The Perfect Theory (5 page)

Hilbert had surrounded himself with some of the most important mathematicians in the world. One of his colleagues had been Hermann Minkowski, who had shown Einstein how his special theory of relativity could be written in a far more elegant, mathematical language—the “superfluous erudition” that Einstein had disparaged a few years before. Hilbert's students and assistants—such as Hermann Weyl, John von Neumann, and Ernst Zermelo—would be leading figures in twentieth-century mathematics. Along with his group at Göttingen, Hilbert had grand plans: to construct a complete theory of the natural world based on first principles, just as in mathematics. He saw Einstein's work as an integral part of his project.

During Einstein's short visit to Göttingen in June of 1915, Einstein lectured and Hilbert took notes. They discussed and argued back and forth about the details. Einstein was strong on the physics and Hilbert on the mathematics. But they didn't make any progress. Einstein, still wary of mathematics and still shaky in his understanding of Riemannian geometry, found it difficult to completely understand Hilbert's detailed, technical points.

Shortly after Einstein ended what seemed like a fruitless visit, he began to doubt his new theory of relativity. He already knew that it wasn't truly general—when he and Grossmann had finished their papers in 1913, it was clear to him that the law of gravity
still
didn't fit. And some of his predictions were off. For example, his theory predicted a drift for Mercury, very much as Le Verrier had observed almost fifty years before, but it wasn't
exactly
right. It was still off by a factor of two. Einstein had to look at his equations again.

Over a period of just three weeks, Einstein decided to ditch the new law of gravity that he had proposed with Grossmann, which didn't obey the general principle of relativity. He wanted a law of gravity that would be true in any reference frame, much as he had already done with the other laws of physics. And he wanted to use the new Riemannian geometry that he had learned from Grossmann. Every few days he would tweak what he had done before, writing down a law, relaxing some assumptions while imposing others. And as he did so, he shed some of the physical prejudices that had held him back and delved deeper and deeper into the mathematics that he had learned. He realized that even though his physical intuition had served him well throughout his spectacular career, he had to be careful not to let it cloud the bigger picture coming out of the mathematics.

Finally, by the end of November, he realized he had done it. He had finally discovered a general law for gravity that satisfied the general principle of relativity. On the scale of the solar system it was accurately approximated by Newtonian gravity, exactly as it should be. Moreover, it predicted Le Verrier's precession of the perihelion of Mercury bang-on. And it predicted that as light rays passed by a heavy object, they would be bent even more—in fact twice as much as he had originally predicted when he first thought of the idea in Prague.

Einstein's completed general theory of relativity offered an entirely new way of understanding physics, one that superseded the Newtonian view that had held sway for centuries. His theory provided a set of equations that came to be known as the “Einstein field equations.” Although the idea behind them, relating the geometry of Gauss and Riemann with gravity, was beautiful—“elegant,” as physicists would want to call it—the detailed equations could look like a mess. They were, in practice, a set of ten equations of ten functions of the geometry of space and time, all nonlinearly tangled up and intertwined such that, in general, it was impossible to solve for one function at a time. They all had to be tackled together, head-on—a truly daunting prospect. Yet they held much promise, for their solutions could be used to predict what would happen in the natural world, from the motion of a bullet or an apple falling off a tree to the movement of planets in the solar system. The secrets of the universe, it seemed, were to be found by solving Einstein's equations.

On November 25, 1915, Einstein presented his new equations to the Prussian Academy of Sciences in a short three-page paper. His new law of gravity was radically different from what anyone had ever proposed before. In essence, Einstein argued that what we perceive as gravity is nothing more than objects moving in the geometry of spacetime. Massive objects affect the geometry, curving space and time. Einstein had finally arrived at his truly general theory of relativity.

But Einstein was not alone. Hilbert had been mulling over Einstein's Göttingen lectures and had, without Einstein realizing, made his own attempts at coming up with new gravitational equations. Completely independently, Hilbert had come up with exactly the same gravitational law. On the twentieth of November, five days before Einstein's presentation to the academy in Berlin, Hilbert presented his own results to the Royal Society of Sciences in Göttingen. It seemed as if Hilbert had scooped Einstein.

