SEVEN
Sharpening the Tools
GROSSMANN’S FIRST STEP was to help Einstein catch up with the mathematics he’d missed in their student days by skipping so many classes. If empty space had curves in it, they would need some way of detecting this. Einstein was astounded when Grossmann—this man seemingly knew everything!—showed him that many of the necessary tools had already been worked out.
The mathematical approaches Grossmann showed Einstein built on what cartographers trekking over our planet, measuring latitudes and longitudes, had long before realized. When eighteenth-century surveyors measured points from wooden observational towers that were dozens of miles apart, even if the ground in between seemed a flat, snow-driven waste, they were able to tell from the size of the angles how curved or not the surface really was.
On flat plain, any huge rectangle that was staked out would have all of its interior angles a strict, clean 90 degrees. On much more curved surfaces, rectangles are “pushed” upward in the middle, so the angles at their corners are swollen to more than 90 degrees.
The earth’s surface is a continuously curved surface, and while this is so gradual that travelers can’t detect it with the naked eye, the curvature can produce amazing effects. Imagine, for example, that there is perfectly flat ice from Finland to the North Pole. Two skaters from a small town in Finland are made to stand a mile or two away from each other and then, when a signal is given, are told to skate on absolutely straight lines due north.
At first they think this is going to be easy. From their experience on the flat frozen lakes near their homes, they know that two skaters who start out in parallel can seemingly continue that way indefinitely.
But now, as they continue farther and farther away from home, assiduously keeping to their instructions—checking their compasses and making sure they don’t veer to the side by even one inch—they see themselves being “pulled” together as each gets closer and closer to the pole, until at the very top of our planet, they crash into each other.
From their perspective, that would be inexplicable. How can two men who start out miles from each other and very carefully stay parallel end up colliding? But from high enough above the earth—from a giant balloon, say, looking down on those two tiny figures gliding along—it would be obvious. What the skaters experience as an ineluctable pull toward each other is not due to a mysterious force. Because the basic shape of the earth is a sphere, any travelers following straight, parallel lines on the curved surface will necessarily collide.
This phenomenon is identical to the one Einstein imagined in his thought experiment with the eerily approaching apples—it’s just that one doesn’t appear to take place on a surface, while the other one does. In Einstein’s day, hardly anyone believed that these strange effects and curved paths could apply outside the surface of our planet—that outer space, which seemed empty, might also have some hidden structure that could influence how the objects within it moved. Rather, everyone assumed that the distant space in which planets and stars existed was as Newton had imagined it: flat, empty—a bare, dark stage set, before the actors appear.
Grossmann now explained to Einstein that a few mathematicians had already dared to look beyond that widely shared assumption. Several decades before Abbott wrote his Flatland fable, these intrepid few had begun to imagine that our planet might exist within wider geometries than we could possibly see. To the Hungarian army officer János Bolyai, writing in 1820, the idea was so exciting that after working through the logical possibilities, he wrote, “created a new universe from nothing!” To the academic German mathematician Carl Friedrich Gauss, who explored these ideas on and off for decades, “the theorems of [curved geometries] appear to be paradoxical, and to the uninitiated absurd; but calm, steady reflection reveals that they contain nothing at all impossible.”
But when none of these elite mathematicians found the experimental evidence to back up these possibilities, the field dwindled. Abbott did pick up some knowledge of these abortive efforts when he studied at Cambridge, and there were occasional mentions in literature, but most physicists couldn’t take this seriously. Mathematicians who continued to play with these possibilities were generally thought to be wasting their time. Even Einstein had joined in the mockery, writing to Marić back in 1902, “Grossmann is getting his doctorate on a topic that is connected with fiddling around with non-Euclidean geometry. I don’t know what it is.” Now, though, in 1912, Einstein changed his view. “I have become imbued with a great respect for mathematics!” he admitted.
THE LONG-FORGOTTEN TOOLS the pioneering mathematicians of the previous century had developed for studying these geometries of curved space were indeed tremendous—and they were also perfectly suited for the task Einstein and Grossmann had at hand. This was especially clear in an idea that one of Gauss’s protégés, the mathematician Bernhard Riemann, had demonstrated in an 1854 lecture—one the elderly Gauss attended—in which he noted that creatures who lived on any sort of surface would be able to work out how much it was curved at any particular location. This idea developed what the cartographers had already noted: If triangles bulged outward, whatever surface they existed on was like that of our spherical earth. If triangles shrank inward, the surface was concave—and all that could be seen without stepping off the surface. Mr. A. Square, living in a two-dimensional universe, could have used these procedures to deduce that he was living on a flat surface even before the visiting sphere lifted him up to let him see it from above.
If we followed Gauss and Riemann’s procedures carefully enough, we, too, Einstein realized—by measuring angles across great distances—could tell if something was making our three-dimensional space bulge or shrink. No one could detect it without such measuring equipment, since to our unaided senses the space right in front of us obviously seems flat. Humans have no capacity to “see” higher dimensions; not even Einstein. But with our calculations, we would be able to tell if there were “curves” there.
