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by Michio Kaku

  This bending, of course, is invisible, and from a distance, Newton’s picture appears to be correct. Think of ants walking on a crumpled sheet of paper. Trying to follow a straight line, they find that they are constantly being tugged to the left and right as they walk over the folds in the paper. To the ants, it appears as if there is a mysterious force pulling them in both directions. However, to someone looking down on the ants, it is obvious that there is no force, there is just the bending of the paper pushing on the ants, which gives the illusion that there is a force. Recall that Newton thought of space and time as an absolute reference frame for all motions. However, to Einstein, space and time could assume a dynamic role. If space is curved, then anyone moving on this stage would think that mysterious forces were acting on their bodies, pushing them one way or the other.

  By comparing space-time to a fabric that can stretch and bend, Einstein was forced to study the mathematics of curved surfaces. He quickly found himself buried in a morass of mathematics, unable to find the right tools to analyze his new picture of gravity. In some sense, Einstein, who once scorned mathematics as “superfluous erudition,” was now paying for the years in which he cut the mathematics courses at the Polytechnic.

  In desperation, he turned to his friend, Marcel Grossman. “Grossman, you must help me or else I’ll go crazy!” Einstein confessed, “Never in my life have I tormented myself anything like this, and that I have become imbued with a 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.”

  When Grossman reviewed the mathematical literature, he found that, ironically enough, the basic mathematics that Einstein needed had indeed been taught at the Polytechnic. In the geometry of Bernhard Riemann, developed in 1854, Einstein finally found the mathematics powerful enough to describe the bending of space-time. (Years later, when looking back at how difficult it was to master new mathematics, Einstein noted to some junior high school students, “Do not worry about your difficulties in mathematics; I can assure you that mine are still greater.”)

  Before Riemann, mathematics was based on Euclidean geometry, the geometry of flat surfaces. Schoolchildren for thousands of years had been grilled in the time-honored theorems of Greek geometry, where the sum of the interior angles of a triangle equals 180 degrees, and parallel lines never meet. Two mathematicians, the Russian Nicolai Lobachevsky and the Austro-Hungarian János Bolyai, came extremely close to developing a non-Euclidean geometry, that is, in which the sum of the angles of a triangle can be more or less than 180 degrees. But the theory of non-Euclidean geometry was finally developed by the “prince of mathematics,” Carl Friedrich Gauss, and especially his student, Riemann. (Gauss suspected that Euclid’s theory might be incorrect even on physical grounds. He had his assistants shine light beams from atop the Harz Mountains, trying to experimentally calculate the sum of the angles of a triangle formed by three mountain-tops. Unfortunately, he got a negative result. Gauss was also such a politically cautious individual that he never published his work on this sensitive subject, fearing the ire of conservatives who swore by Euclidean geometry.)

  Riemann discovered entirely new worlds of mathematics—the geometry of curved surfaces in any dimension, not just two or three spatial dimensions. Einstein was convinced these higher geometries would yield a more accurate description of the universe. For the first time, the mathematical language of “differential geometry” was working its way into the world of physics. Differential geometry, or tensor calculus, the mathematics of curved surfaces in any dimension, was once considered to be the most “useless” branch of mathematics, devoid of any physical content. Suddenly, it was transformed into the language of the universe itself.

  In most biographies, Einstein’s theory of general relativity is presented as fully developed in 1915, as if he unerringly found the theory fully formed by magic. However, only in the last decades have some of Einstein’s “lost notebooks” been analyzed, and they fill in the many missing gaps between 1912 and 1915. Now it is possible to construct, sometimes month by month, the crucial evolution of one of the greatest theories of all time. In particular, he wanted to generalize the notion of covariance. Special relativity, as we saw, was based on the idea of Lorentz covariance, that is, that the equations of physics retain the same form under a Lorentz transformation. Now Einstein wanted to generalize this to all possible accelerations and transformations, not just inertial ones. In other words, he wanted equations that retained the same form no matter what frame of reference was used, whether it was accelerating or moving with constant velocity. Each frame of reference in turn requires a coordinate system to measure the three dimensions of space and the time. What Einstein desired was a theory that retained its form no matter which distance and time coordinates were used to measure the frame. This led him to his famed principle of general covariance: the equations of physics must be generally covariant (i.e., they must maintain the same form under an arbitrary change of coordinates).

  For example, think of throwing a fishing net over a tabletop. The fishing net represents an arbitrary coordinate system, and the area of the tabletop represents something that remains the same under any distortion of the fishing net. No matter how we twist or curl up the fishing net, the area of the table top remains the same.

