Coming of Age in the Milky Way - Timothy Ferris (2003)


Chapter 10. EINSTEIN’S SKY

I want to know how God created this world. I am not interested in this or that phenomenon, in the spectrum of this or that element. I want to know His thoughts, the rest are details.


Once the validity of this mode of thought has been recognized, the final results appear almost simple; any intelligent undergraduate can understand them without much trouble. But the years of searching in the dark for a truth that one feels, but cannot express; the intense desire and the alternations of confidence and misgiving, until one breaks through to clarity and understanding, are only known to him who has himself experienced them.


           In much the same way that Newton’s account of gravitation and inertia advanced physics to the point that it could embrace a moving Earth in a heliocentric solar system, Einstein’s relativity enabled physics to deal with the much higher velocities, greater distances and more furious energies encountered in the wider universe of the galaxies. If Newton’s domain was that of the stars and planets, Einstein’s extended from the centers of stars to the geometry of the cosmos as a whole.

To bring about so great an expansion of the scope of science, Einstein was obliged to abandon Newton’s conceptions of space and time. Newtonian space and time were inflexible and inalterable; they formed a changeless proscenium arch within which all events took place and against which all could be unambiguously measured. “Absolute space, in its own nature, without relation to anything external, remains always similar and immovable,” Newton wrote. “… Absolute, true and mathematical time, of itself, and from its own nature, flows equably without relation to anything external.”1 Einstein determined that this assumption was both superfluous and misleading. The special theory of relativity revealed that the rate at which time flows and the length of distances gauged across space vary, according to the relative velocities of those measuring them. The general theory of relativity went on to portray space as curved, and derived from spatial curvature the phenomena that Newtonian dynamics had attributed to the force of gravity.

Einstein grew up in an age when the classical conception of space, if not of time, was already coming unraveled. In order to explain how “absolute” space could have any reality—and, more to the point, how light and gravitational force could be conveyed across the empty space separating the stars and planets—Newton and his followers had postulated that space is pervaded by an invisible substance, an aether. The word was borrowed from Aristotle’s term for the celestial element of which the stars and planets were made, and like its namesake this new, updated aether was wonderful stuff. Lucid and friction-free, static and unchanging, it not only permitted the unimpeded motion of the planets and stars but actually wafted right through them—like a breeze through a grove of trees, as the English physicist Thomas Young put it.*

The appealing idea that space is pervaded by an aether began to run into trouble once it became possible to make precise measurements of the velocity of light. That light travels at a finite velocity had been appreciated since the 1670s, when the Danish astronomer Olaus Römer detected periodic variations in the time when Io, innermost of the four bright moons of Jupiter, went into eclipse: The eclipses came earlier than expected when Jupiter was relatively close to the earth and later when Jupiter was farther away. Römer realized that the discrepancy must be caused by the time it takes light to travel across the changing distance from Jupiter to Earth. From what was then known of the absolute distance of Jupiter, he was able to calculate the velocity of light to within about 30 percent of the accurate value (which is 186,272 miles per second).

Galileo had once tried to determine the velocity of light. He stationed two men with shuttered lanterns on hilltops about one mile apart, then timed the interval that elapsed between the instant when the first man opened his shutter and the second, responding to this signal, opened his shutter, sending a light beam back to the first. Römer’s finding made it clear why Galileo had failed; the interval he had attempted to measure (without a clock!) was less than a hundred thousandth of a second.

Römer’s result also suggested a way of measuring the velocity of the earth relative to absolute space: If light were propagated by a stationary aether, the absolute motion of the earth relative to the aether could be detected by measuring variations in the observed velocity of light. Imagine that the earth were a sailboat on an aether lake, and think of the light coming from two stars on opposite sides of the sky as ripples spreading from two stones dropped in the lake, one ahead of the boat and one behind. If we were standing on the deck of the boat and we measured the velocity of each set of ripples, we would find that those radiating from the stone dropped ahead would appear to be moving faster than those coming from behind. By measuring the difference in the observed velocity of the ripples coming from ahead and behind, we could calculate the speed of the boat. Similarly, it was assumed that the velocity of the earth’s motion could be determined by observing differences in the velocity of light waves coming through the stationary aether from stars ahead and behind.*

To measure this “aether drift”—as it was called, though what was thought to be drifting was not the aether but the earth—would of course be a delicate matter, since the velocity of the earth amounts to but a tiny fraction of the velocity of light. But by the latter part of the nineteenth century, technology had advanced to a sufficient degree of precision to make the task feasible. The critical experiment was conducted in the 1880s by the physicist Albert Michelson (who devoted his career to the study of light, he said, “because it’s so much fun”) and the chemist Edward Morley.

Aether drift theory held that if the velocity of light was constant relative to a stationary, all-pervading aether, then when the earth in its orbit was moving away from star A and toward star B, the observed speed of the light coming from star ? would be higher than that of the light coming from star A.

The Michelson-Morley apparatus, set up in a basement laboratory at Western Reserve University in Cleveland, Ohio, was based on the principle of interferometry. A beam of light was split and the two resulting light beams were reflected at right angles, then recombined and brought to a focus at an eyepiece. The idea was that the earth’s motion through the stationary aether would show up as a change in the interference pattern produced when one of the light beams, the one that had to travel into the aether wind, was retarded relative to the other beam. As Michelson explained the principle to his young daughter Dorothy, “Two beams of light race against each other, like two swimmers, one struggling upstream and back, while the other, covering the same distance, just crosses and returns. The second swimmer will always win, if there is any current in the river.”3 Since we know the earth is moving, there had to be some current—provided that, as Michelson and most other physicists then believed, there was such a thing as an aether that delineated the frame of reference of absolute Newtonian space.

To minimize exterior vibrations, the interferometer floated on a pool of mercury. To alter its orientation relative to the motion of the earth, it rotated on its mercury pool. Michelson spent days peering through the slowly moving eyepiece of the interferometer, looking for the telltale change in the interference patterns that would betray the earth’s motion through the aether. To his intense disappointment, he saw no such change at all. The conclusion was as inescapable as it was repugnant to Michelson: There was no detectable “aether drift.”

