Einstein's War

by Matthew Stanley

  Einstein thought that the equivalence principle, then, demanded some connection between rotation and acceleration. He wasn’t quite sure how to apply his own principle, though. He complained to Besso about his lack of progress: “Every step is devilishly difficult.” His first results on rotation were published in February 1912 in an uncharacteristically tentative paper.

  Einstein still sought physical understanding of what was happening in relativity. When he pondered the bending of light predicted by his thought experiment, he wanted to know what caused the bending. Why did it happen? Perhaps it was rotation. Circular motion could cause Coriolis forces, which gave a sideways push to something moving on a rotating body (these are what cause hurricanes to spin). Maybe a spinning lab would create Coriolis forces that bent light? That went nowhere.

  Einstein then considered an analogy to another phenomenon where light beams were deflected. Light beams were bent as they entered glass or water because they moved more slowly there than they did in air. This was the basic principle of optics that made lenses and eyeglasses work. And in fact, if he was willing to vary the speed of light in relativity he could explain light deflection. But this was precisely the point on which he had attacked his competitors, and Abraham struck back immediately. He gleefully declared that Einstein had delivered a coup de grace to relativity: “Those who, like the author, have repeatedly had to warn against the siren song of this theory, can only greet with satisfaction the fact that its originator had now convinced himself of its untenability.”

  The key to Einstein extricating himself from this trap was yet another thought experiment. This one involved someone riding a rotating disk, like a carousel at a playground. If the carousel is spinning very, very fast, Alice riding at the outer edge might be moving near the speed of light (the playground has excellent safety equipment). Alice will then experience all the expected effects of special relativity—clocks will run slow, masses will grow, and distances will shrink—compared to what Bob sees while standing on the ground nearby.

  Alice and Bob love circles and, as all circle lovers do, decide to measure the circumference (the distance around the edge) and diameter (the distance across) of the carousel. This will let them calculate that beautiful number pi, the ratio of those two numbers. And they will surely get exactly the same number (roughly 3.14159 . . .) that they get every time they calculate that ratio for any circle, ever. The fact that pi is always the same is the very symbol of geometric perfection and has been known since at least the ancient Greeks. Its stability is the guarantee that geometry is universal and (as Einstein liked to say) beyond the merely personal.

  So they are not anticipating any difficulties. They will both measure the same diameter: neither of them are moving toward or away from the center, so there are no relativistic effects on that measurement. But they are moving relative to each other along the edge of the carousel. That movement means that length contraction will occur, and Alice’s meter rod will shrink compared to Bob’s when they lay them along the circumference. Since they are measuring the circumference with different-size rods, Alice and Bob will come to different conclusions about the circumference of the carousel. When they calculate pi, they hear Pythagoras spinning in his grave: their answers are different.

  Einstein’s rotating-disk thought experiment. Alice and Bob measure the circumference differently because of their relative motion.

  ORIGINAL ILLUSTRATION BY JACOB FORD

  Relativity had corrupted the soul of geometry. The unshakable foundations of mathematics that so impressed the twelve-year-old Einstein had now been ruptured. Euclid’s geometry could no longer hold in a relativistic universe. Moving observers will find that they disagree about the shape of a circle, or how many degrees were in a triangle. Einstein had violated the sacred truths that had set him on this path in the first place. Nonetheless, he had to follow the road he had set himself upon. Even worse than ruining his sacred text was his realization that geometry seemed to be the key to expanding relativity: he grudgingly accepted that perhaps, after all, Minkowski had been onto something important.

  As he arrived in Zurich in the summer of 1912, Einstein realized he had overreached. His thought experiments had left him with the vague sense that Euclidean geometry was no longer adequate. But it was one thing to say that his theory required a new kind of geometry; it was something else entirely to create the mathematics that could describe it. He was sure the physics was correct, but he was unsure how to translate that into the crisp, rigid equations he needed.

  Einstein knew there were alternative kinds of geometry different from Euclid, so-called non-Euclidean geometries that tackled problems like having a variable pi. He also knew that he wasn’t very good at it, despite passing his math classes back in college. So he turned again to the very person who had kept him afloat during those classes: his old friend Marcel Grossmann, now himself a mathematics professor at Zurich.

  * * *

  IMMEDIATELY AFTER MOVING into his new apartment, Einstein rushed over to Marcel’s house. Bursting in, he exclaimed, “Grossmann, you must help me or else I’ll go crazy!” Marcel had many years of experience calming Einstein down, and began explaining the mathematics he would need. The foundations for what Einstein needed had been laid down by Carl Friedrich Gauss back in the early nineteenth century. Gauss was a fantastically skilled mathematician who invented many of the quantitative methods essential for modern science. Among his other innovations, he pondered what non-Euclidean geometries might look like. For Euclidean geometry, you usually imagine that you are standing on a flat table where you draw your triangles and circles. Gauss wondered what would happen if you were standing on a curved surface, like the top of a hill or the inside of a tube. The answer is that strange things happen—for instance, a triangle drawn on a globe can have 270 degrees instead of the 180 degrees demanded by Euclid.

