Einstein's Clocks and Poincare's Maps
Page 20
Even in mathematics, machines and mechanomorphic structures were vital for Poincaré. Back in 1889, Poincaré had laid into the logicist purveyors of “teratological” functions. He came back to the theme in August 1900, this time while addressing the mathematicians gathered for their international congress in Paris. Again he pitted the logicists against the intuitionists. While he judged both important for the development of mathematics, there was no doubt on which side he stood. One mathematician (a logician, in Poincaré’s division) could cover pages and pages of print in order to demonstrate unequivocally that an angle could be divided into any number of equal parts. By contrast, Poincaré called his audience’s attention to the Göttingen mathematician Felix Klein: “He is studying one of the most abstract questions of the theory of functions: to determine whether on a given [abstract mathematical] Riemann surface there always exists a function admitting of given singularities [roughly: points where the function becomes infinite]. What does the celebrated German geometer do? He replaces his Riemann surface with a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for.” Now Klein knew perfectly well that this reasoning was not rigorous, but (says Poincaré) “he finds in it, if not a rigorous demonstration, at least a kind of moral certainty. A logician would have rejected with horror such a conception.” More precisely, the logician would never have been able to formulate the intuitionist’s thought.58 But formulate such machine thoughts Poincaré certainly did: in pure mathematics, in geodesy, in philosophy. In the midst of it all, he pressed ahead with studies of electricity and magnetism, bringing his ideas about electromagnetic clock coordination into the heartland of physics itself.
On 10 December 1900, just two days after Poincaré had sent Lallemand and Maurain on their Quito telegraphic longitude mission from the Gare Saint-Nazaire, Poincaré stood at the lectern in the University of Leiden, outlining his ideas on space, time, and ether. He and other luminaries had gathered to honor H. A. Lorentz. Lorentz was for Poincaré (and, I should add, for Einstein) a singular figure among physicists. In many ways, it was Lorentz who had helped exemplify the new professional category, “theoretical physicist.” For many physicists, especially those outside Britain, it was Lorentz who had put Maxwell’s theory of electricity and magnetism into a comprehensible form. Instead of following the British tradition of reducing all matter to etherial flows, vortices, stresses, and strains, Lorentz had a starker doctrine: there were two kinds of things in the world, electric and magnetic fields (states of the ether) on the one hand; and material, charged particles moving through the ether, on the other. Fields could act on particles, particles could create fields. But Lorentz had done still more, as he struggled to explain the seeming failure of experimentalists to reveal the motion of objects—including the earth—through the vast ether that was supposed to pervade the Universe.
For some time Poincaré had, like all other up-to-date physicists, been thoroughly familiar with Lorentz’s theory. He had also been respectfully critical of it when he lectured on it at the Sorbonne in 1899, believing it to be the best available theory. Even with Lorentz seated in the Leiden audience in 1900, Poincaré said, it disturbed him that Lorentz’s account violated the principle of action and reaction. A cannon retreats immediately when it shoots a cannonball, but the ether and atoms had the odd property (according to Lorentz) that an atom would act on the ether but that the material receiver of the atom’s action would only react later. What happened in the meantime? The ether seemed too insubstantial to carry momentum. Poincaré told Lorentz and the assembled that he could mitigate this objection. “But I disdain that excuse because I’ve got one a hundred times better, Good theories are flexible. . . . [I]f a theory reveals to us certain true relations, it can dress itself in a thousand diverse forms, it will resist all assaults and that which is its essence will not change.” Lorentz had created one of those flexible theories, one of the truly good ones: “I will not apologize” for criticizing the theory, he said. Instead, he only regretted that he had so little to add to Lorentz’s old ideas.59 Despite the disclaimer, Poincaré went on to transform the physical significance of Lorentz’s theory.
At the root of Lorentz’s old ideas had been a theoretical account of a seemingly intractable difficulty in ether physics. Was the ether simply dragged along by transparent matter? If ether was hauled along by the earth’s atmosphere, that could explain why experiments never seemed to reveal earthly motion through the ether. Unfortunately, a mid-nineteenth-century experiment by the French physicist Armand-Hippolyte Fizeau seemed to rule that out. By shining light through water flowing at different speeds, he could demonstrate that if the ether was dragged, it was only very partially. But if the ether was not dragged by matter (and so was fixed once and for all in the Universe), then we ought to be able to detect our motion through it. This is just what the American experimenter, Albert Michelson, set out to uncover by using an extraordinarily precise optical “interferometer” that he had invented (see figure 4.4).