At first, few theorists were prepared to abandon the aether hypothesis, and several tried to reconcile it with the null outcome of the Michelson-Morley experiment. Their efforts gave rise to the bizarre idea that the experimental apparatus—and, indeed, the entire earth—contracted in the direction of its motion by just enough to cancel the effects of their velocity through the aether. “The only way out of it that I can see,” said the Irish physicist George FitzGerald, “is that the equality of [light] paths must be inaccurate.”4 In other words, the two beams of light seemed to be of equal length, because their length was distorted by the very motion of the earth they were intended to detect. As FitzGerald put it, “The block of stone [holding the apparatus] must be distorted, put out of shape by its motion … the stone would have to shorten in the direction of motion and swell out in the other two directions.”5 The Dutch physicist Hendrik Antoon Lorentz independently arrived at the same hypothesis, and worked it out in mathematical detail.

This, the “Lorentz contraction,” was to emerge in a different form as a key element in the special theory of relativity. The French physicist Henri Poincaré, one of the few leading scientists to take the Lorentz contraction seriously, came close to developing it into a form that was mathematically equivalent to Einstein’s theory; Poincaré spoke presciently of “a principle of relativity” that would prescribe that no object could exceed the velocity of light.6 But most researchers found it odd to the point of desperation to suggest that the velocity of the earth causes the entire planet to contract, like an orange squashed between a titan’s hands, and Lorentz himself soon set the idea aside. “I think he must have been held back by fears,” the physicist Paul Dirac speculated, years later. “… I do not suppose that one can ever have great hopes without their being combined with great fears.”7

Enter Einstein. He was born in 1879, in Ulm, where Kepler had once wandered in search of a printer, the manuscript of the Rudolphine Tables under his arm. A strong-willed but dreamy boy, Einstein did not begin speaking until he was three years old, and he forever retained something of the brooding intensity owned by the silent child. Intuitively antiauthoritarian, he rebelled against outside discipline, a habit that infuriated many of his teachers. (Years later he would joke that “to punish me for my contempt for authority, Fate made me an authority myself.”)8

At the age of sixteen Einstein escaped from the confines of the Luitpold Gymnasium in Munich—where his Greek instructor told him, “You will never amount to anything,” thus unwittingly earning himself a place in history—by persuading a doctor to write a note stating that the school regimen was pushing him to the brink of a nervous breakdown.9 He failed his college entrance examination, spent a year in preparatory school, and was graduated from the Federal Polytechnic Institute in Zurich in 1900 with respectable but unexceptional marks, having habitually cut classes to play the violin, languish in the cafés, and idle on Lake Zurich aboard rented sailboats with his fiancée, Mileva Marie, one of the few female students at the Polytechnic.

Unable to get a job as a scientist or even as a high school science teacher, Einstein advertised himself as a tutor in mathematics and physics, appending the invitation, “Trial lessons free.”10 The few who responded found him to be a bewildering teacher, cheerful and bright but inclined to romp down arcane avenues of inspiration with a fleetness that left them far behind. Eventually, Einstein found steady employment, as a “technical expert, third class,” in the Swiss patent office in Bern. He married Mileva in 1903 and they had a son, the first of two, in 1904. (Their first child, a daughter, was born out of wedlock and is thought to have died in infancy, perhaps of scarlet fever; no letters between Einstein and Mileva on this point have been found.) His hopes of getting a raise, the better to support his wife and family, were rewarded when in 1906 he was promoted to technical expert, second class.

With his mane of black hair, his limpid, penetrating gaze and his devotion to literature and music and philosophy, Einstein in those days resembled a poet as much as a scientist. Nor was he especially well informed about the progress of physics: His efforts to keep abreast of the scientific literature were impaired by the fact that the technical library generally was closed when he got off work. His technical writings, though occasionally interesting, were in general limited to the sort of speculations about infinity and entropy that may be found in the notebooks of a thousand other postgraduates.

Einstein was indifferent to convention and quick to laugh, a natural enemy of pomp and ceremony. When a friend prevailed upon him to attend festivities at the university in Geneva honoring the 350th anniversary of its founding by Calvin, he marched among the berobed professors in the academic procession wearing an old straw hat and rumpled suit, having no more suitable clothing, and recalled that at the banquet afterward he “said to a Genevan patrician who sat next to me, ‘Do you know what Calvin would have done if he were still here? … He would have had us all burned because of sinful gluttony.’ The man uttered not another word.”11 He was, in short, a Bohemian and a rebel and a high-spirited young man, but nobody’s candidate for scientific distinction.

Yet in 1905, Einstein’s thoughts began to crystallize, and in that year alone he wrote four epochal papers that transformed the scientific landscape. The first, published three days after his twenty-sixth birthday, would help to lay the foundations of quantum physics. Another was to alter the course of atomic theory and statistical mechanics. The other two enunciated what came to be known as the special theory of relativity. When Max Planck, the editorial director of the German Annals of Physics, looked up from reading the first relativity paper, he knew at once that the world had changed. The age of Newton was over, and a new science had arisen to replace it.

In retrospect all is clear, and veins of scientific genius may be found running through the musings of the young Einstein. He had been a quietly religious child, who at the age of eleven composed little hymns in honor of God that he sang on his way to school. But at about age twelve, as he recalled many years later, he

experienced a second wonder of a totally different nature, in a little book dealing with Euclidian plane geometry, which came into my hands at the beginning of a school year. Here were assertions, as for example the intersection of the three altitudes of a triangle in one point, which—though by no means evident—could nevertheless be proved with such certainty that any doubt appeared to be out of the question. This lucidity and certainty made an indescribable impression upon me.12

Einstein later saw this conversion, from traditional religion to what he called his “holy” Euclid text, as involving two ways of striving for the same deliverance:

It is quite clear to me that the religious paradise of youth, which was thus lost, was a first attempt to free myself from the chains of the “merely-personal,” from an existence which is dominated by wishes, hopes and primitive feelings. Out yonder there was this huge world, which exists independently of us human beings and which stands before us like a great, eternal riddle, at least partially accessible to our inspection and thinking. The contemplation of this world beckoned like a liberation…. The road to this paradise was not as comfortable and alluring as the road to the religious paradise; but it has proved itself as trustworthy, and I have never regretted having chosen it.13

