Einstein's War

by Matthew Stanley

  All of this was summed up in one simple, elegant equation (note that there are several different ways this can be written):

  Guv = 8πTuv

  On the left we have the curvature of space-time, the way it is stretched or distorted. On the right we have the arrangement of matter and energy. The presence of matter and energy creates curvature; curvature controls how matter and energy move. The two sides feed and depend on each other. As equations go, this is fantastically concise and clear—assuming you are familiar with non-Euclidean geometry, of course. This short expression actually represents ten separate equations all tangled together. What counts as “simple” in Einstein’s world can be somewhat complicated.

  There was still quite a bit of work necessary for Einstein to make his theory clear and persuasive. The cost of uniform laws of nature was this new bizarre universe that was not easily understood or accepted. This required some more empirical evidence—something that could be seen, something more tangible than equations on a page. Mercury’s orbit was good, but that was known before he had created the theory. He wanted something new and unexpected to convince waverers like Planck—either the redshift or light deflection. Those were beyond what he could do himself.

  He knew he could always lean on his oldest ally, Michele Besso. He hoped they could discuss the new theory in person soon in Switzerland. He was planning a visit, but the border was almost constantly closed. He did not mention that the reason he would be in Bern was for a meeting of the Anti-War Council, to which he had been elected. “In these times everyone must do whatever he can for the community as a whole, even if it is only slight and ineffectual.” He finally had a victory with relativity; a victory against the war would have to wait.

  CHAPTER 7

  To Cross the Trenches

  “Your theory still seems to be almost entirely unknown in England.”

  IT WAS HARD to move around Europe in those years. Hard for people, certainly—trenches ran continuously from the North Sea to the Swiss border. Hard for ideas too. The papers on which scientific theories were written were trapped by the barbed wire and the naval blockade, unable to escape their home countries. We think of equations and hypotheses as immaterial, transcendental, unrestricted by mundane concerns like borders and checkpoints. Einstein learned the hard way how false this was. His triumph with general relativity in 1915 was unknown in most of the world. The articles announcing his theory were halted by machine guns and artillery just as decisively as the advancing armies were. For his theory to cross borders, Einstein needed allies and hard work. Relativity was tough to learn not just because of the formidable mathematics but because science was deeply entangled in the war.

  Besides cutting lines of communication, the endless fighting questioned what it meant to be a scientist at all. This new kind of total war put pressure on the most essential points of what scientists should do, write, and think. Could Einstein pursue the truths of the universe while mired in questions of politics and patriotic strife?

  By 1916, Einstein needed help, and he knew it. It is usually said that Einstein finished the theory of relativity in November 1915. This is true, in a sense. By then he had found his field equations—the mathematical statements that summed up the principles and concepts on which the theory was based. But theories are more than just equations, and there was work to be done in understanding, refining, and applying. By January he wanted nothing more than to sit down with his friends and talk about relativity. Unfortunately he had few colleagues who both understood his work and—perhaps more important—with whom he felt he could talk freely against the background of the Great War.

  Two of the members of this rare group worked in the Netherlands, an island of neutrality in a continent at war. They were professors in Leiden, site of the oldest university in the Netherlands and a scientific and cultural center for centuries. A bustling city crossed by small canals about fifty miles north of Belgium, it was the professional home to Einstein’s friends Paul Ehrenfest and Hendrik Lorentz. Einstein’s intimate correspondence with them had tracked his scientific success and failures for years. It was to Ehrenfest that he declared, “[I was] beside myself with joy and excitement for days” after finally cracking the covariance problem (the key to making relativity truly universal). To Lorentz he confessed that all his papers on gravity had so far been a “chain of wrong tracks.” Joining Einstein’s relativistic clique in Leiden was Willem de Sitter, the director of the Leiden Observatory, striking with his pointed beard and prominent ears. De Sitter was a skilled mathematical astronomer, and his interest in problems of gravitation pulled him into Einstein’s orbit. However, it was not just his friends’ skill in physics that Einstein relied on. He also reveled in their like-minded politics. In imperial Berlin he never felt he could talk openly. One of his favorite parts of visiting the noncombatant Netherlands was that he could “walk around without a muzzle.”

  In January 1916, Einstein was struggling with certain formulations in relativity. He more than once regretted skipping mathematics classes back in college. Lorentz, he hoped, could help. Einstein had confidence in the basic formulas, but his own presentation and derivation of them, he felt, was “abominable.” He had been impressed with Lorentz’s ability to explain covariance and hoped to absorb some of the elder scientist’s wisdom. He worked hard to support his friends’ interest in his theory, even providing them with step-by-step calculations to make sure everything was understood. At the end of January he sent a walkthrough of relativity to Ehrenfest, several pages of dense handwritten formulas. He assured Ehrenfest that he would have no more problems with relativity now that it was all laid out nicely. Einstein asked him to show the material to Lorentz and then return it—it was, he felt, the best summary of his theory yet. He needed the letter back to use it as the basis for a publishable paper.