Figure 4.4 Hunting the Ether. With a remarkable series of experiments, Albert Michelson sought to measure the earth’s motion through the elusive ether. In the 1881 device shown here, he launched a beam of light from a that was split by a half-silvered mirror at b: one-half of the ray reflected off d and into the eyepiece e. The other half of the ray penetrated the mirror at b, reflected from c, and was then bounced from b to the eyepiece e. At the eyepiece the two rays interfered with each other, showing the observer a characteristic pattern of light and dark. If one wave was delayed—by so little as a part of a wavelength of light—this pattern would visibly shift. So if the earth really was flying through the ether, then the “ether wind” would affect the relative time it took for the two beams to make their round-trips (the relative phase of the two waves would shift). Consequently, Michelson fully expected that if he rotated the apparatus, he would see a change in the interference patterns of the two rays. But no matter how he twisted his staggeringly sensitive instrument, the dark and light patterns did not budge. To Lorentz and Poincaré, this meant that the interferometer arms—like all matter—were contracted by their rush through the ether in just such a way as to hide the effect of the ether. To Einstein it was one more suggestive piece of evidence that the very idea of the ether was “superfluous.” SOURCE: MICHELSON, “THE RELATIVE MOTION OF THE EARTH AND LUMINIFEROUS ETHER,” AMERICAN JOURNAL OF SCIENCE, 3RD SERIES, VOL. XXII, NO. 128 (AUGUST 1881), P. 124.
Indeed, Michelson thought he had the ether cornered. For if there truly was an ether wind, then the round-trip time for a light beam should change, depending on whether the light was being sent across or into the wind. But rotating the apparatus showed not the slightest twitch of the interference dark spot; to an extraordinary degree of accuracy, it seemed that no motion through the ether could be detected by optical means. Though Michelson saw his efforts to find the ether as a dismal failure, other physicists, including Lorentz, were moved to theorize.
Taking Michelson’s null result into account, in 1892 Lorentz assumed the existence of a static ether, and introduced the startling notion that any object moving through the ether contracted in its direction of travel. Bizarre as this “Lorentz contraction” sounded, the gambit worked, in the sense that a judicious choice of this contraction factor exactly compensated for the effect of the putative ether wind. Lorentz’s contraction explained why high-precision experiments—even the extraordinary Michelson interferometer—would be powerless to uncover the effects of the etherial breeze on optical phenomena.
Remarkably, Lorentz’s contraction hypothesis was not enough. In the course of demonstrating that all optical phenomena could be described in approximately the same way, he introduced in 1895 a second innovation, a fictional “local time.”60 Lorentz’s idea was that there was
one true physical time, ttrue. True time was the appropriate time to use for objects at rest in the ether. For any object moving in the ether, it proved useful for Lorentz to introduce this fictional time (a mathematical artifice) in terms of which the laws of electricity and magnetism for that object would artificially resemble the laws for an object sitting still in the ether. This helpful quantity (tlocal) depended on the speed (v) of the object plowing through the ether, the speed of light (c), and the location of the object (x); tlocal was just ttrue minus vx/c2. Why did Lorentz choose this local time? Only because it gave a sharp, if purely formal, result: local time allowed a real object moving in the ether to be redescribed as a fictional object at rest in the ether. Local time for Lorentz was but a mathematical convenience.
Poincaré had viewed Lorentz’s theory and its assumptions about length contraction and “local time” with reserve. Even in his Sorbonne course of 1899, he did not connect local time to his technological-philosophical definition of simultaneity; in fact he remarked that corrections due to Lorentz’s “local time” were so small for the earth moving through the ether (the correction was only three-billionths of a second for two clocks separated by a kilometer) that he would simply ignore the correction.61 Meanwhile, Poincaré’s immersion in telegraphic longitude determination intensified. Not only was he in the thick of planning for the Quito expedition during 1899–1900, he was, in 1899, also president of the Bureau of Longitude. If he had somehow stood back from the details of the simultaneity procedures before, now he was fully engaged.
On 23 June 1899, for example, Poincaré wrote to the Royal Astronomer about the confounding discrepancy between the French and British measurements of the Paris-Greenwich longitudinal difference. Could the British begin a new effort immediately? William Christie responded 3 August pleading for more time, but promising he would prepare detailed, published analyses of their procedures and instruments, noting that “it is desirable too that the French results for the longitude Paris-Greenwich should also be published in detail.” Christie copied other letters to Poincaré about the problem, letters Christie had penned to Poincaré’s Bureau colleagues. To Loewy, Christie had speculated that the error source might be a discrepancy between leveling the star-sighting instrument based on a plumb line and on a spirit level. Apparently the French Army had implied that the fault might lie with British clocks; no, Christie replied, not possible. Instead he suggested that both Greenwich and Paris repeat the Paris-Greenwich procedure, only now between two piers inside Greenwich and again inside Montsouris. Presumably that would isolate any error accrued during the signal transmission under the English Channel. Poincaré replied immediately (9 August 1899) that he looked forward to receiving the British publications as soon as possible and that he hoped that they would contain the detailed calculations and the methods by which the data were reduced; in return, the French would open their books as both sides urgently tried to cut the embarrassing longitude discrepancy between Paris and Greenwich.62
In the midst of these longitudinal measurements and troubleshooting, Poincaré came face-to-face with Lorentz at the December 1900 meeting. Preparing for the event, Poincaré reinterpreted, in a strikingly new fashion, Lorentz’s “local time.” First (though he did not cite it) Poincaré introduced his 1898 “Measure of Time” assertion that simultaneity could be given between clocks synchronized by the exchange of electromagnetic signals. That was the argument he had produced at the intersection of longitude and metaphysics. Now he went further, moving the technological-philosophical idea of electrically coordinated clocks into physics itself, forming a triple intersection. For the first time he pursued his clock synchronization method for clocks moving through the ether. Poincaré’s sudden understanding: when executed while moving through the ether, the telegrapher’s procedure for synchronizing clocks gave Lorentz’s fictional local time, tlocal.
