The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)
Chapter 2. GEOMETRY IN THE NATURAL ORDER
Over most of the last two and a half thousand years in the European or Western tradition, geometry has been studied because it has been held to be the most exquisite, perfect, paradigmatic truth available to us outside divine revelation. Studying geometry reveals, in some way, the deepest true essence of the physical world.
—PIERS BURSILL-HALL, “WHY DO WE STUDY GEOMETRY? ”
What is geometry? Many think of it as simply a course they took in high school—a collection of techniques for measuring the angles between lines, calculating the area of triangles, circles, and rectangles, and perhaps establishing some measure of equivalence between disparate objects. Even with such a limited definition, there’s no doubt that geometry is a useful tool—one that architects, for instance, make use of every day. Geometry is these things, of course, and much, much more, for it actually concerns architecture in the broadest sense of the term, from the very smallest scales to the largest. And for someone like me, obsessed with understanding the size, shape, curvature, and structure of space, it is the essential tool.
The word geometry, which comes from geo (“earth”) and metry (“measure”), originally meant “measuring the earth.” But we now put it in more general terms to mean measuring space, where space itself is not a particularly well-defined concept. As Georg Friedrich Bernhard Riemann once said, “geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space,” while giving “only nominal definitions of these things.”1
Odd as it may sound, we find it useful to keep the concept of space rather fuzzy because it can imply many things for which we have no other terms. So there’s some convenience to that vagueness. For example, when we contemplate how many dimensions there are in space or ponder the shape of space as a whole, we might just as well be referring to the entire universe. A space could also be more narrowly defined to mean a simple geometric construct such as a point, line, plane, sphere, or donut—the sorts of figures a grade school student might draw—or it could be more abstract, more complex, and immensely more difficult to picture.
Suppose, for instance, you have a bunch of points spread out in some complicated, haphazard arrangement with absolutely no way of determining the distance between them. As far as mathematicians are concerned, that space has no geometry; it’s just a random assortment of points. But once you put in some kind of measurement function, technically called a metric, which tells you how to compute the distance between any two points, then your space has suddenly become navigable. It has a well-defined geometry. The metric for a space, in other words, gives you all the information you need to divine its shape. Armed with that measurement capability, you can now determine its flatness to great precision, as well as its deviation from flatness, or curvature, which is the thing I find most interesting of all.
Lest one conclude that geometry is little more than a well-calibrated ruler—and this is no knock against the ruler, which happens to be a technology I admire—geometry is one of the main avenues available to us for probing the universe. Physics and cosmology have been, almost by definition, absolutely crucial for making sense of the universe. Geometry’s role in all this may be less obvious, but it is equally vital. I would go so far as to say that geometry not only deserves a place at the table alongside physics and cosmology, but in many ways it is the table.
For you see, this entire cosmic drama—a complex dance of particles, atoms, stars, and other entities, constantly shifting, moving, interacting—is played out on a stage, inside a “space,” if you will, and it can never be truly understood without grasping the detailed features of that space. More than just a passive backdrop, space actually imbues its constituents with intrinsically vital properties. In fact, as we currently view things, matter or particles sitting (or moving) in a space are actually part of that space or, more precisely, spacetime. Geometry can impose constraints on spacetime and on physical systems in general—constraints that we can deduce purely from the principles of mathematics and logic.
Consider the climate of the earth. Though it may not be obvious, the climate can be profoundly influenced by geometry—in this case by the essentially spherical shape of our host planet. If we resided on a two-dimensional torus, or donut, instead, life—as well as our climate—would be substantially different. On a sphere, winds can’t blow in the same direction (say, east), nor can the ocean’s waters all flow in the same direction (as mentioned in the final chapter). There will inevitably be places—such as at the north and south poles—where wind or current direction no longer points east, where the whole notion of “east” disappears, and all movement grinds to a halt. This is not the case on the surface of a single-holed donut, where there are no such impasses and everything can flow in the same direction without ever hitting a snag. (That difference would surely affect global circulation patterns, but if you want to know the exact climatological implications—and get a seasonal comparison between spherical and toroidal living—you’d better ask a meteorologist.)
The scope of geometry is even broader still. In concert with Einstein’s theory of general relativity, for example, geometry has shown that the mass and energy of the universe are positive and hence that spacetime, the four-dimensional realm we inhabit, is stable. The principles of geometry also tell us that somewhere in the universe, there must be strange places known as singularities—thought to lie, for instance, in the center of black holes—where densities approach infinity and known physics breaks down. In string theory, to take another example, the geometry of weird six-dimensional spaces called Calabi-Yau manifolds—where much important physics supposedly takes place—may explain why we have the assortment of elementary particles we do, dictating not only their masses but the forces between them. The study of these higher-dimensional spaces, moreover, has offered possible insights into why gravity appears to be so much weaker than the other forces of nature, while also providing clues about the mechanisms behind the inflationary expansion of the early universe and the dark energy that’s now driving the cosmos apart.