Even Einstein’s lack of precocity looked in hindsight like a gift in disguise. Einstein felt that he had “developed so slowly that I only began to wonder about space and time when I was already grown up. In consequence I probed deeper into the problem than an ordinary child would have done.” Whatever the cause, he certainly possessed unusual powers of concentration: Like Newton, who credited his insights into deep problems to his habit of “thinking of them without ceasing,” Einstein was implacably tenacious in pursuing a line of thought once it had captured his attention.* And, like Galileo, he combined a taste for fundamental philosophical questions with an appreciation of the importance of testing his ideas empirically: “Direct observation of facts,” he said, “has always had for me a kind of magical attraction.”5

The intellectual odyssey that led Einstein to the special theory of relativity—and from there to the general theory, which was to deliver theoretical cosmology from its infancy—began when he was no more than five years old. He was sick in bed, and his father showed him a pocket compass to keep him amused. He asked what made the compass needle point north, and was told that the earth is enshrouded in a magnetic field to which the needle responds. He was astonished. It seemed, he recalled many years later, “a miracle” that an invisible, intangible field could govern the behavior of the very real compass needle. “Something deeply hidden,” he thought, “had to be behind things.”16

He learned what that something might be a few years later, when he read a textbook description of James Clerk Maxwell’s theory of the electromagnetic field. Maxwell had built his field theory on the experimental work of the English scientist Michael Faraday. The two men, as Einstein later noted, were related to each other much as were Galileo and Newton—“the former of each pair grasping the relations intuitively, and the second one formulating those relations exactly and applying them quantitatively.”17 Faraday, a blacksmith’s son, had been an apprentice to a London bookbinder. He read popular-science books in his spare time, and when a friend took him to hear a series of public lectures by the chemist Sir Humphry Davy, Faraday took notes on the lectures, printed them, bound them in leather, and sent them to Davy, who responded by hiring him as a laboratory assistant at the Royal Institution of Great Britain. There Faraday remained for the next forty-six years, eventually succeeding Davy as the institution director. He was an Edisonian figure, his white hair parted in the middle over wide-set eyes and slab-flat cheekbones, his shoulders stooped with work and his large hands buried in laboratory apparatus, though he smiled as habitually as Edison scowled.

In the course of more than fifteen thousand experiments, Faraday found that electricity and magnetism are conveyed by means of invisible lines of force arrayed in space—i.e., by fields. (Students today who sprinkle iron filings on a paper resting on a horseshoe magnet to watch the filings trace out the magnetic field lines are replicating an old Faraday experiment.) His gift to science amounted to a fundamental shift in emphasis, from the visible apparatus, the magnet or electrical coil, to the invisible field that surrounds it and conveys the electrical or magnetic force. Here began field theory, which today explores processes ranging from the subatomic to the intergalactic scale and portrays the entire material world as but a grand illusion, spun on the loom of the force fields. Einstein was to be its Bach.

But though Faraday established the existence of electrical and magnetic fields, he lacked the mathematical acumen required to write a quantitative description of them. This was left for Maxwell. Thin-boned as a bird, with trusting, farsighted eyes and a choirboy’s fragile countenance, Maxwell was at home in mathematical castles inaccessible to Faraday. A methodical thinker, he first studied electricity and magnetism by reading Faraday’s papers—this on Lord Kelvin’s advice, in order to introduce himself to the fields through Faraday’s eyes—and only thereafter subjected them to the arc lamp of his mathematical skills. The result, Maxwell wrote Kelvin in 1854, was to “have been rewarded of late by finding the whole mass of confusion beginning to clear up under the influence of a few simple ideas.”18

This was the beginning of the abstraction of the field concept, a step that would spell the end of purely mechanistic science and lead to the nonvisualizable but far more flexible mathematical flights of relativity and quantum physics. Faraday read the papers Maxwell sent him with the bemusement of a tone-deaf man listening to Beethoven’s quartets, understanding that they were beautiful without being able to appreciate just how. “I was almost frightened when I saw such mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well,” Faraday wrote Maxwell. In another letter he asked, touchingly and tentatively:

When a mathematician engaged in investigating physical actions and results has arrived at his conclusions, may they not be expressed in common language as fully, clearly, and definitely as in mathematical formulae? If so, would it not be a great boon to such as I to express them so?—translating them out of their hieroglyphics.19

Maxwell obligingly rendered some of his explanations of field theory into the mechanical cogwheels and sprocket formulations that Faraday could understand, but it was when stripped to bare equations that his theory flew. With fuguelike balance and power, Maxwell’s equations demonstrated that electricity and magnetism are aspects of a single force, electromagnetism, and that light itself is a variety of this force.* Thus were united what had been the separate studies of electricity, magnetism, and optics.

When the young Einstein encountered Maxwell’s equations they struck him, he said, “like a revelation.” Here was a precise and symmetrical account of the invisible field that governed the compass needle. It animated space, could “weave a web across the sky” as Maxwell had put it, and its differential equations etched the outlines of that web with exquisite balance and precision.

“What made this theory appear revolutionary,” Einstein recalled, “was the transition from forces at a distance to fields as fundamental variables.”20 It was no longer necessary to invoke the idea of an aether transmitting light across space; the electromagnetic field in itself could do the job. This had not been appreciated by the older, classical physicists, Maxwell himself among them. Theirs was a putatively hardheaded, mechanical world view, in which the fields taken by themselves appeared too insubstantial to be real. It was by their consensus that the aether hypothesis had glided on, a ghost ship alive with Saint Elmo’s fire, well after Maxwell’s equations and the Michelson-Morley experiment had emptied the wind from its sails. Einstein, caring little for tradition, abandoned the aether and focused his attention on the field.

Yet if one adhered to both Maxwell’s equations and Newton’s absolute space, the result was a paradox. This the giants of physics understood; it was one of their reasons for underestimating the importance of Maxwell’s field equations. Einstein, ignorant of their wisdom, discovered the paradox for himself, at the age of sixteen. He was at the time enrolled in a preparatory school at Aarau, in the Swiss Oberland, where he enjoyed taking walks along the river oxbows. (Years later he would write a paper defining how rivers meander.) One day, Einstein asked himself what he would see if he were to chase a beam of light at the velocity of light. The answer, according to classical physics, was that “I should observe such a beam of light as a spatially oscillatory electromagnetic field at rest. However, there seems to be no such thing, whether on the basis of experience or according to Maxwell’s equations.”21Velocity was inherent to light; it was, after all, by way of its velocity that light had revealed to Maxwell its identity as an electromagnetic field. Yet if we live in an absolute, Newtonian space demarcated by the aether, it should be possible to catch up with a light beam and thus rob it of its speed. Something had to give, in either Newton’s physics or Maxwell’s.

