Einstein's Clocks and Poincare's Maps

by Peter Galison

  From Paris, Poincaré echoed back Einstein’s silence. Einstein was no doubt a young unknown to him in 1905; there is really no need to explain the absence of any reference to Einstein in Poincaré’s 1906 treatise, “On the Dynamics of the Electron.” But later, although he and Einstein separately published often on questions of time, space, and the relativity principle, Poincaré’s silence continued for seven more years. With Einstein’s name on everyone’s lips—including Lorentz’s, Minkowski’s, Laue’s, and Planck’s—this surely is not happenstance. To Einstein, Poincaré must have seemed beside the point: another older physicist who in 1905 was unable to grasp the import of Einstein’s banishment of the ether or his placement of time, without making the true/apparent distinction, at the starting point of the theory. To Poincaré, Einstein must have seemed derivative, perhaps a provider of heuristic arguments to derive the Lorentz transformations, but one who failed even to address fundamental issues of physics: the ether and the structure of the electron.

  Yet just as Einstein was in no deep sense derivative, Poincaré could not be dismissed as a mere conservative. On the contrary Poincaré proudly hailed the electrodynamics of moving bodies as a new mechanics when he spoke at Göttingen in 1909; even in St. Louis in 1904 he had heralded dramatic changes throughout physics. At the same time, Poincaré’s 1909 presentation at Lille conveyed a certain wistfulness as he invoked a classical physics whose marble columns had begun to crack: “If some part of science appears solidly established, it would certainly be Newtonian mechanics; we lean on it with confidence, and it does not seem that it could ever be weakened. But scientific theories are like empires, and if Bossuet was here, he would no doubt find eloquent accents in which to denounce their fragility.”3 Jacques-Bénigne Bossuet, teacher of the Dauphin, son of Louis XIV, had written for the royal charge his 1681 Discourse on Universal History. Long central within the canon of French literature, Bossuet’s History occupied a featured place in a 1912 volume that Poincaré assembled with colleagues from the Académie Française. Aimed at educating the public in matters both scientific and literary, the volume excerpted just that part of Bossuet’s message that cautioned a prideful humanity: “So you see pass before your eyes, as if in an instant, I say not kings and emperors but those grand empires that had made the whole universe tremble? When you see the Assyrians, ancient and modern, the Medes, the Persians, the Greeks and the Romans appear before you one after the other and fall, one on top of the other, so to speak, this terrifying fracas makes you feel that there is nothing solid among men and that inconstancy and agitation are the proper fate of things human.”4 For Poincaré, scientific theories were like Bossuet’s grand empires. For decades he had mined the confrontation among empires of time, crossing the conventions that moderated the collision with a new philosophy and physics. Yet as he peered over the two great structures to which he had devoted so much of his life—the Newtonian empire and the French one—Poincaré now saw them through the eyes of a man who had registered the fragility of both.

  Einstein and Poincaré finally met—for the first and last time—at the Solvay Conference held in Brussels late in 1911. They were two scientists infinitely close and yet infinitely distant. Both had been fascinated since youth by the problem of the electrodynamics of moving bodies. Both had productively built their theories at the crossroads of technology, philosophy, and physics. Both understood the immense power of Lorentz’s work and both underscored the group structure of Lorentz’s transformations. Both seized upon the principle of relativity as a constitutive foundation of physics. Perhaps most dramatically, both insisted that time in moving frames had to be interpreted by way of clocks synchronized by light-signal exchange. But the distance between the two scientists was as dramatic as their proximity. Poincaré viewed his contributions as a form of world repair, an adjustment, a turning, a rewriting of Lorentzian physics into a new mechanics that he viewed with exhilaration and trepidation. For the young Einstein, repair held little appeal. Tearing down the old was a bracing pleasure. While Poincaré maintained the ether as crucial in his 1909 Lille address, Einstein began a talk of his own at almost exactly the same time with a specific reference to a physicist (not Poincaré) who had assessed the ether’s existence to “border on certainty.” Then Einstein knocked the author’s assertion into the trash.5

  For many reasons it is not surprising that the encounter between the fifty-seven year old and the thirty-two year old did not go well. Maurice de Broglie: “I remember one day at Brussels, while Einstein was explaining his ideas, Poincaré asked him, ‘what mechanics are you using in your reasoning?’ Einstein answered: ‘No mechanics’ which appeared to surprise his interlocutor.”6 “Surprise” is perhaps an understatement. For Poincaré, whose conception of physics came down to mechanics, be it old or new, “no mechanics” was an impossible response. Abstract mechanics, after all, was what Poincaré had held up to his fellow Polytechnicians as constituting the very essence, the “factory stamp” of that unique training for the world of Third Republic France.

