**Essay**
# Henri Poincaré and the Theory of Relativity

### Scott Walter

scott.walter [at] univ-lorraine.fr

### In *Albert Einstein, Chief Engineer of the Universe: 100
Authors for Einstein*, 162-165.

Edited by Jürgen
Renn, Berlin: Wiley-VCH, 2005.

In the month of June, 1905, the theory of relativity
came to light in the form of two scientific papers, one by the French
mathematician Henri Poincaré (1854-1912), the other by a young patent
examiner in Bern, Albert Einstein (1879-1955). The two scientists had
never met, although Einstein was familiar with elements of both the
science and the philosophy of Poincaré, whose name was celebrated in
scientific circles.
Much like Einstein, Poincaré established his scientific credentials at
an early age. In 1880 he proved the existence of a large class of
automorphic functions he named Fuchsian functions. In doing so, he made
an innovative employment of non-Euclidean geometry, as he noticed that
the same relation exists between Fuchsian functions and non-Euclidean
(hyperbolic) geometry, on one hand, and between certain elliptic
functions and Euclidean geometry, on the other hand.
The following year, Poincaré was named Assistant Professor of
Analysis at the University of Paris, becoming Professor of
Mathematical Physics in 1886, and a member of the French Academy of
Science in 1887. His scientific contributions remained little known
outside of mathematics until 1889, when he was awarded the Grand Prize
of King Oscar II of Sweden for his study of a thorny question in
celestial mechanics known as the "three-body problem": how do three
masses behave under the influence of gravitation? The revised
version of his study is a milestone in the history of both celestial
mechanics and dynamics, although some of its more profound insights
lay fallow for decades. For instance, Poincaré provided the first
mathematical description of what is now known as chaotic motion (cf.
June Barrow-Green, *Poincaré and the Three-Body Problem*, 1997).
Other results in the prize paper were rapidly assimilated, including
Poincaré's Recurrence Theorem, which states (roughly) that a closed
mechanical system with finite energy (like that of three planets
gravitating in empty space according to Newton's law) will return
periodically to a state very close to its initial state. The
consequences of this theorem were significant for the molecular
foundation of the Second Law of Thermodynamics (according to which
entropy increases over time for any closed system). Under the reading
argued for by Ludwig Boltzmann (1844-1906), the Second Law is
consistent with Poincaré's theorem, but must be understood as a
probabilistic truth, i.e., one allowing for the occasional period of
decreasing entropy.
Poincaré lectured on all aspects of physics, in an elegant, abstract
style quite different from that practiced elsewhere. With the aid of
student note-takers, he published fifteen volumes of lectures, four of
which were translated into German, including his course on the theory
of electromagnetism by James Clerk Maxwell (1831-1879). After ten
years, Poincaré relinquished his chair in mathematical
physics for another in mathematical astronomy and celestial mechanics,
although on occasion he would still lecture on questions of physics.
Poincaré's lectures of 1899 provide an example of this, as they took
up recent theories that promised to address certain lacunae of
Maxwell's theory, including an adequate explanation of the
electrodynamics of moving bodies. Whereas Maxwell's theory dealt with
continuous macroscopic fields, the theory of Hendrik A. Lorentz
(1853-1928) was based on the notion of elementary charged particles
called electrons, the existence of which had been experimentally
confirmed two years earlier.
Impressed with the ability of Lorentz's theory to explain a curious
splitting of spectral lines in a strong magnetic field (the Zeeman
effect), Poincaré considered this theory to be the best one available,
and in 1900 took it upon himself to eliminate what he saw as its major
defect: the contradiction of Newton's Third Law (for every action
there is an opposite and equal reaction). In doing so, Poincaré noted
that in order for the principle of relative motion to hold, it was
necessary to refer time measurements not to the "true time" of an
observer at rest with respect to a universal, motionless carrier of
electromagnetic waves known as the ether, but to a "local time"
devised by Lorentz as a technical shortcut. For Poincaré, local time
had a real, operational meaning: it was the time read by the
light-synchronized clocks of observers in common motion with respect
to the ether, corrected by the light signal's time of flight, but
ignoring the effect of motion on light propagation.
There was more to this exchange of light signals than the utilitarian
synchronization of clocks, as Poincaré noted in a philosophical essay
entitled "The measurement of time" (1898). Passing over in silence
twenty-five centuries of metaphysical debates, Poincaré proposed that
his readers examine how working scientists seek to establish the
simultaneity of two events. The definition of time adopted by
astronomers, for example, was merely the most convenient one for their
purposes, and in general, he wrote, "no given way of measuring time is more
*true* than another." The notion of the simultaneity of two
events, Poincaré concluded, is not determined by objective