Einstein was acquainted with another electrodynamic paradox as well, one that had turned up literally in his own backyard, in the iron and copper dynamos that his father and his uncle Jakob had built in an electrical shop behind the family home in the Munich suburbs. The principle of the dynamo, established by Faraday, was that the field created by a whirling magnet will generate an electrical current in a surrounding web of wire. This finding had tremendous practical potential: The energy of a steam engine or a flowing stream could be harnessed to produce electricity that could then be exported via electrical lines to power machinery and illuminate city streets miles away. Although the Einstein family never managed to make much of a living from it, dynamo design was on the forefront of contemporary technology, and giant steam-driven dynamos were being commissioned and built at considerable expense.* Yet their performance could not be predicted with exactitude so long as the behavior of the electromagnetic field within the dynamo was so poorly understood. Under existing theory, the moving field was to be explained according to one set of rules if viewed from the perspective of a dynamo’s rotating magnet, and another if viewed from the stationary electrical coil. Every dynamo housed a whirling mystery.

The situation was economically embarrassing for the industrialists. It bothered Einstein as unaesthetic. “The thought that one is dealing here with two fundamentally different cases was for me unbearable,” he recalled. “The difference between these two cases could not be a real difference but rather, in my conviction, only a difference in the choice of the reference point.”23

Such questions were still on Einstein’s mind when he completed his year of preparatory school, but if he hoped to find guidance in solving them at the Polytechnic Institute, he was soon disappointed. His physics professor, the capable but conservative Heinrich Friedrich Weber, was fascinated by dynamos, owed his chair to the philanthropy of the dynamo builder Werner von Siemens, and was sufficiently devoted to the study of electricity that he repeatedly submitted himself to electrical shocks of one thousand volts and more of alternating current—this as part of an effort to determine how much voltage a human being could endure. Yet Weber, steeped in the traditions of classical physics, never lectured on Maxwell or Faraday. Einstein soon lost interest and started cutting Herr Weber’s classes. He read physics on his own and conducted experiments in the Polytechnic’s superb laboratories. One of his experiments resulted in an explosion that badly injured Einstein’s hand and nearly wrecked the lab.

Professor Weber responded by doing what he could—which was a great deal—to prevent Einstein’s getting a job after graduation. Thus stigmatized, Einstein went nowhere. The distasteful experience of cramming for the comprehensive final examinations had in any event left him unable to think about science for a full year, and he spent his time reading philosophy and playing the violin. When he did resume the study of physics, it was with little encouragement from the outside world. He submitted a thesis on the kinetics of gases to the University of Zurich, but no doctorate was forthcoming. He wrote a few scientific papers, but they were almost worthless. And yet, though regretting that he was a disappointment to his parents, Einstein remained serenely self-confident. “I have a few splendid ideas,” he wrote to his friend Marcel Grossmann, “which now only need proper incubation.”25

It was with the help of Grossman’s father that Einstein got the patent office job, and while we may shake our heads at the spectacle of so great a man in so slight a position, Einstein remembered it as “my best time of all.”26 He enjoyed contemplating the mechanical gadgets that came before him for review, found that writing critiques of patent applications helped him learn to express himself succinctly, and reveled in the companionship of his friend Michele Angelo Besse, with whom he discussed philosophy, physics, and everything under the sun. “I could not have found a better sounding-board in all of Europe,” he said.27

At Besso’s urging, Einstein read the works of the Austrian physicist and philosopher Ernst Mach, one of the few leading scientific thinkers to critique the mechanical paradigm on which rested belief in a Newtonian space pervaded by an aether. “The simplest mechanical principles are of a very complicated character” Mach wrote (his italics). “… They can by no means be regarded as mathematically established truths but only as principles that not only admit of constant control by experience but actually require it.”28 A scalding critic of Newtonian space in general and of the aether hypothesis in particular, Mach sought to replace such “metaphysical obscurities,” as he called them, with more economical precepts anchored firmly in the sense data of observation. Space, Mach argued, is not a thing, but an expression of interrelationships among events“All masses and all velocities, and consequently all forces, are relative,” he wrote.29 Einstein agreed, and was encouraged to attempt to write a theory that built space and time out of events alone, as Mach prescribed. He never entirely succeeded in satisfying Mach’s criteria—it may be that no workable theory can—but the effort helped impel him toward relativity.

The emergence of the special theory of relativity was as unconventional as its author. The 1905 paper that first enunciated the theory resembles the work of a crank; it contains no citations whatever from the scientific literature, and mentions the aid of but one individual, Besso, who was not a scientist. (At the time, Einstein knew no scientists.) Einstein’s first lecture in Zurich explaining the theory was delivered not in a university but in the Carpenters’ Union hall; he went on for over an hour, then suddenly interrupted himself to ask the time, explaining that he did not own a watch. Yet here began the reformation of the concepts of space and time.

With the special theory of relativity, Einstein had at last resolved the paradox that had occurred to him at age sixteen, that Maxwell’s equations failed if one could chase a beam of light at the velocity of light. His did so by concluding that one cannot accelerate to the velocity of light—that, indeed, the velocity of light is the same for all observers, regardless of their relative motion. If, for instance, a physicist were to board a spaceship and fly off toward the star Vega at 50 percent the velocity of light, and while on board measure the velocity of the light coming from Vega, he would find that velocity to be exactly the same as would his colleagues back on Earth.

To quantify this strange state of affairs, Einstein was obliged to employ the Lorentz contractions. (At the time he knew little of Lorentz, whom he was later to esteem as “the greatest and noblest man of our times … a living work of art.”)30 In Einstein’s hands, the Lorentz equations specify that as an observer increases in velocity, his dimensions, as well as those of his spaceship and any measuring devices aboard, will shrink along the direction of their motion by just the amount required to make the measurement of light’s velocity always come out the same. This, then, was why Michelson and Morley had found no trace of an “aether drift.” In fact the aether is superfluous, as is Newton’s absolute space and time, for there is no need for any unmoving frame of reference: “To the concept of absolute rest there correspond no properties of the phenomena, neither in mechanics, nor in electrodynamics.”31 What matters are observable events, and no event can be observed until the light (or radio waves, or other form of electromagnetic radiation) that brings news of it reaches the observer. Einstein had replaced Newton’s space with a network of light beams; theirs was the absolute grid, within which space itself became supple.

