But although he was tenacious, he was not mindlessly stubborn. When he finally decided his Entwurf approach was untenable, he was willing to abandon it abruptly. That is what he did in October 1915.
To replace his doomed Entwurf theory, Einstein shifted his focus from the physical strategy, which emphasized his feel for basic principles of physics, and returned to a greater reliance on a mathematical strategy, which made use of the Riemann and Ricci tensors. It was an approach he had used in his Zurich notebooks and then abandoned, but on returning to it he found that it could provide a way to generate generally covariant gravitational field equations. “Einstein’s reversal,” writes John Norton, “parted the waters and led him from bondage into the promised land of general relativity.”66
Of course, as always, his approach remained a mix of both strategies. To pursue a revitalized mathematical strategy, he had to revise the physical postulates that were the foundation for his Entwurf theory. “This was exactly the sort of convergence of physical and mathematical considerations that eluded Einstein in the Zurich notebook and in his work on the Entwurf theory,” write Michel Janssen and Jürgen Renn.67
Thus he returned to the tensor analysis that he had used in Zurich, with its greater emphasis on the mathematical goal of finding equations that were generally covariant. “Once every last bit of confidence in the earlier theories had given way,” he told a friend, “I saw clearly that it was only through general covariance theory, i.e., with Riemann’s covariant, that a satisfactory solution could be found.”68
The result was an exhausting, four-week frenzy during which Einstein wrestled with a succession of tensors, equations, corrections, and updates that he rushed to the Prussian Academy in a flurry of four Thursday lectures. It climaxed, with the triumphant revision of Newton’s universe, at the end of November 1915.
Every week, the fifty or so members of the Prussian Academy gathered in the grand hall of the Prussian State Library in the heart of Berlin to address each other as “Your Excellency” and listen to fellow members pour forth their wisdom. Einstein’s series of four lectures had been scheduled weeks earlier, but until they began—and even after they had begun—he was still working furiously on his revised theory.
The first was delivered on November 4. “For the last four years,” he began, “I have tried to establish a general theory of relativity on the assumption of the relativity even of non-uniform motion.” Referring to his discarded Entwurf theory, he said he “actually believed I had discovered the only law of gravitation” that conformed to physical realities.
But then, with great candor, he detailed all of the problems that theory had encountered. “For that reason, I completely lost trust in the field equations” that he had been defending for more than two years. Instead, he said, he had now returned to the approach that he and his mathematical caddy, Marcel Grossmann, had been using in 1912. “Thus I went back to the requirement of a more general covariance of the field equations, which I had left only with a heavy heart when I worked together with my friend Grossmann. In fact, we had then already come quite close to the solution.”
Einstein reached back to the Riemann and Ricci tensors that Grossmann had introduced him to in 1912. “Hardly anyone who truly understands it can resist the charm of this theory,” he lectured. “It signifies a real triumph of the method of the calculus founded by Gauss, Riemann, Christoffel, Ricci, and Levi-Civita.”69
This method got him much closer to the correct solution, but his equations on November 4 were still not generally covariant. That would take another three weeks.
Einstein was in the throes of one of the most concentrated frenzies of scientific creativity in history. He was working, he said, “horrendously intensely.”70 In the midst of this ordeal, he was also still dealing with the personal crisis within his family. Letters arrived from both his wife and Michele Besso, who was acting on her behalf, that pressed the issue of his financial obligations and discussed the guidelines for his contact with his sons.
On the very day he turned in his first paper, November 4, he wrote an anguished—and painfully poignant—letter to Hans Albert, who was in Switzerland:
I will try to be with you for a month every year so that you will have a father who is close to you and can love you. You can learn a lot of good things from me that no one else can offer you. The things I have gained from so much strenuous work should be of value not only to strangers but especially to my own boys. In the last few days I completed one of the finest papers of my life. When you are older, I will tell you about it.
