Symmetry and Relativity in 1905

Transcrição

Arch. Hist. Exact Sci. 59 (2005) 437–544
Digital Object Identifier (DOI) 10.1007/s00407-005-0098-9
How Einstein Made Asymmetry Disappear:
Giora Hon and Bernard R. Goldstein
Communicated by J.D. Norton
Contents
I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
II. Einstein’s usages of the term symmetry in 1905 . . . . . . . .
1.
The dissertation (April 1905) . . . . . . . . . . . . . . .
Case 1: isotropy . . . . . . . . . . . . . . . . . . . . . .
Case 2: analogy . . . . . . . . . . . . . . . . . . . . . .
Case 3: geometrical usage . . . . . . . . . . . . . . . .
2.
“On the electrodynamics of moving bodies” (June 1905)
Case 1: indifference . . . . . . . . . . . . . . . . . . . .
Case 2: two algebraic usages . . . . . . . . . . . . . . .
Case 3: physical usage . . . . . . . . . . . . . . . . . .
Case 4: algebraic usage . . . . . . . . . . . . . . . . . .
Case 5: rejecting asymmetry . . . . . . . . . . . . . . .
3.
The central claim: making asymmetry disappear
by appealing to a physical argument . . . . . . . . . . .
4.
Conclusion . . . . . . . . . . . . . . . . . . . . . . . .
III. Background. The term symmetry and its “relatives”:
duality, parallelism, and reciprocity . . . . . . . . . . . . . .
1.
Heinrich Hertz (1857–1894) . . . . . . . . . . . . . . .
2.
Oliver Heaviside (1850–1925) . . . . . . . . . . . . . .
3. August Föppl (1854–1924) . . . . . . . . . . . . . . . .
4.
Emil Wiechert (1861–1928) . . . . . . . . . . . . . . .
5.
Wilhelm Wien (1864–1928) . . . . . . . . . . . . . . . .
6.
Hendrik Antoon Lorentz (1853–1928) . . . . . . . . . .
7.
Summary . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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I. Introduction
“It is well known that Maxwell’s electrodynamics – as usually understood at
present – when applied to moving bodies, leads to asymmetries that do not seem to
438
G. Hon and B. R. Goldstein
adhere to the phenomena.”1 This is how Albert Einstein (1879–1955) opened his epochmaking paper entitled “On the electrodynamics of moving bodies” [Zur Elektrodynamik
bewegter Körper] whose centenary has been celebrated in 2005. This paper is an outstanding contribution to science in general and to physics in particular: it transformed
physics by undermining the received conceptions of the nature of space and time and,
perhaps more importantly, it introduced a novel way of thinking that resulted in a revolution in the practice of physics.2 Halfway through the paper, better known as “the
theory of special relativity”,3 Einstein concludes that “the asymmetry mentioned in
the Introduction ... disappears.”4 Making asymmetry disappear has proved to be one of
the significant moves that Einstein took in his annus mirabilis of 1905. Our primary
goal is to clarify what Einstein meant by this asymmetry in Maxwell’s electrodynamics
and then to uncover the way he made it disappear. Further, what were the consequences
of rejecting asymmetry? Did Einstein thereby restore symmetry in electrodynamics? In
contrast to many historical accounts that focus on the role of the principle of relativity
in building the special theory, we will be concerned with the ways Einstein used the
term symmetry in his relativity paper. The result of our research is surprising: Einstein
removed the asymmetry in electrodynamics successfully but, at the same time, he did
not apply symmetry as a heuristic or methodological tool in physics; nevertheless, he
invoked the term symmetry in a number of contexts following, by and large, contemporary usage of mathematical reasoning in physics. Hence, we will argue, it is a mistake to
think that Einstein introduced symmetry into physics. While Einstein did use the term
symmetry and applied it in the relativity paper, he did not treat the relativity principle
itself as a symmetry principle. Explicit recognition that the principle of relativity can be
characterized as a symmetry principle came later with the growing interest in the role of
the concept of symmetry in physics. But this development was not of Einstein’s making:
for him symmetry was just a tool in the physicist’s toolbox, to be applied under very
specific circumstances.
Symmetry as used in physics at the turn of the last century was not just a mathematical property (either geometrical or algebraic) for, as we will see, it often expressed
a physical property as well. A straightforward example is the homogeneity of space –
a feature that is called isotropy. However, we will mostly be concerned with the property of interchangeability: in the 1880s Hertz and Heaviside recast Maxwell’s equations
1
Beck 1989, 140 (slightly modified); Einstein 1905c, 891: “Daß die Elektrodynamik Maxwells – wie dieselbe gegenwärtig aufgefaßt zu werden pflegt – in ihrer Anwendung auf bewegte
Körper zu Asymmetrien führt, welche den Phänomenen nicht anzuhaften scheinen, ist bekannt.”
In this paper we quote the translation of Anna Beck (1989). In some cases the standard translation (Einstein [1905b/1923]/1952, 37–65) is misleading; we are also aware of another translation
(Miller [1981]/1998, 370–393), but have not used it.
2
See Laue’s appraisal in 1907, noted in Klein et al. 1993, 74 n. 11. At the time Einstein
himself used the term “revolution” only for the paper on the light-quantum hypothesis (1905a):
see Klein et al. 1993, 31 and 32 n. 4. It is for this latter work that Einstein eventually received the
Nobel Prize.
3
For the history of the names that were given to the theory, see Stachel et al. 1989, 254.
4
Beck 1989, 159; Einstein 1905c, 910: “... daß die in der Einleitung angeführte Asymmetrie ...
verschwindet.”
439
for electricity and magnetism in two different forms which they both called “symmetrical”. This recasting of Maxwell’s equations was generally accepted, but some later
authors substituted other terms for it, namely, duality, reciprocity, and parallelism. We
will demonstrate that this sense of the term symmetry – and its “relatives” – continued
to be invoked right up to the time of Einstein’s relativity paper. However, this trajectory
has been overlooked in the secondary literature on physics.
We contend that as a rule Einstein followed the common usage of symmetry. However, right at the outset of the relativity paper he explicitly identified an asymmetry – to
be removed later – using this term in a way that, we will argue, diverged from standard
practice. But the argument for removing this asymmetry did not lead him to embrace
symmetry as a core concept of physics. For Einstein the problem had been misunderstood and the solutions that had been presented in symmetrical form were inadequate. In
sum, we claim that Einstein did not consider the principle of relativity to be a symmetry
principle: the removal of the asymmetry was not followed by an explicit use of the term
symmetry in the theory of electrodynamics.
In Part I we frame the questions that will be addressed later in the paper. In Part II we
document Einstein’s usages of the term symmetry in 1905 both in the dissertation and the
relativity paper, and we compare these usages with contemporary practice. We will see
that Einstein does not deviate from standard practice except for the two occurrences of
the term asymmetry in the relativity paper. Our main conclusion comes at the end of Part
II where we argue that Einstein’s use of the term asymmetry is a response to a passage in
Hertz. By deriving Maxwell’s equations in symmetrical form, Hertz resolved his difficulty. Einstein, however, sees the issue differently. The problem does not lie in the form
of the equations, but in the theory that produces two inconsistent explanations of the
same phenomenon. “Judged from the magnet, there was certainly no electric field present. Judged from the electric circuit, there certainly was one present.”5 Einstein refers
to this incompatibility as “asymmetry”. In Part III we will examine the use of the term
symmetry and its “relatives” in the period from 1880 to 1905 for the relation between
electricity and magnetism. This provides the background against which Einstein’s usage
stands out as idiosyncratic.
In any careful historical analysis one expects close attention to be paid to an author’s
terminology together with other contemporary usages, and to examine his (or her) sources
as well as the audience that was addressed. We maintain that context is essential to the
proper practice of writing history. One may differ on what constitutes the appropriate
context, but it surely includes the meaning that a word or a passage had at the time it
was written.6 For this reason we seek to locate some distinctive terms, retrieve their
meanings, and place them in a certain relation to their origins and consequents by focusing our attention – in this work – on the various usages of the term symmetry and their
respective contexts. We are, therefore, concerned here not with a critique of anachronistic interpretations of a given text, but with identifying the links that connect a series of
texts that have not been seen to be related. History of science is the attempt to ascertain
5
Einstein 1920; see n. 149, below.
In this essay we restrict the sense of context to intellectual and linguistic matters; we do not
deal with social context.
6
440
what answers have been given to an evolving set of questions and in what order. In our
view, the history of a term should reflect its meanings as the application of the term
changes over time – in different contexts, different problems arise and, in turn, different
answers are given. We are looking at the ways the term symmetry was used and therefore
we italicize most of its occurrences in this essay. We are of the opinion that a discussion
of symmetry as a concept is too wide-ranging and runs the risk of anachronism which
we seek to avoid.
By the turn of the last century the term symmetry was associated with a family of
concepts, each of which had its distinctive applications in different contexts, mathematical as well as physical. In his dissertation (dated April 1905; formally submitted in July
19057 ) Einstein initially applied the term symmetry in the standard way to a new physical
situation in hydrodynamics. In his relativity paper (dated June 1905) he developed a critical attitude towards the use of symmetry, probably in response to well known research
papers in the domain of electrodynamics and a popular textbook in this discipline. We
will argue that the works which most significantly shaped Einstein’s attitude towards the
use of symmetry in physics were those of Heinrich Hertz (1857–1894), Oliver Heaviside
(1850–1925) – whose ideas only reached Einstein through intermediaries such as August
Föppl (1854–1924) – and Wilhelm Wien (1864–1928); works by other physicists, for
example, Woldemar Voigt (1850–1919), are also likely to have played a role.
By subordinating the laws of nature to the unifying methodological principle of relativity, Einstein was able to recast the electrodynamics of moving bodies into an entirely
new and productive theory, based neither on assumptions concerning the material nature
of electrons nor on their dynamics; rather, the theory is based on principles which render
it truly general. Indeed, Einstein concluded the first part of his relativity paper, namely,
the kinematic part, by emphasizing the role of principles: “We have now derived the
required propositions of the kinematics that corresponds to our two principles, and will
now proceed to show their application in electrodynamics.”8 Clearly, Einstein gave
principles precedence over laws of nature. However, to the best of our knowledge, he
never raised symmetry to the status of a principle and he never regarded the principle of
relativity as a symmetry principle.
Although Einstein did not define the term symmetry, we will argue that his various
usages were neither aesthetic9 nor group theoretic. Rather, he conveys the impression
that he was working within a well established tradition. However, in one crucial case he
rejected the evolving practice of symmetrizing the equations for electricity and magnetism and it was in this context that he reconsidered the physical issue at stake, leading to
his unifying principle of relativity. We seek to identify this tradition (or traditions) for the
use of symmetry that Einstein opposed. We note that at the time there was no comment
at all (as far as we can tell), critical or otherwise, on the issue Einstein raised concerning
7
Stachel et al. 1989, 170, 202–203.
Beck 1989, 156; Einstein 1905c, 907: “Wir haben nun die für uns notwendigen Sätze
der unseren zwei Prinzipien entsprechenden Kinematik hergeleitet und gehen dazu über, deren
Anwendung in der Elektrodynamik zu zeigen.”
9
See Holton [1973]/1988, 384; Pais 1982, 138, 140; Miller [1981]/1998, 122. For an extensive
discussion, see Section II.2, Case 5, below.
8
441
the appropriateness of appealing to symmetry for the relation between electricity and
magnetism. In our view, Einstein objected to invoking the term symmetry for describing
the relations of physical phenomena, but he had no objection to the mathematical use of
symmetry in facilitating the solution of complex equations. In short, for Einstein the use
of symmetry in a mathematical sense was not sufficient where a physical argument is
required. In 1905 the road to fully grasping the power of the heuristic features concealed
in the concept of symmetry vis-à-vis physical situations was still in the future and, in
the years immediately following the publication of the relativity paper, symmetry as
a guiding principle was not applied extensively in physics. Indeed, to the best of our
knowledge, Einstein himself invoked symmetry very rarely, endowing it with neither
heuristic nor methodological power. The modern reader may think that Einstein introduced the principle of relativity on grounds of symmetry, but the historical fact is that
Einstein does not refer explicitly to the principle of relativity as embodying symmetrical
features.
Our principal question is: Why does Einstein use the term symmetry in the ways he
does? Or, to put it another way, How do Einstein’s usages of the term symmetry compare
with the usages of this term by contemporary physicists and mathematicians? To answer
this question we begin with a few remarks that will lead to our analysis.
1. What are the contexts for Einstein’s use of the term symmetry in his dissertation and
in his relativity paper of 1905? We show that at the turn of the last century symmetry
had already become a technical term applied by practitioners both in mathematics
and in physics. To grasp the meaning of these usages, it is best to examine the
passages where Einstein appealed to this term.
2. The only physicists mentioned in the paper are James Clerk Maxwell (1831–1879),
Heinrich Hertz,10 and Hendrik Antoon Lorentz (1853–1928), but no specific works
are cited. Hence, to locate Einstein’s sources one is left with his later recollections,
those of his friends, and occasional remarks in letters that Einstein wrote at the time
(of which only a few survive). These sources tend to give a broad picture of Einstein’s interests rather than offering the source for anything in particular. To recover
the context for Einstein’s usages we apply a method that we have used successfully
in the past where authors fail to indicate their sources, namely, we will trace certain
key terms and expressions in contemporary and near contemporary usage.11
Einstein’s letters in the period before 1905 contain some suggestive remarks on
issues in physics that engaged his attention. For example, in a love letter, dated
November 1901, Einstein mentions reading a book by the Göttingen physicist, Woldemar Voigt, but does not indicate what, if anything, in that book on theoretical
physics struck him as particularly important: “from which book I have already
10
For the phrase “Maxwell–Hertzschen Gleichungen”, see Einstein 1905c, 907, 908. See also
n. 184, below, for the passage where Hertz derived these equations, and n. 217, below, for discussion of Drude’s and Wiechert’s accounts.
11
See, e.g., Goldstein 1991; Hon and Goldstein 2005.
442
learned quite a lot,” he remarks without elaboration.12 In an earlier love letter, dated
August 1899, Einstein reveals that
I ... am at present studying again in depth Hertz’s propagation of electric force.... I am
more and more convinced that the electrodynamics of moving bodies, as presented
today, is not correct, and that it should be possible to present it in a simpler way. The
introduction of the term “ether” into the theories of electricity led to the notion of a
medium of whose motion one can speak without being able, I believe, to associate a
physical meaning with this statement.13
Evidently, the problem which Einstein succeeded in solving in 1905 had already
been on his mind in 1899. Indeed, he later spoke of “years of groping” [nach jahrelangem Tasten]14 before arriving at a satisfactory solution for the problem which he
had recognized in Hertz’s analysis of the fundamental equations of electrodynamics
for bodies in motion. In all probability, it was Hertz’s characterization of the issues
at stake in his paper of 1890, “Ueber die Grundgleichungen der Elektrodynamik
für bewegte Körper,” that motivated a series of fundamental discussions to which
Einstein’s relativity paper of 1905 belongs, as is evident from the title of his paper.15
Einstein returns to this problem in a letter dated December 1901.16 But we have
12
Beck 1987, 184; Stachel et al. 1987, 321: “aus welchem Buch ich schon manches gelernt
habe.” The editors of The Collected Papers of Einstein suggest Voigt 1895 and Voigt 1896 (Stachel
et al. 1987, 321 n. 5), but it might be Voigt 1901 (Voigt dated his preface to this second edition of
his Elementare Mechanik October 1900).
13
Beck 1987, 131; Stachel et al. 1987, 226: “Ich ... studiere gegenwärtig noch einmal aufs
Genaueste Hertz’Ausbreitung der elektrischen Kraft.... Es wird mir immer mehr zur Überzeugung,
daß die Elektrodynamik bewegter Körper, wie sie sich gegenwärtig darstellt, nicht der Wirklichkeit
entspricht, sondern sich einfacher wird darstellen lassen. Die Einführung des Namens ‘Äther’ in
die elektrischen Theorien hat zur Vorstellung eines Mediums geführt, von dessen Bewegung man
sprechen könne, ohne daß man wie ich glaube, mit dieser Aussage einen physikalischen Sinn
verbinden kann.” In a note the editors claim that Einstein is referring to Hertz 1890b. For sceptical
attitudes to the ether hypothesis see, e.g., Drude 1894, p. 9 (cited in Darrigol 1996, 256): “Gerade
so gut, wie man einem besonderen Medium, welches dem Raum überall erfüllt, die Vermittlerrolle
von Kraftwirkungen zuweist, könnte man auch dasselbe entbehren und dem Raum selbst diejenigen physikalischen Eigenschaften beilegen, welche dem Aether jetzt zugeschrieben werden. Man
hat sich bisher vor dieser Anschauung gescheut, weil man mit dem Worte ‘Raum’ eine abstrakte
Vorstellung ohne physikalische Eigenschaften verbindet. Da die Einführung des neuen Begriffes
‘Aether’ durchaus ohne Belang ist, wofern man nur das Princip der Nahekräfte festhält, so soll in
dieser Darstellung von der bisher üblichen Bezeichnung, d. h. der Einführung des Wortes ‘Aether’,
Gebrauch gemacht werden.” For Föppl’s view, see nn. 280 and 284, below.
14
Einstein 1920; Janssen et al. 2002, 280 n. 34.
15
See, e.g., Wien 1904a: “Über die Differentialgleichungen der Elektrodynamik für bewegte
Körper.” See also Cohn 1900: “Über die Gleichungen der Elektrodynamik für bewegte Körper,”
and idem 1904: “Zur Elektrodynamik bewegter Systeme.”
16
Beck 1987, 187: “I am now working very eagerly on an electrodynamics of moving bodies,
which promises to become a capital paper.” Stachel et al. 1987, 325: “Ich arbeite eifrigst an einer
Elektrodynamik bewegter Körper, welches eine kapitale Abhandlung zu werden verspricht.” Cf.
Norton 2004, 45–46.
443
not found any occurrence of the terms symmetry or asymmetry in Einstein’s extant
letters or papers that were written prior to 1905.17
3. We proceed to examine the usages of certain technical terms associated with symmetry and their contexts ca. 1905, without claiming that Einstein was aware of all these
matters. In fact, Einstein may have read things he did not later recall, simply assimilating, or responding to, the ideas in those books or articles. In any event, we seek
to determine those sources that were available at the time and that, at least potentially, could have played a role in clarifying for Einstein the range of applications
of symmetry.
4. In the next stage of our argument we address the question: to what extent do Einstein’s usages follow those of his contemporaries and to what extent has he broken
new ground? In other words, are Einstein’s usages of the term symmetry significant
specifically for setting up the relativity argument? The answer, we will argue, is
no! Symmetry, or rather the removal of asymmetry, did not play a principal role.
However, Einstein’s reconsideration of the traditional usage of the term symmetry
in electrodynamics, indeed, his idiosyncratic usage of asymmetry, facilitated his
methodological insight that a theory could be separated from its formalism. Thus,
the central claim is that there is a difference between a theory and its formalism:
mathematical equations are mere formalism, devoid of physical interpretation; it is
the theory that endows the formalism with physical meaning. We will see that this
distinction allowed Einstein to introduce the theory of special relativity while still
retaining Maxwell’s system of equations.
Moreover, we argue that the context for Einstein’s use of asymmetry in the relativity paper is the recasting of Maxwell’s equations for electricity and magnetism
into explicitly symmetrical form, beginning with Hertz and Heaviside. Thus, by
removing asymmetry Einstein is eliminating the “project” of making the equations
for electricity and magnetism symmetrical (and the kind of issues raised by Wien
in 1904 at the end of his paper, discussed in Section III.5, below). Hence Einstein’s
goal was to eliminate symmetry in this sense, not to reintroduce it in some other
way.
5. We focus on the term symmetry and its “relatives”, duality, parallelism, and reciprocity, as understood by physicists ca. 1905. In this way we seek to elaborate some
thought-provoking remarks by Gerald Holton.18 To the best of our knowledge there
is no survey of symmetry as used in physics ca. 1905 or in textbooks and research
publications that were consulted by physicists at the time, even if written at an earlier
date.
We begin by listing all the occurrences of the term symmetry in the dissertation
and in the celebrated relativity paper of 1905, and we then discuss each instance in
17
The editors of The Collected Papers indicate that, in the earliest extant essay by Einstein
(written in 1895 when he was only 16 years old), one argument is based on symmetry considerations. Nevertheless, the term symmetry is not invoked: see Stachel et al. 1987, 5–9; Beck 1987,
4–6.
18
Holton (1973)/1988, 380–385.
444
its respective context. Einstein published five papers in this annus mirabilis,19 but –
apart from his dissertation – he only referred to symmetry explicitly in the relativity
paper. We classify the usages of the term symmetry in these two papers according
to their essential features. With this typology serving as background, we proceed to
examine what Einstein meant by “asymmetry” and how he made it disappear.
II. Einstein’s usages of the term symmetry in 1905
Before turning to an analysis of each usage in its context, some general comments
are in order. As noted above, in his relativity paper Einstein notoriously did not refer to
any literary source, either theoretical or experimental: the few footnotes scattered in the
paper are just for the purpose of clarification. When he did refer to a contemporary theory,
it was to show that his kinematic results are in agreement with the electrodynamics of
moving bodies of the respected theory proposed by Lorentz.20 In 1905 Einstein was only
26 years old, the author of a few publications, and little known by the leading physicists
of the day, whereas Lorentz’s reputation had already been secured by his formulation of
particle-based electrodynamics some ten years earlier. Indeed, Lorentz had shared the
Nobel Prize with Pieter Zeeman (1865–1943) in 1902 in recognition of their research
on the influence of magnetism upon radiation phenomena.21 In the German collection
of papers on relativity a footnote was added to Einstein’s relativity paper of 1905 that
Lorentz’s paper (1904c), on the electromagnetic phenomena in a system moving with
any velocity less than that of light, was not known to Einstein at the time he wrote his
relativity paper.22 This negative remark which, of course, is missing in the original German version of the paper, is the only clue given to the reader about the background to
Einstein’s work. One is therefore baffled by Einstein’s disregard of the usual academic
convention for citing sources. A few years later Einstein defended his procedure in a
paper published in Annalen der Physik:
It seems to me to be in the nature of the subject, that what is to follow might already have
been partially clarified by other authors. However, in view of the fact that the questions
under consideration are treated here from a new point of view, I believe I could dispense
with a literature search which would be very troublesome for me....23
19
See Stachel 1998.
Beck 1989, 166–167; Einstein 1905c, 917.
21
Another indication of Lorentz’s prestige at the time is the publication of a Festschrift in his
honor: see Recueil de travaux 1900.
22
Einstein [1905c/1923]/1952, 38. For the original German see Blumenthal (ed.) 1922, 26 n.
2: “Die im Vorhergehenden abgedruckte Arbeit von H. A. Lorentz war dem Verfasser noch nicht
bekannt.” Moreover, shortly before his death, Einstein wrote to Carl Seelig in February 1955 that
prior to 1905 he had read Lorentz (1895), but none of Lorentz’s subsequent papers (see Einstein
Archives, Jerusalem, Seelig, folder II, MS 39–069; cf. Stachel et al. 1989, 259 n. 43).
23
Pais 1982, 165; Einstein 1907a, 373; see Stachel et al. 1989, 416. Wien (1904a) includes a
brief survey of the literature which may well have provided Einstein with an excuse to dispense
with a discussion of the “state of the art” in electrodynamics ca. 1905 in his relativity paper.
20
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1. The dissertation (April 1905)
Einstein’s dissertation, “A New Determination of Molecular Dimensions,”24 presents
us with a different case. In keeping with the usual academic standards, here Einstein
refers explicitly to a source. In fact, in a footnote he cites a work by Gustav Kirchhoff
(1824–1887) whose series of lectures on mathematical physics served as the point of
departure for Einstein’s physical analysis. Thus we have reliable evidence for a text that
Einstein consulted.25
Einstein considers the motion of a solvent in the vicinity of a suspended solid body
which plays the role of the dissolved molecule. At issue are the hydrodynamic equations
of the motion of the solvent in these circumstances. As is customary in mathematical
physics, several presuppositions are introduced to simplify the calculations: Einstein
assumes that the solid body is spherical in shape and that the liquid is homogenous
so that its own molecular structure need not be taken into consideration.26 The next
standard step in hydrodynamic analysis is to treat the motion of the liquid as a superposition of three different kinds of motion: (1) a parallel displacement without a change
in the relative positions of the liquid particles; (2) a rotation without a change in the
relative positions of the liquid particles, and (3) a dilatational motion in three mutually
perpendicular directions.27
Case 1: isotropy
With reference to the three basic kinds of motion discerned in hydrodynamics Einstein says:
Due to the symmetry [wegen der Symmetrie] of the motion of the liquid, it is clear that
the sphere can perform neither a translation nor a rotation during the motion considered,
and we obtain the boundary conditions
u = v = w = 0 for ρ = P , ....28
It should be clarified that “the motion considered” is the third one, dilatation, the only
motion that gets modified by the presence of the rigid spherical body. Note further that u,
v, and w are the three functions of the velocity components of the dilatational motion of
24
Einstein 1905b.
See Einstein 1905b, 189. Einstein refers again to the same text of Kirchhoff in footnotes to
both the paper on the movement of small particles suspended in stationary liquids (received by the
Annalen der Physik, 11 May 1905), and the paper on Brownian motion (received by the Annalen
der Physik, 19 December 1905): see Stachel et al. 1989, 230, 342. Thus it is evident that Lecture
26 in Kirchhoff’s Mechanik was an important source for Einstein. See Kirchhoff 1883, 369–387.
26
See Pais 1982, 90, for a complete list of the presuppositions.
27
Kirchhoff 1883, 123–124, 369–374; Einstein 1905b, 187; Beck 1989, 106.
28
Beck 1989, 107 (boldface added); Einstein 1905b, 188: “Zunächst ist wegen der Symmetrie
der betrachten Flüssigkeitsbewegung klar, dass die Kugel bei der betrachteten Bewegung weder
eine Translation noch eine Drehung ausführen kann, und wir erhalten die Grenzbedingungen:
25
u = v = w = 0 für ρ = P , ....
446
the liquid; P is the radius of the sphere and ρ expresses the principal axes of dilatation,
being equal to the square root of the sum of the squares of the three components of the
coordinate system whose axes are parallel to the principal axes of the dilatation.
Symmetry here expresses the feature that there is no privileged direction: the three
equations of motion are exactly analogous for each of the three orthogonal directions.
This feature is usually called isotropy. According to William Thomson (1824–1907) and
Peter G. Tait (1831–1901), isotropy is defined by two elements: resistance to compression and resistance to distortion. Thomson and Tait identify the quality of homogeneity
as a prerequisite for isotropy: “A body is called homogeneous when any two equal, similar parts of it, with corresponding lines parallel and turned towards the same parts, are
indistinguishable from one another by any difference in quality.” They then emphasize
the physical nature of this quality:
The substance of a homogeneous solid is called isotropic when a spherical portion of it,
tested by any physical agency, exhibits no difference in quality however it is turned.29
Thomson and Tait consider indifference (which they also call “indistinguishable”) as
a way to depict this feature of homogeneity. It is significant that physical agencies are
involved in determining this feature; in other words, on this definition isotropy is physical
rather than geometrical. This is, then, the feature that exhibits equal physical properties
or actions – e.g., refraction of light, elasticity, and conduction of heat or of electricity –
in all directions. To return to Einstein’s dissertation: given the conditions of the problem,
that is, the presuppositions indicated at the outset of the work and the geometry of the
situation which Einstein laid down, the isotropic feature restricts the kind of motions
that the rigid spherical body can undergo. The limitation thus imposed allows Einstein
to obtain the boundary conditions he needed for proceeding with the calculation of the
dilatational motion of the liquid.
As we have noted, Einstein follows very closely the text of Kirchhoff to which he
in fact refers. An examination of Kirchhoff’s work shows that he applied symmetry to
hydrodynamic calculations, although not in the lecture which Einstein cites.30 In his
essay of 1869 on the motion of a rotational body immersed in a fluid whose rotational
axis is not fixed in one plane, Kirchhoff begins with a very complex system of equations
of motion. He stipulates that the liquid is without friction, incompressible, and homogeneous and, since he refers to Thomson and Tait, it stands to reason that he regards
the fluid as isotropic.31 Having set the problem, he proceeds to apply symmetry to simplify the equations. The forces, according to Kirchhoff’s argument, vanish due to the
symmetrical distribution of the mass of the rotating body with respect to the axis of
29
Thomson and Tait 1867, 517–518; see also 519. For the second edition, see Thomson and
Tait 1883, 217; see also 216, 218–219. The opposite term is eolotropy (sometimes called anisotropy): “a substance which ... exhibits differences of quality in different directions ...” (ibid., 217).
Thus, in the case of eolotropy, electric, optical, or other physical qualities are affected by change
of position, as when the refractive property of a transparent body is not the same in all directions.
30
Kirchhoff also applied symmetry extensively in his discussion of elasticity; see Kirchhoff
1883, 389–390, 395.
31
Kirchhoff [1869]/1882, 376.
447
rotation.32 In a later step of the calculation he sought to reduce the number of constants
of the equations of motion when the body in question exhibits symmetry properties.33
And then he makes another crucial move when he notes that if the surface of the body
is symmetrical in relation to some plane, say, the xz-plane, that is, when some point on
the body whose coordinates are x, y, z, corresponds to another point on the body whose
coordinates are x, −y, z, then the distribution of mass may be calculated on the basis
of this presupposition with the result that the number of equations of motion is reduced
substantially.34 Kirchhoff concludes the analysis with a general claim that the resultant
equation is not restricted to bodies of rotation; rather, it applies to all bodies as long as
they are symmetrical in relation to two or more pairs of points located in two orthogonal
planes with respect to the x-axis.35
Kirchhoff applies different notions of symmetry to the fluid as the medium and to
the bodies which are immersed in it. The fluid is isotropic, while the symmetry which
the bodies exhibit determines the way the masses are distributed in them with respect to
the axis of rotation, and this physical feature fixes the number of equations required for
the description of the motion of the bodies. However, in both cases, Kirchhoff’s usage
of symmetry is essentially geometrical. To be sure, the claim is physical: the forces, Kirchhoff argues, cancel each other, but note that this is due to the geometry of the problem.
This is one of the ways Kirchhoff’s analysis of 1869 succeeds in reducing the number
of equations. He elaborated this result in Lecture 19 of his Mechanik – the same book to
which Einstein refers – where in § 3 he demonstrates the symmetrical quality of a solid
of revolution.36
In his paper, “On hydrokinetic symmetry” (1884), Joseph Larmor (1857–1942), Lucasian Professor of Mathematics at Cambridge from 1903 to 1932,37 singled out this
result of Kirchhoff as remarkable. The simplified form of the set of equations
applies to any solid which is symmetrical with respect to two planes through its axis at
right angles to one another, and also with respect to two other such planes through the
same axis. Thus it applies to a right prism or pyramid standing on a square or regular
hexagonal base. Such a solid may be said to have the character of a solid of revolution.38
32
Kirchhoff [1869]/1882, 377: “Um die Integration der Differentialgleichungen, die unter
diesen Voraussetzungen gelten, zu ermöglichen, wird dann weiter angenommen werden, dass die
Kräfte, die auf den Körper wirken, verschwinden, dass die Oberfläche des Körpers eine Rotationsfläche und die Vertheilung der Masse in ihm symmetrisch zur Rotationsaxe ist.”
33
Kirchhoff [1869]/1882, 388: “... es soll jetzt untersucht werden, wie diese Zahl sich verringert wenn der Körper gewisse Symmetrien darbietet.”
34
Kirchhoff [1869]/1882, 389: “Es werde nun angenommen, dass die Oberfläche des Körpers
symmetrisch in Bezug auf die xz–Ebene ist, d. h. dass, wenn x, y, z die Coordinaten eines Punktes
derselben sind, sie auch den Punkt x, −y, z enthält.” See also pp. 390–391.
35
Kirchhoff [1869]/1882, 392: “... diese Gleichung ... auch gilt, wenn der Körper kein Rotationskörper ist, sobald er nur symmetrisch ist in Bezug auf zwei oder mehr Paare auf einander
senkrechter Ebenen, die durch die x-Axe gehn.”
36
Kirchhoff 1883, 243.
37
On Larmor as a Lucasian Professor of Mathematics, see Warwick 2003.
38
Larmor [1884]/1929, 77 (italics in the original). This follows, almost verbatim, Kirchhoff
1883, 243.
448
Larmor sought to extend this result “to solids whose cross-sections are any regular figures, and to any regular solids, respectively.”39 For this purpose, he considers more
complex systems that have more intricate symmetrical properties.
Larmor begins by observing that the theory for the motion of a solid in an ideal,
infinite, frictionless, incompressible fluid without circulation is based on obtaining the
expression for the kinetic energy of the system. This expression is a homogeneous quadratic function of the six velocity components of the solid that involves twenty-one
coefficients, whose values depend on the shape of the solid and the distribution of its
mass. Larmor uses the notation u, v, w for the components of velocity of translation,
and p, q, r for those of the velocity of rotation of the solid immersed in the fluid. He
assumes the single condition that the solid retains the same relations to the axes when
the axes of x and y are turned around the axis of z through a definite angle α in either
direction. He then transforms the equation for the kinetic energy of the system to the
new axes. Given these conditions, the transformed equation must be equal for +α and
−α, and the form of the equation, which Larmor marks as (2), becomes,
2T = A(u2 +v 2 ) + Cw 2 + A (p 2 + q 2 ) + C r 2 + 2L(up + vq)
+2M(vp − uq) + 2N wr,
where A, C, A , C , L, M, N are the remaining coefficients.40 Larmor then makes the
following insightful observation:
Let us suppose that there exists another axis situated anyhow, which possesses the same
kind of symmetry as the axis of z, i.e. which has the character of helicoidal symmetry.
Turning the axes of x and y round that of z does not affect the form of (2), as up + vq
and vp − uq are obviously invariants for such a transformation. Let us therefore suppose
them placed, so that the other axis of helicoidal symmetry lies in the plane xz; then by
turning the axes of x and z round that of y through a certain angle φ, the new axis of x
will be this axis of helicoidal symmetry, and the expression for the energy will remain of
the type (2), but with the axes permuted. The process is conducted as before, and easily
shows that the energy must be of the form
2T = A(u2 + v 2 + w 2 ) + A (p 2 + q 2 + r 2 ) + 2L(up + vq + wr),
and therefore (since each of these three terms is an invariant for all rotations), that the
solid is what Sir W. Thomson calls an isotropic helicoid.
Larmor adds:
It follows, therefore, that if a solid possesses the character of helicoidal symmetry about
any two intersecting axes, it is an isotropic helicoid; and, as a particular case, if it possesses the character of perfect symmetry about two intersecting axes, it possesses the
perfect isotropic character of a sphere....
39
40
Larmor [1884]/1929, 78 (italics in the original).
Larmor [1884]/1929, 77–78.
(3)
449
As examples of perfect isotropy we have all regular solids (with their edges and corners symmetrically blunted, so as to avoid discontinuous motions in the fluid), and all
other bodies with the same characteristics of symmetry.
These results of Larmor in 1884 – following on the works of Thomson and Tait, and
Kirchhoff – indicate that symmetry as a geometrical property was assumed to have physical consequences for the motion of a body in a fluid, thus establishing a link between
the geometry of the body and its physical properties, captured by the term isotropy.
The usage of this physical sense of symmetry became fairly common in physics ca.
1900; hence, no innovation is involved in appealing to it. Thus Einstein’s application of
this kind of symmetry may not have originated in his reading of Kirchhoff. We note that
Einstein appeals to symmetry in an argument which follows Kirchhoff closely despite
the fact that Kirchhoff did not apply symmetry to these basic hydrodynamic equations.
Furthermore, we suggest that Einstein made the assumption of isotropy explicit – calling
it symmetry – a move which Kirchhoff had not taken. In other words, Einstein extended
the meaning of symmetry to include isotropy, thereby articulating the presupposition
that the equations of motions for the three orthogonal components of the motion of the
solvent are the same. But, as we have just seen in the case of Larmor, such a move had
been already made in the literature 20 years earlier.
Einstein’s extension of the meaning of symmetry appears to be entirely reasonable
in view of discussions by Voigt at the turn of the century (to which Einstein may have
referred in one of his letters).41 In fact, Einstein’s first steps in setting up the problem he
addresses in the dissertation follow very closely the analysis that Voigt presents concerning the mechanics of deformable body in the revised edition of 1901 of his Mechanik
(1889).42 In the section that deals with the fundamental equations for an incompressible
fluid with friction – which is the case that Einstein addresses in the dissertation – Voigt
makes a connection between symmetry and isotropy:
In the first section of this part we indicated that, for each system of deformations within the
imperceptibly small region B,43 a coordinate system, X0 , Y 0 , Z 0 (the so called principal
system of dilatation axes), exists, in relation to which the angular changes yz0 , zx0 , xy0 and
hence also their velocities (yz0 ) , (zx0 ) , (xy0 ) disappear, which is connected with the fact
that the deformations group themselves symmetrically about this system of axes....
41
See n. 12, above.
Voigt 1901, 346 ff. (§ 29. Unendlich kleine stetige Verrückungen in einem deformirbaren
Körper; Deformationen.)
