Einstein's Clocks and Poincare's Maps

by Peter Galison

  Clocks First

  Göttingen mathematician and mathematical physicist Hermann Minkowski trained his spotlight early and directly on Einstein’s clocks. Long before 1905, Minkowski had built his career by applying geometry where no one else had thought to apply it—to the apparently unvisualizable field of the theory of numbers. (It should be said that the younger Einstein resolutely ignored Minkowski, even when he was enrolled in the great mathematician’s ETH course.) When Minkowski saw the work of Lorentz, Poincaré, and Einstein, he again looked to geometry. Now Minkowski singled out Einstein’s attack on classical time as the key to the puzzle, the key that, in combination with his own formulation of a four-dimensional geometry of “spacetime,” opened up a new understanding of physics. Calling the new physics “radical,” he even more strongly dubbed it “mightily revolutionary” in his private drafts. In his much studied lecture, “Space and Time,” Minkowski made it clear that it was Einstein who had liberated the multitude of physical times from a fictional existence: “time, as a concept unequivocally determined by phenomena, was deposed from its high seat.” Minkowski asserted plainly that it was Einstein who had shown that there was no coherent meaning of “time” by itself, only “times,” plural and dependent on reference frame.74

  In pursuing his four-dimensional world, Minkowski drew on Poincaré who, back in 1906, had also considered a four-dimensional spacetime. Poincaré had remarked that amidst all the changes of time and space from one reference frame to another, there was one quantity that remained unchanged. An analogy: Suppose you mark the location of an observatory and nail the map to a board with a single nail driven through the position of your house. Rotate the map 45 degrees clockwise and you change the horizontal distance from your house to the observatory while you simultaneously alter the vertical distance between house and observatory. But obviously rotating the map in this way does nothing to change the actual distance between home and observatory. That is, if the horizontal separation is A, the vertical separation, B, and the distance C, then rotating the map will change A (if the map is turned so that the house and observatory are aligned vertically, for example, there is no horizontal separation). Similarly a rotation will change B, the vertical separation. But rotating the map cannot change the house-to-observatory distance, C. Minkowski showed that the relativistic transformations of space and time could be rigorously considered a distance-preserving rotation in the four-dimensional space made up of ordinary space and time. Just as distance in Euclidean geometry remained the same (despite a rotation), so in relativity there was a new distance that remained untouched by the transformations of space and time separately. [Spacetime distance squared] was always equal to [time difference squared] minus [space difference squared].

  In a powerful speech in Cologne on 21 September 1908, Minkowski took up the mathematics of Poincaré, but reordered their interpretation in words that instantly seized the imagination of the many physicists: “Henceforth space by itself and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve independence.” For Minkowski reality lay not in what we could grasp with our ordinary senses (space by itself, time by itself), but in the distances of this four-dimensional fusion of space and time. Invoking a striking imagery of projections and objects in four-dimensional spacetime, Minkowski would have called to mind among his Gymnasium-educated audience Plato’s cave scene in the Republic. There Plato described how a prisoner, constrained to view but the shadows of objects dancing on the walls, would find it painful to learn to peer directly at the full three-dimensional objects behind the prisoner’s head, much less at the light that lit them. Minkowski insisted that in the old physics of “space” and “time,” scientists had been similarly misled by appearance. In speaking of space and time separately, physicists were, like Plato’s prisoners, contemplating nothing more than shadows projected onto three dimensions. The full and higher reality of a four-dimensional “absolute world” would reveal itself only through the liberation of thought, and specifically through the insights mathematicians could provide.75

  At first Einstein resisted Minkowski’s formulation, seeing in it needless mathematical complexity. But as he ventured deeper into the theory of gravitation, the notion of spacetime proved ever more essential. Meanwhile, all around him, physicists took up Minkowski’s idiom as their road into relativity, a path many found more accessible than Einstein’s own.76 A few rejected Minkowski’s view that reality lay in four dimensions and that physics itself should be completely reformulated so it described this higher-dimensional world. Among those dubious about a four-dimensional physics figured, ironically, Poincaré himself, who already in early 1907 had preemptively dismissed the utility of such a project:

  It seems in fact that it would be possible to translate our physics into the language of geometry of four dimensions; to attempt this translation would be to take great pains for little profit. . . . [I]t seems that the translation would always be less simple than the text, and that it would always have the air of a translation, that the language of three dimensions seems the better fitted to our description of the world although this description can be rigorously made in another idiom.77

  As he had done so often, Poincaré pointed out new lands, identified the passes to them, and then chose to make his own stand on terra cognita. Once Einstein began his pursuit of four-dimensional space-time set in non-Euclidean geometry, he never let go.

