Einstein's War

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by Matthew Stanley


  Remember that the core of the hole argument was Einstein’s loyalty to Mach. He had been convinced that coordinate transformations—which could represent the different measurements two observers make in different locations—had to have a physical, tangible meaning. That was the whole point of Machian positivism, that measurements were attached to specific activities done with physical measuring devices.

  As he looked back on his earlier arguments, though, Einstein realized he was now seeing things a little differently. He was no longer convinced that the meaning of the numbers had to be quite so physical. Perhaps the changes induced by the coordinate transformations were only mathematical—only an illusion. If he was willing to give up the Machian notion that all measurements had to be associated with actual physical devices, then they could just be mathematical representations. The differences in measurement could be easily accounted for with the right equations, and every observer could agree on the nature of gravity. Einstein realized that if he released himself from what he called his “fateful prejudice”—Mach’s insistence that every number be a physically possible measurement done with rods and clocks—he could have a generally covariant theory after all. He just had to loosen the Machian goggles through which he had been looking. Relativity could again become the theory he had first imagined—laws of nature truly independent of people, place, or movement. It could transcend the merely personal.

  This was a seismic shift in the way Einstein saw science. Relativity had begun in 1905 by thinking deeply about the actual physical processes of measurement and their significance—positivism. He used those results to dismiss unseen entities such as the ether as metaphysical and superfluous. But now Einstein was willing to give the equations a life of their own, even if they referred to things like space-time that could not be seen and directly measured. The deepest truths of the universe needed to be accessed by abstract mathematics, not by empirical experience. We might call him a rational realist by November 1915—he thought the constructs of mathematics and logic could reveal the true nature of things, and transcended what humans could actually see. He was now more like Plato than Mach.

  We are not used to hearing about geniuses changing their minds. But this is precisely what happened here. In the cartoon versions of the way science works, Einstein should have changed his mind because of a new experiment or a more precise calculation—some new piece of evidence. That’s not what happened, though. Instead he just looked differently at what he already had. The temptation is to try to give a rational explanation for Einstein’s shift, some concrete logical justification. There wasn’t any. Scientists are just like everyone else—they don’t always have a good reason for the decisions they make. Einstein made an intuitive leap without a particularly good reason to do so. He saw a way to make general relativity into the kind of theory he wanted, and he took it. In his own words, Einstein was an “unscrupulous opportunist.”

  * * *

  ONCE EINSTEIN RELEASED himself from his “fateful prejudice,” everything began falling into place. Now, convinced that general covariance was possible, he went back and reexamined all the candidate equations he had discarded in 1912–1913. Since then he had accumulated a couple of new tools to use to evaluate them. He had his new non-Machian way of thinking about coordinates, and he had the Lagrangian methods that he had been working with for the previous year. He unburied one tensor that he had given up on early in the Entwurf process because he couldn’t make it equate to the Newtonian limit. It had covariance and equivalence, but the calculation to reduce it to Newton’s law was so complicated he had thrown in the towel. But now his Lagrangians made that calculation much easier, and he found that it did in fact correspond to Newton’s law. It worked.

  He realized that Hilbert would never have made the same mistake he did, and the Göttingen mathematician might well have already discovered these equations. Desperate to keep priority on his own pet theory, Einstein assembled hastily what he had so far and submitted it to the Prussian Academy on November 4. In a moment of humility unusual for any scientist, he took the podium and began describing where he had gone wrong. He rejected the Entwurf publicly, saying that it had been an “absolutely impossible” approach. He admitted that he had “lost faith” in the Entwurf equations and tried to restart the project from zero. He described how he had missed the correct solution in Zurich and only now realized his mistakes. As he put it to a friend: “I have immortalized the final errors in this struggle in the Academy contributions.”