  Einstein spoke to his Solvay audience about light quanta and quantum discontinuities, following which the extraordinarily distinguished audience began to debate. Lorentz was there; so was Poincaré. Einstein initiated the discussion by reminding his listeners that the theory of quanta in its present form really wasn’t a theory at all “in the ordinary sense of the word,” but a useful tool. Einstein surely did not see his comments as a detailed mathematical description of the type Poincaré would dignify with the term “mechanics.” (Einstein had called his account of light quanta a “heuristic” when he had first published it six years earlier.) Rather, Einstein hoped to offer a starting point for a later, more coherent treatment. In the meantime, he had willingly sacrificed the continuous, causal relations that differential equations captured. But that coolness toward the foundation of an intuitive, mathematically describable mechanics was for Poincaré no small matter. Summarizing the conference from his vantage point, Poincaré spoke without minced words. One hears in his remarks a deeply troubled assessment of where the new physics, and the new physicists, were heading:

  What the new research seems to throw into question is not only the fundamental principles of mechanics, it is something that appeared to us till now inseparable from the very concept of a natural law. Could we still express these laws in the form of differential equations?

  Besides, what struck me in the discussions that we just heard, is seeing the same theory sometimes relying on the old mechanics and sometimes on new hypotheses that negate them; one must not forget that there isn’t a proposition that one can’t easily prove insofar as one inserts into the demonstration two contradictory premises.7

  Here is the pent-up frustration of a master of the “old mechanics,” the consternation of someone who had used differential equations to explore, extend, test, and (against his own intent) upend the stability and visualizability of Newtonian physics. It is the voice of a savant who has pushed toward a “new mechanics” in every way he knew. Similarly, Poincaré had significantly advanced the Lorentz theory, “turning it in every direction” as he had said so often. In the process, bit by bit, he had grappled for years with changed notions of mass, length, and most dramatically, time itself. More than anyone, he insisted both in his mathematics and in his philosophy that one could switch, according to circumstance, from Euclidian to non-Euclidean language. Shortly after Solvay, Poincaré even advanced understanding of the new and unsettling quantum discontinuity.

  In Poincaré we face anyone but a change-abhoring conservative. But he contended that there were better and worse ways of handling innovation. The new physics, Poincaré was saying, had lost its way by abandoning, in its frantic race ahead, the consistent and principled base of any—of all—mechanics. The “honest functions” and the differential equations that vouchsafed causality and intuition had been lost. This was not a dispute over which law best fit the phenomena. It was a gulf that, for Poincaré, left Einstein and his supporters o
n the wrong side of “the very concept of natural law.” Borrowing Poincaré’s own earlier phrase, Einstein’s quantum physics seemed less well suited for science than for a teratological museum.

  Einstein’s reaction to the Solvay encounter with Poincaré was swift and unflattering. A few weeks after Solvay, he confided his views to a friend: “H. A. Lorentz is a marvel of intelligence and tact. He is a living work of art! In my opinion he was the most intelligent among the theoreticians present. Poincaré was simply negative in general, and, all his acumen notwithstanding, he showed little grasp of the situation.”8 There was certainly no meeting of minds between them over the relativity theory. Their split over the quantum widened the chasm.

