Einstein's Clocks and Poincare's Maps

by Peter Galison

  If the asteroid began its orbit at a different position and speed, however, it need not travel in such a simple, repetitive way. For example, imagine that another, identical asteroid crossed the sheet near, but not through, the hole F. One possibility is that the punch-throughs would, orbit by orbit, drill a succession of holes through the sheet that approached F, and, in the fulness of eternity, reach it. Imagine a curve S drawn through successive holes. This curved axis S through F is called stable, if an asteroid starting anywhere on the line (that is, whose punch-throughs begin anywhere on the curve) gradually tends toward an orbit that circulates through hole F every time (see map 1 in figure 2.3). Conversely, a curve through F is called unstable if, in the distant past, an asteroid punching F gradually moves away from F on the curved axis U (see map 2).

  It was Poincaré’s claim in his prize paper that if an asteroid’s punch-throughs fled from a fixed point like F, they would eventually settle down toward another fixed point. Such a result would have described an ordered, bounded world, a world fully matching Poincaré’s hopeful motto of stars hemmed within their limits. Pushed by Phragmén to study his argument more closely, Poincaré dissected the problem during the fall of 1889, and his confidence in planetary stability began to crumble. Meanwhile, publication of the prize paper went forward.

  On Sunday, 1 December 1889, Poincaré confessed to Mittag-Leffler:

  I will not conceal from you the distress this discovery has caused me. In the first place, I do not know if you will still think that the results which remain, that is the existence of periodic solutions, the asymptotic solutions [, and my criticism of earlier methods] deserve the great reward you have given me.

  In the second place, much reworking will become necessary and I do not know if you can begin to print the memoir; I have telegraphed Phragmén. In any case, I can do no more than confide my perplexity to a friend as loyal as you. I will write to you at length when I can see things more clearly.26

  Figure 2.3 Poincaré’s Map. Poincaré’s “stroboscopic” diagrams tracking the intermittent punctures of a sheet so resembled a cartographic representation that they became universally known as “Poincaré maps,” with features commonly referred to as “islands,” “straits,” and “valleys.” Map 1 depicts a stable “axis” S along which successive punctures converge toward the fixed point, F. Map 2 shows an unstable axis U with successive punctures running away from F, and Map 3 portrays the crossing of both an unstable and stable axis at F. In this last case a puncture Co near the curved stable axis S is followed by subsequent hits progressing toward F (while remaining not far from S), and then fleeing from F (while remaining alongside the unstable curved axis U).

  Wednesday, 4 December: Mittag-Leffler wrote Poincaré to report how “extremely perplexed” he was to hear from Phragmén of Poincaré’s assessment of the situation. “It is not that I doubt that your memoir will be in any case regarded as a work of genius by the majority of geometers and that it will be the departure point for all future efforts in celestial mechanics. Don’t therefore think that I regret the prize. . . . But here is the worst of it. Your letter arrived too late and the memoir has already been distributed.” Write a letter to me, he continued, in which you explain that, based on your correspondence with Phragmén, you found that the expected stability was not, in fact, proven for all cases, and that you will send me a corrected manuscript. Mittag-Leffler added that a good word for Phragmén would be apt—a chair was open at the university. “It is true that my adversaries, acquired through the success of the Acta, will make a scandal out of this, but I will take it with tranquility because it is no embarrassment to be mistaken at the same time as you, and because I am firmly persuaded that you will eventually unravel the most hidden mysteries of this extraordinarily difficult question.”27