So it’s not just idle boasting when I say that geometry has been an invaluable tool for unlocking the universe’s secrets, right up there with physics and cosmology. Moreover, with the advances in mathematics that we’ll be describing here, along with progress in observational cosmology and the advent of string theory, which is attempting a grand synthesis that has never before been realized, all three of these disciplines seem to be converging at the same time. As a result, human knowledge now stands poised and raring to go, on the threshold of remarkable insights, with geometry, in many ways, leading the charge.
It’s important to bear in mind that whatever we do in geometry, and wherever we go, we never start from scratch. We’re always drawing on what came before—be it conjectures (which are unproven hypotheses), proofs, theorems, or axioms—building from a foundation that, in many cases, was laid down thousands of years before. In that sense, geometry, along with other sciences, is like an elaborate construction project. The foundation is laid down first, and if it’s built correctly—placed on firm ground, so to speak—it will last, as will the structures built on top of it, provided they too are engineered according to sound principles.
That, in essence, is the beauty and strength of my elected calling. When it comes to mathematics, we always expect a completely true statement. A mathematical theorem is an exact statement that will remain an eternal truth and is independent of space, time, people’s opinions, and authority. This quality sets it apart from empirical science, where you do experiments and, if a result looks good, you accept it after a satisfactory trial period. But the results are always subject to change; you can never expect a finding to be 100 percent, unalterably true.
Of course, we often come across broader and better versions of a mathematical theorem that don’t invalidate the original. The foundation of the building is still sound, to continue our construction analogy; we’ve kept it intact while doing some expansion and remodeling. Sometimes we have to go farther than just remodeling, perhaps even “gutting” the interior and starting afresh. Even though the old theorems are still true, we may need entirely new developments, and a fresh batch of materials, to create the overall picture we seek to achieve.
The most important theorems are usually checked and rechecked many times and in many ways, leaving essentially no chance for error. There may be problems, however, in obscure theorems that have not received such close scrutiny. When a mistake is uncovered, a room of the building—or perhaps a whole wing—might have to be torn down and reassembled. Meanwhile, the rest of the structure—a sturdy edifice that has stood the test of time—remains unaffected.
One of the great architects of geometry is Pythagoras, with the well-known formula attributed to him being one of the sturdiest edifices ever erected in mathematics. The Pythagorean theorem, as it’s called, states that for a right triangle (a triangle, that is, with one 90-degree angle), the length of the longest side (or hypotenuse) squared equals the sum of the squares of the two shorter sides. Or as schoolchildren, former and present, may recall: a2 + b2 = c2. It’s a simple, yet very powerful statement that amazingly is as relevant now as it was when formulated some 2,500 years ago. The theorem is not just restricted to elementary school mathematics. Indeed, I use the theorem just about every day, almost without thinking about it, because it has become so central and so ingrained.
To my mind, the Pythagorean theorem is the most important statement in geometry, as crucial for advanced, higher-dimensional math—such as for working out distances in Calabi-Yau spaces and solving Einstein’s equations of motion—as it is for calculations on a two-dimensional plane (such as the sheet of a homework assignment) or in a three-dimensional grade school classroom. The theorem’s importance stems from the fact that we can use it to figure out distances between two points in spaces of any dimension. And, as I said at the outset of this chapter, geometry has a lot to do with distance, which is why this formula is central to practically everything we do.
I find the theorem, moreover, to be extremely beautiful, although beauty, admittedly, is in the eye of the beholder. We tend to like things that we know—things that have become so familiar, so natural, that we take them for granted, just like the rising and setting of the sun. Then there’s the great economy of it all, just three simple letters raised to the second power, a2 + b2= c2, almost as terse as other famous laws like F = ma or E = mc2. For me, the beauty stems from the elegance of a simple statement that sits so comfortably within nature.
2.1—The Pythagorean theorem is often pictured in two dimensions in terms of a right triangle with the sum of the lengths of the sides squared equaling the length of the hypotenuse squared: a2 + b2 = c2. But, as shown here, the theorem also works in three dimensions (a2 + b2 + c2 = d2) and higher.
