrelativistic Galilean transformation it is an invariant; but if generalized relativistically it becomes either the scaler “proper time τ ,” or alternatively the zeroth component (divided by c) of the space-time four-vector (ct = x0, x1, x2, x3). Analogously, the authors claim, the classical (Newtonian) notion of mass generalizes either to the scaler “rest mass m” or alternatively to the zeroth component mr = E/c2 of the momentum four-vector. In fact, the well-known equations dt = γvdτ and mr = γum manifest this analogy in a conspicuous way, which suggests calling τ “the rest time” and m “the proper mass,” as Arthur S. Eddington actually did.26 From the mathematical point of view both sides of the controversy can be equally well defended, provided the two generalizations are equally maintainable, and it is at this point that philosophical considerations come into play.
To understand this issue we have to recall that until not so long ago philosophers regarded the development of science as a linear continuous process of ever-increasing accumulation of knowledge. Even far-reaching innovations in so-called “scientiﬁc revolutions” were ultimately, according to this view, only results of articulations and exten-
25 R. P. Bickerstaff and G. Patsakos, “Relativistic Generalization of Mass,” European Journal of Physics 16, 63–68 (1995).
26 A. S. Eddington, The Mathematical Theory of Relativity (Cambridge: Cambridge University Press, 1924, 1965), p. 30.
56

RELATIVISTIC MASS
sions of existing theories. In the 1960s this so-called “Received View” was challenged by Thomas S. Kuhn, Paul K. Feyerabend, and others, who claimed that the development of science is a sequence of disconnected different canons of scientiﬁc thought, inﬂuenced to a great extent by external factors.27 The various stages in this sequence are characterized by what Kuhn calls “paradigms” (or later “disciplinary matrices”), which are “universally recognized scientiﬁc achievements that for a time provide model problems and solutions to a community of practitioners.” To adopt a new theory or paradigm means to accept a completely novel conceptual scheme that has so little in common with that of the older, now rejected, theory that the two theories are “incommensurable,” for no objective yardstick exists that makes it possible to compare them. Furthermore, as the meaning of every scientiﬁc term in a given theory depends upon the theoretical context in which it occurs, even the individual scientiﬁc terms of the new theory are incommensurable with the terms of the old one, despite the fact that the same terminology is often retained. Any meaninginvariance even of homonymous terms of different theories is therefore strictly denied.
Two of the most frequently quoted incommensurable terms are the “classical (Newtonian) mass” and the “relativistic rest mass.” Thus, e.g., according to Feyerabend “the attempt to identify classical mass with relative [i.e., relativistic] rest mass” cannot be made because these terms belong to incommensurable theories.28 In another context he says, “That the relativistic concept and the classical concept of mass are very different indeed becomes clear if we consider that the former is a relation, involving relative velocities, between an object and a coordinate system, whereas the latter is a property of the object itself and independent of its behavior in coordinate systems.”29
The thesis of the incommensurability of the classical and the relativistic notions of mass can be defended not only on philosophical grounds but also by physical arguments. It can be argued, following Erik Eriksen
27 T. S. Kuhn, The Structure of Scientiﬁc Revolutions (Chicago: University of Chicago Press, 1962, 1970). P. K. Feyerabend, Problems of Empiricism—Philosophical Papers, vol. 2 (Cambridge: Cambridge University Press, 1981).
28 According to Feyerabend, Problems of Empiricism, “two theories will be called incommensurable when the meanings of their main descriptive terms depend on mutually inconsistent principles.”
29 P. K. Feyerabend, “Problems of Empiricism,” in R. G. Colodny, Beyond the Edge of Certainty (Englewood Cliffs, N.J.: Prentice-Hall, 1965), p. 169.
57

CHAPTER TWO

and Kjell Vøyenli, that in particular it would be wrong to regard, as Okun

does, the relativistic rest mass as the only legitimate notion of mass and

as identical with the classical notion of mass and that, instead, both the

classical and the relativistic concepts of mass have to be acknowledged each in its own right.30

The argument is based on the principle of conservation of momentum

and, in the classical case, on the Galilean transformation and, in the

relativistic case, on the Lorentz transformation. In both cases, masses

are implicitly deﬁned by those constant positive quantities mj that in a collision of n incoming particles with velocities u1, . . . , un and p outgoing particles with velocities un+1, . . . , un+p, relative to a reference frame S, satisfy the equation

n

n+p

mjuj =

mk uk ,

j=1

k=n+1

(2.43)

where the total number of particles n + p, but in the relativistic case not necessarily n and p separately, is assumed to be invariant.31 In the special case n = 2 and p = 1, so that

m1u1 + m2u2 = m3u3

(2.44)

measurement of the velocities, assumed to be not parallel, obviously determines the mass-ratios, e.g., m1/m3.
Equation (2.43) can also be written in the form

n+p
εjmjuj = 0,
j=1

(2.45)

where εj = +1 for j = 1, . . . , n, i.e., for incoming particles, and εj = −1 for j = n + 1, . . . n + p, i.e., for outgoing particles.
In order to ﬁnd out how the mass-ratios measured in S are related to the mass-ratios measured in a reference frame S that is moving with velocity v relative to S, we have to distinguish between the classical and the relativistic case. Quantities with a prime ( ) will refer to S .

30 E. Eriksen and K. Vøyenli, “The Classical and Relativistic Concepts of Mass,” Foundations of Physics 6, 115–124 (February 1976).
31 Eriksen and Vøyenli consider not only ordinary particles, i.e., so-called tardyons (with velocity u < c), but also luxons (u = c) and tachyons (u > c). As is well known, whether a tachyon is an incoming or an outgoing particle depends on the reference frame. We conﬁne our discussion to tardyons.
58

RELATIVISTIC MASS

In classical physics, let equation (2.45) be valid in S and

εjmj uj = 0

(2.46)

be valid in S . Since according to the Galilean transformation uj = uj + v, we obtain from equation (2.45)

εjmjuj = −v εjmj.

(2.47)

In the case of three particles, equation (2.46) shows that u1, u2, and u3 are linearly dependent and therefore deﬁne a plane. Hence, the left-hand
side of equation (2.47) is a vector in this plane. Since v can be chosen not
to lie in this plane equation (2.47) splits into the two equations

εjmjuj = 0

(2.48)

εjmj = 0,

(2.49)

the second of which expresses the conservation of mass. Equations (2.46) and (2.48) show that

mj = ηmj,

(2.50)

where η is a constant for all particles. Equation (2.49) guarantees the invariance of the mass-ratios. Hence, the selection of a certain particle as unit mass in every reference frame determines that the mass of every particle is an invariant.
In relativistic physics, where m now stands for the relativistic mass, formerly denoted by mr, replacement of the Galilean by the Lorentz transformation changes equation (2.47) into

εjmj(1 − v · uj/c2)uj = −v εjmj,

(2.51)

which again implies that each side equals zero. Again, the equation

εjmj = 0

(2.52)

expresses the conservation of mass. Correspondingly, equation (2.50) has to be replaced by the general mass transformation equation

mj = η(1 − v · uj/c2)mj,

(2.53)

where η is the same constant for all particles. If γv denotes (1 − v2/c2)−1/2 and γu and γu denote the corresponding quantities, the Lorentz transformation leads to

γv(1 − v · u/c2)γu−1 = γu−1

(2.54)

59

CHAPTER TWO

and equation (2.53) reads

mj γu−1 = (η/γv)mj γu−1.

(2.55)

By virtue of equation (2.55) the invariant mass m0j of the jth particle, deﬁned by m0j = mjγu−1, satisﬁes the equation

m0j = (η/γv)m0j,

(2.56)

which shows the invariance of the mass-ratios. Again, the selection of a certain particle as unit invariant mass in every reference frame determines the invariant mass of every particle and (2.56) implies that

η = γv.