  The next day Mittag-Leffler swung into action, reporting to Poincaré that he had telegraphed Berlin and Paris to demand that they not distribute a single copy of the flawed journal. In Paris only Charles Hermite and Camille Jordan had received copies; Karl Weierstrass had one in Berlin. To Jordan, for example, Mittag-Leffler wrote to say that an error had slipped in and had to be fixed—could he please leave his copy to be picked up by a “domestique” who would whisk it away. “Please don’t say a word of this lamentable story to anyone,” he urged Hermite, “I’ll give you all the details tomorrow.” Tracking down the other copies one by one, Mittag-Leffler began to hope that all copies were recoverable. “I am very glad,” he confided to Poincaré, “that Mr. Kronecker [an equally famous German mathematician and arch enemy of Weierstrass] had not received a copy.” But even Mittag-Leffler’s allies began to bridle at the campaign. When Weierstrass replied to Mittag-Leffler’s sugar-coated letter, his discontent showed: “I confess to you, furthermore, that I take the matter not so lightly as you, Hermite and Poincaré himself.” Weierstrass icily noted that in his country, Germany, it was axiomatic that prize essays were printed in the form in which they had been judged. Weierstrass added that the stability question was not peripheral to Poincaré’s essay; rather, as Weierstrass had indicated in a report that was to have served as an introduction to Poincaré’s essay, it was central. What, Weierstrass demanded, remained in Poincaré’s paper of its entire positive program?28

  Poincaré rewrote the paper. What remained (or rather what he created to plug the gap), was something altogether outside the range of possible motions he—or anyone else—had ever considered. Chaos, not stability, reigned in this new universe. Here is what happened. Suppose, along with Poincaré, that a line of stability and line of instability cross at the fixed point F. (This is not hard to imagine. Consider a saddle: a marble sent down straight from the pommel along the direction of the horse’s spine will oscillate back and forth until it settles in the middle of the saddle: a stable point. But a marble propelled down to the right or left will fall off, never to return: an unstable point.) Now, suppose our asteroid begins near but not on the stable axis; it will gradually move toward F until it gets quite near, at which point it will fall under the sway of the unstable axis and begin to wander away from F. This is depicted in figure 2.3, map 3, as the series of punch-throughs, C0, C1, C2, . . . C7, C8,. . . . So far, Poincaré had no problem.29

  But suppose the stable and unstable axes cross somewhere else, for example at H, which Poincaré dubbed a homoclinic point. H will then be, by assumption, both a point on the stable axis S (driving the subsequent punch-throughs of the asteroid toward F) and a point on the unstable axis U (so an asteroid beginning near F will, slowly at first and then more quickly, puncture its consequents [successive holes] in sequence away from F on U through H). Now suppose that an asteroid flies through H; since it must remain on S, it beats its way, in each subsequent crash through the sheet (H1, H2, H3, etc.) along S toward F. But since any point on the unstable axis always remains on U, since H is also on U, all the consequents of H also have to be on U. So the extended U-axis must hit every consequent of H that we just plotted: H1, H2, H3, and so on. One way this might happen is shown in figure 2.4a.

  Now consider a different asteroid, C, near H but on U (figure 2.4b). Because H is on S, C (being near S) will start to move toward F. This is just what we saw in figure 2.4a. At the same time, since C starts on U, its consequents (C, C1, C2, C3 and so on) have to stay on U as U is extended. So the series C, C1, C2 moves toward F. But eventually, as the consequents of C approach F, the subsequent C’s begin to beat a retreat along the unstable axis U as in figure 2.3, map 3. Since U crosses S at H, the C’s, in their retreat along U, also eventually cross S (here illustrated by C6). The U-axis, after crossing S at H3, has to zoom all the way back to C4 to hit it, and then the U-axis has to head back once again toward F to intersect S at H4. Notice that because the extended U-axis to and from C6 has hit the S-axis (at the two points labeled X) we have two new homoclinic points and the map becomes vastly more complicated yet.

  Figure 2.4 Chaos in Poincaré’s Map. When the stable and unstable axes cross, complexity c
an follow. Indeed, it was precisely the possibility of this chaos-inducing crossing that Poincaré left unresolved in his first submission of the prize essay. As described in the text, this leads to the enormously complex extension of the unstable axis that begins in (a) and then is developed more fully in (b). Even the behavior of (a) is just the beginning of the complexity that would follow as the new crossing points of S and U (at the points marked X) are taken into account. Understandably, Poincaré despaired of ever being able to draw the “latticework” that a more complete representation would have to show.