During the weeks following the announcements, relations between Hilbert and Einstein were strained. Hilbert wrote to Einstein claiming he didn't remember the bit in one of his lectures in which Einstein had discussed his attempts at building the gravitational equations, and by Christmas, Einstein was satisfied that there hadn't been any foul play. As Einstein said, in a letter to Hilbert, to begin with
“there has been between us something like a bad feeling,” but he had come to terms with what had happened, so much so that “I once more think of you in unclouded friendship. . . .” They would indeed remain friends and colleagues, for Hilbert stepped back from claiming any credit for Einstein's magnum opus. In fact, until he died, Hilbert always referred to the equations that both he and Einstein had stumbled upon as “Einstein's equations.”

Einstein had completed his trek. He had gradually succumbed to the power of mathematics to reach his final equations. From then on he would let himself be guided not only by his thought experiments but also by the mathematics. The sheer mathematical beauty of his final theory stunned him. He described his equations as “the most valuable discovery of my life.”

 

Eddington had been receiving the slow trickle of offprints coming out of Prague, then Zurich, and finally Berlin from a friend, the astronomer Willem de Sitter, from Holland. He was intrigued, hooked by this completely new way of looking at gravity in a difficult language. Even though he was an astronomer, and his job was to measure and observe things and try to interpret them, he was up to the challenge of learning the new mathematics of Riemannian geometry that Einstein had used to write up his theory. And it was well worth looking into, especially since Einstein had made quite clear predictions that could be used to test his theory. In fact, an eclipse was predicted to occur on the twenty-ninth of May, 1919, an ideal opportunity for such a test, and Eddington would be the obvious person to lead a team of observers.

There was only one problem, and a massive one at that. Europe was at war, Eddington was a pacifist, and Einstein was in cahoots with the enemy. Or so Eddington's colleagues wanted him to believe. As the war reached its climax in 1918, the risk of the German army completely engulfing the British and the French grew, leading to a renewed wave of conscriptions. Eddington was called up to fight, but he had something else on his mind.

While Eddington had become an enthusiastic advocate of Einstein's new gravity, he faced the antipathy of his colleagues. In an attempt to dismiss German science as having no worth, one of his colleagues declared,
“We have tried to think that exaggerated and false claims made by Germans today were due to some purely temporary disease of quite recent growth. But an instance like this makes one wonder whether the sad truth may not lie deeper.” And while Eddington had the support of the Astronomer Royal, Frank Dyson, to lead the eclipse expedition, he had to escape being sent to jail for refusing to fight. The British government convened a tribunal in Cambridge to look into Eddington's stance. As the hearing proceeded, the tribunal viewed him with increasing hostility. Eddington was going to be refused exemption until Frank Dyson stepped in. Eddington was a crucial player in the eclipse expedition, Dyson said, and furthermore, “under present conditions the eclipse will be observed by very few people. Prof. Eddington is peculiarly qualified to make these observations, and I hope the Tribunal will give him permission to undertake this task.” The eclipse intrigued the tribunal, and Eddington was once again given an exemption for “national importance.” Einstein had saved him from the front.

 

From Einstein's theory there was a prediction: that the light emitted from distant stars would bend as it passed close to a massive body such as the sun. Eddington's experiment proposed to observe one such distant cluster of stars, the Hyades, at two different times of the year. He would first accurately measure the positions of the stars in the Hyades cluster on a clear night, with nothing obscuring his view and nothing in the way to bend their light rays. Then he would measure their position again, this time with the sun in front. It would have to be done during a total eclipse, when almost all the bright light of the sun would be blocked by the moon. On the twenty-ninth of May, 1919, the Hyades would lie right behind the sun and conditions would be perfect. A comparison of the two measurements—one with the sun and one without—would show if there was any deflection. And if that deflection was about four-thousandths of a degree, or 1.7 arcseconds, it would be just as Einstein claimed. Such a clear and simple goal.

