The Genesisofthe Theory of Relativity

S´eminaire Poincar´e 1 (2005) 1 – 22
S´
eminaire Poincar´
e
The Genesis of the Theory of Relativity
Olivier Darrigol
CNRS : Rehseis
83, rue Broca
75013 Paris
The most famous of Albert Einstein’s papers of 1905 is undoubtedly the one concerning
the theory of relativity. Any modern physicist knows that this theory imposes a strict and general
constraint on the laws of nature. Any curious layman wonders at the daring reform of our ancestral
concepts of space and time. As often happens for great conceptual breakthroughs, the theory of
relativity gave rise to founding myths whose charm the historian must resist.
The first of this myth is that Einstein discovered the theory of relativity in a single stroke
of genius that defies any rational analysis. Some of Einstein’s reminiscences favor this thesis, for
instance his allusion to a conversation with Michele Besso in which he would have suddenly realized
that a reform of the concept of time solved long standing paradoxes of electrodynamics. One could
also argue that the historical explanation of a deep innovation is by definition impossible, since a
radically new idea cannot be derived from received ideas. In the case of Einstein’s relativity the
rarity of pre-1905 sources further discourages historical reconstruction, and invites us to leave this
momentous discovery in its shroud of mystery.
This romantic attitude does not appeal to teachers of physics. In order to convey some sort
of logical necessity to relativity theory, they have constructed another myth following which a
few experiments drove the conceptual revolution. In this empiricist view, the failure of ether-
drift experiments led to the relativity principle; and the Michelson-Morley experiment led to the
constancy of the velocity of light; Einstein only had to combine these two principles to derive
relativity theory.
As a counterpoise to this myth, there is a third, idealist account in which Einstein is supposed
to have reached his theory by a philosophical criticism of fundamental concepts in the spirit of
David Hume and Ernst Mach, without even knowing about the Michelson-Morley experiment, and
without worrying much about the technicalities of contemporary physics in general.
A conscientious historian cannot trust such myths, even though they may contain a grain of
truth. He must reach his conclusions by reestablishing the contexts in which Einstein conducted
his reflections, by taking into account his education and formation, by introducing the several
actors who shared his interests, by identifying the difficulties they encountered and the steps they
took to solve them. In this process, he must avoid the speculative filling of gaps in documentary
sources. Instead of rigidifying any ill-founded interpretation, he should offer an open spectrum of
interpretive possibilities. As I hope to show in this paper, this sober method allows a fair intelligence
of the origins of relativity.
A first indication of the primary context of the early theory of relativity is found in the very
title of Einstein’s founding paper: “On the electrodynamics of moving bodies.” This title choice
may seem bizarre to the modern reader, who defines relativity theory as a theory of space and
time. In conformity with the latter view, the first section of Einstein’s paper deals with a new
kinematics meant to apply to any kind of physical phenomenon. Much of the paper nonetheless
deals with the application of this kinematics to the electrodynamics and optics of moving bodies.
Clearly, Einstein wanted to solve difficulties he had encountered in this domain of physics. A survey
of physics literature in the years 1895-1905 shows that the electrodynamics of moving bodies then
was a widely discussed topic. Little before the publication of Einstein’s paper, several studies with
similar titles appeared in German journals. Much experimental and theoretical work was being
done in this context. The greatest physicists of the time were involved. They found contradictions
between theory and experience or within theory, offered mutually incompatible solutions, and

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S´eminaire Poincar´e
sometimes diagnosed a serious crisis in this domain of physics.
Since Heinrich Hertz’s experiments of 1887-8 on the electric production of electromagnetic
waves, Maxwell’s field theory was the natural frame for discussing both the electrodynamics and
the optics of moving bodies. In order to understand the evolution of this subject, one must first
realize that the theory that Maxwell offered in his treatise of 1873 widely differed from what is
now meant by “Maxwell’s theory.”
1
Maxwell’s theory as it was
Like most of his contemporaries, Maxwell regarded the existence of the ether as a fundamental and
undeniable fact of physics. He held this medium responsible for the propagation of electromagnetic
actions, which included optical phenomena in his view. His theory was a phenomenological theory
concerned with the macroscopic states of a continuous medium, the ether, which could combine
with matter and share its velocity v. These states were defined by four vectors E, D, H, B that
obeyed a few general partial differential equations as well as some relations depending on the
intrinsic properties of the medium. In the most complete and concise form later given by Oliver
Heaviside and Heinrich Hertz, the fundamental equations read
× E = −DB/Dt ,
× H = j + DD/Dt
· D = ρ , · B = 0 ,
(1)
where j is the conduction current and D/Dt is the convective derivative defined by
D/Dt = ∂/∂ − × (v× ) + v( · ).