  As a result of all this complexity, far from settling into bounded behavior, the unstable axis, and therefore any asteroids on or near it, will wander all over the Poincaré map, creating a motion so extraordinarily complex that Poincaré himself proved unable to depict it. When he eventually expanded the prize essay for publication in his New Methods of Celestial Mechanics, he struggled to describe the figures that resulted:

  When we try to represent the figure formed by these two curves and their infinitely many intersections, each corresponding to a doubly asymptotic solution, these intersections form a type of trellis, tissue, or grid with infinitely fine mesh. Neither of the two curves must ever cut across itself again, but it must bend back upon itself in a very complex manner in order to cut across all of the meshes in the grid an infinite number of times.

  “I shall not even try to draw it,” he added, yet “nothing is more suitable for providing us with an idea of the complex nature of the three-body problem.”30

  A hundred years after the vexed publication of this prize essay, Poincaré’s exploration of chaos was all the rage, celebrated as the dawn of a new science, a revolutionary advance over the simple predictions of classical science. Some late-twentieth-century physicists, philosophers, and cultural theorists hailed the sciences of complexity (as they came to be called) as a form of “postmodern physics,” while powerful computers spewed Poincaré maps minutely illustrating what their inventor had despaired of seeing on paper. Some of these maps revealed new physical phenomena; others graced art galleries.31 But in 1890, Poincaré proposed no revolution in the nature of science. Confronting a potentially damaging scandal over the prize, he patched up the gap in his argumentation and, in exploring the new dynamics, found what he had neither sought nor desired—a crack in the stability of the Universe.

  Far from waving a radical banner, Poincaré showed that, although the number of such chaotic orbits was infinite, the likelihood that an asteroid would find itself in an unstable regime was insignificant as measured against the likelihood that its orbit would be in a stable one. Having lost an absolute stability, Poincaré had to settle for a probabilistic one: “One could say,” he wrote about two years after the revised paper, that “the [unstable orbits] were the exception and the [stable ones] the rule.” Instead of trumpeting a break from stability, Poincaré stressed the power of the new qualitative methods to explore classical celestial dynamics: “[T]he true aim of celestial mechanics is not to calculate the ephemerides, because for this purpose we could be satisfied with a short-term forecast, but to ascertain whether Newton’s law is sufficient to explain all the phenomena.”32 For Poincaré, the true trial of Newton’s physics would come by probing its qualitative features: “From this point of view [of ascertaining the sufficiency of Newton’s law], the implicit relations which I have just spoken of can serve just as well as explicit formulas.”33 At stake for Poincaré were the basic, underlying relations of things, not the formulae and positions that astronomers busied themselves calculating to ever-higher decimal places.

  Convention

  In Poincaré’s attention to structures rather than things, there is a striking analogy to his use of non-Euclidean geometry. For many of Poincaré’s contemporaries, the non-Euclidean geometries explored throughout the nineteenth century marked an arresting break. Euclidean geometry had, for centuries, practically defined proper reasoning from sure starting points to inevitable conclusions. Since the eighteenth century, philosophers reading (or, perhaps, misreading) Kant, had enshrined Euclidean geometry as knowledge built into the very fabric of the mind. Some scientists and philosophers saw non-Euclidean geometry as a radical shift in the very definition of knowledge, a sign of hopeful modernity that broke completely with intuition; others feared the dreadful loss of certainty. Poincaré maintained a much more pragmatic stance. On the one hand, he insisted in 1891 that if the Euclidean axioms could be known before any experience, we humans could not so easily imagine others. On the other hand, the axioms of Euclidean geometry could not be merely experimental results. If they were, we would be constantly revising them. Since there are no perfectly rigid bodies that we must use to instantiate straight lines, we would soon “discover” that geometry was simply false: we would find, for example, triangles with angles that did not sum exactly to 180 degrees. Poincaré insisted that our choice of geometry is guided by experimental facts but is ultimately open to choice, subject to our need for simplicity.

