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understand your new paper, the solution given by you is entirely different from mine.”<br />
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<br />
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<br />
somewhere,” such as at a secluded inn, he suggested to his son. “What do you think?”<br />
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<br />
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<br />
come to Switzerland at the turn of the year in order to see my dear boy.” 76<br />
Finally, he replied to Hilbert and declined his invitation to visit Göttingen the next day. His letter did not hide his anxiety: “Your analysis interests<br />
me tremendously . . . The hints you gave in your messages awaken the greatest of expectations. Nevertheless, I must refrain from traveling to<br />
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<br />
my impatience.” 77<br />
Fortunately for Einstein, his anxiety was partly alleviated that week by a joyous discovery. Even though he knew his equations were not in final<br />
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<br />
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<br />
calculations using his revised theory.<br />
The answer, which he triumphantly announced in the third of his four November lectures, came out right: 43 arc-seconds per century. 78 “This<br />
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<br />
was so thrilled he had heart palpitations, as if “something had snapped” inside. “I was beside myself with joyous excitement,” he told Ehrenfest. To<br />
another physicist he exulted: “The results of Mercury’s perihelion movement fills me with great satisfaction. How helpful to us is astronomy’s<br />
pedantic accuracy, which I used to secretly ridicule!” 79<br />
In the same lecture, he also reported on another calculation he had made. When he first began formulating general relativity eight years earlier,<br />
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<br />
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<br />
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<br />
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<br />
a prediction that would have to wait for the next suitable eclipse, more than three years away, to be tested.<br />
That very morning, November 18, Einstein received Hilbert’s new paper, the one that he had been invited to Göttingen to hear presented.<br />
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<br />
clearly designed to assert the priority of his own work:<br />
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<br />
difficulty was not in finding generally covariant equations ...for this is easily achieved with Riemann’s tensor . . . Three years ago with my friend<br />
Grossmann I had already taken into consideration the only covariant equations, which have now been shown to be the correct ones. We had<br />
distanced ourselves from it, reluctantly, because it seemed to me that the physical discussion yielded an incongruity with Newton’s law. Today I<br />
am presenting to the Academy a paper in which I derive quantitatively out of general relativity, without any guiding hypothesis, the perihelion<br />
motion of Mercury. No gravitational theory has achieved this until now. 80<br />
Hilbert responded kindly and quite generously the following day, claiming no priority for himself. “Cordial congratulations on conquering perihelion<br />
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<br />
produce its note of apology about why it does not radiate.” 81<br />
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<br />
relativity. The title he picked for his piece was not a modest one. “The Foundations of Physics,” he called it.<br />
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<br />
climactic fourth lecture at the Prussian Academy. Whatever the case, the calculations he had done the week earlier, on Mercury and on light<br />
deflection, helped him realize that he could avoid the constraints and coordinate conditions he had been imposing on his gravitational field<br />
equations. And thus he produced in time for his final lecture—“The Field Equations of Gravitation,” on November 25, 1915—a set of covariant<br />
equations that capped his general theory of relativity.<br />
The result was not nearly as vivid to the layman as, say, E=mc 2 . Yet using the condensed notations of tensors, in which sprawling complexities<br />
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<br />
been, on T-shirts designed for proud physics students. In one of its many variations, 82 it can be written as:<br />
The left side of the equation starts with the term Rmn, which is the Ricci tensor he had embraced earlier. The term g mn is the all-important metric<br />
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<br />
Einstein tensor and can be written simply as G mn—compresses together all of the information about how the geometry of spacetime is warped and<br />
curved by objects.<br />
The right side describes the movement of matter in the gravitational field. The interplay between the two sides shows how objects curve<br />
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<br />
curve, and curved space tells matter how to move.” 83<br />
Thus is staged a cosmic tango, as captured by another physicist, Brian Greene:<br />
Space and time become players in the evolving cosmos. They come alive. Matter here causes space to warp there, which causes matter over<br />
here to move, which causes space way over there to warp even more, and so on. General relativity provides the choreography for an entwined<br />
cosmic dance of space, time, matter, and energy. 84<br />
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<br />
it be inertial, accelerated, rotational, or arbitrary. As he proclaimed in the formal presentation of his theory that he published the following March in