The Hunt for Vulcan

by Thomas Levenson

  Then, after about eight months of letting his mind wander, he persuaded himself to confront gravity one last time.

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  David Hilbert was the most influential German mathematician of his day. He remains famous both for his own work and for his list of twenty-three “Hilbert Problems”—questions unsolved as of 1900 collected in a sally aiming to shape twentieth-century mathematical research. For Einstein, Hilbert, a professor at the University of Göttingen, had special significance: he was one of the very few first-rank mathematicians who took an interest in his work. He did what no one in Berlin had thought to ask: he invited Einstein to give a series of six in-depth lectures on the current state of his thinking.

  In those talks, delivered in late June and early July, Einstein still believed that his work of the previous two years was largely satisfactory. Never mind that he couldn’t quite bring his new general theory into perfect agreement with the special one, nor that the equations he announced in 1914 did not yield the correct orbit for Mercury. The transformation of gravity into geometry was, he remained convinced, correct in all but details. That was what he said in Göttingen, and those lectures had, he felt, left Hilbert ready to accept the necessity of his new approach to gravitation.

  He was right. Hilbert believed him, so much so that he began working on his own version of a theory of gravity compatible with special relativity. It’s not clear when Einstein realized he had a competitor working on the same problem he’d wrestled with for eight years, nor when he decided to reexamine his work with a newly critical eye. It can’t have been later than September 30, when he told his friend and supporter Freundlich that his theory was in deep trouble. The trigger was his sudden grasp of a question he’d faced as far back as 1912, one posed within an idealized representation of a rotating system. Analyzing acceleration in that rotating frame of reference spat out results that seemed to violate the equivalence of acceleration and gravity—the founding principle of the entire effort. It was, he told Freundlich “a blatant contradiction”—fatal for the theory as it stood. That same flaw, he added, would explain the theory’s inability to generate an accurate orbit for Mercury. Worst of all, he couldn’t see a way forward. “I do not believe that I myself am in the position to find the error,” he wrote, “because my mind follows the same rut too much in this matter.”*1

  What a cry for help! Freundlich didn’t reply (or at least no letter survives), and in any event he was no deep theorist. It didn’t matter. Sometime over the next week, Einstein figured out how to proceed. As soon as the solution started to form in his brain, he went almost completely silent. From the 8th of October, he wrote just four letters—two brief notes to organizations, one to a friend in Zurich, mostly about family, and one substantial scientific memo to an older researcher whom Einstein greatly admired, the Dutch physicist Hendrik Lorentz, in which he discussed some of his emerging ideas about gravitation. For the rest, Einstein seems to have devoted all his time to thought and calculation.

  It was, he would recall, the most intense labor of his life.

  Even though there is no record of the detailed sequence of his work over the next six weeks, the broad outline of this final push is clear. In one of Einstein’s notebooks from his prewar collaboration with Grossman, he had worked through some ideas that added up to an almost-complete version of what would become his final result. In 1913 he’d played with them briefly, and then rejected them. Now, two years on, he was ready to take another look.

  Albert Einstein in Berlin in 1914

  For the next several weeks he used that earlier approach to solve the biggest objection to his prior work, to show that acceleration and gravitation remain equivalent under all circumstances. October ended. Einstein was almost all the way there, and he knew it. On Thursday, November 4, he made his way along Unter den Linden and presented the Prussian Academy with the first of four updates on his progress. He still hadn’t fully fleshed out the new theory—the biggest piece missing was the final, correct equation for the gravitational field. He hadn’t calculated a specific result for any of the key tests of the idea. But the work made sense now; it was, at last, internally consistent. Equally important, he showed that an approximate solution to his field equations reproduced Newton’s laws of motion—just as it must, given how spectacularly Newton’s system of the world had accounted for almost all of the motions of the solar system.

  —

  The next Thursday he went back to the Academy with an update that contained what turned out to be a mistake, one he would fix two weeks later. He returned home and continued to think and calculate. A week passed. The theory was becoming robust enough to compare to reality. Never mind the problems remaining in his math, he would write. They were formal concerns, not physical ones, and, he wrote, “I am satisfied” not to worry about the fine print “for the time being.”

  Instead, he turned to the heart of the matter: he placed “a point mass, the sun, at the origin of the co-ordinate system.” Next, he calculated the gravitational field such a point mass would produce. Analyzing events within that field, the first novelty appeared almost immediately: the bending of starlight around the sun, just as in his earlier theory—but with this difference: a ray of light passing through the sun’s gravitational field would deflect by 1.7 arcseconds, double the number his 1913 theory predicted.

  That was the prelude, an undercard bout. The main event was at hand, a demonstration that his theory captured something of reality that no other idea could explain. Two weeks earlier, he’d shown that Newton’s gravity emerged naturally from a first approximation—a kind of low-resolution image—of his new mathematical account of gravitation. He repeated the analysis, and extended it to the obvious next question: what emerges from solutions to a second order, effectively, an exploration at higher resolution? A page of mathematical argument followed, yielding a new equation with one term altered from the Newtonian approximation.