(2.57)

Equations (2.53), (2.55), and (2.56) now read

mj = γv(1 − v · uj/c2)mj

(2.58)

mj γu−1 = mjγu−1

(2.59)

m0j = m0j

(2.60)

and show that, unlike the mass mj, the mass m0j is an invariant and that the invariant mass equals the mass in the rest frame of the particle. Since according to equation (2.52) the sum of the relativistic masses mj is conserved and the sum of the rest masses m0j is not, whereas according to equation (2.49) the sum of the classical masses mj is conserved, it would be wrong to identify the rest mass of a particle with its classical mass. Further, according to equation (2.53) the relativistic mass-ratios are not invariant, whereas in accordance with equation (2.49) the classical massratios are, so it would of course also be wrong to identify the relativistic mass with the classical mass. “At this stage one might think that the three concepts of mass are three different physical quantities that may be dealt with on an equal footing. This would, however, be another misconception. The relativistic and the classical concepts of mass are intimately associated with two contradictory theories that deal with the same subject matter. Hence the classical and relativistic concepts are rival, contradictory concepts.”32 These words are obviously only a restatement of the incommensurability thesis described above.
Those who consider the new theory a generalization or extension of the old one so that the new has a range of applicability that includes

32 Eriksen and Vøyenli, Foundations of Physics 6, 123–124 (February 1976).
60

RELATIVISTIC MASS
that of the old, in agreement with the mathematical generalizations noted by Bickerstaff and Patsakos, clearly reject the incommensurability doctrine. So do, in particular, those who regard Newtonian mechanics as the low-velocity limit of relativistic mechanics, and so certainly do those who, like Okun, declare that “there is only one mass in physics, m, which does not depend on the reference frame,” and that m in the equation E2 − p2c2 = m2c4 “is the ordinary Newtonian mass,” or even more explicitly, that “the mass of a body . . . is the same, in the theory of relativity and in Newtonian mechanics.”33
On the other hand, those who like Tolman regard relativistic mechanics as a “non-Newtonian” theory, established on principles independent of classical physics and declare that “mr is THE mass” in relativity, obviously endorse the incommensurability doctrine, even if they are not aware of it. Our analysis of the m vs. mr debate thus leads us to the conclusion that the conﬂict between these two formalisms is ultimately the disparity between two competing views of the development of physical science.
33 Okun, Physics Today 42, 31–36 (June 1989).
61

p CHAPTER THREE p
The Mass-Energy Relation
It is certainly no exaggeration to say that the mass-energy relation,
usually symbolized by E = mc2, is one of the most important and empirically best conﬁrmed statements in physics. Although initially conceived as a purely theoretical theorem without any practical applications, E = mc2 eventually became the symbol that marks the beginning of a new era in the history of civilization—the age of nuclear energy with its promises and dangers for the human race. As we are interested in this relation only within the context of our study of the notion of mass, we ignore all these far-reaching implications and focus our attention on the conceptual issues involved. We have to admit, however, that because of its epoch-making consequences the discovery of the massenergy relation is itself an important event in the history of physics. It is therefore interesting to note that the very ﬁrst proof of this relation— Einstein’s 1905 derivation—has been criticized as being a logical fallacy involving a vicious circle.
The ﬁrst to claim that “the reasoning in Einstein’s 1905 derivation of the mass-energy relation is defective” was Herbert E. Ives.1 Ives’s claim described in chapter 13 of Concepts of Mass was recently rejected as unjustiﬁed, but had enjoyed rather widespread endorsement.2 The alleged circularity in Einstein’s reasoning was even interpreted as indicative of his genius when it was said: “Ives has shown (beyond any doubt) that this [Einstein’s] derivation is circular. That is, Einstein implicitly postulates the energy-mass relation in his proof. This may be in a way a tribute to Einstein’s genius, for he seems to intuitively know answers before he derives them.”3
1 H. E. Ives, “Derivation of the Mass-Energy Relation,” Journal of the Optical Society of America 42, 540–543 (1952).
2 See, e.g., H. Arzelie`s, E´ tudes Relativistes: Rayonnement et Dynamique du corpuscule charge´ fortement acce´le´re´ (Paris: Gauthier-Villars, 1966), pp. 74–79; A. Miller, Albert Einstein’s Special Theory of Relativity (Reading, Mass.: Addison-Wesley, 1981), p. 377; U. E. Schro¨ der, Spezielle Relativita¨tsthoerie (Thun: H. Deutsch, 1981), p. 118; K. J. Ko¨ hler, “Die Aequivalenz von Materie und Energie,” Philosophia Naturalis 19, 315–341 (1982); C. A. Zapffe, A Reminder on E = mc2 (Baltimore: CAZLab, n.d.), p. 46.
3 A. F. Antippa, “Variations on a Photon-in-a-Box by Einstein,” UQTR-TH-8 (Quebec: Universite´ du Que´bec a` Trois-Rivie`res, May 1975), pp. 1–52.
62

THE MASS-ENERGY RELATION

In order to understand the origin of the circularity claim we shall brieﬂy review, for the convenience of the reader, Einstein’s ﬁrst derivation of the mass-energy relation (in the notation of chapter 13 of COM).4
A body B at rest in an inertial frame S and of initial energy content E0 is supposed to emit two equal quantities of radiant energy in opposite directions, each of amount E/2, so that it remains at rest with decreased energy content E1. Energy conservation requires that

E0 = E1 + E.

(3.1)

Let E0 and E1 be the energies of B before and after the emission, respectively, as measured in a reference frame S that is moving relative to S with a constant velocity v in a direction making an angle φ with the direction of the emitted radiation. From the relativistic transformation equation of radiant energy (proved in Einstein’s very ﬁrst paper on relativity) and the energy conservation principle it follows that

E0

=

E1

+

1 2

Eγv[1

+

(v/c)

cos

φ]

+

1 2

Eγv[1 − (v/c) cos φ], (3.2)

where γv = [1 − v2/c2]−1/2. Hence, by subtraction,

(E0 − E0) − (E1 − E1) = E(γv − 1).

(3.3)

Since E0 − E0 and E1 − E1 are differences in “the energy values of the same body referred to two reference systems moving relatively to each
other, the body being at rest in one of the two systems . . . it is clear that the difference E − E [i.e., E0 − E0 and E1 − E1] can differ from the kinetic evergy T [i.e., T0 and T1, respectively] of the body, with respect to the other system, solely by an additive constant C, which depends on
the choice of the arbitrary additive constants of the energies E and E .
Hence, Einstein concluded,

E0 − E0 = T0 + C E1 − E1 = T1 + C

(3.4) (3.5)

and because of (3.3)

4 A. Einstein, “Ist die Tra¨gheit eines Ko¨ rpers von seinem Energieinhalt abha¨ngig?,” Annalen der Physik 18, 639–641 (1905); “Does the Inertia of a Body Depend upon Its Energy Content?,” in A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity (New York: Dover, 1952), pp. 69–71. The original paper is reprinted in The Collected Papers of Albert Einstein (Princeton: Princeton University Press, 1989), vol. 2, pp. 312–314; English translation in the translation project, also published by Princeton University Press, pp. 172–175 (document 24).

63

CHAPTER THREE

T0 − T1 = E(γv − 1).

(3.6)

Finally, since in the nonrelativistic limit, where the kinetic energy equals

1 2

mv2,

m

being

the

Newtonian

mass

of

the

body,

T0 − T1 =

1 2

mv2

,

(3.7)

and, neglecting quantities of the fourth and higher order, the expansion of E(γv − 1) yields

E(γv

−

1)

=

1 2

(v/c)2

E,

(3.8)

where v is constant, it follows from the last three equations that

E = c2 m

(3.9)

or in words: “If a body gives off the energy E in the form of radiation, its mass decreases by E/c2.” Generalizing this result Einstein declared: “The mass of a body is a measure of its energy content.”5
The paper referred to at the beginning of this derivation (Einstein’s very ﬁrst paper on relativity) is of course his famous article “Zur Elektrodynamik bewegter Ko¨ rper” (“On the Electrodynamics of Moving Bodies”).6 Precisely two years later Max Planck published his essay “Zur Dynamik bewegter Systeme” (“On the Dynamics of Moving Systems”), which, as the title indicates, deals with problems similar to those in Einstein’s ﬁrst relativity paper.7 Planck also showed that “through every absorption or emission of heat the inertial mass of a body changes, the difference is mass being always equal to the quantity of heat . . . divided by the square of the velocity of light in vacuo,” and added the remark that Einstein had already arrived at “essentially the same conclusion by applying the relativity principle to a special radiation process, but under the assumption permissible only as a ﬁrst approximation that the total energy of a body is composed additively of its kinetic energy and its energy referred to a system in which it is at rest.”