(2)
In a linear medium, the “forces” E and H were related to the “polarizations” D and B by the
relations D = E and B = µH, and the energy density (1/2)( E2 + µH2) of the medium had the
form of an elastic energy. For Maxwell and his followers, the charge density and the conduction
current j were not primitive concepts: the former corresponded to the longitudinal gradient of the
polarization or “displacement” D, and the latter to the dissipative relaxation of this polarization
in a conducting medium. The variation DD/Dt of the displacement constituted another form of
current. Following Michael Faraday, Maxwell and his disciples regarded the electric fluids of earlier
theories as a na¨ıvely substantialist notion1
The appearance of the convective derivative D/Dt in Maxwell’s theory derives from his un-
derstanding of the polarizations D and B as states of a single medium made of ether and matter
and moving with a well-defined velocity v (that may vary from place to place): the time derivatives
in the fundamental equations must be taken along the trajectory of a given particle of the moving
medium. The resulting law of electromagnetic induction,
× E = −DB/Dt = −∂B/∂t + × (v × B) ,
(3)
contains the (v × B) contribution to the electric field in moving matter. By integration around a
circuit and through the Kelvin-Stokes theorem, it leads to the expression
d
E · dl = −
B
dt
· dS
(4)
of Faraday’s law of induction, wherein the integration surface moves together with the bordering
circuit. When the magnetic field is caused by a magnet, the magnetic flux only depends on the
relative position of the magnet and the circuit so that the induced current only depends on their
relative motion.
In sum, the conceptual basis of Maxwell’s original theory widely differed from what today’s
physicists would expect. Electricity and magnetism were field-derived concept, whereas modern
1 J.C. Maxwell, A treatise on electricity and magnetism, 2 vols. (Oxford, 1973); H. Hertz, “ ¨
Uber die Grund-
gleichungen der Elektrodynamik f¨
ur bewegte K¨
orper,” Annalen der Physik, 41 (1890), 369-399.

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The Genesis of the Theory of Relativity
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electromagnetism treats them as separate entities. A quasi-material ether was assumed. The fun-
damental equations (1) only correspond to our “Maxwell equations” in the case of bodies at rest,
for which the velocity v is zero and the convective derivative D/Dt reduces to the partial deriva-
tive ∂/∂t. One thing has not changed, however: the theory’s ability to unify electromagnetism and
optics. In a homogenous insulator at rest, Maxwell’s equations imply the existence of transverse
waves propagating at the velocity c = 1/√ µ. Having found this electromagnetic constant to be
very close to the velocity of light, Maxwell identified these waves with light waves. The resulting
theory automatically excludes the longitudinal vibrations that haunted the earlier, elastic-solid
theories of optics.
Within a few years after Maxwell’s death (in 1879), a growing number of British physicists
saluted this achievement and came to regard Maxwell’s theory as philosophically and practically
superior to earlier theories. The Germans had their own theories of electricity and magnetism, based
on electric and magnetic fluids (or Amperean currents) directly acting at a distance. They mostly
ignored Maxwell’s theory until in 1888 Heinrich Hertz demonstrated the emission of electromagnetic
waves by a high-frequency electric oscillator. After this spectacular discovery was confirmed, a
growing number of physicists adopted Maxwell’s theory in a more or less modified form.
Yet this theory was not without difficulties. Maxwell had himself noted that his phenomeno-
logical approach led to wrong predictions when applied to optical dispersion, to magneto-optics,
and to the optics of moving bodies. In these cases he suspected that the molecular structure of
matter had to be taken into account.