  Seven more steps, and he had it: an equation that he could use to analyze the orbit of a planet tracking round its star, still sitting in the center of his coordinate system. If he knew by observation just a handful of parameters, he would then be able to generate a prediction for the perihelion advance for any body in orbit around its central star.

  —

  November 1915, between the 11th and the 18th.

  Einstein gathers the data for Mercury. He writes down its period. He enters its orbital parameters and its nearest approach to the sun. He injects the speed of light into his mathematical apparatus. He does the arithmetic. As he completes each step of the operation, numbers emerge. He peers at the result….

  —

  November 18, 1915.

  Masking his emotions behind the required decorousness of scientific communication, Einstein revealed almost no sign of any excitement in his presentation to the Prussian Academy. “The calculation for the planet Mercury yields,” he told his audience, “a perihelion advance of 43 arc seconds per century, while the astronomers assign 45″ +/- 5″ per century as the unexplained difference between observations and the Newtonian theory.” Belaboring the obvious, he added that “this theory therefore agrees completely with the observations.”

  Such neutral tones could not conceal the explosion thus detonated. Decades of attempts to save the Newtonian worldview were at an end. Vulcan was gone, dead, utterly unnecessary. No chunk of matter was required to explain Mercury’s track, no undiscovered planet, no asteroid belt, no dust, no bulging solar belly, nothing at all—except this new, radical conception of gravity. The sun with its great mass creates its dent in space-time. Mercury, so firmly embraced by our star’s gravitational field, lies deep within that solar gravity well.*2 Like all objects navigating space-time, Mercury’s motion follows that warping, four-dimensional curve…until, as Einstein finally captured in all the abstract majesty of his mathematics, the orbit of the innermost planet precesses away from the Newtonian ideal.

  It was said of Newton that he was a fortunate man, because
there was only one universe to discover, and he had done it. It had been said of Le Verrier that he discovered a planet at the tip of his pen. On the 18th of November, 1915, Einstein’s pen destroyed Vulcan—and reimagined the cosmos.

  —

  In private, among friends, Einstein allowed himself to feel his victory. The equations themselves had simply cranked out the correct orbit. Put the numbers in, and out pops Mercury—as if, to use his own word, by magic. Einstein felt all the pure wonder of that perfect match between theory and reality. Working at his desk, some time in the week before he rose before the Academy, the correct answer appeared as he cranked through the final steps. That was when, he told a friend, his heart actually shuddered in his chest—genuine palpitations. He wrote that it was as if something had snapped within him, and told another friend that he was “beside himself with joy.”

  The sun’s great dent in space-time dictates the path Mercury must take—an ellipse, precessing around its major focus in precise agreement with observations first quantified by Le Verrier in 1859.

  Much later, Einstein tried again to describe what he felt at that first, private instant of great discovery. He couldn’t. “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,” he wrote, “are known only to him who has experienced them.”

  * * *

  *1 I hope it doesn’t make me a terrible person to say how much I love this admission. Einstein himself never had any doubt of his own fallibility, but it does help the rest of us to be reminded that even the best minds can chase their own tails. At least it helps me.

  *2 Farther out, the sun’s influence on local space-time moderates, and the orbits of the other planets approach the Newtonian approximation (though with modern instruments, it is possible to detect a relativistic component in the orbits of several solarsystem bodies).

  Postscript

  “THE LONGING TO BEHOLD…PREEXISTING HARMONY”

  Three weeks into the era of general relativity, Vulcan was gone forever. After half a century in which it had been at once necessary and absent, it was finally revealed to be pure fiction. Its repeated “discovery” was nothing more than an object lesson in how easy it is to see what ought to be rather than what is.

  Still, Einstein’s dispatch of a ghost planet bedeviling Mercury didn’t quite finish the job he’d set himself. On November 25, 1915, he returned to the center of Berlin for the fourth Thursday in a row. At the Academy, he rose to present his final theory of gravity. No errors remained, no unnecessary assumptions, no special observers. The work was done. He came to a close, paused, perhaps, to make conversation as necessary, and left.

  The glow lingered. A few days later, he told Besso that he was “content, but a little worn out.” He allowed a bit more to show through in a letter to a physicist friend. Study the equations well, he wrote, for “they are the most valuable discovery of my life.” In its most compact form, that discovery boils down to a single equation, now called Einstein’s, a single line of symbols from which all else flows:

  On one side lies space-time; on the other lies matter-energy: together, the two halves of the universe. The equation defines their relationship—most simply, it shows how matter and energy together tell space-time (the universe) what shape to be, and how space-time tells matter-energy (all that the universe contains) how to move. The result is a universal theory, an account of the shape of the cosmos, its evolution, and even, potentially, its ultimate fate.

  —

  In late 1915, almost all the world had no idea that an intellectual revolution had just been won. World War I had three more bitter, brutal years to run. Even those few who truly grasped the implications of general relativity could not evade the war’s reach. Karl Schwarzschild devoured general relativity as soon as Einstein’s published lectures reached him, at the Eastern Front. In February 1916, still on active service, he worked out the first exact solution to Einstein’s field equations—a result that pointed to what we now call a black hole. Einstein was unconvinced that such a weird possibility would have any real physical significance, but he presented the paper to the Academy as Schwarzschild’s proxy.