  Einstein hoped to go to Leiden to visit them in person early that year despite wartime restrictions. One of Einstein’s friends jokingly asked if relativity might allow Leiden to come to him, instead of vice versa. Although the Netherlands was neutral in the war, the border was often closed. The Dutch had watched Belgium crushed by the kaiser, and they were not at all confident that their neutrality would be respected. The fighting felt close—one could hear artillery fire from many parts of the country. Their army was mobilized throughout the war just in case. Ehrenfest belonged to a home-guard unit that was never called up. Traffic across the border was carefully controlled by both sides.

  Even to Einstein with his Swiss passport, a visit was challenging to plan. Worse, the police had been investigating Einstein as a politically suspect person. He had foolishly written some of his pacifist correspondence on a postcard, and an alert mail carrier notified the police. Their official investigation concluded in January 1916 that he had only recently become politically active but was definitely “a supporter of the peace movement.” They decided to let him travel for the moment (his movements were eventually restricted in 1918). But even when cooperative, the German bureaucracy was a formidable opponent. Einstein wondered if maybe he should just wait until peace came. Surely the war could not go on much longer.

  For most Germans there were few signs that the war would be ending soon. Karl Schwarzschild, finished with the leave where he had heard Einstein speak, had already arrived at his second military post of the conflict. His abilities were first put to use constructing weather forecasts in Belgium and then calculating artillery trajectories on the Russian front. The earthworks and explosions of the battlefield could hardly have been more different from the calm parks and palaces that surrounded his former institute in Potsdam.

  That said, for a theorist like Schwarzschild, a fortification was a decent place to do science. The German trenches were famously well appointed. Advancing British troops incredulously reported concrete floors, heating stoves, running water, and electric lights. Those deep, comfortable bunkers provided German soldiers almost complete protection from bombar
dment and shelter from the east European winter. Schwarzschild would have had plenty of opportunities to scribble and calculate. He did not need a laboratory, just time and focus.

  Schwarzschild’s trenches represented the farthest reach of the theory of general relativity. He was able to keep up on scientific developments back in Berlin without too much difficulty. Soldiers were able to receive regular mail shipments, and he was able to travel there in person when his unit was on leave. Upon receiving copies of Einstein’s papers on general relativity, he sat down to study this strange new theory. He quickly discovered Einstein’s “abominable” derivations and wondered if he could do better.

  The core issue was the calculation of the orbit of Mercury, which Einstein’s theory matched with observation so nicely. Schwarzschild immediately noticed something important. Einstein’s calculation of the orbit of Mercury was only an approximation. This is a standard practice for physics problems that require large amounts of calculation. When it becomes clear that an exact answer will be labor-intensive, there is a strong incentive to find a shortcut. Generally the shortcut is an answer to a problem that is just about the same as the one you are really interested in, but much easier to solve. If you can present an argument that just about the same is good enough, then your colleagues will often give you the benefit of the doubt. And this was indeed where Einstein found himself. He had a good enough explanation for the wobble in Mercury’s motion. Schwarzschild set himself the task of finding the precise answer, starting from Einstein’s equations, sheer mathematical skill, and a bit of chutzpah.

  A German trench similar to where Karl Schwarzschild performed his calculations

  BRETT BUTTERWORTH COLLECTION

  In short order he was able to demonstrate the needed precession of 43 arc-seconds per century. He was pleased that the derivation was just a few pages long and was impressed that such a concrete, observational prediction came out of the abstract heights of Einstein’s principles. Schwarzschild wrote to the theory’s author to alert him to the achievement. The calculations were literally done when he had a break between firing cannons at the tsar’s armies. Fighting on the eastern front was fairly quiet at the time, with the Russians having been pushed out of German territory over the course of the year. One of his letters to Einstein suggests that he found mathematical physics a welcome distraction: “As you see, the war is kindly disposed toward me, allowing me, despite fierce gunfire at a decidedly terrestrial distance, to take this walk into this your land of ideas.” Einstein was surprised. The calculation was so simple. How had he missed it?

  Einstein was even more startled by Schwarzschild’s next letter a week later. This short note emerged from the trenches to do something that Einstein was unsure was even possible—it presented an exact solution to the equations of general relativity. The significance of this requires a bit of explanation. General relativity is described by mathematical entities called differential equations. These are a little stranger than the equations you learn to solve in high school algebra. In an equation like x + 3 = 7, we say the solution for x is 4. But for a differential equation the solution is a whole other equation—in fact, usually a whole group of equations. If someone hands you a random differential equation, it is pretty straightforward to find an approximate answer—a good enough answer. Unfortunately, if you want an exact solution there is often no easy way to find the answer. It is even possible that there is no exact solution. Finding exact solutions to complicated differential equations is a trying task.

