Einstein began to wonder about the wider setting in which all the energy and all the mass—all the “things” of the universe—moved around. Something must be channeling them, guiding them. That appears impossible in the flat, empty space around us—but what if there was some explanation for how mass and energy moved around in this apparent void? What if space wasn’t quite as empty, and flat, as it seemed?
To sensible thinkers, this quest seemed impossible. We know that the curve of an ocean swell can send a boat swerving to one side. But that makes sense because the swell is just the surface of a larger, three-dimensional body of water. That body is what the surface of the water is curving around. If Einstein’s suspicions were right, however, and space is somehow curved, the question becomes:
What, possibly, is it curving around?
To understand Einstein’s solution—and the confidence he gained from it, as well as the terrible errors to which it led him—it helps to turn to a quiet Victorian schoolmaster named Edwin Abbott. It was he who found that although it’s impossible to visualize a dimension higher than the three we live amidst, it is possible to get a hint of how we might in fact be existing, unaware, within such a higher-dimensional universe.
Part II
“THE HAPPIEST THOUGHT OF MY LIFE”
Einstein and Mileva Marić with their first son, Hans Albert, in Bern, around 1904
INTERLUDE 1
The Romance of Many Dimensions
In 1884 Edwin Abbott, then the head of the City of London School, did something that in Victorian society was more embarrassing for a distinguished schoolmaster than stepping out in the street without a hat. He published a novel that had a hero who was only eleven inches long and lived his entire life on a vast sheet of paper, “on which straight Lines, Triangles, Squares . . . Hexagons, and other figures move freely about, on or in the surface, but without the power of rising above or sinking below it.” This world was called Flatland, and its author, as Londoners learned when the book was first published, was “A. Square,” Abbott’s pseudonym.
The book was a social satire, and it proposed an ingenious way to imagine a physical world that we can’t see.
The lowliest of the beings living in Flatland are the straight lines, whose sharp, piercing tips have to be avoided at all costs. One social level above them are the workers: long, narrow triangles eleven inches on their main sides—beings of little education and dangerous if provoked, but usually docile enough to do what their betters tell them to. One level above them are the middle-class professionals—doctors, teachers, and other respectable fellows. They have the shape of squares, and the book’s humble narrator is one of them. Another level up are the elite, who have yet more sides—pentagons, hexagons, and the like. At the very peak of society are the priestly circles, who glide wherever they wish along the surface, with lowly lines and pointy triangles taught to steer clear of them.
When the story begins, Mr. A. Square is fairly content with this flat world, though he is troubled by a dream he once had of some strange other world where all creatures líved on a single, one-dimensional line, existing like tiny trains restricted forever to a single track. Those poor beings could understand the idea of moving forward and back, but unlike Mr. A. Square, they could not conceive of there being an additional, “second” dimension that allowed movement from left to right. When Mr. A. Square crossed into their line, they could see just fragments of him as different points along his two-dimensional body entered and then departed their one-dimensional world.
A. Square’s dream made it clear to him that visitors from higher dimensions possess a greater power than those in lower dimensions. If a being such as A. Square reached into the line he visited and plucked one of the creatures from its position, the locals left behind in “Lineland” would have no idea where their fellow had disappeared to. Then if A. Square put the Lineland creature back but in a different position, they would be mystified as to how he could appear in a new location without having traveled through the intervening space in any way they could tell.
When A. Square woke up from his dream and saw he was back in proper Flatland, he was content for a time. He was a prosperous enough man, with his own, impressively two-dimensional home: one with an opening for himself and his sons, as well as—for Flatland was a sexist society, and women were considered inferior—an extra, much smaller door that his wife and any other women were to glide in through.
Mr. A. Square’s house
All would have remained fine, but then, as Mr. A. Square remembered from prison later:
“It was the last day of our 1999th year of our era. The pattering of the rain [which strikes only at the wall of their houses, for there’s no concept of such a thing as a roof] had long ago announced nightfall; and I was sitting in the company of my wife, musing on the events of the past and the prospects of . . . the coming century.”
There was a strange sound in their house, and then suddenly, “[to] our horror . . . we saw before us a Figure!” It hadn’t glided in through one of the two doors that led into the house. Rather, in some way that neither A. Square nor his wife could fathom, it had just suddenly appeared in their room. The strange visitor quickly began to transform, from being a very small circle to a larger one. A. Square’s wife was terrified, declared she had to go to bed, and glided as quickly as she could out of the room. Mr. Square was left alone with the stranger. With suitable politeness, he asked where his esteemed visitor had come from. The stranger said, “I came from [the Third Dimension]. It is up above and down below.”
A. Square was confused. Surely, he told the visitor, he must mean that he came from north or south, or possibly from left or right. But the visitor was insistent: “I mean nothing of the kind. I mean a direction in which you cannot look.”