“Local time,” Poincaré insisted, was just the “time” clocks showed in a moving reference frame when they were coordinated by sending an electromagnetic signal from one to the others. No mathematical fiction, this was what moving observers would actually see:
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 translatory motion and consequently believing that the signals travel at the same speed in both directions, they limit themselves to crossing the observations, by sending one signal from A to B, then another from B to A. The local time t' is the time indicated by watches set in this manner.63
Here is the electrical simultaneity procedure that we know twice: from Poincaré longitude-finder and from Poincaré-philosopher. Poincaré-physicist inserts the procedure into a moving frame of reference.
Here, more precisely, is Poincaré’s argument. Consider a moving frame containing clocks A and B that is moving to the right through the ether at a constant speed v. Because of this motion, the light signal from A to B (say) encounters a headwind as it beats into the ether wind; the signal speed is the speed of light c, minus the wind speed, v (like an airplane encountering a headwind as it flies from Europe to the United States). The return light signal from B to A boosts its speed by an ether tailwind: its speed is therefore light speed plus ether wind speed, (c + v).
Because speeds differ in the two directions, a naive one-way clock coordination conducted by moving physicists would go wrong. For example, if B sets her clock using a signal from A, she is using a signal that is “really” going at c + v (relative to the ether). B gets the signal too soon relative to a “correct” signal that would travel at c. The further away B is from A, the bigger the time discrepancy between the moment she gets the tailwind-boosted signal and the moment she would have received it were it sent at the stately speed of c. So B needs to set her clock back a bit—less if she is close to A, more if she is farther away. Poincaré observed that the offset correction to account for the too-speedy signal (–vx/c2) yielded just the fictional “local time” correction of Lorentz’s “local time.”64 Poincaré’s message: Clocks moving in the ether must be coordinated by the launch of electromagnetic signals, as before. But coordination in a moving frame demands an offset of the clocks to compensate for the effect of the ether wind.
Lorentz acknowledged the force of Poincaré’s criticisms in a letter he posted on 20 January 1901—not a word, however, about Poincaré’s interpretation of “local time.” On other matters Lorentz readily ceded: there was a problem with the principle of reaction, and as far as Lorentz could tell that might forever be the case if physicists wanted to account for the experiments; if the ether was absolutely rigid and immovable, then one could not coherently talk about forces acting on it. So ether, as Lorentz had long argued, could exert force on electrons, but electrons could not exert force on the ether. Nor could one part of the ether act on another—any attempt to speak this way was, in effect, invoking “mathematical fictions.” No doubt such fictions were useful for calculating the ways in which the ether eventually acted to move electrons. But they were fictions nonetheless. Alternatively, Lorentz speculated, it might be possible to say that the ether was infinitely massive, in which case electrons could act on it without putting it into motion: “But that way out seems to me pretty artificial.”
Lorentz had produced an extraordinary theory, one that had utterly transformed physics by dividing the world between a vast unmoveable ether and material electrons. Immensely successful, the theory accounted for a myriad of experiments from spectral lines to the explanation of simple optics like reflection. But as Lorentz struggled to extend the theory, he found himself reaching for a variety of tools that he readily allowed were artificial. Lorentz supposed that matter contracts as it plunges through the ether, that the principle of action and reaction is violated, and that the theory needed the mathematical fiction of “local time.”
Poincaré approached Lorentz’s theory differently. As always when treating theories, Poincaré’s procedure w
as to isolate the building blocks of the theory and then to manipulate the theory’s most useful parts in order to advance. Here, at Leiden in 1900, he joined Lorentz’s local time with his own conventional interpretation of simultaneity—the longitude finder’s and philosopher’s convention. Without fanfare, he showed how Lorentz’s local time could be physically interpreted—the telegrapher’s convention now set in an etherial wind.
None of Poincaré’s alterations of the theory sullied his assessment of Lorentz. On the contrary, in January 1902, Poincaré nominated Lorentz for the Nobel Prize, explaining to the Swedish authorities how Lorentz had seized the failure of previous physicists to find the ether or explain their failure to do so. Poincaré: “It was evident that there must have been a general reason; Mr. Lorentz discovered that reason and put it in a striking form by his ingenious invention of ‘reduced time’.” Two phenomena that occurred in different places could appear to be simultaneous even when they were not. Everything happened as if the clock in one of the places was running behind the other, and as if no conceivable experience could allow the discovery of this discordance. As Poincaré saw it, the failure of experimentalists to observe the movement of the earth through the ether was just one such futile attempt.65