In addition to the theorem itself, which is without a doubt a cornerstone of geometry, equally important is the fact that it was proved to be true and appears to be the first documented proof in all of mathematics. Egyptian and Babylonian mathematicians had used the relation between the sides of a right triangle and its hypotenuse long before Pythagoras was even born. But neither the Egyptians nor the Babylonians ever proved the idea, nor do they seem to have considered the abstract notion of a proof. This, according to the mathematician E. T. Bell, was where Pythagoras made his greatest contribution:
Before him, geometry had been largely a collection of rules of thumb empirically arrived at without any clear indication of the mutual connections of the rules. Proof is now so commonly taken for granted as the very spirit of mathematics that we find it difficult to imagine the primitive thing which must have preceded mathematical reasoning.2
Well, maybe Pythagoras is responsible for the proof, though you might have noticed I said the theorem was “attributed” to him, as if there were some doubt as to the authorship. There is. Pythagoras was a cultlike figure, and many of the contributions of his math-crazed disciples, the so-called Pythagoreans, were attributed to him retroactively. So it’s possible that the proof of the Pythagorean theorem originated with one of his followers a generation or two later. Odds are we’ll never know: Pythagoras lived primarily in the sixth century B.C. and left behind little, if anything, in the way of written records.
Fortunately, that’s not the case with Euclid, one of the most famous geometers of all time and the man most responsible for turning geometry into a precise, rigorous discipline. In stark contrast to Pythagoras, Euclid left behind reams of documents, the most illustrious of them being The Elements (published around 300 B.C.)—a thirteen-volume treatise, of which eight volumes are devoted to the geometry of two and three dimensions. The Elements has been called one of the most influential textbooks ever penned, “a work of beauty whose impact rivaled that of the Bible.”3
In his celebrated tome, Euclid laid the groundwork not just for geometry but also for all of mathematics, which depends inextricably on a manner of reasoning we now call Euclidean: Starting with clearly defined terms and a set of explicitly stated axioms, or postulates (the two words being synonymous), one can then employ cool logic to prove theorems that, in turn, can be used to prove other assertions. Euclid did just that, proving more than four hundred theorems in all, thereby encapsulating virtually all of the geometric knowledge of his era.
Stanford mathematician Robert Osserman explained the enduring appeal of Euclid’s manifesto this way: “First there is the sense of certainty—that in a world full of irrational beliefs and shaky speculations, the statements found in The Elements were proven true beyond a shadow of a doubt.” Edna St. Vincent Millay expressed similar appreciation in her poem “Euclid Alone Has Looked on Beauty Bare.”4
The next crucial contribution for the purposes of our narrative—with no slight intended to the many worthy mathematicians whose contributions are being overlooked—comes from René Descartes. As discussed in the previous chapter, Descartes greatly enlarged the scope of geometry by introducing a coordinate system that enabled mathematicians to think about spaces of any dimension and to bring algebra to bear on geometric problems. Before he rewrote the field, geometry was pretty much limited to straight lines, circles, and conic sections—the shapes and curves, such as parabolas and hyperbolas, that you get by slicing an infinitely long cone at different angles. With a coordinate system in place, we could suddenly describe very complicated figures, which we otherwise would not know how to draw, by means of equations. Take the equation xn+ yn= 1, for example. Using Cartesian coordinates, one can solve the equation and trace out a curve. Before we had a coordinate system, we didn’t know how to draw such a figure. Where we had been stuck before, Descartes offered us a way to proceed.
And that way became even stronger when, about fifty years after Descartes shared his ideas on analytic geometry, Isaac Newton and Gottfried Leibniz invented calculus. Over the coming decades and centuries, the tools of calculus were eventually incorporated in geometry by mathematicians like Leonhard Euler, Joseph Lagrange, Gaspard Monge, and perhaps most notably Carl Friedrich Gauss, under whose guidance the field of differential geometryfinally came of age in the 1820s. The approach used Descartes’ system of coordinates to describe surfaces that could then be analyzed in detail by applying the techniques of differential calculus—differentiation being a technique for finding the slope of any smooth curve.
The development of differential geometry, which has continued to evolve since Gauss’s era, was a major achievement. With the tools of calculus in their grasp, geometers could characterize the properties of curves and surfaces with far greater clarity than had been possible before. Geometers obtain such information through differentiation or, equivalently, by taking derivatives, which measure how functions change in response to changing inputs. One can think of a function as an algorithm or formula that takes a number as an input and produces a number as an output: y = x2 is an example, where values for x go in and values for y come out. A function is consistent: If you feed it the same input, it will always produce the same output; if you put 2 in our example, you will always get 4. A derivative is how we describe the changes in output given incremental changes in input; the value of the derivative reflects the sensitivity of the output to slight changes in the input.
