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Minkowski never got the chance to school Einstein on his poor mathematical taste. He died in 1908, just forty-four years old, cut down by appendicitis. Einstein could ignore the implications of the space-time perspective with impunity—for a while—and so he did. There was plenty of other physics to consider, and daily life as well.
That life torqued after 1905, unsurprisingly. While he remained at the patent office for several more years, the inevitable move from a bureaucrat’s job to an intellectual’s got a hesitant start in 1907, when he became a part-time adjunct at the university in Bern. It took until 1909 to land his first full-time academic job, at the University of Zurich. It was a stopgap appointment, an untenured post that left Einstein at the bottom of the totem pole. But in a little over a year he received an offer of a full professorship, with all the status and security that implied. The difficulty was that the invitation came from the German University in Prague in early 1911—in effect the far frontier of German-speaking Europe. In a common academic’s story, he made the leap anyway: the post trumped geography.
Neither he nor his wife, Mileva Marić, ever really liked Prague. Einstein complained to a friend shortly after his arrival that the locals showed “a peculiar mixture of class based condescension and servility, without any kind of goodwill to their fellow man.” The city itself was one of “ostentatious luxury side by side with creeping misery on the streets. Barrenness of thought without faith.” Even so, he found some compensations. A family connection recalled that he loved sitting in cafés along the river, drinking coffee and talking with friends. The elite fraction of the Jewish community there enjoyed a salon society, and it appears that at least once, perhaps more, he found himself in the same room as Franz Kafka.*1 For Marić, though, Prague had no virtues at all. It was the old story as told by a friend of the family: “She was left at home with the children and became more and more discontented every day.”
For Einstein, Prague’s saving grace was the chance to work without interruption. No patents to read, and all the freedom universities afford those in their senior ranks. After he first imagined a relativistic theory of gravity in 1907, Einstein had shifted his attention to the vexing riddles of the quantum realm. He made very little progress for the next several years, until his move to Prague left him with little beyond a rueful appreciation of the sheer nastiness of the problem. His office overlooked the grounds of an insane asylum. He was thinking of quanta when he described the inmates he watched from his windows as the “madmen who do not study physics.”
So he switched mysteries. Settled in his new home, he returned to the equivalence principle and what it could do to help him extend relativity. He knew that the outline of the idea he had given three years before was inadequate. He now found a way forward, one that required him to think deeply about what gravity might do to light.
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To capture the flavor of Einstein’s new approach, return to a version of one of his thought experiments, that rocket last seen accelerating in empty space. This time, its designers have cut a window in its fuselage, so that if someone shines a flashlight into the ship at rest, that beam of light will travel straight across to the opposite wall.
Now imagine the rocket takes off, accelerating as it goes. The craft moves just a little in the time light takes to travel from one side of the compartment to the other. As the rocket moves, that flash strikes the far wall just a little lower than the point at which it entered through the window. To an observer within the rocket, light is bending, actually curving downward. Accelerate more, and each ray of light bends more sharply. That’s under acceleration—and if you accept the equivalence principle, then, as acceleration bends light, so must gravity.*2
In this thought experiment a beam of light shoots through a window in a rocket accelerating upward. To an observer outside the rocket, the light travels in a straight line. But for someone in the rocket, the light enters at one point up the side of a rocket, and hits the other side farther down, following a curving track to get there. Within the accelerating frame of reference, light bends. By the equivalence principle, the same thing happens in a gravitational field.
The next stop followed logically. Given the relationship between light and time in special relativity, it came as no surprise to Einstein that the behavior of light in this new setting would lead to an effect on the flow of time as well. To simplify his reasoning, return to the rocket. Imagine there’s a clock set in the nose of the ship, at the top, and a second clock at the bottom bulkhead, back by the engines. At rest, the two clocks remain synchronized by a flash of light—a signal—that the clock at the bottom sends to the clock at the top, once per second. Both clocks keep good time, and—more important—the same time. It gets interesting, though, when the rocket’s engines start and the ship begins to accelerate.
While the clock at the bottom flashes light, the rocket moves, faster and faster. In the time that the flash of light takes to travel from the bottom of the ship to the clock at the top, the ship has risen, just a bit. The distance the signal must travel grows, which means, of course, that the bottom clock’s light signal takes more time to reach the top clock than it does when the rocket is at rest. The same holds for the next flash, and the one after that. Anyone checking would see that each light pulse arrives in just a little more than a second, as ticked off by that top clock. That means that the clock at the bottom is running slower than the clock at the rockets nose. Once again, by the equivalence principle, clocks in a gravitational field must behave in exactly the same way that the rocket’s clocks do. A clock placed where gravity tugs more strongly, closer to the center of the earth, must run more slowly than one perched a little farther from the earth’s center. The tick of time runs more slowly on the flat plain around Berlin than it does on top of the mountains near Zurich that the young Einstein used to climb.*3
This thought experiment turns on similar reasoning to that which Einstein used to analyze simultaneity. Acceleration increases the distance each successive signal from the bottom clock must travel to reach the top clock. An observer sitting beside that top clock would thus observe the lower clock to be running slowly. Again, by the equivalence principle, the same would hold true under gravity, with time slowing as a gravitational force grows stronger.