He ended with a small apology for seeming so distracted: “I am often so engrossed in my work that I forget to eat lunch.”71
Einstein also took time off from furiously revising his equations to engage in an awkward fandango with his erstwhile friend and competitor David Hilbert, who was racing him to find the equations of general relativity. Einstein had been informed that the Göttingen mathematician had figured out the flaws in the Entwurf equations. Worried about being scooped, he wrote Hilbert a letter saying that he himself had discovered the flaws four weeks earlier, and he sent along a copy of his November 4 lecture. “I am curious whether you will take kindly to this new solution,” Einstein asked with a touch of defensiveness.72
Hilbert was not only a better pure mathematician than Einstein, he also had the advantage of not being as good a physicist. He did not get all wrapped up, the way Einstein did, in making sure that any new theory conformed to Newton’s old one in a weak static field or that it obeyed the laws of causality. Instead of a dual math-and-physics strategy, Hilbert pursued mainly a math strategy, focusing on finding the equations that were covariant. “Hilbert liked to joke that physics was too complicated to be left to the physicists,” notes Dennis Overbye.73
Einstein presented his second paper the following Thursday, November 11. In it, he used the Ricci tensor and imposed new coordinate conditions that allowed the equations thus to be generally covariant. As it turned out, that did not greatly improve matters. Einstein was still close to the final answer, but making little headway.74
Once again, he sent the paper off to Hilbert. “If my present modification (which does not change the equations) is legitimate, then gravitation must play a fundamental role in the composition of matter,” Einstein said. “My own curiosity is interfering with my work!”75
The reply that Hilbert sent the next day must have unnerved Einstein. He said he was about ready to oblige with “an axiomatic solution to your great problem.” He had planned to hold off discussing it until he explored the physical ramifications further. “But since you are so interested, I would like to lay out my theory in very complete detail this coming Tuesday,” which was November 16.
He invited Einstein to come to Göttingen and have the dubious pleasure of personally hearing him lay out the answer. The meeting would begin at 6 p.m., and Hilbert helpfully provided Einstein with the arrival times of the two afternoon trains from Berlin. “My wife and I would be very pleased if you stayed with us.”
Then, after signing his name, Hilbert felt compelled to add what must surely have been a tantalizing and disconcerting postscript. “As far as I understand your new paper, the solution given by you is entirely different from mine.”
Einstein wrote four letters on November 15, a Monday, that give a glimpse into why he was suffering stomach pains. To his son Hans Albert, he suggested that he would like to travel to Switzerland around Christmas and New Year’s to visit him. “Maybe it would be better if we were alone somewhere,” such as at a secluded inn, he suggested to his son. “What do you think?”
He also wrote his estranged wife a conciliatory letter that thanked her for her willingness not “to undermine my relations with the boys.” And he reported to their mutual friend Zangger, “I have modified the theory of gravity, having realized that my earlier proofs had a gap ...I shall be glad to come to Switzerland at the turn of the year in order to see my dear boy.”76
Finally, he replie
d to Hilbert and declined his invitation to visit Göttingen the next day. His letter did not hide his anxiety: “Your analysis interests me tremendously . . . The hints you gave in your messages awaken the greatest of expectations. Nevertheless, I must refrain from traveling to Göttingen for the moment ...I am tired out and plagued by stomach pains . . . If possible, please send me a correction proof of your study to mitigate my impatience.”77
Fortunately for Einstein, his anxiety was partly alleviated that week by a joyous discovery. Even though he knew his equations were not in final form, he decided to see whether the new approach he was taking would yield the correct results for what was known about the shift in Mercury’s orbit. Because he and Besso had done the calculations once before (and gotten a disappointing result), it did not take him long to redo the calculations using his revised theory.