2
Flashback: The optics of moving bodies
Maxwell’s idea of a single medium made of ether and matter implied that the ether was fully
dragged by moving matter, even for dilute matter. Whereas this conception worked very well
when applied to moving circuits and magnets, it was problematic in the realm of optics. The first
difficulty concerned the aberration of stars, discovered by the British astronomer James Bradley
in 1728: the direction of observation of a fixed star appears to vary periodically in the course of a
year, by an amount of the same order as the ratio (10−4) of the orbital velocity of the earth to the
velocity of light.2
The old corpuscular theory of light simply explained this effect by the fact that the apparent
velocity of a light particle is the vector sum of its true velocity and the velocity of the earth (see
Fig. 1). In the early nineteenth century, the founders of the wave theory of light Thomas Young and
Augustin Fresnel saved this explanation by assuming that the ether was completely undisturbed
by the motion of the earth through it. Indeed, rectilinear propagation at constant velocity is all
that is needed for the proof.3
Fresnel’s assumption implied an ether wind of the order of 30km/s on the earth’s surface,
from which a minute modification of the laws of optical refraction ought to follow. As Fresnel knew,
an earlier experiment of his friend Fran¸cois Arago had shown that refraction by a prism was in
fact unaffected by the earth’s annual motion. Whether or not Arago had reached the necessary
precision of 10−4, Fresnel took this result seriously and accounted for it by means of a partial
dragging of the ether within matter. His theory can be explained as follows.
According to an extension of Fermat’s principle, the trajectory that light takes to travel
between two fixed points (with respect to the earth) is that for which the traveling time is a
minimum, whether the medium of propagation is at rest or not. The velocity of light with respect
to the ether in a substance of optical index n is c/n, if c denotes the velocity of light. The absolute
velocity of the ether across this substance is αu, where α is the dragging coefficient and u is the
absolute velocity of the substance (the absolute velocity being that with respect to the remote,
undisturbed parts of the ether). Therefore, the velocity of light along the element dl of an arbitrary
2 J. Bradley, “A new apparent motion discovered in the fixed stars; its cause assigned; the velocity and equable
motion of light deduced,” Royal Society of London, Proceedings, 35 (1728), 308-321.
3 A. Fresnel, “Lettre d’Augustin Fresnel `a Fran¸cois Arago sur l’influence du mouvement terrestre dans quelques
ph´enom`enes d’optique,” Annales de chimie et de physique, 9(1818), also in Oeuvres compl`
etes, Paris (1868), vol. 2,
627-636.

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S´eminaire Poincar´e
Figure 1: Stellar aberration. Suppose that the position of a fixed star in the sky is judged by the
orientation of a narrow straight tube through which it can be seen. If the earth is moving with
respect to the fixed stars at the velocity u, the latter sweeps the distance uτ during the time τ that
the light from the star takes to travel from the beginning to the end of the tube. Therefore, the
true light path makes a small angle with the direction of the tube. When the velocity of the earth is
perpendicular to the tube, this angle is θ ≈ tanθ = u/c. Owing to the annual motion of the earth,
the apparent position of the star varies with a period of one year.
trajectory is c/n + (α − 1)u · dl/ds with respect to the substance (with ds = dl ). To first order
in u/c, the time taken by light during this elementary travel is
dt = (n/c)ds + (n2/c2)(1 − α)u · dl .
(5)
Note that the index n and the dragging coefficient in general vary along the path, whereas the
velocity u has the same value (the velocity of the earth) for the whole optical setting. The choice
α = 1 (complete drag) leaves the time dt and the trajectory of minimum time invariant, as should
obviously be the case. Fresnel’s choice,
α = 1 − 1/n2
(6)
yields
dt = (n/c)ds + (1/c2)u · dl ,
(7)
so that the time taken by light to travel between two fixed points of the optical setting differs only
by a constant from the time it would take if the earth were not moving. Therefore, the laws of
refraction are unaffected (to first order) under Fresnel’s assumption.4
In 1846, the Cambridge professor George Gabriel Stokes criticized Fresnel’s theory for making
the fantastic assumption that the huge mass of the earth was completely transparent to the ether
wind. In Stokes’ view, the ether was a jelly-like substance that behaved as an incompressible fluid
under the slow motion of immersed bodies but had rigidity under the very fast vibrations implied
in the propagation of light. In particular, he identified the motion of the ether around the earth
with that of a perfect liquid. From Lagrange, he knew that the flow induced by a moving solid
(starting from rest) in a perfect liquid is such that a potential exists for the velocity field. From his
4 Cf. E. Mascart, Trait´e d’optique, 3 vols. (Paris, 1893), vol. 3, chap. 15. Fresnel justified the value 1 − 1/n2 of
the dragging coefficient by making the density of the ether inversely proportional to the square of the propagation
velocity c/n (as should be in an elastic solid of constant elasticity) and requiring the flux of the ether to be conserved.
As Mascart noted in the 1870s, this justification fails when double refraction and dispersion are taken into account.