considerations, but is a matter of definition.
Likewise, Poincaré wrote on a different occasion, there is no absolute
space, and it is senseless to speak of the actual geometry of physical
space. Contrary to Poincaré's doctrine, astronomers thought they could
measure the curvature of space, at least in principle, and thereby
determine the geometry of physical space. For Poincaré, any physical
measurement necessarily entailed *both* geometry and physics,
because the objects of geometry-points, lines, planes-are
abstract, and any external use of geometry requires a more-or-less
arbitrary identification of these objects with physical phenomena, for
example, that of equating straight lines with light rays. According to
Poincaré's conventionalist philosophy of science, scientists are often
confronted with open-ended situations requiring them to choose between
alternative definitions of their objects of study. In virtue of this
freedom of choice, which marks the linguistic turn in philosophy of
science, Poincaré was often thought to be upholding a variety of
nominalism, an error he vigorously denounced. The choice scientists
have to make, Poincaré explained, is not entirely free: in
establishing a convention, scientists are "guided by experimental facts."
The conventionalist philosophy of science gained greater recognition
upon publication of a collection of Poincaré's essays, *La
science et l'hypothèse* (1902), a work promptly translated into
several languages. Among its early readers were the members of the
"Olympia Academy" in Bern, made up of Einstein and his friends
Conrad Habicht and Maurice Solovine. Einstein also read Poincaré's
memoir of 1900 on Lorentz's electron theory, with the operational
definition of local time via clock synchronization, although he may
well have read this only after writing his first relativity paper of
1905. Einstein's paper features a penetrating analysis of the notion
of simultaneity, wholly consistent with that of Poincaré, up to and
including the procedure for synchronizing clocks in different
locations by exchanging light signals. This particular analysis
effectively reinforced the idea that Einstein's was a *kinematic*
theory, and thereby more fundamental than any given theory of
mechanics or electrodynamics.
Einstein's approach to relativity contrasts sharply in this respect
with that of Poincaré, who neglected any mention of clock
synchronization in his own relativity paper of 1905 (although he
reviewed the topic a few years later). Nonetheless, Einstein's
mathematical results agree precisely with those of Poincaré; in
particular, both scientists derived the relativistic velocity
composition law from the coordinate transformations connecting two
inertial systems (christened Lorentz transformations by Poincaré).
Likewise for the empirical consequences of the two papers, which are
identical, with one exception.
A significant difference between the relativity papers of
Einstein and Poincaré concerns the status of gravitational phenomena.
This subject was neglected by Einstein, although it had been
identified by Poincaré in his September 24, 1904 address to the
scientific congress of the World's Fair in Saint Louis as a potential
spoiler for the principle of relativity. As Professor of Mathematical
Astronomy and Celestial Mechanics, Poincaré could hardly pretend that
gravitation did not exist, and instead formulated a pair of
relativistic laws of gravitation, the first of their kind.
The laws themselves were observationally on a par with that of Newton,
but hardly compelling on physical grounds; they retain interest,
nonetheless, in virtue of their form, and the way in which Poincaré
derived them. Poincaré introduced a four-dimensional space, in which
three (spatial) dimensions are real, and the fourth (temporal)
dimension is imaginary, such that coordinate rotations about the
origin correspond to Lorentz transformations. Three years later, the
mathematician Hermann Minkowski (1864-1909) extended Poincaré's
geometrical approach in his theory of spacetime, and
used this to express two relativistic laws of gravitation of his own.
Neither Einstein nor Poincaré let himself be impressed by Minkowski's
sophisticated spacetime theory, but shortly after Poincaré died,
Einstein changed his mind, and with the help of his friend, the
mathematician Marcel Großmann (1878-1936), like himself a former
student of Minkowski's, adopted it in view of a radically new
approach to gravitational attraction. Three years later, in November
1915, Einstein discovered the field equations of general relativity,
according to which the geometry of spacetime is curved in the presence
of matter. Had Einstein finally disproved Poincaré's doctrine of
space? Poincaré, that "astute and profound" thinker, Einstein wrote,
was right, "sub specie aeterni" (Geometrie und Erfahrung, 1921).
Nevertheless, Einstein continued, his own point of view was required
by the current state of theoretical physics.

**Bibliography**
Olivier Darrigol. *Electrodynamics from Ampère to Einstein*
(Oxford University Press, 2000).
Peter Galison. *Einstein's Clocks, Poincaré's Maps* (Norton,
2003).
Scott Walter. Breaking in the 4-vectors: the four-dimensional
movement in gravitation, 1905-1910, in Jürgen Renn and Matthias
Schemmel (eds.), *The Genesis of General Relativity, Volume 3,
Gravitation in the Twilight of Classical Physics: Between Mechanics,
Field Theory, and Astronomy,* 193-252 (Springer, 2007).