  Yet Poincaré returned to Paris from Solvay in 1911 deeply impressed by Einstein. That November, Einstein, having only recently moved to Prague, was a candidate for a position at his alma mater, the Swiss Federal Institute of Technology. Setting aside any disquiet at Einstein’s disturbing radicality, Poincaré intervened in his favor, assuring physicist Pierre Weiss that Einstein was “one of the most original minds I have known.” Youth mattered little; the senior scientist judged Einstein to have “already taken a very honorable rank among the leading scholars of his time. What we must above all admire in him, is the facility with which he has adapted to new conceptions and from which he knows how to draw the consequences. He does not remain attached to classical principles, and, in the presence of a problem of physics, is prompt to envision all the possibilities. This translates immediately in his mind into the prediction of new phenomena, susceptible of being one day verified by experiment. I do not want to say that all these predictions will remain impervious to the judgment of experiment when that judgment becomes possible. As he searches in all directions, one must, on the contrary, expect that the majority of the paths in which he embarks will be dead ends; but one must, at the same time, hope that one of the directions that he has indicated will be the right one; and that suffices. It is just so that one must proceed. The role of mathematical physics is to pose questions properly, it is only experiment that can resolve them.” This was a letter of the highest praise. “The future will show more and more the value of Mr. Einstein,” Poincaré concluded, “and the university that finds a way to secure this young master is assured of drawing from it great honor.”9

  Beyond this statesmanly letter, it is impossible to say precisely what effect Poincaré’s one meeting with Einstein had on the senior physicist. Poincaré’s health was deteriorating, and he may at this time of enormous productivity also have had intimations of his own mortality. It may also be that Poincaré’s later reflections upon Einstein’s jarring new vision of physics at Solvay prompted the mathematician to think further on the value of the provisional, heuristic, result-oriented efforts that Einstein had employed to such effect. Just a few weeks after recommending Einstein to Weiss, on 11 December 1911, Poincaré wrote to the founding editor of Circolo matematico of Palermo. Still working on the three-body problem with which he had launched his career decades earlier, Poincaré reported to his correspondent that for two long years he had been struggling with the problem without much progress. He now had to pause, at least temporarily. “It would be fine if I could be sure of being able to take it up again; at my age, I cannot vouch for that, and the results obtained, liable to put researchers on a new and unexplored track, seem to me too full of promises, despite the disappointments that they have caused me, for me to resign myself to sacrificing them.” At fifty-seven, Poincaré was hardly old, but just a few years before, he had needed major prostate surgery. Would the editor be willing to publish an incomplete work, one that would state the problem and report partial results? (He would.) “What embarrasses me is that I will be obliged to put in a lot of figures, precisely because I could not arrive at a general rule, but I only accumulated particular solutions.” As Poincaré had so often insisted, visual-geometrical intuition could go where skeletal algebra could not yet tread.

  Poincaré judged these particular solutions to his lifelong problem “useful.” They were more than that; the paper contributed foundational ideas to the establishment of topology, a new branch of mathematics. Soon a young American mathematician, George D. Birkhoff, proved the crucial conjecture that lay at the core of Poincaré’s exploration.10

  Perhaps tacit echoes of Einstein may be audible in the last speech Poincaré gave on relativity, though the presentation never mentioned Einstein’s name. On 4 May 1912, Poincaré spoke on “Space and Time” to an audience at the University of London. In forceful terms, he once again repeated: “The properties of time are therefore merely those of our clocks just as the properties of space are merely those of the measuring instruments.”11 Over the past years, the ether had grown ever thinner in Poincaré’s writings, as its role in the theory dwindled. Now, however, the all-pervasive substance simply evaporated into silence. No rejection—but no mention of it, either. In his peroration, Poincaré let the old mechanical “principle of relativity” fall away, replaced by the “principle of relativity according to Lorentz.” Events simultaneous according to clocks coordinated in one frame of reference would not be simultaneous if measured by clocks coordinated in another.

  Does this mean that Poincaré had abandoned the ether or become a thoroughgoing Einsteinian? No. Having begun his presentation by asking if his earlier conclusions about space and time now needed to be revised in light of recent developments, he replied: “Certainly not; we had adopted a convention because it seemed convenient and we had said nothing could constrain us to abandon it.” But conventions are not God-given.