5 A. Einstein, “Die Masse eines Ko¨ rpers ist ein Mass fu¨ r dessen Energieinhalt,” Annalen der Physik 18, 641 (1905).
6 A. Einstein, Annalen der Physik 17, 891–921 (1905). Collected Papers, vol. 2, pp. 276–306 (English translation, pp. 140–171). English translation also in Einstein et al., The Principle of Relativity, pp. 35–65.
7 M. Planck, “Zur Dynamik bewegter Systeme,” Berliner Sitzungsberichte 1907, pp. 542– 570; Annalen der Physik 26, 1–34 (1908); Physikalische Abhandlungen und Vortra¨ge (Braunschweig: F. Vieweg, 1958), vol. 2, pp. 176–209.
64

THE MASS-ENERGY RELATION

Ives, having read this paper by Planck, contended that “what Planck characterized as an assumption permissible only to a ﬁrst approximation invalidates Einstein’s derivation.” In other words, according to Ives, equations (3.4) and (3.5) are unwarranted, and in order to ﬁnd the correct relationships, use has to be made of the equations

T0 = m0c2(γv − 1) T1 = m1c2(γv − 1)

(3.10) (3.11)

for the kinetic energy, which Einstein had proved in section 10 of his ﬁrst relativity paper. As described in chapter 13 of COM, Ives now reasoned as follows: Subtracting (3.11) from (3.10) yields

T0 − T1 = (m0 − m1)c2(γv − 1),

(3.12)

which, in view of (3.3) gives

(E0

−

E0)

−

(E1

−

E1)

=

(m0

−

E m1)c2

(T0

−

T1)

(3.13)

or considered “as the difference of the two relations,”

E0

− E0

=

(m0

−

E m1

)c2

(T0

+

C)

(3.14)

and

E1

−

E1

=

(m0

−

E m1)c2

(T1

+

C),

(3.15)

which, if compared with (3.4) and (3.5), show, according to Ives, that “what Einstein did by setting down these equations (as ‘clear’) was to introduce the relation”

E/(m0 − m1)c2 = 1,

(3.16)

which “is the very relation the derivation was supposed to yield.” The really important issue here is not so much the historical question
of whether Einstein’s ﬁrst derivation was a petitio principii or not but rather the question of principle as to whether the derivation is—or can be supplemented in such a way that it will be—rigorously valid. More speciﬁcally, the issue is whether, contrary to Planck’s remark, equations (3.4) and (3.5) can be shown to be strictly correct, or equivalently, since equation (3.3) is undisputable, whether equation (3.6) can be rigorously maintained. That it cannot, generally speaking, was argued in 1973 by

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CHAPTER THREE

Mendel Sachs.8 Sachs claimed that if the body is not a structureless particle but, e.g., a γ -ray emitting nucleus, in which electrostatic forces contribute to the binding of the constituent nucleus, changes in the electromagnetic conﬁguration energy, relative to the reference frame in which the body is moving, had to be taken into account. Hence, the correct equation should read

(T0 − T1) + (I0 − I1) = E(γv − 1),

(3.17)

where I0 and I1 are the electromagnetic conﬁguration energies in the excited and de-excited states of the nucleus, respectively.
The issue was taken up again more recently by John Stachel and Roberto Torretti.9 True, they admit, had Einstein really made use of equation (3.10) or (3.11), he would have indeed committed a circulus vitiosus, for “he had as yet no grounds for assuming that the dependence of the kinetic energy on internal parameters can be summed up in a rest mass term.” But he did not. They also admit that what Einstein regarded as evident (“it is clear that the difference . . .”) needs an explanation. They justify Einstein’s derivation by taking into account the internal energy of an isolated body in equilibrium and at rest in an inertial system and applying the relativity principle, according to which this state must be the same when the body is moving in a uniform motion with velocity v relative to that system. That their justiﬁcation is not a trivial matter can be seen from the fact that Willard L. Fadner criticized it on the grounds that it assumes the possibility “for an observer to measure the rest properties of a body when the observer is moving at a velocity v relative to that body,” a conceptual difﬁculty, which Fadner claims to have eliminated.10
Einstein seems never to have responded to the circularity claim. After all, Ives’s paper was published only three years prior to Einstein’s death. Nor does Einstein seem to have been satisﬁed with his 1905 derivation or, for that matter, with any other of his various derivations of the massenergy relation. Aware of the fundamental importance of this relation, he regarded it as unsatisfactory that in spite of many strenuous efforts he did not succeed in establishing a general proof of the relation, that

8 M. Sachs, “On the Meaning of E = mc2,” International Journal of Theoretical Physics 8, 377–383 (1973).
9 J. Stachel and R. Torretti, “Einstein’s First Derivation of Mass-Energy Equivalence,” American Journal of Physics 50, 760–763 (1932).
10 W. L. Fadner, “Did Einstein Really Discover ‘E = mc2’?,” American Journal of Physics 56, 114–122 (1988).
66

THE MASS-ENERGY RELATION
is, a proof without premises that are valid only in special cases.11 As early as in the introduction to his 1907 derivation he declared that to the question of whether there exist other special cases that would lead to conclusions incompatible with the relation, “a general answer . . . is not yet possible because we do not yet have a complete world-view that would correspond to the principle of relativity.” He remark that only “ein vollsta¨ndiges dem Relativita¨tsprinzip entsprechendes Weltbild” could do full justice to the signiﬁcance of this relation seems to indicate that he assigned not only a purely physical-technical signiﬁcance to the mass-energy relation but also a deep philosophical meaning, a perception that as we shall see further on, proved true. That he also always strived for greater generality by narrowing down the range of the postulated premises can be gathered from the introductory remarks to his last published derivation (1946): “The following derivation of the law of equivalence, which has not been published before, has two advantages. Although it makes use of the principle of special relativity, it
11 It would be a psychologically and methodologically interesting research project to compare Einstein’s various derivations of the mass-energy relation, which are listed here in chronological order: (1) “Ist die Tra¨gheit eines Ko¨ rpers von seinem Energieinhalt abha¨ngig?,” Annalen der Physik 18, 639–641 (1905); Collected Papers of Albert Einstein (Princeton: Princeton University Press, 1989), vol. 2, pp. 312–314; “Does the Inertia of a Body Depend upon Its Energy Content?,” A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity (London: Methuen, 1923; New York: Dover, 1952), pp. 67–71; Collected Papers (English translations), vol. 2, pp. 172–174. (2) “Prinzip von der Erhaltung der Schwerpunktsbewegung und die Tra¨gheit der Energie,” Annalen der Physik 20, 627–633 (1906); Collected Papers, vol. 2, pp. 360–366; “The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy,” Collected Papers (English translation), vol. 2, 200–206. (3) “U¨ ber die vom Relativita¨tsprinzip geforderte Tra¨gheit der Energie,” Annalen der Physik 23, 371–384 (1907); Collected Papers, vol. 2, pp. 413–427; “On the Inertia of Energy Required by the Relativity Principle,” Collected Papers (English translations), vol. 2, pp. 238–251. (4) Section 14 in “U¨ ber das Relativita¨tsprinzip und die aus demselben gezogenen Folgerungen,” Jahrbuch der Radioaktivita¨t und Elektronik 4, 411–462 (1907); Collected Papers, vol. 2, pp. 433–484; “On the Relativity Principle and the Conclusions Drawn from It,” Collected Papers (English translations), vol. 2, pp. 252– 311; “Einstein’s Comprehensive 1907 Essay on Relativity, Part II” (translation by H. M. Schwartz), American Journal of Physics 45, 811–817 (1977). (5) (unpublished) “Manuscript on the Special Theory of Relativity (1912–1914),” Collected Papers (1995), vol. 4, pp. 9–101; “Elementary Derivation of the Equivalence of Mass and Energy,” Bulletin of the American Mathematical Society 41, 223–230 (1935). (6) “An Elementary Derivation of the Equivalence of Mass and Energy,” Technion Yearbook 5, 16–17 (1946); Concise derivations can also be found in his books (7) U¨ ber die spezielle und die allgemeine Relativita¨tstheorie (Braunschweig: F. Vieweg, 1917 and numerous later editions), section 15, as well as in (8) The Meaning of Relativity (Princeton: Princeton University Press, 1921) (4th edition, p. 45).
67

CHAPTER THREE
does not presume the formal machinery of the theory but uses only three previously known laws: (1) the law of the conservation of momentum, (2) the expression for the pressure of radiation; that is, the momentum of a complex of radiation moving in a ﬁxed direction, (3) the well-known expression for the aberration of light.”
We shall not discuss each of Einstein’s derivations or the relations among them separately in detail but wish to point out that, generally speaking, they can be classiﬁed as variants to one of the three different approaches: (I) the study of a symmetric emission (or absorption, in his 1946 derivation) of two identical physical objects (e.g., photons) with respect to two inertial reference frames in relative uniform motion; (II) the study of the motion of a single physical object in a cavity or box, subject to the principle of the conservation of the center of mass or of linear momentum, with respect to a single inertial reference frame; and (III) the study of the relation between energy, work, and momentum of a single object in motion with respect to a single inertial reference frame. Furthermore, all the derivations contain explicitly or implicitly, e.g., via the Lorentz transformation, some reference to electromagnetic radiation, which introduces the velocity of light c into the expression E = mc2.12
Einstein’s ﬁrst (1905) derivation of the mass-energy relation was discussed in extenso at the beginning of the present chapter. It clearly belongs to class (I) of the just mentioned classiﬁcation, the two physical “objects” being the two equal quantities of radiation emitted by the body B and dealt with in the reference frames S and S . It became the paradigm for the construction of numerous variants, each of which was claimed by its respective author to be more elementary and based on fewer assumptions that all those that preceded it.
An interesting example is Fritz Rohrlich’s 1990 “elementary derivation of E = mc2,” which, as its author claims, could have been carried out as early as 1842 when Christian Johann Doppler discovered the effect carrying his name, provided the photon and its particle-like properties had been known at the time. Following Einstein,13 Rohrlich assumes that
12 Even in his (almost) group-theoretical derivation of the Lorentz transformations, which he presented in his lectures on relativity at the University of Berlin, Einstein had to refer to the velocity of light. See “Relativita¨tsvorlesung Winter 1914–1915” in his Notebook, Collected Papers, vol. 6 (document 7), pp. 44–66, especially pp. 49–51.
13 Einstein, according to T. F. Jordan, intended originally to make use of his proposed notion of “light quanta” or “photons,” as they were later called, as early as March 1905 but
68

THE MASS-ENERGY RELATION
a source remaining at rest in a reference frame S emits two photons.14 Conservation of momentum requires them to have equal and oppositely directed momenta, hence equal frequency ν and equal energy hν; conservation of energy requires that the source suffer a loss of energy

E = 2hν.