  In 1912, aware that Riemann’s mathematics was the correct language for gravitation, and guided by the law of general covariance, Einstein searched within Riemannian geometry for objects that are generally covariant. Surprisingly, there were only two covariant objects available to him: the volume of a curved space and the curvature (called the “Ricci curvature”) of such a space. This was of immense help: by severely restricting the possible building blocks used to construct a theory of gravity, the principle of general covariance led Einstein to formulate the essentially correct theory in 1912, after only a few months of examining Riemann’s work, based on the Ricci curvature. For some reason, however, he threw away the correct theory of 1912 and began to pursue an incorrect idea. Precisely why he abandoned the correct theory was a mystery to historians until recently, when the lost notebooks were discovered. That year, when he essentially constructed the correct theory of gravity out of the Ricci curvature, he made a crucial mistake. He thought that this correct theory violated what is known as “Mach’s principle.” One particular version of this principle postulates that the presence of matter and energy in the universe uniquely determines the gravitational field surrounding it. Once you fix a certain configuration of planets and stars, then the gravitation surrounding these planets and stars is fixed. Think, for example, of throwing a pebble into a pond. The larger the pebble, the greater the ripples on the pond. Thus, once we know the precise size of the pebble, the distortion of the pond can be uniquely determined. Likewise, if we know the mass of the sun, we can uniquely determine the gravitational field surrounding the sun.

  This is where Einstein made his mistake. He thought that the theory based on the Ricci curvature violated Mach’s principle because the presence of matter and energy did not uniquely specify the gravitational field surrounding it. With his friend Marcel Grossman, he tried to develop a more modest theory, one that was covariant just under rotations (but not general accelerations). Because he abandoned the principle of covariance, however, there was no clear path to guide him, and he spent three frustrating years wandering in the wilderness of the Einstein-Grossman theory, which was neither elegant nor useful—for instance, it failed to yield Newton’s equations for small gravitational fields. Although Einstein had perhaps the best physical instincts of anyone on Earth, he ignored them.

  While groping for the final equations, Einstein focused on three key experiments that might prove his ideas concerning curved space and gravity: the bending of starlight during an eclipse, the red shift, and the perihelion of Mercury. In 1911, even before his work on curved space, Einstein held out hope that an expedition could be sent to Si
beria during the solar eclipse of August 21, 1914, to find the bending of starlight by the sun.

  The astronomer Erwin Finlay Freundlich was to investigate this eclipse. And Einstein was so convinced of the correctness of his work that at first he offered to fund the ambitious project out of his own pocket. “If everything fails, I’ll pay for the thing out of my own slight savings, at least the first 2,000 marks,” he wrote. Eventually, a wealthy industrialist agreed to provide the funding. Freundlich left for Siberia a month before the solar eclipse, but Germany declared war on Russia, and he and his assistant were taken prisoner and their equipment was confiscated. (In hindsight, perhaps it was fortunate for Einstein that the 1914 expedition was unsuccessful. If the experiment had been performed, the results would not, of course, have agreed with the value predicted by Einstein’s incorrect theory, and his entire program might have been disgraced.)

  Next, Einstein calculated how gravity would affect the frequency of a light beam. If a rocket is launched from the earth and sent into outer space, the gravity of the earth acts like a drag, pulling the rocket back. Energy is therefore lost as the rocket struggles against the pull of gravity. Similarly, Einstein reasoned that if light were emitted from the sun, then gravity would also act as a drag on the light beam, making it lose energy. The light beam will not change in velocity, but the frequency of the wave will drop as it loses energy struggling against the sun’s gravity. Thus, yellow light from the sun will decrease in frequency and become redder as the light beam leaves the sun’s gravitational pull. Gravitational red shift, however, is an extremely small effect, and Einstein had no illusion that it would be tested in the laboratory any time soon. (In fact it would take four more decades before gravitational red shift could be seen in the laboratory.)

  Last, he set out to solve an age-old problem: why the orbit of Mercury wobbles and deviates slightly from Newton’s laws. Normally, the planets execute perfect ellipses in their journeys around the sun, except for slight disturbances caused by the gravity of nearby planets, which results in a trajectory resembling the petals of a daisy. The orbit of Mercury, however, even after subtracting the interference caused by nearby planets, showed a small but distinct deviation from Newton’s laws. This deviation, called the “perihelion,” was first observed in 1859 by astronomer Urbain Leverrier, who calculated a tiny shift of 43.5 seconds of arc per century that could not be explained by Newton’s laws. (The fact that there were apparent discrepancies in Newton’s laws of motion was not new. In the early 1800s, when astronomers were puzzled by a similar wobbling of the orbit of Uranus, they faced a stark choice: either abandon the laws of motion or postulate that there was another unknown planet tugging on the orbit of Uranus. Physicists breathed a sigh of relief when in 1846, a new planet, christened Neptune, was discovered just where Newton’s laws predicted it should be.)

  But Mercury was the remaining puzzle. Rather than discard Newton, astronomers in time-honored tradition postulated the existence of a new planet called “Vulcan,” circling the sun within the orbit of Mercury. In repeated searches of the night sky, however, astronomers could find no experimental evidence for such a planet.

  Einstein was prepared to accept the more radical interpretation: perhaps Newton’s laws themselves were incorrect, or at least incomplete. In November 1915, after wasting three years on the Einstein-Grossman theory, he went back to the Ricci curvature, which he had discarded back in 1912, and spotted his key mistake. (Einstein had dropped the Ricci curvature because it yielded more than one gravitational field generated by a piece of matter, in seeming violation of Mach’s principle. But then, because of general covariance, he now realized that these gravitational fields were actually mathematically equivalent and yielded the same physical result. This impressed upon Einstein the power of general covariance: not only did it severely restrict the possible theories of gravity, it also yielded unique physical results because many gravitational solutions were equivalent.)

