Henri Poincaré and the Theory of Relativity
Edited by Jürgen Renn, Berlin: WileyVCH, 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 (18541912), the other by a young patent examiner in Bern, Albert Einstein (18791955). 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 nonEuclidean geometry, as he noticed that the same relation exists between Fuchsian functions and nonEuclidean (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 “threebody 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 BarrowGreen, Poincaré and the ThreeBody 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 (18441906), 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 notetakers, 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 (18311879). 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 (18531928) 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 lightsynchronized 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 twentyfive 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 moreorless 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 openended 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 fourdimensional 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 (18641909) 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 (18781936), 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 A. Walter. Breaking in the 4vectors: the fourdimensional movement in gravitation, 19051910, 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, 193252 (Springer, 2007).