  And again and again, Einstein came back to the problem of timekeeping. In 1910, for example, he once more insisted that time could not be grasped without clocks. “What is a clock?” he asked. “By a clock we understand any thing characterized by a phenomenon passing periodically through identical phases so that we must assume, by virtue of the principle of sufficient reason, that all that happens in a given period is identical with all that happens in an arbitrary period.”78 If there is no reason to think otherwise we should assume the Universe to be constant. If the clock is a mechanism that drives hands around in circles, then the uniform motion of those hands will mark time; if the clock were nothing but an atom, then time would be marked by its oscillations. By itself Einstein’s remark about the significance of “clock” extended the series of philosophical investigations of time by Mach, Pearson, or Poincaré himself in “The Measure of Time.” Now, however, a clock need not be a macroscopic object at all: it could even be an atom. At the moment Einstein penned these words, he was writing for a scientific publication, but it is not hard to see how these considerations came to count almost immediately as philosophy. Certainly Einstein’s time concept was read that way by the new breed of scientific philosophers that included Moritz Schlick and Rudolf Carnap of the Vienna Circle, or their Berlin ally, Hans Reichenbach.

  On 16 January 1911, Einstein appeared before the Naturforschende Gesellschaft in Zurich. Once again, the increasingly well-known young scientist outlined his reasoning, from the establishment of the Lorentz theory to the starting assumptions of relativity and the procedure for coordinating clocks. Clearly delighting in his example, Einstein explained how “the thing at its funniest” emerged if one imagined a clock—better still, a living clock, an organism—launched into near light-speed round-trip travel. On return, the being would have scarcely grown older, while those remaining at home would have aged through generations. Though previously skeptical, Einstein even tipped his hat toward Minkowski, whose “highly interesting mathematical elaboration” had revealed a method that made relativity theory’s “application substantially easier.” It was praise that came too late for Minkowski to appreciate; he had died suddenly in 1909. But Einstein now, like Minkowski, began celebrating the appeal of representing “physical events . . . in a four-dimensional space” and physical relations as “geometrical theorems.”79

  Kleiner, Einstein’s old thesis examiner and sometime supporter, then took the floor to laud his former, and formerly difficult, student:

  As far as the principle of relativity is concerned,
it is being called revolutionary. This is being done especially with regard to those postulates that are uniquely Einsteinian innovations in our physical picture. This concerns most of all the formulation of the concept of time. Until now we were accustomed to view time as something that always flows, under all circumstances in the same direction as something that exists independently of our thoughts. We have become accustomed to imagine that somewhere in the world there exists a clock that categorizes time. At least one thought it permissible to imagine the thing in such a way. . . . It turns out that the notion of time as something absolute in the old sense cannot be maintained, but that, instead, that which we designate as time depends on the states of motion.80

  By contrast, for Kleiner the relativity concept itself was hardly “revolutionary”; it was a “clarification,” perhaps, but not something “fundamentally new.” If the good professor Kleiner mourned anything in Einstein’s physics, it was the loss of the ether. True, he conceded, the concept had grown more and more incomprehensible. But without it wouldn’t there be “propagation in a medium which is not a medium”? Worse yet, didn’t the abandonment of the ether leave us with formulae that lacked any “mental image”? In reply, Einstein allowed that the ether might, in the age of Maxwell, have had “real value for intuitive representation.” But the value of the ether concept vanished when physicists gave up trying to picture ether as a mechanical entity with mechanical properties. After the demise of the truly intuitive ether, the notion had become for Einstein nothing but a burdensome fiction.

  That January day, just about every speaker directly or indirectly brought up Minkowski’s “Space and Time” speech, which clearly had paved the way for a wider acceptance of Einstein’s theory. Einstein’s exchanges with one member of the audience (a 1904 graduate of the University of Zurich) threw the status of the theory into relief:

  Dr. [Rudolf] Lämmel: Is the world picture resulting from the conceptions of the relativity principle an inevitable one, or are the assumptions arbitrary and expedient but not necessary?

  Prof. Einstein: The principle of relativity is a principle that narrows the possibilities; it is not a model, just as the second law of thermodynamics is not a model.

  Dr. Lämmel: The question is whether the principle is inevitable and necessary or merely expedient.

  Prof. Einstein: The principle is logically not necessary: it would be necessary only if it would be made such by experience. But it is made only probable by experience.

  For Poincaré, too, principles were made probable by experience, but principles were precisely what was expedient; they could be held against the grain of experience only at the cost of immense inconvenience. “Principles,” Poincaré had famously written in Science and Hypothesis, “are conventions and definitions in disguise.” For Einstein, principles were more than definitions, they were pillars, supports of the structure of knowledge. And this despite the circumstance that our knowledge of principles could never be certain; our hold on them was necessarily provisional, only probable, never forced by logic or experience.

  For Ernst Meissner, then a Privatdozent at ETH in physics and mathematics, Einstein’s work on time presented a model for a vast and critical reassessment of every concept of physics. Each one would have to be interrogated to identify those that remain invariant when one switches frames of reference:

  Meissner: The discussion has shown what is the first thing to be done. All physical concepts will have to be revised.

  Einstein: The main thing now is to set up the most exact experiments possible in order to test the foundation. In the meantime, all this brooding is not going to take us far. Only those consequences can be of interest that lead to results that are, in principle, accessible to observation.

  Meissner: You have brooded over this, and discovered the magnificent time concept. You found that it is not independent. This must be investigated for other concepts as well. You have shown that mass depends on the energy content, and you have made the concept of mass more precise. You did not carry out any physical investigations in the laboratory—you were brooding instead.81

  Ah yes, Einstein replied. But think of the fine predicament in which that brooding about time has put us.