(3.18)

Viewed from a frame S , which moves uniformly relative to S so that the source is seen to move with velocity v in the same direction as one of the photons, conservation of momentum and the (classical) Doppler effect require that

p0 = p1 + (hν/c)(1 + v/c) − (hν/c)(1 − v/c),

(3.19)

where p0 and p1 denote, respectively, the momentum of the source before and after the emission. The source’s loss of momentum in S is therefore

p0 − p1 = p = (2hν/c2)v.

(3.20)

Since momentum is the product of mass and velocity or p = mv and v remains constant, the loss in momentum can be accounted for only as a change m in mass. Hence

m = 2hν/c2.

(3.21)

If E0 and E1 denote, respectively, the initial and the ﬁnal energy of the source as measured in S , then clearly

E0 = E1 + hν(1 + v/c) + hν(1 − v/c), and the loss in energy of the source relative to S is

(3.22)

E0 − E1 = E = 2hν = E,

(3.23)

changed his mind because he thought that the idea of “light quanta” is “more revolutionary and less ﬁnished than relativity.” T. F. Jordan, “Photons and Doppler Effect in Einstein’s Derivation of Mass Energy,” American Journal of Physics 50, 559–560 (1982).
14 F. Rohrlich, “An Elementary Derivation of E = mc2,” American Journal of Physics 58, 348–349 (1990). Rohrlich ﬁrst published this derivation in his book From Paradox to Relativity—Our Basic Concepts of the Physical World (Cambridge: Cambridge University Press, 1987). In his otherwise very laudatory review of this book Victor F. Weisskopf called Rohrlich’s proof of E = mc2 “a ﬂawed derivation” but without stating why he regarded it as ﬂawed. It was also criticized by R. Ruby and R. E. Reynolds in their “Comments” on it in American Journal of Physics 59, 756 (1991), as going beyond the conceptual framework of Newtonian physics. But their critique was rebutted by Rohrlich, ibid., 757.
69

CHAPTER THREE

where the last equality follows from (3.18). Comparison of (3.21) with (3.23) yields

m = E/c2.

(3.24)

Rohrlich completes his derivation by arguing that if the total mass of the source is supposed to be used up by emitting photons, equation (3.24) implies

E = mc2.

(3.25)

Rohrlich was not the ﬁrst to use the Doppler effect for a derivation of the mass-energy relation. Apparently unknown to him, Daniel J. Steck and Frank Rioux had done so about ten years earlier, in 1980, the only difference being that the latter had used the relativistic formula of the Doppler effect.15 Thus in their derivation, Rohrlich’s equation (3.19) and those that follow from it read

p0

−

p1

=

hν c

1 + v/c 1/2 − hν

1 − v/c

c

1 − v/c 1/2 1 + v/c

=

2hν c2

γvv =

E c2

γvv,

(3.26)

which with the correspondingly modiﬁed equations

yields again

p = mv

m = mγv

m = E/c2.

(3.27) (3.28)

Steck and Rioux were also not the ﬁrst to apply to Doppler effect to the derivation of the mass-energy relation. Unknown to them—for they stated “in this note we describe a simple derivation of the massenergy equivalence equation that we have not seen previously in the literature”—their derivation, though couched in a different terminology, had been presented seventy years earlier by Paul Langevin.16 In a lecture delivered on March 26, 1913, Langevin explained, though without

15 D. J. Steck and F. Rioux, “An Elementary Development of Mass-Energy Equivalence,” American Journal of Physics 51, 461–462 (1983).
16 P. Langevin, “L’inertie de l’e´nergie et ses conse´quences,” Journal de Physique the´orique et applique´e 3, 553–591 (1913); reprinted in Oeuvres Scientiﬁque de Paul Langevin (Paris: CNRS, 1950), pp. 397–426.
70

THE MASS-ENERGY RELATION
using the term “Doppler principle,” the “variation de masse . . . par l’e´mission du rayonnement” by analyzing the energy content of two equal quantities of radiation emitted in opposite directions, as viewed by two observers in relative motion to each other and implying thereby essentially the relativistic formula of the Doppler effect. Albert Shadowitz reformulated this derivation slightly by means of the relativistic Doppler effect and, calling it “a derivation of P. Langevin,” introduced it into the textbook literature in 1968.17
This “derivation of P. Langevin” should not be confused with another of Langevin’s derivations of the inertia of energy, presented by him in a 1920 lecture at the Colle`ge de France but never published. It would have been irretrievably lost were it not that Jean Perrin attended the lecture and reviewed it in his book on the foundations of physics written for the general reader.18 In contrast to the 1913 version, Langevin’s 1920 derivation is based only on the principle of conservation of energy and the two fundamental postulates of special relativity, i.e., the principle of relativity and the invariance of the velocity of light. A modernized version published recently by Y. Simon and N. Husson clearly demonstrates the important role that relativistic considerations play in this derivation.19
In sharp contrast Rohrlich, as we have seen, declared that his derivation “assumes only nineteenth-century physics.” An enthusiastic reviewer of his essay explicitly declared: “Thus we see that the energymass relation can be derived without the help of the theory of relativity.”20
In a similar vein, Ralph Baierlein, who proposed a derivation of the mass-energy relation not much different from Rohrlich’s, said of it that “it makes no use of Lorentz transformations or other results from the special theory of relativity” and added that “by 1873 Maxwell knew everything necessary to derive the equation E = mc2. All that was missing was a context of inquiry that would have led him to search for a connection between energy and inertia.”21
It is certainly true that the relation between energy and inertia or mass had been a topic of speculation among philosophers and of scientiﬁc
17 A. Shadowitz, Special Relativity (Philadelphia: W. B. Saunders, 1968), p. 90. 18 J. Perrin, Les E´ le´ments de la Physique (Paris: Albin Michel, 1929), pp. 380–391. 19 Y. Simon and N. Husson, “Langevin’s Derivation of the Relativistic Expressions for Energy,” American Journal of Physics 59, 982–987 (1991). 20 V. P. Srivastava, “A Simple Derivation of E = mc2,” Physics Education 26, 214 (1991). 21 R. Baierlein, “Teaching E = mc2,” The Physics Teacher 29, 170–175 (1991).
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CHAPTER THREE
research among physicists, especially among the proponents of the electromagnetic theory of mass, long before the advent of the theory of relativity. Thus, for example, Gustave Le Bon, the director of the Bibliothe`que de Philosophie Scientiﬁque in Paris, complained in his correspondence with Einstein that his anticipation of the equivalence between energy and mass, as stated in his book L’E´ volution de la Matie`re, has never received the credit it deserves because the Germans habitually ignore scientiﬁc contributions of other nations.22 In his reply Einstein conceded that the idea of a fundamental identity between mass and energy had been anticipated long ago but only the theory of relativity has cogently proved this equivalence. Asking Le Bon for his proof of this equivalence Einstein added, in response to Le Bon’s accusation of the Germans, that for violations of intellectual rights only individuals and not nations can be held responsible.23 Having been unable to understand Le Bon’s argumentation Einstein asked him to discuss the matter with Paul Langevin of the Colle`ge de France.
In physics, the electromagnetic theory of mass, according to which inertia is ultimately an electromagnetic induction effect, and especially the conception of an “electromagnetic momentum,” led physicists, such as Max Abraham and Henri Poincare´, to suggest a possible relation between inertia and energy. What was probably the most publicized prerelativistic declaration of such a relation was made in 1904 by Fritz Haseno¨ hrl.24 Using Abraham’s theory, Haseno¨ hrl showed that a cavity with perfectly reﬂecting walls containing electromagnetic radiation behaves, if set in motion, as if it has a mass m given by m = 8Vε0/3c2, where V is the volume of the cavity, ε0 the energy density at rest, and c the velocity of light. In 1921 Philipp Lenard, who became the leading protagonist of “German physics” during the Nazi
22 G. Le Bon, L’E´ volution de la Matie`re (Paris: Flammarion, 1905). Letter from Le Bon to Einstein, dated June 17, 1922. Einstein Archive reel 43-311.
23 “L’ide´e que masse et e´nergie soit la seule ve´ritable substance, e´tait de´ja proclame´e par beaucoup d’auteurs. Mais c’est seulement la the´orie de relativite´, qui permet a` donner une ve´ritable preuve de cette e´quivalence. Si vous vouliez m’e´crire votre manie`re de conclure, je serais tre`s reconnaissant a` vous. Finalement je vous assure, que les crimes contre la proprie´te´ intellectuelle sont des affairs personelles et non nationales.” Letter from Einstein to Le Bon, dated June 17, 1922. Einstein Archive, reel 43-313.
24 F. Haseno¨ hrl, “Zur Theorie der Strahlung in bewegten Ko¨ rpern,” Annalen der Physik 15, 344–376 (1904); Wiener Sitzungsberichte 113, 1039–1051 (1904). “Zur Theorie der Strahlung in bewegten Ko¨ rpern,” Annalen der Physik 16, 589–592 (1905), which contains the correction m = 4Vε0/3c2.
72