With that, Einstein fought through to the last step in his chain of reasoning. The invention of special relativity had already altered the conception of time. No longer an absolute, it became simply that which any given clock actually measures, given its motion relative to an observer. The one consolation was that the different clocks could be reconciled, using Minkowski’s mathematics of space-time, to bind one observer’s findings to another’s. But with his work in Prague, Einstein complicated those limited certainties. If gravity affects clocks, that means that time must vary from place to place. It bows to circumstance, whether one finds oneself at the Dead Sea or on Everest, or even merely in the basement or on the third floor. That each place, every place, has its own unique flow of time was a new vision, and not a comfortable one—not then, and not now. But by no later than the middle of 1911, Einstein saw where any extension of relativity theory must lead. Gravity bends time.
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With that insight, Einstein came to realize that there was a feedback loop in his emerging view of gravity. Getting there took a complex, subtle chain of reasoning. In his theory, as in Newton’s, gravity performs work—in the physicist’s sense—making objects move. In Newton’s version, all the force required to do that work depends on the amount of mass involved; that’s what his famous equation means. But Einstein knew from E=mc2 that energy and mass are equivalent, two faces of that single entity, mass-energy. The next thought was in some ways obvious after the fact—but at the time it represented a breakthrough. Any change in the amount of potential energy contained within a gravitational system would alter the total amount of mass-energy present—and hence the intensity of the gravity acting on the objects involved. That is: Ei
nstein now realized that gravity can impose its own effect on itself, that every change in the conformation of the system alters the system’s gravitational behavior. That, finally, forced him to confront the fact that his mathematical task had just grown much more difficult: any theory of gravity consistent with special relativity would have to model that interplay between energy and mass—which, in technical terms, means that it would have to account for a nonlinear process.
That was a blow, at least for Einstein’s hopes of coming up with a mathematically simple theory of gravity: nonlinear equations are notoriously more difficult to solve, so much so that it’s a standard tactic to try to convert nonlinear problems into linear expressions—which Einstein now knew he could not do. Still, this was the observation that allowed him to enter the next phase, to go beyond what he’d learned about the behavior of gravity in the wild to the fundamentals, a law that would allow him to model the way a relativistic gravitation had to work. Throughout his first winter and into spring in Prague, he made little headway. He told one friend in the spring of 1912 that he had come up against enormous obstacles, and another that “I have been working furiously on the gravitation problem…every step is fiendishly difficult.”
Here, at last, Minkowski’s description of four-dimensional space-time came to the rescue. Minkowski’s chief motive in creating that concept lay with his desire to clarify the implications of special relativity. His scheme retained one vital characteristic held over from earlier ideas: its four-dimensional cosmos that served as the container for whatever mass and energy might be doing within it. It was the stage on which history happened, unmoved and unmoving in itself.
Einstein’s great advance came when he thought about what it had to mean that acceleration and gravity affect the flow of time: space-time would have to bend too, as one of its dimensions (time) flexes under the influence of gravity. With that realization, Einstein’s thinking took on the elegant sweep of his best work. His new catechism: gravity is a property of matter and energy together (not matter alone, as in Newton’s view)—and gravity bends time. Taken together, those two facts led to this conclusion: the total amount of mass and energy determines the strength of the gravitational field in any particular location, and hence the amount any given region of space-time will flex. That warping of space-time in its turn has to affect the paths that matter and energy may take through the cosmos. Space-time is no stage, merely the box the universe comes in; rather, Einstein now realized, it is active, dynamic, shaped by what it contains. As he later put it, he had at last grasped a crucial truth: “the foundations of geometry have physical significance.”
That vision didn’t mean Einstein was done. But this was the essential step. With it, he’d managed to take a pure physical insight—gravity and acceleration are equivalent—and refine it to the point where he now knew what a full, rigorous mathematical account would have to include. He still didn’t know enough to get there himself; he didn’t even know what subdiscipline within the math universe would meet his needs. But he knew a guy.
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Marcel Grossman was both a first-class mathematician and one of Einstein’s oldest friends. They had met as undergraduates at the ETH in Zurich, and Grossman had been known to lend his classmate his notes when Einstein played hooky. Their reunion came when Prague, a backwater, found it couldn’t hold on to a talent like Einstein. The ETH sent out feelers early in 1912, and by summer, the deal was done. Less than two years after leaving Switzerland, Einstein returned in triumph as the professor of theoretical physics at one of the top technical universities in Europe. Grossman was already there as professor of mathematics. At their reunion Einstein begged: “Grossman, you must help me, or else I’ll go crazy.”