The derivative is not just some abstract notion; it’s an actual number that can be computed and tells us the slope of a curve, or of a surface, at a given point. In the above example, for instance, we can determine the derivative at a point (x = 2) on our function, which in this case happens to be a parabola. If we move a little bit away from that point to, say, x = 2.001, what happens to the output, y? Here, y (if computed to three decimal points) turns out to be 4.004. The derivative here is the ratio of the change in output (0.004) to the change in input (0.001), which is just 4. And that is, in fact, the exact derivative of this function at x = 2, which is another way of saying it’s the slope of the curve (a parabola) there, too.
The calculations, of course, can get much more involved than the foregoing when we pick more complicated functions and move into higher dimensions. But returning, for a moment, to the same example, we obtained the derivative of y = x2 from the ratio of the change in y to the change in x because the derivative of this function tells us its slope, or steepness, at a given point—with the slope being a direct measure of how y changes with respect to x.
To picture this another way, let’s consider a ball on a surface. If we nudge it to the side a tiny bit, how will that affect its height? If the surface is more or less flat, there will be little variation in height. But if the ball is on the edge of a steep grade, the change in height is more substantial. Derivatives can thus reveal the slope of the surface in the immediate vicinity of the ball.
Of course, there’s no reason to limit ourselves to just a single spot on the surface. By taking derivatives that reveal variations in the geometry (or shape) at different points on the surface, we can calculate the precise curvature of the object as a whole. Although the slope at any given point provides local information regarding only the “neighborhood” around that point, we can pool the information gathered at different points to deduce a general function that describes the slope of the object at any point. Then, by means of integration, which is a way of adding and averaging in calculus, we can deduce the function that describes the object as a whole. In so doing, one can learn about the structure of the entire object. This is, in fact, the central idea of differential geometry—namely, that you can obtain a global picture of an entire surface, or manifold, strictly from local information, drawn from derivatives, that reveals the geometry (or metric) at each point.
Gauss made many other noteworthy contributions in math and physics in addition to his work on differential geometry. Perhaps the most significant contribution for our purposes relates to his startling proposition that objects within a space aren’t the only things that can be curved; space itself can be curved. Gauss’s view directly challenged the Euclidean concept of flat space—a notion that applied not only to the intuitively flat two-dimensional plane but also to three-dimensional space, where flatness means (among other things) that on very large scales, parallel lines never cross and the sum of the angles of a triangle always add up to 180 degrees.
2.2—One can compute the area bounded by a curve by means of a calculus technique, integration, which divides the bounded regions into infinitesimally thin rectangles and adds up the area of all the rectangles. As the rectangles become narrower and narrower, the approximation gets better and better. Taken to the limit of the infinitesimally small, the approximation becomes as good as you can get.
These principles, which are essential features of Euclidean geometry, do not hold in curved spaces. Take a spherical space like the surface of a globe. When viewed from the equator, the longitudinal lines appear to be parallel because they are both perpendicular to the equator. But if you follow them in either direction, they eventually converge at the north and south poles. That doesn’t happen in (flat) Euclidean space—such as on a Mercator projection map—where two lines that are perpendicular to the same line are truly parallel and never intersect.
2.3—On a surface with positive curvature such as a sphere, the sum of the angles of a triangle is greater than 180 degrees, and lines that appear to be parallel (such as longitudinal lines) can intersect (at the north and south poles, for instance). On a flat planar surface (of zero curvature), which is the principal setting of Euclidean geometry, the sum of the angles of a triangle equals 180 degrees, and parallel lines never intersect. On a surface with negative curvature such as a saddle, the sum of the angles of a triangle is less than 180 degrees, and seemingly parallel lines diverge.
In non-Euclidean space, the angles of a triangle can either add up to more than 180 degrees or to less than 180 degrees depending on how space is curved. If it is positively curved like a sphere, the angles of a triangle always add up to more than 180 degrees. Conversely, if the space has negative curvature, like the middle part of a horse’s saddle, the angles of a triangle always add up to less than 180 degrees. One can obtain a measure of a space’s curvature by determining the extent to which the angles of a triangle add up to more than, less than, or equal to 180 degrees.
Gauss also advanced the concept of intrinsic geometry—the idea that an object or surface has its own curvature (the so-called Gauss curvature) that is independent of how it may be sitting in space. Let’s start, for example, with a piece of paper. You’d expect its overall curvature to be zero, and it is. But now let’s roll that sheet up into a cylinder. A two-dimensional surface like this, according to Gauss, has two principal curvatures running in directions that are orthogonal to each other: One curvature relates to the circle and has the value of 1/r, where r is the radius. If r is 1, then this curvature is 1. The other curvature runs along the length of the cylinder, which happens to be a straight line. The curvature of a straight line is obviously zero, since it doesn’t curve at all. The Gauss curvature of this object—or any two-dimensional object—equals the product of those two curvatures, which in this case is 1 ╳0 = 0. So in terms of its intrinsic curvature, the cylinder is the same as the sheet of paper it can be constructed from: perfectly flat. The zero intrinsic curvature of the cylinder is a result of the fact that one can form it from a sheet of paper without any stretching or distortion. To put it another way, the distance measurements between any two points on the surface of a sheet—whether the sheet is flat on a table or rolled into a tube—remain unchanged. That means that the geometry, and hence the intrinsic curvature, of the sheet stays intact regardless of whether it’s flat or curled up.