It wasn't actually that simple. The few places on the Earth where one could witness the total eclipse were remote and far apart. The astronomers would need to travel quite far, in a world that had just come out of a devastating war, to set up their equipment. Eddington, along with Edward Cottingham from the Greenwich Observatory, set up shop on the island of Príncipe. A backup team of two astronomers, Andrew Crommelin and Charles Davidson, was dispatched to a village called Sobral in the interior of the Brazilian Nordeste, a poor, dusty region near the equator.

Príncipe is a small island in the Gulf of Guinea, a Portuguese colony known for its cocoa. A lush green island, hot, humid, and periodically peppered with tropical storms, it had a few large
roças,
or plantations, spread out where a few Portuguese landowners used the local inhabitants to farm the land. For decades it had supplied the cocoa beans to the Cadbury corporation. At the beginning of the twentieth century the cacao plantations were accused of using slave labor and lost their contracts, destroying Príncipe's economy. When Eddington arrived, the island was slipping into oblivion.

Eddington set up his apparatus in a remote corner at the Roça Sundy, where he was looked after by the landowner. Between daily tennis matches on the only court on the island, he waited for the day of the eclipse, praying that the recurring rainstorms and gray skies wouldn't sabotage his mission. Cottingham primed the telescope, hoping that the heat wouldn't distort the images.

On the morning of the eclipse, it rained heavily and the sky was completely impenetrable until, less than an hour before totality, it started to clear. Eddington and Cottingham caught their first glimpse of the eclipse that was under way with part of the sun already obscured. By 2:15 in the afternoon, the sky was clear and Eddington and Cottingham could take their measurements—sixteen photographic plates of the sun with the Hyades cluster lurking in the background. By the end of the eclipse, the sky was beautiful, clear of any cloud. Eddington telegraphed a message to Frank Dyson:
“Through cloud. Hopeful.”

The cloudy start to the experiment in Príncipe may have saved the day. In Sobral in the Brazilian Nordeste, there was a perfectly clear and hot day on which the eclipse could be followed right from the very start. Crommelin and Davidson were surrounded by the jubilant locals to witness the historic event and were able to take nineteen plates to complement the sixteen taken by Eddington and Cottingham. Exultant, they also telegraphed back:
“Eclipse Splendid.” At the time, they didn't realize that the good viewing conditions and hot clear weather in Brazil had sabotaged their main experiment. The heat had warped the apparatus so much that the measurements on the photographic plates were rendered useless. It was only with backup observations with a smaller telescope that the expedition to Sobral was able to contribute data to the experiment.

The astronomers were unable to return home quickly, and it was only in late July that the various photographic plates began to be analyzed. Of the sixteen plates that Eddington had recorded, only two had enough stars to measure the deflection properly. The value they got was 1.61 arcseconds with an error of 0.3 arcseconds, consistent with Einstein's prediction of 1.7 arcseconds. When the plates from Sobral were analyzed, the results were worrying. The value measured was 0.93 arcseconds, far from the relativistic prediction and very close to the Newtonian prediction, but these were the same plates that had been deformed by the heat. When the backup observations from Sobral, undertaken on the smaller telescope, were analyzed, the deflection came out at 1.98 arcseconds with a very small error of 0.12 arcseconds. Einstein's prediction, again.

 

On November 6, 1919, the team of explorers presented their results to a joint meeting of the Royal Society and the Royal Astronomical Society. In a series of talks led by Frank Dyson, the different measurements from the eclipse expedition were laid out in front of an audience of their distinguished peers. Once the problems that had faced the Sobral expedition were taken into account, the speakers showed that the eclipse measurements spectacularly confirmed Einstein's prediction.

J. J. Thomson, the president of the Royal Society, described the measurements as “the most important result obtained in connection with the theory of gravitation since Newton's day.” He added, “If it is sustained that Einstein's reasoning holds good—and it has survived two very severe tests in connection with the perihelion of Mercury and the present eclipse—then it is the result of one of the highest achievements in human thought.”