43
This region is the same region which Einstein marks as G in the dissertation (1905b, 187;
Beck 1989, 105): “... around this point there is demarcated a region G that is so small that within it
only the linear terms of this development must be taken into consideration.” Cf. Voigt 1901, 347.
42
450
But, since in fluids, as in all isotropic bodies, the same physical values apply in all
directions, for every system of velocities of deformation and the resulting pressure forces,
the systems of the principal dilatations and of the principal pressure axes must coincide.44
Voigt argues that one may proceed with the analysis of an incompressible fluid with
friction on the assumption that the fluid behaves like an isotropic body. In an earlier
work, while discussing the dynamics of an ideal fluid, he remarked that
Now, if both of the symmetry elements coincide, as, for example, is always the case when
the rigid body is homogenous, it can be inferred that when both the directions and velocities of translation or rotation of a body are of equal value there result equal values of the
entire kinetic energy [lebendigen Kraft].45
And further on, in a discussion on the elastic forces in an isotropic medium, he adds that
Even though ... the following developments with respect to the crystal groups are formulated on a general level, i.e., no special or simplifying assumptions about their symmetry
relations are introduced, yet it seems that the application or limitation to considerations
of isotropic bodies is desirable.46
Hence we see that Voigt explicitly assumed a connection between isotropy and symmetry.
Another example of the usage of symmetry in physics is found in Voigt’s book, Die
fundamentalen physikalischen Eigenschafter der Krystalle in elementarer Darstellung
(1898), whose first chapter is entitled: “Die physikalische Symmetrie homogener krystallinischer Materie.”47 This is a book on the physics of crystals, and it differs from that
of Arthur Schönflies (1852–1928), published some seven years earlier, Krystallsysteme
und Krystallstructur, which is essentially an algebraic study of crystallography (we will
return to it later). In this treatise Voigt offers definitions of geometrical and physical
symmetry (making use of the term isotropy) and, in the first chapter, he assesses the
relation between these two kinds of symmetry. He first sets up the geometrical meaning
44
Voigt 1901, 475: “Wir haben im ersten Abschnitt dieses Theiles gezeigt, dass für jede System
von Deformationen innerhalb des unmerklich kleinen Bereiches B ein Coordinatensystem X0 , Y 0 ,
Z 0 , das sogenannte Hauptdilatationsaxensystem, existirt, in Bezug auf welches die Winkeländerungen yz0 , zx0 , xy0 und demnach auch ihre Geschwindigkeiten (yz0 ) , (zx0 ) , (xy0 ) verschwinden, womit
zusammenhängt, dass die Deformationen sich symmetrisch um dieses Axensystem gruppiren....
Da aber, wie bei allen isotropen Körpern, so auch bei Flüssigkeiten alle Richtungen unter
einander physikalisch gleichwerthig sind, so müssen in ihnen für jedes System von Deformationsgeschwindigkeiten und dadurch hervorgerufenen Druckkräftern das Hauptdilatations- und
das Hauptdruckaxensystem zusammenfallen.”
45
Voigt 1895, 280–281: “Fallen nun die Symmetrieelemente beider zusammen, was z. B.
stets stattfindet, wenn der starre Körper homogen ist, so kann man schließen, daß nach gleichwertigen Richtungen gleiche Translations- oder Rotationsgeschwindigkeiten des Körpers auch gleiche
Werte der gesamten lebendigen Kraft ergeben.”
46
Voigt 1895, 337: “Wenn nun auch ... die folgenden Entwickelungen bezüglich der Krystallgruppen meist durchaus allgemien gehalten, also keine speziellen vereinfachendenAnnahmen über
ihre Symmetrieverhältnisse eingeführt werden sollen, so erscheint doch mitunter die Anwendung
oder Beschränkung der Betrachtungen auf isotrope Körper erwüscht.”
47
Voigt 1898, 5.
451
of symmetry with respect to crystals by defining Symmetriecentrum, Symmetrieebene,
Symmetrieaxe and finally Spiegeldrehaxe.48 After explaining the geometrical meaning
of these terms, he poses the question: “From what features can we infer the laws of
the physical symmetry of crystals?” [Aus welchen Merkmalen können wir die Gesetze
der physikalischen Symmetrie der Krystalle erschliessen.] He thus makes it clear that
he expects some relation to hold between the geometry of the crystal and its physical
properties. In particular,
In the great majority of cases the symmetry of the crystalline form can be directly equated
with that of the [crystal’s] physical behavior. Hence, the directions that are of equal value
crystallographically are also of equal value physically....49
Here we have an unambiguous connection between geometrical symmetry and physical symmetry. Later in the book the expression, physikalische Symmetrie, occurs fairly
often.50 Isotropy, in the sense of physical symmetry, also occurs, but it seems that this
term is assigned to a body and not necessarily to a crystal; in any event, it is rarely
invoked.51
Returning to the issue at hand, we note that Einstein avoids both the term isotropy
and the expression physical symmetry.52 But this is also the approach that Voigt took
in 1895 in his study of the motion of rigid bodies. In the introduction to the elements
of symmetry in physical laws governing crystals, he states the following fundamental
hypothesis:
The symmetry of physical behavior is never less than the symmetry of the growth phenomena, which usually expresses itself in [the symmetry] of the crystalline form, so that
directions that are of equal magnitude crystallographically are also of equal magnitude
physically.53
This seems to be precisely the sense in which Einstein is using symmetry in this case,
that is, the same physical behavior is exhibited in each orthogonal direction. In any event,
for us the crucial point is that the literature in physics at the turn of the century abounds
with usages of symmetry, both geometrical and physical, and physical symmetry was
used in the sense of isotropy.
48
Voigt 1898, 7. Here Voigt builds on his earlier analysis of the symmetry elements that a
crystal possesses, as expressed in physical laws: see Voigt 1895, 129ff.
49
Voigt 1898, 9: “... lässt sich in den bei weitem meisten Fällen die Symmetrie der Krystallform direct derjenigen des physikalischen Verhaltens gleichsetzen, sodass also krystallographisch
gleichwerthige Richtungen auch physikalisch gleichwerthig sind....”
50
See, e.g., Voigt 1898, 108.
51
See, e.g., Voigt 1898, 44.
52
As far as we can tell, the first occurrence of isotropy in Einstein’s published papers is in
Einstein and Laub 1908a, 535.
53
Voigt 1895, 128: “Die Symmetrie des physikalischen Verhaltens nie geringer ist, als die Symmetrie der Wachstumserscheinungen, die sich meist in derjenigen der Krystallform ausdrückt, so
daß also Krystallographisch gleichwertige Richtungen jedenfalls auch physikalisch gleichwertig
sind.”
452
For another prominent example from that time, let us consider the views of Henri
Poincaré (1854–1912) who, in 1899, drew an explicit distinction between symmetry
and isotropy in the context of discussing the Zeeman effect, as interpreted according to
Lorentz’s theory. He argues:
Our equations ought to stay the same whatever the orientation of the plane of the wave,
since the medium is isotropic. Moreover, the medium is not just isotropic without [mirror]
symmetry (as, for example, oil of turpentine), it is symmetrical as well as isotropic.54
Poincaré considers here an isotropic medium, with no distinction between left and right.
To demonstrate that isotropy does not imply that the medium admits a mirror symmetry, Poincaré calls attention to the example of oil of turpentine, a medium that is both
homogeneous and isotropic as well as optically active – which means that it rotates the
plane of polarization of a beam of polarized light in a preferred direction. (The direction
differs with different types of the oil.) To avoid such a failure of mirror symmetry in the
case of the Zeeman effect, Poincaré says that we must also require that the equations
governing the effect are unchanged when we transform, say, from a right-handed to a
left-handed coordinate system. Hence, for this effect, Poincaré stated that “the equations
ought not to change when one replaces the system of axes by a symmetrical system with
respect to the origin.”55 In a prior section, where he analyzes isotropy in plane waves,
Poincaré implies that the term symmetrical designates mirror symmetry:
The medium is not just isotropic, it is symmetrical; therefore, our equations ought not
to change when one replaces our system of axes by a symmetrical system (the plane of
symmetry being the xz-plane, for example).56
Symmetry, then, for Poincaré refers to the relation of pairs of coordinate axes, or of positions with respect to a coordinate system, under whose transformation the properties of
the system are preserved, and it is not the same as isotropy (as Thomson and Tait defined
it).
In Lorentz’s lectures of 1909 on the theory of electrons he asserts that
True isotropy, i.e. perfect equality of properties in all directions, can never be attained by
a finite number of separated particles. It is only when we are content with the explanation
of triplets, that no difficulty arises from this circumstance, because in this case equality
of properties with respect to three directions at right angles to each other will suffice for
our purpose.57
54
Poincaré [1899]/1954, 455: “Nos équations doivent rester les mêmes quelle que soit l’orientation du plan de l’onde, puisque le milieu est isotrope; de plus le milieu n’est pas seulement
isotrope sans symétrie (comme l’essence de térébenthine par example), il est isotrope et symétrique.”
55
Poincaré [1899]/1954, 455: “Les équations ne doivent donc pas changer quand on remplace
le système des axes par un système symétrique par rapport à l’origine.”
56
Poincaré [1899]/1954, 453: “Le milieu n’est pas seulement isotrope, il est symétrique; nos
équations ne doivent donc pas changer quand on remplace notre système d’axes par un système
symétrique (le plan de symétrie étant le plan des xz, par example).”
57
Lorentz [1909]/1916, 120.
453
Clearly, for Lorentz isotropy is a well defined physical concept and he does not suggest
that it implies mirror symmetry.
While Poincaré sought to adhere to the distinction between isotropy and symmetry,
we have seen that Larmor and Voigt did not make a comparable effort. It is our view that
the absence of a privileged direction may have been called symmetry at the time, a term
used in physics in a variety of contexts without being sharply defined. For these reasons
we conclude that symmetry in Case 1 has the sense of isotropy.
Case 2: analogy
The second instance of the application of symmetry in the dissertation occurs when
Einstein makes use of the feature of linearity. He first stipulates that the functions u, v,
and w must satisfy the hydrodynamic equations and should therefore include internal
friction while neglecting inertia. On the basis of Kirchhoff’s work, he then establishes
a relation between the hydrostatic pressure and the velocity components. The system
of equations that describes the motion of the liquid is assumed to be linear. Exploiting
the feature of superposed solutions, Einstein then obtains three constants that relate the
velocity functions, u, v, and w, to the hydrostatic pressure, p, and ρ; he originally had
the sum of these three constants, A, B, and C vanish due to the incompressibility of the
liquid. Through a process of substitution in the relevant equations Einstein shows that the
formula he obtained, relating the constants of the velocity components to the hydrostatic
pressure, satisfy the boundary condition, ρ = P for the velocity component, u, that is, u
vanishes in this new set of equations. This result is consistent with the claim concerning
the boundary conditions which Einstein made at the outset of the calculation.58 At this
juncture Einstein remarks:
We see that u vanishes for ρ = P . For reasons of symmetry [aus Symmetriegründen]
the same holds for v and w. We have now demonstrated that equations (5) satisfy equations
(4) as well as the boundary conditions of the problem.59
As noted, the boundary conditions are those that were stipulated at the outset, that is,
the coordinate components of the velocity of the liquid vanish when ρ is set to equal P .
Einstein shows that the equations he has developed with the linearity restriction indeed
obey these boundary conditions, but now he has the additional equations that correlate
the pressure p with the constants of the velocity components. Apparently, the appeal to
symmetry here refers to the equations, that is, to the analogous character of the equations
for u, v, and w – they are symmetrical in the algebraic sense of substitution of corresponding elements in the equations. In other words, since one computes the values for u,
v, and w in the same way, the result should be the same in each case. While this second
58
There is a typographical error in the dissertation which Einstein corrected in the version he
published in the Annalen der Physik in 1906. See Stachel et al. 1989, 203 n. 19.
59
Beck 1989, 110 (boldface added); Einstein [1905b]/1989, 191: “Man erkennt, dass u für
ρ = P verschwindet. Gleiches gilt aus Symmetriegründen für v und w. Es ist nun bewiesen,
dass durch die Gleichungen (5) sowohl den Gleichungen (4) als auch den Grenzbedingungen der
Aufgabe Genüge geleistet ist.”
454
application of symmetry appears to be different from the first usage which is physical,
it is nonetheless dependent on the geometrical situation and the physical assumptions
pertaining to the three axes of motions.
It is somewhat surprising that Einstein appeals here to symmetry, where all that is
required is a straightforward substitution. It may be the case that Einstein alludes to the
first usage of symmetry to which the algebraic usage ultimately relates. However, if he
had assumed this first instance of symmetry as well as the boundary conditions in this
second instance, he would have been involved in a circular argument. The solutions of
the equations for v and w are completely analogous with the solution for u.
Case 3: geometrical usage
Later in the dissertation, Einstein considers the liquid and the suspended sphere
together and calls these combined elements “the mixture”. The equation, he writes,
“gives the impression that the coefficient of viscosity of this inhomogeneous mixture ...
is smaller than the coefficient of viscosity k of the liquid.” Einstein argues that this is
not so, and continues:
For reasons of symmetry [Aus Symmetriegründen], it follows that the principal dilatation directions of the mixture are parallel to the directions of the principal dilatations A,
B, C, i.e., to the coordinate axes.60
In effect, the geometry of the physical situation is such that the dilatation direction of
motion of the mixture is the same as that of the liquid. This usage of symmetry clearly
refers back to the first instance but now the geometrical aspect of symmetry is enhanced.
This allows Einstein to proceed with the calculation which gives the result:
k ∗ = k(1 + ϕ)
(1)
where k ∗ is the coefficient of viscosity of the inhomogeneous mixture and ϕ denotes the
fraction of the volume that is occupied by the spheres. The equation, then, determines
the ratio between the coefficients of viscosity of a liquid with or without the suspended
sphere. Einstein concludes the section:
If very small rigid spheres are suspended in a liquid, the coefficient of internal friction
increases by a fraction that is equal to the total volume of the spheres suspended in unit
volume, provided that this total volume is very small.61
60
Beck 1989, 115 (boldface added); Einstein [1905b]/1989, 196: “Aus Symmetriegründen
folgt, dass die Hauptdilatationsrichtungen der Mischung den Richtungen der Hauptdilatationen
A, B, C, also den Koordinatenrichtungen parallel sind.”
61
Beck 1989, 117; Einstein [1905a]/1989, 198: “Werden in einer Flüssigkeit sehr kleine starre
Kugeln suspendiert, so wächst dadurch der Koeffizient der inneren Reibung um einen Bruchteil,
der gleich ist dem Gesamtvolumen der in der Volumeneinheit suspendierten Kugeln, vorausgesetzt,
dass dieses Gesamtvolumen sehr klein ist.”
455
These three instances of symmetry in the process of calculating the “real [wahren]
sizes of molecules” were part and parcel of a new technique that Einstein introduced in
his dissertation. As he notes at the outset of this research work, the sizes of molecules
were determined by applying the kinetic theory of gases and no attempt had been undertaken to establish them on the basis of hydrodynamic principles. Einstein explains that
“this is no doubt due to the fact that it has not yet been possible to overcome the obstacles
that impede the development of a detailed molecular-kinetic theory of liquids.”62 Given,
however, the internal friction of the solution and the pure solvent, the diffusion of the
dissolved substance within the solvent, and provided that the volume of the molecule of
the dissolved substance is large compared with the volume of the molecule of the solvent
(so that the solvent can be treated as a continuum), one could – according to Einstein
– overcome the impediments and calculate hydrodynamically the molecular radii and
Avogadro’s number. In order to determine these two unknowns, Einstein had to come
up with two equations. The equation that expresses the ratio of the two coefficients of
viscosity is the first equation, and the second arises from combining the expression from
Stokes’s law for the terminal velocity of a sphere moving in a liquid and Van ‘t Hoff’s
law for the osmotic pressure.63
Einstein’s brilliant mastery of the complexity of the mathematical methods in hydrodynamics was not error-free. In view of the experimental work of Jean Perrin (1870–
1941), Einstein published a note in 1911 in the Annalen der Physik indicating that the
original equation (1) for the ratio of the coefficients of viscosity is offset by an amount
2.5 times the volume ϕ. The correct equation, replacing equation (1), is given as:
k ∗ = k(1 + 2.5ϕ).64
(2)
In the context of our argument it is important to note that this miscalculation did not
arise from an erroneous application of symmetry. It is also worth noting that the calculation, which explicitly appeals three times to symmetry, is applicable to a wide range
of phenomena concerning the deformation and fluid properties of particles suspended in
liquid. In fact, it has been asserted that the dissertation is the most cited paper of Einstein
with wide applications in industry and ecological research.65
Assessing these three instances of symmetry, we may conclude that the first instance
is physical, essentially expressing the feature of isotropy; the second is algebraic in the
62
Beck 1989, 105; Einstein [1905b]/1989, 186: “Die ältesten Bestimmungen der wahren
Grösse der Moleküle hat die kinetische Theorie der Gase ermöglicht, während die an Flüssigkeiten
beobachteten physikalischen Phänomene bis jetzt zur Bestimmung der Molekülgrössen nicht gedient haben. Es liegt dies ohne Zweifel an den bisher unüberwindlichen Schwierigkeiten, welche
der Entwickelung einer ins einzelne gehenden molekularkinetischen Theorie der Flüssigkeiten
entgegenstehen.”
63
For analyses of the physics of the problem, see Stachel et al. 1989, 170–182, and Pais 1982,
88–92.
64
Einstein 1911, 592. Cf. eq. [1], above. It is perhaps of interest that Einstein did not find
the error himself. He relates that he asked his sometime collaborator, Ludwig Hopf, to check the
calculations and, indeed, Hopf found a mathematical error which Einstein describes as one that
affects the result considerably (Einstein 1911, 591). On Hopf, see Pais 1982, 485.
65
See Pais 1982, 89–90.
456
sense that properties of equations that express the physics of the system are at issue, and
symmetry refers to the analogous form of these equations; and, finally, the third is unmistakably geometrical, referring to the geometry of the coordinate systems. The symmetry
that Einstein discerns in the equations for the velocity components of the dilatational
motion originates in the symmetry of the physical conditions, thereby giving the phenomena priority over the formalism. Einstein accepted the extension of the meaning of
the term symmetry to include both geometrical isotropy and algebraic substitution where
physical elements are involved. Moreover, while we have here illustrations of three types
of symmetry, the algebraic symmetry, the second instance, can be seen as expressing the
physics of the geometrical arrangement of the liquid and the sphere immersed in it. We
will later see that algebraic symmetry can express purely mathematical properties that
are not based on physical considerations. This, however, is not the case here.
We now proceed to examine the relativity paper which was received by the Annalen
der Physik on June 30, 1905 – exactly two months after Einstein dated his dissertation –
and published in September 1905.66
2. “On the electrodynamics of moving bodies” (June 1905)
There are six instances of symmetry in the relativity paper, an abundance which is
noteworthy in view of the absence of such usages in any of the papers that Einstein published in 1905 other than his dissertation. We have organized them in the order of their
importance, beginning with a simple case of “indifference” and culminating with the
radical move of making asymmetry disappear. For purposes of our discussion we have
combined two usages into a single instance: in this case Einstein adds an explanatory
footnote to the text and appeals to symmetry in both text and footnote; however, as we
shall see, the two usages are distinct.
Case 1: indifference
Having postulated the two principles that underpin his theory, namely, the principle
of relativity and the principle of the constancy of the velocity of light, and then discussing
in general terms the relativity of lengths and times that follow from these two principles,
Einstein proceeds to § 3: “The theory of transformation of coordinates and time from a
system at rest to a system in uniform translational motion relative to it”. He begins the
section by setting up the coordinate systems: the X-axes of the two systems coincide
and their Y - and Z-axes are parallel. The two coordinate systems are each provided with
a rigid measuring rod and a number of clocks such that the two systems are exactly
alike.67 He continues:
66
Stachel et al. 1989, 306.
Einstein 1905c, 897: “Jedem Systeme sei ein starrer Maßstab und eine Anzahl Uhren beigegeben, und es seien beide Maßstäbe sowie alle Uhren beider Systeme einander genau gleich.”
67
457
The origin of one of the two systems (k) shall now be imparted a (constant) velocity v in
the direction of increasing x of the other system (K), which is at rest, and this velocity
shall also be imparted to the coordinate axes, the corresponding measuring rod, and the
clocks. To each time t of the system at rest K there corresponds then a definite position of
the axes of the moving system, and for reasons of symmetry we may rightfully assume
that the motion of k can be such that at time t (“t” always denotes a time of the system at
rest) the axes of the moving system are parallel to the axes of the system at rest.68
Es werde nun dem Anfangspunkte des einen der beiden Systeme (k) eine (konstante)
Geschwindigkeit v in Richtung der wachsenden x des anderen, ruhenden Systems (K)
erteilt, welche sich auch den Koordinatenachsen, dem betreffenden Maßstabe sowie den
Uhren mitteilen möge. Jeder Zeit t des ruhenden Systems K entspricht dann eine bestimmte Lage der Achsen des bewegten Systems und wir sind aus Symmetriegründen
befugt anzunehmen, daß die Bewegung von k so beschaffen sein kann, daß die Achsen
des bewegten Systems zur Zeit t (es ist mit “t” immer eine Zeit des ruhenden Systems
bezeichnet) den Achsen des ruhenden Systems parallel seien.69
Einstein designs the two coordinate systems such that they are internally identical,
indeed they are “exactly alike” as he explicitly demands. The only difference between
them is that one is set into motion, inertial motion that is, with no dynamic considerations
whatsoever. The situation is entirely idealized and no forces are involved. As the editors
of the Collected Papers of Einstein rightly observe, “Einstein evidently was attempting
to assure that initially identical measuring rods and clocks in the two inertial frames
measure equal spatial and temporal intervals, respectively, once the two frames are in
relative motion.”70 It is arbitrary to say which frame is moving and which remains at
rest; in sum, neither frame is privileged – the two physical situations are indistiguishable or, one might say, it is a matter of indifference. This is all assumed in the way the
problem is set up and the appeal to symmetry refers to the fact that the physical situation
is indifferent with respect to either frame. This usage of symmetry is reminiscent of the
way Thomson and Tait defined homogeneity (see Section II.1, Case 1, above). Again,
the point of this usage is to call attention to the fact that the setup is symmetrical – it is
not a consequence; rather it is built into the system – to be used, as we will see, in Case
3 (Section II.2, below). The expression: “for reasons of symmetry, we may rightfully
assume...”, is intended to convey the idea that, given the setup, symmetry (and homogeneity of space) are assumed. With the conditions stated by Einstein, the relevant axes
of the two identical systems would remain parallel to one another and in fact no appeal
to symmetry is required.
68
69
70
Beck 1989, 146 (boldface added).
Einstein 1905c, 897 (boldface added).
Stachel et al. 1989, 308 n. 14.
458
Einstein then concludes that
To every system of values, x, y, z, t that determines completely the place and time of an
event in the system at rest, there corresponds a system of values ξ, η, ζ, τ that fixes this
event relative to the system k....71
The problem is now to find the system of equations which connects these two sets of
values. It is noteworthy that Einstein first appeals to the linear property of the equations
which he infers from the homogeneity of space as well as time.72 Recall the way the
system of equations of motion is treated in hydrodynamics where, as we have seen, the
fluid is considered homogeneous to ensure, together with some additional conditions,
that the fluid is isotropic; the similiarity is striking. The point is worth stressing: Einstein
presupposes homogeneity of space (and time) much as he presupposes the two postulates of the theory. This presupposition is essential for securing the validity of some of
the symmetries that are part of his argument.73 The conditions assumed in the problem
guarantee from the outset the parallel relation of the two coordinate systems after one
of them has been set into a translational motion.
It is of interest that in Einstein’s review article on the relativity principle and the
conclusions drawn from it (completed late in 1907), the derivation of the transformation
equations differs from that given in the original relativity paper of 1905. In § 3, headed,
“Transformation of coordinates and time,” Einstein quickly disposes of the idealized
case he presented in the original paper, but this time he does not appeal to symmetry. He
then proceeds to deal with the general case:
yet in general these planes [the coordinate planes of the two systems] will not be perpendicular to each other. However, if we choose the position of x -axis in such a way that
it has, with reference to S, the same direction as the translational motion of S has with
reference to S, then it follows for reasons of symmetry that the S-referred coordinate
planes of S must be mutually perpendicular.74
In the general situation the planes of the coordinate systems are initially not perpendicular to each other. Hence, in choosing a certain position for one of the coordinates such
that its translational motion has the same direction as the system of reference, it is not at
all obvious that the coordinate planes of the two systems would then be perpendicular to
71
Beck 1989, 146; Einstein 1905c, 898: “Zu jedem Wertsystem x, y, z, t welches Ort und Zeit
eines Ereignisses im ruhenden System vollkommen bestimmt, gehört ein jenes Ereignis relativ
zum System k festlegendes Wertsystem, ξ, η, ζ, τ , und es ist nun die Aufgabe zu lösen, das diese
Größen verknüpfende Gleichungssystem zu finden.”
72
We stress that Einstein attributes homogeneity not only to space but to time as well. It seems
to us an extraordinary assumption to endow time with the property of homogeneity.
73
For another set of presuppositions in the relativity paper concerning the logical properties
of the definition of the synchronization of clocks, see Holton [1973]/1988, 194.
74
Beck 1989, 258–259 (boldface added); Einstein 1907b, 418 (boldface added): “doch werden
diese Ebenen im allgemeinen nicht aufeinander senkrecht stehen. Wählen wir jedoch die Lage
der x -Achse so, daß letztere – auf S bezogen – die gleiche Richtung hat, wie die auf S bezogene Translationsbewegung von S , so folgt aus Symmetriegründen, daß die auf S bezogenen
Koordinatenebenen von S aufeinander senkrecht stehen müssen.”
459
each other. This is a generalization of the simple, idealized case from 1905 and invoking symmetry under these general circumstances makes the argument transparent. The
imposition of symmetry secures the physical condition such that the coordinate systems
retain their alignment.
In concluding this case, we call attention to the fact that later in this same section of
the paper, Einstein introduces K , a third coordinate system, to the setup and once again
appeals to the symmetry of the arrangement.
Case 2: two algebraic usages
Einstein develops the theory of special relativity in two parts: kinematic and electrodynamic. This is one of his many innovations, namely, establishing electrodynamics on
kinematic principles. As we have noted, after he derived the required propositions of the
kinematics from first principles, he proceeded to apply them in electrodynamics. The
first step he takes in Part II (§ 6) of the paper is to address the transformation of the Maxwell–Hertz equations for empty space. Einstein applies the transformation equations –
obtained earlier in the kinematic part – to the Maxwell–Hertz equations, referring to the
electromagnetic processes observed in the coordinate system which moves with velocity v. He then reconsiders the equations resulting from this calculation in light of the
relativity principle which demands that the Maxwell–Hertz equations for empty space
be valid for both coordinate systems, the moving system k and the stationary system K.
This amounts to the claim that the vectors of the electric force (in Einstein’s notation:
X , Y , Z ) and the vectors of the magnetic force (L , M , N ) of the moving system k
satisfy two sets of equations which must coincide in structure, apart from the symbols
representing the vectors. A consequence of this result, which is the immediate product of
applying the relativity principle and its associated transformation equations, is that these
two sets of equations – one for electricity and one for magnetism – must coincide up to a
velocity dependent factor which Einstein designates as the function ψ(v). This function
replaces the functions in each system of equations and is independent of location and
time.
Einstein inserts ψ(v) into the two sets of equations and obtains
X = ψ(v)X,
Y = ψ(v)β[Y − (v/V )N ],
Z = ψ(v)β[Z + (v/V )M],
L = ψ(v)L,
M = ψ(v)β[M + (v/V )Z],
N = ψ(v)β[N − (v/V )Y ].
Where
√ V stands for the velocity of light in vacuum, and β the relativistic constant:
1/ [1 − (v/V)2 ]. Einstein then continues:
If we now invert this system of equations, first, by solving the equations just obtained and,
second, by applying the equations to the inverse transformation (from k to K) which is
characterized by the velocity −v, we obtain, if we take into account that the two systems
of equations so obtained must be identical,
ψ(v) · ψ(−v) = 1.
Further, it follows from reasons of symmetry1 that
460
ψ(v) = ψ(−v);
thus
ψ(v) = 1, ....75
Bildet man nun die Umkehrung dieses Gleichungssystems, erstens durch Auflösen der soeben erhaltenen Gleichungen, zweitens durchAnwendung der Gleichungen auf die inverse
Transformation (von k auf K), welche durch die Geschwindigkeit −v charakterisiert ist,
so folgt, indem man berücksichtigt, daß die beiden so erhaltenen Gleichungssysteme
identisch sein müssen:
ϕ(v) · ϕ(−v) = 1.
Ferner folgt aus Symmetriegründen1
ϕ(v) = ϕ(−v);
es ist also
ϕ(v) = 1, ....76
Notice that at the place in the text where he appeals to symmetry Einstein directs
the reader to a footnote which we will presently discuss. But first we remark that the
application of symmetry in the text here has nothing to do with either geometry or physics: no axis or directionality is involved. In spite of the fact that a function dependent
on velocity is under consideration, the move is purely mathematical, to be precise –
algebraic. Einstein’s specific goal in this case is to eliminate the function ψ(v) from the
equations. Thus, in spite of the fact that he indicates by −v the transformation from k to
K, the crucial demand is that the two sets of equations be identical under the inversion
of the algebraic sign of the velocity dependent factor, ψ. ”For reasons of symmetry”
here simply means that the function ψ(v) is the same for positive and negative values.
Since this is clearly an algebraic feature, we designate this usage as algebraic.
Turning now to the footnote, we find that Einstein proposes to illustrate his mathematical maneuver by a concrete case. He remarks
75
Einstein 1905c, 909 (boldface added). In the original paper Einstein used ϕ for the function
dependent on v, although earlier in this section he called this function ψ (see p. 908). Einstein
did not correct this inconsistency in the first reprint edition of the paper in 1913, but did so in
the second edition of 1922 (the translation is based on the version of 1922). Note that ϕ(v) is the
function Einstein used in section 3. See Blumenthal 1913, 36 and 42; and Blumenthal 1922, 33
and 40.
76
461
< footnote>1 If, e.g., X = Y = Z = L = M = 0 and N = 0, then it is clear for reasons
of symmetry that if v changes its sign without changing its numerical value, then Y too
must change its sign without changing its numerical value.77
< footnote>1 Ist z. B. X = Y = Z = L = M = 0 and N = 0, so ist aus Symmetriegründen klar, daß bei Zeichenwechsel von v ohne Änderung des numerischen
Wertes auch Y sein Vorzeichen ändern muß, ohne seinen numerischen Wert zu ändern.78
We follow Einstein’s example and set the relevant variables to zero in the above equations, and we get
L = 0,
X = 0,
Y = −ψ(v)β[(v/V )N ], M = 0,
Z = 0,
N = ψ(v)β[N ].
We now solve for N and insert the resulting expression in the equation for Y :
Y = −ψ(v)β[(v/V )N /ψ(v)β]
Since ψ(v)β appears in both the numerator and the denominator, it cancels out. Hence
Y = −(v/V )N .
The calculation demonstrates that Einstein’s claim holds: if we change the algebraic sign
of v, e.g., from plus to minus, without changing its numerical value, Y would indeed
reverse its algebraic sign – in this case from minus to plus – while retaining its numerical value. To make sure that this is indeed the case for other variables, let us consider
X = Z = L = M = N = 0 and Y = 0. We then get the following equations:
X = 0,
Y = ψ(v)β[Y ],
Z = 0,
L = 0,
M = 0,
N = −ψ(v)β[(v/V )Y ].
We solve for Y and insert the resulting expression in the equation for N :
N = −ψ(v)β[(v/V )Y /ψ(v)β].
Again, since ψ(v)β appears both in the numerator and the denominator, it cancels out.
Hence
N = −(v/V )Y .
As before, replacing v by −v results in a change in the algebraic sign of N without
changing its numerical value.
Thus, the appeal to symmetry in this footnote is different from that in the text. While
both are algebraic, they appeal to different features. The instance of symmetry in the
77
78
462
text appeals to the symmetry of the Maxwell–Hertz equations as they undergo the relativistic transformation, whereas the instance of symmetry in the footnote has to do with
the permutation of the variables. The latter may clearly be seen in a later work by the
mathematician, Hermann Minkowski (1864–1909), who cast the Maxwell–Hertz equations in a compact mathematical way so that the symmetry of the equations could be
immediately seen through the permutation of indices.79 We will return to this type of
symmetry later on.
In concluding Case 2, we remark that at the end of § 6, near this usage of symmetry,
Einstein refers to asymmetry. As we will argue, this usage of (a)symmetry has nothing
to do with the usage of symmetry we have just described. Clearly, Einstein appealed to
a variety of meanings of symmetry.
Case 3: physical usage
As we have noted at the end of our analysis of Case 1 in Section II.2, Einstein
appeals once again to the symmetry of the arrangement of the coordinate systems that
he announced at the beginning of § 3. He obtains a set of transformation equations which
involves a certain function ϕ(v) which is not yet known. By calculating the transformation from one system to another with respect to a spherical wave emitted from the
origin of the coordinate system common to both systems at the time of emission, Einstein
demonstrates that the set of equations he found for the transformation is compatible with
the two fundamental principles of the theory. The transformation depends, however, on
this unknown function ϕ(v) and Einstein proceeds to determine it.
For this purpose he introduces a third coordinate system, K , which is also in parallel
translational motion to k but in this case the motion is in the opposite direction, that is,
−v. A twofold application of the transformation equations vis-à-vis the three coordinate
systems allows Einstein to get a set of relations for ϕ(v) and ϕ(−v) to which he adds
the identity transformation [die identische Transformation], namely,
ϕ(v)ϕ(−v) = 1,
based on the observation that K and K are at rest relative to each other. He now explores
the physical consequences of ϕ(v) by examining the property of the length, l, of a rod in
the system K that moves perpendicular to its axis with a velocity v relative to the system
k such that one end lies at the origin of this system and its other, when in motion, at
x1 = vt, y1 = l/ϕ(v), and z1 = 0.
He continues:
The length of the rod, measured in K, is thus l/ϕ(v); this establishes the meaning of the
function ϕ. For reasons of symmetry it is obvious that the length of a rod measured
79
Minkowski 1908, 58–59: “Man bemerkt bei dieser Schreibweise sofort die vollkommene
Symmetrie des ersten wie des zweiten dieser Gleichungssysteme in Bezug auf die Permutationen
der Indizes 1, 2, 3, 4.”
463
in the system at rest and moving perpendicular to its own axis can depend only on its
velocity and not on the direction and sense of its motion. Thus, the length of the moving
rod measured in the system at rest does not change when v is replaced by −v.80
Die Länge des Stabes, in K gemessen, ist also l/ϕ(v); damit ist die Bedeutung der Funktion
ϕ gegeben. Aus Symmetriegründen ist nun einleuchtend, daß die im ruhenden System
gemessene länge eines bestimmten Stabes, welcher senkrecht zu seiner Achse bewegt ist,
nur von der Geschwindigkeit, nicht aber von der Richtung und dem Sinne der Bewegung
abhängig sein kann. Es ändert sich also die im ruhenden System gemessene länge des
bewegten Stabes nicht, wenn v mit −v vertauscht wird.81
Before we explain this usage, note that Einstein is not concerned here with the
physical issue of directionality of motion; rather, his goal is to obtain, by means of a
mathematical procedure, another relation for ϕ(v) in order to eliminate this auxiliary
function from the transformation equations. We have already seen Einstein employing
such a technique in Case 2 of Section II.2 where he eliminated the auxiliary function ϕ.
Indeed, since ϕ(v) is not dependent on directionality, Einstein can set
l/ϕ(v) = l/ϕ(−v), or ϕ(v) = ϕ(−v),
so that now, given the identity transformation (above), ϕ(v) must equal 1, and the celebrated transformation equations of the relativity theory immediately follow.82
A close reading of Einstein’s original text reveals that he is using the verb “vertauschen” to indicate replacing v with −v. As we will elaborate in the next case (and again
in Section III.1), Vertauschung is the very term that has been associated with symmetry
in the domain of algebra.83
Notwithstanding the algebraic mode, this “interchange” in Einstein’s paper is not
just a formal change of symbols. Here the concern is with coordinate systems and their
geometrical relations: axes of two coordinate systems to which a third one has been
added are involved. We therefore claim that the interchange is also geometrical.
Moreover, Einstein argues that the length of the rod does not depend on the direction
of its motion, but only on its velocity: replacing v with −v does not change the value
of the expression for y1 , which means that the length of the rod remains the same. This
claim is clearly connected to the presupposition of the homogeneity of space (and time)
that Einstein set up earlier in § 3, as we have seen in Section II.2, Case 1. Thus the
symmetry here is also related to the isotropic nature of space.
In this case Einstein applies symmetry in a way which draws on its usage in several
domains (algebraic and geometrical as well as physical), thereby extending its range of
80
82
Notice that in his review article of 1907, Einstein explicitly introduced another constraint,
namely, “ϕ(v) = −1 is obviously out of the question.” (Einstein 1907b, 420 n. 2; see Stachel et
al. 1989, 442; Beck 1989, 260.)
83
For the use of vertauschen in the context of interchanging the variables in an equations, see
Voigt 1896, 639. Cf. Boltzmann [1891–1893]/1982, 2: 123.
