Why Beauty is Truth Page 23

  Pondering this, the mathematicians Henri Poincaré and Hermann Minkowski were led to a new view of the symmetries of space and time, on a purely mathematical level. If they had described these symmetries in physical terms, they would have beaten Einstein to relativity, but they avoided physical speculations. They did understand that symmetries of the laws of electromagnetism do not affect space and time independently but mix them up. The mathematical scheme describing these intertwined changes is known as the Lorentz group, after the physicist Hendrik Lorentz.

  Minkowski and Poincaré viewed the Lorentz group as an abstract expression of certain features of the laws of physics, and descriptions like “time passing more slowly” or “objects shrinking as they speed up” were thought of as vague analogies rather than anything real. But Einstein insisted that these transformations have a genuine physical meaning. Objects, and time, really do behave like that. He was led to formulate a physical theory, special relativity, that incorporated the mathematical scheme of the Lorentz group into a physical description not of space and a separate time, but of a unified space-time.

  Minkowski came up with a geometric picture for this non-Newtonian physics, now called Minkowski space-time. It represents space and time as independent coordinates, and a moving particle traces out a curve—which Einstein called its world line—as time passes. Because no particle can travel faster than light, the slope of the world line can never get more than 45° away from the time direction. The particle’s past and future always lie inside a double cone, its light cone.

  Geometry of Minkowski space-time.

  That took care of electricity and magnetism, two basic forces of nature. But one basic force was still missing from this description: gravity. Attempting to develop a more general theory that included gravity, and again relying on the principle that the laws of nature must be symmetric, Einstein was led to general relativity: the idea that space-time itself is curved and that its curvature corresponds to mass. From these ideas emerged our current cosmology of the Big Bang, in which the universe grew from a tiny speck some 13 billion years ago, and the remarkable concept of a black hole, an object so massive that light cannot escape its gravitational field.

  General relativity traces back to early work on non-Euclidean geometry, which led Gauss to the idea of a “metric,” a formula for the distance between any two points. New geometries arise when this formula is not the classic Euclidean one derived from the Pythagorean theorem. As long as the formula obeys a few simple rules, it defines a meaningful concept of “distance.” The main rule is that the distance from one point A to another C cannot decrease if you pass through a point B in between. That is, the direct distance from A to C is less than or equal to the distance from A to B plus that from B to C. This is the “triangle inequality,” so called because in Euclidean geometry it states that any side of a triangle is shorter than the other two put together.

  The Pythagorean formula holds in Euclidean geometry, in which space is “flat.” So when the metric is different from the Euclidean one, we can attribute that difference to some kind of “curvature” of space. You can visualize this as a bending of space, but that’s not really the best picture because there must then be a bigger space for the original one to bend in. A better way to think of “curvature” is that regions of space are either compressed or stretched, so that from the inside they seem to hold less, or more, space than they do from outside. (Fans of the British TV series Doctor Who will be reminded of the Tardis, a spaceship/time-machine whose inside is larger than its outside.) Gauss’s brilliant student Riemann extended the idea of a metric from two dimensions to any number, and he modified the idea so that distances can be defined locally—just for points that are very close together. Such a geometry is called a Riemannian manifold, and it is the most general kind of “curved space.”

  Physics happens not in space but in space-time, where—according to Einstein—the natural “flat” geometry is not Euclidean but Minkowskian. Time enters into the “distance” formula in a different way from space. Such a geometric setup is a “curved space-time.” It turned out to be just what the patent clerk ordered.

  Einstein struggled a long time to devise his equations for general relativity. He first investigated how light moves in a gravitational field, and this led him to base his later research on a single fundamental principle, the equivalence principle. In Newtonian mechanics, gravity has the effect of a force, pulling bodies toward each other. Forces cause accelerations. The equivalence principle states that accelerations are always indistinguishable from the effects of a suitable gravitational field. In other words, the way to put gravity into relativity is to understand accelerations.

  By 1912, Einstein had convinced himself that a theory of gravity cannot be symmetric under every Lorentz transformation; that kind of symmetry applies exactly, everywhere, only when matter is absent, gravity is zero, and space-time is Minkowskian. By abandoning this requirement of “Lorentz-invariance” he saved himself a lot of fruitless effort. “The only thing I believed firmly,” he wrote in 1950, “was that one had to incorporate the equivalence principle in the fundamental equations.” But he also recognized the limitations even of that principle: it should be valid only locally, as a kind of infinitesimal approximation to the true theory.

  By 1907, Einstein’s friend Grossmann had become a geometry professor at ETH, and Albert was persuaded to take a position there too. Not for long—after a year he left for Berlin and later went to Prague. But he kept in contact with Grossmann, and this paid off handsomely. In 1912, Grossmann helped Einstein to work out what kind of mathematics he should be thinking about:

  This problem remained unsolvable to me until . . . I suddenly realized that Gauss’s theory of surfaces held the key for unlocking the mystery . . . However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way . . . My dear friend the mathematician Grossman was there when I returned from Prague to Zurich. From him I learned for the first time about Ricci and later about Riemann. So I asked my friend whether my problem could be solved by Riemann’s theory.