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The Genesis of the Theory of Relativity
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Figure 2: Fizeau’s experiment. After reflection on a semi-reflecting blade, the light from the source
S is divided into two beams. The upper beam travels against the water stream in A B , crosses the
lens L , is reflected on the mirror M, crosses L again, travels against the water stream in AB, and
returns to the semi-reflecting blade. The lower beam does the symmetrical trip, which runs twice
along the water stream. The phase difference between the two beams is judged from the interference
pattern in O.
recent derivation of the Navier-Stokes equation, he also knew that this property was equivalent to
the absence of instantaneous rotation of the fluid elements. Consequently, the propagation of light
remains rectilinear in the flowing ether, and the apparent position of stars in the sky is that given
by the usual theory of aberration.5
In order to account for the absence of effects of the earth’s motion on terrestrial optics,
Stokes further assumed that the ether adhered to the earth and had a negligible relative velocity
at reasonable distances from the ground.
To sum up, before the middle of the century, there were two competing theories of the optics
of moving bodies that both accounted for stellar aberration and for the absence of effects of the
earth’s motion on terrestrial optics. Fresnel’s theory assumed the stationary character of the ether
everywhere except in moving refractive media, in which a partial drag occurred. Stokes’ theory
assumed complete ether drag around the earth and irrotational flow at higher distances from the
earth.
In 1850 Hippolyte Fizeau performed an experiment in which he split a light beam into two
beams, had them travel through water moving in opposite directions, and measured their phase
difference by interference (see fig. 2). The result confirmed the partial drag of light waves pre-
dicted by Fresnel. Maxwell knew about Fizeau’s result, and, for a while, wrongly believed that it
implied an alteration of the laws of refraction by the earth’s motion through the ether. In 1864, he
performed an experiment to test this modification. The negative result confirmed Arago’s earlier
finding with improved precision. As Stokes explained to Maxwell, this result pleaded for, rather
than contradicted the Fresnel drag. Yet Maxwell remained skeptical about the validity of Fizeau’s
experiment. In 1867 he wrote:
This experiment seems rather to verify Fresnel’s theory of the ether; but the whole question of the state of the
luminiferous medium near the earth, and of its connexion with gross matter, is very far as yet from being settled
by experiment.
In this situation, it was too early to worry about an incompatibility between the electromagnetic
theory of light and the optics of moving bodies. In 1878, one year before his death, Maxwell still
judged Stokes’ theory “very probable.”6
5 G.G. Stokes, “On the aberration of light,” Philosophical magazine, 27(1845), 9-55; “On Fresnel’s theory of the
aberration of light,” ibid., 28 (1846), 76-81; “On the constitution of the luminiferous ether, viewed with reference
to the phenomenon of the aberration of light,” ibid., 29 (1846), 6-10. This result can be obtained from Fermat’s
principle, by noting that to first order the time taken by light to travel along the element of length dl has the form
dt = (1/c)ds − (1/c2)v · dl (v denoting the velocity of the ether), so that its integral differs only by a constant (the
difference of the velocity potentials at the end points) from the value it would have in a stationary ether.
6 H. Fizeau, “Sur les hypoth`eses relatives `a l’´ether lumineux, et sur une exp´erience qui paraˆıt d´emontrer que

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6
O. Darrigol
S´eminaire Poincar´e
Figure 3: The Michelson-Morley experiment. The light from the source S is divided into two beams
by the semi-reflecting blade R. After reflection on the mirrors M1 and M2, the two beams return
to R. Their interference pattern is observed through the telescope T.
In the 1870s a multitude of experiments confirmed the absence of effect of the earth’s motion
on terrestrial optics. In 1874, the author of the best of those, Eleuth`ere Mascart, concluded:
The translational motion of the earth has no appreciable influence on optical phenomena produced by a terrestrial
source, or light from the sun, so these phenomena do not provide us with a means of determining the absolute
motion of a body, and relative motions are the only ones that we are able to determine.
Mascart and other continental experts interpreted this finding by means of Fresnel’s theory. British
physicists mostly disagreed, as can be judged from a British Association report of 1885 in which
a disciple of Maxwell criticized “Fresnel’s somewhat violent assumptions on the relation between
the ether within and without a transparent body.”7
In 1881 the great American experimenter Albert Michelson conceived a way to decide between
Fresnel’s and Stokes’ competing theories. Through an interferometer of his own, he compared the
time that light took to travel the same length in orthogonal directions (see fig. 3). If the ether was
stationary, he reasoned, the duration of a round trip of the light in the arm parallel to the earth’s
motion was increased by a factor [l/(c − u) + l/(c + u)]/(2l/c), which is equal to 1/(1 − u2/c2).