Observers in motion experience a slowing in the passage of time, as well: An astronaut traveling at 90 percent of the velocity of light would age only half as fast as her colleagues back home, so that at, say, a twentieth class reunion of interstellar astronauts, those who had served the most aboard relativistic spacecraft would be the youngest. Mass, too, is rendered plastic within the framework of the light beams; objects approaching the speed of light increase in mass. The effects of relativistic time dilation, mass increase, and change in dimension are minute at ordinary velocities like that of the earth in its orbit or the sun through space (which is why it had not been noticed sooner) but become pronounced as speeds increase, and go to infinity at the speed of light. If the earth could be accelerated to the velocity of light (a feat that would require infinite energy to achieve) it would contract into a two-dimensional wafer of infinite mass, on which time would come to a stop—which is one way of saying that acceleration to light speed is impossible.

Nor are these effects illusory, or merely psychological: They are as real as the stone that Dr. Johnson kicked in his famous refutation of Bishop Berkeley, and have been confirmed in scores of experiments. The relativistic increase in the mass of particles moving at nearly the velocity of light is not only observed in particle accelerators, but is what gives the speeding particles most of their punch. Relativistic time dilation has been tested by flying atomic clocks around the world in commercial aircraft; the clocks were found to run slow by just the tiny amount the theory specifies. A NASA ground controller once threatened to dock astronauts in space a fraction of a penny of their flight pay, to compensate for the decrease in the passage of time they experienced as a result of their velocity in orbit.

These and other implications of special relativity initially struck the lay public, and many scientists as well, as uncommonly strange.* But if Einstein’s approach was radical, his intention was essentially conservative. As is implied by the title of his original relativity paper, “On the Electrodynamics of Moving Bodies,” his aim was to redeem the laws of electrodynamics so that they could be shown to work in every imaginable situation, not just in a quiet laboratory in Zurich but in whirling dynamos and on moving worlds hurtling past one another at staggering speeds. The term relativity, coined not by Einstein but by Poincaré and applied to the theory by the physicist Max Planck, is somewhat misleading in this sense; Einstein, stressing, its conservative function, had preferred to call it Invariantentheorie—“invariance theory.”

Relativity nonetheless cast its net wide, embracing the study not only of light and space and time, but of matter as well. The theory derives its catholic impact from the fact that electromagnetism is implicated not only in the propagation of light but also in the architecture of matter: Electromagnetism is the force that holds electrons in their orbits around nuclear particles to make atoms, binds atoms together to form molecules, and ties molecules together to form objects. Every tangible thing, from stars and planets to this page and the eye that reads it, carries electromagnetism in the fiber of its being. To alter one’s conception of electromagnetism is, therefore, to reconsider the very nature of matter. Einstein caught sight of this connection only three months after the first account of special relativity had appeared, and published a paper titled, “Does the Inertia Content of a Body Depend Upon Its Energy Content?” The answer was yes, and ours has been a sadder and wiser world ever since.

In the first paper, as we have seen, Einstein demonstrated that the inertial mass of an object increases when it absorbs energy. It follows that its mass decreases when it radiates energy. This holds true, not only for a spaceship gliding toward the stars, but for an object at rest as well: A camera loses a (very) little mass when the flash goes off, and the people whose picture is being taken become a little more massive in the exchange. Mass and energy are interchangeable, with electromagnetic energy doing the bartering between them.

Einstein, contemplating this fact, concluded that energy and inertial mass are the same, and he expressed their identity in the equation

in which m is the mass of an object, E is its energy content, and c is the velocity of light. In composing this singularly economical little equation, which unifies the concepts of energy and matter and relates both to the velocity of light, Einstein initially was concerned with mass. If instead we solve for energy, it takes on a more familiar and more ominous form, as

E = mc2

Viewed from this perspective, the theory says that matter is frozen energy. This of course is the key to nuclear power and nuclear weapons, though Einstein did not consider these applications at the time and rejected them as impractical once they were proposed by others. In the hands of the astrophysicists, the equation would be used to discern the thermonuclear processes that power the sun and stars.

But for all its protean achievements, special relativity was silent with regard to gravitation, the other known large-scale force in the universe. The special theory has to do with inertial mass, the resistance objects offer to change in their state of motion—their “clout,” or “heft,” so to speak. Gravitation acts upon objects according to their gravitational mass—i.e., their “weight.” Inertial mass is what you feel when you slide a suitcase along a polished floor; gravitational mass is what you feel when you lift the suitcase. There would appear to be distinct differences between the two: Gravitational mass manifests itself only in the presence of gravitational force, while inertial mass is a permanent property of matter. Take the suitcase on a spaceship and, once in orbit, it will weigh nothing (i.e., its gravitational mass will measure zero), but its inertial mass will remain the same: You’ll have to work just as hard to wrest it around the cabin, and once in motion it will have the same momentum as if it were sliding across a floor on Earth.*

Yet for some reason, the inertial and gravitational mass of any given object are equivalent. Put the suitcase on the airport scale and find that it weighs thirty pounds: That is a result of its gravitational mass. Now set it on a sheet of smooth, glazed ice or another relatively friction-free surface, attach a spring scale to the handle, and pull it until you get it accelerating at the same rate at which it would fall (i.e., 16 feet per second, on Earth), and the scale will register, again, thirty pounds: That is a result of its inertial mass. Experiments have been performed to a high degree of precision on all sorts of materials, in many different weights, and the gravitational mass of each object repeatedly turns out to be exactly equal to its inertial mass.

The equality of inertial and gravitational mass had been an integral if inconspicuous part of classical physics for centuries. It can be seen, for instance, to explain Galileo’s discovery that cannonballs and boccie balls fall at the same velocity despite their differing weight: They do so because the cannonball, though it has greater gravitational mass and ought (naively) to fall faster, also has a greater inertial mass, which makes it accelerate more slowly; since these two quantities are equivalent they cancel out, and the cannonball consequently falls no faster than the boccie ball. But in Newtonian mechanics the equivalence principle was treated as a mere coincidence. Einstein was intrigued. Here, he thought, “must lie the key to a deeper understanding of inertia and gravitation.”32 His inquiry set him on his way up the craggy road toward the general theory of relativity.