2.4—A torus, or donut-shaped, surface can be entirely “flat” (zero Gauss curvature), because it can be made, in principle, by rolling up a piece of paper into a tube or cylinder and then attaching the ends of the tube to each other.
Similarly, if we could create a donut or torus by attaching the circular ends of a cylinder together—again doing so without any stretching or distortion—the torus would have the same intrinsic curvature as the cylinder, namely, zero. In practice, however, we cannot actually construct this so-called flat torus—at least not in two dimensions where folds or wrinkles will inevitably be introduced at the seams. But we can construct such an object (known as an abstract surface) in theory, and it holds just as much importance to mathematics as the objects we call real.
A sphere, on the other hand, is quite different from a cylinder or flat torus. Consider, for example, the curvature of a sphere of radius r. It is defined by the equation 1/r2 and is the same everywhere on the surface of the sphere. As a result, every direction looks the same on the surface of a sphere, whereas this is obviously not the case on a cylinder or donut. And that doesn’t change, no matter how the sphere is oriented in three-dimensional space, just as a small bug living on that surface is presumably oblivious to how the surface is aligned in three-dimensional space; all it likely cares about, and experiences, is the geometry of its local, two-dimensional abode.
Gauss—in concert with Nikolai Lobachevsky and János Bolyai—made great contributions to our understanding of abstract space, particularly the two-dimensional case, though he personally admitted to some confusion in this area. And ultimately, neither Gauss nor his peers were able to liberate our conception of space entirely from the Euclidean framework. He expressed his puzzlement in an 1817 letter to the astronomer Heinrich Wilhelm Matthäus Olbers: “I am becoming more and more convinced that the necessity of our geometry cannot be proved, at least by human reason and for human reason. It may be that in the next life we shall arrive at views on the nature of space that are now inaccessible to us.”5
Some answers came not in the “next life,” as Gauss had written, but in the next generation through the efforts, and sheer brilliance, of his student Georg Friedrich Bernhard Riemann. Riemann suffered from poor health and died young, but in his forty years on this planet, he helped overturn conventional notions of geometry and, in the process, overturned our picture of the universe as well. Riemann introduced a special kind of field, a set of numbers assigned to each point in space that could reveal the distance along any path connecting two points—information that could be used, in turn, to determine the extent to which that space was curved.
Measuring space is simplest in one dimension. To measure a one-dimensional space, such as a straight line, all we need is a ruler. In a two-dimensional space, such as the floor of a grand ballroom, we’d normally take two perpendicular rulers—one called the x-axis and the other the y-axis—and work out distances between two points by creating right triangles and then using the Pythagorean theorem. Likewise, in three dimensions, we’d need three perpendicular rulers, x, y, and z.
Things get more complicated and interesting, however, in curved, non-Euclidean space, where properly labeled, perpendicular rulers are no longer available. We can rely on Riemannian geometry, instead, to calculate distances in spaces like these. The approach we’ll take in computing the length of a curve, which itself is sitting on a curved manifold, will seem familiar: We break the curve down into tangent vectors of infinitesimal size and integrate over the entire curve to get the total length.
The tricky part stems from the fact that in curved space, the measurement of the individual tangent vectors can change as we move from point to point on the manifold. To handle this variability, Riemann introduced a device, known as a metric tensor, that provides an algorithm for computing the length of a tangent vector at each point. In two dimensions, the metric tensor is a two-by-two matrix; in n dimensions, the metric tensor is an n-by-n matrix. (It’s worth noting that this new measurement approach, despite Riemann’s great innovation, still relies heavily on the Pythagorean theorem, adapted to a non-Euclidean setting.)
A space endowed with a Riemannian metric is called a Riemannian manifold. Equipped with the metric, we can measure the length of any curve in a manifold of arbitrary dimension. But we’re not limited to measuring the length of curves; we can also measure the area of a surface in that space, and a “surface” in this case is not limited to the usual two dimensions.
With the invention of the metric, Riemann showed how a space that was only vaguely defined could instead be granted a well-described geometry, and how curvature, rather than being an imprecise concept, could be encapsulated in a precise number associated with each point in space. And this approach, he showed, could apply to spaces of all dimensions.