  Today some physicists want to adopt a new convention. It is not that they are constrained to do so; they consider this new convention more convenient; that is all. And those who are not of this opinion can legitimately retain the old one in order not to disturb their old habits. I believe, just between us, that this is what they shall do for a long time to come.12

  Poincaré’s time to come was short. His medical difficulties came more frequently and more severely. Nonetheless, when asked to accept the presidency of The French League of Moral Education and deliver its founding address on 26 June 1912, Poincaré characteristically accepted. For him, scientific prestige was inextricably associated with civic leadership and responsibility. In the midst of battles between anticlerical and clerical movements, in the face of an escalating conflict with the Germans in North Africa, even as the walls of Paris were plastered with partisan appeals, Poincaré sought principles that would undergird a unifying French morality. Against those ready to manipulate hatreds, Poincaré saw discipline as the only defense. Discipline—morality—was all that secured mankind against “an abyss of sufferings.” “Mankind is . . . like an army at war,” an army that must prepare for battle in peacetime, not at the last, too-late moment of engagement with the enemy. Hatred could propel collisions among men, collisions that risked changing their faith. “What will happen if the new ideas which they adopt are those which their former teachers conveyed to them as the very negation of morality? Can this mental habit be lost in one day? . . . Too old to acquire a new education, they shall lose the fruits of the old!”13 In morality, in physics, in mathematics, Poincaré wanted to build dramatic new structures, but he wanted to do so using the old bricks; he would employ, not discard, the legacy of an illustrious past.

  Poincaré underwent surgery again on 9 July 1912, and for a few days friends and family hoped for a recovery. It never came; Poincaré died, following an embolism, on 17 July 1912. Dozens of éloges appeared across the world. Perhaps the most fitting monument was the most anonymous: later that year the Eiffel Tower began radiating its precision time signal. Those pulses bathed the world in an expanding sphere of Hertzian light, fixing simultaneity (and longitude) into Africa and across the Atlantic to North America on the basis of techniques that Poincaré had introduced into geodesy, epistemology, and physics.

  Two Modernisms

  Reflecting back on Henri Poincaré in 1954, Prince
Louis de Broglie, the physicist who had shown that particles could act like waves, lamented that the great mathematician had just missed being the first to develop the theory of relativity in all its generality, “thus gaining for France the glory of that discovery.” “It is impossible to be closer to the thought of Einstein,” de Broglie judged. “And yet Poincaré did not take the decisive step; he left to Einstein the glory of grasping all the consequences of the principle of relativity and, in particular, of establishing, by a profound criticism of the measure of lengths and durations, the true physical character of the relation that the principle of relativity has between space and time. Why did Poincaré not come to the end of his thought? It is no doubt the turn, a little too critical, of his spirit, due perhaps to his education as pure mathematician. . . .” For de Broglie, it was Poincaré’s training as a mathematician that had led him to see science as no more than the informed and expeditious choice, on the basis of convenience, of one theory from among all logically equivalent ones. According to de Broglie, Poincaré failed to tread the better path laid down by the physicist’s intuition.14 By de Broglie’s lights, Poincaré was too much a mathematician, too indifferent to the real world to have formulated relativity as Einstein did.

  My view? De Broglie’s diagnosis is far too narrow. I would argue that Poincaré did come to the “end of his thought,” to an image of knowledge—including his view of mathematical knowledge—that carried with it a nineteenth-century optimism, a Third Republic Polytechnician’s engaged, hopeful vision of a calculable, improvable, rational world. If anything, Poincaré paid too much attention to the real world: when he judged in 1898–99 that the corrections to Newtonian time were in principle necessary but too small to matter, it was because at that moment he was assessing the light-signal “relativistic” errors against the real-world “ordinary” longitude-timing errors. Yet to call Poincaré’s approach “conservative” or “reactionary” is to miss the point; Poincaré’s sight-line aimed directly toward the ideals of a revolutionary Enlightenment that, by century’s end (Poincaré’s time), had grown into institutionalized French empire. All our great constructions eventually crack, Poincaré says on many occasions. But our response to these fissures, these crises, should not be the mysticism or the melancholy of the intellectual elites, but instead a redoubled effort to repair those breaks by the systematic application of reasoned action. As Poincaré saw it, the scientist-engineer could apply analytical reason as readily to the understanding of a coal-mine accident as to planetary motion, as easily to the mapping of the world as to the reconstruction of Lorentz’s theory of moving electrons.