81
464
meanings. In other words, the importance of this case is Einstein’s appeal to symmetry
in a physical sense, building on previous usages in geometry and algebra.
In fact, Einstein may be responding here to a usage that Voigt introduced some ten
years earlier. In his paper on mechanical features of the medium associated with the
Maxwell–Hertz equations, Voigt asserts a relation between the isotropic nature of the
medium and symmetry:
... but for isotropic media, this usually takes place according to symmetry. In crystalline
[media] deviations from this condition which cannot be observed are possible; the same
is true for the electromagnetic system.84
As we have seen, Voigt connects symmetry with physical reasoning. In another juncture
of this paper, when he remarks on the similarity between the equations that describe the
mechanical features of a medium with no forces and the Maxwell–Hertz equations (as he
referred to the electromagnetic theory) he points out that: “In order to make the expressions more symmetrical, one can treat the medium’s inertia, that is ε, as vanishingly
small.”85
This application of symmetry amounts then to drawing physical consequences from
the isotropic character of space with respect to the relations between coordinate systems. That the algebraic change of sign of the parameter v entails no physical difference
for the systems under discussion is a consequence of the isotropy of space which was
presupposed.
Case 4: algebraic usage
In contrast to the cases analyzed so far where Einstein used the expression, aus Symmetriegründen (for reasons of symmetry), in the present case he applies the adjectival
form: in symmetrischer Weise (in a symmetrical fashion). In § 5 – the concluding section
of the kinematic part of the paper – Einstein proves the addition theorem of velocities
[Additionstheorem der Geschwindigkeiten]. The goal is to demonstrate the consistency
of the theory by showing through calculation that “the composition of two velocities
that are smaller than V always results in a velocity that is smaller than V ,” where V is
the velocity of light in empty space.86
84
Voigt 1894, 670 (italics in the original): “... dies findet aber in isotropen Medien nach
Symmetrie stets statt. In krystallinischen sind Abweichungen von dieser Bedingung möglich, die
sich der Beobachtung entziehen; dasselbe gilt im electromagnetischen System.”
85
Voigt 1894, 668: “Dabei kann man noch, um die Ausdrücke symmetrischer zu machen, die
Trägheit, also ε, des Mediums verschwindend klein annehmen.”
86
Beck 1989, 155; Einstein 1905c, 906: “... aus der Zusammensetzung zweier Geschwindigkeiten, welche kleiner sind als V , stets eine Geschwindigkeit kleiner als V resultiert.” Interestingly
enough, in the relativity paper Einstein does not use the modern symbol for the velocity of light in a
vacuum, namely, c, which Abraham and Wien, among other contemporary physicists, had already
used. See Abraham 1903, 114: “c = Lichtgeschwindigkeit”, and Wien 1904a, 642: “Bezeichnen
wir ... die Lichtgeschwindigkeit mit c....” Rather, Einstein retains the symbol V for this velocity, a
notation that can be found, e.g., in the experimental work of Michelson and Morley (1887, 336):
“Let V = velocity of light”, and in Lorentz’s theoretical work (1895, 17, see also 35–38): “wenn
465
Einstein considers the two coordinate systems, namely, k and K, such that k moves
with velocity v along the X-axis of the stationary system K. He then examines the
motion of a point with velocity w relative to system K. Applying the transformation
equations demonstrated in § 3, he obtains an expression for each of the three orthogonal
components of the moving point in relation to K. He then proceeds with the following
calculation where x, y, z, and t are the resulting expressions of the transformed equations
for the original variables of the point: ξ, η, ζ , and τ :
We put
U 2 = (dx/dt)2 + (dy/dt)2 ,
w 2 = wξ2 + wη2 ,
and
α = arctg wy /wx ; 87
α should then be considered as the angle between the velocities v and w. After a simple
calculation, we obtain
√
U = [(v 2 + w 2 + 2vw cos α) − (vw sin α/V )2 ]/(1 + vw cos α/V 2 ).
It is noteworthy that v and w enter the expression for the resultant velocity in a symmetric
fashion. If w too has the direction of the X-axis ( -axis), we obtain,
U = (v + w)/(1 + vw/V 2 ).88
Wir setzen:
U 2 = (dx/dt)2 + (dy/dt)2 ,
w 2 = wξ2 + wη2
und
α = arctg wy /wx ;
α ist dann als der Winkel zwischen den Geschwindigkeiten v und w anzusehen. Nach
einfacher Rechnung ergibt sich:
√
U = [(v 2 + w 2 + 2vw cos α) − (vw sin α/V )2 ]/(1 + vw cos α/V 2 ).
Es ist bemerkenswert, daß v und w in symmetrischer Weise in den Ausdruck für die resultierende Geschwindigkeit eingehen. Hat auch w die Richtung der X-Achse ( -Achse),
so erhalten wir:
mann mit V ... die Lichtgeschwindigkeit in Aether, bezeichnet.” Poincaré too used this notation:
see Poincaré 1900, 272 (Poincaré [1900]/1954, 483): “Si alors V ... est la vitesse de la lumière....”
But in his Encyklopädie article, Lorentz applied the notation c (1904a, 65): “Lichtgeschwindigkeit
im Äther c.” By 1907 Einstein had also adopted what is now the standard practice (1907b, 412):
“das Verhältnis v/c jener Relativgeschwindigkeit zur Lichtgeschwindigkeit im Vakuum....”
87
The editors of the collected papers of Einstein call attention to an error in Einstain’s equation.
The fraction should have been wη /wξ . See Stachel et al. 1989, 308 n. 20.
88
Beck 1989, 154–155 (boldface added).
466
U = (v + w)/(1 + vw/V 2 ).89
With this famous equation for the addition of velocities Einstein demonstrates the
limiting value imposed by the velocity of light. It can be seen immediately that in the
simple case, when w too is in the direction of the X-axis, the composition of the two
velocities that are smaller than V will always result in a velocity that is also smaller
than V . Thus, the velocity of light cannot be exceeded by compounding it with some
velocity less than that of light. This is, of course, consistent with the requirement of
the theory of relativity. As is well known, Newtonian mechanics has no such limiting
value imposed on the composition of velocities. It is significant that Einstein invokes
symmetry to highlight the nature of this result, one of Einstein’s brilliant and original
moves in the relativity paper.
Einstein notes the symmetrical feature of v and w in the sense that these velocities
can be interchanged without affecting the equation for U – the combined velocity of v
and w. In our view, this is just a side-remark: Einstein does not use symmetry here to
advance his argument; hence, the adjectival form of the term. This side-remark has to
do neither with bilateral nor with rotational symmetry nor, indeed, with any other geometrical form of symmetry. Despite the fact that two coordinate systems are involved
and the direction of the velocity may seem essential, Einstein demonstrates that this
is not so. Notice that neither v nor w (or their “frames”) is privileged; they enter the
equation for U in the same way. This unexpected feature of the equation for the addition
of velocities makes the velocities interchangeable [vertauschbar], a characteristic which
Einstein calls “symmetric”. The usage is algebraic: one can permute the values of the
velocities in the equation without affecting the result. This algebraic use of symmetry is
akin to what we have already seen in Section II.2, Case 2, but here Einstein is explicit
about the feature of permutation.
Symmetry in this algebraic sense was fairly common in works by mathematicians,
but rarely used by physicists ca. 1900.90 For example, David Hilbert (1862–1943), one
of the leading mathematicians at the turn of the last century and a driving force in
pure, as well as applied, mathematics in Göttingen, published a paper in 1904 entitled,
“Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen.” In it we find
a certain mathematical argument whose frame is the following:
Due to the symmetry of the expression on the left-hand side, it follows by interchanging
x with y....91
We need not dwell on the mathematical content of the argument; rather, we draw attention to the connection Hilbert makes between symmetry [Symmetrie] and interchange
89
Einstein 1905c, 905–906 (boldface added).
For an example of symmetry used in physics in an algebraic sense, see Voigt 1896, 597. Cf.
Boltzmann [1891–1893]/1982, 1: 42.
91
Hilbert 1904, 55: “Hieraus folgt wegen der Symmetrie des Ausdrucks linker Hand bei
Vertauschung von x mit y....” Cf. Vahlen 1899, 450–462.
90
467
[Vertauschung]: due to the symmetry of a certain expression, x may be interchanged
with y without changing the content of the equation. We suggest that Einstein alludes to
this connection: exchanging v and w does not affect the value of the expression for U.92
In fact, the association of symmetry with permutations had already been in place
for about a century in the domain of algebra. We note that in 1899 Heinrich Burkhardt
(1861–1914), a mathematician at the University of Zurich and one of the examiners of
Einstein’s doctoral dissertation in 1905,93 published a review essay on finite discrete
groups in the first volume of the Encyklopädie der mathematischen Wissenschaften
mit Einschluss ihrer Anwendungen in which he accounted for the connection between
symmetry and permutation. Burkhardt included in his essay references to many papers
published throughout the 19th century. A year later, in the second volume of the Encyklopädie (in the section on analysis of which he was one of the chief editors), Burkhardt
published another review essay (co-authored by L. Maurer) on continuous transformation groups. The Encyklopädie was generally accessible in the German speaking lands,
and these review essays which bring together elements of permutation theory, symmetry,
and continuous transformation groups, constitute the kind of background that could have
facilitated Einstein in acquiring the necessary mathematical concepts and techniques for
establishing his results. We have noted that Einstein did not usually cite his sources and
so we can only identify those available at the time that he might have consulted.
In his essay on finite discrete groups, Burkhardt begins with the theme of permutations and substitutions. He makes the historical remark that already a century ago the
claim had been made that rational functions may not change their values under a certain
interchanges [Vertauschungen] of magnitudes. Augustin Cauchy (1789–1857) built upon
this result by specifying the meaning of substitution as an interchange of certain magnitudes arranged in some order. Burkhardt continues by showing how Cauchy developed
a complete calculus based on this idea that is captured by the concept of group. When
the totality of interchanges [die Gesamtheit der Vertauschungen] has the feature that the
product of any two interchanges remains within this totality, that is, that the totality has
the property of closure, then this totality of interchanges forms a group. Burkhardt then
adds that all groups have the property of associativity, and that if two elements, S and
T , are such that the product ST = T S, then the group has the property of commutativity, i.e., S and T are “interchangeable” [vertauschbar] (which, Burkhardt reminds the
reader, Cauchy called permutable, and Camille Jordan [1838–1922] échangeable). The
substitution which transposes [versetzt] no element is called the identity [identische] and
is designated by 1; and S −1 is called the S inverse [inverse] substitution.94 One case of
a substitution in a particular order is called cyclical (cyklisch in German and circulaire
in French) and is directly relevant for the mathematical treatment of the structure of
crystals.95 Burkhardt notes further that a totality of n! substitutions of n elements is
92
As we have seen, Einstein uses the term vertauscht earlier in the relativity paper in the
context of replacing v with −v: see Einstein 1905c, 902 (§ 3).
93
See Einstein 1905b, 184.
94
Burckhardt 1899, 210.
95
On cyclische Gruppen in crystals, see, e.g., Schönflies 1891, 58.
468
called a “symmetry group” [symmetrische Gruppe].96 So, when Einstein remarks, “it is
noteworthy that v and w enter the expression for the resultant velocity in a symmetric
fashion,” he is alluding to a long tradition of the use of symmetry in algebra, where
symmetry is associated with the mathematical operation of interchange of elements that
leaves the entity operated upon intact. Furthermore, in the relativity case the formula for
the addition of velocities remains the same regardless of the order of compounding the
velocities.
So far we have seen that Einstein’s use of symmetry in Case 4 amounts to extending a
well established concept in algebra to theoretical physics. At the turn of the last century
the idea of an interchange was already a commonplace in mathematical contexts, and
it is plausible to consider the context of this application of symmetry as mathematical;
we thus call it algebraic. We note further that Einstein invokes this usage of symmetry
only as a “side-remark”, pointing to a certain algebraic property, and does not pursue it
further. This, however, is linked to another “throw-away remark” which has fundamental
consequences that Einstein does not pursue. This is the concept of group.
With the formula for adding velocities in hand, Einstein proceeds to consider another
simple case where v and w are in the same direction. Using a third coordinate system, k ,
which moves parallel to k and whose origin moves with velocity w along the axis (the
fundamental axis of the three systems), Einstein applies two parallel transformations.
For the coordinates in k the transformations yield the same result as in § 3 but now, due
to the additional velocity, the value for v is replaced by the quantity,
(v + w)/(1 + vw/V 2 ),
namely, the quantity U that has been established earlier.
The calculation of the sum of the velocities was carried out not only for the purpose
of testing consistency, but also for attaining a striking result which Einstein does not
follow up. It is thus a “throw-away line” whose consequences are left unexplored: “From
this we see that such parallel transformations form a group – as they indeed must.”97
Einstein states this claim enigmatically, leaving the reader to persuade himself of its
correctness. In fact, Einstein demonstrated that all the requirements for forming a group
have been met, but his discussion of this theme is scattered and not brought together.
This may reflect haste in writing the paper, not a result of any confusion on the part of
Einstein.
Let us examine first what is needed for such a proof. Felix Klein (1849–1925) – one
of the chief contributors to this new mathematical theory whose roots go back to the first
half of the 19th century – put forward the following requirements:
96
Burkhardt 1899, esp. 213. See also Hölder 1899, esp. 499–504, and Netto [1882]/1892,
32–33.
97
Beck 1989, 156; Einstein 1905c, 907: “man sieht daraus, daß solche Paralleltransformationen – wie dies sein muß – eine Gruppe bilden.” Einstein uses the term “group” twice in his
earlier paper on the light-quantum – Erscheinungsgruppen and Resonatorengruppen – where the
term “group” does not have its technical-mathematical sense (Einstein 1905a, 133, 135 n. 2). For
the term Erscheinungsgruppen, see Föppl 1894, 353.
469
1. The product [Produkt] or, the combination, A · B = C also belongs to the system
(closure of the system) [Abgeschlossenheit des Systems].
2. The associative law holds, i.e., (AB) · C = A · (BC).
3. There is a unit E such that AE = A and EA = A.
4. An inverse exists, i.e., the equation Ax = E has a solution. 98
There are two essential elements for such a proof: (1) closure, and (2) the inverse property (i.e., that the inverse of a transformation is also in the group). Einstein demonstrates
closure in § 5 immediately before the claim for the group property,99 giving the impression that the group property is a consequence of closure – which would be an error when
applied to continuous groups where the inverse property has to be demonstrated independently.100 Einstein, however, calls attention to both (1) die inverse Transformation,
and (2) die identische Transformation,101 although they are not mentioned in the same
section. In particular, the inverse property is shown in § 6 for the transformation k to K
which is the inverse of the transformation from K to k, due to the motion of the system
in the opposite direction.102 The identity property is demonstrated in § 3 where “a third
coordinate system K which, relative to the system k, is in parallel-translational motion
parallel to the axis such that its origin moves along the -axis with velocity −v.” The
transformation from K to K then “must be the identity transformation.”103
We know today that the claim for the group property is indeed correct and has profound implications but it is not clear that Einstein fully appreciated its significance,
for he does not refer to the group property later in the paper. However, he does invoke
the “addition theorem” in § 9: Einstein applies a transformation to the Maxwell–Hertz
98
Klein [1926]/1979, 315–316 (slightly modified); Klein 1926, 335. In a book on mathematical crystallography and the theory of groups of movements, Hilton (1903, 45) remarks: “A series
of operations is said to form a group when (1) the product of any two of the operations, or the
square of any one, is equivalent to some member of the series (i.e. some operation of the series),
and (2) the series always contains the operation A−1 if it contains the operation A. Every group
contains the identical operation, for A · A−1 = 1. A group is said to be finite or infinite according
as the number of operations it contains is finite or infinite.”
99
Einstein 1905c, 907 (§ 5)
100
See note 117, below, for the importance of the inverse. It is relatively easy to decide if a
certain collection of elements has an identity relation. But, in order to show that these elements
form a group, it is necessary to demonstrate that the inverse property holds, and this is not as easy
to determine.
101
Einstein 1905c, 902 (§ 3), 909 (§ 6).
102
Einstein 1905c, 909.
103
Beck 1989, 150; Einstein 1905c, 901–902: “Wir führen zu diesem Zwecke noch ein drittes
Koordinatensystem K ein, welches relativ zum System k derart in Paralleltranslationsbewegung
parallel zur -Achse begriffen sei, daß sich dessen Koordinatenursprung mit der Geschwindigkeit
−v auf der -Achse bewege ... und es ist klar, daß die Transformation von K auf K die identische
Transformation sein muß.”
470
equations when convection currents are taken into consideration. Here he appeals to the
addition theorem of velocities, proved in § 5, and argues that
with our kinematic principles taken as a basis, the electrodynamic foundation of Lorentz’s
theory of electrodynamics of moving bodies agrees with the principle of relativity.104
This is an important result. Einstein shows that the addition theorem, based on the kinematic principles of the theory, is consistent with Lorentz’s theory.105
Einstein’s discussion indicates that he knew at least some of the mathematical literature on group theory and simply did not wish to digress from the main thrust of the paper
– the electrodynamic part that comes next. The invocation of an algebraic concept, group,
strengthens our claim that the usage of symmetry in this instance is indeed purely algebraic. Despite an extensive search for the sources of Einstein’s remark on transformation
groups, we have not come across any reference to group theory in treatises on physics
prior to 1905 with the notable exception of a series of papers by Schönflies in the 1880s
(1886, 1887, 1889) and his book, Krystallsysteme und Krystallstructur (1891), of which
he gave a summary in his lecture in 1893 to the International Mathematical Congress in
Chicago (Schönflies 1896). Schönflies’s papers appeared in the Mathematische Annalen,
not in journals devoted to physics. He discusses at length the concept of group which he
classifies with respect to the totality of all possible motions: Translationsgruppen, Rotationsgruppen, and allgemeine Bewegungsgruppen.106 His analysis of Bewegung der
Gruppen is mathematical and the references to Jordan and Klein make it amply clear
that Schönflies is working in a mathematical tradition, although he is concerned with a
material object, a crystal.107 In this respect, Schönflies’s book constitutes a stark contrast
to Voigt’s treatise on the physical properties of crystals that we have already discussed.
And, in fact, in 1898 Voigt did not find anything worthy of citation in Schönflies’s group
theoretic approach apart from a single reference to the famous table of the 32 crystal
classes.108
For our argument it is important to note that Schönflies calls attention to the property
of closure, i.e., that the result of an operation on any two members of the group (usually
considered a product) stays within it.109 Indeed, in the preface to his book Schönflies
104
Beck 1989, 166–167; Einstein 1905c, 917: “... wie aus dem Additionstheorem der Geschwindigkeiten (§ 5) folgt ... so ist damit gezeigt, daß unter zugrundelegung unserer kinematischen Prinzipien die elektrodynamische Grundlage der Lorentzschen Theorie der Elektrodynamik
bewegter Körper dem Relativitätsprinzip entspricht.”
105
In 1909 Einstein returns to this point concerning the validity of the addition theorem and the
status of Lorentz’s theory: see Einstein 1909; Stachel et al. 1989, 569–570; Beck 1989, 383–384.
106
Schönflies 1886, 320.
107
Schönflies 1886, 325ff; for Jordan, see Schönflies 1886, 319 and 326 and for Klein, see
108
Voigt 1898, 191; see p. 243 n. 88 for the reference to Schönflies 1891, 146.
109
Schönflies 1891, 54: “Unter einer endlichen Gruppe von Operationen verstehen wir eine
endliche Reihe nicht äquivalenter Operationen von der besonderen Beschaffenheit, dass das Product von irgend zweien derselben stets einer operation der Reihe äquivalent ist.” One such operation is permutation in some definite way which Schönflies calls cyclische Gruppen. See also
471
makes it clear that recently crystallographers have moved away from the empirical study
of crystals towards the deductive method that depends on a single basic law, namely, the
law of symmetry (first stated by René-Just Haüy [1743–1822] in 1815110 ). The theory of
the structures of crystals has then been developed mathematically from this fundamental
hypothesis.111
As we have indicated, the technical literature on group theory in algebra was extensive at the turn of the last century. Burkhardt refers to the fundamental contribution by
Jordan in 1870 where we find the condition that a set of permutations must satisfy to
form a group:
One says that a system of substitutions forms a group ... if the product of any two of the
substitutions of the system is again a member of that system.112
We stress (with Wussing) that the result is closure under multiplication. This is exactly
what Einstein needed for his transformations, a limiting upper bound that will restrict
the addition of velocities. However, Jordan addresses discrete, finite groups whereas
transformations of coordinate systems have to do with continuous operations. We draw
attention to the review essay by Maurer and Burkhardt on continuous transformation
groups that provides precisely the technical means which could have facilitated Einstein’s recognition that the transformations of systems of coordinates form a group.
This essay includes reports on important developments in this domain of algebra, especially the pioneering work of Sophus Lie (1842–1899) and Klein. At the outset of these
researches in the early 1870s, after the publication of Jordan (1870), Lie and Klein
referred to their object of inquiry as a “closed system” [geschlossenes System].113 At
that time Klein used “closed system” and “transformation cycle” interchangeably.114
This technical expression was eventually replaced in the complete theory by “closure of
the system” [Abgeschlossenheit des systems].115 Wussing quotes a passage by Klein on
the early stages of this project:
Then Lie and I decided to elaborate the significance of group theory for different areas of
mathematics. We stated that a group is a class [Inbegriff] of unique operations A, B, C,...
110
Haüy 1815.
Schönflies 1891, iii: “In der Behandlung derjenigen Fragen, welche die Eintheilung der
Krystalle nach den Symmetrieeigenschaften, sowie die Theorie der Structur betreffen, ist man
in den letzten Jahrzehnten mehr und mehr von der empirischen zur deductiven Methode übergegangen. Wir verdanken diesem Schritt die Erkenntniss, dass sich die Systematik der Krystalle
aus einem einzigen Grundgesetz und die Theorie der Structur aus einer einzigen fundamentalen
Hypothese in mathematischer Weise ableiten lässt.
Nach moderner Ansicht tritt die Eigenart der Krystallsubstanz in der Abhängigkeit des physikalischen Verhaltens von der Richtung in die Erscheinung. Das Grundgesetz, welches das physikalische Verhalten regelt, ist das Symmetriegesetz.” Cf. Schönflies 1896, 341.
112
Wussing [1969]/1984, 142; Jordan 1870, 22 (italics in the original): “On dira qu’un système
de substitutions forme un groupe ... si le produit de deux substitutions quelconques du système
appartient lui-même au système.”
113
Maurer and Burkhardt 1900, 403 n. 4. Cf. Hawkins 2000, 16.
114
Hawkins 2000, 16 n. 14.
115
Klein 1926, 335.
111
472
such that the combination of any two operations A, B again yields an operation C of the
class
A · B = C [i.e., closure].
In the course of further investigations on infinite groups Lie found it necessary to require
that for each A the group should contain its inverse A−1 ....116
In fact, in his memoir on groups of rigid motions, Jordan (1869) also tacitly assumed
the inverse operation. According to Hawkins, “they all took it for granted that groups
as they defined them always possess inverses – that closure under composition entails
closure under inversion, as it does for permutation groups.”117 It appears that, at the
outset of investigations into continuous groups, it was assumed that closure sufficed, but
later Lie found that this postulate by itself was inadequate. The four standard postulates
for a group were then defined: closure, associativity, identity, and inverse.
We therefore suggest that in 1905 an interested physicist could have found without
difficulty the relevant mathematical information about group theory. To be sure, Einstein
may have consulted a different text to learn about group theory, for there is no hint that he
was even acquainted with the essays that had appeared by 1905 in the physical section of
the Encyklopädie, edited by Arnold Sommerfeld (1868–1951) and to which prominent
physicists, e.g., Lorentz, contributed.118 Since Einstein did not pursue the technicalities
of group theory, all he needed to make the claim that the parallel transformations form a
group was the realization that these transformations – bounded as they are by an upper
value, namely, the velocity of light in vacuum – satisfy the conditions for forming a
group.
The introduction of group theoretical considerations into the theory of the dynamics of electron begins with Poincaré who, in June 1905 (at the same time Einstein was
composing his relativity paper), delivered a talk at the French Academy of Sciences in
which he briefly presented the kernel of his famous Rendiconti paper of 1906 (dated
July 1905), “Sur la dynamique de l’électron.”119 In his report to the Academy, Poincaré
responds explicitly to Lorentz’s paper (1904c) on the electromagnetic phenomena in a
system moving with any velocity less than that of light.120 He follows closely the results
which Lorentz had presented in his paper and states that his own results essentially agree
116
Wussing [1969]/1984, 228; Klein 1926, 335: “Als dann Lie und ich es unternahmen, die Bedeutung der Gruppentheorie für die verschiedensten Gebiete der Mathematik herauszuarbeiten,
da sagen wir: ‘Gruppe’ ist der Inbegriff von eindeutigen Operationen A, B, C ... derart, daß
irgend zwei der Operationen A, B kombiniert wieder eine Operation C des Inbegriffes ergeben:
A · B = C. Bei seinen weitern Untersuchungen über unendliche Gruppen sah sich Lie genötigt,
ausdrücklich zu verlangen, daß neben A auch die Inverse A−1 in der Gruppe vorhanden sein soll.”
117
Hawkins 2000, 15 n. 13. See also Jordan 1869.
118
Sommerfeld 1904–1922. In 1908 Einstein thanked Wien for calling his attention to a passage
in Lorentz’s article of 1904 in the Encyklopädie: Einstein and Laub 1908b, 549 n. 1. This may
mean that Einstein had not read this essay by Lorentz before 1905.
119
Poincaré [1906]/1954; see also Poincaré [1905]/1954.
120
Einstein claimed to have been unaware of Lorentz 1904c in 1905: see n. 22, above.
473
with those of Lorentz, but then adds that, on some crucial points, he found it important to
modify Lorentz’s view and indeed to bring it to perfection. At this juncture he presents
Lorentz’s equations of transformation which he names after Lorentz.121 Poincaré infers
that if in Lorentz’s theory it is impossible to perceive any change of phenomena due
to physical translations of the coordinate system, there would be no difference in the
equations of the electrodynamic medium under certain transformations: two systems,
one stationary, the other in a state of translation, thus become the exact image of each
other.122 After specifying the details of the transformation, Poincaré adds:
One sees that in this transformation the x-axis plays a special role but, clearly, one can
construct a transformation where this role would be played by any straight line passing
through the origin.
Poincaré realized that Lorentz’s equations are independent of the choice of axes and
concludes:
The set of all transformations, together with the set of all rotations of space, must form a
group....123
Thus, according to Poincaré, to bring Lorentz’s theory to perfection requires appealing
to the principles of group theory.124
Unlike Einstein, Poincaré drew consequences from what he named “the Lorentz
group” [le groupe de Lorentz125 ], characterized as a continuous group where the elements of the group are spatial transformations. In his extended paper of 1906, he obtained
an important result based on group theory, namely, that every transformation of the
Lorentz group can be resolved into a linear transformation that leaves unaltered [qui
n’altère] the quadratic form
x 2 + y 2 + z2 − t 2 .126
As Hawkins remarks:
Poincaré had always been a believer in the importance of groups in mathematics and
especially the continuous groups of Lie, and so it must have pleased him to see a group
hidden away in Lorentz’s calculations.127
121
Poincaré [1905]/1954, 490; Poincaré [1906]/1954, 498–503.
Poincaré [1906]/1954, 495: “deux systèmes, l’un immobile, l’autre en translation, deviennent ainsi l’image exacte l’un de l’autre.” Notice that Poincaré does not say that the two systems
are symmetrical.
123
Poincaré [1905]/1954, 490: “On voit que dans cette transformation l’axe des x joue un rôle
particulier, mais on peut évidemment construire une transformation où ce rôle serait joué par
une droite quelconque passant par l’origine. L’ensemble de toutes ces transformations, joint à
l’ensemble de toutes les rotations de l’espace, doit former un groupe....”
124
Poincaré [1906]/1954, 496: “... c’est ce que je retrouve à mon tour par une autre voie en
faisant appel aux principes de la théorie des groupes.” For the full discussion, see pp. 513–515.
125
Poincaré [1906]/1954, 514.
126
Poincaré [1906]/1954, 515. Cf. Schwartz 1971 and 1972, 1294; and Kilmister 1970, 171.
127
Hawkins 2000, 338.
122
474
This quadratic form, which remains unchanged under relativistic transformation, subsequently became the foundation for the theory of relativity. And Poincaré immediately
uses this result in the analysis of his gravitational hypothesis:
In order to proceed further, it is necessary to ascertain the invariants of the Lorentz
group.128
He then goes on to exploit the invariant properties of this quadratic form under the
Lorentz transformation.
It may be an irony of history that within a few days in June 1905 Einstein indicates
that parallel transformations must be [sein muß] a group, while Poincaré says that the set
of all transformations, including the set of all rotations of space, must form [doit former]
a group. It should, however, be understood that these two expressions of necessity are
based on entirely different reasoning. While Poincaré is well versed in the mathematics of
group theory and discusses the link between group theory and transformation functions
(and alludes to the pivotal role of Lie in the development of the theory of transformation groups),129 Einstein’s interest in group theory is very limited and he emphasizes
the property of closure.130 Again, for Poincaré closure is not the issue; rather, as he
put it: “the transformations which do not change the equations of motion must form a
group.”131 We may further note that Lorentz, who assumes in his theory an absolute, stationary ether that can provide the framework for the transformations, does not introduce
group theory into his theory of electrons.
In Case 4 we have seen that with respect to symmetry the usage amounts to extending a result from algebra to theoretical physics. Thus, the innovation is modest, given
that Vertauschung and symmetry had already been related in the previous mathematical
literature. Moreover, symmetry does not seem to advance the argument and enters only
by way of a side-remark that leads to no new results. The more important innovation,
however, is Einstein’s invocation of a (continuous) group because we have not found
this concept in any earlier work in physics. But then we have seen that Poincaré also
introduced the concept of group into physics at the same time. And again group, like
symmetry, does not seem to do anything to advance Einstein’s argument. The case of
symmetry is one of Vertauschung and states something that is true, but seems irrelevant
to the argument; it only suggests that velocities will have a special role in proving closure
for the transformations that in turn form a group.
128
Kilmister 1970, 175 (italics in the original); Poincaré [1906]/1954, 541 (italics in the
original): “Pour aller plus loin il faut chercher les invariants du groupe de Lorentz”.
129
Poincaré [1906]/1954, 514.
130
In his reconsideration of “Einstein versus Lorentz”, Janssen (2002a, 428) notes that both
Einstein and Poincaré called attention to this group at about the same time. Janssen then adds,
“What this means is that the inverse of a Lorentz transformation as well as the combination of
any Lorentz transformation followed by another are themselves Lorentz transformations.” This is
a true statement, but we note that both properties need to be proved in order to demonstrate that
these transformations form a group.
131
Schwartz 1971 and 1972, 870; Poincaré [1906]/1954, 535–536 (italics in the original): “Les
transformations qui n’altèrent pas les équations du mouvement doivent former un groupe....”
475
Einstein did not refer to any discussion of symmetry in physics, but his usage of
symmetry is neither aesthetic nor purely geometrical. Moreover, group theory, as it was
commonly understood among crystallographers in late 19th century, does not apply in
this case, for their interest was primarily in finite groups whereas for Einstein’s theory
of relativity the properties of a continuous transformation group are required. In this
case, Einstein made a modest extension with respect to the use of symmetry, in contrast
to making a truly innovative link between an established concept in algebra, namely,
group, and theoretical physics.
Case 5: rejecting asymmetry
The two remaining instances of symmetry in Einstein’s paper are in fact a pair and
appear in the negative – asymmetry. This case can be considered the jewel in the crown
of Einstein’s invocation of symmetry in physics. When consulting the following text of
Einstein, the reader should be mindful that he begins his paper with an appeal to the
plural, asymmetries. He then brings § 6 – the first section of the electrodynamic part
of the paper – to a conclusion by invoking the singular form, referring explicitly to the
opening lines of the paper. The use of the plural and singular forms are consequential, as
we will explain. But first the text. The following is the opening paragraph of Einstein’s
paper:
It is well known that Maxwell’s electrodynamics – as usually understood at present –
when applied to moving bodies, leads to asymmetries that do not seem to adhere to the
phenomena. Let us recall, for example, the electrodynamic interaction between a magnet
and a conductor. The observable phenomenon depends here only on the relative motion
of conductor and magnet, while according to the customary conception the two cases, in
which, respectively, either the one or the other of the two bodies is the one in motion, are to
be strictly differentiated from each other. For if the magnet is in motion and the conductor
is at rest, there arises in the surroundings of the magnet an electric field endowed with a
certain energy value that produces a current in places where parts of the conductor are
located. But if the magnet is at rest and the conductor is in motion, no electric field arises
in the surroundings of the magnet, while in the conductor an electromotive force will
arise, to which in itself there does not correspond any energy, but which, provided that the
relative motion in the two cases considered is the same, gives rise to electrical currents
that have the same magnitude and the same course as those produced by the electric forces
in the first-mentioned case.
Examples of a similar kind, and the failure of attempts to detect a motion of the earth
relative to the “light medium”, lead to the conjecture that not only in mechanics, but in
electrodynamics as well, the phenomena do not have any properties corresponding to the
concept of absolute rest, but that in all coordinate system in which the mechanical equations are valid, also the same electrodynamic and optical laws are valid, as has already
been shown for quantities of the first order.... [The] two postulates suffice for arriving at a
simple and consistent electrodynamics of moving bodies on the basis of Maxwell’s theory
for bodies at rest. The introduction of a “light ether” will prove superfluous, inasmuch as
476
in accordance with the concept to be developed here, no “space at absolute rest” endowed
with special properties will be introduced....132
We now turn to the complementary occurrence of asymmetry in § 6:
If a pointlike unit electric pole is in motion in an electromagnetic field, the force acting
on it equals the electric force present at the location of the unit pole, which is obtained
by transforming the field to a coordinate system that is at rest relative to the unit electric
pole. (New mode of expression.)
Analogous propositions apply for “magnetomotive forces.” We can see that in the
developed theory, the electromotive force merely plays the role of an auxiliary concept,
whose introduction is due to the circumstance that the electric and magnetic forces do not
have an existence independent of the state of motion of the coordinate system.
It is further clear that the asymmetry mentioned in the Introduction when considering the currents produced by the relative motion of a magnet and a conductor, disappears.
Questions as to the “seat” of electrodynamic electromotive forces (unipolar machines)
also become pointless.133
Daß die Elektrodynamik Maxwells – wie dieselbe gegenwärtig aufgefaßt zu werden
pflegt – in ihrer Anwendung auf bewegte Körper zu Asymmetrien führt, welche den
Phänomenen nicht anzuhaften scheinen, ist bekannt. Man denke z. B. an die elektrodynamische Wechselwirkung zwischen einem Magneten und einem Leiter. Das beobachtbare
Phänomen hängt hier nur ab von der Relativbewegung von Leiter und Magnet, während
nach der üblichen Auffassung die beiden Fälle, daß der eine oder der andere dieser Körper
der bewegte sei, streng voneinander zu trennen sind. Bewegt sich nämlich der Magnet
und ruht der Leiter, so entsteht in der Umgebung des Magneten ein elektrisches Feld von
gewissem Energiewerte, welches an den Orten, wo sich Teile des Leiters befinden, einen
Strom erzeugt. Ruht aber der Magnet und bewegt sich der Leiter, so entsteht in der Umgebung des Magneten kein elektrisches Feld, dagegen im Leiter eine elektromotorische
Kraft, welcher an sich keine Energie entspricht, die aber – Gleichheit der Relativbewegung bei den beiden ins Auge gefaßten Fällen vorausgesetzt – zu elektrischen Strömen
von derselben Größe und demselben Verlaufe Veranlassung gibt, wie im ersten Falle die
elektrischen Kräfte.
Beispiele ähnlicher Art, sowie die mißlungenen Versuche, eine Bewegung der Erde
relativ zum “Lichtmedium” zu konstatieren, führen zu der Vermutung, daß dem Begriffe
der absoluten Ruhe nicht nur in der Mechanik, sondern auch in der Elektrodynamik
keine Eigenschaften der Erscheinungen entsprechen, sondern daß vielmehr für alle Koordinatensysteme, für welche die mechanischen Gleichungen gelten, auch die gleichen
elektrodynamischen und optischen Gesetze gelten, wie dies für die Größen erster Ordnung
bereits erwiesen ist.... [Die] beiden Voraussetzungen genügen, um zu einer einfachen und
widerspruchsfreien Elektrodynamik bewegter Körper zu gelangen unter Zugrundelegung
der Maxwellschen Theorie für ruhende Körper. Die Einführung eines “Lichtäthers” wird
132
133
Beck 1989, 140–141 (slightly modified, boldface added).
Beck 1989, 159 (slightly modified, boldface added).
477
sich insofern als überflüssig erweisen, als nach der zu entwickelndenAuffassung weder ein
mit besonderen Eigenschaften ausgestatteter “absolut ruhender Raum” eingeführt....134
The complementary passage is:
Ist ein punktförmiger elektrischer Einheitspol in einem elektromagnetischen Felde bewegt, so ist die auf ihn wirkende Kraft gleich der an dem Orte des Einheitspoles vorhandenen elektrischen Kraft, welche man durch Transformation des Feldes auf ein relativ zum
elektrischen Einheitspol ruhendes Koordinatensystem erhält. (Neue Ausdrucksweise.)
