Black holes have now been discovered all over the universe, including one at the center of our galaxy that is a few million times more massive than our sun. “Black holes are not rare, and they are not an accidental embellishment of our universe,” says Dyson. “They are the only places in the universe where Einstein’s theory of relativity shows its full power and glory. Here, and nowhere else, space and time lose their individuality and merge together in a sharply curved four-dimensional structure precisely delineated by Einstein’s equations.”4
Einstein believed that his general theory solved Newton’s bucket issue in a way that Mach would have liked: inertia (or centrifugal forces) would not exist for something spinning in a completely empty universe.* Instead, inertia was caused only by rotation relative to all the other objects in the universe. “According to my theory, inertia is simply an interaction between masses, not an effect in which ‘space’ of itself is involved, separate from the observed mass,” Einstein told Schwarzschild. “It can be put this way. If I allow all things to vanish, then according to Newton the Galilean inertial space remains; following my interpretation, however, nothing remains.”5
The issue of inertia got Einstein into a debate with one of the great astronomers of the time, Willem de Sitter of Leiden. Throughout 1916, Einstein struggled to preserve the relativity of inertia and Mach’s principle by using all sorts of constructs, including assuming various “border conditions” such as distant masses along the fringes of space that were, by necessity, unable to be observed. As de Sitter noted, that in itself would have been anathema to Mach, who railed against postulating things that could not possibly be observed.6
By February 1917, Einstein had come up with a new approach. “I have completely abandoned my views, rightly contested by you,” he wrote de Sitter. “I am curious to hear what you will have to say about the somewhat crazy idea I am considering now.”7 It was an idea that initially struck him as so wacky that he told his friend Paul Ehrenfest in Leiden, “It exposes me to the danger of being confined to a madhouse.” He jokingly asked Ehrenfest for assurances, before he came to visit, that there were no such asylums in Leiden.8
His new idea was published that month in what became yet another seminal Einstein paper, “Cosmological Considerations in the General Theory of Relativity.”9 On the surface, it did indeed seem to be based on a crazy notion: space has no borders because gravity bends it back on itself.
Einstein began by noting that an absolutely infinite universe filled with stars and other objects was not plausible. There would be an infinite amount of gravity tugging at every point and an infinite amount of light shining from every direction. On the other hand, a finite universe floating at some random location in space was inconceivable as well. Among other things, what would keep the stars and energy from flying off, escaping, and depleting the universe?
So he developed a third option: a finite universe, but one without boundaries. The masses in the universe caused space to curve, and over the expanse of the universe they caused space (indeed, the whole four-dimensional fabric of spacetime) to curve completely in on itself. The system is closed and finite, but there is no end or edge to it.
One method that Einstein employed to help people visualize this notion was to begin by imagining two-dimensional explorers on a two-dimensional universe, like a flat surface. These “flatlanders” can wander in any direction on this flat surface, but the concept of going up or down has no meaning to them.
Now, imagine this variation: What if these flatlanders’ two dimensions were still on a surface, but this surface was (in a way very subtle to them) gently curved? What if they and their world were still confined to two dimensions, but their flat surface was like the surface of a globe? As Einstein put it, “Let us consider now a two-dimensional existence, but this time on a spherical surface instead of on a plane.” An arrow shot by these flatlanders would still seem to travel in a straight line, but eventually it would curve around and come back—just as a sailor on the surface of our planet heading straight off over the seas would eventually return from the other horizon.
The curvature of the flatlanders’ two-dimensional space makes their surface finite, and yet they can find no boundaries. No matter what direction they travel, they reach no end or edge of their universe, but they eventually get back to the same place. As Einstein put it, “The great charm resulting from this consideration lies in the recognition that the universe of these beings is finite and yet has no limits.” And if the flatlanders’ surface was like that of an inflating balloon, their whole universe could be expanding, yet there would still be no boundaries to it.10
By extension, we can try to imagine, as Einstein has us do, how three-dimensional space can be similarly curved to create a closed and finite system that has no edge. It’s not easy for us three-dimensional creatures to visualize, but it is easily described mathematically by the non-Euclidean geometries pioneered by Gauss and Riemann. It can work for four dimensions of spacetime as well.
In such a curved universe, a beam of light starting out in any direction could travel what seems to be a straight line and yet still curve back on itself. “This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived,” the physicist Max Born has declared.11
Yes, but what is outside this curved universe? What’s on the other side of the curve? That’s not merely an unanswerable question, it’s a meaningless one, just as it would be meaningless for a flatlander to ask what’s outside her surface. One could speculate, imaginatively or mathematically, about what things are like in a fourth spatial dimension, but other than in science fiction it is not very meaningful to ask what’s in a realm that exists outside of the three spatial dimensions of our curved universe.12
This concept of the cosmos that Einstein derived from his general theory of relativity was elegant and magical. But there seemed to be one hitch, a flaw that needed to be fixed or fudged. His theory indicated that the universe would have to be either expanding or contracting, not staying static. According to his field equations, a static universe was impossible because the gravitational forces would pull all the matter together.