Prior to Riemann, a curved object could only be studied from the “outside,” like surveying a mountain range from afar or gazing at the surface of Earth from a rocket ship. Up close, everything would seem flat. Riemann showed how we could still detect the fact that we were living in a curved space, even with nothing to compare that space with.6 This poses a huge question for physicists and astronomers: If Riemann was right, and that one space was all there is, without a bigger structure to fall back, it meant we had to readjust our picture of almost everything. It meant that on the largest scales, the universe need not be confined by the strictures of Euclid. Space was free to roam, free to curve, free to do whatever. It is for this very reason that astronomers and cosmologists are now making meticulous measurements in the hopes of finding out whether our universe is curved or not. Thanks to Riemann, we now know that we don’t have to go outside our universe to make these measurements, which would be a difficult feat to pull off. Instead, we should be able to figure this out from right where we’re sitting—a fact that could offer comfort to both cosmologists and couch potatoes.
These, in any event, were some of the new geometric ideas circulating when Einstein began drawing together his thoughts on gravity. Early in the twentieth century, Einstein had been struggling for the better part of a decade to combine his special theory of relativity with the principles of Newtonian gravity. He suspected that the answer may lie somewhere in geometry and turned to a friend, the geometer Marcel Grossman, for assistance. Grossman, who had previously helped Einstein get through graduate coursework that he’d found uninspiring, introduced his friend to Riemann’s geometry, which was unknown to physics at the time—although the geometer did so with a warning, calling it “a terrible mess which physicists should not be involved with.”7
Riemann’s geometry was the key to solving the puzzle Einstein had been wrestling with all those years. As we saw in the previous chapter, Einstein was grappling with the idea of a curved, four-dimensional spacetime (otherwise known as our universe) that was not part of a bigger space. Fortunately for him, Riemann had already provided such a framework by defining space in exactly that way. “Einstein’s genius lay in recognizing that this body of mathematics was tailor-made for implementing his new view of the gravitational force,” Brian Greene contends. “He boldly declared that the mathematics of Riemann’s geometry aligns perfectly with the physics of gravity.”8
Einstein recognized not only that spacetime could be described by Riemann’s geometry, but also that the geometry of spacetime would influence its physics. Whereas special relativity had already unified space and time through the notion of spacetime, Einstein’s subsequent theory of general relativity unified space and time with matter and gravity. This was a conceptual breakthrough. Newtonian physics had treated space as a passive background, not an active participant in the proceedings. The breakthrough was all the more spectacular considering that there was no experimental motivation for this theory at all. The idea literally sprang from one person’s head (which is not to say, of course, that it could have sprung from anyone’s head).
The physicist C. N. Yang called Einstein’s formulation of general relativity an act of “pure creation” that was “unique in human history … Einstein was not trying to seize an opportunity that had presented itself. He created the opportunity himself. And then fulfilled it on his own, through deep insight and grand design.”9
It was a remarkable achievement that might even have surprised Einstein, who hadn’t always recognized that basic physics and mathematics could be so intricately intertwined. He would conclude years later, however, that “the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.”10 Einstein’s theory of gravitation was arrived at by such a process of pure thought—realized through mathematics without any prompting from the outside world.
Equipped with Riemann’s metric tensor, Einstein worked out the shape and other properties—the geometry, in other words—of his newly conceived spacetime. And the resulting synthesis of geometry and physics, culminating in the famous Einstein field equation, illustrates that gravity—the force that shapes the cosmos on the largest scales—can be regarded as a kind of illusion caused by the curvature of space and time. The metric tensor of Riemannian geometry not only described the curvature of spacetime, but also described the gravitational field in Einstein’s new theory. Thus, a massive body like the sun warps the fabric of spacetime in the same way that a large man deforms a trampoline. And, just as a small marble thrown onto the trampoline will spiral around the heavier man, ultimately falling into the dip he creates, the geometry of warped spacetime causes Earth to orbit the sun. Gravity, in other words, is geometry. The physicist John Wheeler once explained Einstein’s picture of gravity this way: “Mass grips space by telling it how to curve; space grips mass by telling it how to move.”11
Another example might help drive this point home: Suppose that two people start at different spots on the equator and set out at the same speed toward the north pole, moving along longitudinal lines. As time goes on, they get closer and closer to each other. They may think they are affected by some invisible force that’s drawing them together. But another way to think of it is that the assumed force is really a consequence of the geometry of the earth and that there’s actually no force at all. And that, in short, gives you some idea as to the force of geometry itself.