THE MASS-ENERGY RELATION
regime, republished Haseno¨ hrl’s discovery together with Johann Georg Soldner’s 1801 calculation of the deﬂection of a light ray by 0.84” when grazing the sun, in order to discredit Einstein by calling into question has authenticity concerning the well-known results of the theory of relativity.25
In a rejoinder to Lenard’s article Max von Laue admitted that Haseno¨ hrl might be credited with having made the ﬁrst attempt to construct a dynamical theory of cavity radiation by means of the concept of electromagnetic momentum. “But that every energy ﬂow carries momentum and that conversely every momentum implies a ﬂow of energy is an insight which only the theory of relativity could reach in a consistent way; for only this theory shattered the foundations of Newtonian dynamics.”26 Von Laue also rejected Lenard’s proposal to call the inertia of energy “Haseno¨ hrlsche Masse” as misleading because the concept of “mass” is always identical with the concept of “inertia of energy.”
In a contribution to the well-known Schilpp book on Einstein, von Laue discussed in more detail the impossibility of a nonrelativistic derivation of the mass-energy relation, which he called “the law of the inertia of energy” and declared: “Einstein derived this law relativistically. And, in fact, a rigorous derivation must start from there.”27 This statement by von Laue, namely that only the theory of relativity admits a rigorous derivation of the mass-energy relation, highlights the question of whether or not this notion has been refuted by those who, like Rohrlich, Srivastava, or Baierlein, have claimed to derive that relation without any “use of Lorentz transformations or other results from the theory of relativity.” We believe that the answer lies not so much in the possibility that these derivations are not rigorous as in the fact that they use the expression hν/c for the momentum of a light quantum or Maxwell’s expression for the ratio between momentum and energy of
25 F. Lenard, “Vorbemerkung zu J. Soldner, U¨ ber die Ablenkung eines Lichtstrahls von seiner geradlinigen Bewegung durch Attraktion eines Weltko¨ rpers, an welchem er nahe vorbeigeht,” Annalen der Physik 65, 593–604 (1921).
26 “Dass aber jede Energiestro¨ mung Impuls mit sich fu¨ hrt, und dass umgekehrt aller Impuls auf Energiestro¨ mung beruht, diesen Gedanken konnte erst die Relativita¨tstheorie folgerichtig durchfu¨ hren; denn erst sie ra¨umte mit der ihr widersprechenden Newtonschen Dynamik grundsa¨tzlich auf.” M. von Laue, “Erwiderung auf Hrn. Lenards Vorbemerkung zur Soldnerschen Arbeit von 1801,” Annalen der Physik 66, 283–284 (1921).
27 M. von Laue, “Inertia and Energy” in P. A. Schilpp, ed., Albert Einstein: Philosopher– Scientist (Evanston, Ill.: Library of Living Philsophers, 1949), p. 524.
73

CHAPTER THREE
electromagnetic radiation. After all, as the c in the equation E = mc2 indicates, somehow Maxwell’s theory must have been involved, but Maxwell’s theory is a relativistic one. Hence, one can say of any derivation of the mass-energy relation that refers to it even only implicitly what Einstein has said of his 1946 derivation, that “it makes use of the principle of special relativity, [although] it does not presume the formal machinery of the theory.”
We conclude our discussion of class-I derivations with an analysis of a modiﬁcation of the prototype of these derivations, which is a derivation of the mass-energy relation that its authors, Mitchell J. Feigenbaum and N. David Mermin, call “a purely mechanical version of Einstein’s 1905 argument.”28 In fact, the physical scenario of their derivation differs from that of Einstein’s 1905 paper only insofar as the body B loses energy not, as in Einstein’s argument, by emitting two equal quantities of radiant energy but by emitting two equally massive particles. In order to see whether this modiﬁcation enabled the authors to obtain their result, as they claim, “without ever leaving the realm of mechanics” we ﬁrst have to review their argumentation.
Like Einstein, Feigenbaum and Mermin calculate the energy loss of B from the viewpoint of two inertial reference frames S, the rest frame of B, and S , which moves relative to S with uniform velocity v along a direction making an angle φ with the direction of the emission. In S, E1 denotes the energy of B before the emission, E2 its energy after, and E3 the energy of each of the particles emitted in opposing directions moving with velocity u. Energy conservation requires

E1 − E2 = 2E3(u).

(3.29)

In S the initial and ﬁnal energies of B are denoted by E1(v) and E2(v) and the energies of the emitted particles by E(u ) and E(u ), respectively. Energy conservation for any value of φ requires

E1(v) − E2(v) = E3(u ) + E3(u ).

(3.30)

Since the left-hand side of this equation is independent of φ, the righthand side must be independent of φ as well, although u and u individually depend on v, u, and φ in accordance with the relativistic addition rule of velocities, which can be written in the form

28 M. J. Feigenbaum and N. D. Mermin, “E = mc2,” American Journal of Physics 56, 18–21 (1988).
74

THE MASS-ENERGY RELATION
γu = γuγv(1 − uv cos φ/c2)

and

γu = γuγv(1 + uv cos φ/c2),

(3.31)

where of course for any velocity w the symbol γw is an abbreviation of (1 − w2/c2)−1/2. In Einstein’s 1905 derivation the arbitrariness of φ was an unnecessary feature because the argument could have been carried out taking φ = 0 from the very beginning. Indeed, as equation (3.2) clearly shows, φ cancels out. For the Feigenbaum-Mermin derivation, in contrast, this arbitrariness is of decisive importance because it imposes severe constraints upon the mathematical structure of the function E(u). As Feigenbaum and Mermin show by a clever use of the relativistic velocity addition rule, E(u) must have the structure

E(u) = E0 + k(γu − 1),

(3.32)

where E0 and k are velocity-independent constants characteristic of the particle. Clearly, E(0) = E0 is the energy content of the particle in its rest frame and the constant k determines its kinetic energy

Ekin(u) = E(u) − E0 = k(γu − 1).

(3.33)

Application of the generally valid equation (3.32) to the energy conservation equation (3.29) and use of (3.31) yields

E1(0) + k1(γv − 1) − E2(0) − k2(γv − 1) = 2E3(0) + 2k3(γuγv − 1) (3.34) since by (3.31) γu + γu = 2γuγv. In particular for v = 0

E1(0) − E2(0) = 2E3(0) + 2k3(γu − 1).

(3.35)

Subtracting (3.35) from (3.34) and canceling the common factor γv − 1 gives

k2 = k1 − 2k3γu.

(3.36)

But since by (3.33)

E3kin = E3(u) − E3(0) = k3(γu − 1)

(3.37)

it follows from (3.35) that

k2 = k1 − 2k3 − 2E3kin

(3.38)

Equation (3.33) shows that in the nonrelativistic limit (i.e., u c)

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CHAPTER THREE

Ekin(u)

=

1 2

ku2/c2,

(3.39)

which, compared with the classical equation Ekin

=

1 2

mu2

,

identiﬁes

k with mc2. Adopting the traditional nomenclature, Feigenbaum and

Mermin obtain the equation

m1 − m2 = 2m3 + 2E3kin/c2,

(3.40)

which shows that the loss in mass, m = m1 − m2, of the emitting body B is equal to the sum of the masses of the two emitted particles and their kinetic energies, the latter divided by c2. This is indeed the mass-energy relation applied to the case of emitted particles that carry away mass as well as energy.
Having reviewed the Feigenbaum-Mermin paper, let us now ask whether they have derived the mass-energy relation really “without ever leaving the realm of mechanics.” It is certainly true that no explicit reference has been made to nonmechanical terms—with the exception, of course, of the letter c, which denotes the velocity of light and has been introduced by the relativistic velocity addition theorem. As is well known, this theorem is usually derived as a consequence of the Lorentz transformations. Incidentally, Mermin himself, ﬁve years before he wrote the paper with Feigenbaum, had presented an alternative proof, which shows that the theorem is a direct consequence only of the constancy of the velocity of light.29 Further, the Lorentz transformations are usually derived from the “light postulate,” according to which the velocity of light is a relativistic invariant. But such an invariance denies the possibility of conceiving the propagation of light as a mechanical process in a hypothetical ether. It follows therefore that the relativistic addition theorem, which, as we have seen, plays the key role in the Feigenbaum-Mermin argumentation, exceeds the conceptual framework of the purely mechanical. The problem to be faced here is, of course, the same one that we encountered in our discussion of Tolman’s derivation of the expression for relativistic mass within the framework of his “non-Newtonian mechanics.” Again, a possible, even if only partial, solution can be found in the work of Landau and Sampanthar described in chapter 2.30