Analoges gilt über die “magnetomotorischen Kräfte”. Man sieht, daß in der entwickelten Theorie die elektromotorische Kraft nur die Rolle eines Hilfsbegriffes spielt, welcher
seine Einführung dem Umstande verdankt, daß die elektrischen und magnetischen Kräfte
keine von dem Bewegungsaustande des Koordinatensystems unabhängige Existenz besitzen.
Es ist ferner klar, daß die in der Einleitung angeführteAsymmetrie bei der Betrachtung
der durch Relativbewegung eines Magneten und eines Leiters erzeugten Ströme verschwindet. Auch werden die Fragen nach dem “Sitz” der elektrodynamischen elektromotorischen Kräfte (Unipolarmaschinen) gegenstandslos.135
The (a)symmetry we encounter here does not fit prima facie any of the usages of
symmetry that we have discussed so far, either in the dissertation or in the relativity
paper, for it addresses neither mathematical nor physical characteristics and may be
considered a methodological principle. Since it was put in the very first sentence of this
famous paper, it is not surprising that it has attracted much attention and has prompted
some historians to consider this an aesthetic move. Gerald Holton is perhaps the most
outspoken historian of science to promote this view. Already in 1973 he considered Einstein’s interest in removing asymmetries aesthetic, and cast it into a thematic principle
in Einstein’s pursuit of physics:
[It was his] sensitivity to previously unperceived formal asymmetries or incongruities of
a predominantly aesthetic nature (rather than, for example, a puzzle posed by unexplained
experimental facts) – that is the way each of Einstein’s three otherwise very different great
papers of 1905 begin. In all these cases the asymmetries are removed by showing them
to be unnecessary, the result of too specialized a point of view. Complexities that do not
appear to be inherent in the phenomena should be cast out. Nature does not need them.136
The physicist (who was also Einstein’s biographer),Abraham Pais, follows this approach,
stating the aesthetic origin of the relativity paper:
134
Einstein 1905c, 891–892 (boldface added).
136
Holton 1973, 366. See also Holton 1986, 17, 53. Holton restates this view in the revised
edition of his 1973 book, see Holton [1973]/1988, 193. In a similar vein, Arthur I. Miller (1981,
130) remarks: “Einstein’s tendency to frame arguments such as Mach’s into quasi-aesthetic form,
revealing asymmetries that should not be contained in laws of nature, was characteristic of his mode
of thinking in the relativity paper.” Miller restates this claim in the revised edition: [1981]/1998,
122; cf. Miller 2001, 197–198, 238.
135
478
Einstein was driven to the special theory of relativity mostly by aesthetic arguments, that
is, arguments of simplicity. This same magnificent obsession would stay with him for the
rest of his life. It was to lead him to his greatest achievement, the general relativity, and
to his noble failure, unified field theory.137
Moreover, Holton conceived of Einstein’s entire scientific enterprise, and even his life
style, in terms of the polarity, “symmetries and asymmetries”.138 He stresses that the
impasse which Einstein realized at the outset of his relativity paper is neither experimental nor theoretical. Rather, it leads to asymmetries which Holton then queries:
At first glance it is surely curious that he used the term asymmetry for this apparent
redundancy or lack of universality. Moreover, such terms as symmetry or asymmetry still
referred at that time largely to aesthetic judgments, often thought to be the polar opposites
of scientific judgments. In physics literature, symmetry arguments were quite uncommon
(and even these are easier to find now in retrospect) ... the term itself was rarely used
except in such branches as crystal physics.139
Holton adds that usages of symmetry were rare in physics at the turn of the last century
and even later, so much so that the Sachregister of the encyclopedic volume Physik of
1915 does not have an entry for this concept. According to Holton, symmetry was only
gradually accepted and
few physicists, if any, can have thought in 1905 that there was something of fundamental
importance in the asymmetry to which Einstein pointed.
And if one considers how many troubles there were in electrodynamics at the time, it
must have seemed peculiar indeed to seek out this quasi-aesthetic discomfort, and to put it
at the head. What Einstein’s perception of symmetry at this point does show us, however,
is his remarkable and original sensitivity to polarities and symmetry properties of nature
that later became recognized as important in relativity theory and in contemporary physics
generally.140
In sum, Holton suggests in unambiguous terms that Einstein’s usage of the term asymmetry is extraordinary, a sign of genius of this great scientist. Indeed, this analysis that we
have quoted in extenso appears in a chapter entitled: “On trying to understand scientific
genius”.141
While we acknowledge the pioneering historical work of Holton in drawing attention
to the importance of this opening line in Einstein’s paper, to the best of our knowledge
there has been no systematic discussion of symmetry as it is used by Einstein in 1905.
Moreover, although the instance of asymmetry in this paper has received much scholarly attention, no attempt has been made to link it to the usages that we have already
137
Pais 1982, 140.
Holton [1973]/1988, 374.
139
Holton [1973]/1988, 381, 382–383. Brush (1999, 191) refers with approval to Holton’s view:
“Since Einstein began his 1905 paper with an aesthetic question – the problem of symmetry [sic]
in Maxwell’s equations – it would not be surprising if his followers also gave priority to such
issues.”
140
Holton [1973]/1988, 383–384 (italics in the original).
141
Holton [1973]/1988, § 9.
138
479
discussed.142 To anticipate our result: we claim that Einstein responded to contemporary
discussions in which symmetry (or asymmetry) was invoked. In other words, Einstein
considered both symmetry and asymmetry to be technical terms in physics, and applied
them in ways that were easily understood at the time. On occasion he extended the common usage of symmetry in a reasonable and seemingly “natural” way and, in the case of
asymmetry, he introduced a critique of the common usage. The fact that symmetry (and
asymmetry) has an aesthetic meaning should not confuse us, just as the algebraic notion
of group should not make us think of social cohesion.143 To demonstrate our claim we
begin with a close analysis of Einstein’s move and then turn to detailed expositions and
analyses of usages of symmetry and related terms in the literature of physics, principally
in electrodynamics, prior to 1905.
3. The central claim: making asymmetry disappear
by appealing to a physical argument
In his paper on 19th-century views on induction in moving conductors, Ole Knudsen seeks to supplement Holton’s discussion by presenting examples of the treatment
of the problem, thus delineating some features of its history prior to Einstein’s work.144
We generally follow this approach but in greater detail, for we have found relevant discussions by more players, and we are able to establish a link between Einstein’s usage
of asymmetry and the occurrence of symmetry and asymmetry in texts that are nearly
contemporary with Einstein’s work of 1905. For this purpose we have had to consider
related terms, namely, duality, parallelism, and reciprocity. This background material
may be “unfamiliar” to modern readers of the relativity paper but, as we will show, it
was in the literature of physics at the time.
Knudsen begins his paper by referring to symmetry as a technical term which captures a certain mathematical characteristic of the law of induction as it was formulated
in the early 19th century. He reminds us that Michael Faraday (1791–1867) articulated
the law of induction based on numerous and careful experimental studies. Assuming
that rigid magnetic field lines are connected to the magnet in its translational motion, an
integration of the differential expression of the law exhibits complete symmetry between
the electric and the magnetic field. This is expressed in the fact that the total electromotive force in a conductor is proportional to the number of magnetic field lines moving
across the conductor per unit of time. The situation changed when James Clerk Maxwell
(1831–1879) recast the experimental findings of Faraday into a very productive set of
142
Holton ([1973]/1988, 383) mentions, in parentheses without any discussion, the fact that
there are other occurrences of symmetry in the relativity paper.
143
This is not the place to discuss the general question of appeals to aesthetic considerations in
scientific discussions, but we are persuaded that in physics at the time symmetry did not belong to
that category. For harmony as an aesthetic consideration, see, e.g., Pyenson 1982. Darrigol (1996,
295) accepts the claim made by Paty (1993, 54–55) that Einstein’s argument is epistemological
rather than aesthetic. We, however, will suggest that Einstein’s argument is based on physical
reasoning.
144
Knudsen 1980, 347.
480
mathematical equations, interpreting the electromagnetic field as a physical state in the
ether. In this new framework the ether constitutes a third element in the system, and thus
the action between the conductor and the magnet is no longer a direct one, but mediated
by the ether.145
Knudsen further observes that Maxwell’s mature theory of induction resulted in an
asymmetrical field vector for the electromotive intensity. This is essentially due to two
different contributions to this vector: one component depends only on the time variation
of the magnetic field and thus arises at every point of space; while the other component is
only in the body as it moves through space with a certain velocity. As Knudsen shows, it
follows that a calculation of the transformation of the electromotive force from a system
K in which the magnet is at rest, to K the rest frame of the conductor, assumed to be
moving translationally relative to K, leads to conflicting results. Needless to say, when
we relate K and K to a frame which is at rest with respect to the ether, the contradiction
vanishes.146
Knudsen proceeds to discuss the different approaches taken by British physicists,
mainly represented by Larmor, and continental physicists, like Hertz and Lorentz, with
respect to the dynamic theories of the electromagnetic field. We will cite Knudsen’s
paper, where appropriate, when we discuss individual physicists whose work bears on
this issue. Like Knudsen, we will explore the various ways the problem, to which Einstein responded, was addressed ca. 1900, and we will also be mindful of the explicit use
of technical terminology. But first let us consider Einstein’s argument in the opening
paragraph of the paper.
Einstein begins his paper by calling attention to the fact that Maxwell’s electrodynamics147 – “as usually understood at the present time” – leads, when applied to moving
bodies, to asymmetries which do not correspond to anything in the phenomena. We may
be tempted to think that Einstein directs his criticism, right at the outset of the paper, at
the formalism of Maxwell’s theory, for this formalism is responsible for these asymmetries. Thus it would seem that one needs to address the form of the equations, either to
modify it or replace it, with the goal of making it correspond faithfully to the phenomena.
But we understand Einstein’s expression “as usually understood at the present time” to
mean that the fault lies in the way the theory had been understood, not in the equations.
After all, later in the paper Einstein introduces the Maxwell–Hertz equations lock, stock,
and barrel, and does not question their validity in any respect. The formalism is taken to
be correct without comment. The expression “Maxwell’s theory”, or “Maxwell’s electrodynamics”, refers then not merely to the equations but to the equations together with
a host of assumptions, derivations, and interpretations. Indeed, the evidence in support
of Maxwell’s equations at the turn of the 20th century was overwhelming; they satisfactorily described a large number of phenomena. Einstein, to put it bluntly, “buys”
the formalism, that is, the equations, but he rejects the interpretation(s) outright. The
fact that at the very beginning of the paper Einstein calls attention to an asymmetry that
145
Knudsen 1980, 347–349.
Knudsen 1980, 349–352.
147
Note that here Einstein refers to Maxwell but later in the paper he refers to the equations of
“Maxwell–Hertz”. Cf. n. 10, above.
146
481
results from a theory is an important clue that attention should be directed to the physical
interpretation of the formalism.
If we recast Einstein’s claim positively, it may yield a new perspective: even in cases
where the electrodynamic (or electromagnetic) phenomena arise from the same relatively moving bodies, the theory leads to an asymmetry, for it distinguishes different
cases based on the choice of which body is moving and which is at rest, whereas only
the relative motion is relevant. Einstein takes for granted that the phenomena should be
given precedence over the theory and considers the following thought experiment. In
the first case a magnet moves and a conductor (a wire) is stationary, while in the second
case the conductor moves and the magnet is stationary.
Einstein’s language is worthy of close attention: Asymmetrien ... welche den Phänomenen nicht anzuhaften scheinen (with the verb anhaften: “to cling to”, or “to adhere to”).
We may then paraphrase this expression as: “asymmetries ... which do not seem to be
associated with the phenomena.” Einstein is careful to avoid claiming that an equation
leads to asymmetry (or symmetry) in the phenomena because that would be an improper
mixture of categories. Strictly speaking, equations do not lead to phenomena; rather,
they lead to mathematical descriptions of phenomena which, in turn, can be interpreted
physically – this is the theory – as exhibiting (a)symmetrical features. In this way the
theory (not the equations by themselves) can lead to asymmetry (or symmetry) that does
not correspond to anything in the phenomena. Thus, the asymmetry is an “artifact” of
the theory; it does not “adhere” to the phenomena.
Einstein means that in one perspective a changing magnetic field – i.e., a magnet in
motion – induces in accordance with Faraday’s law an electric field which in turn generates an electric current in the stationary conductor. In another perspective no electric
field is induced, but (in modern terms) the free electrons in the conductor experience a
Lorentz force exerted by the stationary magnetic field and their resultant circular motion
will register as a current in the conductor. This is the asymmetry: the phenomenon is the
same, namely, electric current arising from relative motions of two elements – the current
and the motion being objectively observed – but the descriptions of the phenomenon,
that is, the explanations the theory provides, are different in the two cases.148 Put another
way, in one case there is an electric field that gives rise to the phenomenon of an electric
current whereas in the other there is no electric field to generate the phenomenon: yet
the same phenomenon occurs. In 1920 Einstein reviewed his reasoning in the relativity
paper as follows:
The idea ... that these were two, in principle different cases was unbearable [unerträglich]
for me. The difference between the two, I was convinced, could only be a difference in
choice of standpoint [Standpunktes] and not a real difference. Judged from the magnet,
there was certainly no electric field present. Judged from the electric circuit, there certainly
was one present. Thus the existence of the electric field was a relative one, according to
the state of motion of the coordinate system used, and only the electric and magnetic field
together could be ascribed a kind of objective reality, apart from the state of motion of
148
Cf. Janssen 2002b, 503, where diagrams are provided to illustrate the experimental setup.
482
the observer or the coordinate system. The phenomenon of magneto-electric induction
compelled me to postulate the (special) principle of relativity.149
In effect, Einstein paraphrases his opening remark in the relativity paper, but this time
without the term asymmetry. We take this omission to be an important marker. As we
will see, the meaning Einstein assigned to this word did not catch on, and Einstein rarely
used it. In 1905 the asymmetry referred to two versions of the same experiment that,
according to Maxwell’s theory (as usually understood), led to the presence or absence of
an electric field, depending on the standpoint of the observer. This was “unbearable” to
Einstein. We remark in passing that this judgment is not based on an aesthetic principle,
but on physical reasoning. And accordingly the solution of the problem is not aesthetic
but physical.
Einstein does not address a characteristic feature of causality, that is, different causes
may lead to the same effect. Rather, it is the same cause (the relative motion) that should
have the same effect: either the relative motion produces an electric field or it does not.
One cannot both have and not have an electric field. The motion is the cause and the
current is the measurable effect; the field is a presumed effect that follows from the
equations (and the underlying theory as “usually understood”). The current as a phenomenon (the effect) is not in question; hence, the type of field is the only item at issue.
The resolution of the question concerning the “presumed effect” of the field is to change
the interpretation of the theory – leaving the equations as they were – and to introduce
the relativity postulate.150
There is much to be admired in the lean account that Einstein gives in his research
papers. He expressed his argument tersely and we wish to unpack it. Einstein argues that
the two experiments should lead to the same result, and indeed they do. However, the
descriptions of this same result of the two different experiments – according to Maxwell’s electrodynamics – are not the same, and the asymmetry that Einstein discerns
here is not to be seen in the phenomena. The fault lies in the theory and not in the
phenomena, and Maxwell’s electrodynamics needs to be corrected. Einstein implicitly
argues that the source of the asymmetry in Maxwell’s electrodynamics is the absence
of a relativity principle. That is, he does not address the role of Maxwell’s equations in
149
Einstein 1920; Janssen et al. 2002, 264–265 (italics in the original): “Der Gedanke, dass es
sich hier um zwei wesens-verschiedene Fälle handle, war mir aber unerträglich. Der Unterschied
zwischen beiden konnte nach meiner Überzeugung nur ein Unterschied in der Wahl des Standpunktes sein, nicht aber ein realer Unterschied. Vom Magneten aus beurteilt, war sicherlich kein
elektrisches Feld vorhanden, vom Stromkreis aus beurteilt war sicher ein solches vorhanden. Die
Existenz des elektrischen Feldes war also ein relative, je nach dem Bewegungszustand des benutzten Koordinatensystems, und nur dem elektrischen und magnetischen Felde zusammen konnte,
abgesehen vom Bewegungszustande des Beobachters, bezw. Koordinatensystems, eine Art objecktiver Realität zugestanden werden. Die[se] Erscheinung der magnetelektrischen Induktion
zwang mich dazu, das (spezielle) Relativitätsprinzip zu postulieren.” Cf. Norton 2004, 49, and
Holton [1973]/1988, 382. Einstein made a similar remark in an inaugural address in 1920: see
Janssen et al. 2002, 313.
150
It turns out that the relativity postulate is not sufficient and so the light postulate has to be
added.
483
leading to this asymmetry; he disregards the formalism and looks directly at the nature
of the phenomena. This is physical reasoning at its best.
Again, we see that the phenomena do not show any asymmetries, that is, despite the
differences in the theoretical entities entailed by Maxwell’s equations, no distinction is
observed in the phenomena. By his use of the plural “asymmetries”, Einstein suggests
that this difficulty is manifest in other kinds of experiments, but offers no details. For
Einstein the distinction between an electric field and a magnetic field in the case of
moving bodies is meaningless; there is only an electromagnetic field which may be perceived differently by different observers.151 In a letter to his friend, Marcel Grossmann
(1878–1936), dated 1901, Einstein proclaimed: “It is a glorious feeling to recognize the
unity [Einheitlichkeit] of a complex of phenomena which appear as completely separate
entities to direct sense perception.”152 As we will show, he made the elements that comprise the electromagnetic field indistinguishable and this does not depend on any appeal
to symmetry. To be sure, “indistinguishability” can, in some circumstances, be related to
symmetry as, for example, in descriptions of a homogeneous medium.153 But this does
not apply to the case in hand, for the distinction between electricity and magnetism is
simply inappropriate and leads to the presence and absence of an electric field.
What was the motivation for Einstein to drive a wedge between the theory and its
formalism, to see so clearly that the system of equations could be retained despite a
radical reinterpretation of Maxwell’s theory?154 As we have indicated, Einstein read
Hertz closely and he appears to be an adherent of Hertz’s methodology. In 1892 Hertz
reported in the theoretical part of the Introduction to his Electric Waves that,
I ... endeavoured to form for myself in a consistent manner the necessary physical conceptions, starting from Maxwell’s equations, but otherwise simplifying Maxwell’s theory
as far as possible by eliminating or simply leaving out of considerations those portions
which could be dispensed with, inasmuch as they could not affect any possible phenomena.... Thus the representation of the theory of Maxwell’s own work, its representation as
a limiting case of Helmholtz’s theory, and its representation in the present dissertations –
151
Holton [1973]/1988, 380–381; Miller [1981]/1998, chapter 3. As Holton ([1973]/1988, 221–
222) first noted, Einstein’s thought experiment had already been presented in Föppl (1894) and,
according to Holton ([1973]/1988, 223–224), it is likely that this text influenced Einstein in his
formative years. Norton has indicated that Einstein ignored a second thought experiment described
by Föppl for which this analysis does not work: see Norton 2004, 54–56, 101–102. Since Einstein
did not mention it, we postpone our discussion until Section III.3 where we consider Föppl’s
treatment of the problems in electrodynamics.
152
Beck 1987, 166 (slightly modified); see Stachel et al. 1987, 290–291: “Es ist ein herrliches
Gefühl, die Einheitlichkeit eines Komplexes von Erscheinungen zu erkennen, die der direkten
sinnlichen Wahrnehmung als ganz getrennte Dinge erscheinen.”
153
See n. 29, above, for the definition of isotropy.
154
According to Martinez (2004, 25 n. 10), Walter Ritz (1878–1909) did not pay attention to
this important distinction which Einstein had assumed in his relativity paper. In an earlier paper,
Einstein (1905a, 132, 133) refers several times to Maxwell’s theory of electromagnetic processes
as an example of a wave theory which describes energy as a continuous spatial function. At stake
here is a different issue from the one in the relativity paper and there is no need for Einstein to
distinguish between the theory and its formalism.
484
however different in form – have substantially the same inner significance. This common
significance of the different modes of representation (and others can certainly be found)
appears to me to be the undying part of Maxwell’s work. This, and not Maxwell’s peculiar
conceptions or methods, would I designate as “Maxwell’s Theory.” To the question, “What
is Maxwell’s theory?” I know of no shorter or more definite answer than the following: –
Maxwell’s theory is Maxwell’s system of equations.155
Having stated his belief in the formalism of the equations as the core of the theory, Hertz
proceeds to spell out the different interpretations given to the formalism, claiming that
the difficulty is not necessarily of a mathematical nature.156 And he cautions the reader
that
scientific accuracy requires of us that we should in no wise confuse the simple and homely
figure, as it is presented to us by nature, with the gay garment which we use to clothe it.
Of our own free will we can make no change whatever in the form of the one, but the cut
and colour of the other we can choose as we please.157
Hertz wished to strip from the theory everything (which he calls “garments”) but the
equations, and then, as we will see, he recast them in a symmetrical form. Einstein
took this methodology to heart, but with a twist: in some respects he is more Hertzian
than Hertz. The theory with its interpretative baggage could, and should, be separated
from its purely formal-mathematical representation. Hence, according to Einstein, seeking symmetrical equations – in the sense that the relations between the symbols that
comprise the equations support some structure that exhibits correspondence among the
symbols – may not affect the actual physical issue at stake, thus adding an unnecessary
“garment” that has led physicists astray. The problem that Einstein presents at the outset
of his relativity paper is that the theory leads to two descriptions related to the same
155
Hertz [1893]/1962, 20–21; Hertz 1892, 22–23: “Ich versuchte deshalb mir die unentbehrlichen physikalischen Vorstellungen widerspruchsfrei selbst zu construiren, indem ich von den
Maxwell’schen Gleichungen ausging, im Uebrigen aber die Maxwell’sche Theorie so viel wie
möglich vereinfachte durch Elimination oder einfache Fortlassung aller derjenigen Elemente,
welche ich nicht verstand und welche entbehrlich waren, da sie auf keine möglichen Erscheinungen einen Einfluss üben konnten.... Die Darstellung der Theorie in Maxwells eigenem Werk,
die Darstellung als Grenzfall der Helmholtz’schen Theorie und die Darstellung in den vorliegenden Abhandlungen sind also wesentlich verschiedene Formen für einen wesentlich gleichen
gemeinsamen Inhalt. Dieser gemeinsame Inhalt der verschiedenen Formen, für welche gewiss
noch viele andere Formen gefunden werden können, erscheint mir als der unsterbliche Theil der
Maxwell’schen Arbeit, diesem Inhalt und nicht den besonderen Vorstellungen oder Methoden
Maxwells möchte ich den Namen ‘Maxwell’sche Theorie’ vorbehalten wissen. Auf die Frage,
‘was ist die Maxwell’sche Theorie’ wüsste ich also keine kürzere und bestimmtere Antwort als
diese: Die Maxwell’sche Theorie ist das System der Maxwell’schen Gleichungen.”
156
Hertz [1893]/1962, 22; Hertz 1892, 23: “Die oft gehörte Ansicht, dass diese Schwierigkeit
mathematischer Natur sei, kann ich nicht theilen.”
157
Hertz [1893]/1962, 28; Hertz 1892, 31: “Aber die Strenge der Wissenschaft erfordert doch,
dass wir dies bunte Gewand, welches wir der Theorie überwerfen, und dessen Schnitt und Farbe
vollständig in unserer Gewalt liegt, wohl unterscheiden von der einfachen und schlichten Gestalt
selbst, welche die Natur uns entgegenführt und an deren Formen wir aus unserer Willkür nichts
zu ändern vermögen.” See also idem [1893]/1962, 195, and idem 1892, 208.
485
phenomenon, and that these descriptions are different (introducing different kinds of
fields).158 To be truly Hertzian, one has to come up with a physical argument to account
for this disparity of one phenomenon with two contrasting descriptions. And we claim
that in all likelihood this is the way Einstein read Hertz.
In outline, our argument goes as follow: Hertz’s work is part of the background to
Einstein’s thinking on issues in electrodynamics. We will see (Section III.1, below) that
Hertz explicitly used the term asymmetry in his essay of 1884. However, for Hertz this
asymmetry is purely formal and he then derives Maxwell’s equations in a symmetrical
form. The derivation does not depend on the ether for, according to Hertz, the derivation is consistent with views opposed to Maxwell’s theory. A short time later Hertz
became aware of faults in the derivation, but the formalism nevertheless holds and is
well confirmed. After his successful experimental demonstration of the existence of
electric waves, Hertz opts for the ether as presented in Maxwell’s theory but retains the
equations in their symmetrical form. Our claim is that Einstein was fully aware of these
developments that Hertz expressed in his papers and responded to them.
Consider the opening paragraphs of two of Hertz’s important papers. In the paper of
1884 we read:
When Ampère heard of Oersted’s discovery that the electric current sets a magnetic needle
in motion, he suspected that electric currents would exhibit moving forces between themselves. Clearly his train of reasoning was somewhat as follows: – The current exerts
magnetic force, for a magnetic pole moves under the influence of the current; and the
current is set in motion by magnetic forces, for by the principle of action and reaction a
current-carrier will also move under the influence of a magnet.159
Admittedly, in this paragraph Hertz refers neither to Faraday nor to Maxwell. Further,
for Hertz the issue is whether there are two kinds of magnetic forces. Notwithstanding
these differences, the opening lines are similar in structure to Einstein’s opening paragraph of the relativity paper (Section II.2, Case 5, above). We call attention to the fact
that both Einstein (1905c) and Hertz (1884) begin with discussions of induction and
illustrate this phenomenon with similar experiments presented in bare outline. When we
turn to the opening passage of Hertz (1890a) which explores the fundamental equations
of electromagnetics for bodies at rest, we find:
The system of ideas and formulae by which Maxwell represented electromagnetic phenomena is in its possible developments richer and more comprehensive than any other of
the systems which have been devised for the same purpose. It is certainly desirable that a
158
Darrigol also argues that the issue at stake concerns two contrasting descriptions; see Darrigol 1996, 295: “The theory gives two different descriptions of [the magneto-induction current]
phenomenon according as it is the magnet, or the conductor, that moves in the ether.”
159
Hertz 1896, 273 (slightly modified); idem [1884]/1895, 295: “Sobald Ampère die Entdeckung Oersted’s erfuhr, dass der elektrische Strom die Magnetnadel in Bewegung Setze,
vermutete er, dass auch unter einander elektrische Ströme bewegende Kräfte äussern müssten.
Offenbar war sein Gedankengang nahezu dieser: Der Strom übt magnetische Kräfte aus – denn
ein Magnetpol bewegt sich unter dem Einflusse des Stromes; und der Strom wird bewegt durch
magnetische Kräfte – denn nach dem Prinzip der Reaktion bewegt sich auch ein Stromträger unter
dem Einflusse des Magnetes.”
486
system which is so perfect, as far as its contents are concerned, should also be perfected
as far as possible in regard to its form. The system should be so constructed as to allow its
logical foundations to be easily recognised; all unessential ideas should be removed from
it.... Maxwell’s own representation [of his ideas] indicates that, in this respect, he did not
reach this goal; it frequently wavers between the conceptions which Maxwell found in
existence, and those at which he arrived.... This procedure leaves behind it the unsatisfactory feeling that there must be something wrong about either the final result or the way
which led to it. Another effect of this procedure is that in the formulae there are retained
a number of superfluous, and in a sense rudimentary, ideas which only possessed their
proper significance in the older theory.... One would expect to find in these fundamental
equations relations between the physical magnitudes which are actually observed, and not
between magnitudes which serve for calculation only.160
Hertz shows the way to a critical examination of Maxwell’s theory. In particular, he
insists that the equations should only concern physically meaningful magnitudes, not
artifacts of the theory. One could say that Einstein followed Hertz in his recognition
that something was wrong in the theory, but Einstein had a different “diagnosis” and
eventually came up with a different “cure”.
The formalism, the core of Maxwell’s theory stripped of all its “garments”, does not
include the ether. Note that Einstein writes: Daß die Elektrodynamik Maxwells ... that is,
he does not address the formalism, but the theory as it is commonly interpreted. Thus,
“as usually understood” refers to the assumption of the ether and the further assumption
that electricity and magnetism are distinct phenomena. For Einstein the issue is not the
symmetry or the asymmetry of the equations – which was a central theme for Hertz,
Heaviside, and some other notable physicists – but the removal of asymmetries that are
“artifacts” of the theory and have no objective reality.
How then does Einstein resolve the problem he formulates at the outset of his relativity paper? Note that the problem is new. Both versions of the experiment lead to the
same measurable (and verifiable) current which means that the underlying equations are
correct. But the fact that in one case the theory demands that there should be an electric
field, and in the other case no electric field (what seems to be a basic contradiction),
160
Hertz [1893]/1962, 195–196 (slightly modified); idem [1890a]/1892, 208–209: “Das System von Begriffen und Formeln, durch welches Maxwell die elektromagnetischen Erscheinungen
darstellte, ist in seiner möglichen Entwickelung reicher und umfassender, als ein anderes der
zu gleichem Zwecke ersonnenen Systeme. Es ist gewiss wünschenswerth, dass ein der Sache
nach so vollkommenes System auch der Form nach möglichst ausgebildet werde. Der Aufbau
des Systems sollte durchsichtig seine logischen Grundlagen erkennen lassen; alle unwesentlichen
Begriffe sollten aus demselben entfernt und die Beziehungen der wesentlichen Begriffe auf ihre
einfachste Gestalt zurückgeführt sein. Die eigene Darstellung Maxwell’s bezeichnet in dieser
Hinsicht nicht das erreichbare Ziel, sie schwankt häufig hin und her zwischen den Anschauungen, welche Maxwell vorfand, und denen, zu welchen er gelangte.... Dieser Gang hinterlässt das
unbefriedigende Gefühl, als müsse entweder das schliessliche Ergebniss oder der Weg unrichtig
sein, auf welchem es gewonnen wurde. Auch lässt dieser Gang in den Formeln eine Anzahl überflüssiger, gewissermassen rudimentärer Begriffe zurück, welche ihre eigentliche Bedeutung nur
in der alten Theorie der unvermittelten Fernwirkung besassen.... In [die Grundgleichungen] man
ohnehin erwarten darf, Beziehungen zwischen Grössen der physikalischen Beobachtung, nicht
zwischen Rechnungsgrössen zu finden.”
487
indicates that there is something wrong in the theory apart from the equations. Even
after realizing that something is wrong with a theory, there is usually no direct way to
know what has gone wrong and how to fix it. In Part III, below, we will present evidence
in support of the claim that prominent contemporary physicists discussed symmetry and
asymmetry (as well as a few related terms) for the relation between electricity and magnetism. However, no one – to the best of our knowledge – associated these terms directly
with the phenomena in the way Einstein did.
The formulation of a new physical problem at times requires a new way of looking at
physics and consequently an altogether new solution. In Part II of the relativity paper, that
is, the electrodynamic part, Einstein introduces without much ado the Maxwell–Hertz
equations and immediately proceeds to subject them to the transformations he derived
earlier in § 3. Notice that in the course of pursuing the consequences of these transformations, Einstein appeals to algebraic symmetry – Case 2 in our discussion above (Section
II.2) – but in no way does he link it to the asymmetry that he is about to remove; rightly
so, for these are two different contexts and two different senses of symmetry, and in any
case one is algebraic and the other is physical.
Having obtained the transformed equations, Einstein then interprets them. He imagines a pointlike unit electric pole in motion in an electromagnetic field and examines
the forces to which it is subjected in its motion. In the standard treatment one would
expect a distinction to be made between the electric and the magnetic force. Indeed,
Einstein speaks of analogous [Analoges gilt] treatment of the two forces. But then, in
view of the new physical analysis, the electromotive force merely plays the role of an
auxiliary concept [die Rolle eines Hilfsbegriffes] that is required because “the electric
and magnetic forces do not have an existence independent of the state of motion of the
coordinate system.”161 In other words, electric and magnetic forces are one and the same,
depending only on the state of motion of the system. That is, the distinction between the
electric and magnetic forces is an artificial feature of Maxwell’s formalism; this feature
is not found in the phenomena and the distinction vanishes when the theory of relativity
is applied.
Einstein is now in a position to remind his reader of the problem he formulated
at the outset of the paper. He refers to the specific case he illustrated, hence the use
of the singular form, Asymmetrie. He claims that it simply disappears [verschwindet].
Seen from the perspective of relativity theory the two cases are in fact one.162 With
this conclusion, there is only a single phenomenon and calling it symmetrical was no
longer an issue for Einstein. By removing the asymmetry (singular), Einstein has unified the electromagnetic field and he does not show any interest in describing this result
as symmetrical. Einstein thus established a unified foundation for electrodynamics and
mechanics and, indeed, for the whole of physics. He did not regard the principle that
underlies this unification – that is, the relativity principle – as a symmetry principle.
161
Beck 1989, 159; Einstein 1905c, 910.
The interested reader may wish to pursue the idea that Einstein’s claim that the two experiments are really only one is, in principle, similar to Joseph’s interpretation of pharaoh’s two
dreams: “The dream of Pharaoh is one” (Gen 41:25).
162
488
Considering the range of meanings for symmetry which – as we have seen – Einstein
invoked in two of his papers in 1905, why did he choose the term asymmetry, giving it an
entirely new meaning here? To be sure, the contrast, “with”/“without” an electric field,
gives the asymmetry in this case a coherent meaning,163 but then Einstein could simply
have referred to the situation as a contradiction. This, however, would imply a logical
strictness that was probably more than he intended. Asymmetry, as he uses it, gives him
much flexibility, for the term is not precisely defined.
In fact, we may inquire: What is the symmetry which corresponds to Einstein’s asymmetry? We may say that formerly there was an asymmetry between different frames and
one – the absolute frame – was preferred; afterwards, we might say that there is a symmetry between all inertial frames – a fact expressed by the relativity principle. However,
as we have seen, by removing the asymmetry Einstein does not state that he has restored
symmetry and, in any event, he does not pay any attention to this issue. So the question
persists, why did Einstein choose the term asymmetry to describe contradictory physical descriptions? Following our methodology that words are markers – they indicate a
trajectory – we suggest that Einstein is, as it were, in conversation with Hertz. On this
view, Einstein opens the relativity paper with a critique of Hertz’s approach to electrodynamics. Using Hertz’s own methodology, Einstein implicitly responded to Hertz that
his attempt at recasting Maxwell’s equations into a symmetrical form was ill-conceived:
asymmetry is deeply embedded in Maxwell’s theory and a formal modification of the
equations does not suffice to render the theory symmetrical; in any event, this deep-seated
asymmetry is an artifact that has no counterpart in the phenomena. A radical solution is
called for: recasting the fundamental postulates of the theory. We will discuss at length
the position of Hertz in the next section. For now it may suffice to indicate passages in
Hertz that provide a context for Einstein’s usage.
In 1884 Hertz began an intensive study of Maxwell’s theory. He noted the asymmetrical way in which one deduced the electric and magnetic forces from the vector-potentials
of their currents, respectively. We first display the German text and then the English translation, paying special attention to the usage of the terms – marked in boldface – that we
claim resonate with terms in Einstein’s relativity paper.
Die Vektorpotentiale elektrischer und magnetischer Ströme traten bisher als etwas verschiedenes auf und aus ihnen leiteten sich die elektrischen und magnetischen Kräfte in
asymmetrischer Weise ab. Dieser Gegensatz zwischen beiden Arten von Kräften verschwindet, sobald wir versuchen, die Ausbreitung dieser Kräfte selbst zu bestimmen, d.h.
sobald wir die Vektorpotentiale aus den Gleichungen eliminieren.164
The vector-potentials of electric and magnetic currents have hitherto occurred as quite separate, and from them the electric and magnetic forces were deduced in an asymmetrical
manner. This contrast between the two kinds of forces disappears as soon as we attempt
163
The contrast is not, as we have indicated, between the pair, electric/magnetic, which in any
case is not a proper contrast.
164
Hertz [1884]/1895, 310.
489
to determine the propagation of these forces themselves, i.e. as soon as we eliminate the
vector-potentials from the equations.165
We suggest that there is an “echo” of these usages in the passage by Einstein, quoted in
Section II.2, Case 5, above:
Es ist ferner klar, daß die in der Einleitung angeführte Asymmetrie bei der Betrachtung
der durch Relativbewegung eines Magneten und eines Leiters erzeugten Ströme verschwindet.
It is further clear that the asymmetry mentioned in the Introduction when considering
the currents produced by the relative motion of a magnet and a conductor, disappears.
Notice the usage of the same verb, namely, verschwinden, for describing the result of
the application of some theoretical move to counter the occurrence of asymmetry. The
claim is not that Einstein’s view was foreshadowed by Hertz; rather, Einstein uses the
language of Hertz to make a different point. And Einstein did not have to say “disappear”
to convey the sense of “removed” or “vanished”.166 Einstein’s usage of asymmetry is
odd since the meaning he gives to it is previously unattested.167
As we will see, in Hertz’s case the formalism for removing the asymmetry involves
modifying the equations by eliminating the vector potentials. The asymmetry disappears
and the derivation becomes symmetrical in the following sense:
[Die Gleichungen] verknüpfen daher die magnetischen und elektrischen Kräfte im leeren Raume überhaupt, unabhängig vom Ursprunge der letzteren. Die magnetischen und
elektrischen Kräfte sind jetzt miteinander vertauschbar.