Grossman did. He already knew what Einstein needed: a way to escape what had for two thousand years seemed to be the only way to map the shape of nature: Euclidean geometry. It’s almost impossible to overstate the hold on natural philosophy Euclid’s book, Elements, possessed for most of its history. In more than two millennia no one has found an error in its analyses of planes, surfaces, and solids. The shortest distance between two points on a plane is a straight line, through a point not on a line there is no more than one line parallel to the line, the angles of a triangle add up to 180 degrees. All this and more seemed to be necessarily true—not only in the Elements, but out here, in the real world.
And then it wasn’t. In the early and mid-nineteenth century some of the most audacious thinkers in mathematics discovered they could modify one or another of Euclid’s assumptions, called axioms or postulates—those statements taken to be so obviously true that they required no proof. They found alternate geometries, just as consistent as Euclid’s, but ones in which, for example, no parallel lines exist. Grossman told Einstein that the version invented by Bernhard Riemann could serve his needs: it could analyze how to make measurements at any point on a smoothly changing curved object. When Riemann had created his system, he was thinking like a mathematician, one who focuses on ideas, not things. But for Einstein, this was a revelation: This strange, unsettling vision allowed him to play with space, treating it as something in which matter and energy would move, not on straight lines, but on curving paths.
Most important, he could now answer the crucial question: what, exactly, is gravity? Clearly not Newton’s instantly felt occult force-at-a-distance. Instead, in his still-forming analysis, gravitation is built into the geometry of space-time. Formally, gravity is the local curvature of space-time, the particular shape given to it by concentrations of mass-energy, like the earth or the sun. Mathematical analyses of such dents reveal the precise relationship between the distribution of matter and energy and the nearby shape of space-time. Objects navigating the cosmos—planets in orbit around a star, moons around a planet—are not mysteriously dragged along those paths. Rather, they simply follow the shortest route available to them around the dings and dips in space-time produced by all the matter and energy in their vicinity.
The classic visualization of gravity-as-geometry sees a massive object—like a star—stretching and deforming space-time imagined as a kind of rubber sheet. It’s an imperfect analogy, but it gets to the idea.
A problem remains, at least as far as gaining an intuitive sense of how the shape of space-time generates what we all experience as a force, the one that splashes wine on the floor when you tip over your glass. To get a sense of what’s going on, imagine a vast, seemingly featureless plain. It’s so flat that anyone living there can only perceive two dimensions, length and width, with no discernible changes in altitude. Go for a walk—say, on the most direct track between your home and a distant village—and you find that after a mile or so your steps come harder. It takes a little more effort to keep going. You begin to puff and labor. You clearly sense that you’re being tugged by something—a force you could call gravity. It pulls at you as you walk along what you are sure is a straight line. To anyone watching from a vantage where they can perceive three dimensions, not two, there is a simpler explanation. What feels like a mysterious force is simply the result of taking the shortest path up a hill.
That is: the “gravity” felt by a hiker on that empty plain is nothing more than the measure of a curvature of space, a rise the walker cannot actually see. The analogy is not perfect, as it only deals in space, not time. But it gets to the nub of the matter: we inhabit a locally curved region of space and time created by the mass of the earth. The weight we feel as we stand by our beds in the morning is the sensation of our daily slide down a well in space-time, a warp bending down toward the center of the earth. That sensation is born of the geometry of experience, an exercise in space-time dynamics that, every day, holds our feet to the floor.
It was all there by mid-1913. Einstein had a physical picture, and he was finally working on it with the right mathematical tools. Most important, he had a model, both in the formal quantitative sense and as a mental picture, a way to imagine this stillbizarre notion of gravity as geometry-in-act
ion. Einstein was confident—more, almost celebratory. He wrote to his fellow physicist Ludwig Hopf at the beginning of his work with Grossman, “It is all going marvelously with gravitation. If it isn’t all a trick, I have found the most general equations.”
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It wasn’t all a trick. It just wasn’t quite right. It had taken Grossman and Einstein almost a year to produce what they called a “Draft of a Generalized Theory of Relativity and of a Theory of Gravitation.” That title was fair. For all the time they’d spent on it, that paper was genuinely a draft. As published it contained several important errors, some simply slips of calculation, but others flowing from the fact that Einstein hadn’t yet truly mastered how to marry his physics to the dense and difficult mathematics Grossman had taught him.
But the idea itself was almost there, as Einstein knew, or at least felt very deeply. It was close enough for him to see that it could be tested. The theory made one clear prediction: light as well as matter would have to follow the contours of space-time, which meant that a ray of light passing close to the edge of the solar disk would bend round that gravity well created by the sun’s mask. The effect was big enough to be detectable, Einstein realized, but only during a total eclipse of the sun. Under the new theory that deflection would be .87 seconds of arc—a number well within the reach of experienced eclipse observers.
Einstein kept to himself the other possible validation. But in a document that surfaced more than three decades after his death, it turns out that he and Michele Besso, an amateur of science who was his dearest friend, tried to model a single specific circumstance: the problem of Mercury—what would happen to its orbit, rolling around the steep slopes deep down the well in space-time created by the sun.
The Hunt for Vulcan Page 13