  The story he told the Academy of his search for general relativity was not quite accurate. He described his search as solely a mathematical one, largely ignoring the physical considerations that had guided him. This certainly made for a cleaner story and was easier for the audience to follow. Einstein scholars Michel Janssen and Jürgen Renn call this the “arch and scaffold” strategy. The scaffold of the Entwurf had all kinds of physical reasoning and messy techniques, but it allowed Einstein to build the arch of his field equations. Einstein then dismantled and tried to hide the scaffold, so only the beautiful arch remained. Generally he stuck to this version of the story for the rest of his life, making the quest for relativity sound much neater and more straightforward than it actually was. As often happens, a confusing struggle was edited to make a better story.

  The meeting ended with his paper incomplete, though he felt he had made a huge step forward. Thrilled, he wrote to Hans Albert: “In the last few days I completed one of the finest papers of my life; when you are older I’ll tell you about it.” Three days later Einstein sent a brief note to Hilbert attached to his new equations. He was careful to note that these changes were based on work he had done four weeks before, just in case Hilbert was close. Sommerfeld had let Einstein know that Hilbert had found “a hair” in the soup of relativity, and Einstein was deeply worried that his fix had been anticipated. “I am curious whether you will take kindly to this new solution,” he wrote.

  Amazingly, in the midst of this race with Hilbert, finally having broken the logjam preventing his progress on relativity, Einstein again put his equations down. He had received a letter from the Berlin Goethe League regarding his essay on the war. They said they would indeed like to publish it, if they could only remove a few paragraphs. Calling for new forms of political organization was fine; attacking patriotism itself was perhaps a bridge too far. In his response he expressed his opinion that “the glorification of war” must be opposed by “all genuine friends of human progress.” In the end he agreed to the publication of the edited version.

  He wrote his response on November 11, the day he continued his presentation to the Academy of the new general relativity equations. Dropping the politics and picking up the equations, he began by correcting a few errors he had made the week before. He remained very concerned about establishing his priority, stressing that these “new” results had really been discovered in 1912, he just hadn’t realized it. Letters began flying back and forth between Einstein and Hilbert as each strove to inform the other of any progress that had been made. Einstein particularly noted that he had found something that bore on the ETM theories that Hilbert was so interested in, a tweak that surely did not go unnoticed.

  Hilbert casually let Einstein know that he was planning on presenting all of his work on general relativity at a seminar in Göttingen on November 16, a Tuesday just three days away. He invited Einstein to be present for it, even suggesting two trains that would get him there on time. He would be happy to host Einstein at his home so they didn’t even have to worry about a hotel reservation. Einstein did not respond until Monday, demurring that he was tired out and suffering from stomach pains from his increasingly blockade-damaged diet. He asked Hilbert to keep sending his new work regardless.

  Einstein was now extremely pleased with his equations, and it was time to start testing them on the old stuck bolts that had been bothering him. He sat down to calculate the perihelion shift of Mercury with his new tools—and t
he bolt moved. His calculation said the shift should be 43 arc-seconds (about 1/100 of a degree) per century; astronomers had measured the shift at 45 ±5 arc-seconds. He had it—clear, empirical evidence for his theory. The same day he declined Hilbert’s invitation he wrote to his friend that he had solved the problem of Mercury: “Imagine my joy!” And then to Besso describing his victory:

  In these last months I had great success in my work. Generally covariant gravitation equations. Perihelion motions explained quantitatively. The role of gravitation in the structure of matter. You will be astonished. I worked horrendously intensely; it is strange that it is sustainable.

  It was a huge step. He wrote an anxious letter to Hilbert to let him know (and to remind him that the equations were actually three years old): “Today I am presenting to the Academy a paper in which I derive quantitatively out of general relativity, without any guiding hypothesis, the perihelion motion of Mercury discovered by Le Verrier. No gravitation theory has achieved this until now.” Einstein’s marking of his territory was less than subtle. Hilbert sent his “cordial congratulations on conquering perihelion motion.” He was amazed that Einstein had completed that calculation so fast—it was extremely difficult. Einstein had the advantage over Hilbert because he had already done a very similar calculation back in 1913, so was able to do this one quite rapidly.