[The equations] connect together the electric and magnetic forces in empty space quite
generally, whatever the origin of these forces. The electric and magnetic forces are now
interchangeable.168
The idea that the equations should be symmetrical by retaining distinct yet interchangeable electric and magnetic forces is a constant theme for Hertz, despite various substantial
165
Hertz 1896, 286–287 (slightly modified).
See nn. 298 and 309, below, for the verbs used by Wiechert and Wien, respectively: beseitigen
and fortfallen.
167
For another instance of this kind, note that Einstein (1905c, 910) uses the expression “magnetomotorische Kräfte” (with quotation marks). The editors (Stachel et al. 1989, 309 n. 25) report
a similar usage in Heaviside ([1892a]/1925a, 446 and 546), but they overlooked the occurrence of
this relatively rare expression in Hertz [1884]/1895, 300: “die magnetische (magnetomotorische)
Kraft” (cf. Hertz 1896, 277), which is most likely Einstein’s source. Hence, in addition to Asymmetrie, Hertz 1884 and Einstein 1905c have another technical term in common. But even before Hertz
and Heaviside, the expression, “magneto-motive force”, occurs frequently in an earlier article by
an American physicist, H. A. Rowland (1880, 91, et passim). In all cases (including Einstein’s)
this term represents an analogy with “electro-motive force” which is much more commonly used
(see, e.g., Lorentz 1904a, 89: “Elektromotorische Kräfte”).
168
Hertz [1884]/1895, 310–311 (boldface added); Hertz 1896, 287.
166
490
changes in his theory over the years. This idea is the target of Einstein’s criticism. Moreover, in 1884 Hertz recast the derivation of the equations in electrodynamics into a
symmetrical form without appealing to the ether. In his mature theory, developed in
the wake of the experiments on electric waves, he included the ether. As we have seen,
already in 1899 this was not at all to Einstein’s liking.
We suggest that in Einstein’s view Hertz did not address the right problem. The issue
for Einstein concerned the theory, not just the equations; thus, formal solutions were
not in fact solutions, for they were merely manipulations of formulas. This is implied in
Einstein’s explicit reference in the opening sentence of the paper to Maxwell’s theory
and not to his equations (to which he refers later in the electrodynamic part of the paper).
The problem is that the theory distinguishes between electricity and magnetism whereas,
in fact, these phenomena are different manifestations of the same thing – the electromagnetic field. In a word, indistinguishability is the key concept, not interchangeability.
This is connected with the ether hypothesis and, if one rejects the ether (as Einstein
considered in 1899), it seems that one has rejected Maxwell’s theory at the same time.
But the equations are well confirmed – so how to resolve this dilemma?
Einstein takes Maxwell’s equations in Hertz’s form since they do not depend on
the ether hypothesis (despite Hertz’s later views on this issue). The question is then,
has Hertz eliminated the asymmetry as he claimed to have done? And Einstein replies,
No – asymmetry still remains in the theory even though it has been removed from the
equations. For this argument to work, Einstein has to adhere to Hertz’s terminology, at
least to some extent.
But what term should be used to express the problem that remains in Maxwell’s
theory? Einstein chose asymmetry, the very term which Hertz used to refer to problems
in Maxwell’s theory in the derivation of the equations which he, according to Einstein,
did not succeed in removing. The particular difficulty is that, according to Maxwell’s
theory, one can generate an electric field, and not generate it, by means of the same experiment (seen in two different ways). That is, the term asymmetry in Einstein’s account
is an artifact of his response to his predecessor; it did not have this meaning previously (or afterwards). Einstein’s vocabulary is often idiosyncratic and here it reflects
his engagement with the ideas expressed in Hertz’s publications that led to his critique
of Maxwell’s theory. For Hertz the removal of asymmetry results in symmetry, but for
Einstein the contrary of asymmetry – symmetry – is not an issue: symmetry has become
inconsequential. Thus Einstein’s usage of the term asymmetry is sui generis.
We have argued that Einstein’s argument is physical reasoning at its best. In Part
III we will show that notable contemporary physicists associated asymmetry with the
formalism of Maxwell’s equations, but again Einstein’s approach differs from theirs and
is unprecedented.
In 1889 Hertz addressed the relation between light and electricity and at the end of
his discussion formulated what he called “the ultimate goal of physics”.
In another direction looms the question of the nature of electricity. Viewed from this standpoint it is somewhat concealed behind the more definite question of the nature of electric
and magnetic forces in space. Directly connected with these is the great problem of the
nature and properties of the ether which fills space, of its structure, of its rest or motion, of
its finite or infinite extent. More and more we feel that this is the all-important problem,
491
and that the solution of it will not only reveal to us the nature of what used to be called
imponderables, but also the nature of matter itself and of its most essential properties –
weight and inertia. The quintessence of ancient systems of physical science is preserved
for us in the assertion that all things have been created out of fire and water. For physics
at the present time the question is no longer far [from consideration], is it perhaps not
the case that all things have been created out of the ether? These are the ultimate goals of
physical science in our time.169
In 1905 Einstein eliminated Hertz’s “ultimate” goal for physics by rendering the ether
entirely superfluous.
To the best of our knowledge, there was no response to Einstein’s use of asymmetry
in the first few years immediately after the publication of the relativity paper. One is
forced to conclude that its significance was not appreciated at the time – or even later,
much later, that is, “as usually understood at the time”.
4. Conclusion
In our survey of Einstein’s varied usages of symmetry in 1905 we have found that,
for the most part, Einstein conformed to contemporary practice. In contrast, Einstein’s
asymmetry is unprecedented and we have searched the literature in physics before 1905
in an effort to find the relevant context for it. As we will argue in Part III, Hertz was
the key figure, known to Einstein, who introduced a new usage for the term asymmetry
to which Einstein responded. Following Hertz’s methodology of separating the physical content of a theory from its formalism, Einstein suspects early on (1899) that the
ether may be devoid of physical meaning. Einstein associates the asymmetry that flows
from Maxwell’s theory with Maxwell’s physical presupposition of the ether and not
with the structure of his system of equations (whether it be in symmetrical form or
otherwise). Put differently, Einstein focuses on the physics of the problem and not on its
formal representation or the manipulation that the formalism facilitates. Einstein shows
that electrodynamics does not depend on absolute motion from which it follows that
the ether hypothesis is superfluous.170 Thus, in the relativity paper (1905c), Einstein
169
Hertz 1896, 326 (slightly modified); Hertz [1889b]/1895, 353–354: “In anderer Richtung
liegt nicht ferne die Frage nach dem Wesen der Elektricität. Von hier gesehen verbirgt sie sich
hinter der bestimmteren Frage nach dem Wesen der elektrischen und magnetischen Kräfte im
Raume. Und unmittelbar an diese anschliessend erhebt sich die gewaltige Hauptfrage nach dem
Wesen, nach den Eigenschaften des raumerfüllenden Mittels, des Äthers, nach seiner Struktur,
seiner Ruhe oder Bewegung, siener Unendlichkeit oder Begrenztheit. Immer mehr gewinnt es
den Anschein, als überrage diese Frage alle übrigen, als müsse die Kenntnis des Äthers uns nicht
allein das Wesen der ehemaligen Imponderabilien offenbaren, sondern auch das Wesen der alten
Materie selbst und ihrer innersten Eigenschaften, der Schwere und der Trägheit. Die Quintessenz
uralter physikalischer Lehrgebäude ist uns in den Worten aufbewahrt, dass alles, was ist, aus dem
Wasser, aus dem Feuer geschaffen sei. Der heutigen Physik liegt die Frage nicht mehr ferne, ob
nicht etwa alles, was ist, aus dem Äther geschaffen sei? Diese Dinge sind die äussersten Ziele
unserer Wissenschaft, der Physik.”
170
Cf. Martinez 2004, 8.
492
explicitly rejects the ether hypothesis while relying on Hertz’s formal presentation of
the equations – without any derivation and without any remark on their symmetrical
structure. The presuppositions in Maxwell’s theory lead to two incompatible descriptions of the same phenomenon. Einstein calls this incompatibility asymmetry and it had
to be eliminated. Einstein realized that focusing on the formalism is misleading; at stake
was an issue in physics, not in mathematics, and physical reasoning, not mathematical
manipulations, has to be brought to bear on the problem.
III. Background. The term symmetry and its “relatives”:
duality, parallelism, and reciprocity
We now proceed to present and analyze various discussions in the period between
1880 and 1905 of the relation between electricity and magnetism. This issue arose in the
context of recasting Maxwell’s equations in symmetrical form. We focus our attention on
the treatment of this issue by some principal players notably, Hertz, Heaviside, Föppl,
Emil Wiechert (1861–1928), Wien, and Lorentz. Although symmetry and asymmetry
were often used as technical terms, none of these scientists anticipated Einstein’s view
of asymmetry. We will also indicate some of the ways these scientists responded to each
other.
1. Heinrich Hertz (1857–1894)
Hertz died at the early age of 37; nevertheless, in this short lifetime he became a leading classical physicist and an outstanding philosopher.171 Indeed, his contributions are
highly regarded both for his experimental originality and for his theoretical insights.172
In 1884 Hertz embarked on a detailed and critical analysis of Maxwell’s set of equations.
This major undertaking preceded his famous tour de force: proving experimentally that
electric waves exist. Hertz attempted in this paper “to demonstrate the truth of Maxwell’s
equations by starting from premises which are generally admitted in the opposing system
of electromagnetics [elektrodynamik], and by using propositions which are familiar in
it.”173 That is, Hertz sought to show the validity of Maxwell’s set of equations even if one
starts with the premises of opposing theories, perhaps alluding to the viewpoint of his
171
For extensive lists of primary and secondary sources as well as discussions, see Baird et al.
1998.
172
See, e.g., Pyenson 1982, 141.
Hertz 1896, 289; idem [1884]/1895, 313: “Ich habe im Vorhergehenden versucht, die
Gültigkeit der Maxwell’schen Gleichungen nachzuweisen auf Grund von Prämissen, welche auch
von der gegnerischen Elektrodynamik zugegeben werden und unter Benutzung von Schlussreihen,
welche dieser Elktrodynamik geläufig sind.” Cf. Buchwald 1994, 198. By “opposing system” Hertz
meant any electrodynamics which could accommodate Neumann’s potential law: see Hertz 1896,
276 n. 1 and idem [1884]/1895, 298 n. 2. On Neumann’s potential law, see Archibald 1989.
173
493
mentor, Hermann von Helmholtz (1821–1894).174 In so doing, Hertz did not introduce
the ether in the derivation of the equations. Hertz distanced himself from the assumption
of an ether or, at least, he did not ascribe to it the importance it had in Maxwell’s original
theory.175 At stake then was the issue of derivation.
Hertz’s wish to construct a “bridge” between the opposing views is expressed in the
following principle that guides him throughout the paper:
It has perhaps nowhere been explicitly stated that the electric forces, which have their
origin in the action of [electric and magnetic] induction, are in every respect equivalent to
equal and equally directed electric forces of electrostatic origin; but this principle is the
necessary presupposition of, and the conclusion from, the chief notions which we have
formed of electromagnetic phenomena generally.176
Hertz termed this principle, “the principle of the unity of electric force”, and then elaborated another principle, that of “the unity of magnetic force”.177 Thus, according to
Hertz, “the essential step in this reasoning is the assumption that only one kind of magnetic force exists.”178 Hertz distinguishes then between the electric and the magnetic
force, claiming, however, that each force is unified in the sense that different sources
yield the same kind of force.
Perhaps for our purposes the clearest statement of Hertz’s intentions appears in a
paper which he published in the following year: “On the Dimensions of the Magnetic
Pole in Different Systems of Units.” Here we see that Hertz is particularly sensitive
to laboratory demands: his interests are theoretical but also and, perhaps even more
so, experimental. Hertz explicitly writes that his motivation is to “make the magnetic
and electrostatic systems change places [Platz vertauschen].”179 He sought to establish,
first theoretically and then experimentally, that electrostatic force and electromagnetic
induction are interchangeable. Thus, “in the laws of electric induction we need only interchange the words ‘electric’ and ‘magnetic’ throughout in order to obtain the inductive
174
Hertz was a student in Berlin where his teachers included Helmholtz and Kirchhoff, both of
whom served as examiners of his dissertation in 1880. See Buchwald 1994, 59, 97.
175
For the first instance in Hertz (1884) of ether, see n. 192, below. Hertz then extends the
claim to every ”homogenous medium” [homogene Medium].
176
Hertz 1896, 274 (slightly modified); idem [1884]/1895, 296: “Dass diejenigen elektrischen
Kräfte, welche aus Induktionswirkungen entspringen, nach jeder Richtung gleichbedeutend seien
mit gleichen und gleichgerichtenten Kräften elektrostatischer Quelle, ist ausdrücklich vielleicht
nirgends behauptet worden, aber dies Prinzip ist die notwendige Voraussetzung und Folgerung der
hauptsächlichsten Anschauungen, welche man sich über die elektrodynamischen Erscheinungen
überhaupt gebildet hat.”
177
Hertz 1896, 274; idem [1884]/1895, 297: “... der Einheit der elektrischen und ... der Einheit
der magnetischen Kraft ... bezeichen könnte....”
178
Hertz 1896, 273 (italics in the original); idem [1884]/1895, 295 (italics in the original): “Das
wesentliche Glied in dieser Schlussfolge ist die Annahme, dass es nur eine Art magnetishcer Kraft
gebe....”
179
Hertz 1896, 292; idem [1885]/1895, 316: “... das magnetische und das elektrostatische System genau ihren Platz vertauschen.”
494
actions in the magnetic circuit.”180 Notice that Hertz sought a scheme in which the
elements are interchangeable, though they do keep their identity as magnetism and electricity. Buchwald aptly comments that, beginning with his paper of 1884, Hertz was
engaged in “a struggle to forge a consistent understanding of electrodynamics out of the
contemporary chaos.”181
Against this background, Hertz states that the two forces, namely, electric and magnetic, have usually been deduced in an asymmetrical manner and, given the two principles
announced at the outset of the paper, it is not surprising that he was dissatisfied with
the common approach. He therefore proceeds to eliminate the asymmetry between the
forces and to render them interchangeable: he does so by purely formal, mathematical
means. This is a key text for our analysis; hence, we quote it at length. Note the symbols
that Hertz used (most of which were retained by Einstein in his relativity paper): A is
the reciprocal of the velocity of light in a vacuum; L, M, and N are the components of
the magnetic force; and U, V, W are the electric ones. The translators of Hertz’s paper
changed the notation, replacing the symbol for derivative with the symbol for partial
derivative.182
The vector-potentials of electric and magnetic currents have hitherto occurred as quite
separate, and from them the electric and magnetic forces were deduced in an asymmetrical [asymmetrischer] manner. This contrast between the two kinds of forces disappears
as soon as we attempt to determine the propagation of these forces themselves, i.e. as
soon as we eliminate the vector-potentials from the equations. This may be performed by
differentiating equations (9) with respect to t and removing the differential coefficients
of U, V, W with respect to t by equations (10).
[where equations 9 and 10 are, respectively
L = A(∂V /∂z − ∂W/∂y),
M = A(∂W/∂x − ∂U/∂z),
N = A(∂U/∂y − ∂V /∂x),
X = −A2 dU/dt
Y = −A2 dV /dt
Z = −A2 dW/dt
and L, M, N, X, Y, Z the completely corrected forces and U, V, W are presented by
converging series, with corresponding equations for the magnetic currents.183 ]
It may also be performed by differentiating equations (10) with respect to t, remembering
that, e.g.
A2 d 2 U/dt 2 = . . . = (∂/∂y)(∂U/∂y − ∂V /∂x) − (∂/∂z)(∂W/∂x − ∂U/∂z),
180
Hertz 1896, 277; idem [1884]/1895, 300: “Wir brauchen nur in den Gesetzen der elektrischen Induktion konsequent die Namen ‘elektrisch’ und ‘magnetisch’ zu vertauschen, um zu den
hier gesuchten Induktionswirkungen magnetischer Stromkreise zu gelangen.”
181
Buchwald 1994, 183.
182
This change in notation for Hertz’s equations is already evident in Cohn 1890, 626, and
Drude 1894, 315.
183
Hertz 1896, 284–285; idem [1884]/1895, 308.
495
and removing the functions of U, V, W in the brackets by means of equations (9). In this
way we get six equations connecting together the values of L, M, N, X, Y, Z in empty
space, viz., the following:
A(dL/dt) = ∂Z/∂y − ∂Y /∂z,
A(dM/dt) = ∂X/∂z − ∂Z/∂x,
A(dN/dt) = ∂Y /∂x − ∂X/∂y,
A(dX/dt) = ∂M/∂z − ∂N/∂y
A(dY /dt) = ∂N/∂x − ∂L/∂z
A(dZ/dt) = ∂L/∂y − ∂M/∂x
(12)
These same equations connect together the forces produced by magnetic currents, for
they are obtained by eliminating P, Q, R as well as U, V, W. Hence they connect together
the electric and magnetic forces in empty space quite generally, whatever the origin of
these forces. The electric and magnetic forces are now interchangeable [vertauschbar].184
Die Vektorpotentiale elektrischer und magnetischer Ströme traten bisher als etwas
verschiedenes auf und aus ihnen leiteten sich die elektrischen und magnetischen Kräfte
in asymmetrischer Weise ab. Dieser Gegensatz zwischen beiden Arten von Kräften verschwindet, sobald wir versuchen, die Ausbreitung dieser Kräfte selbst zu bestimmen, d.h.
sobald wir dieVektorpotentiale aus den Gleichungen eliminieren. Das kann einmal geschehen, indem man die Gleichungen 9) nach t differenziert, und die Differentialquotienten
von U V W nach t mit Hilfe der Gleichungen 10) entfernt, zweitens kann es geschehen,
indem man die gleichungen 10) nach t differenziert, sich erinnnert, dass z.B.:
A2 d 2 U/dt 2 = U = (d/dy)(dU/dy − dV /dx) − (d/dz)(dW/dx − dU/dz)
ist, und nun die in Klammern stehenden Kombinationen der UVW mittels der Gleichungen
9) fortschafft. Auf diese Weise erhält man sechs Gleichungen, welche die L, M, N, X, Y,
Z im leeren Raume miteinander verknüpfen, nämlich die folgendern:

 A(dL/dt) = dZ/dy − dY /dz,
A(dM/dt) = dX/dz − dZ/dx,

A(dN/dt) = dY /dx − dX/dy,
A(dX/dt) = dM/dz − dN/dy,
A(dY /dt) = dN/dx − dL/dz,
A(dZ/dt) = dL/dy − dM/dx.
(12)
Dieselben Gleichungen verknüpfen auch diejenigen Kräfte, welche von magnetischen
Strömen erzeugt werden, denn sie werden ebensowohl durch Elimination der P , Q, R als
der U , V , W erhalten. Sie verknüpfen daher die magnetischen und elektrischen Kräfte
im leeren Raume überhaupt, unabhängig vom Ursprunge der letzteren. Die magnetischen
und elektrischen Kräfte sind jetzt miteinander vertauschbar.185
Hertz is explicit about his move: his intention is to eliminate the asymmetry between the forces and to render them interchangeable. According to Buchwald, the forces
“should be symmetrical because they derive in the same manner from infinite series
of interaction potentials that have precisely the same form....”186 As Hertz formulated
it some years later, “electricity in motion produces magnetic force, and magnetism in
184
185
186
Hertz 1896, 286–287 (boldface added; slightly modified).
Hertz [1884]/1895, 310–311 (boldface added).
Buchwald 1994, 194. Cf. D’Agostino 1975, 288–292.
496
motion produces electric force; but both of these effects are only perceptible at high
velocities.”187 Indeed, the vector-potentials, both of the electric and of (what Hertz
called188 ) the magnetic current, which satisfy analogous differential equations, show
themselves to be quantities that are propagated with finite, though high velocity – the
velocity of light. Hertz aimed then at the following symmetrical result: “magnetic currents act on each other according to the same laws as electric currents.”189 It is important
to note that Hertz expected the forces and their governing laws to be interchangeable;
indeed, he tried to capture this idea in his new formalism, but nowhere – to the best
of our knowledge – does he suggest that the two distinct elements should be united.
Symmetry for Hertz appears to embody the notion of interchangeability, and Maxwell’s
equations ought to be recast to exhibit this feature. Thus, in Hertz’s case, the elimination
of asymmetry restores symmetry to the mathematical form of the equations. Despite certain verbal similarities, Hertz and Einstein do not share a common outlook. Hertz seeks
to restore symmetry between electric and magnetic forces; Einstein seeks to render the
electric and magnetic fields indistinguishable and thereby to eliminate the asymmetry
in the description of electrodynamic phenomena.
In order to obtain a symmetrical structure for the equations, Hertz eliminated the
vector-potentials. He did so by differentiating the equations with respect to t by which he
obtained the set of equations in (12) that do not contain components of the vector-potentials. An examination of this set of equations reveals that the structure is symmetrical
in the sense that the components correspond to each other in the same manner and are
readily interchangeable – vertauschbar.190 In subsequent sections of Part III we will
discuss this usage of symmetry in other physics texts of the late 19th century, but a
comparison with Einstein is called for here.
As we have seen, Einstein refers to asymmetry at the beginning of the relativity paper:
“Maxwell’s electrodynamics ... when applied to moving bodies, leads to asymmetries
that do not seem to adhere to the phenomena.” He argued that, on the one hand, from
Maxwell’s theory one would expect asymmetries in the descriptions of the phenomena
but, on the other, no such asymmetries are to be found in the phenomena. For Hertz,
the issue of symmetry does not concern the phenomena directly; rather, it concerns the
structure and the derivation of Maxwell’s equations. There is a clear difference here
between Hertz and Einstein. Hertz identifies and addresses a different problem from the
one Einstein states at the outset of his relativity paper. In Hertz’s case the problem lies in
the way Maxwell’s equations are set up and derived: the distinct phenomena of electricity and magnetism are analogous and the formalism ought to exhibit a correspondence
between them; hence the equations should be symmetrical in the sense that analogous
elements have to correspond and be interchangeable. Einstein, by contrast, argues that
187
Hertz 1896, 318; idem [1889b]/1895, 345: “Bewegte Elektricität übt magnetische Kräfte,
bewegter Magnetismus elektrische Kräfte aus, welche Wirkungen indessen nur bei sehr grossen
Geschwindigkeiten merklich werden.”
188
Hertz 1896, 276; idem [1884]/1895, 299: “Wir bezeichnen die Veränderung einer magnetischen Polarisation als einen magnetischen Strom....” Cf. Darrigol 2000, 236.
189
Hertz 1896, 276; idem [1884]/1895, 299: “Magnetische Ströme wirken aufeinander nach
den gleichen Gesetzen, wie elektrische Ströme.”
190
See nn. 91 and 184, above. Cf. Darrigol 2000, 237.
497
the fault is not in the equations – be they symmetrical or asymmetrical – but in their
interpretation, that is, in embedding them in an ether. Curiously, Einstein seems to arrive
at this position on the basis of Hertz’s own methodology:
1. The equations are valid regardless of the way they were originally determined;
2. The validity of the equations depends on their successful accounting for a wide
range of phenomena; and
3. The equations are valid independent of any specific interpretation of them.191
The third item is of particular interest for us here: Einstein has indicated that an interpretation which distinguishes electric and magnetic forces of moving bodies is false, for
there is only one kind of force involved, and it is simply electromagnetic. But, despite
the radical change in interpretation, the equations are still valid.
Hertz does not call attention to any lack of “analogy” in the consequences that flow
from his symmetrical version of Maxwell’s equations, but this is where Einstein focuses
his attention: the distinction between electricity and magnetism has to be dropped in
favor of “electromagnetic phenomena”. In sum, the contrast is between Hertz’s interchangeability and Einstein’s indistinguishability; while the former calls for a symmetrical structure, the latter seeks unity. The asymmetries of which Hertz and Einstein speak
are therefore conceptually different and ought not to be conflated.
We see then that, although Hertz was motivated by physical reasoning, he sought
analogous treatment of the equations for electricity and magnetism. He observes that
Maxwell’s equations are deduced in an asymmetrical fashion and so he manipulates the
equations formally in order to make them display the symmetry he sought. He adds that
Now the systems of forces given by the equations ... is just that given by Maxwell. Maxwell found it by considering the ether to be a dielectric in which a changing polarisation
produces the same effect as an electric current. We have reached it by means of other
premises, generally accepted even by the opponents of the Faraday–Maxwell view.192
Here Hertz derives the equations; he does not regard them as having an axiomatic status.
Once the formalism was in place, the equations allowed Hertz to deduce consequences
that are experimentally testable.193 The formalism, as Hertz noted, takes on a life of its
own. In 1889, in an essay on light and electricity, Hertz revealed the extent to which
Maxwell’s set of equations had impressed him:
It is impossible to study this wonderful theory without feeling as if the mathematical
equations had an independent life and an intelligence of their own, as if they were wiser
191
For a discussion of Hertz’s methodological position see nn. 155 and 157, above. Cf. Hon
1998, 59–67.
192
Hertz 1896, 288; idem [1884]/1895, 311–312: “Das durch die Gleichungen ... gelieferte System von Kräften ist nun kein anderes als das von Maxwell angegebene. Maxwell gelangte zu demselben, indem er den Äther als ein Dielektrikum ansah, dessen Polarisation bei ihrer Veränderung
die Wirkungen elektrischer Ströme ausübt. Wir sind zu demselben gelangt auf Grund anderer,
auch von den Gegnern jener Faraday–Maxwell’schen Anschauung im allgemeinen anerkannter
Prämissen.” On the difference between the two derivations, see Buchwald 1994, 195, where it is
also noted that Hertz “had neither used the ether nor assumed that ‘force’ propagates.”
193
For the expected new effects, see Hertz 1896, 276; idem [1884]/1895, 298–299.
498
than ourselves, indeed wiser than their discoverer, as if they gave forth more than he had
put into them....194
Hertz’s step can be understood as removing asymmetry but, as we have seen, this is
not the asymmetry with which Einstein (some 20 years later) opens his relativity paper.
We have argued that for Einstein this only meant that Hertz had identified a problem
in electrodynamics whose solution, however, was not properly supported by physical
considerations.
The experiments on electric waves, which made Hertz world-renowned, appear to
have changed his views. Presumably Hertz needed an ether to account for the transmission of the newly discovered electric waves in space. The assumption of the ether plays
a role in Hertz’s second phase of theoretical studies which begins right after the completion of his experimental work in 1888, for the ether offered a physical interpretation
(or model) to account for the electric waves. But, Knudsen notes, the ether as mediator
interferes with the symmetry of the situation.195 As an adherent of Maxwell’s theory,
Hertz then gives the set of equations a new status.
Maxwell arrived at [the equations] by starting with the idea of action-at-a-distance and
attributing to the ether properties of a highly polarisable dielectric medium. One can also
arrive at them in other ways. But in no way can a direct proof of these equations be deduced
from experience. It appears most logical, therefore, to regard them independently of the
way in which they have been arrived at, to consider them as hypothetical assumptions,
and to let their probability depend upon the very large number of natural laws which they
embrace.196
In contrast to his earlier view, Hertz now considers the equations axiomatically. His
point is that it is better to take these equations as postulates rather than claiming that
they can be proved or derived from principles and observations. He felt confident in
making this move because of “the very large number of natural laws” which follow from
the equations.197 The Einsteinian spirit of this methodological “inversion” is startling.
But, of course, it is the other way around: Hertz’s change of heart, namely, abandoning
194
Hertz 1896, 318; idem [1889b]/1895, 344: “Man kann diese wunderbare Theorie nicht
studieren, ohne bisweilen die Empfindung zu haben, als wohne den mathematischen Formeln
selbständiges Leben und eigener Verstand inne, als seien dieselben klüger als wir, klüger sogar
als ihr Erfinder, als gäben sie uns mehr heraus, als seinerzeit in sie hineingelegt wurde.”
195
Knudsen 1980, 349.
196
Hertz [1893]/1962, 138 (slightly modified); idem [1889a]/1892, 148: “Maxwell gelangte zu
[diese Aussagen], indem er von Fernkräften ausging und dem Aether die Eigenschaften eines in
hohem Grade dielektrisch polarisirbaren Mittels beilegte. Man kann auch auf anderen Wegen zu
denselben gelangen. Auf keinem Wege kann indessen bislang ein director Beweis für jene Gleichungen aus der Erfahrung erbracht werden. Es erscheint deshalb am folgerichtigsten, dieselben
unabhängig von dem Wege, auf welchem man zu ihnen gelangt ist, als eine hypothetische Annahme zu betrachten und ihre Wahrscheinlichkeit auf der sehr grossen Zahl an Gesetzmässigkeiten
beruhen zu lassen, welche sie zusammenfassen.”
197
See also Hertz [1893]/1962, 209: “Each one of the previous sections means an increase in
the number of facts embraced by the theory.” Cf. idem [1890a]/1892, 223: “Von den bisherigen
Abschnitten vermehrte ein jeder die Zahl der von der Theorie umfassten Thatsachen.”
499
the attempts at deriving the equations and turning them instead into the axioms on which
the theory is based, is similar to the move that Einstein makes in the introductory section
of the relativity paper where he raises two conjectures to the status of postulates.198 Be
that as it may, this change of view on the part of Hertz also affected his understanding
of the symmetry of the situation: it has become thoroughly formal in exactly the way
that Einstein found objectionable.
In 1890 Hertz published two very influential papers on the fundamental equations of
electromagnetics for bodies at rest and in motion.199 While it would have been natural
for Hertz to proceed from his pioneering work in 1884, he completely ignored it. As
D’Agostino remarks, this is all the more surprising, since “the symmetric form in which
he writes the equations for Maxwell’s electric and magnetic forces in 1884 is exactly the
same as the one he writes in 1890....”200 But Hertz’s point of departure, following his
experimental demonstration of electric waves, was now categorically different from that
of 1884. In fact, as Buchwald and Darrigol report, Hertz became aware that his original
argument suffered from some grave difficulties; in particular, an apparent contradiction
had escaped his notice, namely, his presupposition of the unity of the electric force cannot
accommodate Ampère’s force law.201 The change of methodology and presuppositions
may explain the fact that Hertz disregarded his own work and even ceded priority to
Heaviside:
Mr. Oliver Heaviside has been working in the same direction ever since 1885. From
Maxwell’s equations he removes the same symbols as myself; and the simplest form
which these equations thereby attain is essentially the same as that at which I arrive. In
this respect, then, Mr. Heaviside has the priority. Nevertheless, I hope that the following
representation will not be deemed superfluous.202
Notice that Hertz approves of Heaviside’s approach; in fact, they both do very much the
same thing – the purely formal manipulation of the equations in search of a symmetrical
presentation. In the next section, we will recount Heaviside’s treatment of Maxwell’s
equations and his way of introducing symmetry into them.
198
That is, seen in the context of Hertz, Einstein is not to be credited with a methodological
innovation in this respect.
199
Hertz [1890a]/1892, [1890b]/1892.
200
D’Agostino 1975, 295. Hertz did not include the paper of 1884 in his Untersuchungen über
die Ausbreitung der elektrischen Kraft (1892) where it truly belongs. The equations in (12) in Hertz
[1884]/1895, 311 are the same – to the letter – as equations (1) and (2) in idem [1889a]/1892, 148.
201
Buchwald 1994, 195–199, and 203–214; Darrigol 2000, 238.
202
Hertz [1893]/1962, 196–197; idem [1890a]/1892, 209–210: “In gleicher Richtung hat bereits
seit 1885 Hr. Oliver Heaviside gearbeitet. Die Begriffe, welche er aus den Maxwell’schen Gleichungen fortschafft, sind dieselben, welche auch ich fortschaffe; die einfachste Form, welche diese
Gleichungen dadurch annehmen, ist, von Nebendingen abgesehen, die gleiche, zu welcher auch
ich gelange. In dieser Hinsicht also gehört Hrn. Heaviside die Priorität. Trotzdem wird man, hoffe
ich, die folgende Darstellung nicht für überflüssig halten.” Drude (1894, 316), following Hertz,
also gives priority to Heaviside.
500
Given the equations,203 Hertz searches for solutions of the forces of the electric oscillations204 and examines, in relation to bodies at rest, “further conjectures respecting the
constitution of the ether.”205 However, his discussion of the fundamental equations of
electrodynamics for bodies in motion is more directly relevant to Einstein’s relativity
paper. Hertz opens this study with an explicit statement about his presupposition of the
ether.
From the outset the conception was insisted upon, that the electric and magnetic forces at
any point owe their action to the particular states of the medium which fills the space at the
point.... It was further assumed that the electric and magnetic state of the medium which
fills space could be completely determined for every point by a single directed magnitude
for each state....206
In extending the fundamental equations to include the phenomena of electrodynamics
of moving bodies, Hertz makes ample use of the ether hypothesis. In his view, “the
disturbances of the ether, which arise simulataneously, cannot be without effect; and of
these we have no knowledge.”207 Thus, “arbitrary assumptions” [willkürlicher Annahmen] concerning the motion of the ether will have to be considered. Hertz introduces
the Maxwellian view that the ether can move independently of the motion of ponderable
matter and for this reason one cannot remove the ether from any closed space. This is
part of the theoretical background for extending the equations:
If now we wish to adapt our theory to this view, we have to regard the electromagnetic
states [elektromagnetischen Zustände] of the ether and of the tangible matter at every
point in space as being in a certan sense independent of each other. Electromagnetic phenomena in bodies in motion would then belong to that class of phenomena which cannot
203
The English translator draws attention to the fact that the equations are presented in their
symmetric form: see Hertz [1893]/1962, 200 n. 1. In fact, Hertz’s papers of [1889a]/1892,
[1890a]/1892, and [1890b]/1892 have exactly the same equations as Hertz [1884]/1895. On Hertz’s
experimental and theoretical works, see Buchwald 1994, 203–329, and Darrigol 2000, 253–257.
204
Hertz [1893]/1962, 139ff.; idem [1889a]/1892, 149ff. Cf. Buchwald 1994, 310–324.
205
Hertz [1893]/1962, 201; idem [1890a]/1892, 214: “... die weiteren Vermuthungen über die
Constitution des Aethers anzuknüpfen.”
206
Hertz [1893]/1962, 241 (slightly modified); idem [1890b]/1892, 256: “Von Vornherein und
mit Strenge war die Anschauung zur Geltung gebracht, dass elektrische und magnetische Kräfte in
jedem Punkte ihrer Wirksamkeit besonderen Zuständen des daselbst befindlichen raumerfüllenden
Mittels entsprechen.... Es war weiter vorausgesetzt worden, dass der elektrische und magnetische
Zustand des raumerfüllenden Mittels für jeden Punkt vollständig bestimmt sei durch je eine einzige
Richtungsgrösse....”
207
Hertz [1893]/1962, 241–242; idem [1890b]/1892, 257: “Die gleichzeitig eintretenden Bewegungen des Aethers aber können nach unserer Anschauung nicht ohne Einfluss sein und von
diesen haben wir keine Kenntniss.” This remark is reminiscent of the view which Heaviside
expressed about the ether, see n. 249, below. For Hertz’s view on the ether, see also n. 169, above.
501
be satisfactorily treated without the introduction of at least two directed magnitudes for
the electric and two for the magnetic state.208
For our argument the important point is that Hertz draws a clear distinction between the
electric and the magnetic state even for bodies in motion, despite using an expression
with two aspects, namely, elektromagnetischen Erscheinungen. In this respect, his view
concerning bodies in motion is the same as his view concerning bodies at rest:
We here regard electric and magnetic force as signs of the state of the moving matter in
the same sense in which we have hitherto regarded them as signs of the states of matter
at rest.209
As we have seen, for Einstein the relative motion of the elements produces electromagnetic phenomena. Any attempt to separate the electric from the magnetic state in such
phenomena would result in an artifact of Maxwell’s theory that cannot correspond to
anything in the phenomena. In extending the formal symmetry of the equations to the
case of moving bodies, Hertz consistently pursues the interchangeability of two distinct
elements in the equations.
Hertz’s usages of symmetry in his later writings are similar to the way it was used
in setting up and solving hydrodynamic problems in the latter half of 19th century,210
and invoking the notions of isotropy and eolotropy is consistent with this approach.211
Moreover, he refers to the notion of “common axes of symmetry” with respect to the
optics of crystalline bodies.212 Most characteristic, however, is his usage of symmetry
with respect to moving bodies:
By a suitable orientation of our system of coordinates we can reduce the number of necessary constants; similarly a reduction takes place when there happen to be symmetrical
relations with respect to the system of co-ordinates used. In the simple case, in which the
substance is not only isotropic in its initial state, but also remains isotropic in spite of every
deformation that arises – viz. in a fluid, – the number of the new constants reduces to a
208
Hertz [1893]/1962, 242 (slightly modified); idem [1890b]/1892, 257: “Wollen wir nun dieser
Vorstellung unsere Theorie anpassen, so haben wir in jedem Punkte des Raumes die elektromagnetischen Zustände des Aethers und der greifbaren Materie in gewissem Sinne als unabhängig zu
betrachten. Die elektromagnetischen Erscheinungen in bewegten Körpern gehören alsdann zur
Classe derjenigen, welche sich nicht bewältigen lassen, ohne die Einführung mindestens je zweier
Richtungsgrössen für den elektrischen und den magnetischen Zustand.”