The answer, which he triumphantly announced in the third of his four November lectures, came out right: 43 arc-seconds per century.78 “This discovery was, I believe, by far the strongest emotional experience in Einstein’s scientific life, perhaps in all his life,” Abraham Pais later said. He was so thrilled he had heart palpitations, as if “something had snapped” inside. “I was beside myself with joyous excitement,” he told Ehrenfest. To another physicist he exulted: “The results of Mercury’s perihelion movement fills me with great satisfaction. How helpful to us is astronomy’s pedantic accuracy, which I used to secretly ridicule!”79
In the same lecture, he also reported on another calculation he had made. When he first began formulating general relativity eight years earlier, he had said that one implication was that gravity would bend light. He had previously figured that the bending of light by the gravitational field next to the sun would be approximately 0.83 arc-second, which corresponded to what would be predicted by Newton’s theory when light was treated as if a particle. But now, using his newly revised theory, Einstein calculated that the bending of light by gravity would be twice as great, because of the effect produced by the curvature of spacetime. Therefore, the sun’s gravity would bend a beam by about 1.7 arc-seconds, he now predicted. It was a prediction that would have to wait for the next suitable eclipse, more than three years away, to be tested.
That very morning, November 18, Einstein received Hilbert’s new paper, the one that he had been invited to Göttingen to hear presented. Einstein was surprised, and somewhat dismayed, to see how similar it was to his own work. His response to Hilbert was terse, a bit cold, and clearly designed to assert the priority of his own work:
The system you furnish agrees—as far as I can see—exactly with what I found in the last few weeks and have presented to the Academy. The difficulty was not in finding generally covariant equations ...for this is easily achieved with Riemann’s tensor . . . Three years ago with my friend Grossmann I had already taken into consideration the only covariant equations, which have now been shown to be the correct ones. We had distanced ourselves from it, reluctantly, because it seemed to me that the physical discussion yielded an incongruity with Newton’s law. 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. No gravitational theory has achieved this until now.80
Hilbert responded kindly and quite generously the following day, claiming no priority for himself. “Cordial congratulations on conquering perihelion motion,” he wrote. “If I could calculate as rapidly as you, in my equations the electron would have to capitulate and the hydrogen atom would have to produce its note of apology about why it does not radiate.”81
Yet the day after, on November 20, Hilbert sent in a paper to a Göttingen science journal proclaiming his own version of the equations for general relativity. The title he picked for his piece was not a modest one. “The Foundations of Physics,” he called it.
It is not clear how carefully Einstein read the paper that Hilbert sent him or what in it, if anything, affected his thinking as he busily prepared his climactic fourth lecture at the Prussian Academy. Whatever the case, the calculations he had done the week earlier, on Mercury and on light deflection, helped him realize that he could avoid the constraints and coordinate conditions he had been imposing on his gravitational field equations. And thus he produced in time for his final lecture—“The Field Equations of Gravitation,” on November 25, 1915—a set of covariant equations that capped his general theory of relativity.
The result was not nearly as vivid to the layman as, say, E=mc2. Yet using the condensed notations of tensors, in which sprawling complexities can be compressed into little subscripts, the crux of the final Einstein field equations is compact enough to be emblazoned, as it indeed often has been, on T-shirts designed for proud physics students. In one of its many variations,82 it can be written as:
The left side of the equation starts with the term Rmn, which is the Ricci tensor he had embraced earlier. The term gmn is the all-important metric tensor, and the term R is the trace of the Ricci tensor called the Ricci scalar. Together, this left side of the equation—which is now known as the Einstein tensor and can be written simply as Gmn—compresses together all of the information about how the geometry of spacetime is warped and curved by objects.