  The presentation Einstein mentioned was the first one he delivered orally that fall. This was probably because he hoped to get the attention of one crucial astronomer in the audience—Karl Schwarzschild. Schwarzschild had been the director of the Astrophysical Observatory in Potsdam, outside Berlin, and was well known for his powers in mathematical physics. He was admired worldwide for his research in stellar atmospheres and motions as well as for his outgoing personality. As a loyal Frankfurter he’d enlisted in the German Army immediately after the outbreak of war, despite being over forty years old. He happened to be on leave on November 18 and was in the audience at the Academy—he was good friends with Hilbert and had heard a lot about relativity. Schwarzschild was exactly the person Einstein needed to impress if he was ever going to get astronomers on his side, so he put on his best show.

  He needed more empirical evidence, and that meant persuading astronomers to conduct the completely new predictions of the gravitational redshift and deflection of light. Einstein reworked both of those predictions with his new equations and had a bit of a shock. The redshift was unchanged. The deflection, however, had been totally wrong. With his new methods he found that the correct prediction was about 1.7 arc-seconds, twice what he had predicted in 1911. So if the Brazil eclipse had not been rained out, and if Freundlich’s Crimea eclipse had not been stopped by the war, they would have been looking for the wrong deflection. If their measurements had been accurate, his theory would have been disproven before it even started. We can only imagine Einstein’s sigh of relief as he realized that tragedy had turned into the greatest of luck.

  These few weeks were Einstein’s most productive and exciting time of his entire life. It was the culmination of, essentially, all of his career since the patent office. He wrote to Paul Ehrenfest, “I was beside myself with joy and excitement for days.” The results were all that he had hoped for: “The theory is beautiful beyond comparison.”

  Just two days after he presented his Mercury results, though, Hilbert presented the full set of correct equations for general relativity—which Einstein did not yet have—at the Royal Academy of Sciences in Göttingen. Hilbert credited Einstein with providing a starting point for his work, though his paper suggested that Einstein had merely formulated the right questions and Hilbert had found the answers. He had, somehow, gotten to Einstein’s end goal in a matter of weeks instead of years. Certainly a large part of this was that Einstein had already done much of the work by the time they met in the summer of 1915. Another large part was simply that Hilbert was a better mathematician and was never hamstrung by Einstein’s various dead ends.

  Five days later, Einstein presented his version of the final equations. He was not happy about being scooped by Hilbert. His friend, whom he had named as being the one person in the world who truly understood relativity, had claimed Einstein’s results for his own. Bitterly, Einstein wrote, “In my personal experience I have hardly come to know the wretchedness of mankind better than as a result of this theory and everything connected to it. But it does not bother me.” Hilbert quickly realized he had gone too far, and revised his paper on December 6 to give priority to Einstein. He may have written an apology directly to Einstein as well. On December 20, Einstein wrote a short note hoping to bring their rivalry to a close:

  There has been a certain ill-feeling between us, the cause of which I do not want to analyze. I have struggled against the feeling of bitterness attached to it, and this with complete success. I think of you again with unmixed geniality and ask you to try to do the same with me. Objectively it is a shame when two real fellows who have extricated themselves somewhat from this shabby world do not offer each other mutual pleasure.

  Finding a like-minded person in politics and physics was too precious a thing to waste on a priority dispute. Einstein knew he had trouble making friends. Once he took someone into his inner circle he would go to great lengths to keep them there. If he could still be friends with Haber after Ypres, he could still be friends with Hilbert after Göttingen.