The corresponding fringe shift was about twice what his interferometer could detect. From the null
result, Michelson concluded that Fresnel’s theory had to be abandoned.8
A French professor at the Ecole Polytechnique, Alfred Potier, told Michelson that he had
overlooked the increase of the light trip by 1/
1 − u2/c2 in the perpendicular arm of his interfer-
ometer. With this correction, the experiment became inconclusive. Following William Thomson’s
and Lord Rayleigh’s advice and with Edward Morley’s help, Michelson first decided to repeat
Fizeau’s experiment with his powerful interferometric technique. In 1886 he thus confirmed the
le mouvement des corps change la vitesse avec laquelle la lumi`ere se propage dans leur int´erieur,” Acad´emie des
Sciences, Comptes-rendus, 33 (1851), 349-355; J.C. Maxwell, “On an experiment to determine whether the motion
of the earth influences the refraction of light,” unpub. MS, in Maxwell, The scientific letters and papers, ed. Peter
Harman, vol. 2 (Cambridge, 1995), 148-153; Maxwell to Huggins, 10 Jun 1867, ibid., 306-311; “Ether,” article for
the Encyclopedia Britannica (1878), reproduced ibid., 763-775.
7 E. Mascart, “Sur les modifications qu’´eprouve la lumi`ere par suite du mouvement de la source et du mouvement
de l’observateur,” Annales de l’Ecole Normale, 3 (1874), 363-420, on 420; R.T. Glazebrook, Report on “optical
theories,” British Association for the Advancement of Science, Report (1885), 157-261.
8 A. Michelson, “The relative motion of the earth and the luminiferous ether,” American journal of science, 22
(1881), 120-129.

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Vol. 1, 2005
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Fresnel dragging coefficient with greatly improved precision.9
At this critical stage, the Dutch theorist Hendrik Lorentz entered the discussion. He first
blasted Stokes’ theory by noting that the irrotational motion of an incompressible fluid around a
sphere necessarily involves a finite slip on its surface.10 The theory could still be saved by integrating
Fresnel’s partial drag, but only at the price of making it globally more complicated than Fresnel’s.
Lorentz therefore favored Fresnel’s theory, and called for a repetition of Michelson’s experiment of
1881after noting the error already spotted by Potier. Michelson and Morley fulfilled this wish in
1887 with an improved interferometer. The result was again negative, to every expert’s puzzlement:
while Fizeau’s experiment confirmed Fresnel’s theory, the new experiment contradicted it.11
3
Lorentz’s theory
When in the early 1890s Hertz and Heaviside perfected Maxwell’s electrodynamics of moving
bodies, they noted that it was incompatible with Fresnel’s theory of aberration, but decided to
postpone further study of the relation between ether and matter. Unknown to them, Lorentz had
long ago reflected on this relation and reached conclusions that sharply departed from Maxwell’s
original ideas. Unlike Maxwell’s British disciples, Lorentz learned Maxwell’s theory in a reinter-
pretation by Hermann Helmholtz that accommodated the continental interpretation of charge,
current, and polarization in terms of the accumulation, flow, and displacement of electric particles.
In 1878 he gave a molecular theory of optical dispersion based on the idea of elastically bound
charged particles or “ions” that vibrated under the action of an incoming electromagnetic wave
and thus generated a secondary wave. For the sake of simplicity, he assumed that the ether around
the molecules and ions had exactly the same properties as the ether in a vacuum. He could thus
treat the interactions between ions and electromagnetic radiation through Maxwell’s equations in
a vacuum supplemented with the so-called Lorentz force.12
Using lower-case letters for the microscopic fields and Hertzian units, these equations read
× e = −c−1∂b/∂t ,
× b = c−1[ρmv + ∂e/∂t] ,
· e = ρm ,
· b = 0 ,
(8)
f = ρm[e + c−1v × b] ,
where ρm denotes the microscopic charge density (confined to the ions) and f denotes the density
of the force acting on the ions. Note that there are only two independent fields e and b since
the constants
and µ are set to their vacuum value. Although from a formal point of view these
equations can be seen as a particular case of the Maxwell-Hertz equations (1), they were unthinkable
to true Maxwellians who regarded the concepts of electric charge and polarization as emergent
macroscopic concepts and believed the molecular level to be directly ruled by the laws of mechanics.