29 N. D. Mermin, “Relativistic Addition of Velocities Directly from the Constancy of the Velocity of Light,” American Journal of Physics 51, 1130–1131 (1983).
30 B. V. Landau and S. Sampanthar, “A New Derivation of the Lorentz Transformation,” American Journal of Physics 40, 599–602 (1972).
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THE MASS-ENERGY RELATION
Let us also recall in this context that from as early as 1910, beginning with Waldemar von Ignatowsky followed by Philipp Frank and Hermann Rothe, physicists and mathematicians realized that the (structure of the) Lorentz transformations, and hence of the relativistic velocity addition theorem as well, can be derived without invoking the light postulate or any other reference to electromagnetic phenomena merely by using general principles, such as the principle of relativity or the isotropy and homogeneity of space.31 Of course, such group-theoretical derivations can involve only a limiting velocity α in lieu of c. The price to be paid for not invoking the light postulate or any other equivalent assumption is as Wolfgang Pauli phrased it: “Nothing can, naturally, be said about the sign, magnitude and physical meaning of α. From the group-theoretical assumption it is only possible to derive the general form of the transformation formulae, but not their physical content.”32 The fact that for α = ∞ these equations degenerate into the Galilean transformations of Newtonian physics and the mass-energy relation E = mα2 becomes meaningless can be interpreted as an indication that this relation is an exclusively relativistic result. Conversely, it can also be said that the mass-energy relation E = mc2 or the velocity-dependent equation of inertial mass can replace the second postulate in the logical construction of the special theory of relativity.33 As long as α remains ﬁnite, its indeterminacy affects the numerical relation between mass and energy but not the conceptual content of this relation.
The preceding derivations of the mass-energy relation belong to class (I) in the classiﬁcation described earlier. The ﬁrst derivation belonging to class (II) is Einstein’s 1906 second derivation. Like his ﬁrst, it is based on
31 For bibliographical references up to 1964 see H. Arzelie`s, Relativistic Kinematics (Oxford: Pergamon, 1966), pp. 80–82. Important more recent group-theoretical derivations of (generalized) Lorentz transformations are: G. Su¨ ssmann, “Begru¨ ndung der LorentzGruppe allein mit Symmetrie- und Relativita¨tsannahman,” Zeitschrift fu¨ r Naturforschung 24a, 495–498 (1969); V. Gorini and A. Zecca, “Isotropy of Space,” Journal of Mathematical Physics 11, 2226–2230 (1970); A. R. Lee and T. M. Kalotas, “Lorentz Transformations from the First Postulate,” American Journal of Physics 43, 434–437 (1975); J.-M. Levy-Leblond, “One More Derivation of the Lorentz Transformation,” American Journal of Physics 44, 271–277 (1976).
32 W. Pauli, The Theory of Relativity (New York: Pergamon, 1958), p. 11. 33 For more details and a simple group-theoretical derivation of the (general) Lorentz transformations see M. Jammer, “Some Foundational Problems in the Special Theory of Relativity,” in G. Toraldo di Francia, ed., Problems in the Foundations of Physics, Proceedings of the International School of Physics ‘Enrico Fermi’, Course LXXII (Amsterdam: NorthHolland, 1979), pp. 202–236.
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CHAPTER THREE
Maxwell’s theory of the electromagnetic ﬁeld, which, if supplemented by J. H. Poynting’s theorem (1884), predicts that electromagnetic radiation of energy E falling on an absorbing body exerts a pressure on it and transfers to it a momentum equal to E/c. This effect was experimentally conﬁrmed by Petr N. Lebedew in 1890 and with greater precision by Ernest F. Nichols and Gordon F. Hull in 1901.34
With the exception of this item from Maxwell’s theory, Einstein’s second derivation uses only the principles of mechanics as its title “The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy” indicates.35 It considers a “rigid hollow cylinder Z, “freely ﬂoating in space,” of mass M and length L. If the electromagnetic radiation E is emitted at time t = t1, say, from the left interior wall of Z and reaches the opposite wall at time t = t2, so that (approximately) t2 − t1 = t = L/c, conservation of momentum requires Z to recoil to the left with a velocity u given by Mu + E/c = 0, and hence over a distance x1 = u t = −L E/Mc2. If then, as Einstein assumes, E in any form of energy is transferred back to the left wall by a massless carrier, Z will recoil to the right over a distance x2 = mL/M, where
m is the mass associated with E. According to the center-of-mass conservation principle the total displacement of Z has to be zero. But since this total displacement is x1 + x2 = −(L E/Mc2) + ( mL/M), it follows that m = E/c2 is “the necessary and sufﬁcient condition for the law of the conservation of motion of the center of gravity to be valid.” Einstein was of course well aware that both the equation for
t and the nonrelativistic expression Mu for the momentum of Z were valid only “apart from terms of higher order.” He admitted therefore that this derivation is correct only “in ﬁrst approximation.” This deﬁciency was certainly one of the motivations for his continuing search for more accurate derivations. Furthermore, he soon realized that the notion of a rigid body is incompatible with the theory of relativity.
The notion of which this derivation hinges is the concept of momentum of radiation or radiation pressure, which is a necessary consequence of Maxwell’s electromagnetic theory and, as such, implicitly a relativistic conception. Replacing the radiative emission by a purely
34 For the history of this effect, which dates back to at least 1708, see E. Whittaker, A History of the Theories of Aether and Electricity (London: Thomas Nelson, 1910, 1951), vol. 1, pp. 273–276.
35 A. Einstein, “Prinzip von der Erhaltung der Schwerpunktsbewegung und die Tra¨gheit der Energie,” Annalen der Physik 20, 627–633 (1906).
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THE MASS-ENERGY RELATION
classical mechanical recoil process would not have led to the massenergy relation. It is therefore erroneous to contend, as R. T. Smith did, that Einstein’s 1906 derivation is “purely classical and has nothing to do with relativity.”36
Just as Einstein’s 1905 derivation became the prototype of numerous modiﬁed versions belonging to class I, so his 1906 derivation initiated a long series of class II variants, of which each was intended to be more rigorous than all those that preceded it. Since Adel F. Antippa’s detailed survey of class II derivations is readily available a brief summary of this development will sufﬁce.37
In his contribution to the Schilpp book, Max von Laue reformulated Einstein’s 1906 derivation with only one minor change.38 He added to the physical scenario two bodies or disks, one at each end of the cylinder, one of which transfers E back from right to left. He thus replaced Einstein’s “imagined massless carrier,” which he regarded as physically unrealistic by a mechanical process. Another disturbing feature of Einstein’s 1906 derivation is his assumption of the rigidity of the cylinder, an assumption which, in his third (1907) derivation, he showed to be incompatible with the relativity of simultaneity. This deﬁciency in the 1906 derivation was criticized in 1960 by Eugene Feenberg, who pointed out that “the recoil generates an elastic wave traveling with ﬁnite velocity from the source point; the far end does not begin to move until the radiation has been absorbed, and then the ﬁrst motion is away from the source.”39 It is only after some time, when the elastic waves are damped out by dissipative processes that the cylinder is ﬁnally at rest, having undergone the displacement. However, as Feenberg shows, these complications do not invalidate the correctness of the mass-energy relation.
In the early 1920s, in the wake of an international wave of general interest in the theory of relativity, Max Born was invited to deliver
36 R. T. Smith, “Classical Origins of ‘E = mc2’,” Physics Education 27, 248–250 (1992). 37 A. T. Antippa, “Variations on a Photon-in-a-Box by Einstein,” UQTR-TH-8, Universite´ du Que´bec a` Trois-Rivie`res, pp. 1–48; “Inertia of Energy and the Liberated Photon,” American Journal of Physics 44, 841–844 (1976). See also the earlier survey on some of Einstein’s derivations by W. Kantor, “Inertia of Energy,” American Journal of Physics 22, 528–541 (1954). A thorough analysis of Einstein’s 1906 and 1907 derivations as well as their elaborations by Planck and von Laue has also been given by A. I. Miller in his Albert Einstein’s Special Theory of Relativity (Reading, Mass.: Addison-Wesley, 1981), pp. 353–367. 38 Van Laue in P. A. Schilpp, ed., Albert Einstein, pp. 524–527. 39 E. Feenberg, “Inertia of Energy,” American Journal of Physics 28, 565–566 (1960).
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CHAPTER THREE

a series of public lectures on relativity at the University of Frankfurt. Both in his lectures and in his book,40 which is an elaboration of these

lectures, he strictly followed Einstein’s 1906 derivation when explaining

the mass-energy relation. However, when he was asked to republish his

book in an English version in the early 1960s he took fully into account,

following Feenberg, the time intervals during which the elastic move-

ments, excited by the emission and by the absorption of E expanded

over the whole cylinder (or tube, as he called it) and during which “also

all elastic vibrations have died out and only the displacements of the whole tube are left over.”41 Still retaining the approximation t = L/c

for the ﬂight duration of E, Born showed that all these corrections do

not impair the mass-energy relation. That even—in order to correct the

T equation—the introduction of an additional inertial frame, relative

to which the tube is at rest during the interval between the emission

and the absorption of E, does not affect the mass-energy relation, was shown by Carl J. Rigney and Roy H. Biser.42