A. Square thought this must be an attempt at a joke, but the visitor again was adamant: “Sir; listen to me. You are living on a Plane. What you style Flatland is the vast level surface . . . [on] the top of which you and your countrymen move about, without rising above or falling below it.”
To prove his point, the visitor said that he was going to move from below Flatland, travel through it, and then hover over the top. As that happened, what A. Square saw astonished him. From being a big circle, the visitor became a smaller circle, and then a smaller one, ending up as just a tiny dot.
The sphere arriving
The process then reversed. We know that the visitor was a sphere who had moved up through the surface of Flatland and then come back again. Mr. A. Square, however, had only been able to see a series of sideways cuts. He was flummoxed. It was no surprise to him when the creatures in the one-dimensional Lineland were startled to see a fresh line suddenly appear in their midst. That was because they didn’t understand that they actually existed on the wider, two-dimensional Flatland. But Mr. A. Square was convinced that was as far as matters went. He couldn’t imagine that he himself actually existed within a broader three-dimensional space.
The visitor realized he needed to give a further demonstration. Mr. Square kept his account books in a large room (the study) in his house. The strange visitor asked Mr. Square to close and lock the door to that room. He then said he would rise up into a third dimension, which existed invisibly “above” Flatland. From there he would descend into the locked room (which had no roof, of course, for in a two-dimensional world, no such thing would exist) and take the account books.
Mr. Square didn’t believe him. It was true that in his own dream about Lineland, he’d been able to reach in and grab things that to the lowly line creatures seemed suddenly to disappear. But that was because he could travel all around them in the exciting two-dimensional Flatland and they were stuck in a limited one-dimensional space. But nothing similar could happen here, he was sure, for what could possibly exist beyond Flatland? As the visitor got smaller and then disappeared, Mr. Square sprang into action.
“I rushed to the [study] and dashed the door open. One of the tablets was gone. With a mo
cking laugh, the Stranger appeared in the other corner of the room, and at the same time the tablet appeared upon the floor. I took it up. There could be no doubt—it was the missing tablet. I groaned with horror, doubting whether I was not out of my sense.”
And at that moment, A. Square was finally ready to grasp the truth. The strange visitor explained.
“What you call Space is really nothing but a great Plane. I am in [true] Space, and look down upon the insides of the things of which you only see the outsides. You could leave the Plane yourself. A slight upward or downward motion would enable you to see all that I can see.”
The stranger went ahead and lifted him “up.”
“An unspeakable horror seized me,” Mr. Square remembered. “There was a darkness; then a dizzy, sickening sensation.”
The sphere told him to open his eyes and try to look steadily. And when he did that:
“I looked [down], and, behold, a new world! . . . My native city, with the interior of every house and every creature therein, lay open to my view in miniature.”
He finally saw that the entire world he’d known before was composed solely of little geometric shapes, sliding around on the surface of a flat sheet. When he had been living down there, he hadn’t recognized it, for this only made sense when he moved up to the higher dimension. It’s a general principle: what seems odd to a creature living in a given dimension makes every bit of sense if that creature can envisage it from the next higher one. Creatures disappearing from Lineland and reappearing in new locations were baffling to the residents of that straight-line kingdom but made sense when viewed from Flatland. Similarly, what A. Square experienced with his visitor—objects magically disappearing from locked rooms—made sense once he realized he existed not just in the Flatland he was used to and could see, but that it was just part of a far greater Sphereland, which he had never been able to imagine.
Once back home, however, A. Square couldn’t get his family or anyone else to grasp what he’d seen. As time went on, he also realized, to his distress, that he was starting to forget his eye-opening experience: “About eleven months after my return from Spaceland, I tried to see a Cube with my eye closed, but failed; and though I succeeded afterwards, I was not then quite certain (nor have I been ever afterwards) that I had exactly realized the original. This made me more melancholy than before.”
A. Square’s story did not end well. He was eventually brought before the High Council, where he found out that the priests of Flatland knew that they existed in only two dimensions. But since they didn’t want to let the news get out—from fear of rebellion—and since in their eyes Mr. A. Square was not to be trusted, our intrepid explorer got locked up.
“Seven years have elapsed and I am still a prisoner,” he says on the book’s final page. His only hope is “that these memoirs, in some manner, I know not how, may find their way to the minds of humanity . . . and may stir up a race of rebels who shall refuse to be confined to limited Dimensionality.”
The analagy of Flatland is, of course, to our own world. Abbott wanted Englishmen to question the ruling class’s privileges, which were so taken for granted that they often seemed entirely invisible. The straight-line fragments living in Lineland couldn’t see that there was a wider two-dimensional world beyond them. The squares and pentagons and triangles living in Flatland couldn’t see that there was a wider three-dimensional world beyond them.
This is why readers shouldn’t feel bad for being unable to visualize curved space. No one can visualize it, not even an Einstein. Abbott simply wanted to say that even our greatest scientists might be as blinkered as the civilization of A. Square’s Flatland. As Abbott was also a devout Christian, he didn’t mind if readers saw parallels with religious beliefs—the arrival of the Logos in John 1:1, the miracles, the Ascension—that could seem impossible when limited to three-dimensional space.