The underlying idea was so simple, and so beautiful, that Einstein later felt comfortable explaining it to his younger son, Eduard. Imagine, he said, that a small caterpillar is crawling around a large tree trunk. The caterpillar can’t tell that the tree trunk under him is curved and that he’s taking a curving path through space as he crawls. Only we, looking at the trunk from farther away, can see that taking place. The reason that Einstein spent so much time in his study, he explained to his son, was that he was trying to find a way for the caterpillar, caught in those paths, to work out if the world he was on was actually curved.
Einstein was still playing intellectual “golf” on the side, but Grossmann helped a lot as his occasional tennis partner. “I am now working exclusively on the gravitation problem,” Einstein wrote the once suspicious but now admiring physicist Arnold Sommerfeld in Munich, “and I believe that, with the help of a mathematical friend here, I will overcome all difficulties.”
Grossmann and Einstein made a good pair, even though they enjoyed playing up their differences. Grossmann was “not the kind of vagabond and eccentric I was,” Einstein noted later. In their nearly two years together at ETH, Einstein lived in rumpled, comfortable clothes; Grossmann always wore a proper suit and a crisp, high-collared white shirt. Whereas Einstein teased that he’d stayed away from mathematics because “[it] was split into numerous specialties, each of which could easily absorb our short lifetime,” Grossmann suggested that physics was ridiculously simple, saying that there was only one useful insight it had been able to teach him. Before studying physics, Grossmann said, “when I sat on a chair, and felt the trace of heat left by [the person before me] I used to shudder a little. That is completely gone. For on this point physics has taught me that heat is something completely impersonal.”
Einstein’s notebook from this period survives—a small, brown, cloth-covered volume filled with his neat inked handwriting, all the letters slightly angled to the right. On the first page, he doodles with recreational puzzles, drawi
ng a system of train tracks and shunted railcars to help work through them. But then he gets into his serious calculations. After several pages, the plaintive words “zu umstaendlich”—“too complicated”—appear when Einstein finds himself stuck, trying to list curvatures in ways that would make sense from whatever direction an observer approached a surface. At another point, the name “Grossmann” reassuringly appears—just at the place where his friend has brought in a key idea to help.
In 1913 Einstein and Grossmann presented their preliminary findings in a paper with an appropriate two-part structure: Grossmann signed the mathematical part, and Einstein the physical part. But Einstein’s skill was improving. By the end of that year, he had arranged to take a full-time post in Berlin the next year. Grossmann had given him as much help as he could.
From here on, Einstein was on his own.
COMPLETING ALONE WHAT he and Grossmann had begun together was the hardest work of Einstein’s life. “Compared with this problem, [1905’s] original theory of relativity is child’s play,” Einstein wrote. “No one who had not gone through the torments, false hopes, could know what it entailed.”
His colleagues saw how immersed he was. “Einstein is stuck so deep into gravity that he is deaf to anything else,” Arnold Sommerfeld reported to a colleague. But Einstein kept at it as the months rolled on —“Never in my life have I tormented myself like this,” he observed—because he felt something far greater than E=mc2 was waiting to be discovered. “Nature is only showing us the tail of the lion,” he wrote to his old forensic medicine friend Heinrich Zangger in Zurich. “But I have no doubt that the lion belongs to it, even though, because of its colossal size, it cannot directly reveal itself to the beholder.”
There was a further complication. The move back to Zurich in 1912 had done nothing for his and Marić’s marriage. Partly it was the sexism of the era, pushing the educated, intelligent Marić into a life focused on the home. Also, fatally, though still living with Marić in Zurich, Einstein had become entranced by a distant relative in Berlin, Elsa Lowenthal, a widow who had two grown daughters.
A trained actress with beautiful blue eyes, Lowenthal was well connected in Berlin’s art world. She spoke French fluently, far better than Einstein (which wasn’t especially hard; one sympathetic Frenchman he met reported that not only did he mangle the language by enunciating far too ponderously, but he also often mixed in some German). Lowenthal shared Einstein’s appreciation of music and theater, yet she also knew him well enough to be amused when he mocked her more pompous friends. And having been educated in the arts, not the sciences, she had no reason to feel diminished if scientific visitors only briefly acknowledged her before turning to Einstein.
At one point in 1912, Einstein realized he had to stop all contact with Lowenthal, and wrote her a letter telling her so: his wife had begun to understand that she was not just a distant relative but a threat. But Einstein also included his return address, and when, early in 1913, she casually wrote to him on the pretense of wanting advice on finding a good popular guide to relativity, he couldn’t resist starting their correspondence again.
Marić was furious when Einstein accepted the offer to move from Zurich to Berlin, for she knew it meant her husband would be closer to this woman who threatened their family. Their young sons had no idea what was going on, and when the family did transport all their belongings again, arriving in Berlin in the spring of 1914, they seemed delighted by the huge, modern city. But for Einstein and Marić, the days of delighting in new moves, of sitting on their balcony, looking at the Alps, and holding each other, were now impossibly far away. Friends saw how suspicious, cold, and easily hurt they were. In those first Berlin weeks in 1914, Einstein cruelly told Marić that he would put up a minimal front of friendliness only if it was “completely necessary for social reasons,” even though the impending breakup was clearly his fault.