  Gauss, and later his student Bernhard Riemann, developed elaborate mathematical systems that would describe the forms of geometry that would exist on all kinds of surfaces. These geometries relied on the notion of the “curvature” of the surface at a particular point. A flat table, for instance, has no curvature; a water pipe has some curvature; a drinking straw has more intense curvature. The curvature can even change from point to point on a surface—imagine a half-rolled carpet. The triangles you draw on each of those surfaces will follow different rules, and non-Euclidean geometry is all about understanding those new rules.

  So if Einstein was going to apply these mathematical tools to Minkowski’s four-dimensional space-time, he had to decide what it meant for space-time to have “curvature.” If you imagine someone walking along a sheet of space-time, curving or bending that sheet will distort the measurements of space and time they make of the world around them. And distorted measurements are precisely what relativistic effects like time dilation and length contraction consist of—those strange phenomena were just how humans perceived the curving of four-dimensional space-time. Our flawed perceptions like to divide things up into space and time, but the universe sees them as one continuum where stretching time results in squeezing space. As Minkowski said, we see only shadows.

  Einstein’s spinning-disk thought experiment convinced him that this curvature would be induced by acceleration (since spinning is just a special kind of acceleration). Alice riding the carousel would experience space-time curving around her. And by the equivalence principle, acceleration and gravity have to be indistinguishable. If acceleration curved space-time, then gravity curved space-time. Gravity bends light, then, not because of a pulling force, but because light simply follows the natural curve of space-time.

  The warping of space-time by gravity. Larger masses create more severe curvatures.

  ORIGINAL ILLUSTRATION BY JACOB FORD

  To talk about things like the path of light through space-time, Einstein needed a way to define distances in four dimensions. Minkowski had found the interval
of space-time, which worked well for noncurved space. It was invariant, meaning that all inertial observers would always agree on its value. But for curved space-time, Einstein learned that he needed a new kind of mathematical object called a tensor. This is a complicated mathematical entity that had a property crucial for Einstein. It is generally covariant, meaning that different observers, moving in whatever strange ways, will all agree on its value. This was essentially a promotion of the first postulate of special relativity. Beyond saying there are no privileged inertial reference frames, covariance demands that there be no privileged observers no matter what. Something written as a tensor would be invariant for everyone. Covariance was what would give a truly universal physics independent of the traps of perspective and individuality.

  And this universal physics would have to be written in tensors, so it could be recognized no matter one’s position or movement. Unfortunately for Einstein, the mathematics of tensors was, to say the least, complex. A single four-dimensional tensor had ten separate elements that one had to keep track of, all of which could be constantly changing. Worse, the tensors he needed were nonlinear, meaning that the result of an equation actually changed the value of the numbers originally put into the equation, creating a kind of feedback loop that made calculating them very difficult.

  Grossmann showed Einstein that there were several kinds of tensors: the Riemann tensor, the Ricci tensor, and so on. Pure mathematicians such as Tullio Levi-Civita in Italy had spent decades developing elaborate mathematical systems based on these tensors, known as the absolute differential calculus. The systems were developed by mathematicians for mathematicians without the slightest sense of possible practical application, and Einstein was stunned by the complicated equations he had to deal with. Few physicists ever bothered to learn such esoteric things. He often complained about not being good at math (which has given comfort to generations of frustrated high school algebra students), but that was just in comparison to the company he kept. He was very good at math. If you spent all your time around world-class mathematicians such as Grossmann, though, just being “very good” began to feel inadequate. Einstein wrote, “Never in my life have I tormented myself anything like this, and . . . I have become imbued with great respect for mathematics, the more subtle parts of which I had previously regarded as sheer luxury! Compared to this problem the original relativity theory is child’s play.” He thanked Grossmann for having “saved me,” just like he had back in college.

  Einstein asked Grossmann to stay on the project and help him through the mathematics. Grossmann was careful to avoid any responsibility for the physics. By mid-1912 they had a good sense of what they were looking for. In order to make a general theory of relativity, they needed to find equations that fulfilled several criteria:

  They had to be generally covariant. They had to express the laws of nature in a form (probably a tensor) that would make sense to any observer in the universe, no matter how they were moving. This had to include all of the strange features of special relativity as well.

  They had to include the equivalence principle. They had to make gravity and acceleration indistinguishable, and predict things like the gravitational deflection of light and the gravitational redshift.

  They had to contain the great laws of classical physics, the conservation of momentum and energy. To Einstein these were nonnegotiable. Whatever general relativity ended up looking like, it had to preserve the things we already knew were true.