It was around the time of the publication of Flatland that speculations about different geometries entered popular culture. In the Sherlock Holmes stories, the dastardly criminal Professor Moriarty is an expert in mathematics and would probably know of non-Euclidean geometries. In Dostoevsky’s Brothers Karamazov, when Ivan tries to explain the problem of evil to his simpler brother Alyosha, he says, “I have a Euclidean earthly mind, how could I solve problems that are not of this world? And I advise you never to think about it either, my dear Alyosha, especially about God, whether He exists or not. All such questions are utterly inappropriate for a mind created with an idea of only three dimensions.”
To most physicists, however, the issue of whether different geometries actually exist remained meaningless. Ivan Karamazov was a character from Dostoevsky’s imagination. Professor Moriarty didn’t exist. Scientists could get on with their work, unperturbed by the visions that had troubled the once contentedly bourgeois A. Square.
The hidden world that these beings had glimpsed, however, was exactly what Einstein would need to confront if he was going to solve the problems he’d begun struggling with at the Patent Office after he’d finally recovered from the exertions that had led to his E=mc2.
FIVE
Glimpsing a Solution
BY 1907 TWO years had passed since Einstein had published his series of papers—two years since he had united the realms of mass and energy, showing they could be seen as just a single category of interconnected “things,” transforming when they need to in crisp accord with his equation E=mc2.
Einstein’s theories were powerful, to be sure, but they left open the question of why the unity in the universe didn’t go further, and that question remained unresolved in 1907. All those “things” of mass and energy exist in a surrounding realm of “empty space.” Why should God—or whatever forces set up the universe—have decided that there should be two utterly unrelated categories: “things” on one side, and “empty space” on the other? If energy and mass were interrelated, why wouldn’t things and space be as well?
To Einstein, still rooted in a religion where one single deity created everything, it made no sense. So he got back to work.
The new project that Einstein began in the Patent Office in 1907 would yield a new theory. This one would be called general relativity, in contrast to the more restricted work he’d published in September and November 1905, which dealt with special relativity and its consequences. Einstein’s second, broader effort would revolutionize physics in ways we are still grappling with today. This period of his life would lead him to creative heights that far surpassed his E=mc2—but it would also, ultimately, lead to his fall.
GENIUS OPERATES in indirect fashion. At work, Einstein liked to close his eyes, to tune out the scraping of fountain pens in his office and the constant tsk-tsking of Herr Haller as he patrolled the worktables, so that he could think more clearly. But at one point in 1907, he’d kept his eyes open as he was reflecting, and either had seen some workmen climbing on a ladder to the edge of a nearby roof, or just imagined them on the roof. In some unfathomable mix of neurons firing, he later recalled, suddenly “there came to me the happiest thought of my life.”
Einstein began to think about falling from a house roof. If the house was very high, once you fell past the edge, neither you nor anyone else falling with you would be able to tell, without looking at the surroundings or feeling the wind, if you were moving or not. If you were holding hands with your partners and then let go, your partners would remain in the same position, as seemingly “stationary” as you were. You’d feel weightless, and so would they.
This would be your perspective as you fell. But if someone on the ground were looking up, not only would that person see you quickly plummeting downward, but he himself, of course, wouldn’t be weightless. He would weigh just as much as he did before you slipped off the roof.
Why, Einstein wondered, should the person on the ground feel gravity and you suddenly not feel it? Gravity couldn’t have suddenly disappeared around you when you slipped off the roof.
There had to be a way of unders
tanding this better, and Abbott’s book Flatland provides a start. Many of the characters in the book exist embedded in higher dimensions than they recognize, which means there are guiding “curves” in their own dimensions that explain what otherwise might seem mysterious. Consider the lowly beings who live in one-dimensional Lineland, existing like tiny trains on a narrow track. Their greatest geniuses would be perplexed if they found that after great travels, which constantly went straight ahead, they somehow ended back exactly where they had begun. But that would make perfect sense to an observer from a higher dimension, such as Mr. A. Square, who saw that the train track the Lineland beings lived on was actually curving in two-dimensional space and formed a circle. “We are,” as Abbott put it in Flatland’s introduction, “all liable to the same errors, all alike the Slaves of our respective Dimensional prejudices.”
The conclusion is straightforward. If objects move through higher dimensions, they can be guided in ways that seem incomprehensible to them. Here on earth, in our three-dimensional universe, we think that an invisible force of gravity is stretching up from the center of our planet and pulling us downward. But what if what’s really going on is that when we fall, we’re gliding along some curved pathway in space—a curve that’s impossible for us to sense directly, but that mathematical analysis might be able to reveal? That would be a fantastic link between Things and Space: some sort of twist or channel existing in Space that Things slide along as they move.
Einstein's Greatest Mistake Page 5