  Einstein’s revision of time seized the attention of some of his most illustrious contemporaries. Max von Laue declared in his 1911 text on the relativity principle that it was the “groundbreaking work” of Einstein that had, with the single stroke of his radical critique of time, solved the puzzle of Lorentz’s real but undetectable ether.82 Max Planck, the dean of German physics (who had introduced the quantum discontinuity into physics), went further. Speaking in 1909 at Columbia University in New York, he told the assembled: “It need scarcely be emphasized that this new view of the concept of time makes the most serious demands upon the capacity of abstraction and the imaginative power of the physicist. It surpasses in boldness everything achieved so far in speculative investigations of nature, and even in philosophical theories of knowledge: non-Euclidean geometry is child’s play in comparison.”83 Planck’s words lofted Einstein’s reputation, broadcasting Einstein’s own sense that time was the pivot point of his work. The relativity of simultaneity, Einstein remarked soon afterward, “signifies a fundamental change in our concept of time. [It] is the most important, and also the most controversial theorem of the new theory of relativity.”84

  Emil Cohn, who had been thinking about light-coordinated clocks at least as early as 1904, returned to the question in 1913 in a brief, popular book (The Physical Aspect of Space and Time) to confront simultaneity in a thoroughly Einsteinian way (he referred to the “Lorentz-Einstein Relativity Principle,” no mention of Poincaré). Including a photograph of a wire and wood model of clock coordination, Cohn drew dozens of clocks and rulers to emphasize at each step of his argument that the Einsteinian kinematics was physical, procedural, and altogether visualizable in terms of coordinated public clocks: “The synchronization of a Strasbourg and a Kehl clock (that have been previously checked to run at similar rates) can and ought to be set in this way: Strasbourg sends at time 0 a light signal to Kehl, which is reflected; it gets back to Strasbourg at time 2. The clock in Kehl then is correctly set if, at the moment that the signal arrives there, its clock shows time 1 (and if not it should be corrected to do so).” Einstein liked Cohn’s presentation and said so in print.85

  Not for a moment did Einstein himself stop “brooding” about time; in 1913 he too published a new and strikingly simple argument for the relativity of time. Imagine, he said, that two parallel mirrors make up a “clock” where each tick is defined by the traversal of a burst of light from one mirror to the other (figure 5.12a). Now suppose that this light clock is moving to the right (figure 5.12b). To the stationary observer, the up-and-down motion of the flash of light appears to make a sawtooth pattern, much the way the ball of a running basketball player would follow such a trajectory as seen by the spectators. Here’s the point: the inclined trajectory of the moving clock (as seen by the stationary observer) is obviously longer than the perpendicular trajectory of the rest observer’s own clock. But by assumption the speed of light is the same in every frame of reference, so the angling trajectory of light travels at c. (This is not true for the basketball case, since the spectators would see the incline motion of the ball to be faster than the simple up-and-down motion seen by the player.) Since light has further to travel along the incline than it does along the perpendicular, it takes longer (D is greater than h). It follows that one tick for the moving observer (which appears to the stationary observer as following an incline) is registered as taking more than one click for the stationary one (which goes straight up and down).86

  So as far as the stationary frame is concerned, everything that takes place in the moving frame runs slowly. However Einstein presented his theory, the core lesson was the same: Absolute time was finished. In its place he offered a simple, practical procedure: Synchronize clocks by the exchange of light. Everything else in the theory fo
llowed from it alongside the fundamental assumptions of relativity and the absolute speed of light.

  Radio Eiffel

  When center-issued electromagnetic signals arrived at distant points, whether in the next room or a hundred kilometers distant, it was not only Einstein and Poincaré who defined them as simultaneous. Not at all. On the basis of the exchange of electrical signals, railroad planners scheduled trains, generals roused troops, operators telegraphed business deals, and geodesists drew maps. Indeed, precisely during Einstein’s early patent office years, preparations were being made to send time coordination signals by radio waves—the American Navy began experimenting with low-powered time signals from New Jersey in September 1903, with the order coming down to broadcast from Cape Cod (Massachusetts) and Norfolk (Virginia) in August 1904. Nor was radio time just a matter for the Americans. There was an intense burst of activity surrounding radio coordination systems in 1904 both in Switzerland and in France as workers tested, developed, and began deploying new radio time systems. The director of the French journal La Nature himself took up his pen to record new developments in the distribution of time by wireless. Reporting on experiments conducted at the Paris Observatory, he noted that with the aid of a chronograph, distant synchronization now appeared possible to within two- or three-hundredths of a second. Wireless technologies promised to distribute time everywhere in Paris and its suburbs, supplanting not only the antique steam system but also the cumbersome land lines of telegraphically communicated electric time. Radio time had advanced science through more precise determinations of longitude; now radio would free time from the physical burden of wire. At last, simultaneity could be broadcast to ships at sea and even into “the ordinary household.”87 Patents soon began arriving in Einstein’s office with schemes for radio time synchronization.88