  One of Poincaré’s more broadly directed pieces dealt with simplicity, convenience, and the hypotheses of geometry. What are the hypotheses? In what sense are they true? For Poincaré geometry was nothing other than a group, that is, a set of objects along with an operation that has certain properties. One property of the operation is that it should be reversible. The integers ( . . . -3, -2, -1, 0, 1, 2, 3, . . . ) have this property under addition and subtraction; adding a whole number can always be reversed by subtracting the same number. The group should also have among its possible actions that of identity, the operation that leaves a given object unchanged; here, adding zero does just that. And combining operations should lead to an element still in the group: adding 5 and adding 8 gives the same effect as adding 13. We learn which groups are useful to us by our encounter with the world but—as Poincaré often insisted in his philosophical writings—the idea of the group itself was a tool with which we were born. Not surprisingly, one particularly interesting group for us humans was the movement of rigid bodies in space. Poincaré argued that we have picked ordinary Euclidean geometry from among many other choices because the group underlying it corresponds, in a simple way, to the group of rigid-body motions in space—solid objects move in our real world. Could one have chosen otherwise? Of course, Poincaré says: we have only chosen the most convenient geometry. Does that mean that the other geometries are false? Not at all. No more (according to Poincaré) could one promote Cartesian coordinates (measuring a point’s position by the usual x and y axes) as true and derogate polar coordinates (measuring a point’s position by its distance from the origin and the angle of that radius line) as false. Once again, Poincaré emphasized that there are free choices in representing the world, choices fixed not by something completely exterior, but rather fixed by the simplicity and convenience of our knowledge. “I won’t insist any further; because the goal of this work is not the development of these truths which are starting to become banal.”34

  In fact, Poincaré was so committed to the role of conventional choice that he argued that the selection of a geometry was like the choice between French and German: Poincaré contended that one can choose this or that language or idiom to express the same thoughts. Imagine intelligent ants who lived on the surface of a saddle, and defined straight lines as the shortest distance between two points. Ant Mathematician says: “The sum of the angles of a triangle is less than two right angles.” We (from our human perspective) would describe the same situation in “Euclidean” rather differently, because we would see the ants’ triangles as shapes with curved sides. Human Mathematician might say: “If a curvilinear triangle has for its sides arcs of circles that, if produced, would cut orthogonally the fundamental plane, the sum of the angles of the curvilinear triangle will be less than two right angles.” Both statements capture the same situation but in different idioms. So there is no contradiction. Such a correspondence shows that the theorems of ant-saddle geometry are no less consistent than our ordinary geometry;
they might even be useful: a saddlelike “geometry being susceptible of a concrete interpretation, ceases to be a useless logical exercise, and may be applied.” That is the punchline: Various geometries are simply different ways of presenting relations among things; which we use depends on convenience.35

  All through Poincaré’s mathematics, his philosophy—and as we will see, his physics—lay this productive theme: find the group structure of what can be altered, and choose the most convenient representation. Yet while making those free decisions never lose sight of the fixed points, the invariants—those bits of the world left unchanged by our choices.

  WHERE, THEN, WAS Poincaré by 1892? The rising star of French mathematics, he had made his name through work that made dramatic use of non-Euclidean geometry, advances in qualitative approaches to differential equations, and a stunning new approach to the three-body problem. In philosophy, geometry, and dynamics, he had begun to work out a vision of knowledge that had two simultaneous goals. De-emphasizing particulars, he found the free choice available in formulating the problem. The choice of a coordinate system was not given by nature but made by us for our own convenience. Similarly, particular approximation schemes were to be selected by us for specific purposes; even the choice of a specific geometry had no absolute importance. Poincaré’s view: use Euclidean geometry when it is useful, when it pays to employ a non-Euclidean geometry, use that. In differential equations or in the physical systems they represented, there were always many ways of choosing variables—to describe, for instance, the flow lines of water down a stream. What was significant were the underlying relations that remained unchanged even after such changes in description: the vortices in a flow of water, the knots, saddle points, or spiral endpoints of geometrical lines. Similarly, the length of a line remains fixed when we rotate coordinates.