The power of that example hit me with full impact when I was a first-semester graduate student learning about general relativity for the first time. It was no secret, of course, that gravity shapes our cosmos and that gravity was, indeed, its principal architect in terms of the big picture. On smaller scales, in the confined venue of most physics apparatus, gravity is extremely feeble compared with the other forces: electromagnetic, strong, and weak. But in the grand scheme of things, gravity is pretty much all there is: It is responsible for the creation of structure in the universe, from individual stars and galaxies up to giant superclusters stretching a billion light-years across. If Einstein was right, and it all came down to geometry, then geometry, too, was a force to be reckoned with.
I was sitting in a lecture class, pondering the implications, when a series of thoughts occurred to me. I had been interested in curvature since college and sensed that in light of Einstein’s insights, it may be a key to understanding the universe, as well as an avenue through which I might make my own mark someday. Differential geometry had provided tools for describing how mass moves in a curved spacetime without explaining why spacetime is curved in the first place. Einstein had taken those same tools to explain where that curvature comes from. What had been seen as two separate questions—the shape of a space under the influence of gravity and its shape under the influence of curvature—turned out to be the same problem.
Taking it a step further, the question I pondered was this: If gravity comes from mass telling space how to curve, what happens in a space that has no mass whatsoever—a space we call a vacuum? Who does the talking then? Put in other terms, does the so-called Einstein field equation for the vacuum case have a solution other than the most uninteresting one—that is, a “trivial” spacetime with no matter, no gravity, and no interaction and where absolutely nothing happens? Might there be, I mused, a “nontrivial” space that has no matter, yet whose curvature and gravity are nonzero?
I wasn’t yet in any position to answer these questions. Nor did I realize that a fellow named Eugenio Calabi had posed a special case of that very question more than fifteen years before, though he had approached it from a purely mathematical standpoint and wasn’t thinking about gravity or Einstein at all. The best I could do then was to marvel, open-mouthed, and wonder: “What if?”
It was a surprising question for me to ask, in many ways, especially given where I’d come from—starting on a trajectory that was as likely to have taken me to the poultry trade as it was to have led me to geometry, general relativity, and string theory.
I was born in mainland China in 1949, but my family moved within a year to Hong Kong. My father was a university professor with a modest salary and a wife and eight children to feed. Despite his taking three teaching jobs at three universities, his total earnings were meager, affording neither enough money nor food to go around. We grew up poor, without electricity or running water, taking our baths in a river nearby. Enrichment, however, came in other forms. Being a philosopher, my father inspired me to try to perceive the world through a more abstract lens. I remember as a young child overhearing the conversations he had with students and peers; I could feel the excitement of their words even though I couldn’t grasp their meaning.
My father always encouraged me in mathematics, despite my not getting off to the most promising start. When I was five, I took an entrance exam for a top-notch public school but failed the mathematics part because I wrote 57 instead of 75 and 69 instead of 96—a mistake, I now tell myself, that’s easier to make in Chinese than in English. As a result, I was forced to go to an inferior rural school populated by a lot of rough kids who had little patience for formal education. I had to be rough to survive, so rough that I dropped out of school for a time in my preteen years and headed a gang of youths who, like me, wandered the streets looking for trouble and, more often than not, found it.
Personal tragedy turned that around. My father died unexpectedly when I was fourteen, leaving our family not only grief-stricken but destitute, with a slew of debts to pay off and virtually no income. As I needed to earn some money to support the family, an uncle advised me to leave school and raise ducks instead. But I had a different idea: teaching mathematics to other students. Given our financial circumstances, I knew there was just one chance for me to succeed and I placed my bets on math, double or nothing. If I didn’t do well, my whole future was done, leaving nothing to fall back on (other than fowl husbandry, perhaps) and no second chances. In situations like that, I’ve found, people tend to work harder. And though I may have my shortcomings, no one has ever accused me of being lazy.
I wasn’t the best student in high school but tried to make up for that in college. While I was a reasonably good student in my first year, though by no means exceptional, things really picked up for me in the second year when Stephen Salaff, a young geometer from Berkeley, came to teach at our school, the Chinese University of Hong Kong. Through Salaff I got my first taste of what real mathematics was all about. We taught a course together on ordinary differential equations and later wrote a textbook together on that same subject. Salaff introduced me to Donald Sarason, a distinguished Berkeley mathematician who paved the way for me to come to the university as a graduate student after I had completed just three years of undergraduate work. Nothing I’d encountered in mathematics up to that point rivaled the bureaucratic challenges we overcame—with the help of S. S. Chern, the great Chinese geometer, also based at Berkeley—in order to secure my early admittance.
2.5—The geometer S. S. Chern (Photo by George M. Bergman)
Arriving in California at the age of twenty, with the full range of mathematics lying before me, I had no idea of what direction to pursue. I was initially inclined toward operator algebra, one of the more abstract areas of algebra, owing to my vague sense that the more abstract a theory was, the better.