Einstein’s first insight into the question came one day in 1907, in what he later called “the happiest thought of my life.” The memory of the moment remained vivid decades later:

I was sitting in a chair in the patent office at Bern, when all of a sudden a thought occurred to me: “If a person falls freely he will not feel his own weight.” I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.33

To appreciate why this seemingly straightforward picture should have so excited Einstein, imagine that you awaken to find yourself floating, weightless, in a sealed, windowless elevator car. A diabolical set of instructions, printed on the wall, informs you that there are two identical such elevator cars—one adrift in deep space, where it is subject to no significant gravitational influence, and the other trapped in the sun’s gravitational field, plunging rapidly toward its doom. You will be rescued only if you can prove (not guess) in which car you are riding—the one floating in zero gravity, or the one falling in a strong gravitational field. What Einstein realized that day in the patent office was that you cannot tell the difference, neither through your senses nor by conducting experiments. The fact that you are weightless does not mean that you are free from gravitation; you might be in free fall. (The “weightlessness” experienced by astronauts in orbit is precisely of this sort: Though trapped by the earth’s gravitational field they feel no weight—i.e., no effect of gravitation—because they and their spaceship are constantly falling.) The gravitational field, therefore, has only a relative existence. One is reminded of the joke about the man who falls from the roof of a tall building and, seeing a friend looking aghast out a window on the way down, calls out encouragingly, “I’m okay so far!” His point was Einstein’s—that the gravitational field does not exist for him, so long as he remains in his inertial framework. (The sidewalk, alas, is in an inertial framework of its own.)

The same ambiguity applies in the opposite situation: Suppose that when you awaken you find yourself standing in the elevator car, at your normal weight. This time the instructions say that you are either (1) aboard a elevator stopped on the ground floor of an office building on Earth, or (2) adrift in zero-gravity space, in an elevator attached by a cable to a spaceship that is pulling it away at a steady acceleration, pressing you to the floor with a force equal to that of Earth’s gravitation—at one “G,” as the jet pilots say. Here again, you cannot prove which is the case.

Einstein reasoned that if the effects of gravitation are mimicked by acceleration, gravitation itself might be regarded as a kind of acceleration. But acceleration through what reference frame? It could not be ordinary three-dimensional space; the passengers in the elevator in the New York skyscraper, after all, are not flying through space relative to the earth.

The search for an answer required brought Einstein to consider the concept of a four-dimensional spacetime continuum. Within its framework, gravitation ts acceleration, the acceleration of objects as they glide along “world lines”—paths of least action traced over the slopes of a three-dimensional space that is curved in the fourth dimension.

A forerunner here was Hermann Minkowski, who had been Einstein’s mathematics professor at the Polytechnic Institute. Minkowski remembered Einstein as a “lazy dog” who seldom came to class, but he was quick to appreciate the importance of Einstein’s work, though initially he viewed it as but an improvement on Lorentz. In 1908 Minkowski published a paper on Lorentz’s theory that cleared away much of the mathematical deadwood that had cluttered Einstein’s original formulation of special relativity. It demonstrated that time could be treated as a dimension in a four-dimensional universe. “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 an independent reality,” Minkowski predicted.34 His words proved prophetic, and the special theory of relativity has been viewed in terms of a “spacetime continuum” ever since. Einstein initially dismissed Minkowski’s formulation as excessively pedantic, joking that he scarcely recognized his own theory once the mathematicians got hold of it. But he came to realize that if he were to explore the connection between weight and inertia, he would do well to travel farther up the trail Minkowski had blazed.

Minkowski’s spacetime continuum, though suitable for special relativity, would not support what was to become general relativity. Its space was “flat”—i.e., euclidean. If gravitation were to be interpreted as a form of acceleration, that acceleration would have to occur along the undulations of curved space. So it was that Einstein was led, however reluctantly, into the forbidding territory of the noneuclidean geometries.

Euclidean geometry, as every high school math student is taught, has different characteristics depending upon whether it is worked in two dimensions (“plane” geometry) or three (“solid” geometry). On a plane, the sum of the angles of a triangle is 180 degrees, but if we add a third dimension we can envision surfaces such as that of a sphere, on which the angles of a triangle add up to more than 180 degrees, or a saddle-shaped hyperbola, on which the angles add up to less than 180 degrees. The shortest distance between two points on a plane is a straight line, but on a sphere or a hyperbola the shortest distances are curved lines. In the noneuclidean geometries a fourth dimension is added, and the rules are changed in a similarly consistent manner to allow for the possible curvature of three-dimensional space within a four-dimensional theater. Two categories of curved space then can be imagined (or at least calculated): spherical, or “closed” space, in which three-space obeys geometrical rules analogous to those of the two-space on the surface of a sphere, and hyperbolic, or “open” space, analogous to the surface of a three-dimensional hyperbola. (One can also work out a flat, euclidean four-dimensional geometry, but in that case the rules do not change, just as plane two-dimensional geometry obeys the same rules if the planes happen to be the sides of a three-dimensional cube.)

Triangles in flat two-dimensional space have interior angles that always add up to 180 degrees. But when two-space is curved into a third dimension, the angles always total either less than 180 degrees (if the curvature is hyperbolic, or “open”) or more than 180 degrees (if the curvature is spherical, or “closed”). Similarly, the geometry of the three-dimensional universe may be either flat (euclidean), or open or closed (noneuclidean), when viewed in the context of Einstein’s four-dimensional spacetime continuum.