  That was Schwarzschild’s last meaningful scientific accomplishment. In the filth of the battlefield, illness was almost as great a threat as a bullet—and that spring he contracted a rare skin disease. It took him two months to die. Einstein privately deplored Schwarzschild’s too-patriotic politics—but publicly eulogized the loss of his ferociously powerful mind.

  Walther Nernst was another Einstein colleague enmeshed in the war. The chemist had made the pilgrimage to lure Einstein away from Zurich in 1913. In August 1914, he launched into a more farcical journey, having his wife drill him in proper military bearing before racing westward in his private motorcar to see if he could serve as a courier for German troops on the road to Paris. A fifty-year-old bespectacled professor being of no great use on the front, he soon returned to Berlin. But he sent his two sons into the army, and by 1917, both were dead. The revulsion Einstein felt for war fever could evoke brutal contempt for his thus-afflicted friends, but some disasters were too much, even for him, to pass over with a detached “I told you so.” On learning of Nernst’s boys, he said, “I have forgotten how to hate.”

  Such deaths stand in for millions. In Europe’s charnel house there was almost no spare intelligence left in the scientific world to think about the geometry of space-time. That placed general relativity in an awkward position: solving the problem of Mercury’s orbit was a powerful argument that the new theory was correct. But the ultimate test of any new result lies with its predictions: whether it reveals some previously undetected phenomenon to be confirmed (or not) by observation or experiment. General relativity made several such predictions, including one readily testable with the technology at hand: its claim that the sun’s mass bends light by an amount double that predicted by Newton’s theory, 1.7 arcseconds compared to .87. Thus, once again, the decisive test of a physical claim would come at a total eclipse of the sun.

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  There was little chance of mounting an expedition while Europe continued to grind itself to death in the trenches. But the war couldn’t last forever, and late in 1917, a handful of British scientists began to plan for the next available eclipse. It would come on May 29, 1919, following a track across the south Atlantic. In the spring of that first year of the peace, a pair of two-man teams set out, one heading to Sobral on the Brazilian mainland and the other bound for Principe, a tiny island off the West African coast. The Principe team, astrophysicist Arthur Eddington and his assistant, reached the island on April 23. They made control images, night-sky pictures of the star field to be compared with the same stars that would be visible around the eclipsed sun. Under both Newton’s and Einstein’s theories, the positions of those stars would change between the control and the eclipse photographs; the question was by how much.

  May 29 greeted the observers with the familiar eclipse torture: dawn came with a torrential rainstorm. The rain eased by noon, but it wasn’t until 1:30, well into the partial phase, that the astronomers got their first sight of the sun. For the next several minutes clouds thickened and cleared as totality approached, and Eddington recalled that “We had to carry out our program of photographs in faith.” The team took sixteen exposures, but only the final six held out much promise. Four of those six had to be developed back in England, and of the remaining two, just one had seen clear enough skies to permit preliminary analysis in the field. It took Eddington four days, but at last, on June 3, he was able to make his first comparison of the test images to the star positions recorded in the eclipsed sky.

  He found the answer he’d sought: a deflection of 1.61 seconds of arc, plus or minus .3—close enough to Einstein’s predicted result to claim confirmation of the general theory. He would later remember that moment as the greatest of his life. In public he w
as more circumspect. His telegram back to England from Principe read simply “Through cloud. Hopeful. Eddington.”

  Einstein himself never doubted the outcome. Two friends visited him that summer, Paul Oppenheim and his wife, Anna Oppenheim-Errara. Einstein was under the weather, so he greeted them from his bed. As they talked, a telegram arrived from Lorentz with promising news, though not final confirmation. Anna Oppenheim-Errara remembered the scene more than seventy-five years later. Einstein was in his pajamas. She could see his socks. The telegram was brought in; Einstein opened it, and said, “I knew I was right.” Not, Oppenheim-Errara insisted, that he felt, or believed he was correct. “He said, ‘I knew it.’ ”

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  By now, in the daily business of our warped cosmos, Vulcan barely registers, even as an antiquarian curiosity. Only a few have some vague memory of the story—mostly physicists and astronomers with a historical bent. For them, Vulcan is a cautionary tale: it’s so damn easy to see what one wants or expects to find. Le Verrier himself comes off particularly poorly in these tellings, so certain of the implications of his analysis of Mercury, so eager to taste the glory of Neptune’s discovery once more, that he transformed an inoffensive amateur country doctor into a savant in the rough. The others, like Watson, carrying to his grave the certainty that he had seen the long-sought missing planet—they can all serve as warnings that desire has no place in the rigorous, rational, implacably empirical world of science. It’s such a temptation to see the past as not just past, but as less clever than the present. Perhaps that’s why Vulcan’s believers seem somehow comic. Like Edison with his jackrabbit, one waits for them to turn around and to find everyone rubbernecking to catch the joke.