By July 1914, it was too much. Marić couldn’t live like this, with her husband so clearly enamored of someone else. She still thought their marriage could be saved, but she had too much pride to stay. Einstein was caught, for in his heart their marriage was over—he’d even begun to refer to Lowenthal’s children as his stepdaughters—yet he also wanted to keep seeing his sons. In the end, the warmhearted Besso traveled from Zurich to help Marić and the boys move back to Switzerland. Einstein didn’t insist on a divorce and agreed that he would send her half his salary. Sobbing at the Berlin train station as he watched his children go, Einstein then found a small apartment for himself with just enough space for the boys to visit.
The breakup was exhausting, as was his continuing work, and that was not all: the month after he and Marić had split, war had broken out in Europe. Conditions in Berlin rapidly deteriorated. Soon food was limited, there were cuts in electricity and fuel, and a frantic nationalism took over. To his old friend Besso he wrote, “When I talk to people I can sense the pathological in their state of mind.” To a friend in the Netherlands he elaborated, “I am convinced that this is some kind of mental epidemic.”
Einstein’s life was in chaos, but how could he let his explorations go? He had to solve the problem of gravitation that he had been struggling with on and off since 1907; he had to unveil the universe’s innermost secret.
And then, in November 1915, he cracked it.
EIGHT
The Greatest Idea
WHAT EINSTEIN DISCOVERED, in the chill of wartime Berlin, was the greatest breakthrough in understanding the physical universe since Newton: an achievement for all time. If Einstein had never been born, almost certainly someone else would have come up with E=mc2, and not much later than he did, in 1905. The Frenchman Henri Poincaré and the Dutchman Hendrik Lorentz, for example, were at most a few years behind him. But no one else had come close to what Einstein achieved in 1915. Although the details are subtle,* the core can be represented as follows.
Think of truly empty space as like a vast trampoline surface. It’s flat; there’s no curvature, no dips or rises. If you flick a tiny ball bearing along the trampoline surface, it doesn’t distort the trampoline at all, and just travels in a straight line.
Now place a small rock on that surface. Its weight makes the trampoline sag downward. Flick the ball bearing again, and if it passes anywhere near that rock, it will veer slightly toward it because of the sag. The mass of the rock makes the trampoline distort, and that distortion shifts the path of other objects—such as the ball bearing—that come anywhere near, as shown:
That was Einstein’s vision; that was the theory that explains where the distortion of space is coming from. The warping—the curvature he had struggled to define since thinking about the adventurer in the elevator—comes from all the things, the stuff—all the mass and energy—that are scattered through space! Wherever mass or energy is located, they distort the space around them, just like the rock pushing down on our trampoline. Put a small mass somewhere new—move the mass of your hand a few inches sideways through the air, for instance—and it’s as if you’re pressing on invisible rubber sheets, and there really now is a slightly different configuration of space around it. Have a large mass arrive somewhere new—have the entire earth rush forward in its orbit—and it’ll produce far bigger distortions in the invisible space around us.
It was a brilliant, bold idea, and in many ways a parallel to Einstein’s earlier work on locating the tunnel between the two domed cities of M and E. Just as energy and mass were connected through an invisible linkage, Einstein realized, so were those two things interwoven with the space they occupied. There was a unity to the universe, he had always believed, and now he was one step closer to describing it.
Einstein’s theory about the distortion of space represented a watershed in the history of physics, yet it was only half of what he had discovered. For in pinpointing the effect that things had on the space around them, he’d also gained new insight into how that influenced other things in their vicinity.
After all, what happens whe
n a trampoline is bent and sagging? Those distortions in its geometry make the objects near them veer and shift. A ball bearing on the sagging trampoline isn’t tugged by any mysterious force from the rock. It’s simply following the most straightforward path from its perspective.
The idea makes intuitive sense. Create a distorted geometry at some point, and that will lead any things that are nearby to follow a new, otherwise inexplicable path. As we saw, that is why the two Finnish skaters would find themselves ineluctably drawn together as they approach the North Pole. They’re skidding on a two-dimensional surface, which curves around our three-dimensional planet. That is also why the two apples let loose in the free-floating room slowly start moving toward each other if there’s a source of gravity below. They’re skidding along on a three-dimensional space, which—by Einstein’s idea—must be the curved surface of an invisible-to-them four-dimensional space that it wraps around. The unhappy explorer floating between them is simply seeing them tumble along that curve.
In Einstein’s radical reconception of space, there’s no need to imagine an additional force of gravity; rather, gravity is simply the result of the warping of space. The snowy North Pole is not sending out an invisible force tugging the skaters closer to each other. Unless something pushes them away, objects always follow the most straightforward channels stretching ahead of them. One doesn’t even have to envisage the icy north or falling rooms. Watch a surfer rise several feet up on the sea. If the ocean swell under him were invisible, this rise upward would be a great mystery, as would the subsequent glide downward. The moment you see that water, however, all is evident.
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