  “Ricci” is Gregorio Ricci-Curbastro, the coinventor, along with his student Tullio Levi-Civita, of calculus on Riemannian manifolds. The Ricci tensor is a measure of curvature, simpler than Riemann’s original concept.

  Other sources have Einstein saying to Grossmann, “You must help me, or else I’ll go crazy!” Grossmann delivered. As Einstein later wrote, he “not only saved me the study of the relevant mathematical literature, but also supported me in the search for the field equations of gravitation.” In 1913, Einstein and Grossmann published the first fruits of their combined labors, ending with a conjecture about the form of the required field equations: the stress-energy tensor must be proportional to . . . something.

  What?

  They didn’t yet know. It had to be another tensor, another measure of curvature.

  At that point they both made mathematical errors, which set them off on a lengthy wild goose chase. They were convinced, correctly, that their theory had to yield Newtonian gravity in a suitable limiting case—flat space-time, small gravity. They deduced from this some technical constraints on the sought-for equation, that is, constraints on the nature of the required “something.” But their arguments were fallacious and the constraints did not apply.

  Einstein was convinced that the correct field equations should determine the mathematical form of the metric—the distance formula in space-time, which determines all of its geometrical properties—uniquely. This is simply wrong: changes in the coordinate system can change the formula while having no effect on the intrinsic curvature of the space. But Einstein was unaware of the so-called Bianchi identities, which clarify the lack of uniqueness, and apparently so was Grossmann.

  It was every researcher’s nightmare: an apparently watertight idea, which seemed to lead in the right direction but was actually leading them up the garden path. Eradicating such mistakes is desperatel
y hard, because you’re convinced they are not mistakes. Often you don’t even realize what assumptions you are tacitly making.

  At the end of 1914, Einstein finally realized that the field equations cannot determine the metric uniquely because of the possibility of choosing a different coordinate system, which has no physical implications but changes the formula for the metric. He still did not know the Bianchi identities, but now he didn’t need them. He finally knew that he was free to choose whichever coordinates were most convenient.

  On 18 November 1914, Einstein opened up a new front in his war with the gravitational field equations. He had gotten close enough to his final formulation to start making predictions. He made two. One—really a “postdiction,” made after the event—explained a tiny change already observed in the orbit of the planet Mercury. The “perihelion” position, where the planet comes closest to the Sun, was slowly changing. Einstein’s new theory of gravity told him how fast the perihelion should be moving—and his calculation was spot on.

  The second prediction required new observations to verify or falsify it—which was excellent news, because new observations are the best tests of new theories. According to Einstein’s theory, gravity should bend light. The geometry of this effect is simple, and it concerns geodesics—the shortest path between any two points. If you stretch a string tight and hold it in mid-air, it forms a straight line; this happens because in Euclidean space a straight line is a geodesic. If, however, you hold the two ends of the string against a football and pull it tight, it forms a curve lying on the surface of the ball. Geodesics on a curved space—the ball—are themselves curved. The same happens in a curved space-time, though the details are slightly different.

  The physical circumstances in which this effect might show up are also straightforward. A star, such as the Sun, will bend any light that passes nearby. The only way to observe this effect, at that time, was to wait for an eclipse of the Sun, when the Sun’s light no longer drowned out the light from stars whose position in the sky was close to the Sun’s edge. If Einstein was right, the apparent positions of those stars should shift slightly, compared to their positions when they were not aligned with the Sun.

  The quantitative analysis of this phenomenon is less straightforward. Einstein’s first attempt, in 1911, predicted a shift of just under a second of arc. Newton would have predicted a similar amount, based on his belief that light is made from tiny particles: the force of gravity would attract the particles, causing their paths to bend. But by 1915 Einstein had deduced that in his new theory, the light should bend by twice that amount, 1.74 seconds of arc.

  Now there was a real prospect of deciding between Newton and Einstein. On 25 November 1914, Einstein wrote down his field equations in their final form. These Einstein equations constitute the basis of general relativity, the relativistic theory of gravity. They are written in a mathematical formalism known as a tensor—a kind of hyped-up matrix. Einstein’s equation tells us that the Einstein tensor is proportional to the rate of change of the stress-energy tensor. That is, the curvature of space-time is proportional to the quantity of matter present. These equations obey a kind of symmetry principle, but it is a local one. In small regions of space-time, they have the same symmetries as special relativity, provided the local effect of curvature is taken into account.

  Einstein noted that his calculations of the motion of the perihelion of Mercury and the deflection of light by a star remained unchanged by the minor modifications that he had made. He presented his equations to the Prussian Academy, only to discover that the mathematician David Hilbert had already submitted the identical equations but had claimed far more for them than just a theory of gravity. In fact, he had claimed that they included the electromagnetic equations, which was a mistake. It is fascinating to see, yet again, a top mathematician coming very close to beating Einstein to the punch.