Using his equations and averaging over a macroscopic volume element, Lorentz obtained
the first electromagnetic theory of dispersion. In 1892, he realized that he could perform similar
calculations when the dielectric globally moved through the ether at the velocity u of the earth.
He only had to assume that the ions and molecules moved through the ether without disturbing
it. Superposing the incoming wave and the secondary waves emitted by the moving ions, he found
that the resulting wave traveled at the velocity predicted by Fresnel’s theory. The partial ether drag
imagined by Fresnel was thus reduced to molecular interference in a perfectly stationary ether.13
9 A. Michelson and E. Morley, “Influence of the motion of the medium on the velocity of light”, American journal
of science, 31 (1886), 377-386.
10 It seems dubious that Stokes, as an expert on potential theory in fluid mechanics, could have overlooked this
point. More likely, his jelly-like ether permitted temporary departures from irrotationality.
11 H.A. Lorentz, “De l’influence du mouvement de la terre sur les ph´enom`enes lumineux,” Archives n´eerlandaises
(1887), also in Collected papers, 9 vols. (The hague, 1934-1936), vol. 4, 153-214; Michelson and Morley, “On the
relative motion of the earth and the luminiferous ether,” American journal of science, 34 (1887), 333-345.
12 Lorentz, “Over het verband tusschen de voortplantings sneldheit en samestelling der midden stofen,” Konin-
klijke Akademie van Wetenschappen, Verslagen (1878), transl. as “Concerning the relation between the velocity of
propagation of light and the density and composition of media” in Collected papers (ref. 11), vol. 2, 3-119.
13 Lorentz, “La th´eorie ´electromagn´etique de Maxwell et son application aux corps mouvants,” Archives
n´
eerlandaises (1892), also in Collected papers (ref. 11), vol. 2, 164-321.

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S´eminaire Poincar´e
Notwithstanding with their global intricacy, Lorentz’s original calculations contained an inter-
esting
subterfuge.
In
order
to
solve
equations
that
involved
the
wave
operator
∂2/∂x2 − c−2(∂/∂t − u∂/∂x)2 in a reference frame bound to the transparent body, Lorentz in-
troduced the auxiliary variables
x = γx ,
t = γ−1t − γux/c2
(9)
that restored the form of the operator in the ether-bound frame for
γ = 1/ 1 − u2/c2 .
(10)
He thus discovered the Lorentz transformation for coordinates (up to the Galilean transformation
x = ¯
x − ut, where ¯x is the abscissa in the ether frame).14
A few months later, Lorentz similarly realized that to first order in u/c the field equations
in a reference frame bound to the earth could be brought back to the form they have in the ether
frame through the transformations
t = t − ux/c2 , e = e + c−1u × b , b = b − c−1u × e .
(11)
In other words, the combination of these transformations with the Galilean transformation x =
¯
x − ut leaves the Maxwell-Lorentz equations invariant to first order. Lorentz used this remarkable
property to ease his derivation of the Fizeau coefficient and to give a general proof that to first
order optical phenomena were unaffected by the earth’s motion through the ether.15
It is important to understand that for Lorentz the transformed coordinates and fields were
mathematical aids with no direct physical significance. They were only introduced to facilitate
the solution of complicated differential equations. The “local time” t was only called so be-
cause it depended on the abscissa. The true physical quantities were the absolute time t and
the fields e and b representing the states of the ether. In order to prove the first-order invari-
ance of optical phenomena, Lorentz considered two systems of bodies of identical constitution, one
at rest in the ether, the other drifting at the velocity u. He first noted that to a field pattern
e0 = F (x, y, z, t), b0 = G(x, y, z, t) for the system at rest corresponded a field pattern e, b for the
drifting system such that e = F (x, y, z, t ), b = G(x, y, z, t ) (the abscissa x being measured in a
frame bound to the system). He then noted that e and b vanished simultaneously if and only if
e and b did so. Consequently, the borders of a ray of light or the dark fringes of an interference
pattern have the same locations in the system at rest and in the drifting system. The change of
the time variable is irrelevant, since the patterns observed in optical experiments are stationary.
We may conclude that Lorentz’s use of the Lorentz invariance was quite indirect and subtle.