209
Hertz [1893]/1962, 243 (slightly modified); idem [1890b]/1892, 258: “Elektrische und magnetische Kraft betrachten wir hier als Zeichen für den Zustand der bewegten Materie in gleichem
Sinne, in welchem wir sie früher als Zeichen für die Zustände der ruhenden Materie betrachteten.”
210
See, e.g., Hertz [1893]/1962, 139, 140; idem [1889a]/1892, 149, 150. For our previous
discussion of these hydrodynamic problems, see Section II.1, above.
211
Hertz [1893]/1962, 199, 210; idem [1890a]/1892, 213, 223. For example, Hertz
([1893]/1962, 202) points out that “in homogenous isotropic non-conductors the phenomena
are qualitatively identical with those in free ether.” Cf. idem [1890a]/1892, 215: “In homogenen
isotropen Nichtleitern verlaufen die Erscheinungen qualitativ vollkommen wie im freien Aether.”
212
Hertz [1893]/1962, 239; idem [1890a]/1892, 253: “die Coordinatenaxen diesen gemeinschaftlichen Symmetrieaxen”.
502
single one, which then, together with the one magnetic permeability, sufficiently defines
the magnetic properties.213
This is a clear usage of symmetry in the geometrical sense that we have already discussed in hydrodynamic contexts, and Hertz’s reference to a fluid is an important clue.
In fact, Hertz refers explicitly to Kirchhoff’s Mechanik – the same source that, as we
have seen, Einstein cited in his dissertation.214 Hertz no longer refers to the symmetry of
the phenomena; instead, he uses an expression for “replace” [setzen ... an Stelle], rather
than the verb “interchange” [vertauschen], to indicate the relation between the electric
and magnetic force:
If we everywhere replace the word “electric” by the word “magnetic,” and the electrified
test-body by the pole of a magnetic needle, we obtain the definition of magnetic force.215
To conclude our discussion of Hertz’s contribution: Hertz recognized an asymmetry
in Maxwell’s set of equations and sought to present them in a symmetrical form.216
His motivation was a commitment to the interchangeability of electric and magnetic
phenomena. He then manipulated the equations mathematically in order to give them
a symmetrical structure. However, in the wake of his experimental work on electric
waves, Hertz no longer accepted the validity of his derivation of the equations; instead,
he opted for the axiomatic approach, turning the equations into postulates. In solving
the equations, which he displayed axiomatically, Hertz took advantage of symmetry in
the geometrical sense in much the same way that it had been used in hydrodynamics.
At the outset of his relativity paper Einstein called into question Maxwell’s theory
which includes Maxwell’s original set of equations and then, later in the paper, invoked
these equations in their Hertzian form, namely, in the symmetrical form that Hertz presented for the first time in his paper of 1884. And, as we have indicated, Einstein is
explicit about this: he appeals to the “Maxwell–Hertz equations”.217 In fact, Einstein
postulates these equations in the same spirit as Hertz did in 1890, implicitly agreeing
213
Hertz [1893]/1962, 261–262; idem [1890b]/1892, 278: “Durch geeignete Orientirung des
benutzten Coordinatensystems lässt sich die Zahl der erforderlichen Constanten vermindern; eine
Verminderung tritt ebenfalls ein, wenn Symmetrieverhältnisse hinsichtlich des gewählten Coordinatensystemes obwalten. In dem einfachsten Falle, in welchem die Substanz sowohl im Anfangszustande isotrop ist, als auch trotz jeder eintretenden Deformation isotrop bleibt, in einer
Flüssigkeit also, sinkt die Zahl der neuen Constanten auf eine einzige herab, welche alsdann zusammen mit der einen Magnetisirungsconstanten die magnetischen Eigenschaften in ausreichender
Weise definirt.” For another example, see Hertz [1893]/1962, 258; idem [1890b]/1892, 274.
214
See Hertz [1890b]/1892, 277 n. 1.
215
Hertz [1893]/1962, 198; idem [1890a]/1892, 212: “Setzen wir überall an Stelle des Wortes
‘elektrisch’ das Wort ‘magnetisch’ und an Stelle des elektrisirten Hülfskörpers den Pol einer Magnetnadel, so erhalten wir die Definition der magnetischen Kraft.” For vertauschen, see nn. 91 and
184, above.
216
Although Hertz does not call his equations symmetrical, they were explicitly described as
such by some of his successors. See, e.g., Cohn 1904, 1301.
217
Einstein 1905c, 907, 908; cf. n. 10, above. Einstein presented a modified version of Hertz’s
symmetrical form of Maxwell’s equations as had been done by some of his predecessors. For
example, Drude (1894, 315) presents the equations in their symmetrical form (with the partial
503
with Hertz on the axiomatic status of the equations. However, he did not comment on
their symmetrical form – thereby implying that this feature was not essential to the argument of the theory. In his letter of 1899 Einstein already objected to the ether218 which
was part of the theoretical basis for deriving these equations in the first place and later
on, in his relativity paper, he is clearly against invoking the ether. Moreover, Einstein’s
decision with respect to postulating the two principles of his theory (thus giving it an
axiomatic basis) is reminiscent of Hertz’s decision to postulate the equations. But these
lines of influence do not affect Einstein’s view of asymmetry, for it has to be removed
by physical arguments and not by mathematical manipulations. Nevertheless, Hertz’s
pioneering attempt to grapple with the asymmetrical nature of Maxwell’s theory formed
the background against which Einstein could clarify his thoroughly physical approach
to the problem. Hertz was a most effective interlocutor for Einstein: in Hertz’s work
Einstein found the quintessence of 19th-century physics and it served as his point of
departure for inaugurating the physics of a new era.
2. Oliver Heaviside (1850–1925)
Heaviside’s fame rests primarily on his development of the theory of cable signaling. As Oliver Lodge (1851–1940) pointed out, Heaviside recast and remodeled Lord
Kelvin’s theory of the propagation of pulses along cables, discarding the analogy with
conduction of heat on which Lord Kelvin had based his theory. Heat has no propagation, it is merely diffusion, and Kelvin’s theory of cable signaling was entirely based on
diffusion. Instead, Heaviside showed that the propagation of signals along wires was in
all essential respects identical with the propagation of electric waves in free space. His
transmission theory – transmission of waves, like light, not of diffusion phenomena like
heat – requires that the electric and the magnetic energies be equal. While in Kelvin’s
theory the magnetic field was neglected – indeed, it was considered an obstruction – and
only the electric field was taken into account, Heaviside argued that if the two energies
were made exactly equal, one would have true distortionless waves with every feature
of the signal reproduced at the far end, subject to nothing but attenuation.219
In January 1885 Heaviside began a serial publication in The Electrician of his most
fundamental contribution to the study of electrodynamics, “Electromagnetic Induction
and Its Propagation,” in which his critical account of Maxwell’s theory is presented
in vector notation. This version of Maxwell’s theory eventually became standard: it
presents the equations in their most universal form – without regard to any system of
derivative sign) and refers, inter alia, to Hertz ([1890a]/1892). Further, Drude (1894, 316 n. 4)
notes that the algebraic sign depends on the choice of direction of the coordinate system. Wiechert
(1898, 90) also invoked Hertz’s symmetrical presentation of the equations in much the way that
Drude had done.
218
See n. 13, above.
219
Lodge 1925. For Heaviside’s own account of heat diffusion vs. wave propagation, see Heaviside [1892a]/1925a, 434–441.
504
coordinates – by employing symbols, as Heaviside remarked, that only relate to the
intrinsic meaning of the operations involved.220
To the best of our knowledge, Heaviside does not characterize Maxwell’s equations
as asymmetrical; rather, he considers them incomplete. Indeed, he describes Maxwell’s
theory as unclear and remarks that it contains “many obscurities and some inconsistencies.”221 His approach is similar to that of Hertz, for he seeks, as he puts it, “a symmetrical
electromagnetic scheme.”222 In 1892, in the Preface to his collected papers, he reflected:
I here introduce a new method of treating the subject (to which I was led by considering
the flux of energy), which may perhaps be appropriately termed the Duplex method, since
its main characteristic is the exhibition of the electric, magnetic, and electromagnetic
equations in a duplex form, symmetrical with respect to the electric and magnetic sides,
introducing a new form of fundamental equation connecting magnetic current with electric force, as a companion to Maxwell’s well-known equation connecting magnetic force
and electric current.223
For Heaviside the shortcomings in Maxwell’s theory are evident in the lack of correspondence between electric and magnetic phenomena. His own duplex method – in
which Maxwell’s equations are recast in a symmetrical form – brings to light “many
useful relations which were formerly hidden from view by the intervention of the vector
potentials and its parasites.” Heaviside then adds that
there is a considerable difficulty in treating electromagnetism by means of Maxwell’s
equations of propagation in terms of these quantities [e.g., vector potentials], as presented
in his treatise. The difficulty is greatly increased, if not rendered practically insuperable,
when we pass to more advanced cases involving heterogenity and eolotropy and motion
of the medium supporting the fluxes. Here the duplex method furnishes what is wanted
in general investigations, and is the basis of “Electromagnetic Induction”....224
Heaviside makes it clear that he is interested in practical issues, that is, his aim is to
obtain distortionless transmission:
I had introduced in 1885, for purposes of symmetry, the fictitious quality of magnetic conductivity. When its effects upon the propagation of waves in a real conducting dielectric
are enquired into, it is found to act contrary to the real conductivity, so that the distortion
due to the latter can be entirely removed by having duplex conductivity.225
Notice that Heaviside explicitly introduces a “fictitious quality” in order to obtain symmetry – we will return to this important point shortly. In a note to his paper on the forces
in the electromagnetic field that he added in the reprint edition of 1892, Heaviside once
again reflected on his duplex method:
220
221
222
223
224
225
Heaviside [1892a]/1925a, 443.
Heaviside [1893]/1922, vii.
Heavisdie [1893]/1922, 50.
Heaviside [1892a]/1925a, viii.
Heaviside [1892a]/1925a, viii.
Heaviside [1892a]/1925a, x–xi.
505
The method of treating Maxwell’s electromagnetic scheme employed in the text ... may,
perhaps, be appropriately termed the Duplex method, since its characteristics are the
exhibition of the electric, magnetic, and electromagnetic relations in a duplex form, symmetrical with respect to the electric and magnetic sides. But it is not merely a method of
exhibiting the relations in a manner suitable to the subject ... but constitutes a method of
working as well.... The duplex method supplies what is wanted.... The state of the field is
settled by E and H, and these are the primary objects of attention in the duplex system.226
“Symmetrizing” the equations, as Heaviside puts it on another occasion, in the duplex
form was therefore not just a formal matter, but a move principally designed for ease of
applications.227
For Heaviside, the scheme is truly a balancing act: in a distortionless circuit the
leakage of conductance and the resistance of the circuit act oppositely in respect to distortion. The problem was practical, but its solution appeared to be mathematical, or so
Heaviside held.
What was the formal step that Heaviside took? He defines a magnetic current to make
electromagnetic theory symmetrical and thus consistent and formally complete.228 Calling B the magnetic induction, H the magnetic force, and µ the permeability, he argues that
As the rate of increase of the displacement in a non-conducting dielectric is the electric
current, so the rate of increase of B/4π may be called the magnetic current. Let it be G.
Then229
G = Ḃ/4π = µḢ/4π.
(Magnetic current)
(12)
He observes that “in ether, the electric current and the magnetic current are of equal significance.”230 Or, as he restated it some years later, in the duplex form “the electric and
magnetic sides of electromagnetism are symmetrically exhibited and connected....”231
He reports that he could only see the equations clearly and avoid inconsistencies by
changing their presentation. Recasting the equations in a symmetrical form was for
Heaviside an essential move in order to obtain a complete and consistent scheme of
presentation of electromagnetic phenomena.232
After stating the advantages of the symmetrical presentation of the equations, he
introduced a fictitious magnetic conduction current, with dissipation of energy. “There
226
Heaviside [1892b]/1925b, 541–542 n.
In the early 1890s, when Heaviside dealt with the electromagnetic equations in a moving
medium, he manipulated the equations and remarked ([1892b]/1925b, 540): “The fictitious K and
σ are useful in symmetrizing the equations, if for no other purpose.”
228
Recall Hertz’s notion of magnetischen Strom (see n. 188, above). In addition to introducing the notion of magnetic current, Hertz eliminated the vector potentials by differentiating the
equations (see n. 184, above); Heaviside too eliminated the vector potentials which he regarded
as obscure (see n. 224, above).
229
Heaviside [1892a]/1925a, 441 (bold upper case with a dot is a derivative of a vector).
230
Heaviside [1892a]/1925a, 441.
231
Heaviside [1893]/1922, iii–iv.
232
For his view of the complexity of the mathematics involved in electromagnetism, see Heaviside [1892a]/1925a, 206–207.
227
506
is probably no such thing,” Heaviside comments, but then proceeds to argue by analogy
with the electric conduction current. At the outset of his research on electromagnetic
induction and its propagation, Heaviside introduces the equation for Ohm’s law which
represents electric conduction:
C = kE
(Conduction current)
(1)
where C is the conduction current-density, E the electric force, and k the specific conductivity.233 Similarly, he introduces a term, gH, which he called the magnetic conduction
current, and adds it to the magnetic current in eq. (12). Thus,
G = gH + µḢ/4π.
(13)
We focus our attention on this factor, g, which Heaviside introduced by fiat. According
to Heaviside, g is a scalar quantity (corresponding to k in the electric case), but he then
notes that, since a steel magnet may be in a permanent state, it is most unlikely that g
has any existence at all. He candidly remarks that “the conduction magnetic current is
quite imaginary. But we may inquire what would happen in a closed ring of iron under
magnetizing force, on the supposition that g exists.”234
After presenting the electric and the magnetic equations, Heaviside brings the two
forces – the electric and the magnetic – into what he calls “cross connection” of which
there are two kinds.235 The first kind of “cross connection” explicitly refers to the step
that he has taken towards the symmetrical presentation of the equations: “to balance the
displacement current, we have introduced the magnetic current,” and he adds:
But, so far, we have no relations whatever between the electric and the magnetic quantities,
which we must have, in order to make a consistent system.236
There are two relations in Heaviside’s presentation which together render the system consistent: Ampère’s law and Faraday’s law. He casts the first “cross connection” between
the electric and the magnetic quantities into an algebraic formula called “curl”, characterizing it as “the electromagnetic operator” and connecting it with rotation:237
curl H = 4π,
(15)
where is the true current. When viewed purely geometrically, one may consider the
operator as mapping out space in another way than H. Heaviside now indicates that the
derived must be continuous, i.e. circuital, and its divergence is null everywhere. He
thus obtains the second essential operator of his duplex system, namely,238
div = 0;
233
234
235
236
237
238
or, (d1 /dx) + (d2 /dy) + (d3 /dz) = 0.
Heaviside [1892a]/1925a, 429.
Heaviside [1892a]/1925a, 442.
Heaviside [1892a]/1925a, 443, 447.
Heaviside [1892a]/1925a, 443.
Heaviside [1892a]/1925a, 443.
Heaviside [1892a]/1925a, 444.
(17)
507
All that is needed for the presentation of the system is now in place:
In all our equations, from (1) up to (14), not containing any relations between E and
H, those symbols mean the actual resultant electric and magnetic force from all causes.
Hence, in order that the two equations (15) and (21) may harmonise with the preliminary
equations (1) to (14), not only in space where there is no impressed force, but at the places
where such exist as well, we must, whilst still using E and H to denote the actual forces,
deduct from them the impressed forces in using the relations (15) and (21).239
In sum, by assuming the symmetry of the two electromagnetic phenomena, Heaviside
obtains the following set of equations:
curl (H − h ) = 4π = 4π kE + cĖ
curl (e − E) = 4πG = 4πgH + µḢ
(22)
(23)
where h is the impressed magnetic force, c the specific capacity, and e the impressed
electric force. Heaviside adds that
the coefficient g of magnetic conductivity is introduced to show the symmetry, and may
be put = 0. We have now a dynamically complete system.240
Heaviside has thus given Maxwell’s system of equations a symmetrical structure that,
in his view, renders it complete. Notice that once the structure is in place, Heaviside sets
g – the key parameter in the symmetry argument – equal to zero. Thus, although the
motivation to relate magnetic and electric phenomena is physical, the method is purely
formal.241
Heaviside obtained symmetry by purely formal means, as did Hertz. Both physicists
sought to take into account the mutual relations between magnetism and electricity by
giving Maxwell’s system a symmetrical structure. Both achieved this goal by formal
means, that is, by an ingenious manipulation of the equations, and both were pleased
to find clarity and completeness – as well as consistency – in this way.242 However, in
contrast to Hertz, Heaviside proposed two experiments to exemplify the symmetrical
way in which the electric and the magnetic forces are related in his equations. These
experiments are remarkably similar to the thought experiment that Einstein presents
239
Heaviside [1892a]/1925a, 448. Note that by harmonise Heaviside probably means “be consistent”.
240
Heaviside [1892a]/1925a, 449. In the discussion of the flux of energy within the framework of his duplex method Heaviside ([1892b]/1925b, 173) develops Maxwell’s equations by
discarding the potentials and making the equations “parallel”. For “parallelism” see Heaviside
[1892b]/1925b, 173. Heaviside and Hertz approached this issue in very similar ways at roughly
the same time, but Heaviside seems not to expect experimental confirmation because he makes
g = 0, a physical situation that can hardly be confirmed experimentally.
241
For Heaviside’s discussion of this point, see n. 232.
242
For a comparison of the contributions by these two physicists, see Darrigol 2000, 257–258.
508
in the opening paragraph of the relativity paper. Here is Heaviside’s description of the
experiments:
As an example, fix a thin circular iron ring in air. Call the line through its centre perpendicular to its plane the axis. Let there be no current or magnetic force in the first place.
Now shoot a small bullet, having an electrical charge, through the ring, along its axis.
The electric displacement due to the charge will be continually changing; thus there is a
system of electric current in the air accompanying the motion of the bullet. The velocity
of propagation of disturbances in air is so great that, unless the velocity of the bullet be
not a very small fraction of the velocity of propagation, we may neglect the disturbance
in the field of force due to the latter velocity not being infinite, and suppose that the bullet
carries with it in its motion its normal field of force (radiating straight lines) unchanged.
The distribution of displacement current about the moving bullet is then the same as that
of the lines of magnetic force that would come from it if it were uniformly magnetised
parallel to the axis, or line of actual motion in the real case, and the lines of magnetic force
accompanying the displacement currents are circles centred upon the axis, in planes perpendicular thereto, the strength of magnetic force in the air being inversely proportional
to the square of the distance from the centre of the bullet, and directly proportional to
the cosine of the latitude; the equator being the circle on the bullet’s surface in the plane
perpendicular to the axis passing through the centre of the bullet. (With very high velocity
this distribution of displacement current and magnetic force is departed from.) The fixed
ring coincides with the lines of magnetic force during the whole motion of the bullet, and
is therefore solenoidally magnetised thereby, most strongly when the magnetic force is
strongest there, i.e., when the bullet has just reached the centre of the ring, and the current
through the ring is a maximum. The current through the ring may be measured either by
the displacement current through a surface bounded by the ring, or by the rate at which
the ring cuts the lines of electric force (supposed undisturbed) of the bullet.
Next, fix the charged bullet and move the ring instead, so that their relative motion
shall be as before. There is exactly the same amount of electric displacement through the
circuit added per second as before, in corresponding positions of the bullet and ring, with,
therefore, the same magnetic force in the ring and the same magnetisation.
These are two different experiments which, however, deploy the same elements, an iron
ring and a charged bullet. In the first experiment the charged bullet is shot along the
axis of the ring, whereas in the second the bullet is stationary and the ring is made to
move: Heaviside specifies that the relative motion in the two experiments is the same.
In the first case, the moving charge creates a circular magnetic field which coincides
with the iron ring thereby causing, by its motion, a current in the ring. In the second case,
the roles of the elements are reversed, but once again an electric current is produced in
the ring when it passes the stationary charged bullet placed in its axis. We see that the
same phenomenon is generated in both experiments.
The purpose of these experiments is to exemplify Heaviside’s electrodynamic formalism. Heaviside does not find anything puzzling in the outcome of the experiments:
they do not constitute tests; rather, they provide a vivid illustration of his position.
Heaviside then proceeds to note the difference:
Otherwise, however, there is a great difference in the two experiments. In the first case,
changing electric displacement or electric current all through the dielectric, the greatest
strength of current being at the poles of the bullet; whilst in the latter case the field is
509
practically undisturbed except near the moving ring itself. Compare with the induction of
electric force in a ring in a magnetic field, first when the field is moving, and next when
the ring is moved in the field.243
Heaviside clearly indicates the issue at stake: the analysis for each of the two experiments
is different but the resulting phenomenon is the same. He does not appear to be distressed
by this state of affairs; on the contrary, the experiments serve to confirm his view that he
had properly embedded the symmetrical relation between magnetism and electricity in
the structure of Maxwell’s system of equations. In fact, as we have indicated, he seems to
have designed the experiments expressly to demonstrate that the forces are symmetrical
while their instantiations may be different – indeed, one might say “asymmetrical” – and
Maxwell’s system in the duplex form captures this essential feature. Darrigol has rightly
drawn attention to the similarity between Heaviside’s two experiments and Einstein’s
thought experiment at the beginning of the relativity paper.244
The presentation of these two experiments illustrates a basic feature of Heaviside’s
way of pursuing physics in general and electromagnetism in particular. For our argument it is a matter of historical interest that this method of reasoning struck a chord
in the physics community and is remarkably similar to Einstein’s mode of thinking
that surfaced in the first paragraph of the relativity paper. In contrast to Hertz who
occasionally considers physical cases in his theoretical analysis,245 Heaviside systematically demonstrates his formalism with what he calls “examples” (or “problems”).246
As he explained: “an illustration, which, though less exact, is more easily followed by
the mind’s eye, when we cannot render visible [some physical properties].”247 One such
illustration, the actual displacement of a piston in a cylinder (representing a displacement
in a condenser), reflects Heaviside’s skill in vividly describing phenomena. Moreover,
in defending Maxwell’s electrodynamics against its rivals, he argues for the reality of
what, at first, seem to be artifacts of the theory:
Many other changes are also required to make a consistent system, for one change necessitates another, and we shall ultimately come to something extremely different from Maxwell’s system.
In view of the extreme relative simplicity of Maxwell’s views, and their completeness
without any artificial contrivances to save appearances ... one is almost constrained to
believe that the dielectric current, the really essential part of Maxwell’s theory, is not
merely an invention but a reality, and that Maxwell’s theory, or something very like it, is
243
Heaviside [1892a]/1925a, 445–446.
Darrigol 1993, 311. Darrigol goes on to speculate that “Heaviside had no reason to expect
motion with respect to ether to be always without effect. Instead, he must have thought that as
soon as the relative velocity of the charge and the ring approached c, the result of the first experiment would differ from that of the second, because the electric lines of force would no longer
accompany the motion of the charge.” Note, however, that Heaviside does not refer to the ether
in his discussion of these experiments.
245
See, for example, the ring magnet which loses its magnetism (Hertz [1884]/1895, 275).
246
For Heaviside’s use of “example” in the sense of a thought experiment, see his analyses of
mechanical forces and closed displacement (Heaviside [1892a]/1925a, 459ff and 468ff); for his
discussion of “problems”, see idem [1892a]/1925a, 474ff.
247
Heaviside [1892a]/1925a, 479.
244
510
the theory of electricity, all others being makeshifts, and that is the basis upon which all
future additions will have to rest, if they are to have any claims to permanancy.
Electric displacement is primarily a phenomenon of the ether.... But when electric
displacement occurs in a solid dielectric, if there be, as there must be, mutual influence
between the ether and matter, we may expect the elastic properties of the matter to be
communicated, apparently, to the ether.248
In 1885 Heaviside remarked ironically that the ”ether is a very wonderful thing,”
adding:
[The ether] may exist only in the imagination of the wise, being invented and endowed
with properties to suit their hypotheses; but we cannot do without it. How is energy to be
transmitted through space without a medium? Yet, on the other hand, gravity appears to
be independent of time. Perhaps this is an illusion. But admitting the ether to propagate
gravity instantaneously, it must have wonderful properties, unlike anything we know.249
Motivated only by physical considerations, Heaviside sought to recast Maxwell’s
system of equations in a symmetrical form. His argument in balancing, so to speak, the
system according to his duplex method requires an equation for the magnetic phenomena to correspond to the electric one. Heaviside obtained a symmetrical structure by
purely formal means and without appealing to the ether. Despite the fact that both Hertz
and Heaviside accepted the ether, it seems that the ether played no role in the derivation of their equations (and later on Hertz in fact dismisses his derivation altogether).
The ether is just an auxiliary hypothesis, part of the interpretative content of the theory,
a fact that probably did not escape Einstein’s perceptive eyes. Heaviside illustrates the
equations with two experiments which are reminiscent of Einstein’s thought experiment.
But Heaviside recognizes no difficulty, for he separates completely the phenomena as
observed from their measurement which, he says, “is exactly the same.” He goes on to
characterize the nature of the experiments: “Otherwise ... there is a great difference in
the two experiments.”250
Despite the differences in their education and professional background and, more
specifically, in their use of terminology and notation, Heaviside and Hertz approached
the asymmetry in Maxwell’s equations in very much the same way. In particular, it is
manifest that – as in Hertz’s case – Heaviside’s application of symmetry is completely
different from Einstein’s use of asymmetry, that is, Einstein’s asymmetry is certainly not
the negation of Heaviside’s symmetry.
3. August Föppl (1854–1924)
In 1894 Föppl published An Introduction to Maxwell’s Theory which – together with
Max Abraham (1875–1922) – he reissued ten years later in a thoroughly revised form.251
248
Heaviside [1892a]/1925a, 478.
Heaviside [1892a]/1925a, 433.
250
Heaviside [1892a]/1925a, 445. Cf. Darrigol 1996, 262, 296.
251
It has been accepted that Einstein almost certainly read Föppl 1894: see, e.g., Holton
[1973]/1988, 217–218, 224; and Stachel et al. 1989, 260 and 306 n. 1. Cf. Norton 2004, 54. Exten249
511
The “Abraham–Föppl” account of electromagnetic theory, based on Maxwell’s system
of equations, became a standard textbook and it was widely used by generations of physics students.252 In the opening lines of the third part of the second edition (1904), “The
Electromagnetic Field,” the authors observe under the heading, “The analogy of electric
and magnetic quantities” [Die Analogie der elektrischen und der magnetischen Größen]:
In their characterizations of Maxwell’s theory, O. Heaviside, H. Hertz, and E. Cohn were
also guided by the analogy between electric and magnetic magnitudes. They put the fundamental equations of the electromagnetic fields in a form that highlights [lit. brings
forward] this relationship which is reminiscent of the geometrical law of duality.253
The same sequence of contributors to this theory also appears in Physik des Aethers
by Paul Drude (1863–1906) in the very year that Föppl published the first edition of
his Einführung.254 This view was apparently undisputed both with respect to priority
and to the intention of the authors. What was not clear, however, was the status of this
special form of presentation of the phenomena. Indeed, Föppl and Abraham make this
remark only as a historical note and do not pursue it. Rather, in subsequent passages they
use the term analogy, which indicates similarity, but commits them neither to duality
nor to symmetry. To be sure, they take note of the differences between magnetism and
electricity and, when it comes to correlating formally the two elements, they speak of
Heaviside–Hertzsche Analogieen.255 So what do Föppl and Abraham mean when they
claim that the form of the fundamental equations of the electromagnetic fields is reminiscent of the duality theorem in geometry? To answer this question one has to consider
Föppl’s original intention in 1894 which he apparently abandoned in 1904.
In the preface to his book Föppl openly expressed his appreciation of Heaviside and
Hertz:
I believe Heaviside to be the most eminent of Maxwell’s followers from a speculativecritical perspective, as Hertz – who unfortunately died too early – was from an experimental-confirmation perspective.256
sive discussions of Föppl 1894 can be found in Darrigol 1993, 329–333, and Miller [1981]/1998,
142–153.
252
The English version added Richard Becker as third author. Cf. Holton [1973]/1988, 219 and
Miller [1981]/1998, 148.
253
Föppl and Abraham 1904, 211 (§ 54): “O. Heaviside, H. Hertz und E. Cohn haben sich
bei ihrer Darstellung der Maxwellschen Theorie gleichfalls von der Analogie der elektrischen
und der magnetischen Größen leiten lassen, sie haben den Grundgleichungen des elektromagnetischen Feldes eine Form gegeben, welche diese an das Dualitätsgesetz der Geometrie erinnernde
Beziehung deutlich hervortreten läßt.”
254
Drude 1894, 316. Föppl (1894, 214) makes this point when he discusses the definition of
vector potential.
255
Föppl and Abraham 1904, 225, 239. Symmetrie occurs very rarely in this edition; for one
instance, see eidem 1904, 239.
256
Föppl 1894, vii: “Ich halte Heaviside für den hervorragendsten Nachfolger Maxwell’s nach
der speculativ-kritischen Seite hin, wie es der uns leider so früh entrissene Hertz zweifellos nach
der experimentell-bestätigenden Seite hin war.”
512
Föppl’s goal was to provide a coherent synthesis of the views of these two physicists
and then to build on their work. In particular, he was deeply impressed by the vector
formalism that Heaviside developed and devotes the first part of the book to VectorAlgebra, recommending it as the most suitable technique for the study of Maxwell’s
theory.257 In the conclusion to this part of his book, Föppl cautions the reader that calculus is not physics: “This is quite different when we consider the physical meanings of
these magnitudes.” And he adds that “the question of what are their physical meanings
remains open for the time being.”258 Föppl may have had his own project in mind, for
he proposes to extend results from abstract geometry to the physics of electrodynamics.
One of the striking innovations of Föppl’s discussion is a consequence he draws
from the symmetrical formalism of Heaviside and Hertz. Föppl introduced the concept
of duality into electromagnetism on analogy with its usage in projective geometry. He
claims that
It is one of O. Heaviside’s most important achievements with respect to the development
of Maxwell’s theory that he emphasized this duality very prominently and, for the first
time, put it to systematic use in the construction of the whole theory.259
Although Heaviside – to the best of our knowledge – had never invoked the term duality,
it seems that Föppl read this notion into the duplex form of the equations. Föppl presents
the essential elements of Maxwell’s theory and then moves on to the concept of duality
and its application to this theory. This is an important passage that most likely caught
the attention of Lorentz,260 and we quote it at length:
§ 50. The Duality Between Electric and Magnetic Phenomena
The law of duality or reciprocity is well known in [projective] geometry. According
to it, a reciprocal statement can be given for every theorem of spatial geometry – that does
not refer to metrical relations – by simply interchanging [Vertauschung] the words, point
points
and plane, etc. This is often written in the following form: three planes
that do not have a
plane
points
.
straight line in common, determine only one point , which belongs to the three planes
Two propositions are obtained, depending on whether one chooses the words above
or below the line. This correlation is no coincidence. Rather, it is the result of a strict law
257
Föppl 1894, vi, 3, and 4.
Föppl 1894, 87: “Anders ist est aber, wenn man die physikalische Bedeutung dieser Grössen
in Auge fasst.... Die Frage, welche physikalische Bedeutung ihnen beizulegen sein, bleibt dabei
zunächst vollständig offen.”
259
Föppl 1894, 122: “Es ist eins der wichtigsten Verdienste, die sich O. Heaviside um die Fortbildung der Maxwell’schen Theorie erworben hat, dass er diese Dualität scharf hervorhob und sie
zum ersten Male in systematischer Weise beim Aufbau der ganzen Theorie verwerthet hat.”
260
Lorentz 1904a, 99. Unfortunately, Lorentz refers to § 80 in Föppl’s book (rather than to §
50) but, since the context has to do with duality, we are confident that it is just a typographical
error.
258
513
which has its origin in the general features of space. This law can be proven by means of
the theory of [poles] and polars.261
Duality was a well known concept in projective geometry by the turn of the last century.
It is, for example, discussed in Weber’s Encyklopädie of 1905:
At this point we can already double the success of the labor required for the construction
of this geometry when we notice that true propositions result from interchanging [Vertauschung] the words, “point” and “plane” with “plane” and “point”, in axioms I, II, and
III [that form the foundations of projective geometry] and their implications.262
Thus the law of duality for the plane is proved, [namely,] that all propositions of pure projective geometry of the plane pass into true propositions by interchanging [Vertauschung]
the words, “point” and “line”, with corresponding changes of the terms that refer to their
occurrence.263
Morover, in the Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer
Anwendungen references to earlier discussions of this principle may be found. In
particular:
In three works that appeared between 1824 and 1827, Gergonne attempted to provide
a basis for this fundamental relationship [the principle of duality] between “point” and
“line” in a plane, as well as between “point” and “plane” in space, by showing that the
fundamental propositions of geometry permit the interchanges of these words and that
261
Föppl 1894, 121: “§ 50. Die Dualität zwischen den elektrischen und den magnetischen
Phänomenen
Aus der Geometrie der Lage ist das Gesetz der Dualität oder Reciprocität wohlbekannt. Nach
ihm lässt sich durch blosse Vertauschung der Worte Punkt und Ebene u.s.w. zu jedem Lehrsatze
der Geometrie des Raumes, der sich nicht auf metrische Verhältnisse bezieht, ein reciproker Satz
Punkte
angeben. Man schreibt dies oft in folgender Form an: Durch drei Ebenen
, die keine gerade Linie
gemeinsam haben, ist
eine
,
ein
und nur
eine Ebene
ein Punkt
bestimmt,
die
der
den drei
Punkten
Ebenen
gemeinsam zugehört.
Man erhält zwei Sätze, je nachdem man die oberhalb oder unterhalb des Striches stehenden
Worte wählt. Dabei ist dies Zusammentreffen nichts Zufälliges, es entspringt vielmehr einer strengen Gesetzmässigkeit, die in den allgemeinen Eigenschaften des Raumes ihrem Ursprung hat. Mit
Hülfe der Lehre von den Polaren wird diese Gesetzmässigkeit bewiesen.” For the historical context
of Polarentheorie, see Papperitz 1910, 561–562; for technical details, see Struik 1953, 117–119
(§ 5–5).
262
Weber et al. 1905, 165: “Schon an dieser Stelle können wir den Erfolg der Arbeit, der zum
Aufbau dieser Geometrie nötig ist, verdoppeln, wenn wir bemerken, daß aus den Axiomen I, II,
III und ihren Folgerungen durch Vertauschung der Worte ‘Punkt’, ‘Ebene’ mit ‘Ebene’, ‘Punkt’
richtige Sätze hervorgehen....”
263
Weber et al. 1905, 166–167: “Damit ist das Dualitätsgesetz der Ebene bewiesen, daß
alle Sätze der rein projektiven Geometrie der Ebene durch Vertauschung der Worte Punkt und
Gerade und entsprechende Änderung der auf die Inzidenz bezüglichen Ausdrücke in richtige
Sätze übergehen.”
514
this law of symmetry, as the “principe de dualité”, can be found throughout the entire
geometry of the plane and of space, respectively.264
Joseph Diaz Gergonne (1771–1859) enunciated this principle of duality, namely, that
in projective geometry every theorem in the plane relating points and lines corresponds
to another theorem in which points and lines are interchanged, provided no metrical
relations are involved. Thus, the propositions about points and lines exhibit a kind of
symmetry by which is meant the interchanging of the roles played by the words “point”
and “line”. Indeed, the term Wortumwechslungen (interchanging words) is very close in
meaning to the term Vertauschung (interchanging) which we have discussed at length
(see Section II.2, Case 4, above). Symmetry in the sense of interchanging words or
concepts (or, more generally, elements) while retaining the original logical or structural
relations intact is directly connected to the idea of duality in projective geometry. And
Föppl draws important consequences for electrodynamics from this claim.
Having presented the principle of duality, Föppl now argues:
A completely similar duality is evident in the theory of electricity between electric and
magnetic magnitudes. In general it is possible to deduce from any proposition in the theory of electricity a new proposition which remains valid when the terms, electricity and
magnetism (and related terms), are interchanged.265
Recall Hertz’s remark: “in the laws of electric induction we need only interchange the
words ‘electric’ and ‘magnetic’ throughout in order to obtain the inductive actions in
magnetic circuit.”266 And Heaviside’s version of the same idea is:
I here introduce a new method of treating the subject ... which may perhaps be appropriately termed the Duplex method, since its main characteristic is the exhibition of the
264
Fano 1907, 232, see also p. 234: “Diese grundlegende Beziehung [das Dualitätsprinzip]
zwischen ‘Punkt’ und ‘Gerade’ in der Ebene, sowie auch zwischen ‘Punkt’ und ‘Ebene’ im Raume,
versuchte Gergonne in drei 1824–27 erschienenen Arbeiten dahin zu begründen, daß die Fundamentalsätze der Geometrie eben jene Wortumwechslungen gestatten, und daß sich dieses Symmetriegesetz als ‘Principe de dualité’ durch die gesamte Geometrie der Ebene bezw. des Raumes
hindurchzieht.”
265
Föppl 1894, 121: “Eine ganz ähnliche Dualität zeigt sich in der Elektricitätslehre zwischen den elektrischen und den magnetischen Grössen. Im Allgemeinen kann man aus irgend
einem Satze der Elektricitätslehre einen neuen ableiten, der ebenfalls gültig bleibt, wenn man die
Begriffe Elektricität und Magnetismus (und ebenso die zugehörigen) mit einander vertauscht.”