In order to avoid the complications owing to the nonrigidity of the

cylinder or Einstein’s box, as it is often called, Anthony P. French sug-

gested in 1966 to “unhinge” the box, that is to “ignore completely any

connection between the ends of the box and to regard it as two masses
m1 and m2,” separated by a distance L.43 If m1 at the position x = 0 emits the energy E at the time t = 0 and its mass decreases thereby to m1, then according to the momentum-conservation principle m1 will recoil with the velocity

u1 = −

E/c m1

(3.41)

so that its position at time t ≥ 0 is given by

x1(t) = u1t = −

E/c t. m1

(3.42)

At time t = L/c the m2 absorbs E and increases thereby to m2. Its position at t ≥ L/c is given by

40 M. Born, Die Relativita¨tstheorie Einsteins und ihre physikalischen Grundlagen (Berlin: J. Springer, 1922).
41 M. Born, Einstein’s Theory of Relativity (New York: Dover, 1962), pp. 283–286. 42 C. J. Rigney and R. H. Biser, “Note on a Famous Derivation of E = mc2,” American Journal of Physics 34, 623 (1966). 43 A. P. French, Special Relativity (New York: Norton, 1966; Wokingham, Berkshire, U.K.: Van Nostrand-Reinhold, 1968, 1984), pp. 27–28.
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THE MASS-ENERGY RELATION

x2(t)

=

L

+

E m2c

t

−

L c

.

(3.43)

Finally, if M denotes the total mass of the system, X and X the position

of its center of mass before and after the whole process, respectively,

then

MX = m10 + m2L

(3.44)

and

MX = m1

− E/c m1

t + m2

L+

E/c m2

t

−

L c

.

(3.45)

Since according to the center-of-mass principle X = X , the preceding
equations show that m = m2 − m2 = m2 = m1 − m1 = − m1 satisﬁes the equation

E = c2 m.

(3.46)

By “unhinging” Einstein’s box French discarded, picturesquely speaking, the mantle of Einstein’s cylinder and used only the two end walls for his derivation of the mass-energy relation. Antippa continued this demolition process by taking into consideration only one wall, say the left wall, which he regarded as an atom at rest at the distance D from the origin, i.e., at x = 0, and emitting at time t = 0 a photon of energy content E.44 Before the emission, which decreases the mass of the atom from m to m , the center of mass of the system is at the position

X = D,

(3.47)

and since the atom recoils afterward with the velocity

u

=

−

E/c m

its position X is given by the equation

(3.48)

mX = m (D + ut) + m(D + ct),

(3.49)

where m = m − m . The center-of-mass principle requires that X = X , which leads to the equation

D[(m − m ) − m] − t(c m − E/c) = 0.

(3.50)

44 Antippa, UQTR-TH-8 and American Journal of Physics 44.
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CHAPTER THREE

However, this equation should be independent of the choice of the origin and should be valid for all t ≥ 0, which is possible only if the coefﬁcients of both D and t are identically zero. This implies that

m − m = m = E/c2.

(3.51)

In order to avoid any misunderstanding Antippa concluded his derivation of the mass-energy relation with the statement: “It should be noted that m is the ‘relativistic’ atomic mass including the kinetic energy contribution to the mass of the atom. Also m is not the rest mass lost by the atom, but rather the rest mass lost less the mass associated with the kinetic energy of the atom.”45
As this comment indicates and as a closer inspection of Antippa’s as well as French’s derivations shows, their reasoning is partially based on a petitio principii insofar as the existence of a quantitative relation between mass and energy is presupposed and it is demonstrated only that the coefﬁcient of proportionality between m and E is c2. Their reasoning thus differs from that of the preceding class II derivations in which such a quantitative relation was not presupposed a priori.
Turning now to the derivations of class III we must admit that it is difﬁcult to pinpoint exactly where or when they appeared for the ﬁrst time. For being relativistic generalizations of the classical method of calculating the kinetic energy of a particle they were used implicitly, that is, without being recognized as potential derivations of the massenergy relation, by Einstein, Planck, and von Laue in their early papers on relativity. An example is equation 14 in Einstein’s 1907 article “On the Relativity Principle and the Conclusions drawn from It.”46 Because of their analogy to classical calculations they have been readily adopted by many authors of textbooks on relativity, among them by D. Møller (1952, 1972), A. Papapetrou (1955), D. F. Lawden (1962, 1982), W.G.V. Rosser (1964), H. M. Schwartz (1968), and more recently by R. A. Mould (1994),47 to mention only a few. In principle they differ from their classical analogue only in their use of the relativistic mass m = m0γu instead of the classical mass. In their standard one-dimensional version they proceed as follows. They consider an inertial reference frame S in which

45 Antippa, American Journal of Physics 44, 844. 46 Einstein, Jahrbuch der Radioaktivita¨t und Elektronik 4, 411–462. 47 R. A. Mould, Basic Relativity (New York: Springer-Verlag, 1994). See also W. G. Holladay, “The Derivation of Relativistic Energy from the Lorentz γ ,” and the references listed therein, American Journal of Physics 60, 281 (1992).
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THE MASS-ENERGY RELATION

a massive body is being moved by a force F through a distance dx. The

change in the kinetic energy of the body is

dEkin

=

Fdx

=

dp dt

dx

=

dp dt

dx dt

dt

=

dp

dx dt

= ud(mu) = u2dm + mudu,

(3.52)

where du = dx/dt is the velocity, p = mu the momentum, m = m0γu the relativistic mass of the body, and γu = [1 − (u2/c2)]−1/2. Since dm = m0uduγu3/c2 it follows that dEkin = γu3m0u du, which integrates to

Ekin = m0c2(γu − 1) = mc2 − m0c2

(3.53)

with the constant of integration so chosen that for u = 0, Ekin = 0 as well. Dimensional considerations suggest that we also regard m0c2 as an energy, called the rest energy E0. Hence the total energy E of the body is

E = Ekin + E0 = mc2.

(3.54)

Some authors prefer to derive the mass-energy relation by means of

a relativistic four-vector generalization of classical mechanics without

the need for any integration. Choosing the unit of time so that c = 1,

they apply the fundamental invariant of the Lorentz transformation

ds2 = dt2 − dx2 − dy2 − dz2. Writing the ordinary velocity vector as
u = (u1, u2, u3) = (dx/dt, dy/dt, dz/dt) they obtain ds = γ −1dt, where γ = (1 − u2)−1/2. The velocity four-vector U is then given by U = γ (1, u)

and the momentum four-vector P by P = m0U = (m0γ , m0γ u), where m0

is the nonrelativistic mass. Neglecting the third and any higher power of

u,

they

obtain

P

=

(m0

+

1 2

m0u2,

m0u)

and

reason

as

follows.

Since

in

this

approximation the space components m0u represent the components

of the particle’s momentum and the time component, aside from the

additional constant m0, the kinetic energy in classical mechanics, they
conclude that the relativistic kinetic energy Ekin is given by m0γ − m0 so that m0γ = Ekin + m0, or expressed in the usual time units, m0γ c2 = Ekin + m0c2. Finally, since for u = 0 also Ekin = 0 and mc2 has the dimension of energy, they regard m0c2 as the rest energy E0 and m0γ u2

as the total energy E of the particle, i.e.,

E = Ekin + E0 = m0γ c2 = mc2.