There remained a last challenge for Lorentz: to account for the negative result of the
Michelson-Morley experiment of 1887. As George Francis FitzGerald had already done, Lorentz
noted that the frange shift expected in a stationary ether theory disappeared if the longitudinal
arm of the interferometer underwent a contraction by the amount γ−1 =
1 − u2/c2 when moving
through the ether. In order to justify this hypothesis, Lorentz first noted that in the case of elec-
trostatics the field equations in a frame bound to the drifting body could be brought back to those
for a body at rest through the transformation x = γx. He further assumed that the equilibrium
length or a rigid rod was determined by the value of intermolecular forces and that these forces
all behaved like electrostatic forces when the rod drifted through the ether. Then the fictitious rod
obtained by applying the dilation x = γx to a longitudinally drifting rod must have the length that
this rod would have if it were at rest. Consequently, the moving rod contracts by the amount γ−1.
The Lorentz contraction thus appears to result from a postulated similarity between molecular
forces of cohesion and electrostatic forces.16
14 Ibid. : 297
15 Lorentz, “On the reflexion of light by moving bodies,” Koninklijke Akademie van Wetenschappen, Verslagen
(1892), also in Collected papers (ref.11), vol. 4, 215-218.
16 Lorentz, “De relative beweging van der aarde en den aether,” Koninklijke Akademie van Wetenschappen, Ver-
slagen (1892), transl. as “The relative motion of the earth and the ether” in Collected papers (ref. 11), vol. 4,
220-223.

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Fully explained in the Versuch of 1895, Lorentz’s theory gained broad recognition before
the end of the century. Two other physicists, Joseph Larmor of Cambridge and Emil Wiechert
of K¨
onigsberg, proposed similar theories in the same period. In the three theories, the basic idea
was to hybridize Maxwell’s theory with the corpuscular concept of electricity and to reduce every
optic and electromagnetic phenomenon to the interactions between electric particles through a sta-
tionary ether. Besides the optics of moving bodies, these theories explained a variety of magnetic
and magneto-optic phenomena, and of course retrieved the confirmed predictions of Maxwell’s
theory. They benefited from the contemporary rise of an experimental microphysics, including
the discoveries of x-rays (1895), radioactivity (1896), and the electron (1897). In 1896, the Dutch
experimenter Pieter Zeeman revealed the magnetic splitting of spectral lines, which Lorentz imme-
diately explained through the precession of the orbiting charged particles responsible for the lines.
Being much lighter than hydrogen, these particles were soon identified to the corpuscle discovered
in cathode rays by Emil Wiechert and Joseph John Thomson. Following Larmor’s terminology,
this corpuscle became known as the electron and replaced the ions in Lorentz’s theory.17
4
Poincar´
e’s criticism
In France, the mathematician Henri Poincar´e had been teaching electrodynamics at the Sorbonne
for several years. After reviewing the theories of Maxwell, Helmholtz, Hertz, Larmor, and Lorentz,
he judged that the latter was the one that best accounted for the whole range of optic and electro-
magnetic phenomena. Yet he was not entirely satisfied with Lorentz’s theory, because he believed
it contradicted fundamental principles of physics. In general, Poincar´e perceived an evolution of
physics from the search of ultimate mechanisms to a “physique des principes” in which a few general
principles served as guides in the formation of theories. Among these principles were three general
principles of mechanics: the principle of relativity, the principle of reaction, and the principle of
least action.18
For any believer in the mechanical nature of the electromagnetic ether, it was obvious that
these three principles applied to electrodynamics, since ether and matter were together regarded as
a complex mechanical system. In particular, it was clear that electromagnetic phenomena would be
the same if the same uniform boost was applied to the ether and all material objects. It the boost
was applied to matter only, effects of this boost were expected to occur. For instance, Maxwell
believed that the force between two electric charges moving together uniformly on parallel lines had
to vanish when their velocity reached the velocity of light. Poincar´e thought differently. In his view,
the ether only was a convenient convention suggested by the analogy between the propagation of
sound and the propagation of light. In the foreword of his lectures of his lecture of 1887/8 on the
mathematical theories of light, he wrote:19
It matters little whether the ether really exists: that is the affair of the metaphysicians. The essential thing for us
is that everything happens as if it existed, and that this hypothesis is convenient for us for the explanation of the
phenomena. After all, have we any other reason to believe in the existence of material objects? That too, is only
a convenient hypothesis; only this will never cease to do so, whereas, no doubt, some day the ether will be thrown
aside as useless.
As we will see, Poincar´e actually never abandoned the ether. But he refused to regard it as an
ordinary kind of matter whose motion could affect observed phenomena. In his view, the principle
of reaction and the principle of relativity had to apply to matter alone. In his lectures of 1899 on
Lorentz’s theory, he wrote: I consider it very probable that optical phenomena depend only on the
relative motion of the material bodies present –light sources and apparatus– and this not only to
first or second order but exactly.