  In perhaps the greatest mental concentration of Einstein’s life, he then slaved away at his final equation, shutting out all distractions and working himself mercilessly to see if he could derive the perihelion of Mercury. His lost notebooks show that he would repeatedly propose a solution and then ruthlessly check to see that it reproduced Newton’s old theory in the limit of small gravitational fields. This task was extremely tedious, since his tensor equations consisted of ten distinct equations, rather than the single equation of Newton. If it failed, then he would try another solution to see if that reproduced Newton’s equation. This exhaustive, almost Herculean task was finally completed in late November 1915, leaving Einstein totally drained. After a long tedious calculation with his old theory of 1912, he found that it predicted the deviation in Mercury’s orbit to be 42.9 seconds of arc per century, well within acceptable experimental limits. Einstein was shocked beyond belief. This was exhilarating, the first solid experimental evidence that his new theory was correct. “For some days, I was beyond myself with excitement,” he recalled. “My boldest dreams have now come true.” The dream of a lifetime, to find the relativistic equations for gravity, was realized.

  What thrilled Einstein was that through the abstract physical and mathematical principle of general covariance, he could derive a solid, decisive experimental result: “Imagine my joy over the practicability of general covariance and over the result that the equations correctly yield the perihelion movement of Mercury.” With the new theory, he then recalculated the bending of starlight by the sun. The addition of curved space to his theory meant that this final answer was 1.7 seconds of arc, twice his original value (about 1/2000th of a degree).

  He was convinced that the theory was so simple, elegant, and powerful that no physicist could escape its hypnotic spell. “Hardly anyone who has truly understood it will be able to escape the charm of this theory,” he would write. “The theory is of incomparable beauty.” Miraculously, the principle of general covariance was so powerful a tool that the final equation, which would describe the structure of the universe itself, was only 1 inch long. (Physicists today marvel that an equation so short can reproduce the creation and evolution of the universe. Physicist Victor Weisskopf likened that sense of wonder to the story of a peasant who saw a tractor for the first time in his life. After examining the tractor and peering under the hood, he asks in bewilderment, “But where is the horse?”)

  The only thing to mar Einstein’s triumph was a minor priority fight with David Hilbert, perhaps the world’s greatest living mathematician. While the theory was in its last, final steps before completion, Einstein had given a series of six two-hour lectures at Göttingen for Hilbert. Einstein still lacked certain mathematical tools (called the “Bianchi identities”) that prevented him from deriving his equations from a simple form, called the “action.” Later, Hilbert filled in the final step in the calculation, wrote down the action, and then published the final result by himself, just six days ahead of Einstein. Einstein was not pleased. In fact, he believed that Hilbert had tried to steal the theory of general relativity by filling in the final step and taking credit. Eventually, the rift between Einstein and Hilbert healed, but Einstein became wary of sharing his results too freely. (Today, the action by which one derives general relativity is known as the “Einstein-Hilbert action.” Hilbert was probably led to finish the last tiny piece of Einstein’s theory because, as he often said, “physics is too important to be left to the physicists” that is, physicists probably were not mathematically skilled enough to probe nature. This view apparently was shared by other mathematicians. The mathematician Felix Klein would grumble that Einstein was not innately a mathematician, but worked under the influence of obscure physical-philosophical impulses. That is probably the essential difference between mathematicians and physicists and why the former have consistently failed to find new laws of physics. Mathematicians deal exclusively with scores of small, self-consistent domains, like isolated provinces. Physicists, however, deal with a handful of simple physical prin
ciples that may require many mathematical systems to solve. Although the language of nature is mathematics, the driving force behind nature seems to be these physical principles, for example, relativity and the quantum theory.)

  News of Einstein’s new theory of gravity was interrupted by the outbreak of war. In 1914, the assassination of the heir to the Austro-Hungarian throne touched off the greatest bloodletting of its time, drawing the British, Austro-Hungarian, Russian, and Prussian empires into a catastrophic conflict that would doom tens of millions of young men. Almost overnight, quiet, distinguished professors at German universities became bloodthirsty nationalists. Nearly the entire faculty at the University of Berlin was swept up by the war fever and devoted all their energies to the war effort. In support of the Kaiser, ninety-three prominent intellectuals signed the notorious “Manifesto to the Civilized World,” which called for all people to rally around the Kaiser and ominously declared that the German people must defy “Russian hordes allied with Mongols and Negroes unleashed against the white race.” The manifesto justified the German invasion of Belgium and proudly proclaimed, “The German Army and the German people are one. This awareness now binds seventy million Germans without distinction of education, class, or party.” Even Einstein’s benefactor, Max Planck, signed the manifesto, as did such distinguished individuals as Felix Klein and physicists Wilhelm Roentgen (the discoverer of X-rays), Walther Nernst, and Wilhelm Ostwald.