  In priority disputes such as this, there are two major issues. One is the question of independence: was each person really working independently of the other? Or did someone take another’s work without acknowledging it? In this case it can be difficult to determine. Einstein and Hilbert spent a month communicating directly with each other, informing their rival of every small bit of progress. So we probably cannot consider their work to be independent. As irritated as Einstein would feel about it, there was a sense in which they were collaborators rather than competitors. The second issue is equivalency. Did they really find the same thing? It seems in this case that the answer is “not quite.” Hilbert realized that there was a problem with the way his field equations dealt with energy conservation. He held off publishing his results until 1916, when he felt he had fixed the problem. So Hilbert was first over the finish line but with an incorrect result. If priority is important to us, we can still feel comfortable saying relativity belongs to Einstein.

  * * *

  THE REACTION TO Einstein presenting his masterpiece was mixed. Schwarzschild had been skeptical earlier—he had called evidence for the theory “rather fishy.” But in December he wrote to Sommerfeld quite impressed by the calculation of Mercury’s orbit. That felt like real science to him: “That is something much closer to astronomers’ hearts than those minimal line shifts and ray bendings.” It was scarcely a proof, though. Max von Laue was not particularly moved. He said the Mercury prediction was merely the “agreement of two single numbers.” Surely that was not enough to change the “whole physical world picture in its foundations.”

  Einstein was not dissuaded. Most of his letters that month consist of him taking a victory lap, informing old friends and colleagues of his success both with covariant equations and the empirical support from Mercury. He sent his completed papers to Sommerfeld, writing that “it is the most valuable finding I have made in my life. . . . The result of the perihelion motion of Mercury gives me great satisfaction. How helpful to us here is astronomy’s pedantic accuracy, which I often used to ridicule secretly!” He delighted in how the four papers he submitted to the Academy tracked the final emergence of the theory like a film: “As you read [the papers] the final stage in the battle over the field equations is being fought out before your eyes!” Einstein described his Berlin colleagues as all but one “trying to poke holes in my discovery or to refute the matter . . . Astronomers, however, are behaving like an ants’ nest that has been disturbed.”

  He was also pleased that people were taking the larger implications of the theory more seriously. Mor
itz Schlick, one of the founding fathers of the Vienna Circle, had just completed the first major analysis of the philosophical significance of relativity. Einstein was pleased that Schlick had noticed the importance of Mach and Hume in the theory: “It is very possible that without these philosophical studies I would not have arrived at the solution.” He stressed that the most important features of relativity were that it agreed with all previous experiments and theories (that is, Newton’s laws) and was generally covariant. Now that we had covariance, the true nature of the universe was that of space-time, not our ordinary experience. He declared, rather opaquely: “Thus time & space lose the last vestiges of physical reality. There is no alternative to conceiving of the world as a four-dimensional (hyperbolic) continuum of 4 dimensions.” Time and space, the most elemental tools we use to organize our experiences of the universe, were illusions. In their place was a new kind of reality, just at the edge of human comprehension.

  In the four-dimensional universe of relativity, space and time were a single thing. We limited humans, though, think we see a changing mix of the two—the apparently separate phenomena of time dilation and length contraction were actually just different perspectives on the single entity of space-time. The laws of physics were the same for anyone, anywhere. Gravity was no longer a force between objects, as it was for Newton, but something far stranger. For Einstein, objects naturally moved through space-time on the shortest line between two points (a line called a geodesic). What counted as “shortest” could sometimes be altered by the presence of large masses like planets or stars, similar to how a picture drawn on a piece of cloth becomes distorted as you stretch the fabric. This “shortest” line was essentially still straight in four dimensions, though it could look curved to three-dimensional beings like us (much as how the path of an international flight, straight on the Earth’s round surface, looks like an arc on a flat map). Our brains see that “curve” as an object being pushed or deflected by an invisible force—what we call gravity. Gravity, then, was simply a side effect of our limited perception of objects moving across a four-dimensional continuum. Newton said an apple fell because it was pulled by an invisible force generated by a massive body; Einstein said an apple fell because it was trying to find the shortest path through space-time as distorted by a massive body. Did this make gravity less mysterious? Should it be removed from Eddington’s list of perplexing phenomena?

 

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