By the time Einstein came on the scene, the rules of four-dimensional geometry had been worked out—those of spherical four-space by Georg Friedrich Riemann and those of the four-dimensional hyperbolas by Nikolai Ivanovich Lobachevski and János Bolyai. The whole subject, however, was still regarded as at best difficult and arcane, and at worst almost disreputable.* The legendary mathematician Karl Friedrich Gauss had withheld his papers on noneuclidean geometry from publication, fearing ridicule by his colleagues, and Bolyai conducted his research in the field against the advice of his father, who warned him, “For God’s sake, please give it up. Fear it no less than the sensual passions because it, too, may take up all your time and deprive you of your health, peace of mind and happiness in life.”36

Einstein rushed in where Bolyai’s father feared to tread. With the aid of his old classmate Marcel Grossmann—“Help me, Marcel, or I’ll go crazy!” he wrote—Einstein struggled through the complexities of curved space, seeking to assign the fourth dimension to time and make the whole, infernally complicated affair come out right. He had by now begun to win professional recognition, had quit the patent office to accept a series of teaching positions that culminated in a full professorship in pure research at the University of Berlin, and was doing important work in quantum mechanics and a half-dozen other fields. But he kept returning to the riddle of gravitation, trying to find patterns of beauty and simplicity among thick stacks of papers black with equations. Like a lost explorer discarding his belongings on a trek across the desert, he found it necessary to part company with some of the most cherished of his possessions—among them one of the central precepts of the special theory itself, which to his joy was ultimately to return as a local case within the broader scheme of the general theory. “In all my life I have never before labored so hard,” he wrote to a friend. “… Compared with this problem, the original theory of relativity is child’s play.”37 Nowhere in human history is there to be found a more sustained and heroic labor of the intellect than in Einstein’s trek toward general relativity, nor one that has produced a greater reward.

He completed the theory in November 1915 and published it the following spring. Though its equations are complex, its central conception is startlingly simple. The force of gravitation disappears, and is replaced by the geometry of space itself: Matter curves space, and what we call gravitation is but the acceleration of objects as they slide down the toboggan runs described by their trajectories in time through the undulations of space. The planets skid along the inner walls of a depression in space created by the fat, massive sun; clusters of galaxies rest in spatial hollows like nuggets in a prospector’s bowl.

In marrying gravitational physics to the geometry of curved space, general relativity emancipated cosmology from the ancient dilemma of whether the universe is infinite and unbounded or finite and bounded. An infinite universe would be not just large but infinite, and this posed problems. The gravitational force generated by an infinite number of stars would itself be infinite, and would, therefore, overwhelm the local action of gravity; this prospect so troubled Newton that he resorted to invoking God’s infinite grace to resolve it. Moreover, the light from an infinite number of stars might be expected to turn the night sky into a blazing sheet of light; yet the night sky is dark.* The alternative, however—a finite euclidean universe with an edge to it—was equally unattractive: As Liu Chi posed the question, in China in the fourteenth century, “If heaven has a boundary, what things could be outside it?”39 The difficulty of imagining an end to space had been enunciated as early as the fifth century B.C., by Plato’s colleague Archytas the Pythagorean; Lucretius summed it up this way:

Let us assume for the moment that the universe is limited. If a man advances so that he is at the very edge of the extreme boundary and hurls a swift spear, do you prefer that this spear, hurled with great force, go where it was sent and fly far, or do you think that something can stop it and stand in its way?40

General relativity resolved the matter by establishing that the universe could be both finite—i.e., could contain a finite number of stars in a finite volume of space—and unbounded. The key to this realization lay in Einstein’s demonstration that, since matter warps space, the sum total of the mass in all the galaxies might be sufficient to wrap space around themselves. The result would be a closed, four-dimensionally spherical cosmos, in which any observer, anywhere in the universe, would see galaxies stretching deep into space in every direction, and would conclude, correctly, that there is no end to space. Yet the amount of space in a closed universe would nonetheless be finite: An adventurer with time to spare could eventually visit every galaxy, yet would never reach an edge of space. Just as the surface of the earth is finite but unbounded in two dimensions (we can wander wherever we like, and will not fall off the edge of the earth) so a closed four-dimensional universe is finite but unbounded to us who observe it in three dimensions.

Two-dimensional inhabitants of a finite universe must confront the paradox of an “edge” to their cosmos. But if we add a dimension, curving the plane on which they live into a sphere, their world, though still finite, becomes unbounded. General relativity reveals a similar prospect for the four-dimensional geometry of the universe we three-dimensional creatures inhabit: hence Einstein’s “closed, unbounded” universe.

The question of whether the universe is hyperbolic and open or spherical and closed remains unanswered, as we shall see. But, thanks to Einstein, the problem is no longer clouded by paradox. By introducing the scientific prospect of a finite, unbounded cosmos, Einstein’s general theory initiated a meaningful dialogue between the human mind and the conundrums of cosmological space.

The theory was beautiful, but was it true? Einstein, having been to the mountaintop, felt supremely confident on this score. General relativity explained a precession in the orbit of the planet Mercury that had been left unaccounted for in Newtonian mechanics, and he did not doubt it would survive further tests as well. As he wrote his friend Besso, “I am fully satisfied, and I do not doubt any more the correctness of the whole system…. The sense of the thing is too evident.”41

The wider scientific community, however, awaited the verdict of experiment. There would be a total solar eclipse on May 29, 1919, at which time the sun would stand against the bright stars of the Hyades cluster. The English astronomer Arthur Stanley Eddington led an expedition to a cocoa plantation on Principe Island off west Equatorial Africa to observe the eclipse and see whether the predicted curvature of space in the region of the sun would distort the apparent positions of the stars in the briefly darkened sky. It was a scene of high drama—English scientists testing the theory of a German physicist immediately after the end of the Great War. As the time of the eclipse approached, rain clouds covered the sky. But then, moments after the moon’s shadow came speeding across the landscape and totality began, a hole opened up in the clouds around the sun, and the camera shutters were triggered. The results of Eddington’s expedition, and of a second eclipse observation conducted at Sobral, Brazil on the same day, were presented by the Astronomer Royal at a meeting of the Royal Society in London on November 6, 1919, with Newton’s portrait looking on. They were positive: The light rays coming from the stars of the Hyades were found to be offset to just the degree predicted in the theory.

When Einstein received a telegram from Lorentz announcing the outcome of the Eddington expedition, he showed it to a student, Ilse Rosenthal-Schneider, who asked, “What would you have said if there had been no confirmation?”

“I would have had to pity our dear Lord,” Einstein replied. “The theory is correct.”42*

Subsequent experiments have further vindicated Einstein’s confidence. The curvature of space in the vicinity of the sun was established with much greater accuracy, by bouncing radar waves off Mercury and Venus when they lie near the sun in the sky, and the extent of curvature matched that predicted by the general theory of relativity. A light beam directed up a tower in the Jefferson Physical Laboratory at Harvard University was found to be shifted toward the red by the earth’s gravitation to just the anticipated degree. Maelstroms of energy detected at the centers of violent galaxies indicate that they harbor black holes, collapsed objects wrapped in infinitely curved space that shuts them off from the rest of the universe; the existence of black holes was another prediction of the general theory. And the theory has been tested in many other ways as well—in examinations of entombed dead stars, the whirling of active stars around one another, the wanderings of interplanetary spacecraft well past Jupiter, and the slowing of light as it climbs up out of the sun’s space well—and all these trials it has survived.