While Föppl introduced the technical usage of the term duality from projective geometry into
electromagnetism, the general concept of duality had been applied to electromagnetism since at
least the time of Faraday: see Faraday [1855]/1965, 566–567. Boltzmann ([1891–1893]/1982, 2:
24) also used duality in a context which has nothing to do with the usage of this term in projective
geometry: “Nach der dualistischen Anschauung giebt es zweierlei Elektricitäten, die positive und
die negative. Ein Quantum der ersten wird immer mit positivem, eines der letzteren mit negativem
Zeichen bezeichnet.”
266
See n. 180, above.
515
electric, magnetic, and electromagnetic equations in a duplex form, symmetrical with
respect to the electric and magnetic sides....267
Heaviside originally used the term duplex in telegraphy. He begins his essay “On Duplex
Telegraphy” (1873) with the remark that this is “the art of telegraphing simultaneously
in opposite directions on the same wire.”268 It is not difficult to see how Heaviside later
adapted duplex to interchanging the magnetic and electric elements in Maxwell’s electromagnetism in order to render the theory symmetrical. However, it was Föppl’s insight to
realize that, in fact, the notion of duality in projective geometry captures the symmetrical
features of the equations in both Hertz’s and Heaviside’s case. With his appeal to duality
Föppl has succeeded in formalizing the positions of both Hertz and Heaviside. In fact,
Föppl calls the principle of duality in electromagnetism the Heaviside’sche Princip.269
It appears, moreover, that Föppl entertained the idea that this coherent synthesis
of symmetry in electromagnetism may ultimately have its origin in geometry. For the
principle of duality to apply strictly in this physical context, the relation between electricity and magnetism must be of a different nature than mere analogy. Föppl holds this
view and considers his Heaviside’sche Princip workable. He puts duality to good use
when he indicates that the force and induction of the magnetic field can immediately
be obtained from those of the electric field: “the knowledge gained on one side is also
immediately attained [erzielt wird] on the other [where the “sides” refer to electricity
and magnetism].”270 The principle becomes a leitmotif in Föppl’s discussion:
This emphasis has to be given even more weight, insofar as all inferences that – on the
basis of energy relations, and especially on the basis of Lagrange’s equations – can be
deduced for the group of phenomena on the electric side, can also be immediately applied
on the magnetic side.271
However, Föppl reports a problem in the application of duality. Interestingly enough,
he calls the discrepancy he discerned in the equations “asymmetry”. Although the context is that of Heaviside (who is explicitly mentioned here), it is likely that Föppl took
267
Heaviside [1892a]/1925a, viii. Föppl (1894, 122) follows Heavisde to the letter when he
sets the magnetic conductivity equal to zero.
268
Heaviside [1892a]/1925a, 18.
269
Föppl 1894, 181. For another reference to Heaviside’schen Princip der Dualität, see idem
1894, 349.
270
Föppl 1894, 122: “Weit wichtiger ist aber natürlich der Gewinn, der durch die auf einen Seite
erlangte Erkenntniss sofort auch für die andere Seite erzielt wird.” Föppl returns to the argument of
duality in his discussion of the magnetic force (1894, 181): “Nach der zwischen den elektrischen
und magnetischen Erscheinungen bestehenden Dualität müssen wir erwarten, dass auch an den
von magnetischen Strömen durchflossenen Körpern ponderomotorische Kräfte auftreten.”
271
Föppl 1894, 353: “Auf diese Betonung ist um so mehr Gewicht zu legen, als alle Schlüsse,
die wir auf Grund der Energiebeziehungen, im Besonderen auch auf Grund der Lagrange’schen
Gleichung für die Erscheinungsgruppen der elektrischen Seite ableiten können, sofort auch auf
die magnetische Seite anwendbar sind.”
516
the term asymmetry from Hertz since Heaviside does not use it.272 Let us consider how
this asymmetry arises.
In the opening section of his paper on electromagnetic induction and its propagation,
Heaviside summarizes Maxwell’s theory and explains how vectors and scalars “symbolise” – to use Heaviside’s term – the three distinct properties of Maxwell’s electromagnetic scheme: conductivity, capacity, and permeability. The scheme stipulates that,
for example, one vector magnitude, say C (the conduction current-density), is equal to
another vector that has a scalar coefficient, say kE (the specific conductivity and the
electric force, respectively); this vector equation implies that the two vectors, C and E,
are parallel, that is, they lie in the same direction but need not coincide, and that C is k
times as long as E.273 In developing his own version of Maxwell’s theory, Föppl does
not mention Heaviside explicitly, but he follows Heaviside closely, almost to the letter.
In particular, Föppl paraphrases Heaviside’s claim about the vector equation when he
argues that D, the dielectric displacement, and E, the electrostatic force, are arranged in
space in the same way. Thus, according to Föppl, for isotropic bodies274
D = c·E
(114)
D = (K/4π) · E,
(115)
where the coefficient c is replaced by K/4π and both are scalar magnitudes that depend
on the inductive capacity of the material. The most important deviation from Heaviside is
that Föppl adds “for reasons of symmetry” [Symmetriegründen] to justify the claim that
D and E lie in the same direction [gleich gerichtet]. Clearly, Föppl was more sensitive to
symmetry issues than Heaviside, though here too he was – as we have seen – indebted
to Heaviside’s duplex method.
Now, consider the magnetic counterpart: let B be the magnetic induction, G the
magnetic force, and µ thus defined as the permeability for isotropic medium. Then
B = µ · G.
(128)
Föppl then observes that “an asymmetry [Asymmetrie] arises” when equ. 128 (on p. 123)
is compared to equ. 115 (on p. 91), for there is an additional factor, 1/4π , in the equation
for the dielectric displacement.275 However, he assures the reader that this problem is
merely a “superficiality” [Aeusserlichkeit], and now (on p. 123) he appeals explicitly
to Heaviside, who – Föppl remarks – had already dealt with this problem satisfactorily in his presentation of Maxwell’s equations.276 Indeed, Föppl probably had in mind
Heaviside’s remark:
272
Föppl (1894, 268) refers explicitly to Hertz’s elimination of the vector potentials from the
fundamental equations of Maxwell. Recall that with this elimination – so Hertz argued – he
restored symmetry to the formalism (see n. 184, above).
273
Heaviside [1892a]/1925a, 429–430. Cf. Section III.2, above, esp. n. 233.
274
Föppl 1894, 91.
275
Föppl 1894, 123.
276
Föppl 1894, 122, cf. idem 1894, 88. As was his usual practice, Föppl does not specify any
passage in Heaviside. For Heaviside’s presentation, see eqs. 12 and 13 in Section III.2, above (esp.
517
The excrescence 4π is a mere question of units, and need not be discussed here. The 4π ’s
are particularly obnoxious and misleading in the theory of magnetism. Privately I use
units which get rid of them completely, and then, for publication, liberally season with
4π ’s to suit the taste of B. A. unit-fed readers.... But in the mere algebra it is simply a
matter of putting in 4π’s here and there in translating from rational to ordinary units.277
Irony aside, we note that, although Heaviside uses the term symmetry elsewhere, he does
not apply it here; nor does he invoke the term asymmetry. By contrast, Föppl is willing
to combine Heaviside’s analyis with terms that come from other sources, notably Hertz,
probably due to his conception of duality in electromagnetism; in any event, for Föppl
the “excrescence”, 4π, is a mark of asymmetry. Indeed, when he encounters this discrepancy in a discussion of the energy of the electromagnetic field later in the book, Föppl
once again notes that the symmetry conveyed by the formalism is disturbed; but now he
is explicit about the source of the asymmetry, namely, dimensionality: “The symmetry
of the formulas is again disturbed by the factor, 4π which, in the prevailing system of
units, enters them, but this is simply due to an unfortunate choice of units.”278
Following the chapter on the foundations of Maxwell’s theory, Föppl discusses the
reciprocal relation between electricity and magnetism. He refers to the “complete parallel” [vollständig parallel] between the two phenomena of electricity and magnetism
which – he argues – conforms to what he designates “duality relations” [Dualitätsbeziehungen], but he also calls attention to an important difference:
The electrostatic and purely magnetic processes run, as we indicated, completely parallel to each other, the only important difference being that in nature there are electric
conductors and, hence, true electric charges, whereas [in nature] there are no magnetic
conductors (in the proper sense of the word) and, hence, no true magnetic charges can be
found.279
nn. 231–234). Föppl (1894, 166) returns to this issue in his discussion of the law of induction. Once
again he singles out the factor 4π as the cause of the asymmetry and remarks that it disappears
due to Heaviside’s choice of units. (“Für die früher erwähnten rationellen Einheiten Heavisides
verschwindet dieser Unterschied.”) Föppl (1894, 171) comes back to this point in the presentation
of the two fundamental equations of electromagnetism.
277
Heaviside [1892a]/1925a, 432. Idem [1892b]/1925b, 26: “It seems rather unpractical for
the B. A. Committee [the British Association Committee on Electrical Units] to have selected
108 c.g.s. as the practical unit of E.M.F., instead of 109 .... Mac, tom, bob, and dick are all good
names for units. Tom and mac (plural, max), have sentimental reasons for adoption....” Cf. idem
[1892b]/1925b, 543–544, 575–578.
278
Föppl 1894, 353: “Die Symmetrie der Formeln wird nur wieder wegen der unglücklichen
Wahl des herrschenden Maasssystemes durch den Factor 4π gestört.” Föppl refers in this context
to Hertz’s discussion of “verschwindenden Ringmagnete” – a clear indication that he was aware
of Hertz’s early study of Maxwell’s fundamental equations; see Hertz [1884]/1895, 299: “einen
erlöschenden Ringmagnet”.
279
Föppl 1894, 143: “Die elektrostatischen und die rein magnetischen Vorgänge laufen, wie
sich zeigte, vollständig parallel mit einander, nur mit dem einen wichtigen Unterschiede, dass
in der Natur zwar elektrische Leiter und daher wahre elektrische Ladungen, aber keine magnetischen Leiter (im eigentlichen Sinne dieses Wortes) und darum auch keine wahren magnetischen
Ladungen vorkommen.” The term parallel appears in a similar context in Drude 1894, vii: “...
die Parallelität bei der Darstellung der Eigenschaften des magnetischen Feldes und des elektri-
518
These well known empirical facts probably made Föppl suspicious of the function of
the ether in explaining both electricity and magnetism. Föppl expresses his view of the
ether hypothesis as follows:
Since the notion of an ether displacement ... can also be assumed with equal justification
for the flow of the magnetic induction as much as for the dielectrical displacement, the
suspicion arises that it corresponds to the true mechanisms in neither of these two cases.
However, [the ether] is only introduced as a means of demonstration, with no claim with
regard to its objective validity.280
Although the ether hypothesis works as well for magnetism as for electricity, Föppl
suspects that it does not correspond to the true mechanism of either phenomenon
and has no objective reality – despite its usefulness as a means of demonstration. Be
that as it may, the duality that holds between the two phenomena allows for a strong
reciprocal relation. Hence the term “electromagnetic” [elektromagnetischen] which,
according to Föppl, conveys the “double-aspect” [Doppelgesicht] of these phenomena.281
This reciprocal relation has the advantage of putting the two distinct phenomena on
an equal footing: it does not subordinate either one to the other, and Föppl illustrates
this “double-aspect” with a physical image in which a flowing electric current in a wire
generates a magnetic field in its vicinity. According to an action-at-a-distance theory, the
magnetic phenomena are subordinate to the current, that is, the current is conceived as
the cause and the magnetic phenomena are the effects; however, according to Maxwell’s
theory, any flowing electric current is inherently associated with a magnetic field – the
relation being reciprocal with no distinction between “cause” and “effect”.282 For Föppl
this notion of reciprocity between electricity and magnetism was motivated by his belief
in Maxwell’s theory as well as by his own conception of the duality principle, and it
is a central theme of his book. These relations between two kinds of phenomena are
captured successfully by Maxwell’s theory but, in the absence of cause and effect, they
seemed to require a medium to support them.
In Part 5 of his book Föppl draws on Hertz’s study of the electrodynamics of moving
bodies and appeals to Heaviside’s method of illustrating the formalism with concrete
physical images that have come to be known as “thought experiments”. Under the heading, “The Electrodynamics of a Moving Conductor” [Die Elektrodynamik bewegter
Leiter], which is reminiscent of Hertz’s work of 1890, Föppl begins the analysis with
a discussion on an induced electromotive force that arises through motion. At stake is
schen Feldes und ihre konsequente Zurückführung auf Nahewirkungen.” For Lorentz’s usage, see
Section III.6, below.
280
Föppl 1894, 124: “Da die Vorstellung einer Aetherverschiebung ... mit demselben Rechte für
den magnetischen Inductions- wie für den dielektrischen Verschiebungsfluss in Anspruch genommen werden kann, liegt der Verdacht nahe genug, dass sie in keinem von beiden Fällen dem wahren
Mechanismus des Vorganges entspricht. Sie ist aber auch nur als ein Veranschaulichungsmittel
eingeführt worden ohne jeden Anspruch darauf, dass ihr eine objective Bedeutung zukäme.”
281
Föppl 1894, 143.
282
Föppl 1894, 143–144.
519
the issue of relative and absolute motion in space. It is noteworthy that Föppl states
outright the kinematic axiom, namely, there is no way to determine absolute motion,
for there are no means of comparison for establishing it. However, if we imagine empty
space as filled with a medium, then this notion becomes meaningful: absolute motion is
determined relative to the medium that fills “empty space”. For Föppl “empty space” is
a contradictory notion like a forest without trees,283 but he is not entirely convinced of
the existence of the medium that fills it. As Hertz had already done, he characterizes the
problem of the ether as crucial to the development of modern physics: “The decision on
this question forms perhaps the most important problem of science of our time.”284 For
Hertz the issue of the nature of the ether was paramount but, in his case, the context was
not the same as Föppl’s.285 The problem that Föppl singles out concerns the ether in the
context of induction phenomena and their dependence on relative vs. absolute motions.
This is the very issue which Einstein addresses in the introduction to his relativity paper.
Given the kinematic propositions concerning relative motion, Föppl cautions that
one should not conclude that the two cases – moving the magnet in the vicinity of a
closed electric conductor at rest, or the other way around – are the same. Föppl imagines a third case in which there is no relative motion between the two bodies; thus both
the magnet and the closed electric conductor have the same motion with respect to the
medium. But, Föppl argues, ”experience shows” [Nach den vorliegenden Erfahrungen]
that no effect is detected in such circumstances. This conclusion is important: Föppl
argued that there is no difference between the case when the magnet and the electric
circuit are at rest and the case when they are both in motion with no relative velocity. We
note that these are not actual physical experiments; rather, the arrangements are vivid
images in the style of Heaviside. Föppl then turns to examine other effects that may
depend on absolute motion and proposes various physical situations in addition to the
one we have seen in Heaviside’s illustration: the charged bullet and the iron ring. As
Darrigol remarks, Föppl intended to prepare the reader for the view that “electrodynamic
phenomena did not necessarily depend on the relative motion of material bodies only....
There could be, for other kinds of electrodynamic phenomena, a dependency on absolute
motion.”286
One such case concerns two charged particles that are placed in two parallel trajectories in close vicinity to each other with the same uniform velocity. Heaviside devoted
a whole paper to the electromagnetic effects of a moving charge.287 He then followed it
up with a study of electromagnetic effects which are due to the motion of electrification
through the dielectric, increasing the complexity of the setup by the introduction of an
283
Föppl 1894, 308.
Holton [1973]/1988, 221; Föppl 1894, 309: “Die Entscheidung der soeben berührten Frage
bildet vielleicht die wichtigste Aufgabe der Naturforschung unserer Zeit.”
285
See n. 169, above.
286
Darrigol 1993, 330. For Darrigol’s discussion of Föppl, see idem 1993, 329–332.
287
Heaviside [1892b]/1925b, 490–499, esp. 492: “The main subject of this communication is
the electromagnetic effect of a moving charge. That a moving charge is equivalent to an electric
current-element is undoubted, and to call it a convection-current ... seems reasonable.” Föppl has
a section entitled Convectionsströme (Föppl 1894, 159–161, § 66).
284
520
external field due to another moving charge.288 Föppl adapted this study for his own
purposes, simplifying the setup by (1) considering two charged material points [zwei
elektrisch geladene Körperchen (materielle Punkte)289 ] and (2) arranging the trajectories in parallel lines. Unlike Heaviside, Föppl examines the role of the ether hypothesis
with respect to the wider physical issue of the determination of absolute motion. This
becomes very hypothetical, and Föppl chooses his words carefully. He does not describe
the setup as an “experiment”; rather, he says, “one considers....” [Man betrachte....].290
The two particles are clearly at rest relative to each other, but complex electromagnetic
phenomena arise from the interaction of the two moving particles. Föppl goes on to
eliminate any unnecessary elements from this ideal situation so that, apart from the
medium [Medium] i.e., space filled with ether [äthererfüllten Raume], there are no reference bodies relative to which the motion of the two particles could be detected. Yet,
according to Föppl, the effects of the absolute motion are manifest despite the fact that
it cannot be detected directly. Föppl concludes that “in cases of this type ... the action of
the bodies on each other does not depend solely on their relative motion.”291
He then observes:
These considerations [Betrachtungen] clearly indicate the difficulties with which the treatment of the electrodynamics of a system of moving bodies has to struggle at the present
time.292
What are these difficulties? We have seen that the first consideration – the magnet and
the circuit – suggests that relative motion is sufficient to account for the phenomena,
whereas the second consideration, the case of the two charged particles, suggests that
absolute motion is necessarily involved. So, is there absolute motion or not? Viewed from
a historical perspective, Föppl singles out these considerations as constituting the most
important problem of science of his time. The ether plays a central role even though it has
none in the experiment of the magnet and the conductor (which was already the case in
Heaviside’s experiments). On the other hand, the ether (in the sense of absolute motion)
does play a role in these physical considerations. That is, Föppl discerns a difficulty and
he characterizes it as fundamental. He suspects that the foundations which sustain this
view of nature [Naturanschauung] might lose their validity.293 For our argument it is
important to note that, using Heaviside’s formalism and methodology, Föppl has gone
288
Heaviside [1892b]/1925b, 506–508, esp. 506–507: “Now if the external field be that of
another moving charge.... the mutual energy of a pair of point-charges ... moving at velocities u1
and u2 ,... at a distance r apart.”
289
Föppl 1894, 310. Föppl appears to generalize Heaviside’s “charged bullet”: see n. 243,
above.
290
Föppl 1894, 310.
291
Norton 2004, 55; Föppl 1894, 311: “In Fällen dieser Art hängt die Wirkung der Körper
aufeinander demnach nicht von ihrer Relativbewegung allein ab.” For further analysis, see Norton
2004, 101–102.
292
Föppl 1894, 311: “Diese Betrachtungen zeigen deutlich, mit welchem Schwierigkeiten die
Behandlung der Elektrodynamik eines Systemes bewegter Körper heute zu kämpfen hat.” Cf.
Darrigol 1993, 330.
293
Föppl 1894, 311.
521
well beyond Heaviside; indeed, Föppl articulated the fundamental problem of physics
at that time.
However, as we know from the first paragraph of the relativity paper, Einstein did not
appeal to the physical considerations that Föppl had described; rather, he took as his point
of departure Heaviside’s kind of experiment, namely, the magnet and the conductor. For
Einstein Heaviside’s experiment (reduced to its basic elements) served as a means to
eliminate the issue of symmetry (asymmetry). Einstein’s approach is entirely alien to the
point Föppl had in mind, in spite of (or, indeed, because of) Föppl’s sensitivity to issues
of symmetry. We have seen that Einstein used Hertz’s notation rather than Heaviside’s
vector calculus that Föppl introduced into German physics.294 Nevertheless, it is fair to
conclude that Föppl showed Einstein a way to cast formalism into imagined physical
setups, a lesson that Föppl had learned from Heaviside.
We have indicated that Föppl adopted Heaviside’s vector calculus, turned the duplex
method into a duality principle, and drew consequences from them. Furthermore, in
exploring the issue of induction and its dependence on relative and absolute motions, he
applied Heaviside’s methodology of casting the equations into imagined experiments.
But, in addition, he exercised independent judgment in recognizing the relevance of the
ether hypothesis to this central issue in physics at the time. On all accounts Föppl was
indebted to Heaviside and he did not conceal it. As we have seen, he openly expressed
his admiration for Heaviside but, even when he mentions Heaviside, there are no specific
citations (no book title, let alone pages).
This trajectory (Heaviside to Föppl to Einstein) builds on the consensus among scholars – following Holton’s perceptive remarks – that Föppl was instrumental in making
Einstein aware of the difficulties which arise in induction phenomena through the experiment of the magnet and the conductor.295 We accept this historical insight but claim that
the inspiration for Föppl’s experiments came from Heaviside (which Föppl presented in
abbreviated form). Föppl’s additional contribution (as far as Einstein is concerned) is the
294
Einstein (1907b, 427–428, 460) still uses differential equations in Hertz’s form which he now
calls the Maxwell–Lorentz equations. Einstein and Laub (1908a, 533, 535 and 537) apply Heaviside’s notation but call the equations Maxwell–Hertz. Einstein and Laub (1908b, 542) acknowledge
in a footnote the dual [dualen] treatment of electric and magnetic phenomena, and use Heaviside’s
notation (pp. 543, 544, 547). See Abraham 1903, 116, where Lorentz’s form of the field equations is distinguished from what Abraham calls the Hertz–Heaviside form. Abraham also presents
the form that makes use of the vector potential as exhibited in Des Coudres (1900) and Wiechert
(1900). For discussion of the relative merits of Hertz’s and Lorentz’s versions of Maxwell’s theory
as understood by Abraham in 1904, see Miller [1981]/1998, 148–149.
295
Holton ([1973]/1988, 221) quotes Föppl (1894, 309): “We must not consider [relative motion] as a priori settled that it is, for example, all the same whether a magnet [moves] in the vicinity
of a resting electric circuit or whether it is the latter that moves while the magnet is at rest.” Holton
then takes this passage in Föppl to be Einstein’s experiment in a nutshell. According to Miller
([1981]/1998, 167, n. 14), Einstein’s English was poor before taking lessons, beginning in 1913,
which may well be the reason why Einstein did not read Heaviside directly. But one should not
exaggerate this linguistic issue, for Einstein wrote reviews in 1905 of several articles that had
appeared in English (see Stachel et al. 1989, 118–121, 245–249 [text], 596–597 [references to
the titles in English]). In any event, it seems that Föppl served as the main source for Einstein’s
knowledge of Heaviside.
522
claim that this was the most important issue in physics – which is a different emphasis
from what we find in either Heaviside who did not see the remaining problems in physics (or electrodynamics) in this way, or in Hertz who had recognized the importance of
the problem of the ether but did not conceive of it in the context which Föppl signaled
as crucial for the future of physics. Einstein does not acknowlege his sources, but in
view of our historical analysis we suggest the following chronology: Heaviside (1885)
introduced the experiments of the bullet and the iron ring to provide vivid images for
his duplex version of Maxwell’s theory and regarded them as unproblematic; impressed
by this mode of argumentation, Föppl (1894) recast the experiments into a form which
exhibits the difficulty raised by the relative motion of two bodies (as well as their motions
with respect to the ether), and used it to formulate a central problem in electrodynamics
that looms large for physics in general; and Einstein (1905) finally solved the problem
by rendering the ether superfluous, replacing the distinct electric and magnetic fields of
his predecessors by a single electromagnetic field, and introducing relativity theory.
4. Emil Wiechert (1861–1928)
In 1900 Emil Wiechert (1861–1928),296 a student of Voigt, published an essay on the
fundamental laws of electrodynamics in which he praises the work of Lorentz for introducing a new version of Maxwell’s theory. He refers to the difference between matter
and ether and sets up the mathematical apparatus for discussing optical phenomena in the
ether. The equations he offers originated, as he reports, in a paper by Hertz in 1884.297
Wiechert then modifies these equations by adding a vector-potential to the system and
claims that “Maxwell’s system of equations ... is not symmetric.” He continues:
Initially, this seems to be a disadvantage (which, however, can easily be eliminated [beseitigt]), but as a matter of fact it is not because, when we relate the theory of optics to
the theory of electrodynamics, we are then able to accommodate exactly the asymmetry
[Unsymmetrie] of the electric and magnetic phenomena that we know from experience.298
Notice that Wiechert first alludes to asymmetry in the mathematical formulation and
then suggests that it is in the phenomena. An earlier paper of Wiechert sheds light on
296
Due in large measure to Klein’s efforts to expand the study of mathematics and physics at
Göttingen, in 1898 Wiechert was appointed to the new chair in Geophysics.
297
Wiechert 1900, 552. See also p. 558 where Hertz and Heaviside are cited together. Wiechert’s
paper was reprinted in Annalen der Physik 1901, and thus it reached a wide audience. Abraham
(1903, 116 n. 2) cites both versions. It is likely that Einstein read Wiechert’s article in the version
published in 1901 since, at around that time, Einstein refers to various articles in Annalen der
Physik (see Stachel et al. 1989, 260: “Comments in his letters ... indicate that during those years
[1898–1902] he looked at that journal regularly, and studied a number of articles in it.”). On the
other hand, there is no early indication that Einstein looked at the Lorentz Festschrift (Recueil de
travaux 1900). But see Klein et al. 1993, 61, where Einstein – in a letter to Wien in 1907 – refers
to Wiechert’s article in the Lorentz Festschrift.
298
Wiechert 1900, 553 (italics in the original): “... stellen das angekündigte Maxwell’sche
Gleichungsystem dar. Wie wir erkennen, ist es nicht symmetrisch.
523
this distinction. In 1898 Wiechert put forward hypotheses for a theory of electric and
magnetic phenomena where he made the following remark on the electromagnetic field:
The state of changes of the field – to which, following Maxwell, we attribute the mediation
of electric and magnetic action at a distance – are especially simple when the domain in
question consists of imperceptible matter. We see here the expression of two electrodynamic field-states [Felderregungen]: an “electric excitation”, and a “magnetic excitation”.
The electric excitation appears to have the symmetry of a displacement, i.e., it has a distinctive axis at every point in the field: when reflected in a plane perpendicular to the
axis, it is reversed; and when reflected in a plane through the axis, it remains unchanged.
As was already emphasized by Maxwell, the magnetic excitation behaves significantly
differently insofar as it does not have the symmetry of a displacement but of a rotation.299
Thus, according to Wiechert, the asymmetry is inherent in the phenomena: the symmetry of the electric field is translational, whereas that of the magnetic field is rotational.
This is reflected in the fact that the electric field strength is an ordinary vector whereas
the magnetic field strength is an axial vector. Wiechert does not express any discomfort
with this finding and he assures the reader that, in fact, the asymmetry is advantageous
when it comes to relating optical to electromagnetic phenomena. Now, the contrast with
Einstein’s opening remark in the relativity paper is striking. Both Wiechert and Einstein speak of asymmetry, but while the former sees the asymmetry in the phenomena
expressed in the mathematical formulation of Maxwell’s equations, the latter acknowledges only the asymmetry in the formulation and rejects the asymmetry in the phenomena. No wonder that Wiechert and Einstein drew opposite conclusions: for Wiechert the
asymmetry points to an “exact” connection between optics and electromagnetic phenomena, but for Einstein this is the point of departure for formulating a new physics.
The comparison is historically of great interest because Wiechert points to a precise
source, that is, Hertz’s seminal paper of 1884. So it is plausible to suggest that Einstein
was also responding to Wiechert’s paper since, as we have seen, he refers to Hertz’s
modification of Maxwell’s equations and the term asymmetry is used by Hertz in that
paper.
Das scheint zunächst ein Nachteil (der übrigens leicht beseitigt werden könnte), ist es aber in
Wirklichkeit nicht, denn bei der Einordnung der Theorie der Optik in die Theorie der Elektrodynamik kommen wir so in die Lage, und genau der erfahrungsgemäss bestehenden Unsymmetrie
der elektrischen und magnetischen Erscheinungen anzupassen.”
299
Wiechert 1898, 88: “Die Zustandsänderungen des Feldes, denen wir nach Maxwell die Vermittlung der elektrischen und magnetischen Fernwirkungen zuschreiben, gestalten sich besonders
einfach, wenn das betrachtete Gebiet nicht merklich sinnlich wahrnehmbare Materie enthält. Es
äußern sich dann zwei elektrodynamische Felderregungen, eine ‘elektrische’ und eine ‘magnetische’. Der elektrische Erregung hat allem Anscheine nach die Symmetrie einer Verschiebung, d.
h. sie besitzt an jeder Stelle des Feldes eine ausgezeichnete Axe, wird bei einer Spiegelung in einer
Ebene senkrecht zur Axe umgekehrt und bleibt bei einer Spiegelung in einer Ebene durch die Axe
ungeändert. Wie schon von Maxwell hervorgehoben wurde, verhält sich die magnetische Erregung wesentlich anders, indem sie allem Anscheine nach nicht die Symmetrie einer Verschiebung,
sondern die einer Drehung hat.”
524
5. Wilhelm Wien (1864–1928)
In a love letter, dated September 1899, Einstein (then only 20 years old) reported
that he had recently written a letter to Wien (who had been promoted to professor a few
months earlier):
I too have done much bookworming & puzzling out, which was in part very interesting. I
also wrote to Professor Wien in Aachen about the paper on the relative motion of the luminiferous ether [Lichtäthers] against ponderable matter.... I read a very interesting paper
published by this man on the same topic in 1898. He will write me via Polytechnicum
(when it’s for certain!).300
As we have noted, the title of Einstein’s relativity paper is almost identical to that of
Wien’s article, published a year earlier in Annalen der Physik: “Über die Differentialgleichungen der Elektrodynamik für bewegte Körper.” Following the editors of Einstein’s Collected Papers,301 we note further that the term Lichtäther can play the role of
a distinct marker. Recall (Section II.2, Case 5, above) that in the introduction to the relativity paper Einstein uses this very term when he makes the dramatic move of dispensing
with the ether altogether:
The introduction of a “light ether” [Lichtäthers] will prove superfluous, inasmuch as in
accordance with the concept to be developed here, no “space at absolute rest” endowed
with special properties will be introduced.302
Einstein introduces the term – with quotation marks – which may indicate an allusion to
Wien’s paper of 1898 entitled: “Ueber die Fragen, welche die translatorische Bewegung
des Lichtäthers betreffen.”303 This is the same paper to which Einstein refers in his
love letter.304 Thus, there are several external reasons which support our suggestion that
300
Beck 1987, 135; Stachel et al. 1987, 233–234: “Auch ich habe viel büchergewurmt & sehr
viel ausgetüftelt, zum Teil sehr interessant. Auch hab ich an Professor Wien in Aachen geschrieben
... über Relativbewegung des Lichtäthers gegen die ponderable Materie. Ich habe von diesem Mann
eine sehr interessante Abhandlung vom Jahr 1898 über diesen Gegenstand gelesen. Er wird mir
via Polytechnikum schreiben (wenns gewißt ist!).”
301
On Einstein’s usage of Wien’s term, cf. Stachel et al. 1987, 224, 233–234 nn. 3 and 5.
302
Beck 1989, 141; Einstein 1905c, 892: “Die Einführung eines ‘Lichtäthers’ wird sich insofern als überflüssig erweisen, als nach der zu entwickelnden Auffassung weder ein mit besonderen
Eigenschaften ausgestatteter ‘absolut ruhender Raum’ eingeführt.” Earlier in the paper, Einstein
(1905b, 891) uses the term “Lichtmedium”, in quotation marks.
303
Wien 1904c also has Lichtäther in the title. Cf. Stachel et al. 1989, 306 n. 2, where it is
claimed that the passage in Einstein 1905c, 891, that begins “Beispiele ähnlicher Art, sowie die
mißlungenen Versuche ...”, alludes to the summary of the experimental evidence concerning the
motion of the Earth relative to the ether in Wien 1898, xiii–xviii.
304
For another use of Lichtäther by Einstein, see Beck 1987, 181 (a letter from Einstein to
Grossmann, dated September 1901): “A considerably simpler method of investigating the relative
motion of matter with respect to luminiferous ether [Lichtäther] that is based on ordinary interference experiments has just sprung to my mind.” See also Stachel et al. 1987, 316: “Zur Erforschung
der Relativbewegung der Materie gegen den Lichtäther ist mir nun eine erheblich einfachere Methode in den Sinn gekommen, welche auf gewöhnlichen Interferenzversuchen beruht.”
525
Einstein had read Wien 1904a: (1) previous correspondence with Wien; (2) evidence that
Einstein regularly read Annalen der Physik;305 (3) the fact that Einstein had contributed
several papers, beginning in 1901, to Annalen der Physik; and (4) the similar titles of
Wien (1904a) and Einstein (1905c). Our goal is, however, to show that problems concerning the symmetrical form of Maxwell’s equations were still pertinent in 1904, and
a scientist of the stature of Wien – a Nobel laureate in physics in 1911 – struggled with
them without reaching a definite conclusion. We will argue that Einstein responded, in
part, to difficulties in the theory of electrodynamics which Wien articulated with respect
to its symmetrical form.
Wien opens his paper of 1904 with a brief review of the subject and covers both the
theoretical and experimental sides currently of interest. He faithfully lists the principal
players in the field, referring – in this order – to the theories of Lorentz, Hertz,306 Cohn,
Heaviside, Abraham, and Poincaré, as well as to the experimental results of Michelson
and Morley, Fizeau, and Kaufmann.307 A characteristic feature of this overview is that
Wien calls attention to the principal difficulties that beset electrodynamics. This is very
much in the spirit of his response to Abraham, published later in that year:
Electrodynamics has not yet progressed in its development to such a point that we can
speak apodictically of a theory claiming unique [allein] validity. Yet Dr. Abraham speaks
as if he were in the possession of such a theory.308
According to Wien, electrodynamic theory faces a fundamental challenge:
For the most part, the difficulties with the electrodynamic theory of moving bodies are
due to the fact that, on the one hand, the ether’s capacity for motion was supposed to be
covered by the theory but, on the other hand, it was considered necessary to distinguish
between ether and matter. If, however, one takes the standpoint – first taken by Lorentz
– that all interactions between ether and matter are caused by the elementary charge of
the atoms, almost all of these difficulties disappear [fortfallen], and Maxwell’s system for
stationary bodies – without any [additional] auxiliary hypothesis – completely suffices for
an account of the electrodynamics of moving bodies. Given that the chemical properties
of bodies make it necessary to abandon the continuity of matter and to conceptualize
matter as consisting of elementary quanta in a discontinuous manner, it seems to me that
adopting this system is even more advisable.309
305
See n. 297, above.
A comparison of the equations in Wien (1900, 97) with those in Hertz ([1884]/1895, 311)
reveals that Wien uses the same equations (not reversing the algebraic signs); the only difference
is that Wien uses the symbol for partial derivative rather than Hertz’s “d” (and reversing the order
of the columns – which seems to have no significance). Wien thus appears to be well acquainted
with the symmetrical equations of Hertz.
307
Wien (1898) presents a more extensive survey. Cf. Stachel et al. 1987, 234 n. 5.
308
Wien 1904b, 635: “Die Elektrodynamik ist noch nicht bis zu einer solchen Entwickelung
gediehen, daß in apodiktischer Weise von einer allein Gültigkeit beanspruchenden Theorie gesprochen werden darf. Und doch spricht Hr. Abraham so, als ob er im Besitz einer solchen wäre.”
309
Wien 1904a, 643–644: “Der größte Teil der Schwierigkeiten für die Theorie der Elektrodynamik bewegter Körper rührt davon her, daß man einerseits die Bewegungsmöglichkeit des Äthers
in die Theorie aufnehmen wollte, andererseits eine Trennung von Äthers und Materie für nötig
hielt. Stellt man sich dagegen auf den zuerst von Lorentz eingenommenen Standpunkt, daß alle
306
526
Before we continue our analyis, let us pause to consider Wien’s terminology. In particular, his expression, die Elementarladungen der Atome, is reminscent of Wiechert’s
expression, von elektrischen Atomen, that appeared in an article published back in 1899 –
with the very idea that Wien had in mind, namely, that a charged particle mediates the
interactions between ether and matter:
... the particle that is ejected from the cathode is many times smaller than a hydrogen
atom.... The “electric charge” of each and every material particle represents an electrodynamic linkage with the ether which is firmly based on the properties of the particle
and never changes.... By means of cathode rays only the existence of special negative
atoms is indicated. We have, as far as I can see, no specific evidence that a corresponding
kind of positive atom exists.... Of course, even the boldest hypothesis, that matter can
be completely analyzed into [components of] two kinds of electric atoms, negative and
positive, is nevertheless conceivable.310
By 1904 this elementary charged particle was known as the “electron” and it is not clear
why Wien avoids the term here while using it later in the same paper.311
Be that as it may, Wien accepts Lorentz’s position, that is, the theory in which an
elementary charged particle brings about the reciprocal action between ether and matter,
and proceeds to apply the theory to a particle at rest or in motion. In following Lorentz, Wien explicitly endorses the ether hypothesis. Wien seeks to show that there is an
agreement between the following two considerations: (1) the charged particle – which
he calls “the elementary charge of the atoms” – is at rest and the ether is made to move
in a certain direction; (2) the particle is made to move in accordance with Maxwell’s
equations in the direction opposite to the motion of the ether. This expected equivalence
is probably related to one of Lorentz’s “Hypothèses fondamentales” (on the distinction
between ether and matter) which he stated in 1892:
Wechselwirkung zwischen Äther und Materie nur durch die Elementarladungen der Atome hervorgerufen werden, so fallen fast alle jene Schwierigkeiten von selbst fort und das Maxwellsche
System für ruhende Körper genügt völlig, um auch die Elektrodynamik bewegter Körper ohne
Zuhilfenahme irgend einer Hypothese zu umspannen. Da von der anderen Seite die chemischen
Eigenschaften der Körper notwendig fordern, die Kontinuität der Materie aufzugeben und die
Materie als aus Elementarquanten in unstetiger Weise bestehend anzunehmen, so scheint mir um
so mehr die Annahme dieses Systems geboten.”