This did not accord with what most astronomers thought they had observed. As far as they knew, the universe consisted only of our Milky Way galaxy, and it all seemed pretty stable and static. The stars appeared to be meandering gently, but not receding rapidly as part of an expanding universe. Other galaxies, such as Andromeda, were merely unexplained blurs in the sky. (A few Americans working at the Lowell Observatory in Arizona had noticed that the spectra of some mysterious spiral nebulae were shifted to the red end of the spectrum, but scientists had not yet determined that these were distant galaxies all speeding away from our own.)
When the conventional wisdom of physics seemed to conflict with an elegant theory of his, Einstein was inclined to question that wisdom rather than his theory, often to have his stubbornness rewarded. In this case, his gravitational field equations seemed to imply—indeed, screamed out—that the conventional thinking about a stable universe was wrong and should be tossed aside, just as Newton’s concept of absolute time was.13
Instead, this time he made what he called a “slight modification” to his theory. To keep the matter in the universe from imploding, Einstein added a “repulsive” force: a little addition to his general relativity equations to counterbalance gravity in the overall scheme.
In his revised equations, this modification was signified by the Greek letter lambda, λ, which he used to multiply his metric tensor gμν in a way that produced a stable, static universe. In his 1917 paper, he was almost apologetic: “We admittedly had to introduce an extension of the field equations that is not justified by our actual knowledge of gravitation.”
He dubbed the new element the “cosmological term” or the “cosmological constant” (kosmologische Glied was the phrase he used). Later,* when it was discovered that the universe was in fact expanding, E
instein would call it his “biggest blunder.” But even today, in light of evidence that the expansion of the universe is accelerating, it is considered a useful concept, indeed a necessary one after all.14
During five months in 1905, Einstein had upended physics by conceiving light quanta, special relativity, and statistical methods for showing the existence of atoms. Now he had just completed a more prolonged creative slog, from the fall of 1915 to the spring of 1917, which Dennis Overbye has called “arguably the most prodigious effort of sustained brilliance on the part of one man in the history of physics.” His first burst of creativity as a patent clerk had appeared to involve remarkably little anguish. But this later one was an arduous and intense effort, one that left him exhausted and wracked with stomach pains.15
During this period he generalized relativity, found the field equations for gravity, found a physical explanation for light quanta, hinted at how the quanta involved probability rather than certainty,† and came up with a concept for the structure of the universe as a whole. From the smallest thing conceivable, the quantum, to the largest, the cosmos itself, Einstein had proven a master.
The Eclipse, 1919
For general relativity, there was a dramatic experimental test that was possible, one that had the potential to dazzle and help heal a war-weary world. It was based on a concept so simple that everyone could understand it: gravity would bend light’s trajectory. Specifically, Einstein predicted the degree to which light from a distant star would be observed to curve as it went through the strong gravitational field close to the sun.
To test this, astronomers would have to plot precisely the position of a star in normal conditions. Then they would wait until the alignments were such that the path of light from that star passed right next to the sun. Did the star’s position seem to shift?
There was one exciting challenge. This observation required a total eclipse, so that the stars would be visible and could be photographed. Fortunately, nature happened to make the size of the sun and moon just properly proportional so that every few years there are full eclipses observable at times and places that make them ideally suited for such an experiment.
Einstein’s 1911 paper, “On the Influence of Gravity on the Propagation of Light,” and his Entwurf equations the following year, had calculated that light would undergo a deflection of approximately (allowing for some data corrections subsequently made) 0.85 arc-second when it passed near the sun, which was the same as would be predicted by an emission theory such as Newton’s that treated light as particles. As previously noted, the attempt to test this during the August 1914 eclipse in the Crimea had been aborted by the war, so Einstein was saved the potential embarrassment of being proved wrong.
Now, according to the field equations he formulated at the end of 1915, which accounted for the curvature of spacetime caused by gravity, he had come up with twice that deflection. Light passing next to the sun should be bent, he said, by about 1.7 arc-seconds.
In his 1916 popular book on relativity, Einstein issued yet another call for scientists to test this conclusion. “Stars ought to appear to be displaced outwards from the sun by 1.7 seconds of arc, as compared with their apparent position in the sky when the sun is situated at another part of the heavens,” he said.“The examination of the correctness or otherwise of this deduction is a problem of the greatest importance, the early solution of which is to be expected of astronomers.”16
Willem de Sitter, the Dutch astrophysicist, had managed to send a copy of Einstein’s general relativity paper across the English Channel in 1916 in the midst of the war and get it to Arthur Eddington, who was the director of the Cambridge Observatory. Einstein was not wellknown in England, where scientists then took pride in either ignoring or denigrating their German counterparts. Eddington became an exception. He embraced relativity enthusiastically and wrote an account in English that popularized the theory, at least among scholars.
Eddington consulted with the Astronomer Royal, Sir Frank Dyson, and came up with the audacious idea that a team of English scientists should prove the theory of a German, even as the two nations were at war. In addition, it would help solve a personal problem for Eddington. He was a Quaker and, because of his pacifist faith, faced imprisonment for refusing military service in England. (In 1918, he was 35 years old, still subject to conscription.) Dyson was able to convince the British Admiralty that Eddington could best serve his nation by leading an expedition to test the theory of relativity during the next full solar eclipse.