Too modest to be immodest, Einstein had written when publishing his completed account of general relativity that “hardly anyone who has truly understood this theory will be able to resist being captivated by its magic.”44 But, even if only those mathematicians and physicists who have mastered general relativity are in a position properly to understand it, still we can all appreciate it to some degree, if, while keeping in mind its basic concepts, we contemplate the universe of effortlessly wheeling galaxies deployed across the blossom petals of gently curving space. Einstein’s epitaph could be Christopher Wren’s: If you seek his monument, look around.

*If, instead, the earth dragged the aether along with it, like a ship gathering up seaweed as it plows through the Sargasso Sea, then the aberration of starlight that Bradley had first observed (the effect of moving through starlight like a woman running in the rain) would not occur.

*I have adopted this metaphor from one employed by Einstein’s colleague Banesh Hoffmann.2

*The American mathematician Ernst Straus was treated to an example of Einstein’s tenacity one afternoon while working as his assistant at the Institute for Advanced Study in Princeton in the 1940s. “We had finished the preparation of a paper and we were looking for a paper clip,” Straus writes. “After opening a lot of drawers we finally found one which turned out to be too badly bent for use. So we were looking for a tool to straighten it. Opening a lot more drawers we came on a whole box of unused paper clips. Einstein immediately started to shape one of them into a tool to straighten the bent one. When I asked him what he was doing, he said, ‘Once I am set on a goal, it becomes difficult to deflect me.’”14

*Maxwell found that the speed with which electromagnetic fields are propagated is equal to the ratio between the electrical force exerted by two electrical charges when at rest and the magnetic force they exert when in motion. As this turned out to be nothing other than the velocity of light, Maxwell concluded that light itself is an electromagnetic field. Since popular accounts of the special theory of relativity sometimes convey the mistaken impression that the velocity of light is an arbitrary speed limit, like that set by legislatures for public highways, it is helpful to keep in mind Maxwell’s finding—that the velocity of light results from a fundamental constant in the equations that describe the behavior of electromagnetic fields.

*A sense of the allure of the dynamo was preserved by the American historian Henry Adams in his The Education of Henry Adams. Describing his visit to the “great hall of dynamos” at the Paris Exposition in 1900, he writes, “To Adams the dynamo became a symbol of infinity. As he grew accustomed to the great gallery of machines, he began to feel the forty-foot dynamos as a moral force, much as the early Christians felt the Cross. The planet itself seemed less impressive, in its old-fashioned, deliberate, annual or daily revolution, than this huge wheel.”22

Research like Herr Weber’s was being applied with dispatch to the execution of convicts and the punishment of malingering conscripts. The first electrocution of a criminal in the United States occurred in 1890, less than ten years after the first public power station in America started operating; the method was purportedly humane, but it took the victim fifteen long minutes to die. Shell-shocked German soldiers in the trenches of the First World War were administered jolts of electricity and then sent back to the front; if they returned, they were given still more severe shocks, in a closed circuit of fear and pain that drove some to suicide.24

*Einstein shared this fate with Newton, whose ideas were routinely characterized as incomprehensible. A student who saw Isaac Newton passing in his carriage is said to have remarked, “There goes the man that writ a book that neither he nor anybody else understands.”

*I was treated to an inadvertent demonstration of this effect one day, aboard a DC-3 in a violent storm over the Bahamas, when a doctor’s iron scale, standing about four feet tall, tore loose from its moorings in the aft end of the cabin. The plane then plunged into a downdraft, rendering everything momentarily weightless, and the scale rose into the air and drifted toward me. I fended it off with my feet, thus briefly experiencing its inertial mass absent its gravitational mass. The fact that the menacing object happened to be a weightless device for measuring weight invested the lesson with a certain ironic intensity.

The definitive experiments were conducted by Baron Roland von Eötvös in Budapest in 1889 and 1922. Eötvös suspended objects of various compositions from threads and looked for deviations in these plumb lines caused by differences between their gravitational mass (which was being pulled straight down) and their inertial mass (which was being pulled sideways, by the rotation of the earth). “In no case,” he wrote, “could we discover any detectable deviation from the law of proportionality of gravitation and inertia.” This remains the case today, although one recent reenactment of the experiment did produce subtle anomalies that could not immediately be accounted for.

*The very term “fourth dimension” called to mind the enthusiasms of eccentrics and ecstatics like Charles Hinton, who sought to enhance his appreciation of its subtleties by manipulating 81 cubes that represented the units of a 3-by-3-by-3-by-3-unit euclidean hypercube. Hinton’s career was interrupted—and his subject cast into further ignominy—when he was convicted of bigamy for living out the free-love philosophy of his father, who liked to say that “Christ was the Savior of men, but I am the savior of women, and I don’t envy Him a bit!” Hinton fits dropped dead at a banquet of the Society of Philanthropic Inquiry in Washington, D.C., moments after delivering a toast in honor of femininity.35

*This disturbing puzzle, known today as Olbers’s paradox after the nineteenth-century German astronomer Wilhelm Olbers, was discovered independently by other astronomers, among them Halley, who lectured on it at a Royal Society meeting in 1721. Newton chaired that meeting, but for some reason never wrote about the paradox. The historian of science Michael Hoskin suggests that the old man was napping while Halley spoke.38

Alternately, general relativity allows that the universe might be structured like a four-dimensional hyperbola, in which case it would be both infinite and unbounded. This possibility resurrects some of the difficulties that afflict all infinite-universe models, but they could perhaps be resolved, should the observational data indicate that space is indeed hyperbolically rather than spherically curved.

*Einstein once astonished Ernst Straus by saying of Max Planck, the father of quantum physics, “He was one of the finest people I have ever known and one of my best friends; but, you know, he didn’t really understand physics.” When Straus asked what he meant, Einstein replied, “During the eclipse of 1919, Planck stayed up all night to see if it would confirm the bending of light by the gravitational field of the sun. If he had really understood the way the general theory of relativity explains the equivalence of inertial and gravitational mass, he would have gone to bed the way I did.”43