310
Wiechert 1899, 740–742 (italics in the original): “Sollte die Emissionshypothese überhaupt
beibehalten werden, so blieb nur übrig, zu schliessen, dass die von der Kathode fortgeschleuderten
Theilchen vielmals geringere Masse als die Wasserstoffatome besitzen.... Die ‘elektrische Ladung’
eines jeden materiellen Theilchens bedeutet eine elektrodynamische Verkettung mit dem Aether,
welche in der Eigenart des Theilchens fest begründet ist und sich niemals ändert.... Durch die
Kathodenstrahlen wird nur die Existenz besonderer negativer Atome angezeigt. Wir besitzen, so
weit ich sehe, kein bestimmtes Anzeichen dafür, dass es auch eine entsprechende Art positiver
Atome giebt.... Natürlich ist trotzdem selbst die weitest gehende Hypothese denkbar, dass die Materie sich ganz in zwei Arten von elektrischen Atomen, eine negative und eine positive, auflösen
lässt.”
311
Wien 1904a, 651. See Falconer 2001, 86–91. In 1900 Wien invoked a different set of terms
for this particle: see nn. 323–324, below.
527
The charged particles may be regarded as being of “ponderable matter” to which forces
may be applied; however, I will suppose that in all the space occupied by a particle the
ether is also found, and even that a dielectric displacement and a magnetic force, produced
by an exterior cause, may exist in this space as if the “ponderable matter” did not exist
there. The latter is thus considered as perfectly permeable to these actions.312
The demonstration of this equivalence of descriptions – that is, the motion of ponderable
matter in the form of a charged body and the displacement of the ether as if the body
were permeable to the actions of the ether while remaining stationary – is the motivation
for Wien’s calculation, and he formulates the goal of his paper as follows:
We will thereby obtain at the same time an integral which contains a generalization of the
known electromagnetic process for [bodies] at rest [that applies equally well to an electromagnetic process] for [bodies that have] an arbitrary motion with constant velocity.313
The search is then for an integral for electromagnetic processes which applies more
generally: previously, one only had an expression for bodies at rest; now there is an
expression that applies both to bodies at rest and to bodies with any constant velocity.
The two descriptions exhibit relative motions that ought not to result in any difference –
and this is indeed what Wien shows without, however, exploring physical consequences
that might lead to a theory of relativity, an option which may have intrigued Einstein.
Indeed, in the electrodynamic part of the relativity paper, at the end of § 8, Einstein
states:
All problems in the optics of moving bodies can be solved by the method employed here.
The essential point is that the electric and magnetic forces of light, which is influenced by
a moving body, are transformed to a coordinate system that is at rest relative to that body.
This reduces every problem in the optics of moving bodies to a series of problems in the
optics of bodies at rest.314
312
Lorentz 1892, 230: “Les particules chargées seront regardées comme étant de la ‘matière
pondérable’ à laquelle des forces peuvent être appliquées; cependant, je supposerai que dans tout
l’espace occupé par une particule se trouve aussi l’éther, et même qu’un déplacement diélectrique
et une force magnétique, produits par une cause exterieure, peuvent exister dans cet espace comme
si la ‘matière pondérable’ n’y existait pas. Cette dernière est donc considérée comme parfaitement
perméable à ces actions.”
313
Wien 1904a, 645: “Wir werden damit gleichzeitig ein Integral erhalten, das die Verallegemeinerung eines für die Ruhe bekannten elektromagnetischen Vorganges für eine beliebige
Bewegung mit konstanter Geschwindigkeit enthält.”
314
Beck 1989, 165; Einstein 1905c, 915: “Nach der hier benutzten Methode können alle Probleme der Optik bewegter Körper gelöst werden. Das Wesentliche ist, daß die elektrische und
magnetische Kraft des Lichtes, welches durch einen bewegten Körper beeinflußt wird, auf ein
relativ zu dem Körper ruhendes Koordinatensystem transformiert werden. Dadurch wird jedes
Problem der Optik bewegter Körper auf eine Reihe von Problemen der Optik ruhender Körper
zurückgeführt.”
528
It should be emphasized that this useful transformation from a moving to a stationary
system in Einstein’s relativity theory is carried out without the ether hypothesis.
In § 1 of his paper Wien discusses the relations in Lorentz’s equations for a stationary body with respect to a moving body and reaches two equivalent solutions for the
moving body and the stationary body in the special case when their relative velocity is
constant.315 This he regards as a “remarkable result” [bemerkenswerte Resultat] since,
with it, one may proceed from the solution of the form of the equations for a stationary body to a solution for a moving body.316 Then in the final section of the first part
of his study Wien applies his method of calculation to “longitudinal” [longitudinaler]
and “transverse” [transversaler] mass – a distinction to which he calls attention in the
introductory section of his study, acknowledging, inter alia, the work of Abraham.317
In § 10 of Einstein’s relativity paper: “Dynamics of the (slowly accelerated) electrons”
[Dynamik des (langsam beschleunigten) Elektrons], this terminology is invoked, albeit
for a different reason from those given by either Abraham or Wien.318
In this section Wien reaches what he takes to be an unsatisfactory result:
We therefore cannot infer that exceeding the velocity of light – which would have to occur
for an electron during its longitudinal oscillation when it is already moving at the velocity
of light – is impossible.319
However, he adds that “not much can be said in favor of this possibility [of exceeding the
velocity of light], for ... the acceleration during the longitudinal oscillation is infinitely
small of the second order.”320 Wien was reluctant to accept superluminal velocities –
even though he had not constructed an argument against this possibility.
In the second part of his paper Wien seeks to complete his account of electrodynamics by examining magnetism. While in Part I he was concerned with phenomena
associated with the elementary charge of the atoms as the mediating bodies between
ether and matter, in Part II he focuses on magnetic dipoles. He compares the results
of Hertz and Heaviside for an oscillating dipole, and determines its transversal oscilla315
Wien 1904a, 647: “Nur in dem speziellen Fall, daß vx konstant ist, führt die benutzte Transformation auch zur allgemeinen Integration.”
316
Wien 1904a, 649: “Die beiden Lösungen stimmen also überein und wir haben das bemerkenswerte Resultat, daß wir für unsere Lösung die Form der Gleichungen für bewegte Körper
gar nicht brauchen, sondern von den Gleichungen für ruhende Körper ausgehen können.”
317
Wien (1904a, 644) refers to Abraham (1903). Cf. Abraham 1903, 150–151. Einstein adopts
this terminology in his relativity paper without, however, referring to his source: see Einstein
1905c, 918; cf. Stachel et al. 1989, 270 n. 112.
318
In the relativity paper the term electron is not used until this section. Beck 1989, 167: “In an
electromagnetic field let a pointlike particle endowed with an electric charge ε (called ‘electron’
in what follows)....” Einstein 1905c, 917: “In einem elektromagnetischen Felde bewege sich ein
punktförmiges, mit einer elektrischen Ladung versehenes Teilchen (im folgenden ‘Elektron’
genannt)....”
319
Wien 1904a, 656: “Wir können daher keine Folgerung derart ziehen, daß die Überschreitung
der Lichtgeschwindigkeit, die ja bei einem bereits mit Lichtgeschwindigkeit bewegten Elektron
während der longitudinalen Schwingung erfolgen müßte, unmöglich wäre.”
320
Wien 1904a, 656: “Aber andererseits spricht auch nichts für diese Möglichkeit, denn ... die
Beschleunigung während der longitudinalen Schwingung unendlich klein von zweiter Ordnung.”
529
tion. Wien arrives at a result that diverges from that of Abraham (this developed into a
public controversy between the two physicists321 ). He ends the paper with a characteristic hesitation (on the one hand ..., but on the other....), that is, the status of electricity
and magnetism in contemporary theories is left undecided. Since this final comment is
directly relevant to our argument, we quote it at length:
Finally, I would like to point to a fact that has come to the fore in the course of these investigations, namely, the introduction of magnetic poles was necessary in order to achieve the
univocality [Eindeutigkeit] of the solutions. Although magnetic and electric quanta appear
to be on an equal footing within the symmetrical construction of Maxwell’s equations, the
question of whether we can assume [the existence of] real magnetic elementary quanta is
of utmost importance. In previous theories, we see a tendency to attribute the interaction
between ether and matter solely to the electric elementary quanta, and to ascribe the magnetic properties of bodies to their motions. On such accounts, magnetic quanta are mere
fictions, on analogy with Ampère’s magnetic double layer on a plane that is encased by
a linear current-conductor. This account is supported by the fact that there is no isolated
positive or negative magnetism.
Now, the magnetic quantum of the magnetic dipoles that were introduced by us is
likewise zero, and so we could put forth the claim that those dipoles are likewise merely
fictitious, introduced to represent a special configuration of magnetic force lines which are
due to the movement of the electric dipole. In this connection, however, it is dubious and
arbitrary [to introduce] different conceptions of the electric and magnetic dipoles, which
within the theory appear equally valid in all respects and, in addition, it requires that the
univocality [Eindeutigkeit] of the solutions be abandoned. For, if we view the magnetic
dipoles as merely fictitious, then the two solutions, [given] above, for a transversally moving electric dipole are equally valid in all respects, and this would be extremely awkward
for a theoretical treatment. If, on the other hand, we assume the possibility of the existence
of magnetic quanta which act directly on the ether exactly as [ganz analog] the electric
[quanta] do, we can view those two solutions as different, since it obviously has to make
a difference whether or not a really existing magnetic dipole is active as distinct from the
electric one.322
321
Wien 1904b and Abraham 1904. On the Wien–Abraham debate in Annalen der Physik
(1904), see Klein et al. 1993, 57.
322
For the translation of Eindeutigkeit as “univocality”, cf. Howard 1996, 122; “univocal” is
defined in the OED as follows: “Having only one proper meaning or signification; admitting or
capable of a single interpretation or explanation; of which the meaning is unmistakable; unambiguous.” Wien 1904a, 667–668: “Schließlich möchte ich noch auf den in diesen Untersuchungen
hervorgetretenen Umstand besonders hinweisen, daß die Einführung magnetischer Pole notwendig war, um die Eindeutigkeit der Lösungen zu wahren. Obwohl nun in dem symmetrischen Bau
der Maxwellschen Gleichungen die Existenz magnetischer Quanta ebenso wie die elektrischer
ganz gleichberechtigt erscheint, so ist doch die Frage von besonderer Wichtigkeit, ob wirkliche
magnetische Elementarquanten anzunehmen sind. In den bisherigen Theorien tritt die Neigung
hervor, die Wechselwirkung zwischen Äther und Materie nur den elektrischen Elementarquanten
zuzuschreiben und die magnetischen Eigenschaften der Körper auf Bewegung dieser zurückzuführen. Magnetische Quanten sind hiernach nur Fiktionen nach Analogie der Ampèreschen
magnetischen Doppelschicht auf einer von einem linearen Stromleiter umschlossenen Fläche.
Unterstützt wird diese Auffassung durch die Tatsache, daß es isolierten positiven oder negativen
Magnetismus nicht gibt.
530
Wien’s way of characterizing the carriers of electricity and magnetism is worthy of comment. We have seen in Part I of Wien’s paper – the electricity part – that he refers to
the “elementary charge of the atoms” [Elementarladungen der Atome] as the mediator
of all the interactions between ether and matter. However, in Part II, when he focuses
on magnetism, he refers to “magnetic poles” [magnetischer Pole] as well as to “magnetic and electric quanta” [magnetischer Quanta ebenso wie die elektrischer ...] and
“real magnetic elementary quanta” [wirkliche magnetische Elementarquanten]. Wien’s
terms for electric quanta go back to 1900 when he proposed an electromagnetic view
of nature.323 Moreover, the term Elementarquanten may be dependent on Heaviside
(although expressed differently). In 1900 Wien comments:
We think of the electric elementary quantum [Elementarquantum] as an electrified point.
The forces and polarizations that originate from such a point have been derived by
Heaviside (Electrical Papers, Vol. II).324
Unfortunately, Wien does not give an exact reference, and we can only suggest that he
may have had in mind Heaviside’s discussion on the electromagnetic effects of a moving
charge.325
Wien’s association of the magnetic pole with a “quantum” had not been mentioned
in his work on electrodynamics before 1904. The phrase: “the magnetic quantum of
the magnetic dipoles” [das magnetische Quantum der... magnetischen Dipole] clearly
indicates that Wien thinks in terms of discrete elements that carry magnetism. As far
as we can tell, he was the first to consider a magnetic quantum on analogy with an
electric quantum that had already been employed at the turn of the last century.326
Nun ist das magnetische Quantum der von uns eingeführten magnetischen Dipole ebenfalls
Null und wir könnten die Behauptung aufstellen, daß diese Dipole ebenfalls nur fingiert sind,
um eine besondere Anordnung magnetischer Kraftlinien, die von der Bewegung des elektrischen Dipols herrühren, darzustellen. Bedenklich ist hierbei aber die Willkür in der verschiedenen
Auffassung des elektrischen und magnetischen Dipols, die in der Theorie ganz gleichberechtigt
auftreten, ferner die Notwendigkeit, die Eindeutigkeit der Lösungen aufzugeben. Denn wenn wir
die magnetischen Dipole nur als fingiert ansehen, dann sind die beiden betrachteten Lösungen für
einen transversal bewegten elektrischen Dipol gleichberechtigt, was für die theoretische Behandlung äußerst mißlich sein dürfte. Nehmen wir dagegen die Möglichkeit der Existenz magnetischer
Quanten, die ganz analog den elektrischen direkt auf den Äther einwirken, an, so sind die beiden
Lösungen als verschieden anzusehen, weil es natürlich einen Unterschied machen muß, ob außer
dem elektrischen Dipol noch ein wirklich existierender magnetischer mitwirkt oder nicht.”
323
See, for example, Wien 1900, 98–99, for expressions such as elektrischen Quanten, Elementarquanten and positiven und negativen Quanten.
324
Wien 1900, 101: “Das elektrische Elementarquantum denken wir uns als einen elektrisirten
Punkt. Die von einem solchen bewegten Punkt ausgehenden Kräfte und Polarisationen sind von
Heaviside (Electrical Papers Band II) abgeleitet.” Cf. Wien 1898, v. In 1898 Wien had already
called electric points Quanta (see p. ii).
325
Heaviside [1892b]/1925b, 492–494. Cf. nn. 287 and 288, above.
326
In his celebrated paper of 1900 on the quantum hypothesis, Planck (1900, 245) used the
expression “Das Elementarquantum der Elektricität”. Interestingly enough, this is the only occurrence in this paper of the term “quantum”.
531
Wien drew a consequence of symmetry that was unprecedented (but did not catch
on).327
Wien goes back to the position of Heaviside: to make the equations symmetrical one
has to add fictitious elements for magnetism. That is, the electric side works fine, but the
magnetic side has to be made to fit the requirements of symmetry. In Lorentz’s theory
(with particles) there is no problem on the electric side, but once again the problem
persists on the magnetic side. That is, changing from fluids to particles has not changed
the essential features of this problem. We have noted that Hertz made the equations into
axioms and did not see any difficulty in their symmetrical form, and Heaviside did not
find this fiction at all distressing.328 Yet for Wien this was clearly a matter of discomfort
and a challenge.
The equations serve as Wien’s point of departure, that is, the formalism, and he seeks
to match them with the phenomena. He sees the issue in terms of drawing logical consequences from the symmetry of the equations, that is, the symmetry of the equations
should dictate the physics. He is quite confident in holding this view even though his
overall remarks are indecisive. One disturbing consequence for him is that, for reasons of
symmetry, there should be a “quantum” of magnetism corresponding to the “quantum”
of electricity. This requires the introduction of a magnetic “quantum” – on all accounts a
fictitious entity – and so Wien is reluctant to do so. Although he offers no alternative,329
he is still unwilling to put into doubt the symmetry of the equations that led to this
difficulty. We see then that Wien is explicitly aware of the problems in drawing physical
consequences from the symmetry of the equations.
For Wien the central issue is the attainment of univocal solutions and this, in turn,
depends on the symmetrical form of the equations (i.e., Maxwell’s equations in Hertz’s
formulation). In the final passage of his paper (quoted above) Wien uses the term Eindeutigkeit twice to characterize the solutions. Later in the same year, Wien comments in
his response to Abraham:
Mr. Abraham then claims that my formulas contradict those that are derived from the
theory of electrons. In response I would like to remark, first of all, that in my opinion
there is no theory of electrons par excellence, but only a Maxwellian one, and my formulas are consistent with that theory. In addition, I have not sought to guess [the properties
327
Quantum was rarely used in physics before 1905, but it is found in a letter from Einstein
to Conrad Habicht in 1904. Klein et al. 1993, 26: “Die Beziehung zwischen der Größe der Elementarquanta der Materie und den Strahlungswellenlängen habe ich nun in höchst simpler Weise
gefunden.” Cf. Beck 1995, 17: “I have now found the relationship between the magnitude of the
elementary quanta of matter and the wavelengths of radiation in an exceedingly simple way.” In
Einstein’s paper of 1905 on the light-quantum hypothesis (the only paper he published in 1905
that he called “revolutionary” at the time [Stachel et al. 1989, 134]), the terms Energiequanten
and Lichtquanten were introduced (Einstein 1905a, 133, 144).
328
See, for example, nn. 196, 225, 227, and 234, for the views of Hertz and Heaviside, respectively.
329
Wien informs the reader that he plans to work on a complicated solution that does not make
use of a magnetic dipole. (Wien 1904a, 659: “Auf die kompliziertere Lösung, bei der kein Magnet
mitwirkt, gedenke ich in einer besonderen Untersuchung einzugehen.”)
532
of] the field; rather, I discovered [them]. Since the Maxwellian theory has only univocal
[eindeutige] solutions, [this field] can therefore withstand [any] objections.330
From Wien’s response to Abraham, we see that he regards a “univocal solution” of some
physical formalism – in this case, Maxwell’s system of equations – as sound and hence
able to withstand any objection. The idea is that there is no ambiguity in the description
of phenomena and their relation to the formalism; that the solution, say, of Maxwell’s
equations, only allows a single interpretation or explanation for which the meaning is
unmistakable. The frequent usage of the term indicates that Wien probably intended to
invoke what was then called “the law of univocality” [Das Gesetz der Eindeutigkeit].331
Wien refers to the interpretation of the solutions and argues that the interpretation ought
to be clear and unambiguous.
And yet the puzzle has not been eliminated. On the one hand, the symmetrical construction of Maxwell’s equations results in univocal solutions, but at a considerable
price: the introduction of fictitious elements into the theory. Further, if one were also to
pursue the analogy with respect to the ether, then not only the electric “atom” mediates
between ether and matter, but the magnetic “quantum” should do so too, in which case
the solutions will remain univocal – but again, at a great price, namely, the introduction
of a fiction into the theory. Wien states a dilemma but offers no clue for resolving it. As
far as we can tell, Wien was the only physicist to express any uncertainty or hesitation
about the symmetry of the equations for electricity and magnetism – in contrast to all the
other physicists who expressed a view on symmetry/asymmetry. So Wien stands alone
in putting this issue in the category of an open question.
330
Wien 1904b, 636: “Dann behauptet Hr.Abraham, daß meine Formeln den aus der Elektronentheorie abgeleiteten widersprechen. Hierzu möchte ich zunächst bemerken, daß es für mich keine
Elektronentheorie par excellence gibt, sondern nur eine Maxwellsche und mit dieser sind meine
Formeln in Einklang. Außerdem habe ich das Feld nicht zu erraten gesucht, sondern gefunden, und
da die Maxwellsche Theorie nur eindeutige Lösungen hat, so ist es damit einwandfrei bestimmt.”
For another usage of Eindeutigkeit by Wien, see Wien 1904a, 659. Abraham noticed Wien’s
frequent usage of the term and placed it at the head of his rejoinder to Wien. Abraham 1904,
1039 (italics in the original): “In seiner Erwiderung auf meine Kritik behauptet Hr. W. Wien,
seine Methoden bestimmten eindeutig das Feld eines bewegten leuchtenden Punktes.” Planck
(1900, 239) also uses Eindeutigkeit: “... aber ich lege hier, ... nicht Wert auf den Nachweis der
Notwendigkeit und der leichten praktischen Ausführbarkeit, sondern nur auf die Klarheit und
Eindeutigkeit der gegebenen Vorschriften zur Lösung der Aufgabe.”
331
Petzoldt 1895, 168: “Beides, unser individueller Bestand und jenes Postulat, wie wir die betreffende Voraussetzung auch bezeichnen dürfen, gehören untrennbar zusammen. Letztere besteht
in nichts anderem als in der Annahme der durchgängigen vollkommenen Bestimmtheit oder –
wie wir, um die wichtigste Seite der Sache hervorzuheben, sagen wollen – in der Annahme der
Eindeutigkeit aller Vorgänge.” (“... the above-mentioned postulate ... consists in nothing other than
the assumption of continuous, complete determinacy [for all processes] or – as we might say in
order to emphasize the most important aspect of this issue – in the assumption of the univocality
of all processes....”) For further elaboration of the meaning of this law, see esp. pp. 184–185.
533
For Einstein Wien’s approach was no way to make any progress; he recasts the problem so that Wien’s difficulties disappear.332 Einstein agrees with Wien that the equations
in their symmetrical form are correct and that the problem lies in their interpretation.
Wien expects the symmetry of the equations to correspond to symmetry in the phenomena, but Einstein denies this suggestion for reasons we have already discussed. The
solution is to embed the equations in an entirely new theory where the principle of
relativity demands only a single electromagnetic field (rather than distinct electric and
magnetic fields) without an “ether”. At the beginning of the relativity paper (1905c)
Einstein does not deny the symmetry of the equations for electricity and magnetism –
but the symmetry of the consequences drawn from the theory. In the paragraph at the end
of his paper (1904a), Wien struggles with the same issue (expressed differently) without offering a solution. We may thus conclude that Einstein’s invocation of asymmetry
for Maxwell’s theory can be understood in part as a response to Wien’s claim that the
Maxwell–Hertz formulation is symmetrical with respect to magnetism and electricity.
6. Hendrik Antoon Lorentz (1853–1928)
Einstein refers explicitly in the relativity paper to only three physicists: Maxwell,
Hertz, and Lorentz. While the first two physicists are mentioned in a general way, that
is, the Elektrodynamik Maxwells and the Maxwell–Hertzschen Gleichungen, the reference to Lorentz is somewhat more specific, namely, an allusion – albeit with no exact
reference – to Lorentz’s electrodynamics and optics of moving bodies.333
Lorentz is one of the towering figures in physics at the turn of the last century, and so
it is no surprise that Sommerfeld turned to him to provide separate accounts of the state
of the art of Maxwell’s electromagnetic theory and of electron theory for volume 5 –
Physik – of the Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer
Anwendungen. Lorentz published these two important essays in 1904 but, as we have
noted, there is no evidence that Einstein was aware of these works by Lorentz before
composing his relativity paper.
In the first essay, “Maxwells elektromagnetische Theorie,” Lorentz reviews electromagnetism comprehensively and critically. Of special interest to us is his treatment of
symmetry, a term that by 1904 – as we have shown – abounds in the literature. We have
discerned a consistent pattern in Lorentz’s report and analysis. As befitting a central
and reliable figure in this discipline, he reports on usages of symmetry, but he invariably avoids the term symmetry and uses alternative terminology. Here are a few striking
examples.
We note at the outset that Lorentz gives precise references to the works of Hertz,
Heaviside, Föppl, Wiechert, Ludwig Boltzmann (1844–1906), and Henry Rowland
332
In another famous paper (completed in March 1905, just over three months prior to the
relativity paper) – “On a heuristic point of view concerning the production and transformation
of light” – Einstein (1905a, 137 and 139) cites Wien twice with explicit approval of his work on
“black-body” radiation. Wien (1904a, 665) made use of the term “heuristisch” which also appears
in the title of Einstein’s paper.
333
Einstein 1905c, 916.
534
(1848–1901).334 Lorentz refers (boldface added) to the parallelism between the phenomena of electricity and magnetism that has been introduced into Maxwell’s equations.335 In the section entitled, Der magnetische Strom und die unipolare Induktion.
Dualität zwischen den elektrischen und den magnetischen Erscheinungen, Lorentz
clearly avoids the term symmetry, stating that Hertz sought Parallelismus in his work
of 1884, rather than Symmetrie. Lorentz is careful to mention Heaviside as the physicist
who, together with Hertz, established this view of the electromagnetic phenomena in
the algebraic form of the equations. He also refers to Föppl (almost certainly his source
for the term duality) who – as we have seen – built on the work of Heaviside.336 Indeed,
Föppl himself used both Reciprocität and parallel to characterize what he called Dualitätsbeziehungen, which may have lent support to Lorentz’s decision to depart from
the original terminology of Hertz and Heaviside.337
More corroborating evidence come at the end of this section, where Lorentz reports
briefly on the pioneering work of Rowland (1880). Lorentz indicates that Rowland noted
the reciprocity [Reziprozität] between electric and magnetic phenomena. However, an
examination of Rowland’s paper reveals that, in addition to the term reciprocal, he had
also used the term symmetry in precisely the way Hertz and Heaviside did a few years
later. After setting up the electric and the magnetic systems, Rowland argues that they
are “perfectly symmetrical” but later he refers to the “almost perfect reciprocity between
electric currents and magnetic induction.”338 In 1893 Boltzmann cited Hertz (1884) in
the same way that Lorentz later referred to Rowland, that is, Boltzmann used the term
Reziprozität, rather than symmetry, to characterize the relation between electricity and
magnetism in reporting on the work of Hertz.339
334
Lorentz 1904a, 64–65, 68 nn. 3 and 4, 99 nn. 29 and 30, 101 n. 31, 135 n. 77.
Lorentz 1904a, 85: “... weil in dieser Weise der Parallelismus zwischen den elektrischen und
den magnetischen Erscheinungen auch in der Form der Gleichungen am besten zu Tage tritt.” Cf.
Lorentz 1904a, 112.
336
Lorentz 1904a, 99 n. 29. See also n. 260, above.
337
Föppl 1894, 121: “Aus der Geometrie der Lage ist das Gesetz der Dualität oder Reciprocität
wohlbekannt.” Cf. Föppl 1894, 143.
338
Rowland 1880, 90, 93, 99. On Rowland see, e.g., Buchwald 1985, 102–106.
339
Boltzmann [1891–1893]/1982, 2: 124. As we have indicated, Lorentz cites Boltzmann but
he does not refer to this occurrence of Reziprozität. Surprisingly, after 1905 Lorentz extended the
meaning of reciprocity to Einstein’s usage of the principle of relativity. In his 1906 lectures on
electron theory (published in 1909), Lorentz refers to the “remarkable reciprocity” that Einstein
obtained for stationary and uniformly moving observers. He notes ([1909]/1916, 227) that the
reciprocity consists in the following: “if the observer A describes in exactly the same manner the
field in the stationary system, he will describe it accurately.” Lorentz draws this inference from
the fact that the equations of electrodynamics remain the same under the transformation of the
relativity theory (p. 229): “It would be impossible to decide which of them [the observers] moves
or stands still with respect to the ether, and there would be no reason for preferring the times and
lengths measured by the one to those determined by the other, nor for saying that either of them is in
possession of the ‘true’ times or the ‘true’ lengths.” Lorentz notes that this is the result of Einstein’s
application of the principle of relativity, but fails to mention that a second postulate is involved,
namely, the postulate concerning the velocity of light. Lorentz apparently thinks that Einstein
has described a symmetry (which Lorentz calls reciprocity), but this is simply a description of
335
535
Towards the end of the essay, Lorentz discusses in general terms the various properties of electric and magnetic magnitudes. Characteristically, he avoids the use of the
term symmetry in discussing the relations between these two groups of phenomena and
resorts to the notion of “mirror image” [Spiegelbild].340 Lorentz expresses his reservations explicitly when he reviews electron theories in his second essay in the Encyklopädie,
“Weiterbildung der Maxwellschen Theorie. Elektronentheorie.” He remarks:
Finally, it should be emphasized that in the theory of electrons the parallelism between
electric and magnetic phenomena does not go as far as it does in Hertz’s theory.341
Lorentz pioneered the theory of electrons and, evidently, he did not find it useful to adopt
Hertz’s symmetrical approach to the formalism.
Despite his reluctance to use the term symmetry in the context of the relation between
electricity and magnetism, Lorentz does use this term in the section of his review article
on duality, but this usage has nothing to do with either the algebraic structure of the
equations or the phenomena – it is a geometrical usage. Lorentz develops an argument
and imposes the following constraints:
For example, consider a metal mass M, which has the shape of a rotational body with the
axis OZ, located in a symmetrical magnetic field surrounding OZ;....342
There are many examples of similar usages in the works of Lorentz; here we will cite
one of them that occurs in his review article on theories of the electron:
A body has three planes of symmetry which are perpendicular to each other. One then
designates them as the coordinate planes.343
Lorentz demonstrates his versatility as a great physicist in extending his studies to
crystals. In this context he uses symmetry in the same way that Voigt and Schönflies did.
“relativity”. Clearly, this usage is not attested prior to 1905; it is not part of Einstein’s vocabulary
and, as far as we know, it was not accepted by anyone other than Lorentz.
340
Lorentz 1904a, 135. In this section (§ 41) Lorentz has the expression ”... entweder die
Vektoren der ersten Gruppe translatorischen und die der zweiten Gruppe rotatorischen Charakter
haben, oder umgekehrt”, and in Lorentz 1904b, 216, we find Gruppen von Leitungselektronen
and Gruppe von Teilchen: these expressions are reminiscent of Einstein’s usage of ”group” in his
“light-quantum” paper (Einstein 1905a). Cf. n. 97, above. Lorentz also appeals to the notion of
“mirror image” in his Versuch (1895, 29, 75, 129, 131).
341
Lorentz 1904b, 238 (italics in the original): “Schließlich möge hervorgehoben werden, daß
in der Elektronentheorie, der Parallelismus zwischen den elektrischen und den magnetischen Erscheinungen nicht so weit geht, wie in der Theorie von Hertz.”
342
Lorentz 1904a, 100: “Man betrachte z. B. eine Metallmasse M, welche die Gestalt eines
Rotationskörpers mit der Achse OZ hat und sich in einem um OZ symmetrischen magnetischen
Felde befindet;....”
343
Lorentz 1904b, 239: “Ein Körper habe drei zu einander senkrechte Symmetrieebenen; diese wähle man zu Koordinatenebenen.” See also Lorentz [1892]/1936, 305 (§ 145, boldface
added): “Si P et P sont deux points, l’un dans le premier système et l’autre dans le second, et
qui sont symmétriquement situés de part et d’autre du plan E, on trouvera dans ces points, à tout
moment,....” Cf. Lorentz 1895, 74, 129, 131.
536
Lorentz also uses the term “symmetry relations” [Symmetrieverhältnissen] to indicate
the relation between the physics of the crystal and the form of the equations.344 Nevertheless, he did not accept such a relation in electromagnetism and thought that symmetry
was an inappropriate term for describing the way Maxwell’s equations had been recast
by Rowland, Hertz, and Heaviside.
7. Summary
In Part III we have considered the relation between electricity and magnetism as
described by a number of physicists prior to 1905. Hertz discerned that the form of
Maxwell’s equations is asymmetrical. He then derived the equations in a symmetrical
form and later posited them as axioms. In his methodology, any theory – namely, the
interpretative element – that is consistent with the equations is viable. In particular, his
“mature” theory invokes the ether to account for electric waves. The wedge that Hertz
drove between the theory and its formalism, helped Einstein see that the theory can
be discarded while still retaining the equations. Thus he was able to discard the “ether
hypothesis” and to build the relativity theory on transforming the equations among several coordinate systems.
Heaviside had no qualms about introducing certain fictitious elements into the equations for magnetism to make them symmetrical with the equations for electricity. He
called this the duplex method, and saw great advantages in having formulated the equations for electricity and magnetism in a symmetrical form. He illustrated this symmetry
with two vivid experiments: the charged bullet and the iron ring.
Föppl agreed with Heaviside but sought a deeper reasons for the symmetrical form
of the equations. He recast Heaviside’s results in terms of duality, on analogy with duality in projective geometry. Föppl demonstrated this duality but, where he encountered
difficulties, he referred back to Heaviside. He implemented Heaviside’s method of demonstrating the formalism with vivid experimental situations which he described in terms
of his own ether theory, and pointed to outstanding difficulties that he characterized as
central to the subsequent development of physics.
Wiechert modified Hertz’s equations; he recognized an asymmetry in Maxwell’s
equations, but claimed that this is to be expected since the phenomena are not symmetrical. Moreover, the asymmetrical form is helpful in linking optical and electromagnetic
phenomena.
Wien accepted the symmetrical form of the equations, but then expected the phenomena of electricity and magnetism also to be symmetrical. In particular, if there is
a unit (“atom”) of electric charge (as in Lorentz’s theory), there ought to be a unit of
magnetism – magnetischer Quanta. But this unit of magnetism is fictitious, that is, there
is no evidence that such a thing exists in nature. Wien expressed bewilderment with this
situation and could not resolve it.
In reviewing the discipline in 1904, Lorentz made use of a variety of terms to describe
the relation between electricity and magnetism – both the formal and the phenomenal.
Lorentz avoided the term symmetry in this context (for reasons that are not stated), but
344
Lorentz 1904b, 239. Cf. Lorentz 1907, 299.
537
he had no problem with the underlying concept as long as another word is used. Thus, he
accepted Föppl’s duality and referred to passages in Hertz and Heaviside where they used
symmetry. But he characterized this form as parallelism and recast the relation between
electricity and magnetism as reciprocal. He is consistent in doing so, for he does not
report Rowland’s usage of symmetry and only cites the expression, “reciprocal relation”. Notice that some ten years earlier Boltzmann had referred to Hertz’s symmetrical
presentation of the equations as reciprocal.
In brief, the “story line” is as follows: Hertz (1884) and Heaviside (1885) independently recast Maxwell’s equations in a symmetrical form. Hertz does not repeat his
initial claim of 1884 for having made the asymmetry disappear but keeps the equations
in their symmetrical form. Heaviside calls the symmetry he introduced the duplex form
of the equations. Boltzmann (1893) changes Hertz’s symmetry to reciprocity (citing
Hertz 1884). Föppl (1894) then cites both Hertz and Heaviside, and changes Heaviside’s
duplex to duality – a change he anchored in projective geometry. In his survey of the
literature, Lorentz (1904) accepts Föppl’s duality (citing both Hertz and Heaviside as
well as Föppl), and surprisingly does not use the term symmetry in this context of reporting these works. Wiechert (1900) accepts the term asymmetry (citing Hertz 1884) and
finds no problem with it. Wien (1904) appeals to symmetry but has problems with it.
He notices both the symmetry of the equations and the lack of correspondence in the
phenomena of electricity and magnetism. The entire chain of usages occurs in responses
to either Hertz or Heaviside.
In all cases, except Föppl, it is just a matter of terminology, i.e., it is not clear why one
term is better than another, and we have not encountered any elaboration on the various
usages or discussions of their relative merits. Indeed, no one seems to worry about the
definitions of any of these terms, including symmetry. But Föppl gives a precise definition
based on the analogy with duality in projective geometry – a term which Lorentz accepts
but does not put to any use. We have here a fine illustration of vacillating terminology.
Even those who accepted the symmetrical form of Maxwell’s equations were not
keen on the term symmetry – beginning with Heaviside who later preferred duplex, Föppl
who preferred duality, Boltzmann who preferred reciprocity, and Lorentz who preferred
parallelism and duality. Wiechert accepted the term asymmetry and was aware that Hertz
had derived the equations in a symmetrical form. Wien invoked symmetry but the extent
of its application was left unresolved. Finally, Einstein joins the discussion and dismisses
the entire enterprise, for it is essentially about formalism. In Einstein’s view physical
reasoning has to guide the formalism: in physics there is no other way.
Acknowledgments. We thank Ole Knudsen and Matthias Schwerdt for their assistance and
encouragement, and we are particularly grateful to John Norton for many detailed and insightful
comments. We are also indebted to Uljana Feest for generously providing us with draft translations
of many German texts. One of us [G.H.] benefited greatly from the hospitality and resources of
the Dibner Institute (Cambridge, Mass.) during the drafting of this paper.
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Department of Philosophy
University of Haifa
Haifa 31905
Israel
[email protected]
and
Faculty of Arts and Sciences
University of Pittsburgh
2604 Cathedral of Learning
Pittsburgh, PA 15260
USA
[email protected]
(Received April 1, 2005)
Published online June 10, 2005 – © Springer-Verlag 2005

Symmetry and Relativity in 1905

Transcrição

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