Although Berkeley was strong in many branches of math, it happened to be a world center—if not the world center—for geometry at the time, and the presence of many impressive scholars like Chern began to exert an inexorable tug on me. That, coupled with a growing recognition that geometry constituted a large, rich subject ripe with possibilities, slowly lured me into the fold.
Nevertheless, I continued to expose myself to as many subjects as possible, enrolling in six graduate courses and auditing many others on subjects including geometry, topology, differential equations, Lie groups, combinatorials, number theory, and probability theory. That kept me in the classroom from 8 A.M. to 5 P.M. every day, barely leaving time for lunch. When I wasn’t in the classroom, I was in the math library, my second home, reading as many books as I could lay my hands on. As I couldn’t afford to buy books when I was younger, I was now like the proverbial kid in the candy shop, literally working my way through the stacks from one end to the other. Having nothing better to do, I often stayed until closing time, regularly qualifying as the last man sitting. Confucius once said: “I have spent a whole day without eating and a whole night without sleeping in order to think, but I got nothing out of it. Thinking cannot compare with studying.” While I may not have been aware of that quotation at the time, I embraced the philosophy nevertheless.
So why did geometry, of all the areas of mathematics, come to occupy center stage for me, both in my waking thoughts and in my dreams? Primarily because it struck me as the field closest to nature and therefore closest to answering the kinds of questions I cared about most. Besides, I find it helpful to look at pictures when grappling with difficult concepts, and pictures are few and far between in the more abstruse realms of algebra and number theory. Then there was this fantastic group of people doing geometry at Berkeley (including Professors Chern and Charles Morrey, as well as some young faculty members like Blaine Lawson and fellow graduate students like future Fields Medal winner William Thurston) that made me want to be part of that excitement and, hopefully, add to it.
On top of that, there was a much larger community of people, not only at other campuses, but throughout the world, and—as we’ve seen in this chapter, throughout history as well—who had paved the way for the fertile period I was now fortunate enough to step into. It’s kind of like what Isaac Newton said about “standing on the shoulders of giants,” though Newton himself is one of the foremost giants upon whose shoulders we now stand.
Around the time that I first began thinking about Einstein’s general theory of relativity and the curvature of space in a vacuum, my adviser, Chern, returned from a trip to the East Coast, excited because he’d just heard from the renowned Princeton mathematician André Weil that the “Riemann hypothesis,” a problem posed more than a century before, might soon be solved. The hypothesis relates to the distribution of prime numbers, which don’t appear to follow any pattern. Yet Riemann proposed that the frequency of these numbers was, in fact, related to a complex function since named the Riemann zeta function. More specifically, he suggested that the frequency of prime numbers corresponded to the location of the zeros of his zeta function. Riemann’s assertion has been confirmed for the location of more than a billion zeros, but it still has not been proved as a general matter.
Although this was one of the most prominent problems in all of mathematics—and if I were lucky enough to solve it, it would bring in countless job offers and seal my fame for life—I couldn’t muster much enthusiasm for Chern’s proposition. The Riemann hypothesis just didn’t excite me, and you have to be excited if you’re going to set off on an ambitious project, which had thwarted so many talented people before and would, at a minimum, take years to complete. The lack of passion for this problem would unquestionably have hurt my chances of solving it, which meant there was a real possibility that I could work on the Riemann hypothesis for years and have nothing to show for it. What’s more, I like pictures too much. I like mathematical structures you can look at in some fashion, which is why I like geometry. Plus, I already knew of some areas in geometry where I could achieve some results—albeit not nearly so spectacular.
It’s like going out fishing. If you’re content to bring back small fish, you’re likely to catch something. But if you only care about bringing back the biggest fish ever caught—a mythical creature that is the stuff of legends—you’re likely to come home empty-handed. Thirty-five years later, the Riemann hypothesis remains an open problem. As we say in mathematics, there’s no such thing as 90 percent proved.
So that was part of my thinking when I turned down Chern’s request. But there was more to it than that. At the time, as I’ve said, I was already getting intrigued by general relativity, trying to sort out how many of the features of our universe emerge from the interplay of gravity, curvature, and geometry. I didn’t know where this line of inquiry might take me, yet I had an inkling, nevertheless, that I was embarking on a great adventure, harnessing the powers of geometry to go after the truth.
As a child born of modest circumstances, I never had the opportunity to see much of the world. Yet my passion for geometry was piqued at an early age and grew out of desire to map a land as great as China and travel a sea without knowing the end. I’ve journeyed farther since then, yet geometry still serves that same purpose for me. Only now, the land has become the whole earth, and the sea, the universe. And the little straw bag I used to carry around has been replaced by a small briefcase containing a ruler, compass, and protractor.