That eclipse would occur on May 29, 1919, and Dyson pointed out that it would be a unique opportunity. The sun would then be amid the rich star cluster known as the Hyades, which we ordinary stargazers recognize as the center of the constellation Taurus. But it would not be convenient. The eclipse would be most visible in a path that stretched across the Atlantic near the equator from the coast of Brazil to Equatorial Africa. Nor would it be easy. As the expedition was being considered in 1918, there were German U-boats in the region, and their commanders were more interested in the control of the seas than in the curvature of the cosmos.
Fortunately, the war ended before the expeditions began. In early March 1919, Eddington sailed from Liverpool with two teams. One group split off to set up their cameras in the isolated town of Sobral in the Amazon jungle of northern Brazil. The second group, which included Eddington, sailed for the tiny island of Principe, a Portuguese colony a degree north of the equator just off the Atlantic coast of Africa. Eddington set up his equipment on a 500-foot bluff on the island’s north tip.17
The eclipse was due to begin just after 3:13 p.m. local time on Principe and last about five minutes. That morning it rained heavily. But as the time of the eclipse approached, the sky started to clear. The heavens insisted on teasing and tantalizing Eddington at the most important minutes of his career, with the remaining clouds cloaking and then revealing the elusive sun.
“I did not see the eclipse, being too busy changing plates, except for one glance to make sure it had begun and another halfway through to see how much cloud there was,” Eddington noted in his diary. He took sixteen photographs. “They are all good of the sun, showing a very remarkable prominence; but the cloud has interfered with the star images.” In his telegram back to London that day, he was more telegraphic: “Through cloud, hopeful. Eddington.”18
The team in Brazil had better weather, but the final results had to wait until all of the photographic plates from both places could be shipped back to England, developed, measured, and compared. That took until September, with Europe’s scientific cognoscenti waiting eagerly. To some spectators, it took on the postwar political coloration of a contest between the English theory of Newton, predicting about 0.85 arc-second deflection, and the German theory of Einstein, predicting a 1.7 arc-seconds deflection.
The photo finish did not produce an immediately clear result. One set of particularly good pictures taken in Brazil showed a deflection of 1.98 arc-seconds. Another instrument, also at the Brazil location, produced photographs that were a bit blurrier, because heat had affected its mirror; they indicated a 0.86 deflection, but with a higher margin of error. And then there were Eddington’s own plates from Principe. These showed fewer stars, so a series of complex calculations were used to extract some data. They seemed to indicate a deflection of about 1.6 arc-seconds.
The predictive power of Einstein’s theory—the fact that it offered up a testable prediction—perhaps exercised a power over Eddington, whose admiration for the mathematical elegance of the theory caused him to believe in it deeply. He discarded the lower value coming out of Brazil, contending that the equipment was faulty, and with a slight bias toward his own fuzzy results from Africa got an average of just over 1.7 arc-seconds, matching Einstein’s predictions. It wasn’t the cleanest confirmation, but it was enough for Eddington, and it turned out to be valid. He later referred to getting these results as the greatest moment of his life.19
In Berlin, Einstein put on an appear
ance of nonchalance, but he could not completely hide his eagerness as he awaited word. The downward spiral of the German economy in 1919 meant that the elevator in his apartment building had been shut down, and he was preparing for a winter with little heat. “Much shivering lies ahead for the winter,” he wrote his ailing mother on September 5. “There is still no news about the eclipse.” In a letter a week later to his friend Paul Ehrenfest in Holland, Einstein ended with an affected casual question: “Have you by any chance heard anything over there about the English solar-eclipse observation?”20
Just by asking the question Einstein showed he was not quite as sanguine as he tried to appear, because his friends in Holland would certainly have already sent him such news if they had it. Finally they did. On September 22, 1919, Lorentz sent a cable based on what he had just heard from a fellow astronomer who had talked to Eddington at a meeting: “Eddington found stellar shift at solar limb, tentative value between nine-tenths of a second and twice that.” It was wonderfully ambiguous. Was it a shift of 0.85 arc-second, as Newton’s emission theory and Einstein’s discarded 1912 theory would have it? Or twice that, as he now predicted?
Einstein had no doubts. “Today some happy news,” he wrote his mother. “Lorentz telegraphed me that the British expeditions have verified the deflection of light by the sun.”21 Perhaps his confidence was partly an attempt to cheer up his mother, who was suffering from stomach cancer. But it is more likely that it was because he knew his theory was correct.
Einstein was with a graduate student, Ilse Schneider, shortly after Lorentz’s news arrived. “He suddenly interrupted the discussion,” she later recalled, and reached for the telegram that was lying on a window sill. “Perhaps this will interest you,” he said, handing it to her.
Naturally she was overjoyed and excited, but Einstein was quite calm. “I knew the theory was correct,” he told her.
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