einstein

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 replied 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=mc 2 . 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 g mn 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 G mn—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