(3.55)

This derivation, like any other derivation based on the correspondence, in the limit, with classical mechanics, is vulnerable to a criticism that Einstein expressed in 1935 as follows:

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CHAPTER THREE
Of course, this derivation cannot pretend to be a proof since in no way is it shown that this impulse [momentum] satisﬁes the impulse-principle and this energy the energy-principle if several particles of the same kind interact with one another; it may be a priori conceivable that in these conservation principles different expressions of the velocity are involved. Furthermore, it is not perfectly clear as to what is meant in speaking of the rest-energy, as the energy is deﬁned only to within an undetermined additive constant; in connection with this, however, the following is to be remarked. Every system can be looked upon as a material point as long as we consider no processes other than changes in its translation velocity as a whole. It has a clear meaning, however, to consider changes in the rest-energy in case changes are to be considered other than mere changes of translation velocity. The above interpretation asserts, then, that in such a transformation of a material point its inertial mass changes as the rest-energy; this assertion naturally requires a proof.
Clearly, the validity or acceptability of “new” theoretical constructs cannot be proved by showing that, in the limit, they converge or reduce to their corresponding classical analogues unless it is also shown that they satisfy the theoretical principles for the validity of which they have been contrived. For the convergence, or reduction, to their classical analogues is a necessary but not a sufﬁcient condition for their acceptability.
In the present case these principles are those of the conservation of momentum and of energy. Einstein thus saw the real task of his 1935 essay on the mass-energy relation as demonstrating the following: “If the principles of conservation of impulse and energy are to hold for all coordinate systems which are connected with one another by the Lorentz transformations, then impulse and energy are really given by the above expressions and the presumed equivalence of mass and restenergy also exists.”48
In order to carry out this task Einstein assumed that the relativistic momentum and relativistic energy of a particle or, as he called it, “material point” moving with velocity u relative to a reference frame S are
48 A. Einstein, “Elementary Derivation of the Equivalence of Mass and Energy,” Bulletin of the American Mathematical Society 41, 223–230 (1935). See also F. Flores’s instructive essay “Einstein’s 1935 Derivation of E = mc2,” Studies in History and Philosophy of Modern Physics 29, 223–243 (1998). Chapter 5 of this essay contains a detailed analysis of Einstein’s 1935 derivation of the mass-energy relation.
84

THE MASS-ENERGY RELATION
given, respectively, by pn = munF(u) and E = E0 + mG(u), (n = 1, 2, 3), where m is the rest-mass (or simply mass), E0 is the rest-energy, mG(u) is the kinetic energy of the particle, and F and G are universal even functions of u which vanish for u = 0. The assumption that the same mass constant m occurs in pn and E is later shown to be at least partially justiﬁed. By analyzing both an elastic eccentric collision and an inelastic collision between two particles of equal mass and equal rest energy he showed that the conservation of momentum and energy requires that F(u) = γ (u) and G(u) = γ (u)−1. Einstein thus arrived at the conclusion: “If for collisions of material points the conservation laws are to hold for arbitrary (Lorentz) coordinate systems, the well-known expressions for impulse and energy follow, as well as the validity of the principle of equivalence of mass and rest-energy.”
All the derivations of the mass-energy relation discussed so far have dealt only with the inertial mass of a body. But as we already know and as will be explained soon in greater detail, there is a conceptual distinction between inertial and gravitational mass. The former determines the inertial behavior of a physical object and is used in the equation of its kinetic energy, whereas the latter determines the weight of the body. It may be asked therefore whether a mass-energy relation can also be derived for gravitational mass. That the answer is positive was shown by Einstein as early as in the 1907 essay on special relativity referred to above. When dealing at the very end of this essay with the principle of energy conservation he showed that in addition to the quantity E— the energy value as measured at a given location—the energy integral also contains a term E /c2, where is the gravitational potential at that location. He thus concluded that “to every energy E there always belongs in the gravitational ﬁeld an energy which is as large as the energy of position of a gravitational mass of magnitude E/c2.” In other words, the mass-energy relation has also been proved to be valid for the concept of gravitational mass.
Let us now turn to the philosophical problem concerning the massenergy relation, that is, to the question of what, precisely, is the conceptual meaning of the equation E = mc2. As we shall see, at least two different interpretations have been proposed in the literature on this subject. According to one interpretation the relation expresses the convertibility of mass into energy or inversely of energy into mass, with one entity being annihilated and the other being created. According to another interpretation the equation expresses merely a proportionality between two attributes or manifestations of one and the same
85

CHAPTER THREE

ontological substratum without the occurrence of any annihilative or

creative process.49

The problem of the meaning of E = mc2 became the subject of

lively discussions after the Second World War, that is, after the atomic

bombardment of Hiroshima and Nagasaki had so tragically revealed

the ominous signiﬁcance of the mass-energy relation for the destiny of

humanity. In fact, the ﬁrst public debate on the issue began in 1946 with

C. Roland Eddy’s statement in the widely circulated periodical Science:

“It is evident, from many recent writings on the atomic bomb, that a

serious misconception still persists, not only in the popular press but

also in the mind of some scientists. The idea that matter and energy are

interconvertible is due to a misunderstanding of Einstein’s equation,

E = mc2. This equation does not state that a mass, m, can be converted

into an energy, E, but that an object of mass m contains simultaneously

an energy, E .50

To corroborate the statement that mass is not converted into energy

in a nuclear ﬁssion Eddy considered a symmetrical disintegration of a

nucleus of rest mass M into two fragments, each of rest mass m0 and velocity u. According to the mass-energy relation the energy released is

E = (M − 2m0)c2, and according to the theory of relativity the kinetic

energy

of

each

fragment

is

1 2

E

=

m0c2(γu

− 1)

=

mc2

− m0c2,

since

the mass of a particle at velocity u is m = m0γu. By combining the two

former equations he obtained M = 2m, which shows that the initial mass

equals the ﬁnal mass. Thus, since no mass is lost, he concluded that no

mass can have been converted into energy. In the sequel to his paper

he claimed that this conclusion also holds in the case of a more general

ﬁssion process as well as in the case of the so-called “annihilation” of a

positron and electron if it is recalled that the mass of a photon is hν/c2.

A few weeks later Science published critical responses to Eddy’s arti-

cle. Marshall E. Deutsch declared that, although he agrees with Eddy’s

statement of the law of conservation of mass as far as elementary par-

ticles are concerned, “I must reserve doubts about this law applying to

matter” in general. Referring to exothermic reactions in physical chem-

istry he declared that “except for bodies at a temperature of absolute

zero, as far as mass is concerned, the whole (mass of an entire body) is

less than the sum of its parts (masses of the individual bodies composing

the body)!” Another participant in this rejoinder, Austin J. O’Leary,

49 This substratum was dubbed “massergy” in chapter 13 of COM. 50 C. R. Eddy, “A Relativistic Misconception,” Science 104, 303–304 (1946).
86

THE MASS-ENERGY RELATION
expressed the view that Eddy’s conclusion, “the law of conservation of mass still holds,” is “purely a question of deﬁnition.”51
A particularly strong protest against the misconception of an interconvertibility of mass and energy was voiced by E. F. Barker in the same year. Barker distinguished sharply between the notions of mass and matter and admitted that matter, but not mass, can be created out of energy as, e.g., in the process of pair production, where “mass is conserved, though matter is not.” Analyzing in detail, as an example, the famous 1930 J. D. Cockcroft and E.T.S. Walton experiment of the production of two α-particles by bombarding a lithium atom with a proton, he showed that in this experiment, as in any other nuclear disintegration, mass is not changed into energy nor is energy changed into mass. He thus concluded: “Energy may be transferred from one system to another, either with or without a change in form; mass is always transferred in the process, but is never transformed.”52
The debate about this “misconception” has been revived several times. In 1976, e.g., J. W. Warren complained that numerous modern texts perpetuate this “misconception.”53 He presented a long list of quotations from such books and reported on a poll that he conducted among 147 students of science and engineering in which he asked whether the following statement is correct: “A nuclear power station differs from one burning coal or oil as it converts mass into energy according to the law E = mc2.” Only 32 students, Warren complained, found fault with the expression “converts mass into energy.” Another equally long list of such “misinterpretations” in scientiﬁc publications, including the Encyclopaedia Britannica, was collected by Sir Hermann Bondi and C. B. Spurgin.54 They recommend never forgetting that (i) energy has mass, (ii) energy is always conserved, (iii) mass is always conserved, and (iv) never using the term “equivalence of mass and energy.” Their advice stirred some lively debate. Calling these rules “dogmatic,” Rudolf Peierls takes exception especially to rules (ii) and (iii) for the following reason.55 When talking of the mass of a body one
51 “Comments on ‘A Relativistic Misconception,’ ” Science 104, 400–401 (1946). 52 E. F. Barker, “Energy Transformations and the Conservation of Mass,” American Journal of Physics 14, 309–310 (1946). 53 J. W. Warren, “The Mystery of Mass-Energy,” Physics Education 11, 52–54 (January 1976). 54 H. Bondi and C. B. Spurgin, “Energy Has Mass,” Physics Bulletin 38, 62–63 (February 1987). 55 R. Peierls, “Mass and Energy,” Physics Bulletin 38, 128 (1987).
87