  Several attempts were made to verify Einstein’s prediction that light would be deflected by the gravitational field of the Sun. The first, in Brazil, was spoiled by rain. In 1914, a German expedition went to observe an eclipse in Crimea, but when World War I began they were instructed to return home—fast. Some did. The others were arrested but eventually made their way home unharmed. Naturally, no observations were made. The war prevented observations in Venezuela in 1916. The Americans tried in 1918, with inconclusive results. Finally, a British expedition led by Arthur Eddington succeeded in May 1919, but did not announce its results until November.

  When it did, the verdict favored Einstein over Newton. There was a deflection, it was too big to fit a Newtonian model, and it fitted Einstein’s beautifully.

  In retrospect, the experiments were not as decisive as they seemed. The range of experimental error was quite large, and the best conclusion was that Einstein was probably right. (More recent observations, with better techniques and equipment, have confirmed Einstein’s theory.) But at the time, they were represented as definitive, and the media went ape. Anyone who could prove Newton wrong must be a genius. Anyone who could discover radically new physics must be the greatest living scientist.

  Thus was a legend born. Einstein wrote about his ideas in the Times of London. A few days later, the paper’s editorial page responded:

  This is news distinctly shocking and apprehensions for the safety of confidence even in the multiplication table will arise . . . It would take the presidents of two Royal Societies to give plausibility or even thinkability to the declaration that light has weight and space has limits. It just doesn’t by definition, and that’s the end of that—for commonfolk, however it may be for higher mathematicians.

  But the higher mathematicians were right. Soon the Times was telling the world that “only twelve people can understand the theory of ‘the suddenly famous Dr. Einstein,’ ” a myth that circulated for years, even when large numbers of physics undergraduates were routinely being taught the theory in their coursework.

  In 1920, Grossmann showed the first symptoms of multiple sclerosis. He wrote his last paper in 1930 and died in 1936. Einstein went on to become the iconic physicist of the twentieth century. In later life he grew to tolerate his fame, finding it vaguely amusing. Early on, he seems to have enjoyed interacting with the media.

  But now we must leave Einstein’s career, except to remark that after 1920 his efforts in physics were devoted to a fruitless quest to combine relativity and quantum mechanics in a single “unified field theory.” He was still working on this problem the day before his death, in 1955.

  12

  A QUANTUM QUINTET

  “Almost everything is already discovered, and all that remains is to fill a few holes.” This is discouraging news for a talented young man intending to study physics, especially when the news comes from someone who ought to know: in this case, Philipp von Jolly, a physics professor.

  The date was 1874, and von Jolly’s view reflected what most physicists of the period believed: physics was done. In 1900 no less a luminary than Lord Kelvin said, “There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.”

  Mind you, he also said, “I can state flatly that heavier than air flying machines are impossible,” and “Landing on the moon offers so many serious problems for human beings that it may take science another 200 years to lick them.” Kelvin’s biographer wrote that he spent the first half of his career being right and the second half being wrong.

  But he wasn’t totally wrong. In his 1900 lecture “Nineteenth-Century Clouds over the Dynamical Theory of Heat and Light,” he put his finger on two crucial gaps in the period’s understanding of the physical universe: “The beauty and clearness of the dynamical theory, which asserts heat and light to be modes of motion, is at present obscured by two clouds. The first involves the question, How could the Earth move through an elastic solid, such as essentially is the luminiferous ether? The second is the Maxwell–Boltzmann doctrine regarding the partition of energy.” The first cloud led to relativity,
the second to quantum theory.

  Fortunately, the young recipient of Jolly’s advice was not daunted. He had no wish to discover new things, he said—all he wanted was to develop a better understanding of the known fundamentals of physics. In the search for this understanding, he brought about one of the two great revolutions in twentieth-century physics, and dispelled Kelvin’s second cloud. His name was Max Planck.

  Julius Wilhelm Planck was a professor of law in Kiel and Munich. His father and his mother had both been theology professors, and his brother was a judge. So when his second wife, Emma Patzig, presented Julius with a son—his sixth child—the boy was certain to grow up in an intellectual environment. Max Karl Ernst Ludwig Planck was born on 23 April 1858. Europe was in the usual political turmoil, and the boy’s earliest memories included Prussian and Austrian troops marching into Kiel during the Danish–Prussian War of 1864.

  By 1867 the Plancks had moved to Munich, and Max was being tutored by the mathematician Hermann Müller at the King Maximilian School. Müller taught the boy astronomy, mechanics, mathematics, and some basic physics, including the law of conservation of energy. Planck was an excellent student, and he graduated unusually early, at the age of sixteen.

  He was also a talented musician, but he decided to study physics despite Jolly’s well-intentioned advice. Planck carried out some experiments under Jolly’s supervision but quickly switched to theoretical physics. He kept company with some of the world’s leading physicists and mathematicians, moving to Berlin in 1877 to study under Helmholtz, Gustav Kirchhoff, and Weierstrass. He passed his first examinations in 1878 and obtained a doctorate in 1879 with a thesis on thermodynamics. For a time he taught mathematics and physics at his old school. In 1880, his habilitation thesis, on equilibrium states of bodies at different temperatures, was accepted, and he was qualified for a permanent academic career. He duly secured such a position, but not until 1885, when the University of Kiel made him an associate professor. His research focused on thermodynamics, especially the concept of entropy.