17 Lorentz, Versuch einer Theorie der elektrischen un optischen Erscheinungen in bewegten K¨orpern (Leiden,
1895), also in Collected papers (ref. 11), vol.5, 1-139; “Optische verschinitjnelsen die met de lading en de massa der
ionen in verband staan,” Koninklijke Akademie van Wetenschappen, Verslagen (1898), transl. as “Optical phenomena
connected with the charge and mass of ions” in Collected papers (ref. 11) vol. 3, 17-39.
18 H. Poincar´e, Electricit´e et optique. La Lumi`ere et les th´eories ´electrodynamiques [Sorbonne lectures of 1888,
1890 and 1899)], ed. J. Blondin and E. N´eculc´ea, (Paris, 1901).
19 Poincar´e, Th´eorie math´ematique de la lumi`ere (Sorbonne lectures, 1887-88), ed. J. Blondin (Paris, 1889), I.

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10
O. Darrigol
S´eminaire Poincar´e
Figure 4: Cross-signaling between two observers moving at the velocity u through the ether. The
points A, A , A , B, B , B represent the successive positions of the observers in the ether when
the first observer sends a light signal, when the second observer receives this signal and sends back
another signal, and when the first observer receives the latter signal.
It must be emphasized that at that time no other physicist believed in this acceptance of the
relativity principle. Most physicists conceived the ether as a physical entity whose wind should have
physical effects, even though the precision needed to test this consequence was not yet available.
The few physicists, such as Paul Drude or Emil Cohn, who questioned the mechanical ether, felt
free to violate principles of mechanics, including the relativity principle.20
Lorentz’s theory satisfied Poincar´e’s relativity principle only approximately and did so through
what Poincar´e called two “coups de pouce”: the local time and the Lorentz contraction. Moreover,
it violated Poincar´e’s reaction principle, since Lorentz’s equations implied that the net force acting
on all the ions or electrons should be the space integral of ∂(e × b)/c∂t, which does not vanish
in general. In his contribution to Lorentz’s jubilee of 1900, Poincar´e iterated this criticism and
further discussed the nature and impact of the violation of the reaction principle. In the course
of this argument, about which more details will be given in a moment, he relied on Lorentz’s
transformations (11) to compute the energy of a pulse of electromagnetic radiation from the stand-
point of a moving observer. The transformed fields e and b , he noted, are the fields measured
by a moving observer. Indeed the force acting on a test unit charge moving with the velocity u is
e + c−1u × b = e according to the Lorentz force formula. Poincar´e went on noting that the local
time t = t − ux/c2 was that measured by moving observers if they synchronized their clocks in
the following manner:21
I suppose that observers placed in different points set their watches by means of optical signals; that they try to
correct these signals by the transmission time, but that, ignoring their translational motion and thus believing that
the signals travel at the same speed in both directions, they content themselves with crossing the observations, by
sending one signal from A to B, then another from B to A.
Poincar´e only made this remark en passant, gave no proof, and did as if it had already been
on Lorentz’s mind. The proof goes as follows. When B receives the signal from A, he sets his watch
to zero (for example), and immediately sends back a signal to A. When A receives the latter signal,
he notes the time τ that has elapsed since he sent his own signal, and sets his watch to the time
τ /2. By doing so he commits an error τ /2 − t , where t is the time that light really takes to
−
−
travel from B to A. This time, and that of the reciprocal travel are given by t = AB/(c + u) and
−
t+ = AB/(c − u), since the velocity of light is c with respect to the ether (see fig. 4). The time
τ is the sum of these two traveling times. Therefore, to first order in u/c the error committed in
setting the watch A is τ /2 − t = (t
)/2 = uAB/c2. At a given instant of the true time, the
−
+ − t−
times indicated by the two clocks differ by uAB/c2, in conformity with Lorentz’ expression of the
local time.
Poincar´e transposed this synchronization procedure from an earlier discussion on the mea-
surement of time, published in 1898. There he noted that the dating of astronomical events was
20 Poincar´e, ref. 18, 536.
21 Poincar´e, “La th´eorie de Lorentz et le principe de la r´eaction.” In Recueil de travaux offerts par les auteurs `a
H.A. Lorentz `
a l’occasion du 25`eme anniversaire de son doctorat le 11 d´ecembre 1900, Archives n´
eerlandaises, 5
(1900), 252-278, on 272.

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