The right side describes the movement of matter in the gravitational field. The interplay between the two sides shows how objects curve spacetime and how, in turn, this curvature affects the motion of objects. As the physicist John Wheeler has put it, “Matter tells space-time how to curve, and curved space tells matter how to move.”83
Thus is staged a cosmic tango, as captured by another physicist, Brian Greene:
Space and time become players in the evolving cosmos. They come alive. Matter here causes space to warp there, which causes matter over here to move, which causes space way over there to warp even more, and so on. General relativity provides the choreography for an entwined cosmic dance of space, time, matter, and energy.84
At last Einstein had equations that were truly covariant and thus a theory that incorporated, at least to his satisfaction, all forms of motion, whether it be inertial, accelerated, rotational, or arbitrary. As he proclaimed in the formal presentation of his theory that he published the following March in the Annalen der Physik, “The general laws of nature are to be expressed by equations that hold true for all systems of coordinates, that is they are covariant with respect to any substitutions whatever.”85
Einstein was thrilled by his success, but at the same time he was worried that Hilbert, who had presented his own version five days earlier in Göttingen, would be accorded some of the credit for the theory. “Only one colleague has really understood it,” he wrote to his friend Heinrich Zangger, “and he is seeking to nostrify it (Abraham’s expression) in a clever way.” The expression “to nostrify” (nostrifizieren), which had been used by the Göttingen-trained mathematical physicist Max Abraham, referred to the practice of nostrification by which German universities converted degrees granted by other universities into degrees of their own. “In my personal experience I have hardly come to know the wretchedness of mankind better.” In a letter to Besso a few days later, he added, “My colleagues are acting hideously in this affair. You will have a good laugh when I tell you about it.”86
So who actually deserves the primary credit for the final mathematical equations? The Einstein-Hilbert priority issue has generated a small but intense historical debate, some of which seems at times to be driven by passions that go beyond mere scientific curiosity. Hilbert presented a version of his equations in his talk on November 16 and a paper that he dated November 20, before Einstein presented his final equations on November 25. However, a team of Einstein scholars in 1997 found a set of proof pages of Hilbert’s article, on which Hilbert had made revisions that he then sent back to the publisher on December 16. In the original version, Hilbert’s equations differed in a small but important way from Einstein’s final version of the November 25 lecture. They were not actually generally covariant
, and he did not include a step that involved contracting the Ricci tensor and putting the resulting trace term, the Ricci scalar, into the equation. Einstein did this in his November 25 lecture. Apparently, Hilbert made a correction in the revised version of his article to match Einstein’s version. His revisions, quite generously, also added the phrase “first introduced by Einstein” when he referred to the gravitational potentials.
Hilbert’s advocates (and Einstein’s detractors) respond with a variety of arguments, including that the page proofs are missing one part and that the trace term at issue was either unnecessary or obvious.
It is fair to say that both men—to some extent independently but each also with knowledge of what the other was doing—derived by November 1915 mathematical equations that gave formal expression to the general theory. Judging from Hilbert’s revisions to his own page proofs, Einstein seems to have published the final version of these equations first. And in the end, even Hilbert gave Einstein credit and priority.
Either way, it was, without question, Einstein’s theory that was being formalized by these equations, one that he had explained to Hilbert during their time together in Göttingen that summer. Even the physicist Kip Thorne, one of those who give Hilbert credit for producing the correct field equations, nonetheless says that Einstein deserves credit for the theory underlying the equations. “Hilbert carried out the last few mathematical steps to its discovery independently and almost simultaneously with Einstein, but Einstein was responsible for essentially everything that preceded these steps,” Thorne notes. “Without Einstein, the general relativistic laws of gravity might not have been discovered until several decades later.”87
Hilbert, graciously, felt the same way. As he stated clearly in the final published version of his paper, “The differential equations of gravitation that result are, as it seems to me, in agreement with the magnificent theory of general relativity established by Einstein.” Henceforth he would always acknowledge (thus undermining those who would use him to diminish Einstein) that Einstein was the sole author of the theory of relativity.88 “Every boy in the streets of Göttingen understands more about four-dimensional geometry than Einstein,” he reportedly said. “Yet, in spite of that, Einstein did the work and not the mathematicians.”89
Einstein: His Life and Universe Page 27