  They had to explain why Newton’s law of gravity worked so well. Newton’s theory was adequate for almost all circumstances, and Einstein had to show why we spent a couple centuries thinking it was correct. In physics jargon, we say that Newton’s law must be a limit or approximation of Einstein’s new law, apparently where gravity was fairly weak. Hopefully general relativity’s equations should apply in all cases, but in some of those cases they must look so much like Newton’s law that we can barely see the difference. We say that we should be able to reduce Einstein’s laws to Newton’s.

  There was a natural split among these criteria that suggested two strategies for finding general relativity. The final three are questions of physics, of how the physical world operates, of the actual processes that make everything go. What kind of law could combine Newton’s theory, the conservation of energy, and the equivalence principle? What might nature look like when we mix all of those together?

  The first criterion, however, was a question of mathematics. Mathematicians knew a great deal about tensors and non-Euclidean geometry. Could Einstein and Grossmann find what they needed just through manipulating symbols and following abstract principles of mathematical simplicity and elegance? Scholars who have studied Einstein’s notebook from this period (notably Michel Janssen, Jürgen Renn, John Norton, John Stachel, and Tilman Sauer) refer to these two options as the physical and mathematical strategies for finding general relativity. Einstein and Grossmann pursued both routes, often switching strategies when they hit a roadblock along one.

  They worked ferociously over the winter and spring of 1912–13. Grossmann might suggest a particular tensor, and Einstein would check it against the requirements of physics. This one wasn’t compatible with the equivalence principle; that one didn’t look enough like Newton’s law. Or Einstein would come up with an equation that fulfilled his physics needs and ask Grossmann if there were any tensors that looked like it. Every time, it seemed, there was one piece of the puzzle that wouldn’t fit. The tensors were complex objects and the pair often tore them up and rearranged the pieces—what if we use the first half of this one, or just the last chunk of that one? They tinkered, balancing mathematical beauty against the stringent requirements of the physical world.

  Einstein’s notebook from that winter reads like a mathematical melodrama. We can see his false starts, his notes to himself, his hopeful jottings. The frustration when he simply crossed out an entire page of calculations. One group of equations from late 1912 looked particularly promising, but he couldn’t see how it would reduce to the Newtonian limit. The calculations with this set of equations were so tedious and complicated that they weren’t worth the time to complete. He complained that his brain was “much too feeble” to handle this. He gave up, and moved on to another possibility.

  It did not seem that there were any equations that would fulfill all of Einstein’s needs. Something had to give. By May 1913 he decided that he had found the best field equations (that is, the equations that describe any gravitational fields) that he could. His equations gave him the equivalence principle, the conservation of momentum and energy, and could be massaged into a Newtonian approximation. But he could not make them generally covariant. They could not provide a truly general relativity—not all observers were equal. This was maddening, as this was essentially the initial premise of the entire theory of relativity. The conclusion of his calculations contradicted his starting point. He complained to Lorentz that without covariance “the theory refutes its own point of origin and therefore hangs in the air.”

  Could a house be built that destroyed its own foundations? He was both frustrated and proud. These equations were powerful and could do almost everything he asked of them. “Deep down I am now convinced that I have hit on the right solution, but also that a murmur of outrage will run through my colleagues when the paper gets published.” This version of the theory came to be called the Entwurf, after the label Einstein and Grossmann gave it, meaning something like “outline” or “draft.” Einstein admitted that the Entwurf was “more in the nature of a scientific credo than a secure foundation.” He could feel it was incomplete but was not sure how to proceed. He tried to express his unease to a friend: “Nature is showing us only the tail of the lion. But I have no doubt that the lion belongs to it, even though, because of its colossal dimensions, it cannot directly reveal itself to the beholder. We see it only like a louse sitting on it.”

  Nature was not
always cooperative. Einstein apologized for the “equations of considerable complexity” required by the Entwurf. One of his colleagues, Max Born, expressed admiration at the stunning abstraction of the theory—gravity, light, time, space, and motion all stripped down and synthesized into unvisualizable, unintuitive symbols on a page. Most physicists were skeptical of that magnitude of abstraction. Wasn’t this mere speculation?

  Einstein knew he needed to fight back against that critique, to show how his ideas could connect to the real physical world. The best way to do that was with the tests he had already been pondering. He recruited Besso to help with calculating whether the Entwurf could explain the anomalies of Mercury’s orbit. An exact calculation was impractical, and even their approximate methods were tedious. They did get a number, though. According to these equations, Mercury’s perihelion should advance by 18 arc-seconds per century, less than half of the observed 43 arc-seconds. Not reassuring, but there were still two other tests.

  The predicted redshift was still too small to be seen. The deflection of light, then, would have to be decisive (it helped that the deflection would also distinguish relativity from Nordström’s competing theory). He pushed Freundlich for the possibility of observing the deflection during daytime, and even wrote to George Ellery Hale of the titanic Mount Wilson Observatory in California. Hale wrote back that it was impossible. Einstein would have to wait for the next solar eclipse.