  So when Einstein “finished” general relativity in November 1915, what he actually had was a group of ten differential equations that might or might not have solutions. If they did, he didn’t know what they were. It would be reasonable at this point to voice an objection like this: what good are the field equations, then, if we still need further solutions? This happens a lot in physics. When you articulate a general principle (like F=ma, Newton’s second law) as a differential equation, you are making a statement about what kinds of things are allowed to happen in the world—what kind of processes nature prefers to do and what patterns you might expect to see. It is a claim about how the universe likes to behave in general. But for a specific example of that principle in action, you need a solution. Newton’s second law tells us the general rules for accelerating bodies; a solution to the second law tells us the actual trajectory a specific body will take in a certain situation.

  Imagine that you are a scientist studying highways. Your differential equation might be a statement like “rest stops are always found along highways” or “cloverleaf interchanges appear where highways meet.” Those are important things to know about highways. They are not, however, the same as the map of an actual highway. You know the actual highway will obey your principles, but someone still has to go out there and make the map.

  Einstein wasn’t sure there was a map for general relativity. He was confident the differential equations—the general principles of how space-time should behave—were correct without knowing what an actual piece of space-time might look like. He was just working with approximations, with the good enough answers. Schwarzschild was the first to find an exact solution to Einstein’s equations. He found a possible real world within the kind of universe that Einstein had imagined. Nowadays in the twenty-first century we have many solutions to the Einstein equations. This was the first.

  * * *

  • • •

  WHAT WE NOW call the Schwarzschild solution applies to a very specific physical situation: a perfectly round object sitting by itself. His solution describes the precise shape that space-time takes around such an object, which then lets us find exact trajectories for things moving in that area. Physicists love perfectly round objects because they make many calculations much, much simpler. Spheres have a property called rotational symmetry, which means they look the same from any angle. It is impossible to overemphasize how much easier this property makes physics, and how much physicists like to use it. There is an old joke in which a dairy farmer asks a physicist for help increasing milk production. The physicist pulls out a blackboard and begins, “Assume the cow is a sphere. . . .”

  Fortunately for Einstein and Schwarzschild, most astronomical bodies look a lot like spheres, so this simple solution is extremely helpful. Einstein was enormously impressed (and grateful). He read Schwarzschild’s paper verbatim at the next meeting of the Prussian Academy. Few in attendance would have appreciated the significance. He knew he still needed an empirical test to win over the skeptics. He hoped that Schwarzschild, as an astrophysicist, might be able to help with a test as well. The contribution to the Mercury problem was good, he wrote, though “the question of light deflection is of most importance.” It was on this observation—whether a beam of light was deflected by gravity—that he thought his theory would rise or fall.

  Einstein tried to enlist Schwarzschild in his endless task of winning time for Erwin Freundlich to undertake new deflection measurements. Outside of Schwarzschild and Max Planck (both of whom were somewhat lukewarm), relativity had few enthusiasts. Even Max Born, with whom Einstein got along well, called the theory “frightening” and best “enjoyed and admired from a distance.” Astronomers in particular were generally uninterested in this strange theory that seemed to have little significance for their work. But Einstein knew he would eventually need those same astronomers to search for the empirical evidence for relativity. Freundlich was thus an important ally. Einstein had been working diligently to get him liberated from his tedious ordinary duties at the observatory in order to get the young man focused on Einstein’s own project.

  Freundlich was still despondent about his failure to test the deflection at the solar eclipse in 1914. As an alternative, he thought he might be able to measure the light deflection due to Jupiter’s gravity. It would be spectacularly difficult (about a hundred times harder than measuring the deflection from the sun). Nonetheless, an increasingly anxious Einstein declare
d to Schwarzschild, “It has to work!” Freundlich’s boss, Otto Struve, had no interest in losing his assistant to Einstein’s odd ideas and blocked all efforts. Einstein hoped Schwarzschild, as a fellow astronomer, might be able to persuade Freundlich’s famously truculent supervisor to be more reasonable. With typically Einsteinian bluntness, he described Freundlich as not “a very great talent” but merely the first astronomer to understand the importance of general relativity. And that “is why I would regret it deeply if he were deprived the possibility of working in this field.” Schwarzschild commiserated, but replied that Freundlich had so alienated Struve that the task was hopeless. Also, he informed Einstein, Jupiter would be too far south for precise observations for the next several years. This was only one of Einstein’s many attempts to gain support for Freundlich. He complained to Arnold Sommerfeld that he simply lacked contacts in astronomy. To David Hilbert, he described his fruitless efforts as a “foiled assault on the astronomical fortress defended by an invincible phlegm.”

  In the same letter that Schwarzschild dashed Einstein’s hopes for Jupiter, he noted that he had found a strange feature of his solution to the field equations. If enough mass were packed into a small enough place, a sort of “closed-off” pocket of space-time would form. He thought of this as a mathematical oddity, a quirk of the equations that probably had no physical significance. Later generations of physicists would decide that this closed-off pocket was quite real and give it a special name—a black hole. Neither correspondent realized the implication of this at the time.